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L,A 1 REPORT 24<br />
LECTURE SERTEG ON NUCBAR PHYSICS<br />
First Serie$: TERkINOLOGY; Lecturer: 'EIMII MCMXLLAgb<br />
LECTURE I: The Atom ahd the NuBkeias<br />
" 11: htehtidn of LIght with Matter<br />
It %If! Forces Holding the Nuclei Together<br />
{' IV: Reactions Betweeh Nuclei<br />
V: RadioactJvity<br />
VI: I) 6 -Decay; 11) Neutron Reactions<br />
t' VII: N~~cI.ear Fission<br />
Bhge 1<br />
1<br />
8<br />
17<br />
24<br />
'<br />
I,;;;<br />
t~.%<br />
32a<br />
g<br />
42<br />
Second Series : RADIOACTIVITY; Lecturers : E. ,Seere<br />
E, Teller<br />
F. Bloch<br />
LECTURE VIII:<br />
IX:<br />
X:<br />
The Radioactive Decay Law<br />
48<br />
48<br />
I, The Radioactive Decay LLPN (Cont'd).<br />
11. Pluctuations. 111. Passflpe Through<br />
Matter 55<br />
-Particles and Their Interactions<br />
with Matter 62<br />
XI:<br />
XII:<br />
XI'II:<br />
XIV:<br />
xv:<br />
XVI :<br />
XVI 1 :<br />
XVIII:<br />
The Theory of Stopping Power 66<br />
The Theory of Stopping Power (Cont'd) 72<br />
Interaction of Charged Particles with<br />
Matter 80<br />
Interaction of Radiations with Matter 83<br />
The Properties of Nuclei<br />
Survey of Stable Kucleus 92<br />
Theory of Alpha Disintegration. Theory<br />
of Beta Particles . 100<br />
I. Positron Emission and K capture.<br />
11. Gamma Radiation 105<br />
a7<br />
Copyrighted
.<br />
~ORRECTIONS<br />
, pI 241 - Equation 48 should read<br />
, ..,<br />
p. 254 - Missing only because of an error in numbering and not<br />
because of censorship.<br />
i?-k<br />
'2%<br />
p. 276 - Equationiat top of page should read<br />
. .
Third Series! NEUTRON P€lS?SICS; Lecturer: J.H. Williams; Page 108<br />
LECTURE XIX: Discovery of the Neutron<br />
108<br />
II<br />
It<br />
XX: Neutron Sources (cont'd)<br />
Typical Reacttons f& Production of ~OUt<br />
irons<br />
114<br />
XXI2 Properties of Slow Neutrons 119<br />
'' XXII: Properties of Slow Neutrons (Cont'd) 125<br />
XXIII:<br />
XXIV:<br />
'Fast Neutrons<br />
Properties of Fast Nqutrons<br />
C r 0 s s Se c ti on Mea suremen t s<br />
135<br />
146<br />
Fourth Series: TWO BODY PRORLEN; Lecturer: C,L.Critchfield:<br />
Page 153<br />
LWTURn XXV:<br />
Nuclear Constants<br />
" XXVI! Neutron-Proton Interaction<br />
153<br />
162<br />
I' XXVII: I) Spin-Spin Forces in the Deuteron<br />
TI) Proton-Proton Scattering 168<br />
'I<br />
XXVIII: Three Body Problems<br />
178<br />
I' XXIX and XXX: Interaction of Neutron and Proton<br />
wfth Fields<br />
185<br />
Fifth Series:<br />
THE STATISTICAL TEEORY OF NUCLEAR REACTIONS;<br />
Lecturor: V, F, 4;Weisskopf Page 207<br />
LECTURE XXXI: Introduction 207<br />
'I XXXII: Qualitative Treatment 210<br />
" XXXIII: Cross Sections and Emission Probabilities 218<br />
" XXXIV: The Excited Nucleus 227<br />
?I<br />
XXXV: Resonance Processes<br />
237<br />
'' XXXVI! Non-Resonance Processes<br />
255<br />
Ju<br />
LECTURE XXXVII:<br />
Diffusion Equations and Point Source<br />
Solutions 287<br />
XXXVIIIt Point Source Problems and Albedo 292
LECTURE XXXIX! The Slowing Down of Neutrons Page 304<br />
11 XL: The Slowing Down of Neutrons in<br />
Water 315<br />
2 XLI: Tho Boltzrnann Equation: Carrections<br />
to Diffusion Theory 323
LA 24 (1)<br />
Sentembar 14, 1943.<br />
1<br />
LECTURE SERIES OH NUCLEAR F'HYSICS<br />
FXRST SERIES: TEFblIINOLOOY LECTURER: E, M. MCMILLN .<br />
LECTURE I:<br />
We consider a piece of matter, for instance copper,<br />
If we magntfy it, we first sea that it consists of crystallites.<br />
They in turn are regularly arrmged arrays of 82;oms. In the<br />
special. case of copper, the eeometrical arrangement i s the same<br />
as a close packed arrangement of spheres, the atoms in one plane<br />
making the following pattern:<br />
I<br />
At the center of each atom there fa a nuoleus, Of coursel the<br />
picture is not really like the ne drawn., 'The elsotrons fill<br />
the whole space, there are no botmdariea, the atoms actually<br />
overlap, thelr limits are fuzzy. In a metal like copper it is<br />
not even clear which electron belongs to which atom, In an insulator,<br />
f'or instance the noble gas Argon, each atam holds on to<br />
its own electrons and each electron belongs to 8 definite atom.<br />
.-<br />
In a chemio-al compound, f'or instance carbon dioxide, tfhe e 8<br />
aped by the atams belonging to one molecule, but eaahmoleas<br />
its own electronsr All the chemical properties of' the<br />
are determined by khe outer electrons,<br />
The question arises what arfect the nucleus has on the<br />
behavior af the atom, The effect af the nucleus is due to its<br />
This charge is an integral multiple of the
positive charge which is of the same magnitude as the charge of<br />
2<br />
the electron,<br />
Atoms must be neutral because a positive charge<br />
would result in the attraction of more electrons, whereas a<br />
negative charga would cause the atws to lose electrons. Each<br />
atom has enough (negatively charged) ezectrons to c&npensate the<br />
nuclear charge.<br />
This number of electrons 2s aZso called the atan-<br />
ic number and dot;eminas the chemical properties of tho atom.<br />
Another significant property of the atomic nucleus is<br />
the nucleus mass,<br />
This mass is much greater than the mass Of the<br />
electrons, Even ths lightest nucleus, namely of hydrogen, is 1840<br />
times heavier than that of an electron,<br />
Thus the mans of matter<br />
is essentially conoentrated in the nuclei,<br />
The mass of the<br />
nucleus depends in first approximation on the number of neutrons<br />
and protons within the nucleus,<br />
The neutrons and protons have<br />
approximately equal mass hut this equality does not hold as<br />
exactly as tho equality of positive and negative charges within<br />
an atm. The protons carry one unit of positive charge, The<br />
neutrons are uncharged,<br />
Tho charge of tho nucleus, that is the<br />
atomfc number, is equal to the numb@r of protons Sn ths nucleus,<br />
The total. weipht of the nucleus on tho other hand, is roughly<br />
eW&l to the weight of a proton (or a neutron) multiplied by the<br />
totah nwnbor OS neutrons and protons within the nuclcus, This<br />
number is callcd the mags number,<br />
'Jhus if two nuolei have thcj<br />
same charge, it docs not moan that they haw the same mass number<br />
because a frivcn nwnbep of protons might be associatcd with a dif-<br />
$'@rent nwnber of noutrons in different nucl-ei,<br />
If two nuclei have<br />
tho same nwnbor of protons thay will belong to the same chemical
dlf'fcront nwnbor of neutrons they will bo different isotopcs,<br />
For instance hydrogen has different ktnds af isotapes,' &very<br />
hydpo$on nucleus has one proton but it may havo 0, 1 or 2 noutr~ns.<br />
Therc cx$s.t: wany more nuclaar spcclie$ than atomic<br />
ordor to designata the nucloar spsoiea WQ writa the atomic symbol,<br />
for instanco H in thc cas0 of hydrogon,<br />
This alroady implias<br />
what tho nuclear charpa. is and thercforo shows how many protons<br />
aro found in thc nucleus,<br />
by an uppor index,<br />
Tho mags nwnbor of tho nuolcvs is shown<br />
The following tablc pivcs the numbor of n6uc<br />
trons and prOtons Tor tho known hydrogcn and helium isotopes:<br />
numbar of protons numbcr of new-<br />
3. trons<br />
B k 0<br />
H2 E 1<br />
H3 1 2<br />
3<br />
B<br />
r,<br />
2 4<br />
Thc? approximate S%Z?- of anatqm is IOe8 cm,<br />
DEfferent<br />
abomic spocics havo dlfforent size but the total variation in size<br />
is not mare than a factor of ton times.<br />
One may visuallza the<br />
size of' an atom by expanding our scale of rncasuromcnts in such a<br />
way that 1 om, will appear 8s ,000 milos, Then an atom will bc<br />
about 1 cm, in radiuls.<br />
Tho 8Sze of the atomic nuclous is about<br />
crr~.~ that is, about lo5 times smaller than the atom,<br />
distancc between the sun a.nd earth is about 100 timas the radius<br />
of the sun,<br />
paint of viow of<br />
tho earth,<br />
The<br />
Thus the yluclous is relatively much smaller from the<br />
its outer electrons than thE: sun as viowed from
4<br />
It was stated that tho lightcst nucleus is about 2,000<br />
timos haavior than an olsctron. /Ire know that in 1 gm, of hydropcn<br />
thoro arc 6 x atoms, thoroforo tho mass of one hydrogsn<br />
nucleus is J1/6 x<br />
grms,<br />
Bofore turning. to tho dimension of the cbctron, we<br />
shall d$.scuss the peculiar dirficulty whfch arises in connection<br />
with visualizing tho electronis within the atoms, It is ofton<br />
stated that an atom is likc a solar system. ThG atom of' iron for<br />
Instance has 26 alcctpons. Can we rnOasuro these as in motion in<br />
approximatcly clliptical orbits around thc nuclcus? If we should<br />
take a microtjcopc of vcry high powou, and rcsolution and take a<br />
photograph of thc inside of an atom at a ,r;rivcn instance, me would<br />
rind a disorderly arranprnwnt of elcotrons within tho atom. One<br />
might then try to tako another photograph a vemj short time later<br />
in order to find thc motion of' thc eloctrons within thcir orbits.<br />
If such a photograph is taken no relation is discovercd botwocn<br />
the positions in tho two consccutive pictures, hc tually if tho<br />
resolving powcr OS tho microscopo would have bccn infinity, the<br />
second picture would have boon 'blank, Thc reason fop this situstion<br />
is that the light which ia uacd in taking tho pictures carries<br />
momcntum and one cannot thcrofore takc a photograph without<br />
,giving a kick to the clcctrons, If good resolving power is<br />
wantod, you must uac light of vary short wavo longth and such<br />
lipht carpios particularly hiph rnomsntwn, Gnc might try to '<br />
cornpromiso in following tho orbit of tho okctrons by using a<br />
lowcr rcsalving power and haping that in this way thc: vcloc.ity<br />
of the clcctrons will not bo too much disturbed in taking a
5<br />
J)<br />
0<br />
photograph but the quant 1.t 8 t ivc e itua t ion is such that tho<br />
nccossary unc c r t ain t ie s in position and momentum aro just<br />
to mako it impossible to definc orbits within atoms,<br />
This conclqsion is a consequence of quantum mechanics,<br />
Quantum mochanics is thc mechanics that appliss to all bodies,<br />
for instance to a tennis ball, but in the case of a large body<br />
like a tennis ball tho laws of quantum mochanics approach the laws<br />
of classical: mechanics so closoly that the two bscome practically<br />
undisbinguishablc, Thc word quantum means amount and bccum in<br />
the designation quantum moohanics bocauso many of thc characteristic<br />
laws of quantum mccharzios am as sociatcd with the fact that<br />
ccrta$n quantitioa appear in amounts that arc multiples of a given<br />
cbaracteristlc amwnt;, Tho most basic of these quantitlcs is the<br />
s o-cal Led "quantum of action'' which is dosienatod This<br />
quantity has tho dimansion of a dlstancc times tho momontm,<br />
h Is the measure a$ the limitation on moasuremont, If the position<br />
o$ a particlc (4.e. its distancc from a rcfcrcncs system)<br />
is well knawn, then little can ba known about its momentum and<br />
vice vcrsa, If tho momcntum is known tho position must bo unknown.<br />
The product of the uncortaintics in momonturn and positian is of<br />
the magnitude h, For a big mass this uncertainty docs not moan<br />
much, This 18 so bccause thb momcntum is a product of mass and<br />
velocity and if tho mass is big, the uncm-tainty in tho velocity<br />
bccomos correspondingly small, but for tho lightest partlcl'os, the<br />
electrons, $he uncortainty of momentum causcs such an uncartainty<br />
in velocity that to dcfins anything likc an orbit for an electron<br />
bocomos impossible4 The only statcrnonts that wc can mako about
ht for instance tako many dnstantanoaus photographs of a<br />
4 certain kind of atom in tho way'doscribod abov~, onc might then<br />
suporimpose those photographs and WQ thon find that tho olcctron<br />
position all1 distribute itself mor0 in cortain rogions (noar tho<br />
nucleus) than in ather rogions.<br />
thc probable donsity of tho c'loctrona inside tho atom, ire, the<br />
probability with which the c3cctrons can bc Found in tho various<br />
volwno elemopts insido tho atom.<br />
Such a plcturo gives an idea of<br />
Tho doscription just givon might induce onc to say that<br />
an olcctron is asbip 8s the wholc atom.<br />
6<br />
HOWOVOF, in an individual<br />
moasurcmcnt the position of an electron may be more sharply de-<br />
ffned and in fact electrons do havo thoir own radii which a r much ~<br />
smaller than thc atomic radius.<br />
Tho conccpt of the electron<br />
radius can be undcrstood by cansidaring the clcctric field of an<br />
olectron.<br />
This clectric flold La nccQrding to tho Coulomb Law<br />
inversely proportional to tho squnrc of tho distance from tho<br />
olectron,<br />
If tho olootron were a point, the ffald mar the<br />
electron would bo infinite, It is not belicvcd that such ** in-<br />
finite fields can oxist and it is asswad thnt at somo givcn<br />
distance from tho c ctron, thc Cqulamb Law CORSCS to hold, This<br />
0<br />
distance has bocn estimated<br />
That is the cloc-<br />
tron radius4and tho nuclear radii a m of comparablc size.<br />
COUPSC,<br />
about 30-13 cm.<br />
tho position of tho nuclci IS not quito so hard to dsf'ino<br />
as tha position of olcctrons duc to tho fast that the nuclei hclvc<br />
much bigger mass and accordin@; to that reasoning unccrtaintics<br />
in their momenta, will not cause very great unccrtaintics in their<br />
volocitics and will not wash out thoir orbits to the seme axtent<br />
Of
VIQ<br />
7<br />
In some of the applications of quantum mechanics which<br />
shall encounter XatBr, the wave Eennths associated with par-<br />
ticles will play a fundamental role,<br />
a given particle is h/mv<br />
Gne lonpth associated with<br />
(h = quantum of action; m : mas8 of par-<br />
ticle; v = velocity of particle; mv = momentum of particle),<br />
%'his<br />
length is actually equal to the uncsrbainty a% position associated<br />
with an uncertninty in momentum of the magnitude mv. De Broglie<br />
was the first to postulate that a wave process is associated with<br />
every particle and it is shown that the wevo lencth is h/mv, This<br />
&$sociation between paptlcles and wave proceaaos is surprising but<br />
it has been actually demonstrated by interference experiments in<br />
which electrons or light atams (for instance hydrogen) have been<br />
reflected from crystal surfaces.<br />
The laws of' this reflection pro-<br />
cess are of the same kind as the laws of X-ray r'eflection and have<br />
,<br />
been described In terns of wave intcrferonca,<br />
It is apparant from<br />
the above statements that the de Sroglie wave lcnpth is closely<br />
related to the uncertainty of electron position and also to the<br />
fuzziness of atomic boundaries,
8<br />
LA 24 (2)<br />
September 265 1943<br />
LZCTURE SE~?XYS ON NCTCL’2AR PHYSICS<br />
, -<br />
FIRST S7RI”S : TTR!‘l INQLGGY LECTtJlBR: ?7 .I$ e MCIJ ILLAN<br />
LECTURE 11: ,<br />
1WT’~RACTICN GF LIGHT TflfITK MATTER<br />
%<br />
The list of simple particles mentioned in the last Xec-<br />
turo comprised electrons, protons and neutrons.<br />
$‘fa may add to this<br />
1Lst light.<br />
It is known that,light consists of‘ eloctromagnctic<br />
waves, but if, as has boon mentioned, the wave process is 8580-<br />
ciated with electrons, n.eutrons or protons, which we ordinarily<br />
consider as particles, then it will, not be too suryrising to find<br />
that light, which ordinarily can bo thought Of‘ a8 a wRVO process,<br />
has certain particle properties, iillhon light bohavss like a particle,<br />
we talk about a Xipht quantum.<br />
There is an important dffforonce between light and<br />
proper particles, for instance tho electrons, Tho electrons have<br />
a strong tcmdencp to persist. fivht quanta do not. They pat absorbed<br />
and omitted, It 3.3 not oasy to count them and for this<br />
reasoh it is impracticable to consider them as constituents of<br />
matter, Gn tho other hand, whonever 1icht does got- absorbed or<br />
emitted, a chunk of anergy of a definite size disappears or is<br />
produced in each such process as though light did consist of pap-<br />
ticlcs cBrpying il certain amount of energy.<br />
This enerpy of a<br />
light quantum is hq whore zfis the frequency of the 1Pgh.t considcred<br />
and h 3.8 the smc quantum constant which appears iur the<br />
dc Broglio rolation conncctinp the momentum of a particle with the
i<br />
1<br />
I<br />
wave longth associated tp it.<br />
,<br />
If the light gGantwn posscsses a definite cncrgy hv, on$<br />
will also expect it to bkrry a momentum. According to the theory<br />
1<br />
. of FgJ,B$ivity this momontmn is hz/Jc, where c is the light v~lo-<br />
city.<br />
bt will be noticog that thc ratio of onereg to rnomc;ntum is<br />
I<br />
'' not thc samG hs ih Nowto)-iian mobhanics. If we write for tho kino&<br />
1<br />
tic onergy mv2/2 (m is mass of gartiolo; v, its velocity) and for<br />
the momentum mv, thcn the ratfa of the 2 quantities is V/2,<br />
that ds<br />
one half of the velocityi of the particle rathar than the velocity<br />
itsclf.<br />
I<br />
1<br />
But vcwtonian mbchanics nppXics only as long as the velo-<br />
city<br />
v is small compared to tho velocity of light c.<br />
hv/c of the momentum of a light quantum can be easily soen to Ggreo<br />
with tho de Broglie rclaItion.<br />
Jcngth of light and<br />
I<br />
I<br />
I<br />
In fact, c/z/is equal to<br />
The value<br />
tho wave<br />
WX l momontum<br />
I<br />
is the samo equation whilch also holds for electrons, noutrons and<br />
protons.<br />
i<br />
I<br />
I<br />
In discussinp llight quanta, it was ncccssary to introduco<br />
1<br />
relativity, ulJhile this lbranch of physics, as wall 8s quantum<br />
mechanics mentioned last1 tims, have an elaborate mathematical<br />
foundation, we nced t9 b(c concorncd horc only with a f'ow s;lmp,lo<br />
consoquenccs of thesc tdoories,<br />
1<br />
I<br />
Nuclear physics too will bc<br />
troatod hero on a simildrly simple dcscriptivo plane, in which tho<br />
clcmontary particles, cl/cctrons, protons and ncutrons, aro intro-<br />
I<br />
duced without doscribind in detail tho evidence that thcy arc<br />
I<br />
olementary particles ani withaut discussing the difficulties to<br />
:I)<br />
whlch this concept leads.<br />
~ C I may ndd hcrc to tho list of tho pro-<br />
e<br />
I<br />
1
10<br />
@<br />
per particles, the positron, whoso existonce was prodictod by thr:<br />
combined application' of quantum mechanics and rslativity, Tha<br />
positron has the smb proportios as an electron, exccpt that it<br />
rSes a positive rather than a ncgative chargc, Tho only rcason<br />
why its discovery occurred much latcr than that 0% tho olcctron is<br />
that the positron is not stable in the prosonce of olcctrons. Tho<br />
positron is attrnctcd by an olcctron to which it happons to comc<br />
elosc., The chargcs of the two particles componsato cach other and<br />
their energy is radiatod out in thc fom of Light, Thus an cloctron-positron<br />
pair is annihilated. It is possiblc to produce posi-<br />
trons but in tho prcscnco of clcctrons which arc found in all OPdinary<br />
mattcr thcy can oxist only a wry short tirnc. Actually the<br />
oduction of positrons i s the rcvcrse proccss of tho annihilation;<br />
in cach production procoss, an elactron-positron paiP appears,<br />
thcrcforc, whcn a productioncannihflation cycle is ondcd, we arc<br />
left with the original nmbcr of clcctrons,<br />
In addition to all thcsc particles, wo shall cncountor<br />
two mom half-known k5nds of pmtZclo8, thc moso- and thc<br />
neqtrinos, Tlno mcsotrons RPO known cxperimcntnllg, Thcy are<br />
creatod in spccial praccsscs obscrvcd in cosmic-ray phJTsics but<br />
oy have not bocn obscrvod up to now in ordinary nuclotlr roactions<br />
Like the positrons thcy diaappcar shortly after thcir creation.<br />
They arc of importancc in the theary of nuclonr forccs which at<br />
a<br />
prcsont is widely acccptod.<br />
Tho ncutrinos haw not bacn obscrvod.<br />
Thcy may bo Qonaiderod. at prcscnt RS a pu~o invcntion introduccd<br />
to satisfy thc conservation laws of cncrgy of momcntum in ccrtain<br />
wcll known nucloar roactions fn which thaso consorvation laws arc
11<br />
agparontly violated,<br />
Cnc important conscqucncc of' the thcozy of relativity is<br />
the conncction bctwaon maas and cnergy, A movinp particlc is heavw<br />
fer than the same partielc at rcst. differonce in mass is propartional<br />
to tha kinetic encrgy, Potcntlal cnergy has a similar<br />
irifluwcc on mass. Tho proportion constant batwocn oncrgy and mass<br />
is the square of tho lQht volocity<br />
%is rclatiorz fa difficult to demonstrate under ordinary conditions<br />
it$ which thc kinotic or patantial onurgy pcr gram of matcrial is<br />
rathor small.<br />
In thcsc cascs, duc to thz high value of thc light<br />
volociity, th& encrgy cha.ngas duo to thht; kinetic or patcntfal oncrgy<br />
are very small cornparad to the orlglnal cncrgy of' tho maasca in-<br />
lvcd, but in nuclcar physlca much grcatcr cncrgics are asso-<br />
ciatea with a pivon mass, and hcro thc chnngos in maas bcoomo de-<br />
toctable.<br />
J ~ O may conslidor thc uncrgy relations in a nuclcar roc<br />
aptfon in which two Doutorons (H2<br />
(ao"<br />
uclci) unit0 t o fomn a-particlo<br />
Tho roaction has aCtualSy not bccn observed,' Its occur-<br />
rbncc is vcrg improbable, bccausc conscrvatton of rnomcntun provcnts<br />
qmrt9cla formed from carrying away tho cncrgy rcloasad by<br />
action as kfnctic onorgy,<br />
Thorcforc, tho cncrgy. r?cloasod<br />
crn$tLod in Lhc form of light and all rcactions involving<br />
n or hbsorption of light arc impraba<br />
c as cornparad to thc<br />
rbactions in which only hoavy particles, noutrons, protons or other<br />
nRclc2, arc involtvcd.<br />
*Zt may bc of intcrcst that a collision of<br />
rot; on, Deuteron and W-particle,<br />
dd'itfon to their more systemic
two Deuterons actually Fives H3 + €I1 or He3 + n (y1 stands for 8<br />
neutron) with about aqua1 probability; These reactions are much<br />
m4re probable than the raactlon mentioned above,<br />
I<br />
I mooretically, 2H24 ~e~ .+ light, lis reaction can tse<br />
used as 8 very simple example for the @neFgy-mass equivalence.<br />
The inaas of H<br />
2 is 2,015 maas units while that of He4 is 4,004 mass<br />
uhits, Therafore, the product nucleus He4 is ,026 mass units<br />
j<br />
lightor than the two original €I2<br />
12<br />
nuclei, This d&Fferonce of mass<br />
corresponds to a difference of energy and since in the reaction<br />
thc nuclear mass and enargy have dsoroasod, the differonce in enew<br />
bbcomos available (in tho present case in tho fom of light) as<br />
1<br />
t&e energy of tho roaction. Tho quantitative connection betwean<br />
crkergy and mags works out in such a way that ,001 mass unit OT<br />
a<br />
mil 1 ima s s-unt t c orre spmd s approxima t Q ly to 1 M i 11 ion e le c t r on<br />
V$lts (mors accuratcly ,935 M!l[eV),<br />
i? an cnorpy unit 1,6 x<br />
orgs,<br />
Tho alectron Volt (sV) in turn<br />
It is the encqy an electron<br />
aequircs if it falls through a pot0ntlaL difference of 1 Volt.<br />
The relation between inass and energy is freqvcntly used to find<br />
the oxact value of a nuclotir mass,<br />
r<br />
n~.pl~ar reaction tho ~ R S S of all but one particlpants 8ro known and<br />
id tho snorpy of tho roactian is mcasurod;<br />
This can bo dono if in 8<br />
In practical nucloar physics intoraction of light quarita<br />
withmattor plays an important roJ.~.<br />
i$ photo ionizatiok<br />
In thc figure<br />
Cno way of such intoraction
ative ion, namely tho electron ftself, is also fomed<br />
photo ionization, or the praduction of ions by 138<br />
Ad1 actual obsorvatlon in nuclea.?? physics dcpcncls on ionization,<br />
A typical arranpment for such obsorvationa is sketchod in the f o l m<br />
I<br />
chamber<br />
1<br />
i<br />
galvanomcbor<br />
I<br />
t<br />
battr;ry<br />
The air in tho chamber is ordinarily an insulator and so no CUPrent<br />
can be sont by thhc bnttory thraugh the galvanomcter, but if<br />
ionization occurs in thc cbambcr, ions now moving up to the<br />
t<br />
clbctrodcs will carry a cortaln small amount of curront which can<br />
I<br />
bo; road ius. the galvanomctor. Thus by thc curront one can moasuro<br />
ionization,<br />
1) Ionization can bo produced by charged partiales as wall<br />
as: by lfpht, Actu g thc ionization by f movine electrons,
protons, Deuterons and cb-particlcs, is 8. practical basis of<br />
cxperimcntzl nuclcar physics,<br />
exert an electric ficld and if their velocity is high enough, pro-<br />
ducs ianization,<br />
14<br />
These particles movinp past atoms<br />
Such particlos may. enter a chamber like tha one<br />
sfiown in tho above figwro through a thin Toil<br />
I<br />
They aro stoppod by<br />
a 'thicker foil, Onc of' thc: simplo basic axperimcnts Of' nuoloar<br />
,<br />
physio8 is to find out what is the foil thiclcncss that will stop<br />
a curtain kind of particlc carrying a plvcn onorgy,<br />
Thcro arc other interactions bctwcon light and matt@r,<br />
I<br />
Cqc of thcm 1s the Cormpton cffcct,<br />
This is tho scattering of a<br />
light quantum on a frcc clcctron, that is, one not bound in an<br />
atom.<br />
If'<br />
tho clcctron is not froc but its binding energy is nogli-<br />
gtblc compared to the cncrgy of thc light quantum, WQ may st111<br />
talk of a Gompton effect,<br />
In this offcct;, ths behavior of light<br />
can be satisfactorily dcscribcd by a particle piotzlro and vw thore-<br />
fare draw in the following fipurc thc light quantum as a particle,<br />
impinpin? lipfit quantum<br />
0<br />
scattcrod light quentum<br />
Tho momcntwn and oncrg:y impartcd to thc clcctron by the light;<br />
f<br />
qgnn'cum may bc calculated by laws of conservation of cncrgy and<br />
mbrncntum in rclativistic mechanics. Thc shortor arrow on iho scat-<br />
r<br />
tOrcd lipht qLuonturn indicated that thc light quzntwn has pivon part<br />
of its enorgy and rnornE;ntum to tho elcctron,<br />
This is true if thc<br />
clcctron was originalby at rcst, which in przcticc may be assumed<br />
f
15<br />
almost without sxception, Tho offect was discovered by A, E. Comp-<br />
ton,<br />
EIe observsd that most of the X-ray scattered throuph differ-<br />
e& angles has guffered an enorpy loss which dQpend8d on the anvle.<br />
The enarpy losa agrees quantitativsly with the on0 derived from tho<br />
energy-m.orflontum conservation Zaws, The Compton effect' ea the princri'pnl<br />
means by which "-rays (that; is, high frequency lfpht hd#$<br />
roduced in the Compton effect may be detected by thair<br />
ionizinp power,<br />
'<br />
Thors exfats a third kind of interaction betwocn lipht<br />
anh matter, this is the pair production. In this process one<br />
1P:Fht quantum disappears and an electron-positron pair i s created,<br />
T135.s cffoct can occur only for Ini7.h enor,qy 1iFht quanta, natn~fy,<br />
the enarc$y of tho lipht quanta must su.pply at least tho mass of<br />
the electron and the positron, It may in addition supply an arbi-<br />
I<br />
&aq amount of kinetic encrcy to thcse particlos, G3uantftatLvoly<br />
the light quantum must carry approxiaately at least 1 MeV (nom<br />
accurately, at least 1,024 MeV), Tho proccss can not occur in<br />
empty spaco. It requires tho prcsence of mattor, Tho reason for<br />
is is not tha6 the process requires any other raw material than<br />
the onerqy carried by the lipht quantum but pair production in<br />
empty space would violate tho law al? conssrvatfon of momontun.<br />
pracoss can happcn in tho nciqhborhood of 0. nuclrus. Tho<br />
nqcloua having a preat mass can taka up almost arbitrary amounts of<br />
mobontun, It thcmfarc acts like r4 pla'clarm in the n8ighborhoad of<br />
I<br />
which clcctric flolds can act to p~oducc R pair and transfer any<br />
oxtra nomentum to the nuclousr Tho proccas occurs with proatost<br />
prpbability noar to stronyly ohmgad nvclei In whose neighborhood<br />
i<br />
i
16<br />
hiph electrostatic fields axist,<br />
The highost energy lipht quanta<br />
formed in cosmic rays ~ O S O their energy almost completely by the<br />
pair production PFOCOBS,<br />
The hichor onergyy -rays which occur in<br />
ntwlear proccsses lose cncrgy primarily by the Compton effect end<br />
a r absorbed ~ only to a small cxtcnt; by pair production ~ ~ Q C O S I S ~<br />
1-s tho energy of thoT-.rays pots ~OWOP, photo ionization plays an<br />
increasingly important role in tho energy loss in addition to tho<br />
I
17<br />
U<br />
Septambor 21, 1943<br />
LA 24 (3)<br />
LECTURE SERIES ON - NUCLEAR PmSGCS<br />
.,,<br />
st Scrios: Terminology Lecturer: 5, M, McMillan<br />
LECTVFG 111: FGRCES HGLPIlTC ,THE NUCLTET TOGETEER<br />
As tho bcst known kind of forco acting betweon particles,<br />
wo might first considor elcctrpa~tatic forcos.<br />
live know that par-<br />
tilcles of‘ likc charga rcpol eaoh othor and particles ol unlike<br />
attract each Other, tho f OPCO be in8<br />
to the square 05 the distanco,<br />
oqncept of<br />
from 2nf ini ty<br />
from each other.<br />
It is a<br />
anergy ins toad Of that<br />
inversoly proportionax<br />
simplification<br />
of f OFCO.<br />
to<br />
th@<br />
Thc poten-<br />
thc amount of work naedod to bring the particlos<br />
whore thoy do not interact<br />
and potential anorgy is positive,<br />
dofini tc dis tancc<br />
If thorn is repulsion, work will nood to be done<br />
In casc of attraction less than<br />
no work has to bs donc: and tho potcntial energy is ncgativc,<br />
patcntial snergy is the intagral of the forcc,<br />
For thc invorso<br />
law of clectrostatics, tho potential is proportional to on0<br />
Tho<br />
ovcr thc distance,<br />
Thc fipurc shows such an attractiva potantgal,<br />
e<br />
t h distanoa bctwecn two particles boing applfcd a8 abscissa wid<br />
ttp potential as ordinate,<br />
Tharo is no olcctrostatic<br />
B neutron and a<br />
proton since the formcr does not carry a charge, Actually thore<br />
a6poass t9 bo no stzoablc forco acting botwcon thesc two particlos<br />
unless they apprmch to a distance comparable to<br />
-13<br />
tboir rad%$ which is ab0u.t: 10 ern, Thero a<br />
larpc attrac”c;ve forco suddcnly appsars as shown
18<br />
iguro which shows ths poto<br />
a1 encrgy acting between these<br />
twy9 particles*<br />
The dctailcd dspsndonco OF this pot;ontial on tho<br />
distance 28 npt known but the<br />
B u,<br />
A potential af this typ<br />
and protans, but a190 botwcon two neutrons,<br />
a<br />
ding to tho 1<br />
tanGOS, but<br />
atc shape appears to bo that<br />
IdGrablc, narncly<br />
not; only hotweon neutrons<br />
Two protons will repol<br />
cctrostatics at proat dis-<br />
distancos the same kind of<br />
attraction ~ppsaps 88 bstwocn ncutrons and pro-<br />
tons,. "ha goten 1 bctwcon two protons is 21-<br />
lustrated in th.0 graph.<br />
Gne can sea in this<br />
graph tho dSsCinctton botwson long rapgo forces<br />
as thc clcctrostatic forces wh<br />
such as those due to bhc pot<br />
h act at a pwatcr distancc,<br />
whoso effect 9s noticeable only if tho particles am closc,<br />
cowon cxamplo that might rnPLke this<br />
rcpulsivo forc:: occurring whoiz ripid bodice arc broueht<br />
into contaat and the more slowly va<br />
1<br />
A<br />
If wc now want build from sovcral ncud<br />
pratons, it is ncccssary at lcast thc r;d,gias of those<br />
should touch 80 that tho short rnnpQ nttrnctivc forces<br />
o upcratlvc,<br />
mi8 roquircrnont i s not neccsaary in the<br />
c olcctron Tn thc @.toms, which am held to,vather by thG<br />
e long rangc olcctrostatic attraction pnd in which theref'orc tho
l<br />
19<br />
avcrago distance bctwcca particles is much grcatos, than tho radii<br />
of thcso pnrtiolos? Anothur reason why the noutrons and protons<br />
c o:ns tittat ing the nuclei<br />
hcavy part iclc s mq7<br />
aan<br />
easily<br />
so close togothor is the fact that<br />
have srnc.llcr wavo length,<br />
Accord-<br />
ing to quantum mchmicsI it is not possible to confino 8. particle<br />
within a Sl?s. 11 spacc without effoctivcly pivinp it a similarly<br />
sTall wavc lonpth which according to thc de 3, ogl i D relation moans<br />
c? ,hiph momentum,<br />
Heavy particles may possess such R high momentum<br />
without thoraby obtaining an exccssivc kinctic energy which would<br />
surpass tho nuclmr binding( forcos and disrupt thc nuc1Cus.<br />
Sinco<br />
clcctrons confined to nuclei would have such a high kinotic cncrgy<br />
WCJ muat not expect to encountor than as nuclear constitucnts,<br />
othor words, whcn an clcctron is attractcd to 9 nuclous, thc at-<br />
traction will incroasc its kinetic cnergy and mcrncntum and thoro-<br />
by will dccroasc, its wave lcn,yth, but whcn the clcctron has ap-<br />
pxioachcd to within thc distance of thc nucleer radius its wave<br />
Longth is still not as short as the nuclaar radius and thcroforo<br />
one cnn not think of it as bcing confincd to thc nuclQusa++<br />
nqturc operating<br />
cbctrons do not<br />
rangc forcos may<br />
that such short<br />
hccvicr psrticlos,<br />
mcrc is no cssentifll diffcrcnce bctwccn tho laws of<br />
Sn atoms and nuolci but in the ntorns thct light<br />
as I! rulc comG: cloec enough so that thc short<br />
becomo cffcctivc,<br />
It is not at all improbable<br />
rnngc forces 8C t bctwccn clcctrons as bc twecn tho<br />
Only cosmic ray cloctrons possessing vcry high<br />
may bc squcczcd Into a smnllcr spacc than.<br />
thcy usually occupy in atoms, "'his hEppcns in donsc stars,. The<br />
cncr,q which is necessary to pivc c;loctrons high rnomcntum associad<br />
with thcirmoro close confincrncnt Is dcrivod In this caso from<br />
g F avi t t! ti on.<br />
In<br />
!
energios may come clo~o enough to othcr particles and iy1 those<br />
cascs it is rcasonablc to look for short rRngo potmtinls,<br />
Thc: conclusion is that a nuclous tcnds to bc a more or<br />
I<br />
Ic(ss closcly packed assembly of ncutrbns and protons. Such an as-<br />
@<br />
scmbly is shown schomatically,<br />
prossnt protons, the crirclos noutrons,<br />
T ~ G croaacs re-<br />
ThG close<br />
armngemcnt is due to tho short rango attractions.<br />
On tho othcr hand, tbr~ is also a long 'ringc re-<br />
pu'lsion between tho protons which tends to pull tho nuclaua apart,<br />
Jf'wc: consider ncutrons and protons inixcd in a oLrtain proportion<br />
and incrcasG tho total nutnbcr of particles, wc will cxpock that<br />
I<br />
thjc long rclngo mpulsion will cvcntual-17 pravail and dostroy thc<br />
I<br />
ndclcus,<br />
I<br />
Indocd, tlm short rangc forces may bc considcrod tho<br />
same kind af forcc as cohcsfion.<br />
a surfacc which is txs<br />
mattor Rnd with it tho nuclcar rndius (R),<br />
Thoy will tend to give thc nuclaw<br />
small as poxsibfc, irrcrcnsing tho amount of<br />
Tho stabilLzLng tcn-<br />
I S<br />
d&xy of this forcc will bc proportional to tha surfacc! and thcmfore<br />
to R<br />
2 On thc othcr hand, thc total chargo in such a nuclous<br />
is proporttonal to thc valumc, thRt is to R3 and thc disz.uptivc<br />
. Thc:<br />
forcc is thc squp.rc of this chnrgc, dividcd by thc squarc of tho<br />
drlstmcc bctvcon protons which again is R,<br />
Thus, tho dis-<br />
PUptivc f'QrCo8 Would inCPCLlSO aEl R3 X 5 - ~ 4 '?.nd tl?cy will bo..<br />
R%<br />
comc prodominnnt with increasing R,<br />
, a<br />
Tho structuro as dcscribod abovc in a qunlitativc mnnnor<br />
myst not bo tckcn too literally. ,Thurc is n lot of kinctic cncrgy<br />
iG thc nuclcus 9,nd tho phrtichcs must not bc considcrcd as local-<br />
I<br />
izcd, protonso due to thoir ;nutunJ. rcpulsion, hpvC a tcndoncy
* This<br />
21<br />
to'bc pushed out toward thc surfacc, but &,so thc numbcr of neutrons<br />
and protons pcr un%t volwno tend not to dlffcir too @rcatl$.<br />
will bc particularly notlccd in light nuolci most of which have an<br />
approx9matcly equal number of tlautrons and Qrotons, Thc hcnvicst<br />
nublei hnvc almost twice as many ncutrons AS protons but thcrc: is<br />
mason to bclicvc thet q. nucleus consisting of ncutr-ons alonc (or<br />
clsc protons alonc) would not hold togothGP at all, For B bcttcr<br />
undcrstanding of the situation wc shall havc to explain thc operation<br />
Of' the ss+calXod cxclusion~ nri.clp,la, In order to apply this<br />
prfnciplc in pr'abticd. cndols, wc c?lsb hbvc to msntion thc ,snin of<br />
particlcs,<br />
Mcutrons, protons and also olcctP0,ns carry an angular<br />
rnqmcntwn or spin which in quantum units has thc sfzo of 8, One<br />
rmg think of such c pnrticlc as rotating around nn axis which might<br />
I<br />
'ab oriontcd in spncc in differcnt ways. For E completc description<br />
04 such e particlc, it is not sufffcicnt to say where it is or in<br />
which orbit it movcs, 'nut it is also nccGssary to plvc thc oricntntion<br />
of' its rotntioncl @xis. It is a conscqucncc of tho npplfcntion<br />
quantum mcchanfcs thnt for 8 pnrticlc with thc intrinsic<br />
3<br />
aigular momentum z8 the directional angular momonturn may be descTibed<br />
by simply stating that the spin points upward or downward,<br />
antization of direction is closely related to the quantiaai<br />
tion of? angular monentm, It also should be remarked that, where-<br />
l<br />
a$ the angular momentum of elementary particles can be dbserved,<br />
e<br />
there is no direct way to investi~ate their angular velocity*<br />
There is, however, a further indication or rotation of such partic-<br />
X~EI in their magnetic moiaent.<br />
1<br />
Sven neutrons, whose net charge is
The spin assumes its greatest importance in connection<br />
wi,th the exclusion principle,<br />
I<br />
particles rnuat not be in the sane state.'<br />
This principle states that two like<br />
To show how this prin-<br />
le operates and how it is connected with the apin, let us first<br />
sldetr an imapinary spinless electrdn.<br />
One may ~lassify the<br />
electron orbits around the helium nucleus by their energy,<br />
el+ctron will ocoupy the orbit 0% least energy.<br />
One<br />
Then tha second<br />
can not be assipbed the sms orbit bacausa this would<br />
vibrate the exclusion principle.<br />
The second electrm would have<br />
I<br />
tormove in a lsss stronply bound orbit and the holiw atom would<br />
no% have its particular stability.<br />
If, however, wo now remember<br />
that electrons have a epfn which is capable of pointing in two<br />
d!.,CTarant diwctlans, then it is poissibZe to put two electrons<br />
g in their spin dire<br />
tom, and a very stable arrmgament is obtained,<br />
The third<br />
alectran wotzZd have to agree in it@ spin direction with one of the<br />
twp ele'ctrons and tharofovle the exclusion principle provents it<br />
t
23<br />
I<br />
of protons, Tho simplest nucleus of this<br />
, consisting of two ne-utr<br />
tqo poasible spin orleneations in neutrons and protons, the exalu-<br />
I<br />
inoiple permits aJ.1 partiolss in tho Be4 nucleus to occu<br />
e (namely lowest) orbit, IT now, one mope neutron is added,<br />
I<br />
ip musrt go into a no orbit of higher energy, The resulting<br />
ntkleus would be He 5 but the energy of thtxt next orbit is SO high<br />
that it is not hcZd in at all by its conneation with the other<br />
PQpkicla5 and He5 does not exist, If two neutrons a m added,<br />
ldading to 6 the attraction bctwoen tho two additional noutrons<br />
i
J<br />
thb numbor of neutrons<br />
1<br />
1<br />
I<br />
I<br />
i<br />
-nurnbsr<br />
AI1 known nucZci thah fall in a rathor narrow band which<br />
1<br />
stbrts up at 85O oorrosponding to<br />
trbns and protons,<br />
Later the band<br />
thb inaroassd proportion fn nsutro<br />
oughly equal numbor of nou*<br />
This discussdon makos it rrlora understandablo<br />
number of partfclcs in nuclai is Iimitcd,<br />
!<br />
webe prclsant in a nucleus, the a1acbFostati.c forcss would disrupt<br />
I<br />
thb nucleus; if on tho othor<br />
If' tao many protons<br />
adding protons and<br />
incrcaso the numbcu, of' neutr s, then tho ~xclusbon princlp1.s will-<br />
,<br />
recjuf~g tho plarticlos to po intor such high orbits thab tho short;<br />
rahgo attract,ions aro no longor sufftalcnt; t o hold tho nuclcua<br />
1<br />
1<br />
1<br />
September 23, 1943<br />
LA 24 (4)<br />
1<br />
I<br />
I<br />
I<br />
r '<br />
Fgrat Sorics: T~m~nology *N, McMbllan;<br />
LRCTURZ IV:<br />
RZACTIONS BYT'IZTN T€E f\rtTCLEI<br />
I<br />
n+lof,<br />
edch other,<br />
,<br />
In order to be ablo to diactxss the roactions<br />
we hevc to invodtipato the forces with whaoh<br />
These forcss may bc obtainod from their<br />
I<br />
I
which i s shown in tho figure.,<br />
In: thia figure, tho potential cnorgy is glottcd as a Function of<br />
1<br />
th; distanco of tho nuclei ru<br />
static repulsion bctwccn ths nuclci is tho only forco acting,<br />
this, thc patsntlal incrcaacs as thc invcrsc first powor of thc<br />
e<br />
At prcatcr distancss, >tha olcctro,<br />
At smaller distancos, thc jfstickincss" of thc nuclei<br />
cokcs into play; that is, as soon as thc nuclci touch, thcy attrack<br />
cach othor.<br />
This rasults in a drop 9n potential cnorgy which wo<br />
have reprcscnted as a potcntfal wcll.<br />
tdis potcntlal well is so dcep as to ovor-ccmpcnsato thc clootro-<br />
stktic rcpulsion, and ih thirs ~asc tho potcntlal onorgy bccomcs<br />
1<br />
negative, as has boon actually shown in thc fPgurc.<br />
For not too hoavy nuclei,<br />
Tho strcneth of rcpulsion bctwc;cn nuclci and thcrcfcrc<br />
tUs height to which thc; potential barricr riscs, dopcnda on thc<br />
I<br />
prioduct OS nucloi charpcs, This product is srnR1Tcst for W 2 atoms,<br />
I<br />
I<br />
bu!t cvcn in this caso thc potential. barricr af a fcw hundrcd kV 5s<br />
to/ be cxpcctocri,<br />
This potcnt5al valua is of practical irnportancojh<br />
th;s well-known reaction botwocn two bcavy H nuclQP, which is to bc<br />
In<br />
cd below,<br />
For all othcr pairs of nuclcl, tho potantfa1 bar<br />
ovcn highor: that is, of thc ordcr of a fow million volts,<br />
I<br />
or! for two hsavy nuclei ovon considorably ~OPC*<br />
I<br />
I<br />
Tho stpong rcpulslion bctwoon nucloi cxplaine why nuclcar<br />
a raeictlons arc uncommon under ordinary circumstances, Indocd, tho<br />
I<br />
kipctic cncrgy b$ particlos due t o thcmal motion at room tomperae<br />
!
~<br />
tube is about 1/40 of an electron volt, while energies oncounterod<br />
in chemical reactions arc a few clootron volts.<br />
arb far too littlo to allow nuclei to got closc onough to each 0 t h<br />
so, that thoy may touch and rsaction mey startr<br />
i<br />
These energies<br />
arb known to proceed howcver in the intorior of the sun, where<br />
I<br />
tokporaturcs of the order of' many million degrees arc found,<br />
thb laboratory, it is necossary to impart: to thc nuclei high kinc-<br />
bib cncrgios, in order that they should got close anough t o PBBC~;,<br />
28<br />
Nuclear roactione<br />
In order to accclcrato particles sufficiently, dcvfcos cf<br />
two esspntially different kinds have boon used.<br />
tioally shown in tho figure:<br />
In<br />
Tho Q ~ Q is schbma-<br />
c : . , -<br />
A'<br />
* '<br />
+''*.*,<br />
[4'-&-g#Bw<br />
I<br />
--<br />
c<br />
pabt high volocity to nuolci is tlzo USC of thc cyclotron,<br />
dcirico, tho cnarpy is Fivon to tho nucloi in a sorics of small<br />
Xn that<br />
pu'shcst<br />
The nuclci aro hcld in circular paths by a strong magnotic<br />
fi'old. Tho accelerating clectric ficld is actually an oscillating<br />
f'i'cld which exerts ita accclcrating action, always when the nuclei<br />
rcach a cortafn part of tho cfrclo. Ahi2,c tho particlas @et; ELCccloratod,<br />
they mom in incroasing circlos and arc fSnally ojocted<br />
.as fast particles,<br />
Thc tyoe of interaction betwoon nuclei doscribod in tho
*<br />
I<br />
bc@.nning<br />
of this lccturc Sa quite different from th@ interaction<br />
bo,twccn nucloi and ncutrons,<br />
ThZs intoraction potential i s shown<br />
in bha figure as a<br />
Thcro is af cour80 no cloctrostatic repulsion bctwocn thc neutron$<br />
and 'tie nuclous, This is thc roason why noutrons, though thcy arc<br />
quite common particlcs in thc nuclei, have rcrnainad unknown so<br />
lohe]. A ncutron can approach any nuclcus f'xlccly, can rcact with<br />
that nuclcus (for instanc~, it can be capturod) and so it will<br />
stop being a frcc neutron%, Thus in tho laboratory, a. neutron<br />
may bo sccn just "on the fly", Gnc can obtain them by bombarding<br />
a isuitablc nuclous by a suitablo particlo, for inatancc, a proton.<br />
Thc partfclcs which hnvc bccn actually used as bombard-<br />
*<br />
ing particlcs in nuclear reactions are either uncharged or lightly<br />
I<br />
I<br />
charged particlcs,<br />
i<br />
hikh an cncrgy to ovcroornc tho potcntfal barrier, rind approach<br />
anothor nucleus sufficic,ntly close.<br />
flPat colzmn gfvcs tho namc of particles used in bombarding nuclci,<br />
If that particlo is an atomic nuclcus, its natation appcars in<br />
brhckofs,<br />
I<br />
A heavily charged pnrtiolc: rcquirca much too<br />
f'roquontly used particles arc dosignatod,<br />
In tho following tablc, the<br />
Thc sccond column gives spccinl symbols by whicrh thoso<br />
------- .**---------I---<br />
C_f_Y._.<br />
Whcrc i s anothor purely thcorotfcal rca8on why froc noutrcns aro<br />
dif'flcult to obscrf-c,d '??:ley ere supposed to bc unstablc, md arc<br />
cxpcctcd to transioFm by radioactivc oction into protons ornittIng<br />
antclcctron at tho sarni! tho, but A long thc bcforc this radioactive<br />
trnnsfomation could take placc, a noutron is usually capturod;
* dcutcron<br />
I 28<br />
'r 1<br />
1<br />
proton<br />
d<br />
t%<br />
ncutrcm<br />
! radiation, oP' 7 -pays<br />
I<br />
cloctran<br />
:this Xist, thc nuchi El3<br />
P<br />
n<br />
1'<br />
rr3<br />
and He3 might bc addod, but th@y are<br />
i<br />
pr@ctically unavailable and so thq havo not bcon used In our ox-<br />
pcbbcnts of this khd.<br />
If OMC we.nta to use ncukrons ~8 bombard-<br />
ink particles, onc has to dorivo thorn from nuclcar reactions.<br />
If tkc bombarding particlo in a nuclcar ranction tzi'<br />
chargod, one wlll cxpoct no nuclcar rcaction to occur unlcas tho<br />
r<br />
partiolc has suf'ficiont kinctic oizcrgy to DVCPC~C tho<br />
haryior bctwoGn the two particles, Tbt13~ if tho yfold<br />
of: such a rcaction is plottcd against thc cncrg:,- CY. t?nc bombard-.<br />
in& pnrtfc?!~, ono cx~octs tb.c yield to remain zor? v;,3<br />
cngrgy V mr2 then riscj in somo manner,<br />
tho f?.gvi~ by thc; full ~PVC;,<br />
Yl".C?(1 I<br />
quant<br />
1<br />
I r+<br />
! I<br />
l
.,<br />
29<br />
bombarding particle extends exactly up to thb maximum of the<br />
potential barrier. Due to the wave nature of particles, a ceptain<br />
fuzziness of the orbit is unavoidable and even though the energy<br />
does not suffice to carry the partscle right to the top, there<br />
retnains a small probability that the particle may ri leak through"<br />
the barrier.<br />
limited. extent;,<br />
Therefore, if the particle is piven less and less<br />
energy and is therefore turned back according to classical rnecha-<br />
nlcs at preater and greater distances from the top of the poten?<br />
tial barrier, the probRbiLity that it can "leak through" the bar-<br />
rier decrerses rapidly.<br />
But the fuzziness in quantum mechanics is only of a<br />
Tl?a actual yield to be expacted accoI'dp<br />
ink to qwntwn mechanics is shown as a dotted line,<br />
At high ener-<br />
gies, where the particle can get easily over the potential barrier,<br />
the yield mwy approach a constant value or may even decrease again,<br />
The gield curve may show I;, sharp maximum, This is il-<br />
lustrated by the figure.<br />
I<br />
yie Id<br />
T'<br />
-+ enorgy<br />
To understand this phonoinenon, we must consider<br />
ing and<br />
bohbarded particle aftar they have touched end ha,ve formed a so+<br />
*<br />
cazl-ed compound nucleus. This compound nucleus has quantum levels<br />
an@ A sharp maximum occurs in the yield curve if the energy of the<br />
bornbarding particle is just ripht for the formation of such 8 guan<br />
turn level. Thus the rnaximum is due to pn agreement between the<br />
I
enerpies of tha particlos bof'ore reactioh end tho energy state of<br />
a tho compound nucleus, Such an agreomont is called resonance. The<br />
phenomena due t9 such resonance are analogous to those that nre<br />
encounterod in clasaikal rnocknnics when the frequencies<br />
vibrating strings become equal to each other,<br />
of two<br />
The methemetical<br />
theory of those two situations are similar c?nd the shape of the<br />
yield curve near a resonance maximum is essentially tho same as<br />
,<br />
certzin resonp~nce curvos encountered in classical mechanics,<br />
It may hsppcn thRt in a reaction, the produced particles<br />
have greater intrinsic energy content than the initial particles.<br />
In'this caseJ the reaction is possible only if' this energy def'i-<br />
cicncy is supplied by the kinetic enorgy of the original particles,<br />
men a rtlinimum kinetic cnergy is required for tho reaction to pro-<br />
90<br />
ce@d, and tho yield rcmp.lns zero up to this energy,<br />
The energy at<br />
which tho reaction first becomes possible is called thc threshold<br />
of the reaction,<br />
I<br />
Unlikc tho case of the potential barriers, this<br />
threshold is sharp, and below it tho reectfon cannot proceed even<br />
with B small probabflity,<br />
As CJI<br />
exemplc for 1. sirnplo nuclear reaction, we may con-<br />
aidier the reaction between two E2 nuclei.<br />
There is ono possibility<br />
thflt the two nuclei mag stick to,?ether cnd from an CG-pQrticle.<br />
BuC such an a-particle would have a vary high energy contont,<br />
This enorgy might; be usod for tho omission of radiation, And if<br />
it 'is 80 used, the a-prticlo m2.y stick topether permnnently,<br />
hission of radlction takes$ howcvc~?, too lone 8 time and it is<br />
rl,<br />
more probsblc that before a? m y could bo ernittod, t;heCf;-particlc<br />
r<br />
would Ply apept again, It might fly npart into two deuterons, I.&,<br />
I
one would yct back the oripinal nuclei end tho net effect would bc<br />
rnc~oly a collision in which tho deuterium nuclei have doflccted<br />
cach othep,<br />
Gt'ner reactions c.rc also possible,<br />
tektims consistcd of a noutron and e, proton,<br />
of their colllsion, wo hnva two noutrons and two protons and cithcr<br />
31<br />
Bacli of' tho dou-<br />
Thus, in thc rnomont<br />
one of thcsc four pt-wticles mipht fly off', Thus we might got as a<br />
.<br />
result of' the reaction a proton r.nd F. H3so~ n ncutron and He 3<br />
Both these reactions have been obsorved, Thcty proceed with n con-<br />
$idcrable cvolution of onorgy but thc initial<br />
havc had appreciable kinetic oncrgy, othcrvuiso the reaction bccomcs<br />
quftc improbnblo, sincc tho pmticlcs cannot approach sufficiently<br />
c2ose.<br />
Anothcr possiblc rcaction would be tinc formation of a<br />
nuclcus consisting of two protons rnd of nnothcr nucleus consisting<br />
ofltwo noutrons, Thcsc nucloi arc not stable, Finally, throc or<br />
fo
m<br />
ovsntually by using soparntod Li smplssj in which only Lt7 or<br />
Li6 was prcsont,<br />
Thc pnrticular reaction quotod RbdTFo,<br />
is,/ cndothcmic, approxgmntaly 2llV kinotic cnorgy being nacdbd in<br />
thk prstons for the roaction to proCewd,<br />
in0iabn.t: grotbas over tho t oahold will turn in &qatosh part<br />
ihko kenetie cacrgy of khc houtx"oni<br />
onorg.fr ona might gat neutrons of pivan enorgics,<br />
Tho exccbk onoi?&y of tho<br />
By cbntt-ollfng $he proton<br />
$ ~ . q ronction is<br />
I<br />
actually tho best way to produco f'nfrly monoenergotic noutrons In<br />
th& p~rqa of e, fcw tonths of a MV,<br />
!<br />
'<br />
Tho roaction<br />
I ~09+ ~2- ~30-j- n<br />
a vary copious SOUPCC of ncutrons if deuterons of' 3 MV or rndrO<br />
arb umd.<br />
Such Routorons may easily bo producod i n cyclotrons,<br />
Thb roaction 'is strongly oxotbomnio e.nd givos thcrafaro noutrons<br />
high oncrgiea (around 10 I\!V>, Tho ncutrons so prodwcod<br />
,<br />
do'not CVOP haw a well doffnod Cnorgy, Thc 13' produccd in tho<br />
rohcticln may bo Lcft behind in vmious statos of Qxcitation, and<br />
on#WY Of 8"<br />
pons carry away tho varying mounts of cnargy that havo<br />
used vp in exciting<br />
is omitted In thc f'o<br />
at in tho d+d roaction<br />
VOTY light nuchk obtalncd in this<br />
ntually, *the axcitation<br />
citation occups<br />
tion (fop instan<br />
not; havc exoitad lcvela in thc ncighborhood of 3<br />
y avgilablc in thig rdactl~n~ This one<br />
4 MV, which is<br />
thcrcfora<br />
d off as kinotic onorgy of tho reaction products and one<br />
obtains noutrdns of wolbdcfinod oncrgZcs;
Locturor:<br />
Ii,M,McMillan<br />
I<br />
Historically thc first nuclcar transfomntion obscrvod<br />
wss radioQctivlty, Jt W R ~ naticod that Borne mincrals continunlly<br />
waL idmliificd with €Io* nuclei, tho B -radir.tion with olcctrons,<br />
r.nd ths "y -rndiction with qucnta of olcotrmagnotic radiation.<br />
In a radioaotivc process Q nuclrtua unclorpocs a spsntnncous chnngo,<br />
inlwhich it may cmlt a part of its own substcnco.<br />
1<br />
The v-rndiat;im<br />
ho$cvcr corrcsponds morclg to n scttlinp down of a nuclous from a<br />
,<br />
higher cnor.gT stc,t;c into P.<br />
lowcrI mom stablo stRtcl.<br />
Follows some other rcaction n8d is accompnnicd by xz'o chcrnical<br />
change of thc SubstFnoc.<br />
i<br />
It uszznlly<br />
Whcn an =-ray. is emitted, thc cjcctcd pnrticlo carricts<br />
twb chargos end tlic nuGLous is Icft with two chargcs loss, that is,<br />
it mom8 down by two placos in thc poriodic system,<br />
tlid ~ O Qti C on<br />
, Rfi226w Rag22 + Er04<br />
1<br />
An oxcrmplo is<br />
iniwhidh thc clcmant Radium, which is rinalngous in its proportics<br />
I<br />
@<br />
to Barium, is transfornod into tho noblo gas Radon, This reaction<br />
is balanccd in tho sGmo mnnnor as the artificlel rcaotions mom<br />
tiinad aarlicr but r,n osscntinl dif'rcrcncc is that it OCCUPS spon-<br />
tnrilcously,<br />
FO1l.owing tho abovc rcnction, Radon emits in turn an
tances, thc,stickinosa of thc nuclcus will c,auso tho potential<br />
cqorgy t o bo loworedl md near r = Q wo find a patxntinl wall,<br />
Tklc h0riz;ontnl linc not far from tho top of the wcll rcprosonts<br />
tbic oncrgy lcvcl which tbc a,-partiolc occupics within the nua2aus.<br />
This oncrgy IcvcS in a radioactivo nucbus 3.s highor than tho pow<br />
tcntiaP cnorgy a t vcry high r valucs,<br />
This must bo SO bccnusc WQ<br />
know that at high r vaZuos thc Ot;-particlc may appear carrying 00x1,<br />
s&darablo khnctic cncrgy, which kinotio cnorgy must bc dcrivod<br />
from tho onargy that the eparticlc originally poasossod in tho<br />
nuolous, but in opdor that tho G-particlo should cncapo the nu-<br />
cious (moving .long<br />
tlzc dotted linc In tho figuro) it must cros8<br />
~l rcgion of high potantial oncrgy* According to clnssicnl mocharxtcs<br />
this 1s fmpossiblc,<br />
In qucntum mechnnios thc fuzzinoss Q$ par-<br />
thclcs mnkcs ponotrntion through thc potontin1 barrium? possiblc b\l;<br />
t1sc probability of thc particla lcakSng through is vo:-y<br />
small,<br />
This is true in pnrtfculnr if thc potcnticl barrier is high;<br />
, filnds that the avcragc timc which thc a-pnrticlc fs supposcd to<br />
spcnd in thc nuclcuhs bcforc lcalcing out by tho qumtwn rncchnnical<br />
mcthod is in fair aprccrncfit w ith tho tirncs actur:lly abscrvod,<br />
scpsitivo dcpondcnca of th5-a timc on thc hcipht of Chci potcntial<br />
i<br />
bn:rricr cxplnins why radioactivc decay times vary from a smnll<br />
Ono<br />
The<br />
fsaction of? Q<br />
sccond to scvernl billion ycmse<br />
e<br />
s<br />
i<br />
Tho rnnthomationl law of radioactive docay is basod on<br />
th,o fact that each radisactivu :rti;:;~~~ has a constant probability<br />
of! docmy,<br />
l<br />
It 26 to bc notod t;v+, this probability of dccay is<br />
in,dopondcnt of thc time tBo pwticuI,ar nucleus hns already livod,<br />
It fol,lows thnt in n bunch of radionotfvo nucloi, tho nwnbor of
,<br />
i<br />
nuclei dccaying per sccond<br />
proscnt.<br />
If ono plots tho<br />
35<br />
is proportional to tho numbor of nualci<br />
numbor, of nuclai as a function of tho,<br />
givon by tho formula<br />
N =: N,o<br />
" At<br />
No 53 the origiaal number of neutrons (which were present at t=G)<br />
nqdhis the reciprocnl of the so-called mean life,<br />
s$ood that some nuclei live shorter, 2nd cL few,<br />
I<br />
lqneer, th9.n 'this mem life,<br />
active decay by the hcllE-lffe which is piven by<br />
Pt is under-<br />
considerably<br />
It is more usual to dssoribe the rnc3.w<br />
hclf-life Z 0,6g3/'x<br />
The helf-lifs is the time in which hnlf of the original number of<br />
the nuclei hns decayed.<br />
I<br />
In discussing the ,@*decay we shall consider an artifi-<br />
c i R 1Xy JrPd i oac t 1.ve subs t c.nc e, flr t i Fi c 2 n 1 ly rad i oa c t ive nuc 1 e 1<br />
obey the spme laws as nRturnlly radioactive ones; but instead of<br />
being enoountsred in nature, they haw to be prOdUc8d by bombarding<br />
a hnturally occurring nucleus by a light nucJouB,<br />
In this way,<br />
maby 6 -active substances have been produced while artificial<br />
activities are very rwe,<br />
CG-<br />
A well-known RrtfficSaZ owactive sub-<br />
stance is obt8ined by bombarding the on0 stabla 2sotops of Sodium<br />
by' Deuterium<br />
N c L .I. ~ ~ H2 w .Na24 .f. H1<br />
I<br />
!
In: this equation we have written e" instond of 10 in order to show<br />
that negatlve electron is emitted, Tha nucleus produced in ths<br />
reiaction, Magnesium, has an8 more charge than Sodium r.nd thus the<br />
total charge is qQnsBrvedr<br />
It has beop stnted that in R nuclous thoro is no room fo$<br />
electrons, How is it then possible that the nucleus should emit<br />
Rnt oloctran? It must be postulnted that tho oloctron is c3retlCcd<br />
in tho p-disintegration ~POCBBS, Cther prooosscs are indeed known<br />
in which pnrticlos me created,<br />
olsotrbn-positron pnfrs from radiationc<br />
pr:ocess in<br />
-Pctj.vJ,ty is the<br />
a-p++e-<br />
Gno mample is the crontion of<br />
Tho c:!smentary<br />
creation<br />
roacti~n. Tho enorgy dolivorcsd ts tho olectron is partly duo to<br />
tho mass dfffcrencc botwoen the proton and tho neutron end partly<br />
it is drawn from tho onerey store of the @-active nucleua ilyl which<br />
thje noutron was orig:lnnlly formedl<br />
ex,othormic.<br />
The above simpln process is<br />
It is ags~m~d that tho nsukron iesslf is @-activer<br />
This actfvity howcvor te.kOg R longror tirnc than other prosoasos<br />
h ncutrona can discpponr p.nd thcrefore the @ -activity of<br />
e<br />
froo nwtrona has as yot not boon obsorved,<br />
EC t ion<br />
AnoAthcr kind of<br />
p-nctivo nuolous is fomed in tha re-<br />
~14 + ~2 .-+015 + n<br />
The Ol5 nuclous decays according to the aquntion<br />
015-815 +
37<br />
Q<br />
This timc a positivc electron is emitted, This kind of' activity<br />
is'n littl o rarer than tho one discussed above, It does not OCCUP<br />
in tho natixrafly radioactive series. The olemontary reaction in<br />
thfs case is<br />
This timo thc olcmentnry roaction is endothomic,<br />
This monnsI<br />
thdt tho proton is sttlblc nnd a positron can be emitted only if<br />
tha ehargy storo of tho fl-active nucleus can supply an onergy sur-<br />
f'icriont to overcome tho mass difforonoo bctwcon proton and neutron,<br />
Sometimes it hr7.ppens that n nucl'cus instead OF omitting<br />
a positron absorbs one of its own electrons,<br />
tran will bc onc of' an inner layer in. an atom, tb?tz E\<br />
K-clcctron,<br />
~'?ij s proccss requires R<br />
As Q rule this elac-<br />
so-called<br />
samcwhat smc.1 Icr cnergy than<br />
tho emission of a positron beccuao af th:: mass (an& corrcsponding<br />
snorgy) of the positron and eloctron.<br />
possible for R nucleus to ccpturo a positron,<br />
In principlo it would bo<br />
But even tho capture<br />
of an electron is nn improbablc procsss in spite of the fact that<br />
tho clactron is neap to the nucleus a11 tho tSmc,<br />
It practically<br />
ncvcr happens that 8 positron on its briof' passage nofir a nuoleus<br />
is absorbed,<br />
ns of the a-decay,<br />
Tho mathematical dcscription of the fl -dccay is tho smne<br />
sccond to hundreds of ycarsr<br />
Tho hnlf-livos mngo Tram R<br />
fraction of a<br />
Slownoss of thcfi3-procoss however<br />
ccnnot bc explained by th:: prcsoncc of a potential barrier slnco<br />
tho clcctron has n long cnough wave lcngth to loak thraugh nuclcar<br />
in exceedingly short<br />
long lives of B -pro-<br />
ccssos (oven a fraction of n sccond is 8 long time on a nuclcar
time scale) must be expla.lned by an inherent unwillingness for the<br />
v neutrons P M protons ~ to transform into each other, We must acm<br />
capt the Pact that the elementary creation processes mentioned<br />
above havo a low probability and the correspondine reactions pro-<br />
coed at a slow rate.<br />
This also explains the fact why bombardment<br />
04 nuclei by electrons does not produce typical buclear reactfons;<br />
The only action of electrons on nuclerl which has been observed is<br />
due to the electric Field of the e2ectr.on which ,$ives rise to sirni-<br />
lar excltation vrocessos as absorption of<br />
-rays.<br />
Btransf omations are frequently represented in. nuclear<br />
38<br />
The ordinnto gives the number of neutrons, the abscissa the number<br />
of ppotons. Dots in the diagram rcpresent nuclei, diagonal arrow8<br />
reprosant @ wtransf'omations. Tho Q -transformations actually<br />
limit the repion in this dia pan in which stable nuclei can be<br />
.$ound.<br />
If a nixclous contalns too many protons, it will ernit one<br />
or mom positrons and the nucl.eus ~1.11 thereby return to the belt<br />
of stable nuclei,<br />
sfons will accomplish the same tklinp,<br />
in singlo steps,<br />
If too many neutrons are present, electron omis<br />
These eixissions must happen<br />
Simultaneous emission of two electrons or two<br />
positrons is exceedinqly imprababJ.e, as will be seen from the fact<br />
that the probability of sucli a procsss must be the square of the
39<br />
e<br />
probability of a sinple emission process which we hRve stated is<br />
of itself improbable, No such double process haa ever been obsonred,.<br />
It, can happen howevar t1ia.t; a nucleus nay emit oither 4n<br />
electron or a positron, This occur$ when the nucleus can decrease<br />
its cnorpy efthor by tran.sfoming a neutron or by transforming a<br />
u<br />
proton into a neutron,<br />
Charts of the kind shown above aro useful in foliowing<br />
any kind of quclear reaction. In tho part of a chart shown in tho<br />
figure below, two reactions RPC? shown, In the first a proton is<br />
added; a@--activo nucleus is obtainod and a positron is emitted<br />
subs eq.uet?t ly<br />
-*<br />
-<br />
LA 24 (6)<br />
September 30, 1943<br />
Lr CTURT VI: Z) &DECAY XT, ) NETJTHCN RSAC TI OMS<br />
1) In a nuclear process in wh.lich a,@Hparticlo (oloctron)<br />
if3 emitted, the pt?rticular nuclaua goes T'rom one deflnitc enorgy<br />
state to another dcffnfto enorgy state. Hence ano would cmpcct<br />
l)i<br />
the omitted B-particlo to possess a defdnite kinetic B M B P ~ ~ .<br />
Cxperimentally, howevep, Et is found that the ,@-particles are not<br />
homogcyloous in enorgy<br />
.:<br />
but havo a distribution 8s shown In the<br />
f bllowing f ipro N<br />
Energy
angular rnornentw of L/2 quantum unit,<br />
This oxtraordlnary particlo
ducod by noutrond<br />
88<br />
(1) simplc? cnptura, usually fbilowed by $ha<br />
ma88 @hit Sa~gsr than the: rsrigShR3. hbeXou~3; (2) ine$&tia cob<br />
hielDni the ink~rnal mwgy of tho ~UC~QUS 2s changod,<br />
adbi0h mag bc regarded a8 an absorption of f?<br />
nnd a sztbsoqumt ro-omission of a nsut;ron,<br />
a noutron 5s absolrbod nnd<br />
proton is emitted..<br />
This re-<br />
xzoutron by a nuclcus<br />
(3) !kc; (n,p) reaction<br />
Thi8 raaction pro-<br />
coods w$th fast neutrons, although n slow neutron reaction is known<br />
ogan,<br />
Hqro tha product nuclous has an ntornic number one<br />
smallor t1zr.n tho bombarded nuc1cU-8..<br />
axanplos of this aro<br />
f -?z!c?YH<br />
ni<br />
- L 1 6 -k. n.-<br />
-. , &;a+<br />
(4 1 The b;r:,$ cy} reaction:<br />
Thoso reactions proceed with slaw neutrons, For henvier nuclei,<br />
tho c-particPC must overcome a potontial bhrrisr; honce fast<br />
neutrons are requircd,<br />
(5) The (n, 2n) rcaction in which two<br />
noutrbns are Gmittod requires fast neutrons, (6) For the sake of<br />
complotonoss, elastic collisions are inaludcd,<br />
enorgy of the imadiatod nucleus is unohnngod.<br />
LA 24 (7)<br />
Hero the intarnal<br />
WmJor 5, 1943<br />
ZC'IPURE SERIES 0,N TJUCLEAR PHYSICEJ<br />
ries: Tmninology Lecturer: E,M, McMillan<br />
The word fission mean8 splbtking or brenking apart,<br />
We<br />
am concerned here with a proccse in which a nuclous splits into<br />
two more or 159s oqual parts, Tho product nuclei are Fairly heavy.<br />
h fission processes arc known In which tho initial nuclous<br />
is one af the hoaviost nuclci,
43<br />
The reason f0r the finsS-on procoss is the olectroatatio<br />
botwoen protons,.<br />
In highly charged nuclei the potential<br />
*his repulsion ip SQ gpeat that by breaking apart the<br />
nuclaua can P berate energy,<br />
Tg LJlustrate this, WQ plot the ma88<br />
defect as a function of the atomic numbor far stab<br />
I<br />
mass<br />
defect<br />
atomic number<br />
-*<br />
Fig. 3.<br />
dof'oct ia a rnaasuro for the potential enorgy per particle<br />
MUCZOUS, Ono obtalns it by subtracting from Ch.o nuc1cq~<br />
o nearest intogar (which is the nm.bor of neutron<br />
the nucleus cnd has been cnllod tho mass x1<br />
0<br />
dividing by mass nwnbc3r).<br />
By going from Uranium to the SissSon pro-<br />
ducts, this patontial enargy per unit particle has considera<br />
sed and in the fiasion procees an oncrgy of tha ordor of<br />
be liberated,<br />
for a long t b o but it has been co<br />
i%iZ;y aP auch onorgy liberation ha b~en known<br />
sfon Qccurs when Uranium 28 bornbar d by neutrons, f<br />
ducts aro radboactivc.<br />
idorod4 unlikelyr, Actup~lly fig-<br />
ThesQ nctivitios have been observed, but for<br />
D. oonsldorablc t ims thcy were not Intorpretod RS due to fission pro-<br />
ducts bocaUsc of thc prcconcoived idea that the occu6renco of tho<br />
,
I<br />
fissiori process is practically impossible.<br />
44<br />
Finally Hahn observed<br />
that on@ of' the activities is definitely due to Barium.<br />
nucleba has only little more than half the mass of Uranium, it be-<br />
came evident, thgt tho Uranium nucleus did break into large frqmcn2;s<br />
Physicisbs the8 started to look for highly ionizfng particlast<br />
Since this<br />
GoULd be predicted thai fission di.J.1 give rise to sudh particles,<br />
because the great amounts of energy llberated in the process must<br />
evantually be dissipated in the form of ionization -processes+ The<br />
highly ionizing particles were easily found by letting fission prom<br />
ceed in an ionization chamber in which the ions formed are drawn out<br />
by an electric field. and collected on the electrodes.<br />
Et<br />
The pulses<br />
and corresponding ionizations obtained w0re m m times ~ stl?onger<br />
than thoso produced by =-particles<br />
most strongly ionizing particles,<br />
which up to that time were the<br />
A picture explaining the fission process is based on the<br />
droplet model ,of heavy nuclei.<br />
posed to be held togsther by surface tension,<br />
like an elastic skin.<br />
Liko droplets, heavy nuclei are sup-<br />
Surface tension acts<br />
It can be explained by the tendency of the<br />
constituent particles of the droplet to get in as close touch with<br />
each cther as possible,<br />
many contacts as they would in the interior.<br />
On the surf'ac'e the particles do not make as<br />
Thus a surface means<br />
ve potential energy, ancl there is consequently a tcndono-y<br />
to Form as small surfaces as possible.<br />
The charge af the nuclei acts in the opposite manner from<br />
surface tension in trying to gull the nucleus apart,<br />
As long an<br />
the droplet is spherical (as shown in Fig, 2) the charge, being<br />
symmetrical, will not be able to produce motion.<br />
If, however, the<br />
nucleus assumes an ellipsoidical shape, tho charges will accumulate
the droplqt<br />
sion can be<br />
charged nuclei<br />
.tron capture,<br />
larger tho charge, the smaller elongation w ill cause<br />
to break up4<br />
This pi tum explains therefor0 why fis-<br />
+<br />
expected wlth greatest probability in the most highly<br />
In UranSum and Thorium, fission has been induced by nsu-<br />
If the neutron is captured, its binding energy be-<br />
, come@ available, This energy is a8 a rub eventually. emitted in<br />
the f'om of ?-radiation, For that diatlon, however, consider-<br />
able tbe i s needed.<br />
The first effect oi the binding snergy is to<br />
set the particles within the nucleus into motion,<br />
may sot in and may pivs to the nuclmm, temporarily, an elliptical<br />
shape which 2s well known to<br />
Then fission may re~ult~<br />
234<br />
isotope u<br />
Uranim has two pri ipal isotopGs,<br />
is ve~y rare).<br />
An oscillation<br />
CUP in the oscillation of droplets.<br />
and'U235 (a third<br />
The u238 nucleus ia the parent of the<br />
a<br />
Radiuln series,<br />
nuclod undergo fisiion,<br />
u235 is that of th@ Actinium ser2~s. Both these<br />
In u238 the fission has a threshold of a<br />
littlo mope than 1 M.RcV~ The PPssSon cross-section,<br />
rf, varies<br />
with the enorgy of bombarding neutrons as shown in tb?e following
figure .<br />
I<br />
r<br />
4<br />
46<br />
Fig* 4<br />
The threshold energy Ls the amount of enorgy needed, beyond tho<br />
binding energy of neutron, to carry the nucleus beyond thc stability<br />
point, that is, to give to tho nucleus a suFFiciently ellip-<br />
soidfcal. 8hape.<br />
Since this energy I s only needed to overcome a<br />
potential barrier, tho questton arises why by tho leaking through<br />
process fission does Mot occur spontaneously.<br />
Tho answer is two-<br />
fold:<br />
first, the barrier I s high and broad; End socand, spontan-<br />
cous fissSon does occur, but at an cxcocdingly S!.OW<br />
Orr c<br />
There 18 no threshold 3.n tho fission u":'""',<br />
rato,<br />
mhis nucleus<br />
obtains by nautron cqturc an ovon number of neutrons and sincre<br />
nuclei with an even number of nwtrons have a stronger binding<br />
energy, mor0 enor8y is liberated in this noutron capture procmss<br />
than in the capture of U238.<br />
Therefore, enough energy is available<br />
to 8tmt the fission procsss ev~n if' the neutron brings no kinetic<br />
energy along.<br />
Fission in Thorium bohaves in a way which is similar to<br />
fission in u 238 ,<br />
The ,@-activity of the fission products may bo explainod<br />
a<br />
by considering the plot of tho stable nuclei,
oqual fraponts, the corresponding points indicated by tho solid<br />
line are considerably above the roglon of stable nuclei. The fiesion<br />
products will then get back ta that region by a series of<br />
a -decays, indicated in the figure by small arr3ws- Many chains<br />
of this kind have been found, Most of those activftics were not<br />
known previously. Rowever, soma oF them near tho end of the chafns<br />
and 1yIng close to the region of stability are known artificial<br />
activitfos, These activities have holpod to identify tho mass<br />
numbors of these radioactive chains, The atomic numbers could, of<br />
course, be found by invostigating tho chemical propertios.<br />
While thc product nuclei in fission haw approximakclg<br />
equal weight, there is a marked tendency for a slight dissymmetry,<br />
Tn tho most probably f'lssion process, tho ratio of massc8 of' tho<br />
flssion products is distinctly diffcren'c from unity. In spit0 of<br />
many atitempta, no explanation has bsen found which is quite simple
49<br />
v\rh$gh the nunbsr of atoms has decreased to 1/2 of .their original value. 1% is seen<br />
from the radioaotiae decay formula that ecm l/~ Also one has<br />
from which it follows tha%<br />
AT z h 2<br />
= ,6931 .<br />
The designation mean life os<br />
is justified by the fact that it is actually the<br />
average of the time an atom lives before decaying.<br />
In fact the number of' atoms<br />
whidh have lived for a time between .t; and t 4- dt is J!J(t) Ad%, Multiplying by<br />
t, integrating from 0 to a, and dividing by the original number of %toms, one<br />
gets for the mean life<br />
a<br />
L/?T(O) tN(t)A.dt l/n<br />
0<br />
It; is easy to verify tihat this expression is 1/A<br />
3<br />
which by definitios is equal to<br />
The number of atoms decaying per unit time is defined a8 . activity.<br />
...i... .- 1%<br />
is equal %a. N(t>a , Ro-t;ivities are mensure'd in units of curies. k curie corres-<br />
ponds to 3.7 x lolo disintegrations per second.<br />
The reason for the choice of this<br />
numbor was that it was supposed to be tho ntxnbor of disintegrations pcr socond<br />
occurring in one gram of' radium.<br />
Re0en.t measurements seem to show that the number<br />
of disintegrations In radium is actually somowha% lowor (the latest valtw is<br />
3r46 x 1O1O dfs/sco).<br />
But it is better to retain tihe number 3.7 x 1O1O dis/sec as<br />
a fhod unit.<br />
In modicino a different meaning is often adop-bed for the expression<br />
"radium equiva1ont.l'<br />
Thus a milligram radium equivalent may mean an amount of!<br />
substance whirth gives rise bohind a 10 m load shteld to the same density of<br />
ionization as would be caused by 1, mg of radium.<br />
Tho reason fur such a dsfinit-ion<br />
e pOwer,<br />
Tho change of<br />
the amount of radioactive substanco with time may be quite<br />
is that the biological effects of radioactivo su.bstances aro dm to their ionizing<br />
complicated if tihe radioaotiva substanco is itself a product of radioaotiva decay
.<br />
~r'<br />
I<br />
Own moxe particularly if tihe radioactive, stibstance is a member of a long radio-<br />
'How complicated such chains may bocomo can be il.lustratod by tho<br />
example of bhs Urabium family* The following tablo is esseylfially takon from tho<br />
hemistry and Yhysios, 27Lb Ed$tion, page 315,<br />
Tho da6a given in *he handboak are %he results or' an intort&ianol<br />
agree-<br />
QS+<br />
available information in 1950, In the meani;ime, ohangas aro<br />
hese chatlges have been incorporw6sd in tho following tabla.<br />
These are<br />
-P-"f -<br />
----<br />
Uranium<br />
changes neodod,<br />
JN"W.TIQiL<br />
-- --CrUCC-..ICII.*rY.-"---<br />
TABLE OF TEE RADIOACTIVF, ELRIIRRTS AND THRXR CONSTAPITS<br />
-".-*I -., +- -.-.-<br />
J$arne<br />
Series<br />
--I-<br />
Symbol<br />
R ~ l f period --.-.-- '$ Radiation Iso-bops -"-<br />
Uranium I 01. 4.56 x lo9 yrsI a, v<br />
Uranium X1 u41 24.6 dGyS<br />
49 Th<br />
Uranium X2<br />
u-xz 1,15 mine 49 PR<br />
Uranium IT (Brovium)<br />
UII<br />
2,68 x lO5 yrs. QL, U<br />
Ionium<br />
Io 6,9 x 104 yrs. U Th<br />
Radium Ra 1690 yrsr ad Ra<br />
Radon (Rmlium omanat%un,Niton) RQ 3.85 aaya Rn<br />
Radium A ' RWA 3.0 min. a Po<br />
Radium B Ra-B 26.8 min, 43 Pb<br />
' Radium C Ra-C "35 min 99.97% 4 B$.<br />
Radium C' Ra-C' ~4.4 10-4 s80 a Po<br />
Radium D (Radiolead) Ba-D 16.5 yoars 4 Pb<br />
Radium E Ra-R 5,O days 49 Bi<br />
Radiurn F (Polonium RWF 136 days Cic Po<br />
Radium fi 7 Ran' I O + V e * V 8 * I<br />
qe)e.eae Pb<br />
Pb206<br />
The ohango in tho number of' atoms N1<br />
Ra-C .*Cia.***. ;03'$u* Bi<br />
Ra-C" 1-4 mine 8 T1<br />
- c<br />
Ra SZ" 4 e e s *#e;**** Pb<br />
1<br />
of tho firs+ mornbor of p. radio-<br />
acrtivo sorios oboya tho s'implo differonti41 decay law monkionod in tho boginning<br />
af tho lecture<br />
1)<br />
wharo A1<br />
in 112,<br />
dIV1 = - XlNldt,<br />
is tho docay constant of the first moinbar of tho sorbs, Tho chdnga<br />
bho numbw of atoms of thn second member of tho series is<br />
dflz $Hid% - &32dt
51<br />
@rw of the right hand side i s the inoreaae in +he number of atoms N2<br />
due $0 the deaay gf tho firs% member of tha<br />
wrih for the khird member<br />
tions hold for furtiher mombare of the series.<br />
The system of equations<br />
81 4 &I-,, e"<br />
a33<br />
,-h3t<br />
ents all, "21, a319 etc*, may be dstermirned<br />
the sxpreesions for N1, N2, N3!<br />
into %he dif'feren-bial equa.t;ions,<br />
ha time rata of ohango of these quantities, and second, writing tho<br />
for NlI NZ, N3# @to,, for t= 0, and equating ttheso quahkities 00 tho<br />
a1 amounts of the radionotive subdtancas. In this way a nectoneary and<br />
suf'f%oleM'b numbor of squations is obtained to find all tho ooefflic<br />
'<br />
far<br />
%im@ dependoms of Rl, N2, @toI, ikl 6ome of tho oases of p<br />
tanoe in %he Ra family are tabulated in the book of 1bej. Curie.<br />
conseqwnce Qf'<br />
brim. If'., eeg,, a uranium solut<br />
the equations discussed is the<br />
is IAft standine; 90r a tilw long oompared $0<br />
of tho fir&<br />
daugh.0<br />
wry skate w411 establish itself in which the smo numbe<br />
and are formed por unS time.<br />
This is t<br />
nl, or if the atzio of the nutnbor of uranium atoms t o tho<br />
1 atom,<br />
same as tho rabio of the period of tho uranium aOom 'bo<br />
This simplo concop% of radioaotive equilibrium is basod on<br />
ho origivlal mothcr substanca U1 has tag exceedingly long xifa and i ta<br />
e noglootad during tho growth of the daughter subs'WwQe<br />
If U-XL<br />
is precipitatad out oP o. tiraniurn solution, all tho U-XI will bo fgrmed in the
preoipieate and none in the solution,<br />
52<br />
The solution, however, retains all of the<br />
~ * b substanoe r<br />
U1C<br />
a<br />
deony in the precipitate.<br />
figure,<br />
Subsequently, U-X~ will grow in the solu+ion, wliile it will<br />
This growth and decay are illustrated in the following<br />
U-Xl activfty inprecfpltate<br />
J<br />
U-X1<br />
activity in solution<br />
The decay of U-Xl<br />
-t<br />
in ths precipi6ate follows the simple exponential radioactive<br />
decay law, The grmrth of' U-XI in the solution can be obtacliimcl from the fact?<br />
that ohcirjcai $cy+Ir2-*
e<br />
53<br />
method is used for determining tho half-life of radium. In this case, however,<br />
the oldest radium samples (about 40 pars old) begin to show an adtivlty slightly<br />
smaller than they had originally,<br />
Periods in the range betweon a few seclonds and a year may bo cohven-<br />
ielntly dstormfned by following Srho change of actifvity with time,<br />
Most; of tho<br />
known artificial activities fall into this rango,<br />
This is duo to tho fat% 'chat<br />
activities with very short or very long lives are hard to dctoc'c,<br />
Rvon where<br />
the simplest determination of life tima is feasible, a prooiso dotormination of<br />
tho llfa timo is difficulfj.<br />
Thus tho bost known poriod ol" a naturally activo<br />
olcmonk, that of rad.on, is known to a precision of *05$, wliilo .the poriod oi' tho<br />
woll knovm artificially activo nuclcus P32 is known to .2$,<br />
The usual accuracy<br />
of half-livos is less than 1 or 2$,<br />
This cxporimontal unocrkainty may givo rho<br />
to cons5.d.erablc orrors I% amounts of a radioactivo substanoo aro to bo calculatod<br />
from tho docay formula from moasurcrncnts taken long aft-cr th'ni: timo in which onc<br />
i s intoros+odr<br />
Special method6 ar,i nccdod to moasuro poriods *-diioh arc shortor than a<br />
second.<br />
One motlzod axpplicablo for radioactivo gasos is to let tho gas with a<br />
known vclooity strcam 5 , o. ~ pipc, past a sorios of collecting clcc-brodos, as<br />
shown in %lis figure.<br />
collecting olcctrodcs<br />
Tho ourront collcctcd by t-hoso oloctrwclcs is proportional to -the ionization<br />
caused in their neighborhaod by the radioactive gas and this in turn is proportional<br />
to the number of radioaotive atoms remaining in tho stream.<br />
From this<br />
measurement and the known volocity of %he gas, the dQCQY period may be calculated.
A similar method has been used for known gaseous substances.<br />
But<br />
54<br />
instead of the velocity of the gas stream, tho recoil v9locity of the radioactive<br />
atoms was utilized, tho recoil being due to the a-emission by which the short<br />
lived subs'cance was formed,<br />
This method is not reljable bacauso instead of<br />
single atoms, groups of atoms or crystallities may recoil, thus making elre<br />
recoil velocity uncertain,<br />
A more suooessful method utilizes ooincidence counters.<br />
13-coun<br />
counter<br />
coi L;+----l<br />
cedeno6 circuit<br />
Let us suppose that a parent substace emits 6-rays and tlioreby transF6rms into<br />
ana-active substanoe of vary short periodo The substanoct i s placed nexti to an<br />
and 60 a &-.counter and -the counters are connocted to a coincidence oircuit<br />
which will give counts only if the two counters aro activated within a time t,<br />
apart, , Than .tho number of coinoidanco counts to be expcctod i s<br />
"&<br />
Here Cmax 9s the number of counts to bo expeotod if tic is chosen as a very long<br />
-At<br />
time. This quaritity is multipliod by tho intogral of tho activi.ty Aa Of<br />
unit substance owr %he period t,, Thus C gives the number of counts obbainod<br />
ff only tlzoso &-disintegrations are sffective which occur aikhin B time tc<br />
aftor t;ho @ -disintegrations.<br />
Rapoating this cxparimont for various times tC<br />
e<br />
one can find from .tho variation of C. This oxporimcnt has bean onrried out<br />
for the Ra-C -1 Ra-C' pair of radiosotivn SLibstanccs.<br />
It is claimod that; thSs<br />
mothod is capablo of mcasuring half-lives ~C~WOOM 10" and low7 scoon'&s.
$. 24 (9)<br />
55<br />
Lsoturer:<br />
E* Segre<br />
One applicaC5.m of kke radioactrive decay law is tho de'aarrninatioa of<br />
anium oresI If<br />
originally no lead was prosont in the o m and if tl?coro has remairzod undia%urbed<br />
shoe its form&ion, the lead formed by rildioaotivs decay will give a meamre of<br />
has passed since tho famation of the ore, Another similar method<br />
s the helium formed by raclioac;tjive dacay and remaining in *he ore From<br />
.<br />
(I<br />
P<br />
t e age of tho old@$% oras, th.e preaumablo<br />
Values of tho oraor of 109 years ~i.avo bQoM obtained,<br />
of the earth can be d&@rmlned*<br />
ivo moasurcmen'cs aro usually made by countj.ng disintsgratione.<br />
ooourring in those caun+bs in'woduao errors into bho moasuramontst<br />
nvasfiigato tho influanoo of tho<br />
Tho simplost case is if tho eubstafice undor investigation doos not<br />
rociably during the fimo of .the mcastro<br />
shall QOY18idOF suoh a<br />
uma khat tho aoti<br />
ha probab511by o<br />
w corresponds t<br />
in a second Inahad of the<br />
o find pn wo subdivibc tho scoond in&o<br />
ivision fino anough 10 th& tho probability of find'sng<br />
-.-IC---,<br />
I -.u"I<br />
aioulaLlon o f those probabilikios
*<br />
oaunb irr on0 subdivision 2s nogliglblo. To find OW oounti in such a sub-<br />
div5sSon has a probabili'by m/k, To ind n p~rtiolas ilz kho first n divisions<br />
nono Zn %ho f6Jlowfng k-n qubdivisloii has tho probabili-by<br />
(m/k)n (1 -<br />
56<br />
counts distributcd In any manmr Rmong tha<br />
e<br />
k intervals, wc must<br />
k<br />
is last oxprossion by 2rlio binomla1 caoffiaicht (n> i*c., tho number of<br />
ways in which n intQrvals m n bo choscn among k. Thus wo obtain For pn<br />
I<br />
bcaomo s<br />
, gn z (k) (m/k)n (1 - m/k)lrc'<br />
ot' k go t o infinity, t;hc first facOor in %ho abovo oxprossion<br />
(E) 1<br />
wharoas tho last wil<br />
-m<br />
k(k - L),*.(lc * ~1 1) -,kg<br />
nr<br />
k-+ ns<br />
c^---r-..ir*-il-rr*I .-..u--.<br />
--<br />
tho formula simplifios to<br />
ynn<br />
Pn : - Chm<br />
11 s<br />
lcnwm as Poisson's formula, Xn thu fallowhg figure R plot of p~<br />
8 dofinod only for in.togra3<br />
tcr p$otting thosc in"egrn1 valuas ano can Snterpola%o tho<br />
ro@t of' Iiho ourvo,<br />
It ia usual in dc+orm%nmtions of half-livos to ploG tho<br />
occurronco pn of a glven numbor of counts.por socand n aga:Lns%<br />
n and to cornparo this plot with Poisson's formula, lf n Can bo<br />
Sit, one lias khoroby obtained tho doshwd
value of the counts par second and has at %ha same time oheoked thu propr<br />
57<br />
1)<br />
sta-bistical functioning of tho coun*arr<br />
If m is vary large, the plot of the Poisson formula 1.001~s as IndSoa-bed<br />
in Glza follawing figurer<br />
pn<br />
1<br />
m<br />
The ourvo has now a rathor symmetrical sharp maximum around the value n * m<br />
and goes quickly to zero when the dovia-bion of n trom rn baooms appreciablev<br />
In Chis case, Polssonts formula may be replaced by the Gauss approxhcition<br />
1 . 9 *-(n - m>2/2m<br />
-i&Gz<br />
Pn a *<br />
This formula can be clerZved by assuming %hat<br />
M .. m is small compared t o m,<br />
using the Stirling agproxima$ion for %he fao%.t-orial, baking %he logarithm of Pn<br />
and expanding into a Taylor series near "Clie, maximum n me<br />
The wi4th of the curve in tho above figure oan be defined by choosing a<br />
value w. in such a way that one half the area under %he ourve lies in the reg$on<br />
whara In - rnl is smallor than w. This means that +he deviation O f m from n<br />
has the same probabili-ty of being greater or of being less than wI<br />
ah<br />
-<br />
be shown to be<br />
w 0.6745<br />
Ahothsr hteresbing quantiity is the average value of (D=m)2.<br />
The value<br />
Thia 5s<br />
*<br />
denoting n<br />
gPn (n - d2-<br />
m by x and %he average value of its equars by x2 and, roplaoing<br />
by an inttngral (which is psrmtssiblo for %he Large rn values oonsidered hare)<br />
II.
a<br />
we obtain<br />
a time<br />
t, we have an even chanoo that by repeating the measurement many Cfmes<br />
the kverage result will be betwen m - 0.6745Elfl;;l and. m+0.6745 4- m or outside<br />
that interval. 0,6745 m is called the probable error and m the standard<br />
4- 4-<br />
doviation. Tho standard doviation increases with m, the relative standard<br />
deviation F / m = l/llfir decreases with m.<br />
If wo count 100 event6 the relativo<br />
standard deviation of our measurements 5.s lO$, if WB count 1000 events the r o b<br />
tive standard dsvia%ion is 3.1$,<br />
cCc.<br />
It is sometimes important to know the probability of an error cqual to a<br />
multiple+ times tiha standard devia.t.ione<br />
Theso probabilities are tabulaked, ocg,<br />
in the Handbook of Physics, 27th odi-tion, page 199.<br />
The quostion. sometimes arises what conolusSons can be drawn about tho<br />
docay constant if one finas no counts during tho timo of observation.<br />
I+ foLlows<br />
from Poisson's formula that the probability of finding no counts in an interval<br />
ia which m<br />
counts should hava bocyl oxpocted is<br />
po = e-m<br />
A closoly connoctod quostion is that of the ooincidenca corrections to<br />
be appltsd to obscrved counting rates.<br />
Lek us assme that a counter takes the<br />
time z to rocovmo This means khat af.t;er a count, +ha oouvftar remains insen-<br />
.'*<br />
sitivt: €or a period z If' we observe a counting rate mexp , the true oounting<br />
the exponential factor representing the correctiotn due to the :?act thab counts<br />
are missed if they follou eaoh oCher at intervals smaller than z e Assuming<br />
that the time z is short;, so khat the probabili%y of missing any count due to<br />
the above qffeot is small, one finds that the correcJbd counting rate mcory<br />
2<br />
is
expresaed 'In terms of the experimental countitng rate moxp by<br />
59<br />
Mora involved prob~erns arise .if the substance decays appreciably while<br />
the experiment is performed,<br />
One praatical problem is to find which pariods of<br />
measurement are best suited for tihe d8tf3rmfnatiOn of the half-life of the sub-<br />
stanoe.<br />
One might guess that it will be besf to let the subsl;ance daoay for man&'<br />
half-lives and to take advantage of the great change in counting rate which has<br />
accllrred in the meantime. IIowever, if one waitis too long, there is too grsat a<br />
loss of accuraoy due JGo the reduoed counting rate and tho corresponding greater<br />
influenoo of<br />
fluctuations.<br />
In the following figure the logarithm of the oounking<br />
rate is plotftd against the the,<br />
t<br />
+-<br />
-t I<br />
.c.<br />
Acoording t o the radioac-bive deoay lavr this plot should be a stira%@t l%w,<br />
Zf<br />
one detiemines ifs value at points 1 and 3 one might expect a graater acourag.<br />
in the slopo.<br />
But this 5s true only if the moasuremont at 3 doos not bocoma too<br />
inaccurate as hac been ipdicatod in the figuro by the vor'cical line.<br />
It is found<br />
actually $hat i-b is bsst Go choose tho %vo points 1 and 2 at which caunting rates<br />
are-dstorminad one, or one and a half, half-lives apart.<br />
The fluotuationa whioh we have discusfwd am a consequence of tho radio-<br />
actim docay law,<br />
It is thcreforo of fundamontal intarest to sco whoChor those<br />
fluctuations cmform to tho predictions, This was found to be tihs aase whoncmr<br />
0 maasuramonts vmro porf ormod with tho nooassary prooautions. Oyto mistake whioh<br />
frequently has led $0 spurious doviations from tho prodictud law OS fluotuations
ia to neglock Ghe coincidence correction of the counter,<br />
60<br />
I) C-.'.---W^-----<br />
IIIa PASSAGE OF ?ARTICLES TFIROUGH PUTTER .--u<br />
--- -part iGle s<br />
IBasuring deflections of gE-particles in eleatric and magnetic fields<br />
RuOhorford had established that a-particles are helium nuclei,<br />
The charac.teristic property of the G-rays in which we are hsre inter-<br />
ested, is their defihite range.<br />
This mealzs thaCa-par%icles of a givon velooity<br />
(for instance&-particles emitted by a thin foil of polonium) w ill traverse a<br />
given distance in air before ba'lng stoppod.<br />
This can bo domonstrated in the<br />
7Nilson cloud chamber,<br />
The Wilson cloud chamber is a chamber containing satwated vapor.<br />
Sudden<br />
expansion of the chamber causes superr$atura+ion in consequence of adiabatic<br />
doolingc The droplets ivill then condense around ions formed by the passage of<br />
the &.-partiole khus marking the track of %he &-rays.<br />
The .tiracks seen in the<br />
Vilson chamber are of qukbo uniform lengthe<br />
The stopping 0% C;c=-particl~:s is due to collisions with elac-brotzs.<br />
Since<br />
the mass of the electron ja about 7,400 .times smaller 6han that of the<br />
&-parliiclo,<br />
tl?~ cz; -parti.clo is not deflected by such a oollision appreciably<br />
and its path remains visi.bly dxaigllGe Xowever, the enorw lost in such<br />
co$llsions accv.inll?ates.<br />
1% occurs somctimes tha% an c\; =-particle ooll.ides with a<br />
nucleus and suffers a sti-ong deflection.<br />
Vi1 s on chamb r<br />
Thcso events can be easily seen in the<br />
The i'ollen~ing flgum shows the density of' ionization j along the pa%h<br />
abssi .ssa x<br />
I<br />
is the clj.stanoe<br />
i ,'
travelled in air, The above curve is called the Bragg curve.<br />
It s last and best<br />
detemination 9s due to Livingston and Bollovmyz).<br />
Close to the<br />
end of the rang<br />
the Brag curve a%tains its mximum, making about 6,000 ion paths<br />
pes mm of<br />
air.<br />
At tho end of the range tho ionization declines sharply iio zoroI<br />
Ian$zati.on in air 5s accompanied by loss of energy.<br />
Tach ion path oorres-<br />
ponds to an energy loss of about 32 eV, This figure is fairly indoponden% of<br />
%ha vel.oci2;y ai' thecikpar'tiolo.<br />
Thus the Bragg curve gives no% only a measure<br />
1<br />
I of %ha ionization density but also of tbe onsrey lost per mm of path,<br />
1 It is of groat intorost %O fihd out the energy loss ofc&-purtichs in<br />
other w"cerials khaii air.<br />
Thus iZ; is important to knw %he energy lost by the<br />
a f s vthen eutering a chambor or oounter through a thin foil.<br />
The onerig loss or<br />
stopping power i12 various substwoas is host measured by giving the vaJ.uo of the
*<br />
Octobor 14, 1943<br />
LA 24 (10)<br />
62<br />
LECTURR SERIES ON NUCLBAR PHYSICS<br />
__I-- rr.-nrri..-.rr2.-"., .e. .*IIyu-II-.I- ....--<br />
t<br />
$ocond Scries t Radioactivity Lecturer! E, Scgre<br />
LECTURZ X:<br />
n --.&--.-&"--------<br />
~*PARTICJ.,ES AIfD TIiEIR Ii'JTRRACTIOM WIT73 liIATTER<br />
-.*-I -*--I*..-.-.--*------<br />
In cloud chambcr photographs of ana-omitting source, it is ovidant that<br />
tho tracks oxhibit a, more or loss charactoristic longth, known as Gba range<br />
(v. Rusctti, p. 303).<br />
In tlic main, the onorgy of thc &-particlo is dissipatod<br />
in collisions with olactrons: tho path of liho Ctppartic3.o is not dcf'lcotQd by<br />
them many cncountcrs bacause o f t-ho rclatij.vely groat mass of tho a-partiolc<br />
comparod t o that of thc clectron.<br />
TJpon closo inspection of thc trsxcks, Q ~ C may<br />
soc tho faint- tracks of tho projoctod olcc-ixons, all having their origins in tho<br />
hoavy track of the drpartic1.c<br />
Occasiondly ancX-traclc shpws a f ork-likc<br />
structuro; hero t-hc a:-par%iclc has suPfcrcd a nuclcar collision nrzd my bo<br />
approciably doflswhcd, .tho pa'i;h of tho struck particle bcivig tho other branch of<br />
thu Pork,<br />
A monsu.rcmonti of tho anglcs and rangos (onorq) of tho various<br />
colliding bodics slwvrs that- tho oonsi:rva-i;ion laws of momcn-bum and cncrgy aro<br />
sattsfiad.<br />
Sometimes, one finds that kinetic enora i s apparsni;lg not conserved.<br />
In thoso cases an inelastic collision or nuclear reaction has taken placOr<br />
The many collisions of ana-particle wi%h electrons result in ion production<br />
along its path# The number of ions per u~I.t lcng-bh ts oalled the specific<br />
8ragg curve) of specific ionization VS r angs is shown in<br />
the follovring figure ,<br />
*<br />
range
63<br />
The speolfic ionization inoraases with decreasing velocity of theaqartiole,<br />
1)<br />
attains a maximum near the end of the path and then drops to zeroI (See for 8<br />
, quanfitafive graph, Livings-ton and RQlloway, Phys, Rev, P<br />
54 .- 18, 1938,)<br />
%he<br />
tlze<br />
5 10<br />
range<br />
lL<br />
20<br />
As an example,CZ-par-ticlesfroln polonium ham an energy of 5,298<br />
EBV and a range<br />
of 3,842 cmr<br />
-- .. *-.-<br />
-<br />
Stragglinci If the ranges of' an lni%ially homogeneous beap of a-parfiolos are<br />
measured, one finds valuos dis-tributod about a certain value, the moan range R,<br />
with a deviation of<br />
one or two percent.<br />
A plot of the number, n(x), whioh have<br />
a range greator tliaiz<br />
x<br />
is shown in tho following figure.<br />
n(x><br />
.L<br />
Tho value of<br />
x at whioh n(x) 1 1/2 n(0) is oqual to Rm The int;orsoc%ion with<br />
the x-axis of tho .tangent to tho curve at 3 is tho extrapolated rangob<br />
31) Tho distribtuion of ranges (tho phenomenon of straggliw) may bo<br />
oxplainad by the statistical fluctuation of tha onergy loss per ooll$sion, and
e<br />
64<br />
s% the ohmge of’ theahpartic16 varies several thousand kimss, 1.8.)<br />
He’, He, along its path. Those changes, however, ocour almost<br />
en%irely tn the last few mil[limsts~s of rangec<br />
\5<br />
$he stopping power of a substance 2fthe spaco rate of enorgy loss of the<br />
implngihg partioles as they travc;rse the substBnoe, io@,, if F = stopping power,<br />
i<br />
T = kinetic energy of particle, x<br />
P<br />
-dT/dx,<br />
space aoordina~s,<br />
Consequently the range R ia givon by<br />
R 0 TO<br />
R $ ilx = - jOdT/F =<br />
0<br />
The ralattvo stopping power of a su,bstanoe is dofined as<br />
approximately indopondonti of onorgyo<br />
stiopping power divided by the density of the substanco.<br />
mnp -I.-mCrC- in air<br />
rmgo in subtibst;ame’<br />
Finally, ?;ho mass stopping power is tho<br />
The mss s2;opping pmor<br />
s’d 7
7<br />
e.<br />
65<br />
scatterer at ona fogus. The ohange from tho iniiifal direckion of approach to *he<br />
final dirowbion of the par”cio1a is %he so& ing angle 8, The hypo.tjh&ioal,<br />
distance of closest approach if the parOicJe were not deF’lected is tho so-callad<br />
impact parameter, b or tiha above figure,
*<br />
LRCTURE<br />
OoBober 26, 1943<br />
SERIES OlT NUCLEAR PIIYSIGS<br />
L-c- ., *- cu-<br />
- L<br />
cond Series: Rgdioaotivity Leoturerr F. Bloch<br />
LECTURE XLI TIT? TEEORY OF STOPPING POVP3R<br />
-Cw-.,+.”UlrUICUII<br />
_..I. I. --a-<br />
~ircrs*’hw.-urc-.cI<br />
We shall calculake the energy which oharged particles lase when pas<br />
tter.<br />
This onerm 106s detormines JGhs range of chargod particles (ergr,<br />
es or protons),<br />
Tho onergy loss<br />
s also olosely &nneo%od wSQh ionlza-<br />
d by the charged particlo in %ha substanoa through which it p1as0sC We<br />
shall first discuss the Cheo of st~pp!ing powor as givon by Bohr. Hi<br />
only tho sirnpkst comcep+s of c2a<br />
ioal maohagioso<br />
WB consider B particlo with oharge Zo and volooity Ya 1% ixzteraots<br />
wiOh an atom.<br />
llore particularly we shall examine tho interaction with ona<br />
shckron. ot chicgo e and mas8 M wrt.l-hin that a-koni, Tho e1oc”cron shall ’bo<br />
an aguil!,brium position with olaetic forces and shall haw tha vfibra-<br />
tionax Troqucncy &$..<br />
The olootron was originally at rost.<br />
1’ro shall, aaloulato<br />
the amount of cnci’gy AT that %he s2;ron ob$ains durfng the passage of the<br />
lo* Tho charged partiole % thon have lost 2;ho same amount of onorgy,<br />
it has changod i%a onergy by -AT.<br />
In tho ffguro, tho pasit-ion of tho clcctron la show by tho point<br />
I
Tho rnin5,rp.m<br />
6 '7<br />
value of %his distance, that is, tho dis-banca of closest appfroach, is<br />
equal to b, JGhlB distance is called %he lision parameter. The angle<br />
inGluded by the path of the oharged partiale and by<br />
r is oalled $, Wti shall<br />
system whoso x direrottoll ooilzoides with the path of the<br />
a-partioxti and. whose y diroabion is perpendicular to it-. At ths poiht of<br />
olosast approach of the charged particle t o the electran wo set x s 0,<br />
We<br />
furthermore assme -khat for this point also t = 0.<br />
Then at all other times we<br />
have<br />
x m vfi,<br />
Ifire shall solve %ha problsm of tho energy %;ransfor to the elea%ron In ati<br />
approximate but very shpljeway.<br />
The approximation to be used requires; 1) that<br />
the eLectron be considered 88s al.ly as free during the time of collision;<br />
a'b the displacement of the electron during %he collision shall be amall<br />
to the distanoe of cLoseslt approach b.<br />
In ordar to soo wlmther the first assumption is valid lrm shall consider<br />
$he y-compcwn!; of %he force af interaction<br />
F =<br />
"&<br />
r2<br />
of this component i s zero at t E -mand t tcX) e<br />
It has a WXimUm at<br />
B width of %his maximum (%ha* is %he %imo during which ElJr is more than<br />
s maxArnum valuo) WB oar1 tlzs collision .time<br />
The condition that<br />
the elecr-bron may be considered as fr<br />
is fulfilled if Z is small compar<br />
*he, perrod of vibrat<br />
Q<br />
l/a<br />
Indeed if' tl7i.s is tho case the electron mows<br />
collision through a dis%anco which is small compared to 1Gs vibra-<br />
Gional ampli%u&, i.o., the oleotron remains close, to its equ$liW.um positioh,<br />
librim position tlza elastio for~os are small, *hay may be<br />
elQGtTQn may b9 aonaidorod a3<br />
TO shall diSOUS9 *he
*<br />
the<br />
68<br />
validity of our second assumption later,<br />
Tn order to caloulat-e AT we shall first compute the momentum given t-o<br />
eleciwon by the charged particle. According to our second assumption, %he<br />
alegtron has moved a small distance compared to<br />
b<br />
and the force will be prao-<br />
e same a6 though the eleotron hqs not moved at all,<br />
Under these con-<br />
dition@ one sees that the net change in the x-cbmponen*b of the momebtum of tho<br />
electron, dp, i s equal to zero, Indeed tho momentum in +tihe x direction that<br />
the electron has obtained while the charged particle was approaching, will be<br />
cancelled by a momentwn change in the opposite direc.t;ion thb$ occurs while the<br />
charged particle recedec. Thus the only net momentum change is in the y-dire&ion<br />
and after %he passage of %he charged particle *he electron is left vibrating in<br />
that direction.<br />
This momentum change<br />
pY<br />
is the time integral OS the y-oompanent<br />
Sub s t iky0 tlvlg<br />
one finds<br />
1<br />
sin2 $<br />
dt* kx ---<br />
r = b/sin $j>~d v<br />
Ap, ' = ;+-Ze2 5<br />
Irp<br />
sin $ d $ =<br />
2Ze2<br />
0<br />
The corresponding &ange in kinetic enorgy Is<br />
1 2 ~ 2 ~ 4<br />
AT ='z;;; IPx2 + ApY2) 2 -<br />
lnb2v2<br />
\ I<br />
La holds as long &s b,inc: bg bmBx* If' b<br />
s mado above are not valid.<br />
is ou*side this region, *he<br />
We shall calculate bow the anrgy lost by tho particlo when passing<br />
J<br />
through a distance &x, Lot u.s assme that them are N cleo%rons per conti-<br />
motor oubed. In a cylindrical shell of length ax, inner and outor radii b and
e<br />
69<br />
b + db $here will be 23Fbdbd XN e9leotrons. Tho energy transhrred to these<br />
AS = Rhx<br />
7 --c-<br />
Bfl'zZOB<br />
db c Ndx c- 4 z%4 log % bmax $ .<br />
'ET7 b ' m VZ rnin<br />
bmin<br />
(i<br />
From th5s squal;ion tlm energy loss per aentiiineter &T/Ax<br />
may be obtaifiad. as<br />
soon as $he values of bmin and bmax arb dsbmnined.<br />
DiscussSon of brain<br />
--*-<br />
For b - 0 our approxbdto formulae would give AT a00 which is<br />
Actually tha maximum volnoity a Fxee slootron may obtain in a colli-<br />
v and therefore tho maxiinurn energy loss i s<br />
22<br />
I<br />
r<br />
G<br />
I<br />
ose b in suah a manner tlmt %he calaulafed onarc3 loss shall bo equal<br />
imal energy loss, %hen $his value ~f b<br />
will mark the limit 'beyand<br />
whiah the approximate value of the energy loss can no longor be used.<br />
At %his<br />
1<br />
value of b,<br />
the assumption that tho electron has been displaced during %he<br />
through a distance small oompared to b is no longer valid, This<br />
b we shall choose as bmtn we obtain<br />
2 z v<br />
3<br />
m<br />
(2v)2 = v-<br />
i 9 m b,in2v2 2<br />
(&vr =<br />
I-.<br />
ze<br />
a-<br />
,* binin mv 2<br />
, A treatmen% which is rigorous for small values of b u88s the<br />
d farmula for the soatitwing of the charged partiole by %he ehotron.
In the oase<br />
,><<br />
thab<br />
mv<br />
70<br />
the second JGerm Liz the denominator oan bs neglsc<br />
and the integral reduces to<br />
the simpler form which has been used above. At small valuer$ of b the Inbgral<br />
goes to zero.<br />
This eives a qualitative justification for using %he simpler<br />
integral<br />
b<br />
for outting off the irrbegration at<br />
P 282<br />
bmih<br />
mv<br />
D ~ S C L I S ~ of' ~ bmax O ~<br />
I*^.. -..-,"<br />
c- .%-,.c** .<br />
-L.<br />
For too large values of the collision parameter<br />
onger be considered as free. In fa&, coll'lsl,on time is appro<br />
and this t5me will become great<br />
l/&J when b becomes suf.Piciently large,<br />
vibrational. period of the<br />
For large values of b, that<br />
is, when b/v))l/W<br />
the variation of the Toroe of interaction botwesn charged<br />
and eleotron is slow compared to .t;<br />
tion between tihe charged pa<br />
18 cnd %he eloctroq may<br />
treated by Che so-called adiabatic-, approxi.mat ion,<br />
Ih this approximatLon, ths<br />
is displaced whsn %he chargod particle approaches but whan 4.5,<br />
Ipeocsdes<br />
ron is eased baok Into its original position without- iti taking at$<br />
i<br />
reford ju.stifio<br />
ur integral to the poilit where<br />
and to neglect oontri greater b valuos, Thus we have<br />
b,in<br />
into tho enercy loss formula<br />
obtained Bbove, one fhds<br />
* been<br />
c-dT ;1N<br />
dx<br />
sd out this formula fs un proximste. The o<br />
carried ou-b by Bohr rigorously and he obtains<br />
2. "N.---<br />
4 e $e4<br />
log<br />
1,123 mvz<br />
--.r-.*C L<br />
dx mv2 Ze2U
0<br />
Wcro n donotw tho nhboq af<br />
YIO<br />
may us0 tho samo oxprossion<br />
73<br />
atoms por aubio cantimotor, For tho uppar limit<br />
as vms dcrivod in crlassioal fhcary, namoly,<br />
hmvovor, must bo kangod, This classical. lowor limit has tho‘ folloying s2gnif9-<br />
onaidor tho kinetic onorgy ‘of tho cloo2;ron in t hs framo of<br />
i<br />
%ho ohmgad partiolo ie at rest, thon bbin Is the distance at<br />
he pokential energy between the eleotron and 6he charged parflcle becomes<br />
o tha-t; kfnetiio energy,<br />
electron and charged<br />
Now in quant;um neohanics a closer approach<br />
article than the de Broglie vmv~ length is meaning-<br />
WQ have to use therefore @or the lover 1imi.t;<br />
i<br />
bmin (quankum) = %/m.<br />
’ Thi indeed so if tha lowar limit- in guanhn theory ha? a highohor value than<br />
the previously derived classlical lower 1j.m32;, becauss a5 “.‘1$ approaoh closer<br />
%han %he quantum lwier4.irni.t defraction phenomena will OCCW~ and make an effeo-<br />
tdve approaoh impossible.<br />
If hotmvor,<br />
%he olaaaiaal 10~1er-12rn.S;t; should %urn 0u.t;<br />
to ba greater, then down to %hat lower<br />
limit, classical theory 3s applioablo and<br />
be UBOdr<br />
Thus, %he QlaSSiCal or quantum<br />
sd according to afhethher tho ratio<br />
r small compared to uni%y. Assuming Ze2/1;% = 1, one ob%
*<br />
which factor does 110%<br />
dotailed calculation.<br />
74<br />
follon from o w argwzhnt but must bo obtained by a more<br />
Tho stopping power just g h n is valid If tho following throe condi-<br />
a<br />
tions are fulfilled:<br />
(4 viua>> ro<br />
(b) Z@2/*V{{ 1 * 3.<br />
(4"2))flOj.<br />
Condition (a) means that at the upper limit where wo have made use of the f'rc-<br />
quoncy of the oloctron, tho distanoe of' tho charged parkiclo of tho atom should<br />
'be great compared to the atomic radius<br />
ro+<br />
ThSs condiAiion i s nscegsary in<br />
ic field of tlze charged particle should be homogeneous<br />
the atom whenever %he oscillakos 'urea-tmont of the elaotrou has been made use ofr<br />
Unless this condition is satisfied, the analogy to t5e interaction with ligh%<br />
rea-bmeylt by virtual oscillators breaks downo<br />
Co;iclition (b) mem~ thati<br />
%he quantum Sower-Xfrnlt i.s pater than %ha classical 1o:mr-lj.mit<br />
and that<br />
therefore Clm quan-bum 1m*mr-lImi7b should. be usedr Fina!.ly, condieion (0)<br />
expressss the requirement ti22a-b the upper limi.1; of<br />
b<br />
must be considerably<br />
greater bhan the lower limtt of b aud that therefore there shall bo a region<br />
in which our aBproximationi are valid*<br />
ConditIQna (a) and (c) bofh mean n<br />
ically khat *he veloai.hy of the<br />
harp3 particle must be great compared to the valo<br />
0 f -bhe<br />
with whlch the Gharged particle inferacts.<br />
For condikian (a) this Is<br />
fly because<br />
row is equal to ~ ~ 1 In , condition (c) one may replace<br />
'h@l(that<br />
is, tho sxci'bation energy of the electron) by twice %ha kinetic<br />
energ$ of the electron which is a guanrt,i.(;y of the same order of magnitude,<br />
Then<br />
(e) roduces to
which again means that vQ)vel.<br />
Condition (b) i s also satisfied if “))vel#<br />
IV<br />
ele ptrons +<br />
If v and<br />
vel become comparable, our treatment it3 no longer valid<br />
and an exact treatment for this case has no% yet; been carried out,<br />
The fhoa-<br />
retical difficulties are paralleled by a complicated behavior of tho charged<br />
particles a% relatively slow vslooitles,<br />
If‘ their vsloci-by bocomes equal to *he<br />
veloci-by of %he outermost electroiis in the sluwing dovrn medium, then the oharged<br />
par”cclss. my pull ouli electrons from the slovdng down modlwn, attaoh them .Do<br />
khernseXves, and thus reduce their affootlve charges,<br />
In lator collis5ons “cess<br />
charges may again bo lost.<br />
Nhilo tho chargad particle had its charge roduoed,<br />
itra ionizing powor and anergy Ioss will also bo smaller.<br />
Since various charged<br />
pa.rti.31~5 will spend variaus times in tho staiie of reduced oharge, tlio actual<br />
” r range<br />
of %he individual particles will dif%er, This dSff ;renco of range is<br />
aalled straggling,<br />
If an t&-particle has EVO. enorgy of 72 lcV, then its valoci%y is tho<br />
same ad %hit of an eleokron with 10 eV.<br />
This is %he avei*aqs of the ou-l;ermox+i<br />
electrons.<br />
Thus, vc must expec2; ‘chat our formula of s-bopping pwmr for<br />
&-particles becomes Snapplioable under 72 kV1 ActualLy, sexious deviations<br />
from the formula start at the somawha% higher energy of 200 kV.<br />
‘I/he correspond-<br />
ing limit for deuterons i s at 100 kV and for protons a+ 50 kV,<br />
From the formula of the slxpping povmr an estimate of the range of the<br />
particles can be obtained, For e, we replace the energy 10,s~ dT/dx<br />
hange in kin,e%io energy of<br />
iole -dBlc,n/dx.<br />
The formula for the
*<br />
This<br />
Forge;el%-Sng about the logarithmic factor, one ubtains<br />
PQIX~U~& sliws that the quantiky Elcinx ohanges at a constant rate along %he<br />
16<br />
I<br />
path of %lis par%icle and that therefore %ha range should be proportional to %he<br />
square of the original kinetic onergy.<br />
Actually, the range varies vti=bh a &omem<br />
what lower power of the original elnerpj.<br />
is bhe stopping power formula.<br />
A quantitative evaluatiion of the sDopp2~g povrer formula requires knnwl-<br />
edge of all absorption frequenoies and all oscillator s%rongths,<br />
These quan2;iWs<br />
iven in an analytic form for %he hydrogofi a%om and in this case the<br />
requirad calaula+iions have besn carrisd ouG,<br />
Far heavy atoms oha my ude a orudoly simpLffiec1 sbatis-bical model.<br />
Thia model oonsiders tlze elec%rons as a gas, %he IC<br />
o energy or %he e1sotro)ls<br />
being balanced IIT the avorag9 e1eotirosta”ciu potent<br />
find the oscSlLat ion frequency of th5.s gas by hydrodyix!ni-r; ca’J.oulat<br />
uso +ha% froquenoy instead of<br />
CJi and use trq Ju.,uriZ nuliber of aleow<br />
an atom Zo<br />
instead of tlzs oscillator s%reng-b<br />
The vibrational frequency in tho statiskical<br />
the ‘tsound veloc.i--by’’ divided by the a,tomlc radius,<br />
Tho 1),l.t;ruF(+ sacli.us can be shovn<br />
”.<br />
to vary as 2, 1/3*1 Tho sound velooity is of the saylle ordor of magialtude 8s the<br />
obtain<br />
locity of tho electrons and % 8 praporOiona1 t o Z,L/3, Thus li<br />
dT 4.fFe4z2<br />
L-rl<br />
ckC 7<br />
lag<br />
a<br />
ula, R Is the so-oalle<br />
Rydberg frequency, that is fqR is tho<br />
cw-<br />
*%rhhoaviar &%oms ha smaller atomic dii which %s in contradiohe<br />
experience about- atomic rad.ii as cus d, IIovevar, in<br />
the customary definit-ion the atomic radius is essentia<br />
the orbS.%s of‘<br />
the few ou-kermosl; electrons vrhereas in our presen+ disc:<br />
average distaizcs of electrans from tho nucleus, the av<br />
eleotrons,
ionizlation energy of %ha hydrogen ationi, and<br />
ating it, one may 'aajust 5t in suo<br />
17<br />
is a numerical factor, Inatead<br />
way as t o give agreement bet;aaan<br />
the formula and %he oxparisnce for on0 haavy akom, f instanca for gold. Then<br />
la describes tho dependence of "cho st'ogping pawror on the a'ciomic number<br />
sfaotory vmy.<br />
The formula for tho stopping power of eleoGrons i s tho same a8 the for-<br />
mula for other charged particles rriOh the tilxooption that under<br />
faetor '2 cloos ndc appear :<br />
2 mv<br />
uo to %hc fact that i mining the lowr limit of b t<br />
mw leng.t;h musC 530 uscd vrhich corresponds to kho vclooity of' $he atomic elaotron<br />
e of rsfbronoa wkro tho conbo<br />
chargsd particlo is at rest.<br />
of the onooming partiole, as<br />
ravity of tlint ol@cCran and *ho<br />
olboity is practically equal to tha<br />
that oncoming par.l;ic;le i s heavy,<br />
leu$ If the part le i. s an electron its<br />
s one half the velocity o<br />
&lparticlss and of<br />
oharged oosnSc-ray particles the motion OS the<br />
ticle must be treated aclciording to rtslativis meohmlcs. This l-fill hnve %m<br />
) the region in whieh the elea$ria field aots is oontracfed by<br />
Ths Dim@' during which the<br />
same factor; 2) %he strength of %he electrio f is j*nQrQasQd by<br />
Ore, the momentun ah6 energ *trans-<br />
ron during the c<br />
airr the same as in unrelativis$ic<br />
ver, l3he Limits for t parameter b chakp, The vpor<br />
limit I s horeased by tho reciprocal of<br />
tion, we oar], go ta groat<br />
os tho period of the atornic electron.<br />
In %he lower<br />
a~8 Length must agdnba used vhich is given by
LA-24 (19)<br />
80<br />
LRCTURl3 $ERIE<br />
c-"L-lrr..l*..Ic<br />
Sooond Series : Radioaotivity Lcoturer; R, 'Pallor<br />
---<br />
CIlARGED PARTfCLES GilITI-1 rTATTER<br />
*-"..v-.yL)-..i.-uyCr^w.m.--C~~<br />
Tn the previous leoture it vras sh n that, apart fro<br />
of aner~ loss of a moving hwpd partiole (<br />
nu-orsoly proporiiional to % voloci%y Y of the<br />
is faotor arises beonusa thc momenCum transferred lio an alaotron is<br />
proportional to the "kime of collision'' which In turn is inversely proportional<br />
to .the particle volocity.<br />
Ilgnca n par*iclo<br />
nga should therefom bo wEZO Expcrimntally, hot-ravcr,<br />
WE3/'. The d epanoy ma37 bo removed by a more<br />
tho onergy loss oquc,,tion.<br />
Jt will '00 raoallod that<br />
incldont :7nrtiolo nil1 transfor oi~orgy to tho atomic nloctrrotzs $2 i t s voloo<br />
or than the oloc%rron volooitios,<br />
Approximhly, tho numbor of olootrons<br />
om vhich hwo volooj.-tics swllor than v,<br />
and oar! thorci'oro roooivo<br />
onorgf from tho particle, is simply proporkfonal t o v (provided v i s in tho<br />
oloctron velocities). CDYW~VW<br />
*he ra-b of onorgy loss is n m<br />
go iq proportio<br />
0 to noto %ha% tho,ol)or@ loss per<br />
he, speed and charge of the 9noidelif pnrtiKle,<br />
Fnrticles of' the saw '<br />
ere at n rate proportional *Q square of -their rsspeoCSrre Zfs,<br />
so<br />
at n 4 1vN &?partide (2 = 2) has the same range as o. 11fV p<br />
The enerar vrliiah is talcen from the inaiden'c parbicle my be used to<br />
~JI sleotron from atom (ioniza$fon), 'or rnv only produce electronic exoitation<br />
with subsequent om sion of a 1%
e<br />
81<br />
in detection of CC-particles),<br />
Andhher possdble prooesb, and of considerable<br />
impardanocs in contlsoCion Ivi2rh bl.dag5dn2 erfL"ec6s 02 radiation) is molecular<br />
n with subsequent dissooiation into atom or ibnsi Takiwrg inti0 account<br />
these various processes together with seoondary effects, one finds that the<br />
average energy per ion pair i s roughly 30 eV.<br />
An approximate measure of the<br />
biologtoal effeots of radiatiaiz on a part of the body is the amount of enerfg<br />
absorbed by the particular region.<br />
There are, of course, qunli%ntive di:?farences,<br />
becnuso the density of logs is qukh different along pcH~s of varioua j.i!;ipin@.ng<br />
pnrticles, so thnt recombin?.tion protlesses m%y be of another clznracter in the<br />
cam of, shy, a heavily ionized 4L-fraok fis oompared to a veak fi-track,<br />
Ianiz<br />
, --.*-<br />
by heavily charged partfolesb<br />
.---.-..-- - -I-<br />
We have seen thnt the Bra@ cur~@ of nn a-partiale, o. measure of the ~<br />
specSf5c ionization along the pnth, increases slovrly to ~1. mnximuin and. *hen drops<br />
s1znrpl.y to zero+ rhe corresponding QLITYQ for fis5sion par-3.cles slions only a<br />
_ -<br />
in speolfic ionization along its pn-bh. ::hat :is the exp1:in:ci;ion<br />
the<br />
di.i.??erencol<br />
The ini%ial velocity of A fission pnr2;icJ.o 5.s<br />
apprcxi.lmt@XY<br />
ce<br />
One ccn estiinzte it.s initial ohargs because tho fisflion fragmeDt<br />
vi11 8hod all the electrons having veloci2;ies smaller than its ovm,<br />
As it pro-<br />
ceeds dong its pa th, %he particle vd11 pick up electrons, RS 2, result the<br />
speoific ionization will tend .t;o deorease,<br />
At npproxinctoly 1 MY, +he parthle<br />
becomes neueral; its pnth a6 R ohnrged pnrticla is nhou'b five to ten times<br />
an the distiawo it trnvels as n neutral pc.rticle,<br />
The range is ti'bou$<br />
2 am, in air,<br />
of Elec-t;rons.<br />
*.<br />
ft is difficult,to asoribe a range to eleckrons traveling through<br />
OQ~UW of the excossive straggling.<br />
In contrnst to the es$enfilnlly<br />
il)<br />
strni&t; pc.th of an a-particle, an electron my be considerc2bly dcfleoted in<br />
n oolllsion, so that it w ill have a more or loss rnndon pnth.<br />
Secondly, %he
*<br />
energy spaatrum of #-pa clee emitted from a nucleus is not monochromatic (as<br />
in the aase OS W s ) but s a considerabls spread, The combined effect gives a<br />
range distribution which imitates an exponential absorption curv<br />
There are seyeral prpasslsss by wliioh the enorgy nf a fast moving elea-<br />
tron may be dissipabed. When ran electron passos -through the electric field of a<br />
nucleus, it is QGCQh3ra%0d,<br />
This ameleration gives riae to radiative energy<br />
lasses (bremsstjrahlung).<br />
They are appreciable for elecCron energies of several<br />
Mev.<br />
The electron-electvon radiative lossas are quite small inasmuch as there<br />
can be no dipolo, but oiily quadripole, radiaCion.<br />
The eLeotron suffers energy loases in Its collisions with eleatrons.<br />
An approxima'ce formula has been given for this rate of<br />
energy loss, and we have<br />
aeon it depends on the squam of the atomio numbor of thc nuclei,<br />
In .these<br />
oollislons, all of the atomic electrons par%iaipate and ionization in the trctv-<br />
ersed maDerial i s produced,<br />
Tho ratio of ionizstion loss to radiative loss is<br />
givah roughly by<br />
Rato of ionization loss I ZLmev)<br />
R a . t ; e T ? m L % X -<br />
--.I<br />
8 O r<br />
where Z is the atomic numbor of kl10 rnnt;orial, E is +he kinetic e<br />
million elgoZ;ron volts.<br />
To give an idoa of how much shielding is nocessary to stop<br />
one may give an axample.<br />
4-<br />
particbps,<br />
For aluminum 100 mg/om2 will stop 100 kv betas, and<br />
1 g/cm2 will be adequate for 3EW betas.<br />
1% is approximakely %rue that %he<br />
stopping power per unit muss is the 8aine for all tho olomonts.<br />
. .r-...u.--.--ruCu..-<br />
Reflectian of fi pa_r2;iales.<br />
.._I_<br />
Coefficients of reflec.Gion bstwen 1/10 and 1/2 are, qui%e usual,<br />
Heavy<br />
nuclei by virtue of %heir charge are more Eiffoctive in producing ,largo angle<br />
soattaring than 'light nuclei,<br />
Honce pb reflects fi particles much butter than,<br />
I<br />
say,, paraffin6
e<br />
84<br />
Sn particular, tho first roaction is irnportmn-b booousc it givos information con-<br />
corning tho bindi.l?,g cnorgics cnd forces betwoon clomoqtary pnrticl-os,<br />
Gross-<br />
soctions for this 6ypc of' roactions arc rolakivoly small, being of tho ordor of<br />
m2 or smallor.<br />
Tho two ronatlons clitiod nbovo roquirc ralativoly low<br />
3. -cncrgios, i.c. about 2 IJcv,<br />
Elost nuclcar photoolfccts requiro 5-8 Flov<br />
%t-rcys<br />
The interac6ion OF ?-rays with rnaLter are in the maiD interactians<br />
with electrons.<br />
The phenomena may be described as the photo-electric effect,<br />
Compton scatCaring and pair production.<br />
We shall gee %hat the Parims inter-<br />
ac.S;ions depend difforontly on nuolear charge and the energy regions in which each<br />
effect becomes important is more or lesa well defined.<br />
Photoel@otric effect,<br />
.-A. ---".<br />
__I<br />
A bound electron absorbs energy and leaves the atom.<br />
.IC-<br />
Vost of the<br />
energ? of the %-quantum is given to thu elec.l-rrm, most of the mom@ii%uin is given<br />
to Che residual positive ion,<br />
Fach shell o:? electrons wound a nucleus will con-<br />
erilsute whose onera is smaller thavl the eneru of the ? quantum.<br />
Among them<br />
the shells whose ionization energy is closes% to the energy of the ?-ray<br />
makes<br />
the greatest GOYLtribLi'GiOn,<br />
pray energies are usually greator tlian the ioniea-<br />
Lion energy of the IC electrons so that the innermost; shells have the larpst<br />
effebt,<br />
For sufYicientl*y energetic ?-rays,<br />
tho photoelectric absorptioki<br />
ooefficjant goes 8s Z5/E 7/2 where Z is tho atomic number of the<br />
5<br />
and E the ?-ray enera* This formula indicates a very great differ-<br />
9<br />
ance b0.knreen the respective coefficients for Pb and C,<br />
Them i's actually a<br />
great difference in absorp6ion coefficient in the region whore photoelecbric<br />
effect $8 the main source of absarp'cion,<br />
This is the X-ray regtoi?,.<br />
Compkon Sffect<br />
_LII<br />
magnitude of %he scaWiering of a phokon by an elecGron can be
88<br />
uiicer.t;ainty of 30,000 e1eoi;ron. f~ol-bs, This aoouracy oomparcs favorably with bhat<br />
of good optioal spectrographs,<br />
The method of mass deterclinafion utilizes comparisons bs%ween ions of<br />
marly qual rnwses suak as1<br />
Differanoos of -these inasses are than measured. For acaurate measuraments, 5.C is<br />
best if the ion ooncen6ratioas in the discharge are so matched that %heir intensities<br />
are appraxim,%ely equal.<br />
The scale OS measuring masses used in physids is based on the iaotope<br />
0l6.<br />
Tho mass of' this isollope 3.s S Q ~ equal to accurately, 16 unl%fl, then one<br />
i;l&sS unit oorrosponds to 930 Ibv,<br />
This quantity might be compared<br />
with the mas8<br />
of the olectron whioh aorresponds to ,til Pietr,<br />
In ohemistry, it is usual to set @he atomic weight of the natural iso-<br />
topic oxygan equal t o 16,<br />
+,00031 x low2* grams,<br />
In %his c&s@, the unit of nuclear masses Is 1,66035<br />
In discussing a$omic masses, the mass defect is of'<br />
interest.<br />
1% is defined as 34 - A, whore I! is -the 1x83s rneacursd Ill %he phystoal<br />
scale, and A is the integer number closest to 11 aL?d is called tho mas8 numbor,<br />
A further qc'n-bi-ky frequently used is the packing frac,tiol?, which is defined as<br />
F'I - A<br />
In the following figure thb packing fraction is plot.t;od &gains% the<br />
nuclear chargo,<br />
s
The be& figure of Chis kind is found in %he ar6iole of EEahn and Mattauoh,<br />
a9<br />
Physikalische Zeitschrift, 1440, The packing fraction has a minimum a6 about<br />
&other<br />
method of determininG WSSRS is by balancing reaction energies.<br />
For instance, the mass of the iieu.t;ron bas bean deOarmined in thi5 manner,<br />
mass, of cau.rse, co~ild not be determii?ed in the mass spectrograph because neu-<br />
This<br />
trans caiinot be deflected Sn the electromagnetic fields of tha% apparatus,<br />
the determina+ion of the neutron mass the reactfoil<br />
For<br />
Hi2 t hv -Hll *). 11~ 0<br />
is used, Knowing the mass speotrographic values of the mass for M12 aiid lill<br />
which are 2.0147 aad 1,00812 and ~isi.ng tho minlmwn snargy 2,17<br />
Nev at which Gha<br />
light quantum can cause disiiitegratio:1, one fbds for the mass of the neutron<br />
1.00812.<br />
In many othor CBSBS,<br />
cross-ohecks are possible by using mass spectra-<br />
grapliic values of %he masses and data from nuclear reactions.<br />
Size of Nucleus<br />
e. _”I ..“<br />
-I-*---<br />
.,-<br />
One may de-bermiiie the size of’ atomic nuclei from various reaotions,<br />
some of these, hoviever, like reactioim of capture of c low neutrons, givs errat;ia<br />
values. l’he aatual size of tlza nucleus can be definod 8s the radius to vdiich<br />
guolear forces exteml, and is usually ob6ained by inv6stigating the<br />
deviatfohls from the Rutherfwd scatt-wing or by investigating the potential<br />
barrier effects in c&-Clecays<br />
and ot.her nuclear reaotiotis involving chwged par-<br />
ticles. OKW fj.:icis that nuclear radii arc? given approximately by 1.4 x 10-13~1/3.<br />
This maiis %hat %lie nuclear volume is proportionate to the iiuclear mass.<br />
-..-<br />
l!Tuolear Donsl-by<br />
.I-.<br />
different nuclei.<br />
It follow that nuclear densities are approximately +lie same For
first disoovered in a structure of spsctiral lires which 1s CalledAhyperfine<br />
90<br />
s%ructure.<br />
These structures were first observed at the end of the last. centwy,<br />
but their explanation by Pauli and their more dehiled study goes baolc only two<br />
deoades. The explanation of the hyperfine structure is as followsr The orbital<br />
motion of the eieotrons produce8 maglietic fields,<br />
magnetic moment associated with %he nuclear spins,<br />
These interact with the<br />
The interaotion elzero<br />
rnod5fio s the sprectral lines in a differen% way, aocording to $he orientation<br />
of the nuclear spin in the magnotio field of the electronic mokion; thus tho<br />
lines are split,<br />
From the limber of the Compomnks, the spin may be determined,<br />
while from tho sizo of the splitting, "Fie nuclear magnetic moment may be oaScu<br />
1 at e d<br />
Daiia on magne0ic spins and rnRgnetio moments have been summarized up<br />
to 1939 by llorsching (Zeitschrift fur PhysFk),<br />
A now method for the determination of the spin and magne.l;ic, moments<br />
has been davolopad by Xtarn and Wabi.<br />
This method USOS atomic or molecular<br />
beams passing through static and oscillating magnotio fields,<br />
more direct than the om involviiig the hyperfine sZiructure.<br />
have dotermined by this method, the magnotic moment of the noutrons which, of<br />
course could not be dstermined by the uso of hyperfine structure.<br />
The mothod is<br />
Bloch and Alvarez<br />
The following<br />
table suznmarizos the results for %he neut;ron, the proton and the deukeran:<br />
spin -- I<br />
magnetic moment<br />
.(I.-. --_11 -1<br />
n 1/2 ? "1.93<br />
HJ, 1/2 2b78<br />
I1 2 1 0,865<br />
Tlne spin of %ha iioutron has been assumed as 1/2.<br />
This value is probablo, but<br />
*<br />
not quite oer.l-aj.n, The magnotic momon.t.s are measured in units of<br />
eh<br />
m<br />
is tho mass of %he proton,<br />
The minus sign i n the mp;notic momoiit of
positive aharge.<br />
Vuo le ar s-bat 5. st i as<br />
t; obays Dho %om<br />
he premnt piattxre $hat nucloi with svon mass<br />
,
92<br />
. T11us one.muld have Qo believe by at nuolei oontain posi6rons 1<br />
in addl%$on to sle&ironsr<br />
A rather 8Z;rong argument has been given which shwrs tha% neither elao-<br />
011s c~ii exht in polel.<br />
If' a partiole is danffnsd to' a region<br />
lirmir d$mensiorr$ &q then the momeiihm has an uncerb;ain%y */Aq, tha% is,<br />
$he manreliburn may have any value from zero up 00 abou-b this latter value. l''70vr in<br />
%he nuoleus Aq is a few timas om. From tho GO e spoading momentum uiicsr<br />
tiainty one may aaloulats t<br />
sprea4 in klnetlo energy by the formula<br />
P2<br />
-.I<br />
2m<br />
where p is tha momoiitum, and m fs the 8s of the partxiole in question. I%<br />
.tihe pareicle is 8 proton or a neutmn, $hen tho uncertainty in kfrieOio energy' 1s<br />
of the order of the nuclear bindi<br />
OY~QF~IBS.<br />
If, hovrsvor, the pni..ti,ole v m<br />
Icinetig euerpg mould bs much bigger due to the<br />
small value of $he mass 0% the electron. Aotually, i k i -t;l?
*<br />
92;<br />
The table shovss that nuclei vri.t;h an odd nusnbor of protons ar an add numbor of<br />
mutrans and an odd number of pxotakis WQ particularly few. Thora &re only four<br />
that; these nu.oloi form a simple sories. 3;<br />
bo r@marked -bkx&t; not o'2&r<br />
arc3<br />
there fovmr kinds of nuclei with either an odd numbor of noukrons or an odd<br />
number of protons, bvA also tlicat such i:uclei aro loss abundant;.<br />
In connection wit& tho fa& %ha% nuoloi with an odd number of nsu%rons<br />
or an odd. number of protons are rolativoly rare it may b? montionod that lluclei<br />
vdth avr odd value of 2, i.0.<br />
with an odd nunibor of pi*ck~h-~ ;iovor have more than<br />
tno isotoopos, IZth the exception of tho ljghtas2; Q13i?E':'.n, thoso isotopos dikfor<br />
by two L~QLX'G~OIM.<br />
Similarly, if onc oansidars nuclol with a given odd number of<br />
'?<br />
sh QY?S
95<br />
One might imagine the figure made more complete by cadding the energy of<br />
the nuoleus, that is, the enera needed to pull apart all the constituent nw-<br />
tronl; and protons,<br />
One may plot this energy in the third dimension perpondia-<br />
ularly ta *he plane of Flgufe 2.<br />
The enargy surface obtained in this mannor will<br />
have o valley along the dotted line near which tho s%able nuclei clustcr,<br />
Curve as shown<br />
CuZ;%,Zng the enorgy gurfaco along an isobario line one obtains &n onergy<br />
o o ns t a nt<br />
The straight livlos shown in the f'ig~r~ am energies of ehe various isobario<br />
nuclei,<br />
One of thaso energies will lzavo the smallest value and a corrospondihg<br />
nuoleus will be the only stable isobar. The oorrosponding stablo nualwj is<br />
indioatwd by a dot vshilo the uylstablo ones 4.136 shown as crossesb<br />
shown which indioato the dkoction in which nuclear<br />
Arravs are also<br />
-transformations taka plaoo (I<br />
Tho situation depicted in Figwe 3 holds only for nuclei vrith an odd mss numbor.<br />
Isobars of such nuolsi must either contain an won nwnbor of noutrons and an odd<br />
number of protons os olss ~n ovon numbcr of protons and an odd number of hautrons.<br />
Thare Ss no systoma-tic deviahion bOtWQQn the onorgies of them two kinds of<br />
ass number is won, than tho nuclai my eithor contain an moM<br />
rons atzd an won number of protons, or else, an o4d numbor QS<br />
P<br />
neutrons and an odd number of protons.<br />
Tho formcr kind of nucloi have system-<br />
atioally 8, lawor cncrgy than thc lattor kind,<br />
Thus .tho nuclear onergias will<br />
lie on two roughly parallol curves, tho highor on0 oorrosponding to tho odd-odd<br />
QBSC, the 1mrcr on0 Go tho avon-ovon CRSO. This is shown in Figure 4.
96<br />
e<br />
E<br />
2 odd N odd<br />
,Z even N even<br />
A<br />
constant<br />
Fig* 4<br />
It viE1 be noticzed that In this case more than one stable nuolaus may exisf.<br />
In<br />
the figure,<br />
such nuclei w e shown.<br />
The one on the right h<br />
han the one to the left, But the nuoleus with the higher egergy<br />
oould ljransforrn into %he one of tho loiver energy only by simultaneous emission<br />
or two charges which is an exceedingly i d3&tble process<br />
Emission of one<br />
electron or positron) would carry us t o the upper ourvs and tvauld require<br />
snere rather than produce it. It will also be notiGed from<br />
on the highsr ourve may transform into G nuoleus of ~ Q W energy ~ F &%her<br />
king to the right or the left, The<br />
that suoh a ~iucleus emits positrons and elec%ron$.<br />
This has actually been<br />
ob served for several nuchi *<br />
From the sysbernatics of the nuclei, aonolusiona about nualear forces<br />
r nutlet. It is %Q be explained by<br />
wtrone and protons are similar.<br />
The explanation also uses the Pauli prtlnaiple, which asser.t;s that in each state<br />
9<br />
only one parkicle oan exist8 that is, in a gtvsn orbit within the nuoleus one<br />
may have not more than two neutrons (or 80% more thm two pra.t;ons) differing 9u<br />
the direcfion of their spips. From th quality of the foro s acting 012 mutrons
*<br />
and proGons it- follows that up to a given energy %her@<br />
of orbits available far neutrovls and protons.<br />
97<br />
will be a similar number<br />
PF one then filla up the lowes$<br />
orbits available up to a given epargy, one mi13 have to use B roughly eqLtal<br />
number of neutrons and probons .,<br />
This is t o be explained with the help of ths Coulomb law,<br />
It is<br />
easier 60 add a neutron than a proton Do a nucleus whioh is already heavily<br />
charged,<br />
In lighter nuclei, the Cablomb ~O$?O@E are considarably smaller than<br />
%he nuclear forges,<br />
In heavier nuclei Stho Coulamb energies aS the several pro-<br />
tons due to the longer range of 6liase forces add up and beaoma oveiitually<br />
comparable to the nuclsar energio s<br />
Mass defeob and volwne proportional to mass number,<br />
--I .---- -crlucI. .ua"-<br />
These facts ir?dicaCe forces having a saturation ahsracter, +ha% is, a<br />
situaWan 5.11 which one particle oon'xibutos offectivaly $0 the to'l;aX energy by<br />
interaq+iihg with a llhited number of &hQr particles.<br />
The aame situation is<br />
encoun-borad in C;rystalS<br />
Bj.nding onergies of DeuBaron and a -partiole<br />
-I-.-+.--.*--~ .I* * r *nrrUu.lr- -.-.-<br />
Tho fact +that tlia &-particle has a more khan LO times greater binding<br />
enera than the Douteron sliav~s that tho saturation has not been roached at the<br />
Zatter przrWcle.<br />
In the &-par.t;icle<br />
the saturation i s actually reaohed.<br />
Thls<br />
a% R particle, for instimice a nou<br />
nuoloua with another neutron and %wo otihar protona.<br />
an, can effcc+iv~ly intafact in a<br />
Tho usus1 intorprotation is<br />
that a par%icle oan intQracat with all oChor parfioles which aro parrnit-bod by the<br />
InciplQ t o move in tho same orbiti and %hat tho intorao6ion does no%<br />
dopond yory strongly on tho differcnk spin directions vJ'hIch tho particles my<br />
*<br />
possess in this orbit,<br />
The dc Dislhtogration.<br />
LI*-.--IUl*.-IUUC.<br />
The &-particI.as<br />
omitted by a nuc2ous have a well dcfivlsd onargy and
m<br />
range. A vory small fraction of tlia a;-partido has somotimos a considerably<br />
longor rnngo. Tho main faot of tho uniform onorEy of tho a ' s may bo o::plaiulcd<br />
by assuming that the &particlo has boen srni-blied by a nuoleus of R givon enorjg<br />
and that arnothor nuclous of a givon miergy is left bohind after OhoQCI-dooay,<br />
The anorgy difforonoo is thon oarriod away by tho C;C-part.iolo,<br />
Rnat liar important law ~f tho &wadj.oaativity is tho eonnoctioii bc%?voon<br />
tho range of tho par'ciolos and tho docay aons%mt,<br />
Thts is tho Geiger-Nuttal<br />
relation,<br />
It statos thab the logarithm 0% tho docay oonstrrlit is a linoar funm<br />
tion of<br />
tha railgo,<br />
Actually one finds that tho relation botiwocn tho logarithm<br />
of tho dacay constant and tha rango is sligJitly diffcron% for tho radium,<br />
aot initun, ntid trhoriuh famillo s,<br />
Tlzc throo straight liiio s -&pro sonting the<br />
relations for thoso thrm familios arc liovmvor parallel and their diatalioos from<br />
oach othor ara small,<br />
If oil0 had plottod %ho logarithm of tho doaay oonstmt<br />
as a fuiiotion 0% thc onorEy ratilicr than as o, function of tho rango, tho plot<br />
would again bc in good approximation a straight linc.<br />
Thc oxplana%ion of tho Goigar-1Vuth.l relation is closoly oonnoctod<br />
with the shape of fiho potential anergy aoting botwem tlia d-parbicle and the<br />
frwtion of the nucleus which +:le tr;-parbiole has Just left, At radii bigger<br />
than the nuclear radius ro chmm in the 'figure, the interaction is a Coulomb<br />
repulsion as shoTim in +he same Tigura.<br />
At distances closer than ro some kind<br />
of an at0raction must @xist, otliemvise %hers would bo no rea on why the origMal<br />
hualeus that ha3 emitted the &-particle<br />
had stayed together any length of' timer<br />
In the potential valley at distances smaller than<br />
rQ<br />
the &-partiole<br />
had in tihe<br />
original nucleus an en~~cy.r level at; +lie anergy value El
99<br />
I<br />
Figure 5,<br />
After thea-particle has moved out t$o suffioiently greati r<br />
values BO that all<br />
Znteraations including the Coulomb interaction have become negligible, the<br />
Cjpartiole will PcSsseS; that energy E in %he form of kinetic enera.<br />
If' now *lie product nucltsus is bombarded by&*particLes of energy E,<br />
pure Rutherford scattering is observed, iiidicatinG that at the radtus router to<br />
which the =-particle<br />
can penetrate aooording to classical mcrhanics, the unmodi-<br />
fied Coulomb force still holds,<br />
There arises the quostion:<br />
in what way can the<br />
dr-partiole OY'OSS the potential barrier bebveen rinser and router (see<br />
figure 5 ).<br />
This quss2;ion has been answered by Gurne$r and Condon, and by Garnow.<br />
In qu4ntwn mechanios leakage through a potenthal barrier is possiblec though %he<br />
probability of suoh a process becomes small when the barrier becomes high or<br />
broad, The time of decay may be estimated by imagining the &-partiole oscillating<br />
wi%hin the potential hole, by calculating the number of times the &==partiole<br />
bumps into the barrler per second, sand Pi)mlly, by multiplying with %he<br />
probability that the amparticle penetrates .the barrier in one collision.<br />
The facC that the a-particle can penetrate through tho barrier is<br />
*<br />
conneoted with the &partiole's wave nakuro. An illustration from optPas may<br />
be g5.ven. If light impingos on tho intarface between glass and air from the aide<br />
of the glass un4er a sufficiently glancing angle, total reflection takes place,
m Mm first glaars pLaZ;e this aeaand giargo plate will be reaaked<br />
exponeriCla1ly 4eoaying field 3n air and par+ of' the 13gh-b will be 6rammfttad to
*<br />
104<br />
nuolei are known whioh emit in turn, an alpha, a beba, and then another alpha.<br />
The 4x0 aLphas have quite definite energhs, yet the beta partiele does no$,<br />
Some years ago,' attempts were made to meagure the energy of the betas by using<br />
a very thick Pb w alhd oalorimeter,<br />
ft was hoped at the time tihat all the energy<br />
of the emitted particles would be measured.<br />
However, an amount, aorrespottdfng I<br />
o thw ama under the distribution ourve, vas found.<br />
Bohr made the suggestion th& perhaps energy and mornenWm vms not con..<br />
served ina-4$sintegration;<br />
seoond partiole .IC<br />
still valid,<br />
A spsuial hypathesia made by Pauli was thak a<br />
the neutrine -- was emitted, 80 that aanserva'binn laws were<br />
Attempts ham been made to oorrelate the maximum enera E in a begs<br />
speofirwrn wifih the disintegration aonstantA I as was dona in the uase of<br />
&,-partiales (Geiger-Nutall laws), "he Sargent ourves, a plot 02 log h YS E,<br />
we an attempt in +his direction.<br />
From absorption measurements in aluminum and using a standardized<br />
method of extrapolating tho rangeeenorgy ourve, on0 obtains<br />
R 0.543 E 0.16<br />
,<br />
where E is in million electron volts and R in cmb (Fsathor rolation)r<br />
A satisfaotory theory of beta-disintegration must explain the oon-<br />
tinuous spotrum and give a oorroot rahgowenorgy rolation.<br />
Formi has dewlopod<br />
a thoosy with bho neutrino hypothosisr<br />
In tho nuolaus a neu~ron is aonvar%ed to<br />
a proton 'oogother w9.tjh tho omission of an olootron* Tho rolo of Z;b olao..t;ron 5.8<br />
similar to that of' photon in radiation thoory. Tho probability of omissZon OS<br />
tron with onor#<br />
in tho interval El E # AE for lfght nuole3. is gSvon by<br />
P{E)dB g H%7 t{U$Vmdy12 (Eo - Edl3<br />
whore E is in unl2;s of C the voloaity of light, E, %hc maximum
The noutirop is expe&ed to be<br />
astive with Q mean life of several<br />
hours. Experimental evide'noe 9s hckin oaqse the neutron renot6 relatSvely<br />
quiakly k th any nu'clsus that may be pr<br />
tr Hwm@r, experimefits hhvo boon<br />
proposed to mocasurc3 this disintogratlon oonstaht.<br />
Novombor 18, 1943<br />
LA-24 (18)<br />
Lectmor: E, SO~PO<br />
UCTURE XVXfXt<br />
ON AND K CAPTURE<br />
I<br />
it pr>si%rons, giving riw to 1Jsobars with one<br />
nuclei.<br />
This roaakian OOOU~S if tho avail-<br />
able enorgy is at loast 600 kv, tho she~gy oquivnlsnt of tho gosl.ttronz, that is,<br />
$1 tho mass of $he parent nuolous sxcoods thn2; of tha daugh$er by at lwst tho<br />
mass of tho posttron.<br />
If $he onwgy nvtrilnblo is lees, then tho nuolous will<br />
0<br />
capturo an oloo.t;ran fram the innormost shell and subsaquently emit an X-ray
106<br />
quantum,<br />
This so-called K (electron)-oapturs by the nucleus also ooours at $he<br />
higher energies but la less possible,<br />
2) , Gamma ,Radiat;ion<br />
w. , ,<br />
I<br />
The eleotromgnetio radiations emitted by radioactiva wbstanoes are<br />
oalled ?-rays.<br />
The Y-rays are aharaaterized by %heir frequenoy, or 8s is more<br />
usual by the energy of a quaRtwn expressed in electron volts OF in unsts of<br />
2 The range in energy of 9 -rays from rzsturally radioaotive subs.t.anos.s is<br />
approximately 60,000 eV to about 3 Mev (million tslootiron volts),<br />
By bombarding<br />
Li with pro4jons more energstio ?-,rays ara obtained, values up to 17 MeV have<br />
been faundr<br />
The highest; energios In tho lwb have boen renohed with tho XersG<br />
betalwon,<br />
The theory of ?-ray omission is similar to that of the emission of<br />
light from atoms and moleoulad, oxoept that tho magnitudes of aomo of tho quan-<br />
tities involvad are muoh different,<br />
The probability of a transition from one<br />
onore fitate say m, to another lover enorgy etato n, and tho omission of Q<br />
light quantum is &ven by<br />
whore 4.d is 2fi times %ha frequonoy of the omtfkod radiation, C tho volooity of<br />
light.<br />
P i s tho so-oallod matrix olomont of tho transition and is tho ol0a.Dria<br />
momont avcrhgod over tho oigonfunctians of tlio two statas m and p* Equation<br />
(1) roprosonts tho first torm of an expcnsiqn in powars of co/x whom ro 3s<br />
tho nuclear ra@ius and A tho wavolongth of %he omittbd quantum,<br />
PhysicalLy,<br />
tho flrst torm in tho oxpansbon corrosponds to tho dipole radiation, %ho aoomd<br />
ta %ha guadrupolc radiation, otor In atornib prooossos, is 60 largo compnrod 1<br />
to tho radius of tho atom, SO that the soaond term is vory muohismallor than tho<br />
*<br />
first,<br />
For nuoloar procdsms, huwovor, )r and ro do not diffor as grocxtly; con-<br />
sequantly tho second term mczkos an approciable contribution,<br />
Tho magnitudosr of<br />
the matrix ~ lomont~ vary over o. oansidorably w2da rnngo corrosponding to wido
is ejeoted, then<br />
'kin<br />
Enuolsus a Eg<br />
This prooess is known as internal cot~version, Photographs onfl -speotrograph<br />
show tha2; the ihternal ootlversion electrons, sa is to be expected, form very<br />
homogeheous groups in contradistinotion %o disintegration -alectro~ls which have<br />
a oont$nuous distrj;but%on in energyr<br />
Usually a group of lines of varying inten-<br />
sity is observed cmrresponding to ejoation of electrons from different onergy<br />
levels of the atomv<br />
Oacasionally3. -radiation is acreompanted by fl-disintegration and the<br />
questSon arises, as to whether the radiation precedes or follows %he disintegration,<br />
If' inkernalfy converted eleatrons are presen4i and are analyeed to deter*<br />
mine the energy differences beheen the various groups, a11 anwer o m be given.<br />
For if the spacing corresponds to an Xnray diagram of the paront nuolsus, thovl<br />
%he 5 -radiation precedes %he disintegration, whereas if tho spac32ng is that of<br />
the dawghter nuclous, the cmnvarge is truo.<br />
The ratio of the number of oonvorsion sleotrons to the number of<br />
3, -quan.l;wn is lcnovm as the convorsion coeffiaient,
108<br />
reLatlvely very smdl 80 %hat the me<br />
Zy large, %he exaifed level $8<br />
r8'<br />
life Pf %ha% leveb S,<br />
be mefastable,. Le% us<br />
are in the ground state, %hey<br />
underg -disintegra+ion wi-tih a half- 18mr If on Lhw othw hmd,<br />
nuclei are in the first sxaited level, wbiah has a half-3iie for @)I<br />
L<br />
-eminsfan<br />
en the observed halfmlife of t<br />
sinae the rate defermining fact<br />
proof of suah "isarnsri.o!' nucl<br />
sequentfl remission &a ,found ts be<br />
the Blower ra2;e of<br />
hat they oan be produeed by exaita-<br />
s or e?.tzo-brons, first; t o a higher level, from wbioh tihey reaoh the<br />
by 3 -@mhsS$Ot1 of<br />
gh, ra%e r simple ohemical te thni<br />
rt J.<br />
H+ Williams<br />
POD Was found by Both@ .<br />
(1930), (Naturwiss -#39 753 (1931)). They<br />
published the results of tmme experiments i n whbh the 1igh.t elsmen%s L1, Be,
109<br />
and B were bombarded by polonium alpha particles. A wry penetrating “radiatiopf1<br />
was emSItted which vas more diffioule to absorb by lead than tho shorbst wav~<br />
- lowlgth 1(cY)ays Ohon known, thosa of I‘hC’’, Their orlginal explanation was tha%<br />
the radiation was light of a vary short wavo length.<br />
Today wc know that ths<br />
inoreased absorption of wry short WWo*lQng‘ah @)f -rays by the prooass OS pair<br />
areaLima would alimina$e this explanation.<br />
They war0 lasd to *he aasurnption %ha%<br />
tihs Itradiation” was pho.t;ons on other grouurda, bosSdos tihat of trbsorption measure-<br />
mentis, on0 being that an traoks were obsorued in a cloud chamber,<br />
Carrying their hypothosis of hard to its logical conolusion on<br />
the basis of the Klein-Nishina absorption curve as a f(E >, thoso early<br />
”r<br />
axporimehters gave an enorgy for this “radia+ion” of from 7 to 55 Move Their<br />
wbsorpgion moasurcmants were mado wi%h Gobgar Fountor dotootors,<br />
In a nuoloar<br />
squation they pos%ula%od that the reaotlop involved was<br />
Supposo for an exoroiso wo oheolc tho onorgy balanco of Ghis equation<br />
by subeti,tuting masses in Z;ho oqua2;ion<br />
qBe9 9,01504<br />
$Io4 - 4.00389<br />
-1ZTiRZ<br />
c33 I 13,00761<br />
-‘TUTEZ mass uni+is<br />
Ono mass unit<br />
931 Mov<br />
Enorgy availablo EG-b10,5 Mov,<br />
The problom was next attaohad by Curia and Joliot (Comptos Rondus 194<br />
273, 708 (1932) )., Tho important difforonco was that in Chcso resoarohos, thin<br />
window ionization ohambers instoad of Goigor gountors wore usod as dotootors of<br />
n.lnbul’<br />
tho radiation.<br />
Thoy found %hat if matorial aontainlng €1 were into.rposod
a<br />
botwoora. tho &on Ba SOU~CO and tho ionization Qhambor, tx<br />
'ianixa$ion was obswved.<br />
obsorvod,<br />
2arg~ incroaso in<br />
110<br />
For absorbors of othor kinds only small docroasos 'vmo<br />
Thay showod that this fvloraaso in iolziza~ion was duo %o the prescnco<br />
of H roooils leaving; thc "absorbcw" and ontoring thoir ionization ohambor.<br />
Cloud ohambar studias revaalad tho prosonce of those proton roooils as ~013. RS<br />
Ho racails in n Ha gas.<br />
By moasuring tho lengths of thoso roooil tracks it tlrns<br />
possfl~1a to de'cormine tho onora of the roooils,<br />
Soma of thoso rocoil protons<br />
had enorgias up 00 5 MOT,<br />
No traoks wro absckod for tho primary ltradSationtl<br />
which thoy assmod to bo? -rays.<br />
Curio and Jo1io.t; oxplninod thoso obsorva.Dions<br />
by saying that the onorgy lirnnsfor ocourrad in a Compton pro~~sa~ Tho? "ray<br />
enarm otilovlated in this way turned out to bo 50 Mov.<br />
Unfortunately, tho oar-<br />
oulations for othor roooil pnrticlos on Z;ho basis of this %hoary $avo diffaronf<br />
?-ray enorglrss,<br />
They WO~O highor for hanvior rocoil nuolai,<br />
Chadwiok (Prac, Roy. SOC. A* 136, 693, (1932),) solved %hJs d ilom sa<br />
c-<br />
aonvincingly tha% ho reooived tho Nobol prize, A sohomatio diagram of the<br />
apparatus usod follows.<br />
FFg '<br />
1<br />
-**<br />
7-<br />
I<br />
I<br />
7<br />
VQCUUm paraffin ionization ahambas<br />
l<br />
amp3iFiQr<br />
Tho measured rocoil onorgios of H' and<br />
nualai vmro 5.7 M w nnd 1.2<br />
Mov<br />
cospoctivoly.<br />
On the basis of tho phofon hypotiho sis tho oalculatad ltradlatioatl<br />
onoF6;y would bo 55 MQV and 90 Mov rospocti-valy.<br />
Thus 'Gho photon bypathosis was inconsistant with tho obsarvntionsr<br />
Either tho laws of aonservation of onorgy and momantum or the hypothosis as to<br />
*<br />
the naturo of tho pherzomnon was 'Go bo rojeotod,<br />
Chadwick thorefam assumed<br />
that tho "radiation" was a new pnr%iolo of mas8Nmss of proton arzd having Zero
mutual forces between nuclei are fhe subdoof of Mr.<br />
Artifioial Souroos of Neutrons<br />
3.12<br />
Critohfieldrs leoturesr<br />
Not a source of monoonorgotic neutrons,<br />
This mode of disintegration withdj.particlas as tho bombardikg particle<br />
1<br />
which results in neutron produotioh has beon obsorvod for sovcwal light oloments,<br />
Li7, 1311, NI4,<br />
FL9, Na23, Mg24, A1Z7, but is moa6 intenso from Be, then 3 and Li.'<br />
Those souroes af neuJcrons am assentially v&y woak, ralativo Do<br />
sources of whioh we shall speak lator,<br />
Tt was impoosiblo up bo 1942 to get<br />
sourco8 in OhSa manner using Po as tho G-sourco which give mor0 than ,405 r/goo<br />
or oxpressod in anathor way tho yield is 60 n/lOs stoppod, On the o6hor<br />
hand, if tho &,$s &re obtainod from R, which is intimately mixed with Bo, the<br />
numbor of noutsrons em$tt.l-od per mor par mq. of Rn is of tho ordor of 104/soa.<br />
With a curb of Ra on0 thwofora has an appreciably notivt7 sourcoo<br />
Unfortunalioly<br />
' 3( -rays are prasont hero whore %boy aro abscn9 in tho Po &~ourcos, except<br />
tw 1 h ?.',ha<br />
A complete disousaion of @Len roac%Sons, %ha enorgies of tho neutrons<br />
and of .Oxlo Z X S S O G ~ ?*rays ~ ~ Q ~<br />
is given in Livingstion and Bcthe, Rev', Mod, Phya',<br />
9, pagee 303 to 308, 1937,<br />
*-<br />
Photoe1oct;rio Produa@ion of Nautrons,<br />
With two of the li&% elaman4x H12 and Bo noutrons oan bo produoed<br />
4 '<br />
by tho 7 nmys of' ThC" or aqy ?trays whose onergy OXoQeds fjhg bi'nding enorgy<br />
of tha neutron in tha primary nuolOuar<br />
The probability of $his process is low<br />
and as a oonsequonco tilio yield of those SOUYGOS<br />
-<br />
Examplos - H2+ hW-+H1 .t n'+<br />
Q<br />
is oorrospondingly' lawc<br />
Chadwfak and Goldhabar, {Procr Royv SOC, 151, 479 (1435),) usad tho ihCtt +L*rays<br />
of anergy 2,62<br />
Mov and munsumd tho aa.t;ual mrxgnitudo of tho ionization of the
protons formod in tho above reaction in ordor to dohrmi'ne thoir onorgy.<br />
113<br />
Sinoo E, E ~8 oonsaquonoo of their having approximately cqual<br />
P<br />
0 masses, tho olnergy equation moms<br />
The analysis of %he data gav0 Q "-2.2<br />
MCIV<br />
This moans tho naukm is bound fn*tho H2 nudous wi%h an sncrgy of<br />
212 MWVI<br />
Knowledgo of this binding anorgy is of groat thoorotical importance<br />
sinoo it allows us to dskormine the maso of tho nautwon,<br />
Mass of Nou6ron<br />
.-...I<br />
RowrlOing the pho.i;ordisintagrat;ion equatlon<br />
Mn a 1,00893 f?OOOO5<br />
A tnblo in Livingston and Botho, page 353, swnnarizos tho avidmoo on<br />
?-araac%;iQns up to 39370<br />
A modem usoful SOUFOO of noukroyls usiylg ?-rays on light elemonts<br />
illustrchtas tho advanoos in toohniquo mada in nuolsnr physics in rooenf yomsu<br />
As we vrS11 dlsou~ls IcL~Q~, It is possiblo to produoo an isotopo of<br />
by dou4xwon bombarbont of Br vfhioh has a l/Z Iifo of 80 days nnd omits a "')+-ray<br />
of suffid.en6 anorg to dlssoclake Bo9 and givo nautrons of 16Q-k~ onergyt<br />
This souras hnq boon extromoly usoful RS Q SQUTOO of poln%ively monoonorgo'eio<br />
Y<br />
e
134<br />
December<br />
2, 1.943<br />
Third Series: Neutron Physics Lecturer : J. H. Williams<br />
Neutron Sources (Continued).<br />
Neutrons may he produced by<br />
the bombardment af certain nuclei with protons, deuterons or alpha<br />
particles, that is, a nuclear faaction takes place and @<br />
emitted.<br />
neutron is<br />
To initiate the re~ctim, the bombarding particle mu8t<br />
penetrate the potential barrlar surrotmding the target nucleus<br />
In other words9 .the projectiles must have a sufficiently high<br />
kinetic energy<br />
We may distinguish two methods used to e.ccelerate such<br />
particles, namely direct and resonance acceleratcrs,<br />
Tne first<br />
is long vacuum tube one end of which is at a very<br />
high potential and contains a sourca OS ionsS and the other i s<br />
usually at ground potential and contains the target.<br />
The ions<br />
traverse the vacuum tube acquiring kinetic energy came sponding to<br />
the drop in potential,<br />
The resonance acceleratorg or cyclotron, consists cf a<br />
flat, hollow cylinder<br />
resembling the letter D. An<br />
slepara ted into two sections each<br />
oscilla tine voltage difference<br />
applied between the ~ W O “doesft OS the order of.<br />
and &t a Frequency of several rnapacycles.<br />
a hundred kilovolts<br />
Along the axis, a<br />
‘magnetic field of several thousand. gauss is applied,<br />
The ion<br />
i
*<br />
115<br />
source is on the axis, Under the Froper conditions, the ions<br />
travel in circular orbits of increasing radii and Increase tketir<br />
kinetic energy during each passage from one af the dees to the<br />
ather. The sucees~ of the cyclotron is due to the fact that the<br />
angular frequency of the motion is independent of the energy (ieeo<br />
of the radius of curvature of the orbit). When the voltage frequency<br />
is equal to the angular frequency, resonance conditions<br />
obtain. 1t'l.s possible by such resonance means to attain energies<br />
of the order of several million electron voltso<br />
In this type, the Incomlne; particle is a proton and the emitted<br />
one a neutron. Written in detail:<br />
ZA + HI-+ (Z + I)* + nt<br />
where A is the mass number and 2 the atomic number, Q is the<br />
energy of the reaction, It turns out that Q in th
116<br />
and 1.86 MEV. Thus this method is very convenient for producing<br />
mono-enerpetic neutrons simply by controlling the bombarding voltage.<br />
It is clear from consideration of momentum and energy conservation<br />
that Tor a given bombarding voltage, the energy of the emitted,<br />
neutron depends on the direction of emission, so that neutron6<br />
I<br />
observed in the backward direction of the beam will have less energy<br />
than those in the forward direction, SQ that sne may obtain a range<br />
of neutron energies as a funct$on of angle. Practically, there is<br />
an effective spread of bombarding energies because of the fact that<br />
some of the neutrons will be slowed down in the finite thickness of<br />
the tarpet before they initiate a nuclear reaction. This variation<br />
can be made arbitrarily small- but is usually of the order of 5 to<br />
50 kv for intensity reasons,<br />
I_di,_n_) reaction: Here the typo is<br />
ZA .c ~ 2 - (Z I 1W 4 n1 I Q<br />
The so-called (d,d) reaction is an example:<br />
H2 4 H2 .crlt He3 4 nt + 3.2 MEV<br />
This reaction provides a copioits source of neutrons. Yields of the<br />
order 09 IO8 neutrons/sec are possible with bombarding currents of<br />
the order of lOrA impinging on heavy ice targets. The efficiencies<br />
are of the order of lnh05 deuteronso A desirable feature is that<br />
the reaction has an appreciable cross-section at low bombarding<br />
energies. One difficulty is the preparation of thin tarpets. Often<br />
gas targets are used but the yields are correspondingly smaller.<br />
Another example is<br />
c612 + HI2<br />
~*lb* N13 + n' + Q
117<br />
e<br />
with Q t -,25 lvaEV9 which means that the reaction is endoergic.<br />
The energy spectrum of the emitted neutrons is not simple because<br />
of the C13(d,n)N14 reaction.<br />
ut_eRra_tions muted by neut= &a) re&ctiongA<br />
An example of this type is<br />
n' + N14-He<br />
4<br />
+ B 11<br />
One can obtain cloud chamber photoeraphs of this process to<br />
establish its identity beyond question, but it is a somewhat<br />
tedtous procedure since many photographs must be taken before one<br />
finds evidence for the reaction. Another well known reaction is<br />
and also<br />
nf C Li6- H3 9 He4<br />
330th of these reactions have relatiy?ely very large cross-sections<br />
for thermal neutrons and are very useful for the detection of<br />
neutrons by detecting the ionization produced by the emitted<br />
cparticles. A common procedure 3.3 to line the walls of the<br />
ionization chamber with a layer of boron. Often proportional<br />
counters are used which contain gasious boron trifluoride and the<br />
individual particle pulsas are detected.<br />
&D) _r_q&tian: Symbolically this reaction is<br />
*<br />
This reaction is usually endoergic, the case ZA c,~Nl4 being an<br />
exception.<br />
This type is the simple capture<br />
process in which a neutron is absorbed and a gamma quantvm
I<br />
118<br />
e<br />
omitted.<br />
Process was first observed by Fermi in the production of<br />
artificially radioactive isotopes by slow neutron irradiation,<br />
&.$I<br />
r3Gion:<br />
ZA + nt ...lp ~A-1-t 2n<br />
It 9s clear that this react.ion requires at least an amount of<br />
energy corresponding to the binding energy of a neutron in the<br />
nucleus. Hence fast neutrons are required to make reaction go.<br />
This process ban only be reliably identified if the resulting<br />
nucleus is radioactive,
*<br />
LA-24<br />
December 7,<br />
1843<br />
(21)<br />
119<br />
Third Series: Neutron Physics Lecturer:: J, H, Williams<br />
Slow neutrons are produced by allowing fast neutrons to<br />
pass through paraffin, The collisions of the neutrons with the<br />
protons in paraffin result in the initial energy of the neutron<br />
being divided up between the neutron and the proton. Thus In<br />
successive collisions the neutron loses energy until it finally<br />
winds up with thermal energies I)<br />
The process of slowing down of the neutrons depends on<br />
the collision cross-section between neutrons and protons, We shall<br />
discuss the concept of a collision cross-section, as it has been<br />
introduced in the kinetic theory OF gases.<br />
Let us assume that two gas atoms collWe if they come<br />
closer to each other than the distance s, Assume further that the<br />
relative velocity of the two particles is V. Then, %he particle<br />
under consideration wllZ sweep out per unit time a. volume Ss*V and<br />
the particle will coll%de with all collision particles within this<br />
volume, Therefore the number of collisions par second is $T.s2Vn<br />
where n is the number of collision partners per unit volume.<br />
Assumln'g that a11 of the relative velocity Y is due to the motion<br />
of the incident particle, the distance through which this particle<br />
0 travels between colllsions, that is the mean free path, is given by
1)<br />
I<br />
,&=l/fls%. (in the kinetic energy of gases the incident particle<br />
and its collision partners move with the same average veloqity<br />
which introduces into the farmula or the mean free path a factor<br />
1/42.)<br />
distance x<br />
The probability that a particle should move through a<br />
without making a csllision is given by e -x/k re;<br />
e -xm% Here the colZislon cross-section o t.f$s' has<br />
been introduced,<br />
This cross-section i s in gas kinetic theory the<br />
geometric area within which collisions take place.<br />
the above exponentSal formula may be used to meaaure the cross-<br />
--<br />
.L<br />
La<br />
Experimentally,<br />
section, For this purpose a $'lab of the material made aut of the<br />
collision particles and having thickness x<br />
is placed into a beam<br />
of incident neutrons with intensity Io, A smaller intensity I: will<br />
emerge from the slab since the neutrons which made the collision<br />
are missing from the beam, Then the ratio of the two intensities<br />
Ss given by the formula I<br />
=cox<br />
,/ro<br />
X<br />
I<br />
0<br />
Nuclear cross-sections have usually the order of<br />
magnitude 10 -24,,2<br />
This quantity has been introduced as a unit<br />
of nuclear cross-sections. It is called the barn.
3.21<br />
e<br />
The collislon cross-section and mean free path o f neutrons<br />
in paraffin vary a great deal. with tha energy of the neutrons.<br />
crawwmction Is a few barn9 above a My whereas it i o fifty barna<br />
for thermal energies,<br />
few cm, while thermal neutrons have a mean free path af a few mm.<br />
The<br />
At the hSgh energies the maan free path is a<br />
ThB ~nergy loss d h natttybn in a collision with a probdti<br />
depends on the angle of deflecti6M. rf the ifiitial energy of the<br />
neutrons is E,<br />
collisions i s Eo/en.<br />
then the average energy of the neutrons after<br />
to.reduce fast neutrons to thermal energy.<br />
Thus apnroximately 20 collisions are necessary<br />
The number of slow neutrons that can be obtained in a<br />
given volurqe of paraffin depends on the rate of production of sluw<br />
neutrons and on the rate of absorption of slow neutrons by the<br />
hydrogen in the paraffin,<br />
H' + n'4H2 + htb<br />
This reaction is only one example of the very<br />
common process in which a slow neutron is absorbed by a nucleus and<br />
a )t<br />
active.<br />
-ray is emitted,<br />
This absorption is due to the process<br />
The resulting nucleus Is frequently rad5o-<br />
(This is not the case when neutrons are absorbed by HI<br />
The production of radioactive nuclei provides us with an easy means<br />
of measuring the capture cro9s-section9 iqes9 the part of the calli-.<br />
sion cross-section which is duo to the capture process.<br />
n
*<br />
122<br />
To perform the measurement we use a thin sheat containing n absorb-<br />
ing atoms per om2,<br />
and serr~nd.<br />
We let N nuclei impinpapan the sheet per cm2<br />
c ,<br />
Then the number of r%dio-ac$ivs nuclsP fo~r~lad, per cm2<br />
aqd second. is given by nNo,,<br />
Thus the capture croasmxtlon may be<br />
measured by determining the radio-activity formed b<br />
Some nuclei have very high thermar capture orosstsections.<br />
For instance, the capture cross-sectlan of cadmium is 3,000 barns,<br />
While this cross-section I s very muoh greater than the geometric<br />
cross-gection of the nucleus, tha scattering cross-8ection at car-<br />
responding energies i s only a fev barns, that is, comparable to the<br />
geometric cross-section,<br />
A further peculiarity of the capture cross-sections at law<br />
energies was found by Moon and Tillman (Proc. Roy. Soc,<br />
A =_.467<br />
(1936) ) They observed that the capture cross-section changes<br />
rapidly with the neutron energy showing maxima as can be ~een in<br />
the followinp figure e<br />
The maxima are separated bp distances of the order of 10 electron<br />
0 . vults on the enerpy scale.
123<br />
The crude explanatfan of this phenomenon is resonance.<br />
The word resonknee suggests the analogy with the great energy trans-<br />
Pers which become possible<br />
quencies are equal to each other,<br />
between vibrating systems If the fre-<br />
If the rest energy o f the initial<br />
nucleus, plus the rest energy of the neutron, plus the kinetic<br />
enemy carried in by the netttron, happens to agree with the energy<br />
of a compound nucleus made up from the Initial nucleus and the<br />
neutron, then the rearbIan becomes more probable<br />
For instance I ths<br />
above reaction of neutron cap turs by hydrogen may be written in<br />
form HI .I?' n? ++H$-r-)-H 2 $I hz/ where the carnpound nucleus H2* is<br />
an excited state of the nucleus H2,<br />
Not only capture but a .1~0<br />
scatter5hg may be describd with the help of compound nuclei,<br />
iristance, the scattering of neutrons by protons may be written in<br />
the fallowing form H1 4- h'+H 2 jc -.-E1 .f. nf<br />
For<br />
The various means by<br />
*<br />
which the compaund nucleus may disappear, that is, emission of a<br />
quantum or emission of a neutron, or possibly emission of another<br />
particle, compete with each other, At low naut;'ron energies emission<br />
of a 9 -quantum and consequent neutron capture predominates for<br />
most nuclei.<br />
The first theory of resonmce sarture of neutrons was<br />
given by Breit and Wignsr (Phys. Rev. s...T19 (1936) >; Rather then<br />
discuss the complete theory we shall describe the behavior of the<br />
neutrons<br />
A) if their energy is very close to a resonance<br />
B) if their enorgy is very low.<br />
In this way, we shalL understand the peneral appearance of the<br />
curve for at which has been given above, that is, the general rise<br />
I
1)<br />
of with demeasine energy (8) and the superposed resonances (A).<br />
If condition [A) is PulfllLed, the collision cross-section<br />
is deteranlned by the sinple neiphboring resonance with energy E,.<br />
Then, according to Breit and Wlgner a neutron with energy E .<br />
possesses a collision cross-section<br />
'<br />
Here k is the de Broglie wave length divided by 2'R of the neutron<br />
of energy E and, fir is the same quantity for the case that the<br />
neutron possesses the resonance energy E,, f is the width ~f the<br />
resonance and Fn is that part of the width which is due to the<br />
neutron emission, A datafled explanation of these quantities will<br />
be given in a later lecture.<br />
If E - E,<br />
is great compared to r the co?-.iision cross-<br />
section w ill fall off inve r s e ly as the square<br />
of the resonance.<br />
In exact resollitnc~ only the<br />
sz both sides<br />
the denominatar and at this point (or rather cl-ose tc this point)<br />
the cross-section attains its maximum.<br />
7"-r<br />
2 t~m remains in<br />
IF on the cther hand, the energy osf the neutron.is small<br />
compared to all resonance energies (assump K!.[J~ B?, then the energy<br />
dependence of the cross-section is determined by the variation of<br />
5 with energy. In this region, the cross-section varies as 1<br />
over the velocity of the neutron.<br />
on the spacing of the resonance levels.<br />
level is reached the l/v law breaks down.<br />
The size of this region depends<br />
When the first resonance
December 9,<br />
1943<br />
LA-24<br />
Third Series: Neutron Physics<br />
Lecturer:<br />
Lf the width of the lowest level<br />
is greater than the<br />
energy E, of the lowest level, then the Z/V Law will break clown when<br />
the neutron energy becomes comparable t o the width r<br />
The above statements concerning the region of<br />
the l/v law can be proved from the energy dap-endence of<br />
cross-se~tion as given by the Breit-Wkgner formula (see<br />
lecture).<br />
This energy dependence has the form<br />
validity of<br />
the capture<br />
end of last<br />
One sees that Cr,<br />
d/dE<br />
will behave as<br />
-1/2 ))<br />
log E<br />
l/v if the condition<br />
r *}<br />
is<br />
satisfied, This in turn is proved if either<br />
ar
126<br />
One very important substance in slow neutron physics is<br />
wdmiun. Its absorption crossIsection as a function of energy is<br />
roughly represented by Fig. 1,<br />
Its strong absorption for thermal neutrons is due to a resonance<br />
lying very close to thermal energies, A few tenths of a millimeter<br />
of Cd shielding is sufficient to remove from a distribution of<br />
neutrons the thermal neutrons.<br />
The first experlments, by which information on the Cd<br />
absorption and on the distribution of thermal neutrons was obtafned,<br />
were performed at Columbia, The experimental arrangement is shown<br />
in Fig. 2.<br />
'B<br />
Ra+ Be<br />
Cd
*<br />
e<br />
127<br />
On the left side of' this figure a paraffin I1howitzerlr 28 shown,<br />
which is a block of paraffin shaped as shoyn in the figure and containing<br />
a radium-beryllium scmrce at the indicated positjon, The<br />
paraffin blook emits S ~ Q W neutrons from a11 of its surfaces hut the<br />
arrows indicate a particularly stronp and somewhat better directed<br />
beam of slow neutrons ppoduced by the peculiar shape of the howitzal;,<br />
This neutron beam impinges upon a velocity selector which is showp<br />
in the figure to the right of the howitzer.' This velocity SeZectar<br />
consists of two rotating discs mounted on an axis at a distance D<br />
from each other and two stationary discs located before the first<br />
and behind the second rotating disc, On each of the Tour discs, Cd<br />
segments are mounted in such a way that there ar0 on every disc 50<br />
Cd segments and 50 segments free of Cd, This arrangement is shown<br />
schematically,below the velocity selector in the figure, To the far<br />
right of the figure a boron trifluoride counter is shown which<br />
detects the slow neutrons that have passed through the velocity<br />
selector.<br />
The velocity selector is rotated with a velocity of n<br />
revolutions per second, The Cd strips of the rotating and the<br />
adjaxent stationary discs wSll at a given time owrlap, so that the<br />
,<br />
neutrons can pet through the Cd-free parts of the disc. Shortly<br />
thereafter, the Cd segments on the rotating and stationary discs<br />
will alternate so that no neutrons can get through. IT the velocity<br />
of the neutrons is such that they will find either the flrst or the<br />
second pair of rotating and stationary discs closed, then a minimum<br />
of transmitted neutron intensity will he observed, Assuming that<br />
all neutrons have the same velooity v, this minimum should occur<br />
I
when the relation<br />
r*l<br />
v E lOOn/D<br />
is satisfied, It has been found that the neutrons omitted by the<br />
howitzer have an approximate velocity of 2,500 meters per second,<br />
This corresponds to thermal velocities,<br />
A somewhat more detailed<br />
1<br />
analysis showed thgt the neutron velocities have approximately a<br />
Maxwelltan distribution shown in the figure.<br />
In this figure, the number of neutrons having a given anergy E is<br />
plotted against that energy, The maximum of that curve shifts to<br />
higher energies when tho temperature is raised,<br />
An InvestigatSon of the resonances occurring at a few eV<br />
was first carried out by a somewhat indirect method, This method<br />
takes advantage of the fact that the reaction<br />
has a cross soctian obeying the l/v law up to quite high energies.<br />
The experimental arrangement is shown in Fig. 4.<br />
7°F"<br />
13 absorber<br />
Source *<br />
Cd box<br />
Detector fotl
129<br />
On the left hand side is indicated a neutron source (this might be<br />
a howitzor or another arrangement)# The neutrons emitted by this<br />
source pass through a boron absorber of thickness x,<br />
are carried out with various thicknesses X.<br />
Experiments<br />
On the right hand side<br />
of the figure is seen 8 detector foil which i s made of a substance<br />
mes radioactive under neutron bombardment.<br />
It is in thisz<br />
substance that we want to study tha neutron capture r@sonances.<br />
Tho detoctor foil is enclosed in a Cd box.<br />
. ment are made!<br />
Four kinds of measure-<br />
I. no boron absorber present, no aadmim present,<br />
2. no boron absorber present, cadmium present,<br />
3. boron absorber present, no cadmium present,<br />
4. boron absorber presant, cadmium present.<br />
Comparing measurements 5 and 3rl one Finds what fraction<br />
of the thermal and higher energy neutrona have been absorbed by the<br />
boron,<br />
Comparing measurements 2 and 4, one finds what fraction<br />
*<br />
higher energy neutrons have beon absorbed by tho boron (the ,<br />
thermal neutrons are stopped by the Cd box),<br />
Comparing tho d$fforenco of 1. and 2 with tho difference of<br />
3 and 4, one finds what Traction of the thermal ne'utrons haw been
130<br />
absorbed by the boron,<br />
We plot in Flg. 5 the quantity log Ax, that is, the<br />
logarithm of thc intensity observed in the detector foil as a<br />
of' the thickness of the boron absorber, The lower one of<br />
the two stralght lines in the figure<br />
"j A<br />
l - ~ . ,<br />
-7t<br />
is abtained by taking the ratio of msasurements 4 and 2.<br />
curve, therofcrre, refers to high energy neutronso The higher lino<br />
is obtained by. dividing (3 - r> by (1 - 2),<br />
thoreEdre to thermal noutrons,<br />
The intensity va~5'itions with boron<br />
absorber thickness may be represented by formulae of tho type<br />
-Kx -noox<br />
A, s AOo =IO<br />
ThSa<br />
This curve refers<br />
Tho slope of the curws 6ivl;s the absorption co9fficiont X, The<br />
coef%lc?.cct j . boron ~ is proportion41 to -&ho-1/2 powor<br />
rgy of t3:e :ioii$r0111s, Therofore, one obtains for the<br />
ratio of rssonanco oncrgy to thermal onorgy tho expression<br />
o m can calculate the resonance energy,<br />
b
One limgta an of the mothod $uaL<br />
cribed f s that It;<br />
assumes that neutron<br />
the neutron beam only by<br />
absorption in .boron a ing in boron. This 3.8 a valid<br />
asmmption for small neutron energies where tho capture cross-<br />
Bection in boFon is la~g6, For em ies above a kw, soattaring by<br />
boron disturbs the me8$ur@mentd<br />
1 In this manner, the relsonance captures fop v&rious<br />
have bosn explored, For medium and heavy<br />
found that redonances spacod Irrsgularly at 8, dista<br />
of 10 volts from each other are present and that these! POS<br />
have wzdths of the order of 1 volt. In light elements no such<br />
found, Theso elements shau show m'uoln mom broa<br />
i<br />
ea which are much more widely spaced, if In fact they do not<br />
ovexz<br />
ft i s more difficult ta study tho resonances o f an isatape<br />
which doas not; becoma radioactfve when It captures a neutron but<br />
'<br />
s Into anothor stable isotope,<br />
A rather inadequate mothod<br />
which has been applied in such cases is, based on the comparison of<br />
*<br />
The neutrons originating from tho SOUPGO<br />
and then after having flown a ~ Q C Q distcznce ~ D<br />
132<br />
fall on n noutron absorber<br />
dstoctod in a<br />
nautmn Gountoq for instnnke in BF chamber, Tb souma is not<br />
3<br />
g noutsons a13 tho Limo bot'bnaY dtirimg shopt Sntsrvals of<br />
'clrn~~ let us say of 10 micromconds duration, sepamacd by longer<br />
time interval$, of far instance 100 microsecond$,<br />
The neutron<br />
pulees arnitted by the SQU~CQ are shown in tho uppar half of Fig. 7.<br />
*-* . I C - P U l B @ Of SOUPCe<br />
Tho lower half of this flguro shows the time intorvnls during whYch<br />
tha rroutron detector $s sensitive, It is seen that the ddmtar<br />
will not roact to neutrons unless* they arrive t after they have<br />
been-emittad, Thus only neutrons with 8 voloclty<br />
e<br />
are observed. By varying tho thickness of tho absorber, ona can<br />
find the abssrpt;,ion coefficient for those partlcular neutrons in<br />
the material of the absorber, By varying the time delay A t om<br />
can investigate the depondenco of Po upon neutron anergy,
133<br />
sensitive time of tho datcctor which CQ~QS<br />
L period Jator,<br />
This is<br />
called recycling,<br />
Thls difficulty can bo minZmieod by choosSng a<br />
long rctpeat period for the emission pulscs and the corresponding<br />
sansitive intervals<br />
By tho mothad descrtbed bbovo it WBS possible to explore<br />
the propedies of' $IuW kc&aibns dj? ho abduk $0 cv,<br />
We shall no summarize briefly tha intoraction of neuktons<br />
with some element@. This will bo done in connection with a table<br />
which in part has bson taken from A, Both@, Rev, Mod, Phys. 29
I) a<br />
$><br />
1<br />
H1<br />
ETa s tic soatt ering 20 1.7 Efficient neutron<br />
slow-up &orL<br />
of thermal<br />
rleutrons<br />
Elastic scattering 4<br />
1.7<br />
3x6<br />
Neutron absorbed<br />
alcpha emitted<br />
goo<br />
310<br />
tl<br />
C= Elastic Scattering<br />
3000<br />
H3-4 Elastic scattesing 10<br />
and neutron absopbed<br />
proton emitted<br />
st? Elastic scattering 12<br />
and radiative capture<br />
n<br />
5<br />
6000 Produces active<br />
ZSQtOpe
December 24, 1943<br />
~~-24 (23)<br />
UTURE $E 1 ;HY IC<br />
Third Series: Neutron Physics Lecturer f J ,H, Willlams<br />
Generally speaking, neutrons are produced as fast; neutrons,<br />
There ape three principal react2ons in which fast neutrons are<br />
produced, In the first one an alpha particle enters the nucleus<br />
and a neutron is emitted, The reaction is symbolized by (a&<br />
This reac.tion has mostly historical interest since it was the first<br />
reaction in which the production of neutrons was obsemved.<br />
A second type of reaction in which neutrons are produced<br />
is the reaction in which a deuteron impinges on the nucleus and a<br />
nrsutroi? is emitted, The symbol for this reaction is (d,n>, Twa<br />
reactions of this type have become of great practical importance.<br />
One is the Be(d,n) reactton in which'the Be is the bombarded<br />
ntlcLeus and which gives a very copious source of neutrons. Another<br />
is the D(d,n) in which the deuteron hits a stationary deuteron in<br />
the target. In this reaction rnonoenergeliz neutyons can bs produced.<br />
The energy of the neutrons depends ?yi the energy of the<br />
impinging dm'teron, Nsutronv between 2 1vle!v .?,:i2 7 Mev may be pro-<br />
The third important source of neutrons is. the (p,n) reactions<br />
and in particular the Li(p,n). In this reaction an Li<br />
nucleus is bombarded by protons and neutrons emerge leaving the Be
*<br />
nucleus behind.<br />
This reaction i a endoenergetic, that is, the<br />
protons musk have a certain mlnimm energy in order that the<br />
136<br />
r~ectlpn ahouXd proceed at a18. Monoene~ket3.c neutrons between 10<br />
kflsvolts and 2 million volts may ba produced by this reaction for<br />
the similar range of incident particle energies, $.eoo up to 3.5<br />
Mev.<br />
The Li(p,n) reactlon may be written in somewhat mors<br />
detail ixl the following form<br />
**<br />
~17<br />
where the energy of reaction Q<br />
suspect that instead of the above rea<br />
takes place<br />
where Se7',9<br />
+ HI n1+ ~ e+ 7 Q<br />
is a negative quantity, one might<br />
Li7 H1- n I). Be7'4- Q1<br />
&n excited form of the ;Be nucleus and Q1 Is 8 greater<br />
negative reaction energy, If this second reaction proceeds, the<br />
neutron produced would bQ no Langer monoenergetic.<br />
this possibiliey has been found,<br />
ion the fdllowing rsaction<br />
No evidence of<br />
It is true that the Li7 nucleus<br />
(which i s quite analcgms to the Be nucleus having as many neutY'ons<br />
as the Bo7 'ia:3 ~)cckons 3iiC RS many protons as ths Be7 has neutrons)<br />
has an excitaii atnm of 485 kilovolt<br />
hat the Be'? h<br />
There is some reason to<br />
a similar excited state and the presenoe of<br />
such a state might giv rise to S~OWQT nautrons whenever the Li(p,n)<br />
reaction is used to produce neutrohs ab e 3.1'2<br />
.Measurest of. Fast Nqutrqu<br />
milaion volts,<br />
Ths first method of measuring the energy of fast neutron8<br />
made use of the Wilson Cloud chamber,<br />
The arrangement by which this<br />
be perfosmed is shown In the following f igura i
137<br />
,<br />
On the left hand. side the position of the neutron source i s indi-<br />
cated,<br />
shown.<br />
Qn the right hand side, a Wilson Chamber 3s schematically<br />
The chamber $8 fillad with a gas of small atomic weight<br />
such as hydrogen or helium, The path of the neutron is also ihdicated<br />
in'the f$gure. The heavy line in the chamber is the track of<br />
a recoiling particle, If this track makes the angle B with the<br />
continuation with the path of the neutron then from this angle and<br />
from the energy of the recoiling parlicle, the original energy of<br />
ran may be calculated, In partlcular, if the recoiling<br />
particle is a proton then the energy of the neutron is given by the<br />
2<br />
E, t E, cos Q<br />
1)<br />
Where Er is the energy of the recoiling particle and En is the<br />
energg of the neutron, it is usual to count only the tracks<br />
which go in the forward direction and make a small angle 0 (for<br />
smaller than 10') wi$h the direction of the incident<br />
he factor cos2 Q in the above formula may be<br />
and the energy of' the recoiling proton is aqua1 to the<br />
energy of tha inoident neutron,<br />
One danger of this pmcedure is that the neutron might<br />
have bean scattered In the wall of the chamber and while the protan<br />
seems to go farward, it actually does no$ lie in the continuation
2.38<br />
sf the neut;son which hag hit it,<br />
I<br />
Another difficulty is<br />
cks of the recoiling particfe are rather lo~g and they<br />
d d;n the ch er. The anergy 04 such recoils cannot be<br />
chamber measur<br />
A fwkhor diff icu<br />
tn eva2mtin.g results of cloud<br />
fits as walZ as of other meksuremants of fast<br />
noutrona i s tha* a pr0vious bowledge of the dependende of the<br />
callision cross section on tho anergy f the incLdent neutron is,<br />
A typtoal depsndende of thi8 c<br />
olLowirtg figure t<br />
SS sectlaxi 6 (wh<br />
mtia cross selStion) on the neutrqn energy E,<br />
.li_<br />
E,<br />
Umfortwately sometimes, as far instance, in the case of helltun<br />
ndence is mors involvrad EL^<br />
shown in the figurb:<br />
is shown<br />
e<br />
resonance is nd 2 MeV, ThSs resonance<br />
to d great:<br />
between J and 2 rn112l.m<br />
of this resonance them? many tracks were e<br />
8hQWing the presonce o<br />
Recently pho<br />
ha incident neutron has<br />
, Previous t o the disc<br />
cluud ohamber-s In measuring the energy aP neutrons,<br />
the phQtographic plates eon$cd.ns hydrogen,<br />
ous3.y interpreted as<br />
and 2 MeV,<br />
n USQ~<br />
instead of<br />
cause fani;zat$on and leave @evelopable grains in the waka<br />
The emulslon of<br />
The rocoil protons
,<br />
I<br />
Wilson Chamber only those tracks are $0 be counted which make a<br />
g2e with the direction of the neutron,<br />
The photographic<br />
technique has tho Pollawing advantages a$ aomparod $0 tho Wilson<br />
chamber*<br />
First9 the stopping power of the omuJ-sion is g?eatar9 the<br />
lsnbth of the tr~cks is only a few thoasandths of an Inch and the<br />
.danger that ths tsack IoaVtss the emulsion bafore ending is smaller',<br />
S@c?ond, in tho photo$ra$hi@ plate %Hers heod be much less material<br />
than there is in a Wilson chcmbar; consequently the danger of a<br />
scattered neutron causing Q recoil. i s decreased.<br />
Thirdly, the<br />
photographic plate is continuously sonsitive while the sensftivity<br />
of a cloud chambcr I s rostr$ctcd to the moments of expansson, The<br />
photographic technique has howavar the disndvantage that it becomes<br />
Jnaccurabe in couqting slowor neutrons,<br />
The recoils produosd by<br />
utrans make few grains and since the number of the grains<br />
produced is used in determining the neutron energy, fluctuations<br />
in the number of grains mako such anergy doterminations aifl$icult,<br />
The photographla mothod must be calibrated by using R reaction of<br />
known enorgy<br />
A less straightforward, but perhaps mme trlxstworthy<br />
mothod of rncaszxring nautron enargios makes uso of the ionization<br />
chambers,<br />
Such a chamber is shown 6chemnticnlZy In the following
filled with hydrogen.<br />
ians produced by tho rocolling protons.<br />
140<br />
"fie field wlthln the chamber collects the<br />
0111.the sight hand side the<br />
eonsloction to tho ampltfying grid is shown in which the charge pro-<br />
duced by the ;loris is amplified SQ as to give a measurable effect,<br />
The following figuro shows these amplified current pulsesp each of<br />
which corrosponds t o one recoiling particle according to the angle<br />
which tho recoiling proton makos with tho neutron.<br />
The rocoiling<br />
particle will carry more or Xess energy and the size of the pulse<br />
will vary come rspondingly e<br />
Amp 1. iff e, P<br />
output<br />
I<br />
9<br />
lJlilLnii<br />
l-cI3L"..b<br />
\<br />
O m may plot tho number of pulsos which have more cne'rgy than a<br />
given value E agninst srmplifior output<br />
b?mber<br />
of<br />
pu 3.8 e s<br />
pulse size or energy,<br />
Very small puZsos cannot be counted bacause they arc? obscured by<br />
%ha background or noise oE tho nmplifiey,<br />
By dffforontiating this<br />
curve9 one obtains the number of r(ccoil!-i?g particles of a glven<br />
encrgy.<br />
This I s shown in the Tollawing grnpht<br />
I<br />
Number of recoils<br />
If. tho gas in the ionization chamber was hydrogen then a second<br />
dlfforentiation gives tho number of primary neutrons as plotted<br />
against their energy.
*<br />
Ember of privariea<br />
The figure$ given above refer to the case of monoenergetic incident<br />
neutrons.<br />
If a distribution of neutrons with wrious energias<br />
impinges OR the ionization chamber, figures of the following kind<br />
may be obtained:<br />
Number of pulsos<br />
above a given energy<br />
I<br />
Tn w<br />
Ember of primary<br />
neutrons<br />
I<br />
Inswoad of recording the pulses of various hoig,,t,<br />
one mighu use an<br />
anplificr wfth a bias such as to cut out all pulses las~<br />
than a<br />
certein energy.<br />
The sonsitAvity of such a counter is shown below:<br />
S0nsdtivity<br />
-E<br />
Dslow the bias the<br />
amplifier-dotactor will not respond at all. At
142<br />
higher ener(gies the sensitivity will rise to a plateau and than<br />
docrease.<br />
Instend of using an ionization chamber fillod with hydrogen<br />
gas, one mSght use a cbambor in which the side facing the *<br />
neutron SQU~CO is coated with a thick layer of paraffin.<br />
Such chambers are then fillod with argon,<br />
sufficiently heavy so that argon recoils carry rolatlvely little<br />
energy and tho bias may be tzd3ustod in such il way as to count only<br />
the proton rocoils originating in the paraffin,<br />
such a chamber is shown in tho following figure:<br />
Sen s i t ivity<br />
L..=L<br />
---- - %2<br />
The argon nuclei are<br />
The sensitivity of<br />
Th..ere 2s this time an inzraasing ,rise o f sensitivity above the bias<br />
value,<br />
This is duo to tho fact that hfghor e~iergy neutrons produce<br />
protons which hcive R longer range llsn the paraf?'.k ad. have therefore<br />
'
14 3<br />
0<br />
a<br />
The coun$er Is filleti with hydrogen,<br />
Instead of a ~oll&ting plate<br />
it has a collecting wire near which Parge electric fields are<br />
present,<br />
In this field electrons are aocehrat~d towards the wire<br />
causing further ionization and in thts way greater- pulse result,<br />
With such proportional cQunter8, pulses as srilall as 10 kv may be<br />
detected whllle the ionization chamber described above is insensitive<br />
up to 100 kv or higher.<br />
Ionization chambers also may be filled with BF,,<br />
chambers the neutron reacts with the Bra<br />
products of total kinetic energy about 3 Mav.<br />
pulses are produced.<br />
3<br />
In such<br />
nucleus giving rise to end<br />
Thus sufficiently big<br />
Not only the measurement of the energy of fast neutrons<br />
but also the measurement of their number is a diSficuXt question,<br />
In one method due to Fermi and Amaldi<br />
the neutrons are slowed down<br />
in it water bath of sufficiently big volume surrounding the neutron<br />
source on all sides.<br />
After. beinp slowed down, the neutrons am<br />
detected by rhodium or indium foils distributed in the bath.<br />
From<br />
the known cross sections of thewe foils for slow rleutrons, one may<br />
determine the original number of fanSt neutron?, In order to do that<br />
one has to integrate over the<br />
the bath,<br />
the detectors in<br />
This last operation may be avoided if instead of the<br />
foils the slow neutrons are detected by manganese dissolved in the<br />
bath, BY stir Ping this solution and taking sample afterwards,<br />
one may find the integral of neutron absorption in the bath and from<br />
this quantity the original number of fast neutrons may be calculated.<br />
A theoretically very effective way of counting fast<br />
neutrons and measuring their energy is shown in the following *sketah:
..<br />
On the left hand side, an incident neutron beam is shown.<br />
This<br />
ges on an Iankmtion chamber which an the side of the"<br />
neutron beam has a thin paraffin film. This film is folhwed by a<br />
collimator which" is a plate transverged by naxvowe1: channels point-<br />
' ing in the original direction of the neutron, These channeLs will<br />
A<br />
let through only such protons whose direction colncidss with the<br />
direction of the neutrons, From the number of protons in<br />
the thin paraffin layer, from the colllsion cross section between<br />
protons and neutrons and from the number of pulses of various siaes<br />
izat3.m chamber the number of neutrons with various<br />
._<br />
-<br />
not work, The reason<br />
-c--_m<br />
is that there is too much hydrdgen in other places than in the thin<br />
rotons corne from<br />
We shall discuss the following processes ich occur when<br />
ring, ilzelastic<br />
8, (nJn> process,<br />
processes, These<br />
rmation abaut these<br />
een the neutron
14 5<br />
0<br />
$bt$xbs and thk neutran detector.<br />
if the scatterer is smdll enough<br />
and if thb neub!dn has beed ikfiected in the scatterfhg pyoeess by a<br />
I<br />
big enough angle9 then this neutron in the originlra bbam will miss<br />
the detector and no neutron outside the original beam will have an<br />
appreciable chance to be scattered into the detector.<br />
In this wag<br />
the major p’orlion of the elastically scattered neutrons w ill be<br />
missing from the beam,<br />
neutrons vdll be only partially observed,<br />
Similarly the fnalastically scattered<br />
In the case of the<br />
ally scattered neutrons an argument may be given which<br />
t these neutrons are distributed spherically after the<br />
z<br />
scattering process. In fact the scattering process may be described<br />
by the following equation:<br />
- - Q1+ ZA z(A3.1)’ zA*+ n1<br />
This equalton shows that the first act in an anelastic scattering<br />
probess is the formation of a compound nucleus with the atomic<br />
number A 4-1,<br />
neutron will have forgotim about its original direction of ink-<br />
dence,<br />
to be scattered<br />
When this compound nualeus emits the neutron, the<br />
Thus there is no reason for inelastically scattered neutrons<br />
part Icularly small angles and the scattering<br />
geometry described above will cause most inelastically scattered<br />
neutrons to miss the de te c t OT o<br />
(with It<br />
The ne u t r oqs which are captured<br />
gamma emission) or whrich undergo a (n,&dg or (n,p) reaction<br />
be naturally missing from the original beam.<br />
In the<br />
process the neutrons emitted are again spherically distributed and<br />
will therefore in all probability miss the detector.<br />
Thus practic-<br />
ally wha t eve r reactfon a neutron hag made with the scattering<br />
material the result will be that the neutron w ill not appear in the
I C<br />
14 6<br />
detector<br />
rblated to<br />
The intensity I transmitted by the bcatterer will be<br />
the intensity Xd<br />
df the 6rdgiba3. beam by the formula:<br />
Here small<br />
scatterer,<br />
4n%d<br />
I/J, 0 8<br />
n is the numbep of nuclei<br />
x is the thickness of the<br />
per cubic<br />
scatterer<br />
aJ1 C~QSS sections or the total cross section.<br />
centimeter In the<br />
and d is the sw of<br />
December 16, 1943<br />
LA+24 (24)<br />
Third Series Neutron Physics<br />
--.- LECTURE SJRIES OcN_ELEAR, PHYSIC$<br />
LECTURE XXIVt PROPF,RTIE$ OF'<br />
CROSS SECTION<br />
The total. cross section of a substance for fast neutrons<br />
can be measured by a direct transmission experiment In which the<br />
source,. scatterer, and detector are situated in what is known as a<br />
"good geometry". This good geometry is so defined that the solid<br />
angle subtended by the scatterer at elther the position of ,the<br />
source or of the detector is small.
*<br />
14 7<br />
In<br />
scatterer at<br />
by a neutron<br />
removal from<br />
the detector,<br />
this case the neutrons from the source strike the<br />
practically normal incidence and any process undergone.<br />
in its passage through the scatterer will result in its<br />
the-beam and its consequent failwe to be recorded by<br />
The neutrons counted by the detector will consis$<br />
almost entirely of those which have had no interaction with the<br />
atoms of the scatterer.<br />
A very small percentage of the detected<br />
neutrons wfll consist- of those scattered (elastically or inslasti-<br />
@ally) through very small angles, but in a really good geometry<br />
they will constttute a negligible fraction.<br />
A transmission experi-<br />
ment in such a good geometry wi.11 therefore measure the total cross<br />
section for all processes.<br />
If the incident neutrons are normal to<br />
the scc7.ttf;ererS which may be supposed to have 8 thickness Ax and a<br />
number t~1 atoms per cm3, the ratio of the detected intensities with<br />
and without scatterer will be given by<br />
- 14,<br />
-nom<br />
0<br />
where 6“ 2s the total CTOSS section.<br />
In contrast to this measurement with good geometry, trans-<br />
Mission experiments in poor geometry are performed in order to find<br />
the elastic scattering cross section.<br />
In this case the solid angle<br />
subtended by the scatterer at the positfon of the detector is very
14 8<br />
Q<br />
large,<br />
This makes It possible for neutrons scattered through large<br />
angles to be scattered into the detector, thus compensating<br />
.----<br />
Sauroo<br />
De t e c t ox-<br />
*-.-.- _._""<br />
Scat t oyer<br />
PGGR GCGMSTR'SI<br />
for those scattered neutrons which miss the detector.<br />
I<br />
Since this<br />
compensation is almost exact, scattering processes alone would have<br />
'<br />
j<br />
if the detector is biased so as not to record neutrons below a<br />
cartarn energy, the neutrons which are &elasticallyl scattered will<br />
still contribute to the measured reductions in beam intensity along<br />
with all the other processes with the exception of elastic ssattering<br />
This experiment with poor geometry and biased detector<br />
therefore measures the sum of all cross sections with the exception<br />
of. tke cross section for elastic scattering, To find the elastic<br />
scattering cross section alone, w0 therefore subtract the cross<br />
section as measured in poor geometry with a blased detector From<br />
that measured in good g'eometry,<br />
Capture cross sections are easily measured if the nucleus<br />
resuZt$np from the addition'of the neutron 1s radioactive, The<br />
*<br />
C~OSS section can than be obtained from a measurement of the<br />
!I inttuced activity. When the resulting nuofaus is not radioactive
149<br />
there is na easy means for measurinp the capture CPOSS section, X t<br />
sometimes happens that one isotope of a given element will become<br />
radioaofive upon capture of a neutron while 30me other isotope of<br />
the same element will not, However9 it is found empirically that<br />
the capture cross sections of isotopes of a given element are of the<br />
same order of magnitude, Therefore one can sometimes estimate the<br />
captum cross section of an isotope which does not gfve a radioactive<br />
product.<br />
Dividing the energy range Snto roughly three regions, the<br />
resonance Pegion from 0 to 10 kv, the medium fast region from 10 kv<br />
to 1/2 Mev, and the region of fast neutrons from 1/2 Mev on up, one<br />
gets a behavior of the capture cross section with energy which looks<br />
like this:<br />
crc<br />
0 10 kv<br />
In the resonance repian there are sharp resonance peaks in the cross<br />
sockLon superposed on a l/v background. Tn the medium fast region<br />
the resonances are much broader and are discrete only for( the<br />
lighter elementsp say up to oxygen, For the heavier elements the<br />
broad levels overlap and give a smooth behavior something like l/v,<br />
In the fast region above 1/2 Mev the behavior is more like l/E.<br />
For the very fast neutrons the cross section is determined principally<br />
by the area of the nucleus and the competition with other
e<br />
possible processes like inelastic scattering which becomes more<br />
probsbla<br />
the energy of the neutron$ increases.<br />
The cross section for an (n,& or an(n,p) reaction can be<br />
measured by putting a sample of the dement in question inside an<br />
ionization chamber which w ill count the alpha particles or the<br />
pr~tom directly, when the chamber is placed in a neutron flux.<br />
cross section for an (n,2n) reaction is extr)emely difficult to<br />
pt in the case that the residual. nucleus Os radioactive.<br />
Inelastic Scattering of a neutron is characterized by a<br />
reduction in the energy of the neutron as a result of the scatterXng<br />
process,<br />
In light elements liko hydrogen or helium or carbon this<br />
reduction in energy comes from the no<br />
a1 classical collision of<br />
two particles of comparable maw and. such collisions should really<br />
The<br />
be called elastic, However for the heavier elements lika lead OT<br />
gdd, the neutron enters the nucleus and emerges with reduced<br />
enorgy leaving tho residual nucleus in an axcited state,<br />
excited n~clous then falls down into the ground state with the<br />
emissdan of a @pray, Thus a nuclenr inelastic scattering is always<br />
nied by one or more 'prays,<br />
The<br />
The probability for inaxastic<br />
scattering I~CPQ~S~S<br />
with increasing atomic weight sinco the<br />
nu~loi, havlng many constit nt particles, have a large<br />
variety of closely spaced energy 1<br />
els into which the nucleus can<br />
be raised by the addition af the energy of the incident neutron,<br />
Qualitative fnformation about Inelastic scattering can be<br />
obtained by measuring the activity induced by the inelastically<br />
*<br />
scattered neutrons in various detectors whlch respond to neutrons in<br />
different energy ranges. For example, siLver or rhodium can be used
as radioactive detectors of slow neutrons and aluminum ox silicon<br />
ns detoctars of the f3st neutrons.<br />
By measuring the activities in<br />
thcse Botectors with and without the scatterer one can get some idea<br />
of how the SneLastically scattered neutkons have to be dogrnded in<br />
cnergy. Experiments have also been performed in which theq-ray<br />
intonsdty which accompanies any inelastic scattor$ng is rnm6urad.<br />
The magnitude OS this intensity gives an indication of the cross<br />
section for the inelastic scattering. However the experiment is<br />
complicated by the possible presenco of 3. -rays in the source, Not<br />
much Information about the energy of the inelastically scattered<br />
neutrons can be obtained from the energy of the? -rays since the<br />
excited nucleus will in general emft a variety of ?-rays of<br />
different energy corresponding to the many states into which the<br />
nucleus can fall before reaching the ground state.<br />
Complete infor-<br />
mation would be given by n measurement 00 the spectrm of the<br />
inelastically scattered neutrons but this is F?<br />
very difficult thing<br />
to do.<br />
@ & m C T S RADIATION PROTECTION<br />
Electromagnetic radiotion like x-rays or 3. -rays damage<br />
tissue by praducing fast electrons (through the photoelectric<br />
effect or the Coripton effect) which cause lonization and disruptfon<br />
of the molecular structure of' the tissue, These electrons have R<br />
relatively small ionization per unit; path length and ix correspondingly<br />
long range. Neutrons will in many cases give rtse to<br />
energetic heavy chargod particxes like protons or bclrparticles which<br />
will pave R vory high specific ionimtion but R short range. Fast<br />
protons cuuld be produced by collision of a fast neutron wSLh a
162<br />
r,<br />
hydrogen atom in one of the molecules. Slow neutrons may bo<br />
m@urod nnd give rise to q&tys or in some cases, like those of<br />
Agatnst 3/ -rays, lead Or cancre te walls of suff iciene<br />
thickness give good protection. Lead is of no value against<br />
neutrons, however I Materials aontalning hydrogen, like water,<br />
paraffin, or c~ncrete, are ganerally used bec&use of their effoativeness<br />
in slowing the noutrons down to 8 point where they can be
I L A 24 625) 153<br />
, \<br />
'.. .:, *<br />
December 21, 1943<br />
LEC'IWE SERIZS ON NUCLEAR PHYSIC$<br />
. . , . . . ,<br />
Fourth Series : Tm-Bod.y Problem Lecturer: C. L. Critchfield<br />
..). c -. LXC'MJRE XXV: . NUCLRAR CObJSThflTS . '<br />
,. ,<br />
11.1(,~<br />
, ' %,<br />
,-><br />
I:+<br />
. . . ,<br />
There aye two kind; qf nuclear quanti t3e.s:<br />
'5<br />
Quantize& and<br />
., not qvantized. The quantized' numbers are: mass number,- charge,<br />
',* L '<br />
',, spin, statistics and parity,<br />
The qtla'ntiths that are not quan-<br />
.Fized are the nuclear radius, .the nuclear mass, the magnetic mom-<br />
\<br />
eht and the electrical quadruple moment.<br />
.~<br />
It is believed. that eventually there will exl st a thevy<br />
of ithe quantized nuclear numbers correl.ating these to each ' ather.<br />
SUCH, n theory shall link the varjous numbers t.2 each other and<br />
:, ' posslbly to other 'physical qiiantltl.es. IJp ti nw there has been<br />
: I ).<br />
. ,.d<br />
.,<br />
(,'.*'W.'<br />
. .<br />
I $,'.<br />
, G I<br />
'\i<br />
' \,, . .<br />
( '<br />
, ,<br />
' i<br />
I<br />
mly one sipnificant attempt in thl.s direction,<br />
'.<br />
PaiJ-li tried to<br />
cwrehte spin and sttitfstics by statlng that particles wTth a<br />
half ,integral spin have Fermi-Qirac statist1 cs while particles<br />
vj th an i-Qteger spin have Pose-Einstein statistics.<br />
\, The nani.ng of Fermi-Dirac statist! cs mag he explained<br />
with $he help tj? the example ~f the H2 molecule.<br />
'that the slsins<br />
tho Value 1/25 are Rarallel (trlplet state).<br />
StatLstics, which ap&.es<br />
Let us assu.me<br />
"?,the two protons of^ this mqlecu.lo (which have<br />
Then Fermi-Dirac<br />
to the protons, postulates that an:<br />
'intprchange of the posl$ions of th,e, protons will came the wave<br />
. .<br />
\, \\\ " .<br />
fundtions tq change sign, 1 , -
154<br />
The cmsequence is that a rqtational state mi.th no n0d.e (J = 0)<br />
d?es not occi?r simultaneously.<br />
A31 even Values ~f the rotational<br />
qvantim number J are exclud.ed. Odd J values are alloved and<br />
the lovest state correspmds to 3 .= 1. This state of hydrQgen<br />
is called orthohydrogen.<br />
Let u.s now cmsider the D2 molecule.<br />
Let 'tis again assme<br />
that the spin 0.p the devtrons (which have the value 1) are para-<br />
llel. Then tlie Bose-Einstein statistics whJ,ch applies t? the<br />
deuterons 'pmtijlates that the even'rotatiqnal states OF the rnole-<br />
&le, are alone pTesent while the odd. rqtational states do &?&<br />
xcvr.<br />
d eu. t er on.<br />
This stake of the deuteron molecule is. cn1.led ortho-<br />
There also exists a secsnd stnte fw- both the hyd.rqpen<br />
and. the. deu.term in which tho spins o?' the nuclei are nut parallel<br />
and in nhtch those rdational states are allowed which did nqt<br />
Occur in the ortho states and t.hose rotational states are missing<br />
whjch were present in the ortho-molecil.1-es.<br />
called pnrahydrQFen and. paradeuteriurn.<br />
These states are<br />
The experience ab711.t the<br />
statistics qf nuclei is a strmg.argument j.n favw of the nuclei<br />
heing cmstitu.ted frm neutrons and. protqns rather than from<br />
prqtons and electrons.<br />
Assuming Fermi-Dirac statistics f w all<br />
these elementary par'tj.cles (prqtons, electrqns and neu.trms) ,it<br />
follows that a nucleus cmtajning an qdd number of slvh particles<br />
behaves accwdinp to the Fermi-Di.ra,c statistics whilo a nuclew<br />
cqntaining an even number qf particl4s has Rose-Tinstein statis-<br />
tics.<br />
NTW the deuteron, i-? it vere cmstitu.ted from pr3tons and<br />
electrons, would have to contain two protms and one electron,<br />
that is, an ogd number of particles and. should, therefwe, have
155<br />
Fermi-Dirac statistics, wliereas In reality it, has Bosa-Einstein<br />
es. Xf on the 7 t h ~ hand, prrltqns and neutrons are the<br />
ents of the sru.c$ei, a 4euterqn Is made up of ane pratm<br />
and onemntron, that, of an even number qf particles which<br />
edicts Rose-Finstein statistles,<br />
ong the non-uixmtizec! qii<br />
e radius can hot be determined v<br />
e values have been qbtai<br />
ItJ es characteristie of<br />
accurately.<br />
t an early date in the<br />
nbcl.ear physics by scattering 04' alpha particles.<br />
The pPTncLlple of' this determination I s that when the alpha parr<br />
ticle ap3roaches to within the s vm o<br />
he radii of the scattepinfr<br />
nr: the alpha particle then tho Rutharfwd scatterinp Inw<br />
ceases Ita hold, From thls anomalous catt,er.ing csf alpha particlos<br />
4 2<br />
adfi in tho order of 10 CM have kecm qbtained.<br />
Phe electric quadruple moment of a raucleus for a sharply<br />
deftned qmntity has not Fe~n<br />
determfned up tq now wtth v&y<br />
high<br />
*<br />
Tt has been first dis<br />
d rneaswOd by 1 vest i Pa t Sng<br />
errine strvctvre qf' atgntc spectrum. This hyperfine slrucdire<br />
to the intaractton of' the magnekic moment of the ebecw<br />
e ta the spin<br />
ita1 moment sf' the<br />
e nucleus Differant<br />
cl.ear and eLectTica1 angular morn<br />
wont energies and cavse Q nRrrryc~<br />
s which is as a rille less than one<br />
on descr-1.bed abme gives a<br />
fine strvcturee Small deviations<br />
hcavy nuclei, These<br />
additional term in the ener
expl-anation which depends qn the? sqvare of tho cqsine of the<br />
angle Inclvded between dlroctims of the nuclear ancl el-ectric<br />
956<br />
angvlnr moments, PhysicntLy thl.5 Lima1 term is due to<br />
an olectrlc qiiadruolp moment of tho nucleus whl ch interacts with<br />
the inhqmqgeneity q? thr. electric field prqd,tlced hv t h elcctr-+Is.<br />
~<br />
Thcro is also me rothpr direct detwmj,natjm of the quadruple<br />
moment qf the deilteron which IW<br />
The masses 9"<br />
mns8 spectr9gravhic wwk.<br />
shall Clscuss later.<br />
the nuclei arc knwm qllttp accllr~t~1.y frw~<br />
and megnotic flc3,ds inrhlch focus ions T? a riven !.veJght into a<br />
given pqsi ti qn,<br />
this methqd have been carried qvt, he Ralnbrldgc; fw lipht nticloi<br />
and by Dmqpster for hcrvy nucLoi,<br />
This method uses crqsscd el(3CtTic<br />
Rcccnt acvyate determjnatj 3ns of masses by<br />
BainbridEe was able to obtain<br />
high intensities by f'xx~slnq a beam of groat angular deviations,<br />
In hi)s sot-up the positi9n af the fwvs dopends on the maSs qf<br />
the nucleus vcry accurately as a linear function,<br />
In this v~ny<br />
it was possible to obtain mass determinations Isf high accuracy<br />
by comparing the 'ocus Q? ions OF nzarly the same wei-ght,<br />
Thus In his car1,y 141vk he cwparod the mPss of the hellurn 1qn<br />
1R4.t;h the MBSS qf the triraatqm1.c DHFI.<br />
The actual value deduced<br />
frqm this measurement for the mass of thc dovterm titrncd out to<br />
to oxygen masses g?vcn by Astgn.<br />
carried out vith abixdant deutarium s~urces and evaluntcd. 1NitdZ<br />
the correct helium to qxygen ratio gave very 8ccvratc deutaron<br />
masses.<br />
gics.<br />
More rccent work of Rainbridgo<br />
NL~CIOQT rnaSses aro ~llso used lin balancing reactlon oncr-<br />
A systematic sti4y of" thcsc rmctiqn cnergles lcad Bcthc
n<br />
H<br />
1,60893<br />
1 * 09812 ' He3
158<br />
It is svTp3:ising that by morely d.ovbling thc numbor gf particles<br />
within tho nvC&%s tho bindlng rnergy should be Increased by a<br />
fnotor of 13.<br />
Thc magnotic moment is also knmn vith very hirh accuracy.<br />
The beet detorminations in!<br />
by Rahi nnd his cqllahwfltwq.<br />
e carried out at Cplinmbin University<br />
They ueed an ingenious method.<br />
Tlic meam.vemsnts wwe carriad out on rnolccvlcs like H2 or D2 in<br />
vhich the olqctrona nre paired and the effect ~f tha electron spins<br />
cancel each other.<br />
through 3 magnetic fields,<br />
gmous and d,ePLccts thc: molocular be~m,<br />
genotis.<br />
motion.<br />
h beam of the moloct~le passes c-msequcntly<br />
Thcs first 9.F these fio'tds is inhomo-<br />
Arovnd this field the nuclear spins perform a prwessing<br />
The t!i%rd fScld is Inhoni.1 nous again and brings the<br />
molecular beam back t~ c",<br />
The second field is horn&-<br />
fxus in Tvhich thc molecv1.@s moct inde-<br />
penden$ of tho spin qrientztim Q-' the nuclei, In addftim ts<br />
this arrmgewnt an osci?I.ntj,nr;t mngnetic, fi ~ l is d si1pcrp~secl on<br />
the hamogcnws mngnotic flr57-8. Th$a wcill ating fYo1.d iy.parts<br />
energy to the nucl.ci crlusinp thc nucZear soin tr, jump from ono<br />
stcte into c.nd,he,r.<br />
If this happons thc socmd. inhomogcnavs field<br />
will no lonpar compensate thc crleflnctfm cclv.sct3 by the Fimt in-<br />
11s flcld and any chanpc in winntption cmased by the<br />
oscil-lating f2GI.d Tlrj.11, glvo rlsc to EI diminishwl numbor of mole-<br />
cules arriving in thct "(3c1.t~~ A rhanyc: In tho nuclear spin ~rfcn-<br />
tRtfon w111 occw howevw mly i.p the frcqvency of the mcillating<br />
magnetic field is in resonance with the precession of thc nucl.oi?.r<br />
spin.<br />
This precession depends on the s'tl?cngth of the hr>mogonaus<br />
fiold, on the magnetic moment $.nd on tha value of the nuclear<br />
spin. The last quantity may bo detclrmincd from<br />
for instancc by tho analysis of tho spcctrttm.<br />
other oxpcTiment S,
159<br />
fn this way th.en, tho mngnetic rnomont mag be dc&rminod to a high<br />
accuracy,<br />
Actually *’he procisiqn of tho oxpor2ments is liniitod<br />
only by the 2cngth of tho rcpim of tho hqmogcnous electric fidd<br />
in thEt even the tins of flight of the moleculo in this region is<br />
short.<br />
T ~ P accuracy of sesonmce betwan an, oscillating magnetic<br />
fiold and precession frcqucntly is limited.<br />
Vory accwi?.to values<br />
haye been obtained- by the rs?.ztiva megnitvdes of rnagnct5.c momcnts<br />
proccssitm frcqvcncies of tho H and D nuc1ou.s.<br />
is used.,<br />
A rm1”e complex plctvre l.6 Ilstained I$ the> H2 molecule<br />
Here thE C rotcltjnnal stnte can not bc vsed br.cnuse in<br />
this stc?.tc thc nuclear spins hwe opposite directions and the<br />
magnetic moments cmoonsntce<br />
state (J u 1) is t o be used.<br />
Tkorcfwo thc first rqtatlonal<br />
‘In this state also thc spin of the<br />
, ntzcloi add up to mc and th2To ~ ~ rcsult ~ ~ altqgothcr 1 1 9 mtnt1ona.l<br />
and spin stntcs,<br />
Thc em-rglc3s of them states me nf”ectcd by<br />
tho magnetic Interoeticm hctwwn thc prolma and ’nv tho h13tir)n of<br />
the ~l~ctrons induced by rnoLwvlnr rotatlm.<br />
Retweon the 9 statos<br />
mcntioncd nhovc, two sots of‘ 6 transitions are possihlt? and cor-<br />
respondingly ono dme~ves tvto sets of 6 dips of intensity,<br />
order to oxpl.ain thr. oxnct frcqiii’ncy at iKihJ.ch these dips occur,<br />
assumptjons have to bc! made abmt the intcractiqn hetirvem thc mag-<br />
netic rnommts of the nuc2.ci c7nA t;hc rnngnotic fic??.d pr?dt”.ccd by<br />
In
tho rotntkng molecvla.<br />
16 0<br />
Giving apprqprinte Values to thaso cqndi-<br />
tions the position of. .the 6 dips can be explained.,<br />
Hnvfng deterrnincd the molocvLwr qilantitios that have an<br />
effect on tho nuclear mnkncts thc! position Q?<br />
rnay ba predicted..<br />
in ivhjeh J 3 1 and also thc: spin fs equal to<br />
the dips In D2<br />
The ppodictions<br />
hawever did not cbock wit19 the experimental rasults me agroclroi.nt<br />
could. he obtqincd only bv assuninp an adcljtional term in thc enor-<br />
gy which depands m the sqilare of‘ thc cminc? Inclilc?cd by the nux-<br />
lcar rnqment and the mQlecular crlxis. This fs the kind c>f term that<br />
should be if thr? detttcrm h2.d a quadruple morwnt.<br />
Rabi, Rnrnscy<br />
and. othcrs have indeed beon led to as~rno an electric qvadruple<br />
mmnent for the deutrriurn;<br />
Par this mcslccir3,e /)ne may lnvostipato thc. St2.t~<br />
Tho discovcry or the a1indrupl.e rnowmt<br />
Q?? thr deuterium<br />
csmo as a svrprise bocnusc cal..culntiwls on the htncllng cnorgy<br />
find other properties .of the deu.term were based r>n R<br />
symmetric model of tho deuteron whjclz did not porm!t<br />
of any ailndrvple mnmont.<br />
s.r’.mple<br />
the -xistance<br />
The mt:sciwe sf such t? moment shows<br />
that evon. in the simplest two-hqdv problem r7f nuclcar physics<br />
quadrup1.e f’orces appear,<br />
moments of thc simplest niiclei mmsured in units of nuclear<br />
magnltrons<br />
Thc: follqwinp tahLr? gives thc magnetic<br />
H! 2.785<br />
1<br />
n -1. (73<br />
D2<br />
@<br />
855<br />
Tho negative side attczchcd to the magnetic mmcnt of thc neutron<br />
Is naccssary in qrdor that the pTotqn ann’ the neutron rnagncts<br />
lincd up in a parc?l-lcl way shovld give Dt least npnroximat*:ly<br />
the magnetic momcnt of the deuteron.
L A 24 (26)<br />
162<br />
Fourth Series:! "vc, Body Pmhlem Lecturer: L. C, CritchfiePd<br />
*<br />
The earliest atterpts to bitild a mwe detailed thesrv 9f<br />
orces were cmcerned dith explaining the binding ener-<br />
pies of the lirpht nuclei, particularly of the deuteron.<br />
It was<br />
nded that the bindinp energy of the helium nv.cl.eus was thirteen<br />
ti.ws that ~f the deuterm elthwph tlie nvv?"nr of attractive links<br />
Pmir particles is only six, a t mosts av?pared wjth one<br />
between twg particles,<br />
"ripner minted o3.t that the extraordinar-<br />
ily large bqn.ding in the alpha particle ccxzpared with that of tho<br />
indicated that the kinetic energy of the partlc1.e~ in the<br />
c?evtelt.lon almost compensptes the mutual attractim an@ that thls can<br />
be theGcase if the attractive f'wces have 3. very shwt ranFe. ioerq<br />
the ranpe of Forces is srrlal.ler than the wcive length of the gar-<br />
ticlea.<br />
In the alpha particle, on the Qther hand, the number of<br />
attrinctive bmds is prqportimately preatsr than the increase in<br />
kinetic energy, and a 1mer average energy Is possihle.<br />
EhQrt ranpe fwces between nuclear part4 cles makos possible<br />
many simplifiqatims in the tlieory of the deuteron.<br />
Essentially,<br />
we may say t!vt if ncvtrm and wotm are farther apart than a<br />
distance they d . not ~ inf'ltience each other ap:-reciably, but at<br />
separations less than g strxtFr; attractive ~orces act and there is<br />
.
the expct s%pe of the reeion cannot he important tq the des-<br />
criptisn of the stater Accmd.fnpXg, it my. he nssl?msd that the<br />
patenticll energy is a c?nstc?nt Vas inside the rad.ivs<br />
outside it.<br />
Let r be tho rolative cmrdimte of tlic urqtcm. ?v!th<br />
3rd zero<br />
resmct to the n.evtron, 14 the mass of" &?tm m novtron, znd E<br />
TrTc accordingly let
The solution Is then,<br />
164<br />
their<br />
smoothly<br />
I<br />
I<br />
a8 Vo<br />
Thus the<br />
the<br />
by protons, It i.s tlwn assvmcd thzt i s t1-m
*<br />
been considered,<br />
165<br />
So far, only tho space coomjinates af the particles have<br />
It is well established that; ths ground ntat@ of<br />
the deukepon i s a triplet, i&@J tho spins of neutron and proton<br />
a m parallel,<br />
let state.<br />
Thus the above rough calaW,ation applies to the trip-<br />
Furthermore, the wave fu.nction asawned i s spherical-ly<br />
symmetrical SQ that no quadrupole moment is obtained,<br />
In spite of<br />
these defectsl, however, tha scattering induced by the forces thus<br />
derived will be calculated rand cornparad with experiment.<br />
oomputa<br />
Scattering by short range force$ bS particularly easy to<br />
especially if' the wavewlcngth of th3 colliding particxes 9s<br />
1ong'e.r than tho range of forces,<br />
Under these conditions a flnite<br />
orbital mgular momentum will prohibit the particles from coming<br />
close enough together to be akttracted and there is no scattering<br />
if the particles pass each other at a distance larger than a wave-<br />
length.<br />
Consider a plana wave for the neutron incident upon a pro-<br />
tan. This wave represents a aefinito relative velocity v of<br />
ftcollision" but for all possible values of Z;ho impact parameter and<br />
hsnce all angular rnorqent.<br />
According to the foregoing argument de-<br />
flection will bo expcrdcnced only by those collisions of ZDPO<br />
angu-<br />
lar mmcmtum, ire, in ths spherically symmetric state. h t the in-<br />
cident beam be ropresented by 9.. ikr cos0 &<br />
Tho spherical part<br />
oS'qin absencc of a potential well. can be determined by averaging<br />
over all 0,<br />
yoo = sin kr<br />
-E?--<br />
In tho presence of a field the wave will be strongly refracted fn-<br />
side P = a and, so far as the wavo at 2aqp distavlcos i s con-<br />
Lc<br />
carried, tho effect will be to introduce R phase shift e The<br />
e<br />
general form of tho spherical part wlth potential woll is thon
166<br />
ab<br />
with 0 and 8 to be doteminedl, The wave yv" must represent tho<br />
spherical part of the inoidsnt beam plus a scattered wave, and the<br />
Zatter must have the form Seikr/kr. kence<br />
The number of scattored neutrons per unit volma i s then<br />
2<br />
sin 6<br />
--zET--<br />
k r<br />
and the total number of particles mattered by the proton per second<br />
k<br />
The incidtont cii.n~:=lt eensity postulated iu v klence the effective<br />
cross-seck3.cn :xaaoni;ed for scattorlng 9s<br />
k<br />
Dekormination of .5 can be made approximately a8 follows:<br />
the banbarding energy of the neutron8 is aaawnod to be small campared<br />
with depth of the potential wall, hence the wave function<br />
inside the range of' forces, 3, is substantially the same a8 far<br />
the itable deuteron, To join functions smoothly? therefore<br />
+-
e<br />
d' tho bltidihg energy bf t;ha dezrt6,rop<br />
Thus the binding dnergg of the deuteeon determhed the scattterihg<br />
crosarsection, at least if the callision is suffidiantly slow4 conid<br />
'<br />
parison with experiment, howevsr, showed that tho predictsd CPOSBsection<br />
is a little hi'gh at high enepgy (E>2Mv) and about a factor<br />
10 too small at low ehe~?gyi To explain the discrepancy Wignsr<br />
pointed out this cross-section 19 calbulated kroni what is known<br />
absbt tkie b9ipkek btate of the deutehn Only and that it is posd<br />
il<br />
sible to ohooao a I'biilldlhg'' dnergy, for die sihglat skate ih<br />
such a way as to account for the experimontai $esult$r Since Lhi-oe<br />
out of four collisions between neutron and proton ape triplet colj<br />
lisions and on8 is singlet the result for the complete cross-sac.-<br />
tion is<br />
E is tho energy of collision in ths center of mass system and hence<br />
is onowhalf the bombarding energy if the prokons may bo considered<br />
at mst. Thc value of ET needed to fit experiments is of thhe order<br />
of 0.10 MeV but the sign of it is not determined becauso sin<br />
2<br />
6<br />
is determined from the aquaro of tho logarithmic derivative at a,<br />
By taking into account the effect of the finite kinetic<br />
encrgy inside the distance a the expression for fbecomes, in<br />
next approximation
168<br />
Al%hough the spherically symmetrical solution for the<br />
1)<br />
deuteron must be incomplete because of the existence of the quadru-<br />
pale moment it represqnts the grosg propertias quite well,<br />
Bof'oro<br />
considering refinements of tho theory, therefore, it ia worthwhile<br />
to gtudy, the implicd dependence of tho force between neutron and<br />
proton upon their relative spin orientation, With parallal spins<br />
thsrc is a binding energy, 2.16 Mov and with opposite spins<br />
(singlet state) the binding enorgy is very close t o zero. Thus the<br />
relation between V, and a2 obtained in an approximate way abovo applies<br />
vsry closely for the singlet state, For tho triplet the cxact<br />
relation determining V 5 V1 is, in the units used above,<br />
Thus for 2 = 1 the potential wolf. i s almost twice as doop for the<br />
triplet state as for the singlot. This indicates that the spins<br />
of the nuclear particles play a dominating role in the attraction<br />
between neutron and proton,<br />
LA 24 (27)<br />
December 28, 1943<br />
LECTURE SERXES ON NUCLEAR PHYSICS<br />
Fourth Series: Two Body Problem Leoturer: C,L. Critchfield<br />
LECTUm XXVII:<br />
I) SPIN-SPIN FORCES IN THE DEUTER<br />
ON<br />
PRO,TCN-PROTON SCATTERING<br />
rx)<br />
@*<br />
T.<br />
The theory that has been developed for the deutei-on is<br />
generally satisfactory except for describing the electrical quadrupolo<br />
moment. In order to aocount for the quadrupole moment Sbhwinger<br />
used a spln-dependent interaction similar to that between mag-
netic dipoles,<br />
The wave equation then becomes<br />
169<br />
whore<br />
+ $1<br />
SN is the neutron spin operator (Pauli matrix), 6p tho pro-<br />
ton spin operator andvdetomines the amount of the spin-spin<br />
potential,<br />
;I<br />
Sinco the additional forco introduces components of ? '<br />
In a rion*spherical combination it $s in general impossible t o aasme<br />
spherical symmetry f or the solixtion,<br />
For the singlet staee of' tho<br />
dstiterond however, the dipole-dipole interaction cannot affect the<br />
motion of th.e papticles and its average value is zero BO the same<br />
equation as before ie obtained and the same relation betweon Vo and<br />
a2 holds,<br />
For the triplet state the intoraction does not vanish<br />
but rather tends to change the spin directions of the particles and<br />
hence a380 thoir orbital motion,<br />
The intoraction is invariant to<br />
reflection in thc center of symmetry so the orbital stato into which<br />
the dcuteran might go is limitod to even parity and the only pos83.7<br />
bility l a a D stater Symmetry of the operator In 6j, and bp shows<br />
that tha spins of the neutron and protan remain parallel and tho<br />
solution will bo a mixture of 3S, and 3D, states.<br />
Tha mount of<br />
3D, that i s mixod with the 3S, to obtain the lowost snorgy is detsr-<br />
minod by the zliee of "p which is, in turn, choson to give the Correct<br />
value of the quadrupole rnmont, which is 2.73 x 10 -27 2 om<br />
Thc observed quadrupole mmont is positive and that means<br />
that the charge distribution is "cigar shapedVrs Mathematically the<br />
quadrupole moment is defined as tho average of (1/4#8 2 - r 2 ) over<br />
the charge distribution ob.t.ained from the c~olutlon of tho wave
~<br />
e,<br />
equatlsn.<br />
170<br />
Qualitatively it is readfly demonstrated that when two<br />
parallel dipoles interact the energy will be lowest; when they lie<br />
on sn axis and that suoh a ,coqffguratlon 5s brought about by in*<br />
em6 between D waves and S w8vea. The pwt of the D<br />
wave that varies a8 3 cos2 62<br />
*I 1 will interfere with the S wave<br />
becaqse the spins are tho setme for the two waves,<br />
Along tha axis,<br />
ther8fOm, amplitudes add whereas the parts of the D wave that<br />
are large near 0 z (H/2)(mL>O)<br />
netic quantum numbers and intensities add,<br />
one phase of<br />
do not have identical spin mag-<br />
The result is, with<br />
D wave, a prolate charge distribution about the<br />
spin :axis and with the other phase an oblate charge distribution,<br />
The sign of" is thus chosen to glve bQe prolate distribution and<br />
-13<br />
the valus of y appropriate to a range of force 2,80 x 10 om<br />
(a 1) is Ot775, The corresponding depth of potential well is<br />
v, 8 27 mo2 which is close to that for the depth of the gislglet;<br />
2<br />
W e l l Vor 25 m@<br />
In faot Schwingar points out that if the range<br />
of forces had been taken 2.7 x lo -13 cm the two potentials would<br />
have been the same and the entire dependence upon spin of the<br />
neutron and praton is imparted by the dipole-dipole Interaction,<br />
It may be noted hers that if' this dipole interaction had been due<br />
to the magnetic moments it would be of the opposite sign and its<br />
ed value much smaller than that found above.<br />
3<br />
The amount of D-wave in tho lovest state ia found to be only<br />
4 percent, The charactoristic influence of the dipole forcos on<br />
scattaring, capture and photoelectric disintegration are corroa-<br />
pandih@y mall, mounting in general to 2 percent corrections,<br />
Formulas obtainod in the simple<br />
S wave theoyy are thsrafore
3-71<br />
0<br />
adeqyate for most calcu5ations and fail only 18 the true intcrpretation<br />
of the forces and in predicting the quadrupole moment, The<br />
ble effeot; of :'this evidently vory anal1 adrnlxturo of or'bfion<br />
i>s that gsssntl8lly the same depth of potential well<br />
obtain8 for bgth triplet and singlot state. In other words, the<br />
difference in depth found necessary in the spherical states is<br />
just made up by tho strength of the spin coupling but the '13 state<br />
to which tho 3S is coupled is very pd t;o excite, .<br />
A reasonable thaoq and method of calculation is then available<br />
for the deuteron provided that the energies are not too high.<br />
At high energy of collisian the potential well can no longer be<br />
oonnidered sna1.l compared with a wave length and the shape of the<br />
potential CUTW will matter, This energy is roughly equal to the<br />
depths of the potential wells, say 10 MeV, in the canter of mass<br />
Actually, the noutron-proton cross sectim has been obt<br />
24 Mov (12 Mev in the c tep of mass frame) by Sharp<br />
results are fairly well accounted for by tho simpla theory<br />
g P- waves, Abovo t the effects ?f the shape of<br />
and' of tho response t ions with highor angular<br />
momontum should bo in convinci<br />
\<br />
.II)<br />
TI<br />
The range of forces bbtwepn heutron and proton h8s been taken<br />
to be 2.8 x cm (a l)'* The ape two methods of establishing<br />
this value For the range, The first. to be oonB€dered is the scattering<br />
of protons by protons and detection of deviations from the<br />
Rutherford law.<br />
tho range of foroes fn the deuteron
from khese results assumes that the rangas are the same,<br />
second method<br />
and Ha4<br />
As in<br />
involves<br />
the<br />
the cam Q f<br />
onergies available are<br />
making estlmates<br />
O f the deuteron,<br />
The<br />
172<br />
of the masses of H3, He 3<br />
neutron-proton forces the experimental<br />
capable of f inding onSy the nuclear fQPCeS<br />
exerted between two protons when they collide Zn an S-state,<br />
interaction may therefore again be represented by a "square well"<br />
but calculation of the influence of such a ffw~llt' on the scattorad<br />
distribution I s BQmQWhat more complicated for two charged particles<br />
There are three reasons for the complication : The first is that<br />
the 908. t te ring affect Of the "we 11 i s superposed on that of the<br />
electric fields so that intorference terms appear in addition to<br />
thoss due to nuclear forces alone;<br />
The<br />
the socomd roason is that the<br />
Coulomb farces are long range and modify the inciden2; wave even<br />
at infinite distanco so that the simple cxponentials are not solu-<br />
tions at any distance from the point of collision,<br />
In addi ti on,<br />
the collision of two protons if influenced by the statistics of<br />
the particles and a lsecond klnd of interference is introduced,<br />
The Fermi-Dirac statistics admit collision in a IS stat6 and ox-<br />
3<br />
clude the SI so we are here concerned only with the nuclear forces<br />
between protons Of opposite spin,<br />
is -<br />
The wave equation of the reduced system in the Coulomb field<br />
vIr,e;y)=u . .<br />
M<br />
where v is the relative velocity at infinit@ separation. Let<br />
3<br />
metrical, part of the solutionr Thon the equation for u(r) becomos<br />
dr
have the same asymptotic exponential form RS vi except for a<br />
IE<br />
ohan$Q in sign,<br />
This is the case if<br />
Tho sign and the phase,B are not &yon by tho considorations<br />
buts is a function of velooity only, The canplebe theory shows<br />
The spherical portion of the wavo in a Coulomb field ha3 tho<br />
asymptotic $om<br />
The effect of<br />
tb.e forces between protons will be to intPOdUGe a<br />
constant phaae shift Ko 2n such a way as to leave the incoming part<br />
of u unchanged,<br />
“he tom OP the wave that has been influenced<br />
by tkie nuclear forces is therefore<br />
Uf t ,u3iiK,<br />
The asymptotic form of the perturbed wave then becomes<br />
e<br />
If the particles were not identical tho crogs section would<br />
be gtven by the absolute square *of the curly brackets,<br />
In classi-
caL theory, &he cross section for identical particles is the sum<br />
of the c,ross, seckians a('@) + de+ 0).<br />
is an additianal ponsideration to be mader<br />
175<br />
But in quantum theozy there<br />
If one praton is at ra<br />
and the other at rb the wave function describing the state is<br />
~~('a) *JrG(rblG yl(rb) yz(Pa).<br />
in the triplet,<br />
Therefore (since rate) = r,@+@))<br />
tho saattering<br />
at R-8 which is thesame as at%+B interferes constructively with<br />
that at 8 in the singlet colldsLons, i,e, 1/4 of the'time, and<br />
interferes aastructively in triplet collisions (314). If we call<br />
the quantity in curly brackots f',(0) the cross section for scattering<br />
in unit aslid ang3.e at 8 then becom s<br />
This effect w88 first discussed by Mott,<br />
(g).<br />
+ ' ~ '<br />
The first term is the pure CouZomb scattering# the second the pure<br />
0os(&a,0s2(1/2)0+<br />
cos2(i/z)e<br />
potential scattering and the last the interference tam. TO get<br />
the woss*8ection for scattering between @ and 84-68 multiply by<br />
2flsiln 8dB; and to obtain the cross saction in the laboratory<br />
Ko)<br />
, -<br />
j<br />
1
c<br />
system replace every 0 by 20,<br />
taryf and a$ this angle<br />
176<br />
The largcst effect of the potehtial<br />
At a million volts colllsion energy? is about 1/6 and it becomes<br />
smaller as the energy increasos. The effect of a $ma91 phase<br />
shXft due to nuclear forcos is therefore greatly amplified at<br />
these energies,' Further, due to interference, the sign of KO is<br />
determined .'I<br />
Experiments on proton-proton scattering have been carried out<br />
with good precision by Tuve, Hafstad, Hoydenburg and Herb, Kerst,<br />
Parkinsdn and Plain..<br />
The latter have carried the collisiovl energy<br />
up to 2.4 Mev where the ratio OF cross soctions is 43 and KO is<br />
480* At 1 MeV KO 2 33O,, Both experimental groups accelerated<br />
protons by electrostatic,generators a d scattered them in hydrogen<br />
gas rrlaking counts at several angles to the beam,<br />
Comparison of the result8 obtained wieh those that are expected<br />
frm a square we31 can now be made,: For this purpose,,<br />
however, it i s necessary., in general,, to know both the bounded and<br />
* -<br />
unbounded solutions t o the wave equation in the Coulomb field,..<br />
These solutions are called<br />
F and G respectively,<br />
The phase<br />
shift, KO, has been defined to relate to the asymptotio foms of<br />
Fo and W0 for Sdw'aves which are taken t~ be */2 out of phase at<br />
.-
*<br />
inside<br />
177<br />
large r;. At the boundary of the.well the combination Fo COS<br />
sin KO must fit onto the wave function, Fir that applies<br />
the well<br />
and this determines KO&<br />
Fy = bit G& tan K,<br />
'";L T+ cfo tan K,<br />
If we g.ssume that the Coulomb field has<br />
not affected Fb and a,<br />
greatly we gat tho usual ralation<br />
9.k.Q Mevk<br />
Breit and others have shown that about 0.8 Msv sh6txld be<br />
added as an average effect of the Coulomb repulsion insicto thc WGLL<br />
The more exact calculation gives 11L3 Mev as the dspth of the well<br />
and also shows that good agreement w ith results at all energies is<br />
obtained,<br />
The rough method used would be applicable only at<br />
energies very high compared with the Coulomb rapulsion at<br />
Po = 62/rnC 2 4<br />
Using the exact wave functions a. search for the best value of<br />
Po has been made by Breit et el and the decision roached that<br />
2 2<br />
0 /mc is the most acceptable (a = l), A value of B 0 cy5<br />
gives a noticeably Inferior fit as ala0 does a = 1425, The depth<br />
of welS pertaining to a = 1 is 11,3 Mov in good aereenaent with the<br />
most camful estimates of the depth of tha singlst deuteron well<br />
of the same width, 11,6 MeV;
178<br />
e<br />
December 301 1943<br />
LA 24 (28)<br />
LECTUFG3 SERIES ON NUCLEAR PHYSICS<br />
Fourth Series! Two Body Problem Lecturart C. Lc Critchffeld<br />
pwrum xxvIIr: THREE BODY PROBLEMS<br />
A second method of determining the range of huclear forces is<br />
obtaihed in estimating the binding energy of the nuclei containing<br />
three partic&esBH3 and He3,<br />
The binding energies of these nuclei<br />
are approximately equal, the difference being caused<br />
tho electro<br />
statlc repulsion of the two protons in He 3 +*<br />
Since an attractive<br />
force bet'ween two protons hag been demonstrated in scattoring experiments,<br />
it follows from the equality of bindlng in H3 and Ha3<br />
'chat the attractive force exists between neutrons,, Attractive<br />
forces between all particles in these nuchi and in He4 are<br />
also found necessary in getting agreement between observation and<br />
the estimates,<br />
The method of<br />
c alculn ti ng<br />
binding energies that we shall ap-<br />
PlY<br />
is to<br />
problem"<br />
reduce the 'chreerbody problem to an "equivalent<br />
certain as sump tions and avo raging over tho<br />
two body<br />
third<br />
particlo.<br />
The method was first used in this connection by Feon-'<br />
berg, Consider the nucleus H3 which has a binding energy 16.3 mc2.<br />
Assume that the wave function that describes any two particles in<br />
H3 is the same as for the deuteron* In order to exprcss the douteron<br />
wave function as a single algebraic function tho form of the<br />
two particle function will be assumed to be<br />
whore (2/d)3/2 is essentially the radius of the wave of the deu-
179<br />
tsron,<br />
&rCher asmane that the wave fixnation for a19 three par-<br />
tiales is a symmetric product of throe such deuteron functions in<br />
the three separations r 328 q3 and pZ2;:<br />
PO8<br />
ten<br />
one<br />
hihe<br />
I<br />
%e potential eneygy af' eaah particle ~$32 depend upon the<br />
ition of each of tho other two) but an effective two body PO-<br />
Carrying out tho integral OVBP dT3 (the volma elomont of particle<br />
particles is the sarne as in the deuteron except that the range of'<br />
fomas is a larger.<br />
The total potential is three times that<br />
between any two particlea,<br />
In Q similar calculation the kinetic<br />
sner$y may be averaged and the result is that the fotal kinetic<br />
is three thes that in the d'euterqn,<br />
We shall apply the results obtained above to determining the<br />
equivalent solpkion with a ttsquaro" potential well. Let be<br />
the binding energy of I?; the wave squat$.& far the two body
problem "equivalent" to H3 is then<br />
180<br />
a r 1<br />
-<br />
M<br />
Ar --<br />
-I<br />
mixture of singlet and triplet potentials, The two neutrons is<br />
€I3 are certainly in 8 singlet state and if' the spin of the proton<br />
ia parallel to ons of the neutron spina it forms the triplet with<br />
that neutron and half triplet and half singlet wlth the other.<br />
The expected relatkon between vT and a can be determined from the<br />
deuteron calculations and is a simple avorage of singlet and trip<br />
let relations<br />
a- a<br />
This average is based on the assumption that th8 affect Qf the I3<br />
3<br />
wave in i~l the same bctwoen poirs of particlea in tie triplet<br />
state as in the dmateron.<br />
The other relation between vT and<br />
from the oquivalent two body equation for H 3<br />
a may be obtained<br />
That equation may<br />
bo mad@ formally the sme QS for the triplet deuteron by substi-<br />
tuting<br />
*<br />
wheree is the binding energy of the triplet deuteron. Then the<br />
D<br />
relation between X~V a is the Sam8 as botweon vo and<br />
T
a for the triplet deuteron,,% i . e . q<br />
The approprlato VdUe Of x2 is 0.78 and<br />
This, combinad with the relation obtained from tho deutoron (H2)<br />
has the solution 1<br />
a = le08<br />
for equal d@ptbs of well4 vT, The valus for the range of forcos<br />
thus obtained is hiphcr than *hat; indicated by proton-proton sent-<br />
taring but only by 8 porcsnt,<br />
and the result obtained i s satisfactory,<br />
The method is very rough, of COUPSO,<br />
If a neutron-neutron<br />
force had nob been postulated the agromont would be unacceptable,<br />
There is a dif'feroncso in binding betwaen He3 and H3<br />
about '76 Mov that should bo duo to the CouLomb repulsion of the<br />
ppotons.<br />
DE : i.<br />
We may estimate tho expected rspulsion as follows:<br />
,<br />
oO-(3/2) ( drBf<br />
( Q2/r 1 ( r2dr )<br />
.& , , , ~,, : , , ,<br />
$m-(3/2,(t/re) Q<br />
dr<br />
."(%)<br />
, ,<br />
2 represento the reoiprocal square o f the cxtent of tho wave<br />
1/2<br />
of<br />
o!<br />
f'unctfon and mag be taken approximately aquaa t o MG/fi2<br />
= 0,43 in<br />
units of m 2C,/4..<br />
4<br />
e<br />
This givas Or.65 Mev for the calculated Coulomb<br />
energy in satisfactory apoomont with the oxperimsntal value con-<br />
.sldoring; the approximations mado.,<br />
Similar throe and four body onloulationa have been made
y tho variational method with substantia y the sane result for<br />
102<br />
oarroot accounting Qf the bi 3ng ensrgios or H3, Ha3 and,<br />
There is another ty<br />
tal interest to the thoory of'<br />
i<br />
a-body problem O f fundman-<br />
on, This is the scatter-<br />
neutrons by hydrogen rnoleoufes* The theory bas been worked<br />
out 'by Schwingor and TB Ejr and haa ahown that the scattering cross<br />
of tho molcculs 2s oxtranslg sensitive to whother the sing-<br />
to of tho deuteron i8 real or ''vtrtual".<br />
Experiments on<br />
scRttw4ng of liquid-air notltrons havc been intorproted by tho<br />
thoory to prove concZusivaly that the singlet state is virtual,<br />
ieO*, thePo is bawd SlnglOt Stat@'<br />
Tho groat decSsiv<br />
cause af' an accidental near cnncollation of<br />
tion,<br />
tho exprimant come8 about be-<br />
tams in tho aa1cul.a-<br />
Details of tho cnlculntions will not bo given here but tho<br />
principal feature of them Is adily deacribed. The cnlculnted<br />
aross section for noutron-proton scattering duo to the triplot<br />
-24 2<br />
wo11 aXona is 3*50 x ZQ at low enovlgy whoreas the observed<br />
-24 2<br />
La 53 x 10 ern From Wignor's hy.po$hosis the singlot<br />
soattaring must have a CPO~SS saotion 41,5 x lo2* om2,<br />
If<br />
ia<br />
* orda<br />
tho triplet and q tho singlet CF<br />
0; 11*8 [r;<br />
Scattoring ?Prom the hydrogon molecule will show inter..<br />
etwoon spharical waves coming from tho two nuclei if tho<br />
th of tho neutron Is long compared with tho internuclsar<br />
separatian, At room temporaturs tho wave length i s of the sana<br />
do 8s the scgnratlan. Calcvlation of ths CPOS8
and<br />
mounts to 41 if tho singlet stab i s stablo, ire', bo and<br />
havo tho smo sign, and only to 0,38 if the singlet is unr<br />
ro is thus d otor of ov@r 200 in cross section de-<br />
pending upon tho atabilitg<br />
of tho factor is due to tho<br />
the singlet stnte and thc large size<br />
act that sin6 2s so nearly equal to<br />
0<br />
-3 times sinal* catterinp; fj=* 3 = +J.<br />
5 . 2<br />
The right; side is 53 for tho stablo singlet nnd 20 for an unstGabSe<br />
3 irqb t<br />
Tho oxporimontal proce ro i s to cornpara<br />
Of para-hydrogen with that of orthohydrogen (or rather the USURZ.<br />
1 for llquid air<br />
to the furtho?? oomplicat<br />
ion is subject<br />
tpons may convert<br />
one kind of hydro@n infm the othbr, fn this cam the wavo<br />
function of the final molecule has ono si n if tlzo spin of' one prochanged<br />
and the opposito sign if the spin of the other i s<br />
The interforone 8 then dsstructivo and the CTQSB 880-<br />
tion propoptional to<br />
O"conv rrd<br />
(SW 6, - sin0,) 2
185<br />
09 Qourso the oonvorsion is accompanied by n. change in xwtational<br />
state because of the atatfstics,<br />
%onv<br />
Sclzwirrger an4 Teller have shown<br />
is of the 883110 order of magnitude as<br />
0"<br />
when tho parahydrogen is mostly in the ground state, J<br />
whm tho nautrons have su<br />
state i s *023<br />
00<br />
so that only<br />
0, and<br />
low energy w i l l its CrodS see-<br />
so oxtrornely small for tho cas8 of an unstable singlot<br />
n, %e energy required tb rniso 8 para molecule to ortho<br />
e unablo to oonvort<br />
octron volts, hence liquid air neutrons (,012 ev)<br />
pa to orbho<br />
perimsnt is Chat the acattering cros~ seotion OF ortho-H 2<br />
much larger than that of para hydrogen. This provos conaluaivdly<br />
thak tho singlet deutaron is not stabla,<br />
Exautly the same kind of considerations have boon mado by<br />
Schwingor Tor the possibility that tha noutron has spin f3/2jf\ in-<br />
stead of (L/2;)* *<br />
onergy would then bo a quintet,<br />
%e Icv of the deuteron that lies near zero<br />
In this case however, the fortul-<br />
tous oancekxation of toms goas not OCCUP cnd all ross s~ctfons<br />
in molecular hydrogen am a$' ths same order of' magnitude,<br />
the experiments prove not only that the singlet; state of thc deu-<br />
teron is unstable but ~lso that ths ncuhron spin $8 (1/2)fi a<br />
24 (29 & 30)<br />
Thus<br />
& XXX: INTERAC F NEUTRON AND PROTON WITH<br />
Neutrons and protom interact with the fields that hnve<br />
tablOshQd by classical physies nameiy, tho gravi tati ow~l
1<br />
sB Oravitattonal phenomena am<br />
ar procemes.<br />
zhe<br />
hand, am very 3ngaFtan<br />
captum, emhaion of gnmma<br />
G dissociation are<br />
Trs. addition to thsse we<br />
s. One of thsao fa<br />
its rnagnotlc moment and th&<br />
ts magnetic moment,<br />
s there are<br />
fields that influance tho nut-<br />
e~vatione on th.<br />
tho field of moson8,<br />
o of this thoory ia to m-<br />
e nature of<br />
beta activity is cirawnacrfbcxl to sornc oxtant, Some fundamental ,<br />
sa of the neutran is a PO-<br />
tho deutoran.<br />
Also, RccuraCe
I 180<br />
tea particlea, l/fs is the WBVB function of the ground state<br />
of the deuteron and e, FQP s;tmpLlicity we shaS1 assws<br />
q8 in tho f ~ m<br />
Y?
189<br />
is<br />
1 = c(E2$, H2/8m~) e , E Z / ~ * ~ ~<br />
and the CYOSSI section for photo electrtc disintegration (only one<br />
polarization is effective)<br />
If we had normali2ad the neutron-proton Pmwave function8 to unit<br />
@ndrgY the factor pp would have been unity,<br />
Calculation of M is straightforward. Consider thO die<br />
E<br />
pols moment In the direction, B:: Q,<br />
ME sJy8(@I'/z C O s @ ) ~ d ~<br />
D<br />
neutron and grolj'on and T, oxpressed as a function of E andE<br />
1<br />
becomes<br />
% = /8T/3)(@2/Cio)( fi*/Ite )<br />
(@* 1 -t E/€" )3<br />
For the 3. -rays of Th C ', used by Chadwick and GoSdhaber, the cal-<br />
culated CTOSS soction is 6.7 x lon2' cm2<br />
To the CrOB8 section rE must be added the O~QGS soction<br />
due to elfact of the magnetic v@CtQP,CM, The perturbing enerey<br />
is then p. €i, and the matrix olomont, Mmj contains not only tan
190<br />
s bat a sun over t spina, n o<br />
ts to Et 1s state or poai.t;ive<br />
vel of the doukeron in tho poaj,<br />
is very Large<br />
at low onorgy<br />
relative weakness of t<br />
ron magnetic mamonts to bo
I)<br />
neutron by proton,<br />
198<br />
* (€3, / $B) ( h2 BA/ 2<br />
of photo disintegration ilq simple radiative capture of<br />
Syatam A is composod of one photon and tho<br />
stable deuteron; %he photon is capable a$ two polarizwtLons and tho<br />
n of three, hence eA 9 6,<br />
ono proton cacb of<br />
sect<br />
apturo 18 than<br />
8gstsm B aontains on@ neutron and<br />
Tho mom<br />
3/2 ( 2*d/kc ' (a;.,+ % )<br />
ectxio and magnotic effects have dilfsront dependenciss on<br />
tho onsrgy of the nsutron-proton gygtern.<br />
At htgh energy $he mag-<br />
y low energy,. howover, tho<br />
gnotic offact ppedominatad and<br />
rc: 1,4 x 10 -28 6 E fE?<br />
utron at. 1-00112 tompernture, I/~U c,v,, E 2 X[~O Q~v,.<br />
*24 2<br />
% = .,I8 x 10 Gm<br />
lrYhcn the noutron was diacravared thc way wa8 opened for<br />
ament of a field thoory of beta radiation,<br />
Foml In analogy to a&ocCromagnotie themy,<br />
mi# was<br />
!bo fundamsn-<br />
ptions am:" .<br />
nuc lo on ) .,<br />
ron a,nd proton 8ro two<br />
(2). Energyr spin and stati6tic.s am oansorvod in bsta<br />
111) pRd€Gtlon by tho introdpotio f the Pauli mutrino,<br />
($1 T ~ Q rest maeg OS the nautrino is zero (or nsarly
its spin is +<br />
t with the'<br />
o are radiated when neutron<br />
1<br />
and a neutrino are radi<br />
n.gw to proton an6 8 positron<br />
hangoa to nazltron.<br />
(6) Elslotran and neutrkno sharo tka availab onorgy in<br />
a11<br />
sible pr6partions.<br />
Xn analogy t o the inte<br />
n botween chargad particle8<br />
and light quanta, tha<br />
(26)<br />
numbor of<br />
w = g2<br />
matrix element c<br />
e<br />
The interaction eon8<br />
dypam the amp<br />
at thg YlU(J2OU8, p ep%ikkoa<br />
*iC<br />
and/0,, are tha
194<br />
and, as in tho case of<br />
dou tar on<br />
the photoolcctric disintegration of thc<br />
Tho shapo of the electron spoctrum is thus determinod,,<br />
Exporimen-<br />
tally. tho nwnbor per unit enorgy (or unit momontum) is ma8surods<br />
divided by pE or its equivnlcnt in the Coulomb field and tho square<br />
root of the resulting nmbor plottad against E,<br />
In tho carefully<br />
done work with nuclei that hnvo only ono beta activity tho plot is<br />
a good straight line, tha Intercept of which W, Thc thoorg Sa<br />
thoraforo substantinted by tho exporimcnts,<br />
If the nuclonr charge is lcrgcr than, say 20, or if tho<br />
electron encrgy is particularly low the sine wavc 'Is not a. good<br />
2<br />
approximation for tho clcctron W~VC,. 11 mom oxact value of c<br />
is plus the charge nwnbqr of' tho residual nucleus If a positron<br />
9s emittad and minus the chargo number for cloctrun emission,<br />
Tho probability of omission of an olsctron with energy<br />
botwcon E End E -k dE is
4<br />
195<br />
The total probebi<br />
trino per saoond is gftren by the integral OF P($)dE OV8r @l@ctPoll<br />
'energies<br />
EI ma2 to E = W, . For ~>>mt@ th$s integral 5s<br />
czosaly equal to<br />
+<br />
h z 60Q c fi<br />
w5<br />
Tho decay constant, A, $a proporblonal to the fifth power Of<br />
total, energy relaased by the tranaftion,<br />
Moasuremgnts<br />
the half livas of the light radioaotive aloments makes<br />
a detomination of the ''Fermi conatan<br />
mare'is a class of positron mittera "chat<br />
larlv well suited far deternining gr<br />
JZght radioaotive yzucolol. c<br />
i s p<br />
This clasa oomprisoa the<br />
tha<br />
taining one mope proton than neutron,<br />
ra proton turns into a neutron with the emission of a posI*<br />
tron and, since the nuclear forces am avidontap the sms between<br />
all pairs of paPticlos, -bho wave Punotion. of the neutron in the<br />
final nuoleus should be very neBr3y tho same as the wave funo.t;ion<br />
of tho Initial proton,<br />
In thia<br />
nu~lei, their half lives and cncrglbs roleasad am give<br />
on tho following pago,<br />
nttng two*pOssiblo final statoar<br />
e I.I" 3'<br />
A table of thsse<br />
In some Instances two oncrgiQs are givsn<br />
g the Wansltlon mloasing tho most energy,<br />
The life time Is dotor-<br />
tion goes to tho excited etate ths nucleurj subsequent1<br />
IF tho transir
I<br />
196<br />
MAXIMUM KINETXC<br />
sR$xa<br />
1# I6<br />
93 m , 092, 2.20<br />
52'6.0 8 ;LJ<br />
? b 1<br />
2; 20<br />
%<br />
11.6 I 2m82<br />
1.7, 3,o<br />
3.54<br />
3,. 69<br />
J a b 2 8 fl 3 ,,87<br />
2,4 I 4.33<br />
3188 s 4.43.<br />
e07 s 4.94<br />
We have assmod thkt klucbron and neutrino are emitted<br />
lizLi angular momsntum,<br />
the omitted park5<br />
The rnaxbwn amount OS spin that<br />
8s is, t,horoforo,% . Zn order<br />
the nucleus bs ohmged 1<br />
WOOM rloutro<br />
lsld contain tha<br />
tog,<br />
Rmnl'a oI?t?igi<br />
trnn@tion from o m spin t o anot<br />
locttxon or boCh<br />
to bo am$tted with orbital.<br />
il)<br />
e<br />
93% amp5iLudtm of wavos of highar wngulor momcn-%wn<br />
knallar at'tha ntzozoar radiua than the amplitude of tho
e<br />
s<br />
I<br />
and such transitions are tsmed "forbidden".<br />
197<br />
The half-ligo<br />
of a forbiddon transition is therefore much longer than that of an<br />
allow'ad<br />
appears<br />
transition of tho same cnorgy release,<br />
Certain nuclei produce allowed transitions alkhbugh it<br />
certain that the nuclear apiM changes by$).<br />
This is parti-<br />
cula~3y true of He6 which must have spin 0 but decays at an "al-<br />
lowed'' rate to LIB with spin%,, Xn Fermi's theory such a transim<br />
tion would be forbidden,<br />
To acco<br />
of span k l# Gmow and Teller postulated a spin dependant interac-<br />
tion with tho electron-neutrino fiQld,<br />
t fop transitions with cha.nge<br />
The colculatlons made thus far apply only to allowed<br />
transitions in light nuclei.<br />
To oxtend thorn to heavy nuolQi and<br />
bidden transitions more exaot accouet must be takon of the<br />
wave functions for relativistic motion in a Coulomb fiold and there<br />
are aevcral interesting fcaburas of the solution that are Drnpha-<br />
sized in betadecay theopy.<br />
FJELQ<br />
THEORIES OF NUCLEAR FqRCES<br />
The SUCCBSS of Femfrsr theory of beta decay promptsd an<br />
a't;taqst to explain the forces between nuclear pccrrticles by the<br />
same interaction.<br />
Thus the olectron-neutrino field should play<br />
the same role in t h ~ attraction between nuchar particles as tho<br />
eloctromagnotic field plays in the force8 batween ohargsd particles<br />
At the same time ths anomalous charaoter of the rnagnebtc moments<br />
of neutron and proton might also bo explained by the interaction<br />
with this field (~Viok).<br />
The attempt was not successful, howevc<br />
For two very good reasons, either OF which fs dufficicnt,<br />
the forces calqulated are much too weak t o account for the ob-<br />
First,
sorved strength an6 range of<br />
198<br />
nwlear f'oroes; seaondly, ths theory<br />
gives attraction, In first approximation, only bekwcen neutron and<br />
proton,<br />
The forces betwoen like particles nppaar Zn second appro..<br />
ximsltion and are repulalve,<br />
Two linos of endmtvor have been pwausd in tryZng to con*<br />
struct a field theory of nuclear forces.<br />
One generaliaod the beea<br />
bntoractionr<br />
,<br />
Thia was first don@ by Garnow and Teller, and by<br />
Wentzol, who postulated an intoraction with pure ePoOtran fields,<br />
80 that neutron and praton would be able to croate an 6leotron-<br />
positron pair as well a8 an electron-neutrino pair,<br />
This same<br />
type of "pair emission" forces waa lator extonded to meson pairs<br />
(Mllarshak) on ?;ho asaumption that tho meson has spin<br />
On acw<br />
count of the flexibility in SntroducZng new fields th.ose theories<br />
are able to account for ths nuclear forces,<br />
jsctionablo features of the results, howevor,<br />
There aro aavoral ab-<br />
In the electron*<br />
positron patr tbeory, for exwpl8, special forms of tho states into<br />
which tho cloctrons are arnittod have to be assmod in order to<br />
avoid scattering of slow neutrona by the atomic electrons* In the<br />
meson-pair theory the cnlculatod CPO~S section for scnttaring of<br />
cosmic ray mesons by nuclei is higkor than that observcd excopt<br />
For very weak interaction Eorcos (and a rather long range for the<br />
forms), Them is R general objectton to the mson pair theory<br />
that obsarvatiohs ahow about 30% mare positive mesons at sea SevoZ<br />
than negative onc8,<br />
If they were croatod in pairs tho numbor<br />
might be expected to bo more nearly oqual,<br />
Pending further oxparl-<br />
mental ovidenco of<br />
the spin of tho meson, howeivw, thc possibibity<br />
of a nmson-qpair f
e<br />
Tho othor class of attqnpts to describe nuclear forces<br />
by n field tlzeory is typifiod by more direct ganeralizations of<br />
electramagnotic theory,<br />
sented by Yukawa, in 1935, before: the meson was dlhcavered.<br />
Yukawa found that the range and strength of nuclear forces could.<br />
199<br />
Th@ First theory of this class was pre-<br />
be understood if tho nuclear particles omitted "hsavy quanta" hw-<br />
ing a finito rest ma98 of aboub 200 times the electron rest mass.<br />
This quantum was conoeived as having Bossdstatistics, the sgm~ a8<br />
a light quantum, but for simpl;lcStg the spin was assumed to bo<br />
zerod<br />
hrthermoro tho heavy quantum could have one unit of elac-<br />
tric charas posittve or nogativs and. Yukawa suggested that the<br />
heavy quantum could di sintearate intg elaatron and neutrino glvivlg<br />
beta radiation.<br />
The discovery of tho realmeson, of about tho samo mawt<br />
by Neddormeyor and Anderson stimulated interest in Yukawafs theory,<br />
Then it was proveh that chargod quanta of zwo spin gave repul-<br />
sion between neutron and proton in the 33 deuteron,<br />
To correct<br />
this obvious defect the spin of the quantum was as$umed tic) be<br />
unity,<br />
The free apace wave equation9 of tho meson of spin one are<br />
therefores very sim91ar to those of a light quantum, ike.<br />
MaxwoLZ's<br />
equations, Thus thore will be the vector field quantities and<br />
I H and potentials for tlilose fi@ldsyand &,<br />
The inclusion of a<br />
finite rest mass) however, introduces a characteristic length whioh<br />
we shall denote by l/&where<br />
This Lengthl may be. usad to generalize Maxwell's squat$.ons for froe
Thrw ctlffkwent assumptions about the tma electric<br />
202<br />
charge on the sqson8 are generally<br />
sidered,<br />
These are (3.) Utn-<br />
charged, (2) either positive or neg6ttive (+e),<br />
(3) both charged<br />
and unehapged.<br />
The theories developed on thess assumptions are<br />
referred to as neukra'l, chargad and symmetrical theories raspec-<br />
tiuelg.<br />
Tho requirements on the. 8's and f*s for neutron aad pro;.<br />
ton a$@ somewhat different $'or the different theorlea#<br />
Tt is nscesaary to account for oquality of neutronmneuc<br />
tmn, proton-proton, and neutron roton forGes In the stnglot state;<br />
In the neutral theoryI whiah does not; distinguish in any other way<br />
between neutron and proton W(3 have Ql 2 g2 and fl k f2 for any<br />
nrxcleon.<br />
Consoquently the potential function for a singlat:<br />
q' T2"2 ,3, and the third term v&nighes, Sa<br />
The, potential funotion Po& the triplet state is<br />
vl(P) = (e2 3- 2/3 f2I (@*"/P)<br />
4- dipole-dipale tern,<br />
The ertatlc, g 2<br />
term is repulsive for both states and must be ovw-<br />
come by the =2f2 in the singlet and by the dipOlQAdlpbl@ tsrm in<br />
the triplet,<br />
In the triplet the repulsion is increased by 2/3 f2<br />
Bethe has shown that g m<br />
le6 and triplet binding energle<br />
taken to vanish ana the &in$*<br />
orreot--y ab-<br />
tained by c<br />
ipoke term has been discunssed unrjer the theary of the<br />
deuteron,<br />
In that discussion an attractive atatic potential was
203<br />
e<br />
mcnt with ouscrvations on the deuteron with f = 3,246.<br />
In tho charged meson theory there are. no neutral mesons<br />
postulated and the meson must have either &e4 A neutron can emit<br />
only a negative meson and turn into a proton, and @ proton can<br />
emit only a positive meson turning into a neutronr This has an<br />
important effect on the nrathre of the forces. The neutron and the<br />
proton are Considered to be two states of the same particle, The<br />
property of being one or the'othsr is called the isotopic spin in<br />
analogy to the ordinary spin. A nucleon may have m$= -9 in which<br />
case it is a proton or m ~ & = for a neutron,<br />
analogous to ms * -8 and fig f 8<br />
quantum numbers of the nucleon or eloctron,<br />
This is directly<br />
For the two possible rnapnetic<br />
The significance of<br />
the formulation of isotopic spin is evidenced in constructing<br />
rnanyFperticls wave functlons in aocordance with the Pauli exclub<br />
slon principle,<br />
By thfs principle, and the assumption that neu-<br />
tron and proton reprosent two states of the same particle, the<br />
wave function of the deuteron, for example, must be antisymmetric,<br />
Thus the '3<br />
stat6 of deuteron, which is obviously symmetric in<br />
spin and coordinate space, must be antisymmetric in isotopic spin<br />
spaoe; this means that if' the two particles exchangc charge, as in<br />
the emission and absorption of charged mesons, the complete wave<br />
funotion of' tho 3S changes sign on each exchanp.<br />
hand the complete wave function of the 'S<br />
on the other<br />
will not change sign,<br />
sincc it is already antisymmetric duc,to the spin state and the<br />
isotopic spin function must be symmetric,<br />
The nature of charge exchange is thereforc<br />
the two particle potential function by giving a plus<br />
manifested in<br />
sign to states
204<br />
*<br />
with’even 3 L and a minus slgn to states with 0d.d S +Le The<br />
static Forces are thus attractgve far some states and repulsive lor<br />
others, instead of being alwetys repulsive as in the neutral theory,,<br />
Unfortunately the char$ed kheory gives no forces between like parbecause<br />
of the impossibility of exohange In that case and<br />
cann9.t bs seriously considered 8s a theory of nuclear forces, It<br />
represents the simplest case of QxOhanegs fOPCeB, I’rowever, and we<br />
shall need the, description of such forccsa in the sgrmmetricsal<br />
theoqh<br />
The symnmtrical theory postulates that both charged and<br />
ged mesons are represented by ths field.. Slnce the chargad<br />
part contributes nothing to the forces between like particles the<br />
ohoics Of g, = gp and fY1 * f for the neutral mesons is rulsd out<br />
P<br />
$or, then Che forces between unlike particles and batweon like<br />
Ica in the ’S<br />
would bo equal bQcRusc 0‘2 the neutral rnesoYla<br />
Theladditionnl charged field between unlike particles<br />
wowlD then make them unaqual,<br />
neutron-neutron and nsutron-proton states are<br />
Tho potentials in the singlet<br />
v0 (n,n) = (gn2 - 2fn 2)(e “EZr/p)<br />
= (gngp 2fnfp) + 2(gn<br />
1 2 . cc 2f lf 1) (e-KT/r)<br />
n P<br />
rl, 1 c fn 1 etc,, apply to c d meaoylsl The factor 2 in<br />
ed Bxgression takes acc<br />
The unprirned g and F re<br />
the two kinds of cJhapged<br />
o neutral rn@sms1 as before,<br />
We can equate Vo (n,nS, V, (p,p) and V,(n,p) by taking<br />
= -gp n 3 gpl = @, 2 * gp2<br />
f, = mf rnl rgl fn2 M f 2<br />
0 P P
.I,<br />
205<br />
which has the symmetrical solution: all lg/ are equal and ~ l l<br />
if1 are equal. Only tho phases thesf3 mlmbers diJY@r*<br />
Tho ralative merits of tho neutral and symmetrical<br />
I<br />
theories have been determined by Bathe $0 whom thfs semi-classical,<br />
description of nuclear fields is also due. Bothe Finds that tho<br />
neutral theory with Q = o and f 0 3r24a can account for Che two<br />
body problems satisfactorlky.<br />
Thars is one artifioe neaessary,<br />
however, and this arises becauee of the 8ivergenoe of the dipoledipole<br />
forces at r s 0, Ra the wave f'unction of a particle is<br />
concentrated 1x1 a region of radius r tho rninhum kinetic energy 1s<br />
of the order K2/2mr2.<br />
The averagB dSpole-dipole potontlal, on ths<br />
other hand, is proportional to l/P3*<br />
the radius of the wave function the lower the energy and cohse-<br />
quently there is no finfte state of lowest enwgy,<br />
it is customary tolassme thBt the l/r3 law or attraction cease3<br />
at a certain "cut off" radiuq,<br />
This shows that the smallar<br />
To remedy this<br />
In the neutral theory Bethe finds<br />
it nwessary to cut off at 0,32 the range of force (,32/K) if it<br />
is assumed that the potential remains covlstant at smaller r and<br />
equal to the value at cut off,<br />
The theor$ "can give8 the carreot<br />
upole mmont and also the correct S-states and Slscattering*<br />
The same method applied to the symmotrioa<br />
to a cut off at (1.3) (l&)<br />
heory laada<br />
and to a quadrupole moment of the,<br />
signa Thus tho num"e5cal results, at least for the<br />
D states, are hlghly unsatiafacto<br />
and<br />
Qualitatively tho symmetri-<br />
ea1 theory is to be preferred because the charged meson6 appear<br />
-*<br />
in cosmic rays; they might reaeonably be supposed to disintagpats<br />
into electron and neutrino; and they mako possible, in princip3eI
206<br />
an understanding of Lhs anornolous magnatiQ moments of neutron and<br />
@<br />
proton,<br />
In case the mesons are supposed to emit beta particles<br />
the Fern% constant is a product of the intoraction constants per-<br />
taining to nucleon-meson interaction and meson-beta interactions,<br />
The decay period of a free meson, which has been accurately mea-<br />
sured by Rossl, ’ should depend only on the meson-beta interaction<br />
constant.<br />
There RP~, therefore, thr.ae determinations of the two<br />
constants and they are in violent disagreement,<br />
all tihis the calculated burat producing cross section of ohargsd<br />
mesons in the Rtmosphere is muoh JaPger than the observed oross<br />
see t 1 on,<br />
In addition to<br />
From 8 quantitrtive point of view, the neutral meson<br />
theory is the most promZalng analog of the eloctromagnetio theoq,<br />
Perhaps part; of the reason for this l a that it need not explain<br />
the prop0re%e~ of<br />
the obwrved, chargod mesons4<br />
There i’a some<br />
indication Pram the scattering of High energy neutrons by probons,<br />
Ler, tho cas@ in which the P-wave bocomes important, that the.<br />
nsutral theory predicts 8. too large cross sootion,<br />
The symmetri-<br />
cal theory accounts for tho exparbents on fast neutrons satis-<br />
fczctorily.<br />
s<br />
Zn conclusion it O m on3.y bc said that it is hoped that<br />
further work on scattering CTOSB aectlons at higher energlea,<br />
angular distributions in scattering and photoeloctrlc procsasos,<br />
the naturo, particularly the spin, of the mesons# @to.,<br />
will guide<br />
us in tomulnting n fiold theory of nuclear forces that will ex-<br />
plaSn everything.
LECTURE SE2zLE;S ON NUCEEAR PHY<br />
Lecturer:<br />
V, F,: Weisskopf<br />
The ca>cuI.ation and the theoretic prediction of the properties of<br />
is made difficult chiefly by two reasons: our ignorance as to the<br />
e of $the nuclear<br />
inability to solve the complicated,<br />
bbms which we face in the quantum mechanical treatment of almost<br />
us except the deuteron,<br />
Some attempts have been made, however, to<br />
abt$in Qualitati,ve results from the theory in order to interpret the vast ex-<br />
perhankel material on nucleap reactions which been collected so Tar, In this<br />
,<br />
of nuclali,<br />
account is given of the statistical metbods of describing the behavior<br />
These methods use aa few as possible actual assumptions concerning<br />
ar forces or tha nuclear structure.<br />
marized in the following pointsr<br />
Their main assumptions may be sum-<br />
A) Thb nucleus can be considered as a itcondensed phase" of the neutron-<br />
proton system in the tharaodynainical aense.<br />
"he neutrons and protons form a<br />
*<br />
ich they are dsnsely<br />
ed, so that tihe nuclear matter has definite<br />
t the volume lis proportional to the number of constituent8,<br />
By a@-<br />
aripal form, one can, erefore, introduce a nuclear' radius<br />
-til<br />
where A Is the number of constibuents and ro s -I:*# ko em.<br />
The results from this formula are in fair agroement with values Pram~experbnts<br />
in which the si~s of the nucleus is involved,<br />
Furthermore - and this is the<br />
of assumption A - the characteristic psopsrties of nuclear matter,
208<br />
especially the close proximity of all constituents, are maintained even if the<br />
*<br />
nucleus is excited to energies high enough SQ that St rdght emit one ar more<br />
constituents, This fs valid beclause the bability of emission of the donstituents<br />
is very small, 90 that the nucleus is in a wt3,J.l defhHed state before<br />
having emitted the particle,<br />
It is in a stehe which has essentialky a e s m<br />
properties as .the lower excited states which do not emit partioles,<br />
This is in definite contrast to the situation with excited atmns,, An<br />
e4eotron escapes very rapidky if excited above Sonisa.t,ion snsrgy and no states. of<br />
tb atoms in which the electron is still oloss to the atom could possible be de-<br />
fi#ed in this regiov of excitation,<br />
'Ilia nuclear conditions however, are similar<br />
to that of a liquid drop or a solid body, dnce khe heat energy of such systems<br />
b<br />
may weal be much higher than the work necassary to evaporate one singbe molecub.,<br />
Tho state befare emission of this molccul@ hiis a life tba lQng enough to bo well<br />
dofined,<br />
Emission af a constituent by means of nuclear excitation may then be<br />
divided into two steps:<br />
first the oxcitation to the excited state, wd then the<br />
emission of the particle from the excited state.<br />
The analogous proces# in atoms,<br />
.<br />
however, must be describcd in one single step, since the time interval until the<br />
electron leavus the atom is not long enough to define a regular quantum statc<br />
in which the electron is sti4.k within the atom.<br />
The assumption A is no longer fulfilled for very high excitation<br />
anergies of the nucleus,<br />
as in the condensed phwe<br />
The range of anergy in which the nucleus can be con-<br />
ria from nucleus to nuclaus, but it is a&f@<br />
to assume that the assumption is true f ~ a r rang@ of excitation up to a large<br />
fraction of its total binding energy?<br />
E) According to Ikshn, a nuclear reaotion can be divided into two well<br />
od states.<br />
The first is the formtion by the incident particle of a<br />
compound nucleus in a well defined state; the second is the disintegration of,<br />
the compound system into tha product nucleus and an emitted particle,<br />
The second<br />
I) stagu can be treated as independent of the first atago of the processe 'She basis
209<br />
of this assumption is the classical picture of a nucleus as a system of pasthies<br />
.<br />
with very strong interaction and short range forces,<br />
If the incident partido<br />
comes within the range of the forces, its unsrgy is quickly shared among all<br />
constituents we1L before any r'cemission can ocmr., The state of the compound<br />
nucleus is now no longer dependent on the way it was formed and could hava been<br />
produced by any other particle with corresponding mer& incident on a suitable<br />
nucleus,<br />
Its decay into an emitted particlo and a residual nucleus is thus<br />
independmt of what happened before 4<br />
Bohr's assumption is certainly right for a oase in which them is only<br />
one single quantum stato of: tha compound nucleus which can be formed by tho in-<br />
cident particle conaidering its energy and othdr circumstances @<br />
Then, evidcntly,<br />
the properties of this state are well defined and independent of the way it has<br />
been excited,<br />
It is very probably not a good assumption if tha state of the<br />
cirmpound nucleus which is formed by the incident particles consists of a super-<br />
position of several stationary status,<br />
'fhe course of the process then depends<br />
tgtrongly on thc relativo phases of tho states, and is thmofora not independent<br />
af the initial procoss of excitation.<br />
In the case, howeverl that the density<br />
bf states of the compound nucleus is very high, so that their widths overlap<br />
each otha strongly, a groat many states can bo excited simultaneously by the<br />
incident particle,<br />
This phase relation my be at rdfidom znd the resulting<br />
pPocesscs, themfore, independent of the way of excitation.<br />
Bohrls assumption regains approximate validity for this case,<br />
It is possible that<br />
'his is made more<br />
than probable by the classical picture of strong binding forces used above.<br />
Actually, classical considerations are justified in the region of high level<br />
density wit'n overlapping lcvols where no quantum selectivit, v occur.s4<br />
C) The statistical method assums the existence of averagl: values of<br />
certain mvgnitudes averaged over stctes within R not too wide t3xcitation energy<br />
interval.<br />
energy,<br />
The ?"vexages are supposed to be slowly varying functions of that<br />
The ~gnitudes concerned here are level distances, transition probabili-
tain statas, etc~ The validit<br />
ful applications.*, Xf the energ<br />
210<br />
is assumption m n onZy be proved<br />
rvals of averaging must be taken<br />
LECTURE SERIMS QN<br />
V, %, WGisskopf<br />
Ad<br />
Tbr: garieral chhractar af the energy states of a nucleus om be described<br />
lowing' menner~ Thcre isj 50 far, no definite regularity found estabtho<br />
spacing of tha onorgy Levels, The distance between Lha lowost<br />
of the order of ma.<br />
1 Mdv* Solsttimes they lie Qloser and in<br />
groups. Information ?'bout 1,<br />
ecently measworn@<br />
sible to detarmin B directly, It is interesting<br />
t there is no i<br />
est levels on<br />
dependence of the average dis-<br />
alight tendency to smaller disi<br />
tances for highctr weight,<br />
*<br />
It is 'safe to assw~o that tht, level dclnsity increases with higher<br />
energy as is oxpdCtod of elmost eny mechanical system,<br />
tha stronger the mow particlas the nuolcus consists of.<br />
o sxpox'imental muterial mailable<br />
l'hb inorease<br />
Thdre is,<br />
the slow neutron capture oxporim@nta provide some very soant
evidence on the Xsvsl de&<br />
The states of th nuckeus, W i t<br />
a finite lifetim bacaGsc. of the possibi<br />
231<br />
citation as is explained below,<br />
ception of th ground stat@, have<br />
f radiabivs transitions to lower<br />
states and of cjoction of particles,<br />
(The emission of@rays<br />
aan be naglmtud<br />
hero since its probaki&ity is extremely smalli)<br />
l'he states thus hnva a finite<br />
which is connected with the lifetbe %hof .tho nth skate according t o<br />
the rclakion<br />
the total width<br />
I$! the finite li5etSmo is due to different emissions,<br />
he sm or partial widths# which themselves are propor-<br />
tional to the specific ern$ision<br />
-.. A"<br />
' 2s the radiation width and<br />
litLos* We therefore can write :<br />
he partial width corresponding to<br />
of a particle ci2 whish may be o, proton, a noutron, or anaek-;partiole,<br />
tho emission probabil<br />
per second of a partiole & by the<br />
Thle width of the law lykng levels is purely a radiation width as long<br />
tfon energy is lower thnn the Lowost binding enorgy of a particle,,<br />
The bindbg anorgy Ba of a p&rtieAe 5 i s the energy ncoessar-y to remove it from<br />
the ground state and have the residua9 ru~?leus in its ground stab,,<br />
above I<br />
t&e gram4 state.<br />
The binding energy B,<br />
We there-<br />
ciLation energy Qf Lhs nth love1<br />
of e. proton or a nuutron is found<br />
to be about 8 Mav for nucJ,ei which are not taa near %he lower or upper end of<br />
ic tab&, @hs vor olernents about up t o oxyg do not show .my<br />
I From then OM)- 6 rcge hut thz value my be several<br />
B heavy and of the table (A320Q)<br />
g snergy is smlle<br />
af the weight, of'Uc<br />
arags of 6 MeV for the elemen58<br />
Thsse figures are taken from the total binding onergg Wo of the nucleus;<br />
he eno~gy neoessary to d~ornpoa~~ the nucleus into neutrans and protons,<br />
which is given By tho mss defect* The toba<br />
inding energy 'No is roughLy
212<br />
proporbion&. to *the number A of constitucnka and it is therefare justified to<br />
assume that the 'binding energy BB, OS one particle is W,/A,<br />
which leads to the<br />
figure s rrlant iane,d above &<br />
We know, howcverj from the &-values of p-n reactions that it requires<br />
a8 much as 4 Mav to replaoe n neutron by a proton4<br />
This showe thaC<br />
large flVctuatSans in Sa from the averng,s value must be expected.<br />
With sufficient excitation energy En, a pwticle 8 can be emitted with<br />
different energies, corresponding to the differont states in which the residual<br />
nucleus can bo left behind,<br />
We thsn can subdivide ran in2;o differant partial<br />
widths:<br />
(1)<br />
s the width corresponding to an ornissj-on bf<br />
a with the spatlid<br />
hat the residual nucleus should bs: left in the statead<br />
The widths 4" increase w5th higher excitation cnorgy since, firstly,<br />
icles can be emitted and secondly, mors dfffuront states #&of the<br />
nuclei are possible, so that the number .of t'esms in the bum (1) in*<br />
creases, and finally, it is expected that the probabuity of ekssion of a<br />
partScle increases with its velocity.<br />
At a certain excitation enorgy E<br />
the<br />
width A" becoms larger thsn the level distance and the ldvels overlap,<br />
The nuclear levc?&s in tho region abavc En= 3, can be invostfgatcd by<br />
the bombardment of nuclei with pastLclos<br />
of a given an~x'gyg<br />
If the particle<br />
hits the nucleuy and is c;hsosbod, it forms a compound nuclcus with &m GxCitation<br />
energy Ba-kc J This absorption should be ap ociable only if the energy BzI 44<br />
0, or within thu width of, an excitation level of the wmpound nuclsus,'<br />
monoenergdtic partiole beams, it is possible to investigate, by varying<br />
I<br />
peotrum of the compound nuclsus from 3, upwards,<br />
We expect to<br />
obwrvs ltr@sonanccil absorption when Sa<br />
Resonance levels have been observed wlth<br />
6 aqua1 to an excitation enorgy E,;<br />
tich and protQn bombrzrdmmts on<br />
averago dlstanau of LsvcLs
213<br />
with an excitation energy above Ba becomas vary sntall with increaszng atomic<br />
number so that the energy of the bombarding particle beam cannot be dofined<br />
e<br />
sharply enough to separate them in heavier nuclei. No regonance has been observed<br />
with beams of protons, deuterons, oreparticlcs for elements heavier<br />
than zinc (2 qO),A sr66,<br />
Although it is even more difficult Lo produce monoenergetic neutron<br />
beams of arbitrary energy, there is one energy region in which neutrons can be<br />
used with great success; naniely the rogion of wry low thermtt energies.<br />
aan be slowed down until they rcach thermal or nearly thermal energies,<br />
Nautrons<br />
Their<br />
energy is then very woll defined compared to the poor energy definition of beams<br />
of fast particles.<br />
Their anergy can also be measurad to a high accuracy by means<br />
of absorbers which have, in that region, n known snorgy dependent absorption<br />
(baron) +<br />
Recently methods have been developed to produce neutron bums with very<br />
sharp energy up to 5 eve<br />
Resonance absorption with slow neutrons can, therefore, be expccted even<br />
for heavy nuclei.<br />
Although it is not difficult to produce similar low anergy<br />
beams of chnrged particles, they are useless bceausa they do not ponetretc the<br />
Coulomb bcrrier of the nucleus.<br />
The resonance capture experiments with slow natrons give informtion<br />
as to the shape and width of the sesoniince levels of the compound nucleus and will<br />
be discussed in connoction wfth the quantitative expre~sions. It is found that<br />
thu main contribution to thc width of thus@ lcvels is the radiation width.<br />
This<br />
means that a captured slow neutron will in all probability stay within thc com-<br />
pound nuclcus, sincc the most probably effect is the wdssion of’),-quantwn after<br />
which tho excitation dnolrgy sinks below 3,<br />
and no particle omission is possiblec<br />
Some evidence as to the level density of the compound nucleus cm be<br />
obtained from the number of nuclei which are found to show resonanccj cilpture of<br />
slow neutronsI<br />
This nwnbor indicates the probability that a level lies in c?<br />
re-<br />
gion of a few volts above Be+,<br />
One obtains by that method p, lsvel distance of
2 14<br />
roughly of the order 10 ov at the excitation energy Ba for dements whose atomic<br />
c)<br />
number A is about 100 or higher, It is much larger for lighter elsmcntsj in the<br />
region of Ad60 it is probably of the order of 100 ev.<br />
We now proceed to tho discussion of the cboss swtions of mtud nuclcw<br />
reactions, According to assumption B, we may write for the cram section 6(a,b)<br />
of a nuclear reaction in which a nucl.ous is bombarded wfth a particle g and a<br />
pwtlcle .- b is e jectod:<br />
,<br />
. (2)<br />
Here fa the woss section for the formation of the compmnd nucleus,<br />
and 'Qp the relative probability that the particle b is emitted,<br />
The cross section 6 can bi. written in the following form8<br />
a<br />
ba55i&;l<br />
Here Sa is tho maximum value (g can possibly assum and e 1s tho<br />
a<br />
''sticking probability" of a which necassarily is smaller than unity. For<br />
neutrons whose wave length<br />
is large compared to tho nuclear radius R Sn?$Tx ;B<br />
c*<br />
For charged particles, Sa can bo roughly defined as the cross section for penc-<br />
trstion to the surface of the nucleus and is strongly dependent on the onorgy in<br />
1<br />
the familiar way of the Gamow-penetration of Coulomb barriers,<br />
discussion is found in Section XXXIII*<br />
The quantitative<br />
6 2 3 tlic $robability that a formtion of the compound nucleus takes<br />
plnca, if the particle has reached the nucleus.<br />
As a function of the onergy,e<br />
1<br />
will display the reeonance propartizs, mentioned before.<br />
It w ill be small betwoen<br />
tho resonances and large when B +e is equal or near to an excitation levo1 of the<br />
Q<br />
compound nucleus, From a oertain value 4 on, tho widths of the levels overlap<br />
and 6 becomas a smooth function which, however, must be smaller or oqual to<br />
unity.<br />
Naturally, an average value of f over the resonance is needed if the<br />
e<br />
t<br />
to<br />
incident bdam is not sharp enough in onesgy to separate tho ~JV&,<br />
It is reasonable to assume that e becoms unity if & is small compared<br />
a<br />
the distance between thc? nuclear constituents. Then, classical considorations
215<br />
can be applied, which make it a~em vo y plausible that every ppxticle which comcs<br />
bange of ths nuclear $orccs is incorporatad into a cornpaurld nucleus,<br />
th<br />
is SuAfilled for energies abovs 10 &7t,<br />
I<br />
Thu'second factor in (2<br />
clo is anittad by the compound nucILe<br />
ivc probdbility that a pnrti-<br />
s evidently given by<br />
re p"k i+ the; satin<br />
/<br />
of a particle b, amraged 5ver the compound st~tes<br />
whigh arc exci<br />
atage of the process,<br />
particles c which me ejected,<br />
TLo sun in tho denofibn8.tor shau<br />
c<br />
a<br />
(3 )<br />
ound stabs, for the omission<br />
be extended Over aJJ,<br />
TLG wjig~t&i~n of t k . ~ la furdG2gr:t ;.ob different; from tho oal-<br />
culation of the a&, TLa<br />
fornation of a aompaund nuc3eus and there<br />
are tho probabilities of tho opposite gpoaoss to the<br />
a fundaraefltql relet2on be tw@@n<br />
in<br />
A few qualitative conclusions can be drawn without using the formul2s.<br />
The for neu6t.m emission is always much grsa'csr than the<br />
emission, or the rb for any ChargBd particle, because of the Coulo<br />
barrier, except under unusual condiblons 'chat the bin<br />
is considerably smaller than that om Of bhe neutron, $0 that tho proton has much<br />
more energy. at its disposal; Thus, if a reaction with neubron omissi<br />
geticakxy possible, this is gene<br />
probable than all. the otha<br />
are found only Just abbve the threshold of w<br />
with neutron<br />
when the neutron has very smal energy. me<br />
energy dependent and beconie smaller with larger 2, Nuclear re<br />
s therefore are very Pare far heavr olemcnts,<br />
P<br />
f<br />
Deuteron emissions have never bwn observed,<br />
This, is becauge OS tho<br />
to remove a deuteron from the compound nucleus; it ia equal to the<br />
in4 two constitumts &nus the small. amount of the detu
216<br />
energy* The emission of a neutron alone is therefore by far much more probable,<br />
What can be predicted as to the energy distribution of the emitted<br />
particles? Aftar emission of a particle, the residual nucleus is left in on@ of<br />
its excited states, say with an excitation energy EB or in its ground statat<br />
The energy<br />
where E is the sxcitaticm energy of the compould nucleus which 1s in turn dete?r-<br />
mined by the energy e of the incident particle sr producing the compound<br />
;a<br />
nucle;us: E z Ela.CQaI To every level E of the residual nucleus corresponds<br />
k9<br />
and energy value e Thus tho energy distribution of the particles omitted<br />
bfi<br />
consists of a series of peaks which are a picture of the spectrum of the residual<br />
nucleus, thQ highest energy corrosponding to tho lowest level of the residual<br />
nucleus,<br />
In the special case that the incident particle is the sanie a8 the<br />
emitted one - (n,n) - (p,p) process - we thus obtain<br />
I<br />
B<br />
A - Graund level of initial nucleus<br />
B - Ground level of final nucleus<br />
C - Ground level of compound nucleus<br />
The highest energy corresponds to the elastic scattering, This<br />
b<br />
peak wilb be the strongest, especially in the case of charged particles, sinct:
it contains also those partjdes which are scattered by the outside fields,<br />
217<br />
Actually it is impossible to distinguish between the folLowiizg fQur contributions<br />
to the elastic scattering, which give rise to scattered wave-functlohs which &re<br />
superposed coherentlp)<br />
a) Scattqrbg by the Coulomb field if any<br />
t<br />
b) Scattering.jby the &ape of the nucleus in analogy to the<br />
optical diffractAon<br />
c) Penetration to tho surface of the nucleus and reflection<br />
there<br />
d) The forming of a o<br />
emission with squa<br />
und nucleus wit<br />
It is, howover., important to distinguish d) from the other contributions in ordor<br />
which cqrresponds to the fsrmtgon of the compound nucleus can be dcdined by con-<br />
wctring it with
general (apart from individual fluctuations) proportional to the energ;y.Q<br />
218<br />
and<br />
to the penetration probability through the potential barrier in case of oharged<br />
1 a decreasing function of (Eb.<br />
p&rtiCluS. P(cb) is an increasing function,CJ(E<br />
P<br />
We thdrafore obtain cn energy diatribution in the region in which the 10~018 of<br />
tho residual nucleus become undissolved, with a definite maximum which is in the<br />
case or" uncherged pnrticles very similar to a Maxwellinn distribution,<br />
discussed quantitatively in Section XXufVI.<br />
This is<br />
January 20, 194.4<br />
LECTUW SdRIES ON NUCLEAR PHYSICS<br />
Fifth Series: The Statistical l'hcory Lecturer: V. F, Weisskopf<br />
of Nuclear Reactions<br />
UCTUW xXxI11, gross' Seqtions and Emission Probabilities<br />
Tho values of the cross sections of the formation of a compound nucleus<br />
by impact of a particlo with a nucleus, and the probabilities of tho inverse<br />
process, tha emission of a particle by a nucleus, obey certain general rulss,<br />
which are derived from the geome~rical'propertics of the problem,<br />
Let us consider the cross section $('!of<br />
the formation of a compound<br />
nucleus.<br />
It is appropriate to decompose my cross section 6 of a nuclear process<br />
R<br />
with an angular momQntum fi 1 in raspect to the sccttering center:<br />
into tho partial cross sections @ belonging to tho contributions of the particles<br />
)a-,<br />
The angular monieqtwn in respect to the scattering center of a classical particle<br />
_." (6)<br />
*<br />
moving in a straight beam is given by m v d whore d is the nearest distance of a<br />
straight directisn of motion to the center, Quantum theory admits only integer<br />
multiples of % as values for the angylar momentum and it is justifiod to 'say
IJI<br />
219<br />
that, in a straight beam, ell particles nzoving along lines, whose distnnca d from<br />
.ChQ center lies between<br />
$ave an angular morrantwn .kR.<br />
us all particles of angular momentum<br />
asmhn area perpendicular to the beam<br />
of the size ($?gf'i)$%! A nucloar rtlaction which removes particles of a given<br />
angular mornentum,frorn the beam, cannot remove more thsn come in end thus never<br />
A<br />
ss section 6 larger than that area:<br />
I- .-.". (7)<br />
A more exact derivation of this relation is given below,<br />
It is to bs expected that partioles, whose nearest approach on o.<br />
'. ?<br />
straight lina, d, is largljr than the range W of any forces coming from the -scat-<br />
terer, wul not be scattered at all, Thus 6 150 if d >I2<br />
R<br />
$%4<br />
We obtain the maxjmwn possible cross section of a system if we assume tha$ 6"<br />
d"<br />
has its m;sxhm value for .h
*<br />
that every subwmm consists of a superposition of i331 incoming and n.n outgoing<br />
spherical wave of equal intensity (first and second tvrm in tho sqtaaro brccl,ceb){<br />
If a perticle wave of this form impinges upon a nuclcfus, the superr<br />
position is changed in two ways:<br />
2) the intensity of tho outgois spherical. W&VP<br />
is diminished due to absorption of the parkiclos (the latter may or may not be<br />
reemittcd); b) the phase of the remaining outgoing witv~j is changed,<br />
Both effects<br />
give rise to a scattered wave since tho outggolng wave is na longur ablo to inter-<br />
fere with thc. incoming wave in tho sam way as in (y),<br />
thd only way ,j.n which there<br />
is no radiation from the center.<br />
Thus tha aatual wave function ft has not tho<br />
asymptotic form (LO) but<br />
taken place, *q is srnaller than unity, 2.b<br />
waves b<br />
ht us, docompose (loa) into the incoming subwavrts:<br />
.: f'g,e*.~rjs~<br />
and the outgoing ones<br />
$,; 3 *YE '\/2~<br />
% kr<br />
1<br />
is the phase shift of tho outgoing<br />
il @-~(~~+L.N.,J-*(O? .-__<br />
4<br />
--,.---
c wave :<br />
The d<br />
to a plme wave<br />
221<br />
nce betwoen (loa) wd (10) is the wava which one has to add<br />
) $0 get the actual wav9<br />
/f.'i<br />
* I<br />
it id therefore the scattered<br />
The crcm aectian for elastic scattering, for a single d, can be obtained<br />
by calculeting the current of the 4th suiwe.ve. of #-- )I$ .through a large<br />
sphere and dividing'it by the current of the incident wave.<br />
1<br />
' f ci-qe rig) d(&+i) " ..<br />
This cen also be writthn in tho form<br />
This gives<br />
(14)<br />
It is seen froni (12) and (14) that curhain relations hold bctwaen the<br />
Capture cross section is given by (7).<br />
"he elastic cross soctbn, however, has<br />
a higher upper limit, which is reached if r)'sl and a g 2R/#:<br />
--%<br />
This mnxhm can only be reached if 7 S 1 or if 6')s #<br />
c<br />
This can also be understood in tho? following wzy:<br />
. ,<br />
the maximum effect<br />
of elastic sez*ttering is obtained by a change of phase of #' of tho second term<br />
in the square bracket of (10) (tho outgoing subwave) without reducing its<br />
This is identical, with adding to the plana wc.vy just twice the outavd.<br />
Thus the cross section corresponding to this, is 22 times larger<br />
than the one resulting from absorbing thc outgoing subwave, Hence we gat a<br />
maxhurn cross section far scattering Pour times largar than for absorption,<br />
*<br />
given value of@ "I.<br />
ljJXitL3<br />
1'<br />
**f<br />
'9<br />
The expressions (12) and (14) limit the possible values of @ R (4 for a<br />
4<br />
This is indicatod in Pig. 1, whoro thc upper nnd lower<br />
6''. flf" ,.is plottad against It is seen that it is nccessarily<br />
IC) assues its msurimunl value(& A + i ) ~fi F .<br />
djl<br />
We foLlowing theorem is OS interest, since it connccts the total cross<br />
ith the amplitude of the elastic scattering; the total cross section is
222<br />
e which also can be writton in tho farm:<br />
1<br />
q ftd -.-%J.&”pAfl RX%R+I)<br />
wheso An(1-W aigJ] is the value by which ono has to multiply tho outgoing<br />
subwave of the undisturbed plane wave in order to get the scattored subwavo(b3),<br />
i<br />
Fig, 1: Upper and Jowor limit of ths elastiu cross section for given capture<br />
cross section.<br />
If the scsttering object (nctuczlly tho range of the scattering forces)<br />
is large coni9ared , ~ll’wc?vc?s for which kfi(.gcan be absorbed, but the wava<br />
for which 1 - R/n )) 1 vanish for r 5 R and thus arc not influancud and should<br />
a have gicl ;31 0 If R/j( is a large number, it is a goad approximction to<br />
L
the,fact that a<br />
interference effect Pf a ac<br />
kon out by tho sh&daw,<br />
becomes com-<br />
value vnly for vvry few values Q<br />
cind 5t is masthy lower
a<br />
224<br />
nucleus in the state a; b) the umission by tho compound nuclcus, of a particld<br />
a with an engular mornenturn$,!,, and Iswing the residual nucleus in the stato G.<br />
The two processes a) and b) arc invurse. The first procuss is rmasurod by E! cross<br />
section 6aaf and the secmd by Bn emission probability Gak Wo should c?lso<br />
ifiod the direction of khc apin of the emitted particlo in respocL to<br />
some arbitrary mist We omit this and assume in the following that the spin is<br />
given together with tho angular momentum A e<br />
In the definition of CaGAit i s<br />
assumed that the incoming beam is spread over an energy intervd largo enough<br />
to uxcita many component levels, T;*,B is the average value of the emission<br />
probability, averaged over thsse.lavds which can be excited by the first process<br />
within a snmll. anergy region,<br />
In order to siivd indices, we replace the triple index (ad) byeand<br />
writs and T;r/ , wherever it is practicable, The result of the following<br />
considerations will be the relation<br />
whae D is the avcmgo distcnca between the levels, which can be oxcited, Before<br />
c<br />
going into the detailed dcrivntion, it is useful to understand, why a rslation of<br />
this chwacter is expected to exist,<br />
Let us first assurm that no other process but the one charnctcrized by<br />
can follow the c@porption given by<br />
In this case, the level width is<br />
equal to r& It is to be expected that $a is proportional to the ratio p'<br />
(524<br />
of the width to the distance of lwmls, becam this magnitude measures the rela-<br />
tive portions of the spectrum in which absorption is possiblo.<br />
It is plausible<br />
m&ximum absorption is reached, ifr/ is of the order of unity, The<br />
mimwn value of rf is fik '(2ktl)and<br />
d'D<br />
WQ may expoct P, relation of the typQ<br />
where the constant $s of the order unity, The detailed calculation shows that it<br />
II<br />
is 2rt/f,k+ 1).<br />
This considsration also holds true if tha coriipound nucleus<br />
created by the absorption of the particlo a. ccvl decay in suvoral WF.YS.<br />
-<br />
This ef-
228<br />
fect would broaden the lines but would not change the clbsorption avurege ovw<br />
the lines,<br />
9 Let us now oomparo the probabilities of the two invorsb processes,<br />
The probabilities should be equal if they cndud in states of eque-1 weight.<br />
Since<br />
this is not the cnse, they should ba egucQ aftzr dividing thm by the stntjstical<br />
weight af tho ebd stQteS,<br />
Tho pyobability Wl per second of the creation of tho compound nucleus<br />
by tho particle a if it is enclosed in n big sphsro of radius R is giwn by<br />
_. 4" V (15)<br />
W i " ~ f h .. ~ n ~<br />
whore v is the velocity of the particle and g its wave length.<br />
understood in tho following wey:<br />
This can bo<br />
if 1IJZgCc'aaumos its maxiiiiw valw (2k+f)fl<br />
the particle with an angulzr niorientum )iRis sur@ to be captured on its wcy to<br />
the contw of the big sphere,<br />
The averago probability par unit kbm Is bhen<br />
v/2R shoe it nlay be fowxl anywhere along (z diamter, {Its angular momentum<br />
buing fixed, it mves on lines which havc a given distance ,(,% from thc centor,<br />
The length of these lines is na&y 2R,<br />
In C~SQ @&is<br />
invarsu ~TOCQSSOIS.<br />
6<br />
since R i s assumed very large: a >)R&<br />
snxillsr than its ~.ililximrUn value this probebility 5,s reducod in tho<br />
hg ratio and (15)results.<br />
\Ye now compme the statisticr,l wights ,of tht! end status of the two<br />
If the anergy of tho incident particles lies botween Q and<br />
4<br />
the statistical woight of the end state is -Uc(E) dE, hhoro U,(E) is the<br />
level density of the compound nucleus at tho excitation energy E which will. be<br />
sclcction rules,)<br />
(Only thess Iuvcls are couqtod, which can be excited owing to angular<br />
r/,<br />
Tho probability of the inverso process is ~;t<br />
and the statistical<br />
weighb of its end state is thr! stetistical waigbt of a fro@ particle with the<br />
L<br />
*<br />
cnorgy between 6 and<br />
We thus get<br />
+dIE,and the angulcr nornentm 6 in a sphere of a radius<br />
.- -<br />
-
By expressing cross sootion ca by Q dimensionloss mgnitudc .Q s<br />
which never can be larger than unity, an<br />
Ita,<br />
y introducing the a.vcrngo love1 dis-<br />
momentum and with the rasidual nucleus left in a &finit@ state a, is thus bound<br />
by an upper limit (Q 4 L) :<br />
distance D.<br />
6 (16b)<br />
A few remarks are necessary concorning the significance of the lovol<br />
nuoleus, frorn which the particlc can be omitted.<br />
the levels which con bs exciked by the particle .-<br />
Q wi$h tho angular riionentwn<br />
(and given spin), hitting a nuclous in the state Crj, On the other hand, Lis<br />
tho pnrtial width for the snlissfon of a,<br />
the lovols descrihd cbove (whose nverngo distcnco is D) within a cortdn sml1<br />
energy ragion,<br />
D is defined as the distcncs bQtwccn tho levels of the coritpound<br />
Zt is tho diatcww betwoen<br />
with the SCZI:IC? properties awragod ovar<br />
Expression (160) is not bound to this definition of Db<br />
Far<br />
m%r@le, D could bG dafined as tha tweragu distance between<br />
levels of tho<br />
coiuponont nucleus in this energy region; is then understood 11s the averngu<br />
over all thGse levels, (for ciost of whiohra So), tho rolation (ab&) is then<br />
d, In Cha following we will, howuver understend by D End ra<br />
average distance and width of the levels which man emit tho particle 8 with<br />
tho properties specified above.
*<br />
227<br />
LECTURE 3E3S1ES ON NUCLEAH PHYSICS<br />
s: The Statistical Theory Lecturer; V, F. W&iskopf<br />
gf Nuclear Reaction8<br />
LECTUM XXXZVe Th~Clted NQCZ~US<br />
st us consider a vluc1tzus in an excited state n in which it i s able t o<br />
icles.<br />
The st&.Ce tz has a finite Llfebsule and the probability of finding<br />
rn is the width of<br />
eus in $his state decays accordinG to the law e cf”t e<br />
the state, In general bhe state n can emit different particles B and these<br />
y be emitted with different anguLar momenta Asand the residual<br />
be left behind in different states&.<br />
!Therefore tho width Tncan<br />
be subddvided<br />
where<br />
%a,<br />
3<br />
the following way:<br />
rn, $ (18)<br />
B ‘the partial width for the n of a with an angular rnomentw<br />
fng the resfdual nucleus in the state ‘@L,<br />
The value of<br />
is determined by intranuclear and extranuclear<br />
and it is useful t o separate them into the following ones: the probabil-<br />
of emission of a particle a is proportional to the fol-<br />
1) me final velocity which it acquires after having left the<br />
ftdr having penetrated the p entia1 barrier if there i3 any;<br />
&) Q$ bhe barrier which fs defined as the<br />
f intensity of an outgoing spherical wave, corresponding to<br />
face through the potential barrier to the<br />
t is useful tQ consider as PO<br />
barxier, not only the Coulombfield<br />
ged particles, but also the potential of the Csntrlfugal force<br />
i<br />
;*
e<br />
$'A2&<br />
A.<br />
228<br />
(4 *I)&-# the case of an emission of a particle with an angular momentum<br />
This force is described by a repulsive potential, inversely proportional to<br />
the square of the distance, and acts on a particle in the same way as the Coulomb<br />
repulsion. T,fi&)is a function of the charge and of the angular niomentum of a<br />
and of the energy
e<br />
229<br />
being emittod by the original nucleus as, for example, a statu in which tho<br />
particle has a very high angular momentum in respect to the residual nuclous. In<br />
order to describe the excited status of the nuclexs zbove $which aro defined by<br />
.<br />
our assumption A,<br />
%<br />
it is necessary to introduce a device to oxcludc thu solutions<br />
of (2Q) which correspond to a residual nucleus plus a fmdy moving indqwniient<br />
particle which is not created by an emission of the compound nucleus.<br />
This can be<br />
done by mans of boundary conditions for<br />
which should express tho Pact Lhct,<br />
in case the state n includes frec particles outside the nucleus, thoy must be<br />
represented by outgoing spherical 'CIICVI?~. With these conditions the equation (20)<br />
no longer has a continuous spectrum above E. It then hcs discrete Gigenvalues<br />
which belong to quantum states of finite lifctime, becouso of their c;bility to<br />
onli t particles ,<br />
Let us introduce a radius 8, the nuclaar radius, which is the short-est<br />
distance from the nucioar center at which prnctieslly no nuclear force acts upon<br />
vn<br />
a nuclear particle.<br />
(The particle might, however, at that distance bo under<br />
influence of, say, the Coulomb force, ) Tha uigenfunctions yn haw the following<br />
form for ra>R:<br />
Here X are the eigenfunctions of the states 4: of the residual nuclaus whioh is<br />
a#<br />
-<br />
left behind, if tho pnrticlc a is remowd from tho nucleus; pn Is the wave<br />
function of the particle R outside the nucleus, 4; (Ea) is a solution of tha<br />
following one body equation of the pcrticle a in tho space outside of the nucleus:<br />
A
where YQ is e lcrge distilncc! zt which Ghl: kinotic onorgy of c".lJ,'.snrticlos mitt4<br />
is inuck lt?rger than tho potontieil crwrgy Vpw[rF.) cnd thu cuntrifugnl forcl:<br />
. Tho swlu can b;: expresslud by boundcry conditions:<br />
-c 1 =O<br />
n<br />
24) k,, is givw by (a)<br />
Per Pa =% (24)<br />
n<br />
wiiicti 1 ~ 0 ~ G A<br />
part of kmis positive,,)<br />
aid thct sign of thc squcrc, root must bo<br />
Thew Bonditions and, of courset compctiblu with thu ~YEV~ oquation (20)<br />
Tho equation (20) with thc (camplux) boundary conditions (23) or (24) has soh-<br />
tions fop which '& is nucossarily comp2c.x:<br />
wn =E,- irya ' (25)<br />
oigenvc?.luo rizcan~ thrlt tht? st&b decays dxponentially e~oordi& to<br />
Ft ,<br />
,
so that rn is the rociprocal half life OS the width of the st&e n.<br />
23 1<br />
(Actually kn c..lso is coinplox because of thc fact that (21) cdntakis the<br />
complex VJn. It ~xprossus the fact 'chat the outgoing wave (23) changes in intensity<br />
with re, It increases due to the fact, that the nuclear stato decays expanan+<br />
that the parts farther away aro emitted in an earlier stago. This<br />
change of iintansity czlong thd outgoing wave, howvor, is only a very sn~ll effect<br />
na long a? thc width r" of the levo1 is srimu cornpcr'$d ta tht, 3 t?nyy of thu<br />
eniittod pcrticla, llnc ratio bQtwoen rwal and ink7ghary parts of k, is equal<br />
to tho distance trcmllsd by the emit ed pwLic3.e within tho lifotinie of thc level<br />
ed in wavdungthsX'(k *)-' . This ratio will be wry lar o in all<br />
intarost and we ncgltlct?frorn no n the imginnry part of ka A completely<br />
axcopt, of courss the C~SB TIn < Ba { itntion energy less than boundary<br />
anergy) in which<br />
rn,s<br />
is purely hginary nnd r".: 0,)<br />
ih now express tho lwd wtdth r A in term of't.heQq, n . The veluo of<br />
the rcciprocal life time of the level n and tlwreforc equal to the numbor<br />
of palkticlcs emittQd by the nucleus per second.<br />
following way:<br />
We first subdivide rnin the<br />
rn = E r;&, (26)<br />
a,a,a<br />
whero &is thc: partial width for, tho emgssian of 2 with an angular momentum 1 ;<br />
leaving the nucleus in the state a. Ue surround the nucleus by a sphere wikh (he<br />
radius ro and dotermine the number of pcrticlos 2 wilt% tho above proporties crossn,<br />
ing that sphurv por second, which in turn is equal toGL& Thus WI.? obtain (the<br />
index a d is repincod bycx:)<br />
undor the condition :<br />
9n thz limits 1 e' 1 < r.<br />
Q<br />
(27)<br />
owr all vnricblos<br />
*) If pn is 2, slowly deccying stctc3, so thnt thc<br />
particle dsnsiJdy 'wtxwn'thc boundary R of thu nucleus and ro is wry srrtall com-<br />
pemd to the ddnsik,y hsidt! of tho, nuclwa, wo c m also norrmlizo<br />
in tugral<br />
by the<br />
n<br />
In the oxpression (27) the v;?luc. of *-,n is still dt.pcndont on tha field outside<br />
of thc nuc!-a~s. Since thG latter is wvll known it is usoful to show its influence<br />
1 2<br />
is the vdocit of tho pnYklcle at ro~~~(?~,l<br />
is, in the<br />
usud htiro, x-3 t$m~s tho probability of finding a particlo Q in<br />
om volwnu unit at ro.<br />
'
explicitly in the expression for tha width, ?% thoroforc express n (ro)l<br />
232<br />
by tha value of the s m function at the nuclusr radius Re Vh introducit thd<br />
magnitude Tasg(€)which<br />
is tho ratAo by which the "(intensity x r2)1f diminishes<br />
in an outgoin; wave of a particle 2 with an onwgy 6 and an angular momntum 16,<br />
if it pehotratos thQ field around the nucleus from R to roe ro had boon chosen<br />
outside of dl fictlds of forcrs so that the 1l(intsnsity x r 2 )It is coxistant outside<br />
r,. T,g&)can<br />
be cnllcd the transmission coefficient of thc potuntial barrior.<br />
It cnn be calculntod from tho wave cquation of i? parliclt: with an onorgy and<br />
&n angular momntum .46 in the fiold outsi.de of the nuclcus, whew solution wc<br />
will call Fa<br />
0.<br />
r(r) for this purposu:<br />
[& ."<br />
If we choosu .the p&rticular solution which<br />
has tho asymptotic form<br />
wc get<br />
T ddpend9 on the naturi: of<br />
F-e<br />
F
233<br />
far xcc 1<br />
with x k,R. e v&lue of Ta kcG> for chargod particles is diffic<br />
put* sxactly, ~GC~USQ of tho wry SLQW ca<br />
enoe of the expressions for t<br />
enfunction of the Coulomb tiel 0 region considerad horc. 2t 2s<br />
enorgy E.<br />
e8 stronglr with<br />
An approximate cxprwersion, c<br />
char chargo 2 and with fa2S2ng<br />
cd by using tho N.X.B.<br />
method, 5s<br />
(28)<br />
1<br />
is the charge of the pnrticla<br />
is the anergy barrier height for<br />
The valuas of c<br />
mwgy E of the
*<br />
Expression<br />
254<br />
The Pactw G depends on the value of the aigenfunction C& at the boundary af the<br />
nucleus and thus reprosents tho influenee at the intranuclear structum.<br />
(29) soparntes the offects of tho forcos outside of tho<br />
could therefore ba called partial wgdth witho-ut 'outsidc potential barrier,<br />
This<br />
separation of the barrim effect and eho nuclear effwt is by no mucins completx!,<br />
n<br />
It is to be expected that the value ~f..~~~~(R)&I.so depaqds on the €arm of the<br />
WEV~ outside of Rk sincc tho Inside and the outside p ~ of ~ the t function must<br />
join smoothly,<br />
Under certain conditions, howovw, thore is good reason to expect that<br />
the valuds cP0lf\(@do not dopwd on tho bahwior outsidc of the nucleus, 130-<br />
cause of tho strong forces inside of the nucleus, the averagw momentum pi of a<br />
particle inside ,of the nucleus is very high and corresponds to an unwgy of 20 to<br />
30 Lev.<br />
The average monientum is an indication in what space intcrvnl the wave<br />
functSon chenges ita veluc considorably. :!e dofinr: an c2Verc?gu wL?v'c length<br />
xi =<br />
i<br />
and wo can say that bhc? W~VB function insidc the nuclous chcngus its<br />
value by an cmount of the ordm of its avcrco.gc size, if tho coordimtc, of one<br />
portich is chnngvd by h+<br />
1'<br />
much smallm than pi:<br />
If the momontum po outside of the nuclom surfacu is<br />
tho wave function insidw the nucluus must be joined at t h nucltmr ~ surface to<br />
e, tvaw function, whoso villuo chnngos wry slowly cornpard to conditions insido,<br />
ft 1s thOn a good approxhtian in thu duturmination of tho TGVQ function inside<br />
to mswrs: that<br />
t<br />
for all pmtiolos. The vcluGs for<br />
aa.4<br />
arc not very difforunt from thc: truu on@*<br />
guts with this boundary condi4ion<br />
Sinco this condition does not contain
any reforanoe to the properties outaide, tho genwal behavior of 4,<br />
(R) is to<br />
oc" i" 235<br />
I<br />
I<br />
be oxpsdtod the sane for chargod and uncharged particles and also the szmc for<br />
ciiffsranC Rt s,<br />
Particularly, the aqprago vnlucs<br />
neighboring luvols<br />
should bo practically indopndcnt of tho energy 6. a of tho outgoing particlu.<br />
It<br />
will depend only on the properties OS tho excited nuclous, whosl: excitation<br />
enmgy is Ba+f4, an enorgy<br />
- c.<br />
valud 1 aaLof<br />
validity aP (31) end docs not<br />
the pnrtizl<br />
which is much lcrgcr then<br />
chcngo ap2wclablgr in thct region.<br />
in the region of<br />
'l'hi. avcrcgi:<br />
width OVM neighboring 1C;v@ls, is thcn Given by<br />
whcre C(3) is a funation of tho total excitation energy i= Ijk 4- ea only, It<br />
dQcs not vary strongly in an onorgy intaval which is sm11 compnrod to 38a .<br />
Furthergoru , At is c.,pproximRtuly uqu&Q for tho different pcrticles which can bd<br />
arnittcd by Chc! conpound nucleus,<br />
and for t hG diffsront vnLuos of o.ngular monicntum<br />
of thew pwtic1c.s.<br />
The uaofulnoss of this approximation con bc illustratod in n few<br />
oxahplea:<br />
thc herap ixirtial width for the emission of a neutron with<br />
is gj.von by:<br />
F = c(EC}dq (33)<br />
which is ~ssuntidly proportional to the square root of thc outgoing noutron<br />
mwgy in the rGgion in which (31) is valid,<br />
Tm ?.wrap cross scction over neighboring lcvols for the capture of s-<br />
n~utrons can be obtained from (16):<br />
%= ---- D<br />
This relation ShQw thu wll lcnotvn<br />
slow s-natrons ( 1x0).<br />
(34)<br />
1 l ( a '<br />
'/u law for thc ccvpturo cross sdction of<br />
Thc particl width for thG emission of neutrons with 1 # O is givcn by
236<br />
It is thus proportional ta e itg for mill cmrgies. The emission<br />
prababllitics of chargud pcrticlcs haw tho trmsnzission cosffiaicnt TaL(G)<br />
as thG dominwring factor.<br />
z'hu factor C appccring in tha ncutron widths cannot bo datmniiiud but an<br />
.c...<br />
uppor limit can bo sat: WG know from (17) that % 5 ?&T<br />
Since (33) should bc valid up to cbout<br />
1 bkv wo obtain<br />
for noutrons &to).<br />
Thc: exprimentally d~tr:r&od vnluos (see Scotion 5)$ indiontc that C is not much<br />
smeller then D ea, if the energy units arc chooson in 1ievtss<br />
/<br />
So fc?r tho radiation of the nuclear statos has beon neglectod. It intron<br />
, which we call the rodiation wid?&, !he<br />
duccls another additional width<br />
n<br />
tatcl width A of the lcvol is thm giwn by<br />
rn<br />
Tho radtction width consists of a sum af partial. widthsr<br />
transition8 to spacial hvels f9 below n:<br />
fP<br />
corresponding to<br />
u; is thc! matrix dcmont of the from n t o . P<br />
Tho spoctrwn of khc nuclous can be dividd a) into 2" tlstc?bJ.a" region<br />
from 0 to<br />
to rndiatiori; b) in E 'lroson~.nco~J rvgjon abovc 3 whwe tho stntas arc able 'ca<br />
(36 1<br />
where the lvvcls omit light quanta only and the width is moroly duo<br />
Qmit ptzrticlt\s, but still aro smrt from each otlicr by mor3 then their width so<br />
th2.t thoy form c? discrot
c<br />
January 27, 1944<br />
LA 24 (35)<br />
237<br />
UCTUiU 3El.&3 ON NUCUAd PHYSIC3<br />
Fifth Series: The Statistical Thecrry Leeturcr: V. F, Ykisskopf<br />
of Nuclear Reactions<br />
A) The Brcit-WiPntlr Formu2<br />
The cross sections of nucloar rmctions, which wuro cnlculated from the<br />
emission probabilitios in Section WIII, woro definQd as avorag“ cross sections<br />
over 8 region oI enurgy of the incaming particle, which includas mny resonance<br />
levels.<br />
This suction is dovotod to tho study of t h crosq section BS a function<br />
of energy, if the anergy is skierply definod.<br />
phcnomenc. a3 deskxibed qualitativcby in Zection %XII.<br />
In this casu vro oxpuct resonance<br />
:‘io first investigP.tc tho onmgy relations: if eoCia tho energy of the<br />
incoming particle, the onurgyeb of an emittdd particle is givon by:<br />
% = Ea+D,--Bb+E,-Eg (38 1<br />
whcre E<br />
aru the excitation enorgios of the initial and final nucluus<br />
cb and %3<br />
respcctiwly, an4 Ba and i$, we tho binding energius of a to the initial<br />
nucleus or b b0 the final nuclcus rospoctivafy.<br />
The cnorgyca is thus dependent<br />
on €a d i s in general c.l- not equal to thr: onergye,$<br />
of a particlo b umittcd by<br />
any of the excited statm n of tho compound nucleu$, excqt if is just in tho<br />
middlo of n rosclnancu;<br />
This is tho rxsa if Fa+*Ba is just equal to tho<br />
oxcitation energy of thc compound stat3 n,<br />
The expression of ths cross section of a nuclear reaction in the<br />
od by meit and ?fignor” and later by Both@ and<br />
1) Broit and iligner, Phys, Rw. 519 (1936)<br />
I
e<br />
238<br />
Placeek2), by Peierls and Kr.pur3) and: by Sicgwt4), In tho following it is not<br />
attempted to give an exact dcrivation of it, The anelogy to the rosonancc of<br />
dmped oscillators is uscd, to mkc tho main foaturus of tho formula as plcusible<br />
as possibluc<br />
!le first introduce thr? concept of effective wid6hs. According to (29)<br />
22<br />
tho partial widths depend on the energy of the outgoing particlo by ntonns of<br />
n w<br />
the fsctor kd Tb (Cpg Since the oncrgy E<br />
nuclezr reaction is not nocdsserily equcl.1 t oe<br />
@<br />
fin<br />
of thd outgoing parkicls in a<br />
, it is usoful to introduce the<br />
uffectivo width') (cfj) which is a function of It is obtainod from ths<br />
fd<br />
B'<br />
actuel width by tcking thc energy dcpundant factor at th+ value cfi instead of<br />
This rolntion is not quitit exact<br />
I<br />
bucczuao of tha slight energy depundsnco of the<br />
other factors in (281, namLy<br />
I .<br />
(F) l2 ; however, the rosonanc~ rugion dous<br />
not uxtand furthm than the rugion of validity of (31), in which I<br />
not cnergy dopcndcnt,<br />
fflfl(R)/:s<br />
ht us assume that tho energy af an incoming particle a lies within an<br />
intcrvcil d which i s much s m l l G r thnn thv width of a rosonnnco lcv~l of the<br />
compound nucleug. Furthwnorc we ossumo thct thu wwgy is nmr to the onwgy<br />
En at which thc particle would bo at thd mmir;ium of a rosonmcG5), and compara-<br />
tively far 2.~2.~ from othor rusonenoes so that wc do not wed to consider the<br />
influonco of othm lwds. The cross wction c(E)as 2. function of thd wiergyt$,<br />
B<br />
has than thd cheractwistic L*Gscxw.iicd dqwndonco :<br />
(1937 1<br />
3) Poiorls and K~pur, Boy. Soc, Proc. 166, 2'7'7 (1938),<br />
4) SiegGrt, Phys, f2cv, s, 750 (2.939).<br />
.<br />
5) en should, in accordanco with Suction XXXIV, bo writton a," , cnd<br />
shouLd<br />
haw an indox Ea. In this section we omit this indcx in tho cnljrgy of the<br />
incident particlo and the cnGrgy ab which it is at rusonancG with thG nth<br />
IWCZ.
23 9<br />
1)<br />
a(+ F(0 (43 1<br />
wherc d(E)is thc effective totc21 width of the resonance, which $6 thu sum of<br />
all effuctivc: partial widths:<br />
6&1 E E? (43d<br />
B&<br />
is the energy (given by (38) ) with which tho particle is omitted. n e sum<br />
c,,<br />
is taken over a11 possible index triples b41'<br />
F(6) is 3. slowly varying<br />
function ofe, which we determine later, If is in resonance, 6 ran thc<br />
\ rB<br />
energies E arz tho sarie as they would be if th3 particles wcm cmlttod by the<br />
resonance level n indopondently. Thus 6(gn) =An,<br />
Exprcssion (43) cnn bo understood by analogy Lo the caw of forced<br />
vibrations of 3. dcrriiped oscillator with cz proper frequoncy V<br />
~nce of a pirriodic forcc F cos trt with a frequency v<br />
to bt: due to a fruquoncy dcpendunt friction forcc -mb(v)X<br />
undur the influ-<br />
The damping is asawd<br />
(x is thu dis-<br />
placcmnent of the oscillator),<br />
On0 obtains for thc: displacommt xt<br />
And f.or the avorage energy cbsorbud by the oscillator per unit thw:<br />
In this oxpression, tarms smller than of the order (%-Pv(q<br />
me negluct-<br />
cd (mcond term in (44s)). The timo duoendance of the frw vibration of this<br />
oscillator would bc: x hl COS % ; its intonsity in this case<br />
docays wpponontinlly with thu dscny constcnt o(t/,).<br />
From this wc: cm undmstcnd dxprljssion (43) for thd cross sGct$on, by<br />
*<br />
rdpllacing in (Skb) fraqucncics by cnorgios. d(t/)goi!s ov~r into thc: &fm$ivu<br />
total width o(c)of thc: lmd, whoso v2.1~~: for &= krn is rlso (aftvr division<br />
by 'h ,) the actual dr3wy conatcnt of thc: 1 ~ ~ just 1 , as &(%) is thc: dl;cay
constant of the intensity of the oscillator.<br />
The function F(€) in (43) is an<br />
240<br />
analogue to a frequency dependent 'force F, acting on the oscillator.<br />
with its inverse:<br />
We deterdne the function F( ) in (43) by comparing this cross section<br />
particle with the energy<br />
the deoay of the compound nucleus with an emission of a<br />
within the small interval de . 'ilne probability of<br />
this emission P(e)dG must be proportional to the same resonance factor, which is<br />
contained in c(€), but should give, after integration over the total width of the<br />
level, the total emission probability Tn (the partial width for the emission of<br />
a,<br />
the particle a><br />
c, is an energy interval large compared to the width. This is fulfilled by<br />
since<br />
6@)/2?T<br />
?an(e) &py (bp +<br />
)e<br />
be. (45 1<br />
This relation is right if "'&n(e)and 6" (&)are sufficiently slowly variable<br />
over the resonance region.<br />
It &so would be fulfilled by &tin;S the aotual partial width Fz<br />
instead of the effective partial width ?*?&)as the first factor in (45). It<br />
is evident, however, from the derivation of the energy dependence of 9 "(e) in<br />
4x4<br />
(42) that the emission of a particle with an energy e is proportional to?n(C)<br />
rather than to<br />
:;?e now apply the expression of detailed balance (16) to calcuiate the<br />
4%<br />
* where<br />
function F(4) in (43) from (45). Since the end state of $he'capture process has<br />
the weight unity in this case, dG(E)dE must be replaced in (16) by unity,<br />
obtain instead of (16):<br />
is the wave length corresponding to 6 , From this we get by using (45):<br />
We
U<br />
(6) &"(E) 24 1<br />
4 = ?cs2 (@)P.+ [-Y6&]2 (46)<br />
This is the cross section for capture of the particle<br />
a regardless<br />
what reactiok should follow, Since the bn in the numberator of (461 was writ-<br />
ten in (k3a) as a sum of the partial widths belonging to all processes which can<br />
be initiated by the capture of<br />
a, it is self evident that the cross section for<br />
a special nuclear reaction 5 4 6 is given byz<br />
e q$*?ye)Z(kJ-<br />
-Q(aLlfl) =fix (c-gn)a*l*n(G&gJF (47)<br />
If the resonance en~rgies f" of several levels are near enough to e ,<br />
the different contributions add up coherently;<br />
tensitiea must be sunmd,<br />
be understood from the analogy with a forced oscillation.<br />
to (h4a) is proportional to the complex Factor 1<br />
pears in the form ,/?+-E" +ib% ) '<br />
The cross sectioli of a6 f b3,a:, 1<br />
the wplitudes and not the in-<br />
One then obtains an expression, whose form again can<br />
Tha amplitude according<br />
+ix which here<br />
,.@%P,k)reaction with the initial particle<br />
l<br />
a incident with an energy +Z. and a given angular mornentum1-h upon the initial<br />
nucleus in the state Q,, resl~lting in an emission of b with an angular momentum<br />
1' and with the final nucleus in 10 , is given by:<br />
*<br />
Here f;, are the resonance values:-&n = z,yI--*Ba<br />
Ea , i4n is $he<br />
f the excited state n OS the compound nucleus acting as intermediate<br />
state, &% is the energy of the statem of the initial nucleus;<br />
/d.crs<br />
phase difference of the initial and the final sta,to and it vanishes if cX;.""fl<br />
and 3.' t,jp<br />
is the<br />
are the effective partial widths of n, corresponding to the<br />
emission of a and b respectively, with the residual nucleus left in a, or fl .<br />
f(€) is a slowly varying function of<br />
ea which is small compared to the r@son-<br />
ance contributions of the first terra, . It is introduced to replace tho contribu-<br />
tions of the second term in the amplitude (4ka) and to inc;lude the contributions
states n and<br />
z<br />
(%md n<br />
k, and kb are the wave numbers correspo<br />
8 a or b respectively, Hare the<br />
abed in the *119. E' (6) is the @me<br />
aftm division bY the factors<br />
The numeratar of the ~8sonance terms consists here only of nuclear<br />
s independent of the cnargies af the particles,<br />
In formias (48) and (49) dJ ,stata;s 6)<br />
6)<br />
n of tho nuclei are con*<br />
sidered aa of weight unity. Zf song$ of 'thew stabw are degenerated it is so01'3-<br />
re useful to suni over all substates, Tho tots1 cross section is the sum<br />
of all crow sections from every single qubstate of $% to every one of c In<br />
cwe 03 angular rmmentum degenaracy these summations can be performd and Bethe:<br />
te A.<br />
Tho initial and final states are no denoted with capitallotters<br />
stead of the greek letters (C~fl> in order to indicate that they cow<br />
prisa all magnetic substates. The compound state is called C, I/ 1' andjlj/<br />
are the arbitaL and tba total angular molllenturn of the incident or emitted parti-<br />
ectively;<br />
J is tho total angular momentum of' incoining particle and<br />
cleus and therefore also the total spin of the coirrpound state C,<br />
is the, amplibuds<br />
of the compound state C in regpect tc) the emiesion<br />
tide a, with orbital and total angular momentum L and 3 and with the<br />
residual nucleus fn A,<br />
The sumnation sign 1'<br />
C with the spin J,<br />
c<br />
It is evident that @&A<br />
=o wess/j++j)l j-i(<br />
Indicates that the sum should be taken only over states<br />
The sumr.;stion over J 1s outside dF the absolute square signs<br />
'
which 'chat thd cantr cPlCtefjng with Egerent total<br />
moment<br />
In most cases in which (50) Ls appl%ed, the contributi<br />
4<br />
of B<br />
with a<br />
e exfendsd t<br />
utie glso rgdiative phenGmena sutsh<br />
ar reaction in wkii&i<br />
ia particle<br />
ent upon a nucleus in the srta<br />
and a light quantum 2s wdtted<br />
ending in the stla,
m B)<br />
A fm general conclusions cats be drawn from the ono-Zeva1 formula (47);<br />
If there is only ano einission procoss poadible from tha corrlpound<br />
nucleus, which thi!<br />
can put oh<br />
cwsaPi2.y is the reamissios of the absorbed particle, we<br />
and we get; Par the only poslsibls process, the elastic scat-<br />
elastic scattering.<br />
It thura i.3 bne eorty3uCing ~FQCQSS, say the ofixl-ssion of b, we obtain<br />
which i s off reBo<br />
but, in resonance, weakm
consideration:<br />
246<br />
if the proton would be emitted with highor energy than tho one of<br />
1)<br />
thb ndutron, the energy of the residual nucleus would bG lowsr than thc ona of<br />
the initial nucleus, and the two nuclei would be isobars diffGring by onv unit of<br />
charger nus the initial nucleus would bop-Unstable, which is impossible:. If<br />
tho emitted proton had equal or ldss energy than thc nautron it wwld not be able<br />
to pcnetrat,: the potential barrier,<br />
1% first assumo that thdm is only one level sufficiently near to tho<br />
onurgy to which the com2ound nuclas can be excrited by thd neutron. \'TU then apply<br />
(5b) and uso (33) for thc: partial. widthTaA of the neutron, F'urtherrnoru,<br />
' 1 1<br />
JaIkE since tho noutron can contribute only its intrinsic spin, 932, and<br />
The cross<br />
is thcn:<br />
(55)<br />
which is obtaincd by putting ?( s C&" also according to (33) .!<br />
60<br />
The cross sdction for emissioii of a light quantum or anothm particle<br />
This cross section shows the characteristic €-* dapndmae on the neutron<br />
t<br />
ondrgg (tho so-called l/v law), If thhi? wicith bc is wry Xarg6, tho e'g-factor<br />
is the doterraining factor for tho energy doyndonce. This is the casu obgf,) in<br />
thu boron reaction:<br />
where thc total width is datmnjuld minly by th3 ~Z-entission width,<br />
Tho &-pBr-<br />
ticbs are mittod with an urisrgy of about 2.7 MeV, and am able to pass above<br />
the barrier, The corrwsponding width 72 is therefom vory large and was found<br />
to bt: sbout 200 Kay.<br />
Thus the boron raaction follows tho
of a light quantum, (as in triost Dams of he<br />
246<br />
nuclei) the capture OPOQ~ section<br />
C) Wdrimctnt-<br />
A number of resananc; phonomma haw bwn obsorvod with bombardribnts of<br />
oi,<br />
Staub and Tatc17) haw obasrvGd a strong resonance in th\: elastic<br />
.sf neutrons by helium, Thus rQmni&nce could be assign9d to a doubbt<br />
(54).<br />
itud by p-neutrons and was ana<br />
Hcrb and collaborators kn Wiscons<br />
ed by mdans of a formula sinxLlar to<br />
and Tuve et aJ9) havc found row-<br />
nanco~ by inasuring ' thQ emitted ?*r&y intunsZt7 as a function of thu enorgy of<br />
tho bomlwrdGd protons on Li, F, a. Tho? prays @re cithur crnittod by tho coni-<br />
pound nucl~~ il;scLf or by th\: oxcitud fSna.1 nuclwus which is loft over afta<br />
omission of an &-particlo (thu lattur pos$$blo occurs in F). The obsurve<br />
9 dray intensity shows sharp maxhti if the enorgy of thG bombarding protons is<br />
such that B love1 is a citod in thu COLipQUnd nucl@us. Ttiua tho resonances giva<br />
a picburu of tha syct/rurrri of<br />
leus abovc! the proton binding<br />
nd which, bQoawc! af ssldction rubs,<br />
emit GL-particlc:s are so broad<br />
8) Herb, Kurst, McKibbtm, Phys, itev<br />
&irnot, Hwb, Parkinson, Phys, 1%<br />
Plain, Hwb, Hudson, Ijlarro
.-<br />
247<br />
resonanw. The most striking rcsult of a comparhon batweon thc thrw o;tc.mcnts<br />
is the grcatdr leu4 density in the ha.vi+r nuclei. Tho avoraec love1 distanco<br />
is about 500 Kciv in Li, 200 Kcv in F, and 30 Kev in Al.<br />
thc levds arc? a sum of particle: and radiation widths.<br />
The obsurvad widths of<br />
Thuy ari: 11 Jim in Li,<br />
about 8 Kcv in F and of tho order of on3 or two K m in Al,<br />
Tho luwl density in A1 is so high that ono would not expoct to bo able<br />
to find resalvabld rJsonancas in dumcnts much havier than Al with chargGd<br />
partiall;: beam.<br />
Hdsonanws in huaviw nuchi w~ro found with slow neutrons only,<br />
The formula (56) for thu capture of slow nwtrons can thon be tcsted, and tho<br />
diffmant menitudes: ec +ec and C; can be dotm&wdr<br />
nQutron width C& is usually much smalldr than thd radiation width; b;r, thurc-<br />
ford put d"W3i.c<br />
and thG forriiula ($6) rGduces thcn to:<br />
In this caw, tho<br />
In thc swmd hcpression thc dnurgy f is exprdmUd in GV.<br />
c<br />
In this formula tho Dopplor cffdct has bdzn nugl0ctt.d. &causu of tho<br />
thmnal motion of thc: nucllzi, tho rtilative energy batwcwi neutron and nuclleus is<br />
not uxactly cqugl to thc! noutron cnwgy.<br />
around ths nuutron unargy E witih r?. Gaussian distribution: $,(/2x e*pE(E-!$v&@<br />
whum the Doppld width K is giwn by<br />
K=2(mekT/M) A<br />
The rolatiw wmgy E. is distributed<br />
and m/N is thu ratio of thu msscs af tht, nJutron and tho nucleusI<br />
Sinw (56a) givvs th3 dopondoncr: of d(e)on thu relative enurgy, thc3<br />
cross wction as a function of thct actual. c3nmgy 3 i s givdn by:<br />
c<br />
If thd Dopplm width K is Si&U corAipnrr;rd to thc: natural. line width 6<br />
Doppler 4fmt is nGgligiblt: and (568) can bu us& with as thf: actual nGwtron<br />
(57)<br />
the
c<br />
endrgyr If K i s Larp compared to 6 the ahapo of the absorption<br />
rely by th3 Dopplvr dffwt and WF: gat:<br />
24 8<br />
lino nuar<br />
Howover if the anorgy<br />
is sufficiently far away from rosonznce, G-G>> K<br />
c ,C End d of a reswmnce levul:<br />
1) Thd diruct wiwthod of mnquring bh+ absorption of, or bhG radioactivity<br />
0<br />
produced by, a mor?ochrorm,tic nautron beam with vnriablr: unurgy.<br />
fixpurhmts with<br />
rilonoenergljtic ndutrans hi;w been first performed by AlvarGz by mums of a i,iodu-<br />
lated nwtron SQU~C~ and by a suitably aodulatud amplifier, which registers only<br />
nvutrons which havti arrived at the counter jXr s1 spucificd thzu intvrval sftur<br />
uction,<br />
I3zche-r and Baker have jpiprovod this mthod and wcru abll; to<br />
thc nuutrons,<br />
rang@ of encrghs invdstigstod Pram 'two to ebout 5 cv.<br />
This rwthod gives $&,turi3<br />
diroctly as a functAon of thc anlilrgy e of<br />
Thc quzntity mcaaurdd in an absorption r-iuasuretmnt is actually tho<br />
SWJ of ccepturL. and the scathring cross suction,<br />
For low enorgius and n~ar<br />
2) T~L: boron absorption mthQd, Thu boron absorption 1;idLhod is used for<br />
dupendonct: of the cross soction of th\: (na$ proccss in baron, which is<br />
bld foY thu absorption of ndutrons boronu ThG PrinciPlQ of this<br />
cuthod consists of thd ~follov~ing: n ~,lc?,terial is r.ctivnt;rd by zbsorption of<br />
nc;utrons from p. bJ.m contcining a continuous spcrctrw of neutron ohcr@Gs.<br />
Thd<br />
of thd activation is r&nsurod afto covding the matd%d.<br />
with boron.<br />
ction is pToportionc1 to thc absorption In boron and thdr$:foru pro-<br />
to @;' if f?,c is thu wmgy of thu nwtrons at msonanca. As shbwn<br />
bufom,, JVUY imteriab CapturUs to i! curtain atmt very slow n3utm-M of thwwl
important, and tha intensity is given by:<br />
&lation (58) is! corrdct if C!&n$))W , which ;,i&ns that tho andrgy rc:gions<br />
sffi;ct\zd by thc: Dopplvr uffdot $.r$ coriiplcrtuly e.bsorwd.<br />
4) Thd activity mthodq Tho activity producud by a nuutron barn of<br />
given spoctrurn is riiaasurod, aftJr shiolding off tho 'thzrmal nuutrans.<br />
ndutron spoctrurii is usually obtainud by slowing dovfn high unorgy noutrons in<br />
light mc.terial as carbon or pp.raffin,<br />
thm proportional to<br />
y&<br />
4<br />
This Method givds c mI?surc,<br />
sy<br />
of<br />
de= mr*+~ AC<br />
inddpcndcint of thr; Dapplor df&%,<br />
Tho<br />
TI~G intmsity p~r' ondrgy intdrval de is<br />
5) Tho mcsur,JrtiGnt of thG thdmo.1 ?.bsorption cross scction. 'he thurml<br />
absorption ccn bd usGd for thc: dwtGrrlninntion of f+:vc?l constcat8 only if tho<br />
oncrgy of all highdr bwls is lnrg\: cofiipcrod to thu lowost onQ, and if no luvd<br />
*<br />
bdow zero unurgy is of cny influmct..<br />
Tht: lowst 3 ~ vnust d b6 ~ OQJ onough so<br />
that thu absorption at thmml cinurgids is taking placu in the I'wingN of the<br />
rus0nrZncc: ti&"<br />
In cava the one-levo1 formnula is applicable, WJ obtain:
261<br />
I<br />
-11)<br />
from niethod 2): ec .<br />
frois niuthod 3); Q"~ if ac >I> K or from thick Iqws,<br />
Thus thc! throd unknown qupntitius E. C I &, c cen bd dotwrmindd, JnconsiEstcs<br />
bwxusu of tht, contribution of othJr hvds.<br />
1Lthods 3) 2nd 4) may giw too<br />
largd vnlues bduause thay includc: thc: uffuct of higher 3.~vds4<br />
A tab3.u of thu charactGristic vc+lucis far 6011id lGw3.s of diffurcnt ulu-<br />
iwnts is givsn below*<br />
Only dmonts arc: includcd whurc: the ttviduncc soms to bu<br />
not too contradictory.<br />
Thu nuinburs in peronthcsis e.t tho r&.wncos<br />
indicek thu<br />
muthods usud, as list4 pruviously,<br />
C~SOS w h v tho spin of thu nvutron is addud to, or subtmctud f.rorsi, thu spin oft<br />
tho initial nucleus,<br />
Uopplor width in this cnso is dominmt,<br />
C<br />
Thu value of A in As cannot bw doturriinod bocausc thu<br />
The radiation width, rF which, in thwu cases, is practiocl2.y Gqual to<br />
C<br />
thQ total width A , is vury nd?.rly Qque.1 in all c8.swsc It sham uriusunlly<br />
little fluctuction.<br />
?%is r~letivt! constancy is connuctod with thG fkct thct<br />
rtzprdsants a sum of wqy partial widths corrcsponding to r-zdictive transitions<br />
tQ n gr6F.t nuGibor of lowor stctos.<br />
Tho fhctuntions of single trnnsition proba-<br />
*<br />
bilitias wc nv3ragJd Out, Thd valws of C show somewhat gruahr fluctuations,<br />
"h~y rc(prt.amt e singlir transition probability pad my dlspund mor0 critic,n.lly on<br />
thG propwtius of thu individud ldvobr<br />
Thu vnlu~s C' m-wIi to fulfill the, candition (36) fairly vrLLT,<br />
Tho vc?J.ul:
Bombarded Life time ~,Ccvl Ref . Tr Ref. Enx10 * CW* Ref.<br />
Element of product<br />
AS76 27 h 40 1(2,3 1 32 5 1I3)<br />
Aul'* 2.7 days<br />
d -<br />
669 (1937) '<br />
Phys. Rev. 53, 18 (19401<br />
Phys. Rev, 59; 154 (1.941)<br />
-<br />
* *
*<br />
2k (341<br />
255<br />
UCTWl38 8FiRI;E;S ON NUCLEAR PHYSICS<br />
ies: lzse Statistical Theory Lecturer: V. F. VJeFsskopf<br />
of Nuclear React ions<br />
L!ECTU.dE XXXVI:,<br />
Non Reaonance Processes<br />
AI<br />
If the energy of the incfdent particle is higher than a certain limit,<br />
resontanbes do no longer occur. This can<br />
anergy cannot be made mOnQChrOmatic enough to resolve the levels; it can be due<br />
to the Doppler effect; the D~Qpler width reaches, for example, 50 ev for an<br />
inddent particle energy of 250,000 ev, if the nucleus is 100 th3s heavier than<br />
41s; 50 ev is of the order of or<br />
sonames $or heavy nuclei. Furthermare, at energies of several Mav the<br />
actual width of the level reache8 the order of level distances, This can be seen<br />
in the following way: according to (17) the partial width for the endssSon af 8.<br />
fker leaving the nuoleus in a given state, reaches the value (2k+l)D/211;<br />
&dual nucLeus can be le&<br />
in several excited statas, the corresponding<br />
tance D,<br />
d up and gLve rise to a total width, equal or larger than the level dis-<br />
Thus,, even under ideal conditions of' a monochromatic beam and zero<br />
m, rescmances dbappear at atr energy of several &vts.<br />
In order to compute the cross qrsctions of non-resonance processes, we<br />
use Elohr's assumption 3 (see introductiofi). The CPQSS section c3n then be ex*<br />
* where<br />
pressed by (2)<br />
t~ (a,<br />
b) +2= ra *tlb (2)<br />
is the cross section of tha for,mtion of the compound nuO&eUS, and qb
is the relative probability of the emission of the particle b:<br />
256<br />
Here<br />
is the partial width for the emission of b irrespective of in what<br />
- -<br />
state the residua;l nucleus is left: r averaged over the levels<br />
Ak bB/t<br />
of the compound states excited by a, A- is the averaged total width.<br />
-<br />
The partial widths 7~ (we use again the abbreviated index&; see<br />
page '7, part XXXIII) are connected with the cross section Caof the inverse<br />
process: the formation of a compound nucleus by the particle a of angular<br />
momentum 1 and a nucleus in the state 05. This relation is given by the<br />
expression (16):<br />
The cross section Sa appearing in (2) is Gal with 6 being the ground<br />
state of the initial nucleus. Thus we get for (2):<br />
(2a)<br />
where the summation over 1 and 1' refers to the angular momentum index in the<br />
abbreviated indexa, or,@<br />
respectively.<br />
The consistency of the expressions (2), (3), (16), with the resonarlce<br />
formula (48) is shown by averaging (48) over an energy regionbe, which contain9<br />
many resonances.<br />
If the level distances are large compared to' the widths, this<br />
integration is a sum over the contributions of the energy regions around every<br />
single level.<br />
One then obtains:<br />
The sum is extended over all levels within the energy interval Ae .<br />
This relation is right if the term f(g) in (4.8) is sufficiently small compared to<br />
the resonance contributions.<br />
After introducing the average width 2; over neigh-<br />
boring levels, we obtain:<br />
-1<br />
where D= [c+ (E,)] . is the average level distance. After sumriiing over k! and 1'
*<br />
We<br />
e57<br />
and / j we obtain the sane expression as (2a). This shows that expression (2) for<br />
a non-resonance reaction is in agreement with the resonance formula in the region<br />
in which the latter one is valid (D )) )#<br />
now proceed to the quantitative discussion of the magnitudes occur-<br />
.<br />
ring in (2). We make use of the intimate connection between partial widths r&<br />
and cross sections which is given in (16a).<br />
In some energy regions the cross<br />
sections are better known, in others the widths; either magnitude can be used to<br />
determine the other,<br />
absorbed) particle,<br />
We distinguish three energy regions for the emitted (or<br />
The characteristic magnitude to distinguish the three regionp<br />
is Kawhich is the wave number of the particle near the nuclear surface and which<br />
is defined as follows:<br />
M a<br />
kt,<br />
2 2<br />
11, LOW energies: /Kat .C h, . This region includes all energies<br />
near to the one for which it just passes over the barrier or, if there is no bar-<br />
2<br />
rier, all energies smaller than% k:/zm . It includes also energies which are<br />
2<br />
not sufficient t~ pass over the barrier. In this case ff a is negative .<br />
2 2<br />
111. Penetration region: K, < - k, . This includes the cases of<br />
penetration of barriers with energies much lower than the barrier height.<br />
In the first region, claasical considerations are justified since the<br />
wave length of the particles is small compared to the nucleus and of the order or<br />
*<br />
Smallw of the distance between constituents. 1Ve may therefore assume that every<br />
particle that comes to the surface is absorbed.<br />
nearly equal to its maximum value and get:<br />
Hence we put 8&,,eaqual or
258<br />
The second expression follows from (16a). is a pure number smaller than<br />
unity which goes to unity for high energies and is introduced to take care of the<br />
transition regions,<br />
The second energy region is the same region in which condition (31) in<br />
Section XXXIV is valid.<br />
The value of fia can therefore be obtained from the<br />
expression (28) in a simple way.<br />
the form (32):<br />
It was shown there that (28) can be written in<br />
where C(E) is a function of the excitation energy E of the compound nucleus,<br />
which is independent of the nature (and angular momentum) of the emitted particle,<br />
In case of an s-neutron (1 = 0, T = 1) this expression should join (59a) at ka=k,<br />
and it is therefore tempting to put<br />
where<br />
cCFJ=t(D&t)(P1+1)(1 / k 0 )<br />
may be some function of E only. Since, in the extent of the Region 11,<br />
E does not vary by much, we may assume that the value of is approximately<br />
constant in this region and of the order unity as in Region I. We then get:<br />
Region SI (593)<br />
The second equation comes from (16a).<br />
In the third region (high barrier), the wave length x, will be always<br />
ao small that the potential energy outside the nucleus does not change appreciably<br />
over a distance %a. Thus the wmcalculation is . applicable and it yields
259<br />
Here Pa is the penetration of the barrier, which is, in WKB approximation, given<br />
Here rl and r2 are the radii between which the potential energy V is larger khan<br />
the total energy E.<br />
we obtain Pa= (ka/&)<br />
By comparison of this with the VJKB expression (29) for Ta<br />
Ta.<br />
Thus the regions I1 and I11 join smoothly.<br />
It is interesting to investigate the Region 11 by means of the VKB<br />
method of approximation.<br />
'Illis region is defined by the condition that the wave<br />
length ,&of tho particle outside of the nucleus is long cornpared to the distance<br />
between tho nuclear constituents.<br />
If we assume that 5 is still small enough to<br />
assume the potential fields outside the nuclear surface (centrifugal or Coulomb<br />
force) as slowly varying, it is allowed to apply tho TiKB approximation for the<br />
wave function from the outside up to the<br />
nuclear surface, There however, conditions<br />
change abruptly in distances small compared<br />
to 'f; . It seems to be appropriate to as-<br />
sume that, once a particle has panetrated<br />
this surface, it will not return without<br />
change in energy; in other words, it will<br />
It<br />
v<br />
have formed a compound nucleus,<br />
Thus the<br />
penetration process is equivalent to one<br />
dimensional problem with a potential energy<br />
of the type indicated in Fig. 2.<br />
Here a<br />
particle comes from the left side over a<br />
smooth potential barrier.<br />
At x=O, the<br />
potential abruptly drops to a value which<br />
would give to the particle a wave length%<br />
comparable to the wave length inside of the<br />
nucleus, which is of the order of the Fig. 2
260<br />
distance butween nuclear particles , (Vie call the corresponding wave number: EG)<br />
*<br />
This region oxtends to tho infinite, so that the particle does not return, If a<br />
wave of particles with an energy E (A particles per 'second) comes from the left,<br />
it is partially reflected (even if there is no barrier, because the potential<br />
energy changes abruptly at, x = 0).<br />
The current which is not reflected but pene-<br />
trates to the right (B<br />
particles per second) can be calculated easily by 'vbKI3<br />
where XI<br />
is the point at which the potcwtial energy V becomes larger than the<br />
total energy E of the particle, and Vo is the value of' V at x=O. B/A is tho<br />
ratio of the penetrating particles to thc incident ones,<br />
According to this ex-<br />
pression, the cross section for the formation of the compound nucleus is given by<br />
the maximum possible cross section, multiplied with B/A:<br />
sinc*$
Region I:<br />
The energy is more than 1 or 2 Mev above the barrier,<br />
261<br />
ier .<br />
Region 11:<br />
Region XII:<br />
The energy is less than 1 or 2 Mev above or below the bar-<br />
The energy is more than 1 or 2 Mev below the barrier.<br />
We now apply these considerations to the calculation of neutron cross<br />
sections, The total cross section cn for the formation of a compound nucleus by<br />
neutrons, is given by<br />
momentum 6 1<br />
Cn=<br />
where 6" is the contribution from the angular<br />
f 9, L-<br />
We first consider neutrons of low energy E {
(N<br />
Fi9.3 Neutron CI-GSS Section vs. Erlerjy
as functions of the energy e.<br />
k K<br />
B is given by 0 - 2 ( 21<br />
1 263<br />
The energy is given in units of (kR)&* The curve<br />
, This sum converges rapidly for the values<br />
L=O<br />
of E far which it is plotted so that the upper limit of the sum does not enter.<br />
B is proportional to C,V, and A is proportional to the cross section itself.<br />
It is seen, that, with the plausible value for koR ~ ' 3 , the cross saction(Jln goes<br />
smoothly over into its asymptotic value RR 2<br />
.<br />
For a heavy nucleus of Afi.200, the radius R is near to 9 x SO'13<br />
cm,<br />
so that (kR)2 is a measure of the energy in units of 250 Kev,<br />
Hence we must ex-<br />
pect cn to follow a l/v law for low energies up to about 130 Kev and to assume a<br />
value near flR2S2 3 x low% above that energy,<br />
The cross section 0"for charged particles can be computed as follows:<br />
it is given by (60) as long as their energy is appreciably lower than the barrier.<br />
The IVKB method can be applied to calculate the penetration factor, if the barrier<br />
extends over a region which is large enough, so that the potential energy does not<br />
change appreciably over one wavelungth of the particle,<br />
This is the case for<br />
higher nucloar charges (2 } 20).<br />
If the energy is well abovc the barrier, the<br />
expressions (59a) give the cross sections directly, since there is then no pene-<br />
tration factor to calculate,<br />
sions (59b) should be used.<br />
For energies near the barrier, however, the expres-<br />
The VfKB method in this region is no longer reliable,<br />
since the remaining barrier is necessarily very low and extends over a narrow<br />
region,' In the following comptttations, this region has been left out completely.<br />
The validity of the WKB solution, applied at low energies, has boon sxtonded up<br />
to energies equal to the barrier, where it was joined with the solution valid at<br />
energies much higher than the barrier.<br />
These two solutions join smoothly at that<br />
energy since the WKB expression for the penetrability P of the barrier becomes<br />
unity at the barrier.<br />
The justification df this procedure lies essuntially in the<br />
fact that, the barrier heights for different angular moments of the incident<br />
particle differ by amounts of the order of 1 Mev, so that, for a given energy of<br />
the incident particle, only one or two angular moments lead to a case of a Ftegion<br />
,
.:c) 264<br />
11, Most of the partial cro8s sections gare . thus computed correctly .<br />
The cross section for the ponetration through the barrier is a very<br />
sensitive function of the nuclear radius.<br />
The following tables give tho values<br />
for two sets of nuclear radii, one givon by d * 1.5 x A 113 , the other by R=L3<br />
x A l/3<br />
It seems that the second set with its steeper excitation functions, fits<br />
better to the erpx-imental material available.<br />
-<br />
the rb for dl particles b emitted by the compound nucleus.<br />
the sum f" =x<br />
I3<br />
The calculation of q of expression (2) involves the computation of<br />
A? kP1<br />
They are given by<br />
each term of which can be computed from (59a), (59b),<br />
(60) with reasmatle accuracy, apart from the factor 4 which does not differ much<br />
from unity.<br />
L.? thl final nilclew can be left in many oxcitdd statesfl, the sum<br />
can be exgresscd PS Pri inkegred and we obtain from (16a):
.4<br />
05<br />
06<br />
.7<br />
-8<br />
09<br />
1.0<br />
1.2<br />
124<br />
1.6 1;8<br />
2.0<br />
1<br />
40<br />
i Ca<br />
2s55<br />
6,37<br />
12.3<br />
18.3<br />
26.5<br />
35.1<br />
40,3<br />
50,o<br />
56,2<br />
60.4<br />
64,O<br />
66,2<br />
B 1 6.4'7 I 10.06 15.60<br />
120<br />
a<br />
0<br />
0<br />
,0239<br />
.<br />
298<br />
1.72<br />
6.13<br />
15-7<br />
27.6<br />
53.6<br />
71.7<br />
83.4<br />
93,o<br />
i02,<br />
8-72<br />
6.0<br />
7.1 74,6<br />
106 145<br />
I 18.731 9.70 116.24<br />
d<br />
r<br />
L<br />
LI<br />
.84E<br />
3.42<br />
8.15<br />
15.1<br />
23.2<br />
33.7<br />
4613<br />
54.8<br />
63.8<br />
70.2<br />
73.7<br />
78,3<br />
64<br />
Zn<br />
30<br />
8,28 x5&74 -<br />
I<br />
90<br />
Zr<br />
- - I<br />
TT<br />
Gc/ I P<br />
0<br />
0<br />
. 00468<br />
.099E<br />
. 871<br />
3.90<br />
11,2<br />
20,6<br />
44.7<br />
61.9<br />
71-4<br />
€33.3<br />
86.9<br />
14 2<br />
Ed<br />
60<br />
,00433<br />
32;o 7 i.85<br />
.02<br />
I1<br />
19<br />
12,67 I 121.45<br />
"--<br />
73c<br />
3,04<br />
8,23<br />
16.1<br />
26.7<br />
38,8<br />
49.3<br />
69. 9<br />
79.3<br />
86,9<br />
3.03.<br />
110<br />
7,17<br />
-<br />
----<br />
_13<br />
* 0499<br />
, 531<br />
2.88<br />
8.79<br />
17,3<br />
38, I<br />
88.8<br />
91-6<br />
.15<br />
.37<br />
.41<br />
50<br />
10- 98<br />
.04<br />
.14<br />
I<br />
26.5 44;6<br />
173<br />
v<br />
1 70"<br />
5L<br />
27,O<br />
79.4 67.5<br />
96.9 94.4<br />
I<br />
I<br />
4<br />
0<br />
0<br />
0<br />
0223<br />
,410<br />
2.87<br />
11.7<br />
I<br />
1)
266<br />
where the sua in the denominator is extended over all particles c wh$ch ccqn be<br />
emitted in this reaction. It is evident that the integrand in (64)<br />
TABLE: I1<br />
The cross sections 6 and Ca @re given for 8 typical elements as<br />
P<br />
function of the energy :<br />
cp 3 @ = n(x) '10-26 CWl*<br />
n is the number given in the tables, x is a masure of the energy in units of the<br />
barrier hekht B given in each colwnn: e== x.B . The columns headed by I corres-<br />
pond to a nuclear radius R = ro All3 with ro = 1.3 x lOe13 cm, the colwnns<br />
headed by XI, correspond to ro = 1.5 x<br />
cm,<br />
The cross section q(6~) for the penetration of the deuteron is equal<br />
to the one of the proton for en element with a charge s'mller by a factor (2) 1 413<br />
and with an energy smaller by (h) h<br />
is the probability of emission of b with an energy between<br />
and e +dk. It<br />
describes the energy distribution for the outgoing particle as a function of its
energy e . (This is identical with expression (4) ),<br />
267<br />
*<br />
It is interesting to discuss the connection between the distribution<br />
(67) and the Maxwell distribution of particles evaporating from a hot sphere, as<br />
indicated in the introduction.<br />
The distribution I(E, ) ch be written<br />
According to statistical mechanics, $(E) is the entropy of the residual nucleus<br />
with the energy E.<br />
By writing<br />
and by using the well-kncwmstatistical expression<br />
we obtain<br />
=i/T<br />
Mere T is the temperature at which, according to (67), the residual nucleus has<br />
the average energy E, -. If Cb is a slowly varying function of the energy,<br />
as it is in the case of the emission of neutrons, the emitted particles have a<br />
bxwellian energy distribution, just as particles would have, that are evaporated<br />
by a material with a temperature T.<br />
This toniperature is, however, not the<br />
ntemperature of the compound nucleus before emission11 , but the ftternprature of<br />
a nucleus after emissionfl. Here we understand by Ittemperature of a nucleus in a<br />
state of energy Elf the temperature at which it has an average excitation energy E.<br />
This comes from the fact that the Itevaporation” of one particle constitutes a<br />
relatively large loss of energy which reduces the temperature considerably.<br />
The<br />
*<br />
Maxwell distribution of the emitted particles is determined by the temperature<br />
-<br />
after the emission.<br />
If 6b is a strongly varying function of the energy, as it is in case of<br />
charged particles emitted, the Maxwell distribdtion is deformed by the factor a“b<br />
in (68). Since cT~ is usually a strongly increasing function for charged parti-
e<br />
268<br />
cles, the maxhwn of the distribution will be shifted to higher enqrgies..<br />
The limitations of tho validity of the Maxsv 11 distribution in ni clear<br />
emissions are obvious from the method of derivation: it is assumed (a) that the<br />
level density of the residual nucleus in the region of possible excitation is<br />
high enough to permit statistical treatment, (b) that e (( Eb m, which means<br />
that the energy of the emitted neutron is small compared to its mcurimum available<br />
energy,<br />
Since the level density of nuclei is unknown to a great extent, it is<br />
hard to predict the limits of validity in terns of Mevs,<br />
3t )<br />
In recent experiments<br />
it seems that the level<br />
-<br />
density of the lower lyhg levels is rather small, viz.,<br />
about one or two levels per MeV in the first or second Idlev of epitation energy.<br />
The maximum energy Eb p robably must be at luast 5 - 7 MeV in order to expect<br />
a shape similar to a Mawhian distribution for the emitted neutrons by elements<br />
in the middle or at the heavy end of the periodic system.<br />
The values of tho temperature T, which are expected in nuclear<br />
reactions, can be estimated in the following way:<br />
the average energy E of a<br />
system is a monotonically increasing function of the temperature T,<br />
If the temp-<br />
erature is very high, so that all degrees of freedom are excited, E is propor-<br />
tional to T,<br />
This is certainly not the case in a nucleus, when it is excited by<br />
a normal nuclear reaction, If one considers the nucleus very roughly as n gas of<br />
A particles concentrated in a voluma (G R3), one finds that this ttgasll is de-<br />
genersted if the excitation energy is of the order or less<br />
.3As Meve Heavy nuclei should therefore be considered as highly de-<br />
J<br />
generated tlgasoslt<br />
The stroilg interaction between the particle does not affect<br />
r)<br />
this cor,c!-usion which also holds for an electron gas in metals in spite of their<br />
--u_<br />
-L-.-,uru-I-uI-.c-<br />
WFlkins, T, R, and Kuerti, G., Phys, ibv, z, 1134 (1939).<br />
Wilkina, T. R., Phys, Rev..@, 365 (1941).<br />
Dicke, R, H. and Nhrshall, J., Phys. Rev. Q, 86 (1943).<br />
Dunlsp, H, F, and Little, R, N.,<br />
Phys. Rev, 60, 693 (194l)*
interackion,<br />
269<br />
The function E(T) ha8 ta vanishing derivative$ ( &E/ 4 T)T=o = 0 for<br />
c T= 0. (This means that the specifio heat is aero for T*O.aocording to the third<br />
law of thermodynamics,) The expansion of E(T) near T*O etar‘ts therefore with a<br />
quadratic term, If a system is highly degenerate, we may write<br />
E = bT2 (70 1<br />
and neglect the higher powers of T, which are small in this case, From %his we<br />
and finally for the level density: 7<br />
pE<br />
W(ES =C (711<br />
One can try to adjust the constants a and C, 80 that this equation<br />
reproduces our rather scant knowledge on level densities. A fair choice for A } 60<br />
is e*g,,<br />
cad = [;Z/(WOjJ (MeV)”’, Ceven = 9 Codd? a = 3 35 (A-40)* (MeV )”’<br />
if thB eXCitatiOn energies E are measured in Mev,<br />
Codd and Ceven refers to<br />
nuclei with odd’ or even A respectively.<br />
The expression (71) is adjusted to our knowledge of level densities for<br />
very low excitation energies and for excitations in the region of capture of slow<br />
1<br />
neutrons. In the latter case only levels, whose spin differs by kzfi<br />
1<br />
from<br />
the initial nucleus Eire observed. The level density in the expresslon (711,<br />
however, should include all levels which can emit or absorb the particles b with<br />
the energy Eb Our knowledge of level densities and of the distribution of<br />
angular momenta among the levels is insufficient to make any distinction of that<br />
kind at the present %be,<br />
From (70) an estimate can be given of the nuclear temperature which<br />
Ir<br />
appears In expression (69) and which determines the Maxwell distribution of *he<br />
outgoing neutrons. If the maaimum possible energy of the outgoing neutrons is .<br />
Eo, the temperatures are given in the following table in Mev’s according to
I-<br />
60 150<br />
-1<br />
5 1w-15 76<br />
10 1.63 1.07<br />
15<br />
2.00 le31<br />
I-<br />
230<br />
-<br />
66<br />
.93<br />
1.14<br />
These temperatures can be obtained by bombarding the<br />
nucleus with neutrons of an energy Eo, or by any process leading<br />
to the same excitation energy of the compound nucleus.<br />
270<br />
Expression<br />
(69) shows that the maximum intensity is emitted with an energy<br />
2kT, if'ris a slowly varying fuiiction of E<br />
to be in case of neutron ernissfon.<br />
:i/<br />
which serve to compute the factor 'Y\baccording to<br />
are shown in Fig. 4.<br />
107<br />
c-<br />
LO6<br />
102<br />
J Y. -/-<br />
as it is expected<br />
With the expresston (71) for the level density It is<br />
possible to calculate the functions f ( ebmax defined in (65)<br />
(65). Examples<br />
101 '<br />
1<br />
.rl)<br />
/I I I I<br />
0 2 4 6 8 10 12 14<br />
Fig.4. The functions fn,fp and fvare given as functions of energy<br />
for the compound nucloi Cu, Zr and Sn, These values must be multiplied<br />
by 2 or 0.5 if the residual nuclous is odd-bdd or even-even,<br />
.<br />
respectively. Expression ('71) for the level density Is used in the<br />
c ornput a ti on
271<br />
*<br />
The reaction types, discussed in this section, are ordered aocordtng<br />
to the nature of the incoming particle,<br />
Resonance reactions are excluded, since<br />
they were discussed in Section V,<br />
1. Neutron Reactions<br />
The cross section %(e) for the formation of a compound nucleus for<br />
energies e above the resonance region is given by (63a) and (63b), and is represented<br />
in Fig. 2,<br />
in ,n ) Reactions, This is not a real nuclear reaction. eince the bom-<br />
barded nucleus remains unchanged, It may however be left in an excited state<br />
after reemission of the neutron, in which oase we speak of inelastic scattering<br />
of neutrons by the nucleus,<br />
Once the compount nucleus is formed, every other<br />
particle except the neutron has to penetrate a potential barrier to get out.<br />
The<br />
neutron width, therefore, is much larger than any other particle width, and also<br />
larger than the radiation or fission width if E is sufficiently high. Under these<br />
conditions. 7n N t and the cross section for the (n,n) reaction is given by OB.,<br />
its elf.<br />
There is a difficulty in interpreting the observed (n,n) cross section<br />
by means of the expression (2), due to the presence of elastic scattering.<br />
This<br />
is the part of the n,n reaction in which the outcoming neutron has the same energy<br />
a8 the incornbg one.<br />
As mentioned in Section XXXI, page 8, part of the elastic<br />
scattering is due to effects in which the neutron is deflected by the nuclear<br />
potentials, and thus does not penetrate and form a compound nucleus.<br />
of the scattering is evidently not ineluded in the expression (2).<br />
This part<br />
The observed<br />
cross aection of an (n,n) reaction therefore contains a part which has nothing to<br />
do with the formation of a compound nucleus,<br />
The waves of the scattered neutrons<br />
due to external effects (in which no compound nucleus is formed) combine coheront-<br />
ly with the neutrons, emitted after the formation of the compound nucleus if they
are re-emitted by leaving the nucleus in its original state,<br />
2 72<br />
These two effects<br />
II,<br />
together give the elastic scattering.<br />
The observed cross section of an (n,n) reaction therefore contains a<br />
part which has nothing to do with the formation of a compound nucleus. The ob-<br />
served C(n,n) can be much larger than Cr,, especially at low energies, when the<br />
elastic scattering is a relatively large part of the total scattering. It is, of<br />
course, possible to separate tho elastic from the inelastic scattering by dis-<br />
kinguishing between the scattered neutrons of different energy.<br />
The cross section<br />
qn for inelastic scattering, which Is due to the neutrons which have left the<br />
nucleus in an excited state, is certainly lower than CJI,, because it does not<br />
contain the contribution from the elastic scattering, which corresponds to a real<br />
penek-sfion into the nucleus and a suhsequent re-erni$sion by the compound nucleus<br />
wtLh kko sarw emrgj as it came in.<br />
The pax% cf the elastic scattering coming from the ren1,formation of the<br />
csnpovnd ste
15 rn<br />
2 73<br />
There is only very<br />
little experimental material available on the spQctrum<br />
of the scattered neutrons by heavier elements, Dunlap and Little*) measured the<br />
spactrum of neutrons scattered by Pb with an initial energy of 2.5 MeV and found an<br />
inelastic scattering which indicates several excitation levels of Pb.<br />
The h.2n)-reaction occurs if the energy of the incident neutron is sufficiently<br />
high, the residual nucleus after the re-emission of the incident neutron,<br />
. may still be excited highly enough to emit a second neutron.<br />
This is possible only<br />
if en- Gib is larger than tho binding energy E, of a neutron to the bombarded<br />
nucleus where Gn is the energy of the incident neutron and et,, is the energy of<br />
the first emitted neutron. The probability of (n,2n) reaction is therefore given<br />
by:<br />
where I(€ )de<br />
given by (67).<br />
is the probability of emission of a neutron with the energy f. ,<br />
For energies en which make a (n,2n) reaction possible, the energy<br />
distribution can be well approximated by 8 Maxwell distribution (69) with a temperr<br />
ature T, with which one obtains (with. q, =c rCg2 .):<br />
If A<br />
AE-Gf) - B,<br />
-0 the energy surplus over tho threshold energy of the reaction I- is large<br />
compared to tho temperature T, the (n,2n) reaction should be the dominant reaction<br />
.<br />
and its cross section is then nearly equal to 6,. The (n,n) reaction cross section<br />
should become correspondingly smallerr This should occur for A,E 9 2<br />
or 3 Mev for<br />
e<br />
elements in the middle of the periodic systems. For very high primary enorgies -0<br />
e > 2B<br />
n<br />
(n,3n) reactions will occur.<br />
qunlap and 'Little, Phys. Rev.
2 74<br />
The fn.p) reactions naturally are loss probable than the (n,n) reactions,<br />
c<br />
because of the potential barrier which prevents the proton from leaving the<br />
nucleus as easily as a neutron.<br />
The cross section is given by:<br />
where the fts are given by (65). This cross section assumes measurable values<br />
only if the energy of the incident neutron is several Mev above the reaction<br />
threshold, beoause the outgoing proton needs that much energy to penetrate the<br />
barrier,<br />
The rather steep dependence of C(n,p) on the energy gives rise to an<br />
apparent threshold which is much higher than the actual threshold.<br />
If the nucleus which islcrsated by the (n,p) reaction is a negative elec-<br />
tron emitter, the final product of the,@-dewy is identical with tho initially<br />
bombardeq nucleus.<br />
In this case the reaction<br />
energy Q = Cp - en of tho (n,p)<br />
reaction can be calculated by the energy law.<br />
The energy balance in the nuclear<br />
react ion<br />
n+X=Y+p<br />
is given by<br />
En+ EX = Ey+ Ep<br />
where<br />
and Ey are the binding energies of the nuclei X and Y in their ground<br />
states, The ,&-decay has the following energy balance:<br />
%= %---mn+m c e-+ ht/<br />
P<br />
where my, and mp arc the mass energies of the proton and the neutron and<br />
Eo in-<br />
cludes the mass energy mc2 and the kinetic energy CGin of the electron emiteed;<br />
h U is. the energy of a 7 -ray which may accompany the ,@ =decay.<br />
Be thus get for<br />
the +value:<br />
Q =%- %=mn-mp- e-- hn/ = 0.7 Mev - hiV - G& (72 )<br />
In,&) reactions have an extremely small cross section in heavy nuclei .<br />
]I,<br />
because of the potential barrier, which makes it very improbable that the<br />
&-particle leaves the nucleus, If the neutron energias are extremely high,
is<br />
however, (nt&) reactions may occur in heavy nuclei.<br />
275<br />
There will be more chance<br />
to observe these reactions at very heavy elements, where the binding energy of %he<br />
f -particle is sqallcr ,<br />
The (n,’))<br />
reaction is usually called radiative neutron capture and was<br />
disausscd extensively in the r~~onanc~ region in Section XXXV.<br />
energy region the average value over many resonances is observed.<br />
a.-(n 99 = q (T!,<br />
In the higher<br />
The processes which compete in heavy elements with the radiation, are the re-<br />
emission of the neutron and, in some elements, the nuclear fission.<br />
the latter from these considerat5ons.<br />
of a neutron with an angular inomentum .,df~ is given by<br />
. .<br />
We exclude<br />
The cross section for the radiative capture<br />
r-<br />
c<br />
I<br />
is the average radiation width and -1 r,<br />
the neutron width for the compound<br />
states, which are excited by a neutron with an angular momentm . 1% . If the<br />
primary neutron energy is not too high (less than 1 Mev), the probability of in-<br />
elaotic scattering is relatively low, so that the re-omission of the neutron is<br />
R<br />
essentially tho process inverse to the capture”). stands than in the relation<br />
(16) to 5(”), and me get for the total capture cross section:<br />
This can be calculated with the help of (59b). For small energies (<br />
A!= 0 contributes and we obtain fkom (73) and (59b) by assuming<br />
>> rn<br />
of the incoming neutron,<br />
>> R) only<br />
(Ql<br />
a value of 3*corresponding to 2 MeV, one obtains<br />
-----.---<br />
$E the angular momentum of the bombarded nucleus is difforent from zero, the<br />
neutron can bo re-omitted with an angular momontum different from th<br />
angular monenbum of thE Cap%U&Qd one, ’ The psrfcty rule cal~~ows only ld’-/’I =<br />
2, 4 etc., and a closer investigation shows that the probability of /--/’#O is<br />
;1)1 rather small.<br />
(74 1<br />
With
cT"<br />
* if<br />
6 (n<br />
P) =t$"dz--) v<br />
io+ (&< r&<br />
E is measured in ev. is & magnitude which should be not far from unity:)<br />
An example of the energy dependence of cr(n,y) is given in Fig, 5, which<br />
276<br />
shows expression (73a) with the following choice of constants:<br />
R = 9 x<br />
k$= 3, D/2fl= 45F )<br />
high.<br />
278<br />
Then the advantage of the neutron emission, which comes from the absence<br />
of a barrier is made up by the high energy of the competing proton re-emission,<br />
e<br />
In this case qr, can be smaller than 1 and is determined by 'qn Z= fn/(fn+fp).<br />
The f's are functions of the maximum energy Emax of the respective particles as<br />
described previously. If E, m a f,,<br />
P<br />
particularly if the nuclear charge Z is not too high, (See Fig. 4.) A case of<br />
this sort is found in the Ni62 (p,n) Cu6' reaction which has a threshold of 4.7<br />
%- 1<br />
MeV ,<br />
In general, however, the cross section of a (p,n)-reaction<br />
should be<br />
very closely equal to 3 in Table I1 for different nuclei. Comparison with experiments")<br />
have shown that a nuclear radius R = ro A u3 with ro N 1.3 x lom8 cm<br />
fits better for elements in the middle of the periodic system between Ca and Ag,<br />
The energy distribution of the emitted neutrons, is similar to the<br />
distribution in an (n,n)-reaction.<br />
The highest possible energy occurs when the<br />
final nucleus is left in its ground state, and the emitted spectrum corresponds<br />
to the excitation spectrum of the final nucleus in the way that lower energies in<br />
the spectrum differ fram,the highest by an amount equal. to an excitation energy<br />
of the nucleus.<br />
For high initial. proton energies, this spectrum becomes s ~ l a r<br />
to a i'ulaxwell distribution.<br />
If the mrni;nwn energy E,<br />
of the neutrons of a (p,n) reaction is<br />
larger than the binding energy of a neutron to the final nucleus, the latter can<br />
be left in an excited state high enough to emit a second neutron,<br />
We then obtain<br />
a b,2n)-react=. The probability of this event is given by a srimilar expression<br />
to (71a); rn has to be replaced by WP. The energy distribution I(€) of<br />
the emitted neutrons can be represented here also by a illaxwell distribution and<br />
* 'TWaisskopf<br />
aid Euving, P~Ys. Bev. d, 472, (1940).
&e is the excess energy of the proton over the (p,2n) threshold.<br />
279<br />
The (pa?) reaction is the most important resction in case the proton<br />
energy is below the (p,n) threshold.<br />
The only competing process is then Che re-<br />
emission of the proton r(p,p) reaction] , which is not probable since the<br />
-<br />
proton has to penetrate the barrier again. The cross section is given by:<br />
n<br />
and is vanishingly small above the (p,n) threshold because fn rises rapidly and<br />
becomes larger than all other fts, No resonance effects should be expected in<br />
the (p,?)<br />
reactions since in heavy elements, in the resonance region, the energy<br />
of theof the proton is much too small to penetrate the barrier to an observable<br />
extent,<br />
The (p,p) reaction can be observed if the proton energy is not sufficient<br />
to emit a neutron with appreciable probability,<br />
Since the Butherford scattering<br />
by the Goulomb field is always present and usualiy much stronger than any nuclear<br />
reaction, a (p,p) reaction can only by observed either by registering scattered<br />
+t 1<br />
protons with less energy than the incoming ones or by finding the bombarded<br />
dH:)<br />
nucleus in an exLited state<br />
order to be observed.<br />
which must be a metastable (isomeric) state in<br />
The cross section of a (p,p) reaction is given by:<br />
fl- (p, p) = "r -*r+--$--li.K<br />
s<br />
It rises steeper than cp with increasing energy ep of the incoming proton, as<br />
long as fp < f3, since f<br />
P<br />
also is a strongly rising function of ep* qP,P)<br />
is then essentially proportional to the square of the penetration of the barrier.<br />
3. GC-Particle Reactions<br />
These reactions are very similar &o the proton reactions.<br />
The (CC,n)<br />
II)<br />
3t) R. Wilkins, Phys, Rev. 60, 365 (1941).<br />
R. Dicke and J,. IvIarshall, Phys. Eev, Q, 86 (1943).<br />
3Ht) S, Barnes, Phys, Rev, $6, 414 (1939).
B O<br />
reaction is the most probable one and its cross section is given by an expression<br />
like (75) with Crcr, instead of 5,<br />
threshold energy.<br />
after replacing<br />
(77).<br />
T~ is almost unity except below or near the<br />
The (G,2n) reaction is given by the same expression as (76)<br />
by Q, and so is the (a??) reaction cross section given by<br />
The latter one is unobssrvably small sime, below thd (cX,n) threshold,<br />
the a-particle hardly can ponetrate the barrier.<br />
The (CL$p) reaction has a cross section of the form<br />
- tions especially with low Z and E,.p > en -,<br />
It is hard to estimate the threshold energies for the different<br />
and is always smaller than the (CL,n) r&ction but can have observable cross sac-<br />
a-reactions, because of tho fact that the binding enerm of tP.e a-particle to<br />
the compound nucleus is not wall known, but is neaded to determine the excitation<br />
energy of the compound nucleus.<br />
The binding energy Badcan bo estimated as fol-<br />
lows:<br />
four nuclear particles are added, which have had in the a-particle a<br />
binding energy of 28 Mev.<br />
Thus the .binding energy of the =-particle<br />
should be<br />
BG= 4B - 28 MeV, where B is the average binding energy of the two protons and<br />
two neutrons added. B is not well enough known to make any guess about BG. Recent<br />
rssulis on (CL,Zn)<br />
have shown that, with 15 &lev&-particles,<br />
(G2n) reactions were not observable. This shows that, at least in the investi-<br />
Bnl + %2 - B -2 15 Mev, where B<br />
gated elements:<br />
are the binding energies<br />
"1,2<br />
of the first and second neutron of the (dX;,2n) reaction,<br />
4* Deuteron Reactions<br />
The d-reactions follow in general the same laws as the proton raactions.<br />
They'distinguish themselves from all other reactions because of the very high ex-<br />
* +b)<br />
citation of the cornpcmnd nucleus, The excitation energy is equal to the kinetic<br />
energy of the deuteron plus the binding energies of two particles reduced by the<br />
R. N, Smith, Dissertation, Purdue University, unpublished
0<br />
281<br />
relatively small internal binding energy of the deuteron of 2.15 Niv. It amounts<br />
to about 14-16 MeV plus the kinetic energy, This is one of the reasons why<br />
deuteron-reactions have a relatively large yield.<br />
The (d,n) and (d,2n) reactions obey the same laws as (75) and (76).<br />
bd is naturally smaller than 5 at ths same energy. The ~iiaximum energy €n -<br />
of the outgoing neutron is usually much higher than the deuteron energy because<br />
of the high excitation of the compound nucleus.<br />
Thus, there is no positive<br />
threshold for the (d,n) reaction, and Ykn is always nearly unity.<br />
The threshold of the (d,2n) reaction can be calculated from the<br />
.&'energy<br />
of the produced radioactive nucleus if it is a positron emitter, Ac-<br />
cording to the same principles as in (72), we find<br />
Here ED L 2.15<br />
(d,2n) threshold = €+ $. h t/ + m ,-'InpfED<br />
-. 4.<br />
= %n+ hu+2.85 ~ i v<br />
Ivkv is the internal binding energy of the deuteron.<br />
The (d,p) reaction should be much less probable than the (d,n) reaction.<br />
Actually, however, (d,p) reactions are observed with equal yield as (d,n)<br />
reactions.<br />
This has been explained by Oppenhcimer and Phillipz) and is due to<br />
tho following process: the distance between the proton and the neutron within<br />
the deuteron is relatively large, narnely -3.5 x cp.1 and its binding force<br />
60 of 2 bv is un*mxxily small,<br />
This is why the deuteron becomes strongly<br />
"polarizodlt when it approaches a nucleus; the proton goes to the farther end and<br />
.the neutron to the nearer end,<br />
If the neutron arrives at the nucloar surface,<br />
the proton is skill some distance away and has still a potential barrier to penu-<br />
%rate, The nuclear forces, which then act upon the neutron are much larger than<br />
ED, and the deuteron is likely to break up, when the proton still is relatively<br />
far away from the surface. After breaking up, the proton is only under the infbuence<br />
of the repulsive Coulomb force and leaves the nucleus. The actual
nuclear reaction is a capture of a neutron after the deuteron has been broken 282 up<br />
into its parts by the Coulomb field of the nuclear charge.<br />
The apparent effect,<br />
however, is a (d,p) reaction.<br />
The cross section of the Oppenheimer-Phillips process is much larger<br />
than 6d3 since the latter is given by the probability that the charge reaches<br />
the nuclear surface, whereas, in the Oppenkeimer-Phillips process, it reaches a<br />
point some distance outside of the surface.<br />
(d,/) ) processes do not occur with measurable yield since the compound<br />
nucleuB always is able to emit a neutron with considerable energy,<br />
5 Radiative Processes<br />
Nuclear rwctions can be initiated by the absorption of a ?-ray.<br />
It<br />
is evident that the energy ht., must be larger than the lowest binding energy of<br />
a particle.<br />
The compound state is, in this case, a highly excited state of the<br />
initial nucleus.<br />
The levels are then RO close that no resonance can be expected.<br />
The cross section for the cliffwent processes is given by<br />
o-(?,b) = (T+(fb/t<br />
a<br />
where b is the emitted particle and the sum is extended over a11 competitors.<br />
The most cofmon reaction of this type is the (T,n) reaction.<br />
art3 possible, but LL’F/<br />
pete with either FJ:’Ct:J’*&<br />
mcii venker.<br />
3: :i?utron emission.<br />
fa<br />
(9,~) reactions<br />
Any other particle WGU~! not be able to com-<br />
Exceptj-or.:: ta tAis are found in<br />
very heavy elements where photo-fission occurs.<br />
The excited state is then un-<br />
stable against sp3..tttting into two fragments<br />
The VL.?.W<br />
of G;, Is connected with the transitim probabilities between<br />
nuclear states, and before describing its properties, it is necessary to discuss<br />
other radiative nuclear processes.<br />
*<br />
Radiative trnnsitions between low-lying excited states have been ob-<br />
served in great number in the ? -ray emissions accompanyingB-decay andGdecay,<br />
In analyzing term systems, it was found out, that, transitions occur between
283<br />
states which differ by two units of angular momenta, Aj =. 2 and not only between<br />
I)<br />
states differing by one or no unit, as it is the case with atomic spoctra. It is<br />
known that the change of two units corresponds to a qundrupole radiation which<br />
2<br />
should be smaller in the ratio (R/%) compared to the dipole radiation emitted<br />
in the transitions A j=l,O.<br />
This ratio is about 1000 in tho observed region of<br />
hU-5 x 105 ev whereas the observed quadrupole transitions were about equally<br />
probable as the dipole trcnsitions,<br />
This has been explained by the fact that,<br />
dipole radiation only occurs if the center of charge is moving relative to the<br />
center of mass, whereas quadrupole and higher pole radiation is emitted also in<br />
motions where the two centers stay coincidente In a nucleus, the charge is nearly<br />
evenly distributed over the mass, since protons and neutrons are not distinguished<br />
in their nuclaar motion. Thus the center of mass and charge stay practically<br />
together and this strongly seduces the dipole radiation. According to<br />
the observations, thds reduction is strong enough to reduce it to tho strength<br />
of the quadrugole rp? ie t icn ,<br />
We themfoYY exile^::, that the absorption of 3/ -rays by nuclei, also<br />
follows the law nf quidrdy&3 3r dipole absorption,<br />
The ratio of the transition<br />
probabilities in qur-tGr.:po3.e and dipole radiation is pr o?ot*tJi.oml to 2 1 ', and we<br />
expect therefore ~.,zhl-y qmL*ip.de transitions for the a!uxption of T-rays of<br />
both types of radisiio:i. vjre cqually probable for lejs 3v.a 3<br />
bkv.<br />
Quadrupole rric'.itlt,ior, corresponds in the partick ;-ict,ure of light, to<br />
the emission or ~~JT',~:.LoI: ol' a photon with an orbital arigller momentum of one<br />
unit, whereas in tk- d;.r;~:~e ?%diation, the photon only posses3es an intrinsic<br />
tfsphll of one unit and no orbital momentum.<br />
sponds to the absorption of a p-garticle,<br />
Thus quadrupole absorption corre-<br />
The cross section Tv ,md its energy dependence can then bo estimated.<br />
We use the fact that tho wavelength 'x , even for energies as high as 30 k v is<br />
still larger than nuclear dimensions.<br />
The absorption probability should then be
284<br />
just proportional to the intensity E2 of tho electric field 6 at the nucleus.<br />
*<br />
(The same principle, namely that the probability should be proportional to the<br />
neutron density at tho nucleus, lt3&<br />
to the l/v law for neutron capture,)<br />
intensity for a photon is, firstly, proportional to tho frequency, since E2fi~ 11%'.<br />
Secondly, because of the fact that it is a "p-photon" with en angular momentum<br />
one, its intensity at thc nucleus is proportional to (R/?i2<br />
) (cf, page 7, Section<br />
XWrIV)<br />
This<br />
Thus the probo"bi1ity of absorption is proportional to U3. The cross<br />
section 67 is l/v times the probability which gives here, because of v = c:<br />
0;z = C (hGy<br />
.3k)<br />
where C is a constant<br />
e<br />
This relation is found to be well fulfilled according to (y,n) experi-<br />
ments made by Bethe 2nd @ntner'"*)<br />
who bombardad a number of elements withy-rays<br />
of 17 iv1ev aid of 12 lev.<br />
The constcnt C shows surprisingly little change ovor<br />
the psriodic system and is C = 0.65 x cm2/(Mev)3, This value can also be<br />
brought into connection with the observations on T-rays between low-lying ele-<br />
ments: by cpplying expression (161, the width for the emission of a ?-ray can<br />
D is the level distance in the region of the upper level.<br />
If this is applied to<br />
low-lying lovels of h t/ FJ 1 lvlev with a level distcnce of 0 ~ 0 . 5 hv we obtain<br />
rpw10-3 ev,<br />
This is in good agreement with the known transition probabilities<br />
among low-lying nuclear levels 4 ,<br />
The r?.dini;ion width<br />
.---<br />
++) Weisskopf, Phys. Rev, 3, 318, (1941).<br />
3Mt) Bethe and Gentner, Z. fur Phys. 112, 45, (1939).<br />
of levels of the compound nucleus created by<br />
) H. A, &the, Rav. Nod, Fhys. 2, 229 (1937), see also experiments bylialdmn,<br />
Phys. Rev.
285<br />
neutron cbsorption is a composite hzr-gnitude and canQot be compared directly with<br />
rv . rr represents the sum of all transitions from the level of the energy E<br />
to lower lying levels, whose number is very high.<br />
Formula (78) gives the radia-<br />
tion widtb for a transition from an excited state to the ground stat2 only.<br />
one applies (78) &so to transitions to excited states, the radiation width 'F<br />
If<br />
is givcn by:<br />
PE/*<br />
where r7 is the width 2s a function of the emitted rcdiation and Uis the level<br />
density and E tho excitation energy of the level, whose r; is calculated. If E<br />
is assumed 9 Mev and the expressions (71) are used for the level densities (A-1<br />
100) one obtajae from (79): f, *uO.3 Nev which is rathar close to the observed<br />
values in Table I, uut somewhat large,<br />
The applicetion of (78) to transitions to excited stF.tes is question-<br />
is slow~y *;nriable where D is the leva1 distance, if one varies either the upper<br />
level or the lower levo1 of the transition.<br />
In this case one would get instead<br />
whereW(0) is the level density necr the ground state, which is of the order of<br />
3.5<br />
One then gets for Em9 MEW:<br />
r;?= 2.0 x loa4 D' where D' is the<br />
level distance at the upper level, which is of the order of 40 ev. This gives<br />
definitely too srmll vclues.<br />
The truth lies probably soucwhere batween (79) and<br />
(80)<br />
The dependence of r;l on the excitation energy E is given by (80) as<br />
proportional to E6 and by (79) as very slowly indroasing function of E, Hence it<br />
*<br />
is very difficult to predict the dependence of rr on the excitation energy,<br />
Tho expected spectrum of the emitted radiation fron a state with the<br />
excitation energy E depends nlso on the assumptions made. The intensity I(z/)dv
* energies;<br />
286<br />
as a function of frequency is given by the integrand of (79) or (80). If (80)<br />
is valid, thc distribution is proportional tdd and has a maximum at high<br />
3<br />
the transitions to the lowest stntas nre the most probable ones. If<br />
(79) is valid, the intensity has a maximum soniewhere at intermediate oncrgies; it<br />
is proportional tov5 and also to thG level density at the end state, which de-<br />
creases sharply for the transitions of larger energyr
287<br />
February 15, 1944<br />
LECTURE SERIES ON NUCLEAR PHYSICS<br />
Sixth Series: Diffusion Theory Lecturer: R,F. Christy<br />
LECTURE 37:<br />
DIFFUSION EQUATION AND POINT SOURCE<br />
SOLUTIONS<br />
In the first lectures we will discuss the diffusion of<br />
neutrons which have come into energy equilibrium with their surroundings<br />
(thermal neutrons). Later we will discuss the slowing<br />
down and diffusion of fast neutrons and corrections to the diffusion<br />
theory of thermal neutrons,<br />
The Diffusion Equation<br />
The flux of thermal neutrons is --Dan where n is the<br />
thermal neutron density and D the diffusion constant,<br />
D can be<br />
obtained from simple kinetic theory arguments and equals xv/3<br />
where xis the mean free path and v the neutron velocity,<br />
"standard" velocity of themal neutrons 1s 2.2 h/sec.<br />
related to the total cross-sectiorrU-by<br />
where N is the nwnber of nuclei per cc,<br />
sec, =<br />
is<br />
The<br />
the relation 1: 1/N<<br />
Tho diffusion equation can then be written<br />
div (D grad n) 4- productfon/cc sec. - absorption/cc<br />
$n/ At<br />
The production/cc sec.<br />
position,<br />
is usually denoted by q, a function of<br />
If O-, is the absorption cross-section, the absorption/<br />
cc sec. is equal to Nd'nv; a or, writingh= 1/N Ta G the mean<br />
free path for absorption, the absorption/cc sec equals (v/A)n=dT<br />
where<br />
is the mean life of the neutrouis,
This 1ead.s t o the diffusion equation in the following<br />
288<br />
forms*<br />
If we consider the time independent diffusion equation<br />
- xv CJ2n<br />
3<br />
D# n - t q = o<br />
- A nfq = O<br />
2<br />
where L<br />
=AM3 = DT*<br />
Plane Source in an Infinite Medium<br />
L is called the diffusion length,<br />
Let the source be of strength Qn/cm2 and occupy tho<br />
plane 2; = 0, With q = 0 everywhere except 2; = 0, the equation<br />
I<br />
reads<br />
-L.- d2n n =<br />
dz 2<br />
'zz<br />
which has solutions e and Since the neutron density<br />
must approach zero for z =<br />
n=Ae<br />
fa the correct solution must be<br />
-..<br />
In order to relate A to Q, we note that the flow out from the<br />
--<br />
SOUPC8 per cm2 is<br />
2 xv<br />
r &.v A sQ<br />
3 g)z P O - 3 Ti;<br />
so A = SV<br />
and n = 3LQ 0 - 1<br />
ZG<br />
This equation ,can be used to determine L experimentally
289<br />
r)<br />
for such materials as paraffin or water whero LX3 cm is small<br />
compared to convcnient transverse dimensions of about 1 mater,<br />
A source or” thermal neutrons is not directly available but can be<br />
obtained as follows. A fast neutron source is placed below the<br />
plane z = 0 in some slowing medium. At Z = 0 a removable sheet<br />
of Cd (which selectively absorbs thermal neutrons) is placcd,<br />
Above z = 0, data on n is obtained for the Cd in place and with<br />
no Cd, The difference represents the effect of the Cd which is a<br />
thermal source or sink and follows the above equation.<br />
If the diffusing medium cannot be considered infinite in<br />
the transverse direction and if the source can be considered to<br />
have the transverse distribution cos *3c;/a cos Ty/a where a is<br />
the transverse dimension, then the solution is<br />
Substitution in the diffusion equation<br />
- d2n 2 2<br />
i---<br />
d n d n - n<br />
dx 2 2+---2 ;2 0<br />
dY dz<br />
leads to<br />
r e.-<br />
g+.-<br />
2<br />
a<br />
1<br />
- 1<br />
a b 7<br />
= o<br />
which determines b,<br />
This relation 9s frequently used for the<br />
determination of L in media whore L is relatively long, of ordor<br />
Point Source in an Infinite Medium<br />
Here we write tho diffusion equation with g = 0 in
290<br />
*<br />
spherical coordinatos, considering only spherically symmetric solu*<br />
tions.<br />
if u = rn then u satisfies the<br />
d2u -<br />
7<br />
2<br />
dr<br />
e qu a ti on<br />
= o<br />
The condition that n be finite at r -&gives<br />
and<br />
,=A 8 -r/L<br />
r<br />
The source strength<br />
Doprassion of Neutron Density Dear a Fofl<br />
d simple cxprcssion for the depression 6f the neutron<br />
density near a foil can be obtained only if we assume the foil to<br />
have thc form of a spherical shell, Let its thickness b? t, ab-<br />
sorption cross-section Fa, and its nuclear density be N.<br />
The diffusion equation is<br />
Let n = qT A<br />
I2<br />
- e? -r/L which satisfios the equation and has in<br />
r<br />
general the right form. The number of neutrons absorbed per<br />
second by a foil of radius ro is
291<br />
0<br />
The fractional depression in neutron density at the foil is<br />
i<br />
-<br />
A<br />
e<br />
-r/L<br />
r<br />
0 -<br />
1<br />
qT<br />
14- +Po+ VL<br />
3NCr t<br />
a<br />
which can be considerably larger than the fraction of neutrons<br />
absorbed in a single traversal of the foil which is N C t r<br />
a<br />
a<br />
r
292<br />
February 17, 1944<br />
In the idsf; I.eci;;;:e<br />
iwo considered the effect; of a detector,<br />
such as an indrivm .Y9il7 or? tho neutrcm density in a medium in which<br />
theilmal neut.ron..; we~cj c‘,irCi:!sing. It was shown how, as a first<br />
ap~rzximation, t h i‘a?.l ~ coiild. be replacod by a sphei’r? and fairly<br />
simple roslG.ts obt3t:md.<br />
ta the prohi em.<br />
Let us now consider a mo~e exact approach<br />
S.c,-gpose we ccjimider the element of area (dA) of the fail<br />
surface as a paint absaorber, If p is the coordinate of the<br />
element r3A and r ta the cccrdinate of the point at which we want to<br />
know the neutron densj.tyy, then the effect of a point sinh such RS dA<br />
is given by:<br />
To evaluate G<br />
- L<br />
Ce<br />
- Ir -‘TI<br />
we constdor the following:<br />
The number of netltrnns absorbed per second may be set equal to the<br />
number passing thrmgh the surface of a very small sphere surround-<br />
ing our point sinh, fiToa;r the number of neutrons absorbed per second<br />
equals Ntqnv6A wbers :<br />
N : number of atoms/cs in foil
c cfa<br />
t thickness of foil<br />
= absorption cross-section of foil material<br />
v~ : neutron velocity<br />
The number of neutrons passing into a very, very small sphere is<br />
given by:<br />
293<br />
Therefore :<br />
1_3 4ffXv C = -Nt%n<br />
3<br />
v dA<br />
The effect of the whole foil is then the sum of the effects of each<br />
element. On adding these effects and passing to the limit as the<br />
number of subdivisions is increased we get the total effect of the<br />
foil at any point r to be<br />
If we assume our source to be such that in the absence of the foil<br />
the neutron density in the region in which we are interested is<br />
everywhere constant and equal to one, then the neutron density at<br />
any point is merely given by the sum of the contributions of the<br />
foil and the source without the foil<br />
Thus<br />
*<br />
The more pFeclse approach to our problem therefore is seen to lead
2 94<br />
*<br />
to an integral equation, Since this equation is difficult, if not<br />
impossible, to solve exactly the usual procedure is to have recourse<br />
to approximating the foil by a sphere,<br />
Point Source in a Finitesphere<br />
As another example of the way in which the diffusian<br />
equation may be appliQd Let us consider the problem of a point source<br />
in a finite spherical medium. The general solution in such a case<br />
is:<br />
r r<br />
where A and B are arbitrary constants to be determined, B may be<br />
found ira terns rf A by ems of the relation; n(R) z 0 where R is<br />
the outpi’ ~ai!?-us of’ ’the sp1iere.<br />
4R/L<br />
R t’ -A e<br />
anl! the solution becomes:<br />
The cmstkirt A may then bs determined by .equating tha<br />
* As<br />
Up until now the solutions to the diffusion equation that<br />
we have found have involved only a few terms, In general such representations<br />
will not satisfy the boundary conditions, Instead we<br />
must have recourse to aore complex methods, (e.g. the use of Fourier<br />
Series). an illustration of suoh a problem we can take the case<br />
of a point source embedded in a rectangular medium of length a<br />
in
295<br />
0<br />
both x and y directions but infinite in extent in the and - 3<br />
directions, We take axes such that the source which is at the center<br />
of the medium is at the point (0, 0, 0). The general solution of the<br />
equation c7 2 n- 0 is:<br />
3<br />
To determine bmn we can use the fact that each term separately must<br />
satisfy the differential equation. Hence b, is given by the rela-<br />
tion:<br />
To find the coefficients A, we consider the fact that we have a<br />
point source,<br />
Such a smrce can be represented as 6 (x,y) where 6<br />
stands for the Dirac delta function, Were we are assuming a source<br />
of slow neutrons, though in practice a fast neutron source is used.<br />
Let us expand 6(xpy) in a Fourier series.<br />
We determine the Bmn’S as follows:
296<br />
e<br />
To relate the A's and Bls we must use the fact that the number of<br />
neutrons flowing out of the plane z = 0 per second in a glven mode<br />
must equal the number produced per second in that mode. The latter<br />
is equal to Bmn<br />
2xv JK,,, 2 xu m% nKy<br />
Number flowing out ==T A, COS 7 cos<br />
Hence our solution is:<br />
If the so'u:~ce stxngth were Q instead of 1 the above solution must<br />
be multfplied by a factor of Q. We note that B, decreases rapidly<br />
with increasing m and n and so at great distances the only significant<br />
term in our series will be the first. Thus at great distances<br />
from our source:<br />
This presents a very convenient method of measuring L since the<br />
decrease of n with z enables us to findbll and from bll we can<br />
readily compute L,<br />
solution of the Diffusion Equation in an Iqfinite Medium with a as a<br />
Function of Position.<br />
As another type problem suppose we consider the solution
*<br />
of the diffusion equation with a term due to the production of<br />
neutrons (9) by a rather special method. The equation involved is:<br />
297<br />
If g P 0, the solution is (assuming an infinite medium spherical<br />
symmetry)<br />
-'IL<br />
n=Ae where A<br />
r<br />
is some. constant.<br />
As shown in a previous lecture A can be related to the source<br />
strength Q by:<br />
A .P<br />
4.R Av<br />
Hence :<br />
- 3Q e<br />
-r/L<br />
n-- -<br />
r<br />
4T xv<br />
Suppose now q f (x, y, 2).<br />
We can solve the problem as follows: The solution for a1 point source<br />
has the form given above. Our f'unction q can then be treated as an<br />
infinite number of point sources continuously distributed throughout<br />
the medium. The total effect will then be the sum of the effects<br />
due to each point. If ;r is the point at which we want to how the<br />
neutpon density and rf the point where our source is located we<br />
have :<br />
c<br />
Sometimes the above integral can be readily evaluated and<br />
under such conditions represents a very great simpliEication of the<br />
problem. In utilizing this method it is very important to take<br />
advantage of whatever symmetry the problem offers. Thus if plane
298<br />
e<br />
symmetry were exhibited the appropriate plane solution of the diffusion<br />
equation, should be chosen. If cylindrical symmetry were the<br />
case it would be well to find the solution for such conditions and<br />
then apply the method here followed of integrating to gat the contributions<br />
of the many point sources making up the q function.<br />
A quantity of considerable utility in neutron diffusion is<br />
the albedo or reflection coefficient,<br />
If a finite medium with a<br />
point source were left in empty space, the neutrons reaching the<br />
bounding surface would diffuse out and be lost, Now, if another<br />
medium were surrounding the first one, some of the neutrons reaching<br />
the boundary would be reflected back in, thus increasing the neutron<br />
density in our original medium,<br />
The magnitude of the reflection<br />
effect is measured by the albedo? of the, in this case, outside<br />
medium.<br />
Actually the albedo of a medium will depend, not only on<br />
the medium, but also on the angular distribution of neutrons falling<br />
on the medium.<br />
Albedo of Various Media'<br />
\fire w ill neglect this effect.<br />
For this purpose we must consider the angular distribution<br />
of neutrons at a point. We let be the cosine of the angle<br />
between the velocity of the neutron and the radius vector and write<br />
I<br />
'<br />
nv(rp) = + LAW +p~(rg<br />
rl<br />
*
299<br />
\ ,and nv(r,p) = $A(r) -phA'(r)<br />
Tho total flow in the positive radial direction is called the flow<br />
out<br />
flow out = P nv(r,p),pdr = [Aw - ij x A ~ (rj<br />
The total flow in<br />
the negative &dial direction is called the flow in.<br />
')=flow<br />
in z<br />
Now the albedo of<br />
the inside surface R of the medium is<br />
1 -r/L<br />
We can take n = c. e<br />
r<br />
do - - dn z - 1 e<br />
dr r<br />
($ +<br />
The larger is R the better the albedo and for R<br />
the albedo<br />
approaches that of a plane<br />
2 1<br />
= I -5<br />
l + Z &<br />
3 L
0<br />
If in addition we make the medium non-absorbing, L =a,
in diffusion theory nl n2 at the boundary<br />
Ap dnl Ag<br />
and - ,-, =<br />
3 dr<br />
dn2<br />
3 dr<br />
Instead of one af these we can use the ratio<br />
- -<br />
at the boundary<br />
301<br />
NOW this ratio for a medium can be written in terns of the albedo so<br />
is another possible form of the boundary condition,<br />
Extrapolated End Point<br />
Let us examine the neutron densltg. in a plane semi-infinit6<br />
non-absorbing slab, "e will show that if the neutron density<br />
inside the slab is continued to the edge and beyond it would vanish<br />
at a distance<br />
3<br />
Xbeyond the ed&.<br />
n<br />
/<br />
/<br />
/<br />
/<br />
- /<br />
x-<br />
We have, as before
*<br />
302<br />
The solution deep in the slab w i l l be nv = A(x) = a(l 4- t) where L<br />
is the extrapolated end point. Shce A(x) = 0 for x = -L<br />
At(x) z $<br />
nv(x,r.) = 2<br />
The total flow in the +x direction is<br />
J<br />
whichmust vavlibh at x = 0 since there are no neutrons entering the<br />
slab.<br />
An exact solution of this problem leads to L z .7104x actually.<br />
Angular Distribution of neutrons Emerging from a Surface<br />
As above we may take nv(x) = a(l -k )<br />
I<br />
We may consider these neutrons uniformly distri-<br />
buted in angle and calculate the probability of<br />
escape from the surface at angle p (per unit<br />
range of<br />
nv (p)<br />
).“I 00<br />
= 1, kk+$ a) e-’pwx<br />
= i{y+gy2)<br />
So then the flux from the surface -cos 0 4 cos2 8<br />
z<br />
X
303<br />
Actual Distribution Near a Plane Boundary of a Non-absorbing Medium<br />
Actual angular distribution in n -p +@y* , 0 0 m ;<br />
Tho extrapolation of' the asynptotic solution vanishes at<br />
,7104Xfrom the surface,<br />
Very close (in a mean free path) to the<br />
surface the neutron density decreases and has a singulasitg in<br />
slope at the surface, The value of n at the surface is<br />
of tihe value of the straight line solution at the su~face.<br />
-<br />
.*n<br />
a
304<br />
February 24, 1944<br />
LECTURE SERIES ON NUCLEAR PHYSICS<br />
Sixth Series: Diffusion Theory Lecturer: R, F. Christy<br />
SECTURE 39: THE SLOWING DOWN 0.F NEUTRONS<br />
Fermi’s ARe Eauation<br />
The preceding lectures have all dealt with the diffusion<br />
of thermal neutronsn Lot us now consider how neutrons coming from<br />
a fast neutron soui“c”o :ce s!-med 5~vm- In the discussion the<br />
quantity q will kij *JG::I’~ cox~ve~;.1.eni:, :-I,<br />
1s defined as follows:<br />
It fs obvious that the -i;c-ta_? EL,.<br />
must equal the noo produced per second,<br />
of<br />
q we must have:<br />
o+’ neuti-cnr: c:rs.;sing energy E<br />
Hence from our definition<br />
\<br />
* the<br />
where Q is the total number of neutrons produced per second.<br />
The mechanism by which the neutrons are slowed down is<br />
that of collision with other particles. The amount of kinetic<br />
energy lost by a neutron in a given encounter may be computed by the<br />
methoas of classical mechanics as applied to elastic collisions. In<br />
general, if a neutron hits a particle of mass M the ratio of its<br />
opaiginal en@l”gy (B) to its final energy (E’) is a fWLction both of<br />
mass M and the angle between the orfginal path of the neutron
305<br />
1)<br />
and its path after the collision.<br />
angles we can conclude that<br />
By appropriate averaging over all<br />
where C is a constant which depends only on M.<br />
usually written as:<br />
This relation is<br />
where is the average of .<br />
It can be shown that:<br />
where the approximation is better for heavier nuclei (i.e. for<br />
larger MI* A few sample values are:<br />
for hydrogen: 5 P 1<br />
for deuterium: 5 = 0,725<br />
for carbon: 5 = 0.158<br />
for oxygen: 5 0 = 0.120<br />
Using the above we may find the mean number of collisions<br />
suffered by a neutron in slowing down, Suppose the neutrons start<br />
off with energies of 2 Mev and end up at thermal energies (about<br />
1/30 ev), Then the mean number of collisions:
*<br />
The<br />
treatment now to be presented for slowing down is<br />
somewhat limited in its application - even more so than is ordinary<br />
diffusion theory. In the latter case our only important approximation<br />
was that grad n be small, In slowing down theory it will<br />
be demanded that the number of collisions be large and also that the<br />
mean free path (A) not vary much between collisions, These<br />
assumptions are necessary if we are to be able to replace the essentially<br />
discontinuous process of energy loss by collision by a continuous<br />
process, (It may be noted that the most common slowing dom<br />
substances, such as paraffin and water, fail to satisfy the conditions<br />
very well and so any treatment of paraffin, water, and<br />
similar materials by the methods given here may result in considerable<br />
error.)<br />
306<br />
The starting point in treating slowing down phenomena is<br />
the so-called Fermi Age Equation. We shall. give here a non-rigorous<br />
.justification rather than a formal derivation of the equation. Our<br />
standard diffusion equation was:<br />
By analogy we write:<br />
*<br />
If we consider t in this last equation to be a variable depending<br />
on the past history of the neutrons we can relate this to the neu-<br />
tron energy. Thus: - AhE per collision h number of<br />
3<br />
dt<br />
collisions per second.
307<br />
*<br />
But Adn E per collision =k<br />
V<br />
and number of collisions per second =x<br />
Hence :<br />
Then our original equation becomes:<br />
We introduce the quantity 2: (the ttAgeft) by the ,relation:<br />
On substituting we get:<br />
(Fermi's Age Equation)<br />
We note that the tlAgefl has the dimensions of length squared. It<br />
plays very much the same role in slowing down as the quantity L<br />
2<br />
did<br />
in thermal neutron diffusion,<br />
is a function of the energy (E) to<br />
which the neutrons have been slowed and may be found from the<br />
*<br />
where Eois the energy with which the neutrons leave the aource.<br />
Point Source Solution<br />
As our first example of the use of the Age Equation let us
308<br />
*<br />
calculate the distribution of neutrons In space and energy coming<br />
from a point source. We shall consider an infinite medium with<br />
spherical symmetry. Our equations are:<br />
The equation involving the Laplacian reduces to:<br />
We shall solve this equation by the standard method of separation of<br />
variables :<br />
Let<br />
Then<br />
rq = f(r) 0 (TI<br />
s=fd2<br />
f d%/<br />
fl dT<br />
Since the left hand side is a function of r alone and the right of<br />
7 alone we must have:<br />
Thus we get the two equations:<br />
d'f<br />
ti> +-'k2f = o
309<br />
*<br />
The solution of (2) is: 121 = e -k2”C<br />
The solution of (I) Is:: f = A sin kr+ B cos kr<br />
t Solution is: q D, 5<br />
m e<br />
(A sin kr J- 3 cos kr) e<br />
r<br />
-k2y<br />
where A and B are arbitrary constants depending on k.<br />
stay flnite at r = 0, we must have B z 0.<br />
Since q must<br />
Hence:<br />
q z & sin kr e<br />
r<br />
-k2”r: ,<br />
Since we have no boundary conditions to be satisfied, any real value<br />
of k will satisfy our equations.<br />
The general solution, which is the<br />
sum of‘ all possible solutions, is then found by by multiplying the<br />
above expression by dk and integrating from - fl to -COO<br />
Using the fact that we have a point source we can determine the<br />
‘ function A(k). We know that;<br />
q( T = 0) =: c 6(r><br />
where C is a constant to be determined later whiczh depends only on<br />
the source strength &. b(r) is a delta function such that:<br />
b(r) =: 0, r # 0 and<br />
c<br />
30<br />
r2 h (r)dr t 1<br />
Substituting in the above we get:<br />
500<br />
r Ch(r) P A(k) sin kr dk<br />
-00<br />
As the integrand is symmetric (as will be seen from later results)<br />
written as:<br />
I
3 10<br />
rCb(r)<br />
a 2 5<br />
cx;,<br />
A(k)sin kr dk<br />
0<br />
Multiplying by<br />
- 1 yields<br />
Now the Fourier Sine Transform tells us that<br />
fla<br />
If f(x) =fi pI (u) sin xu du<br />
then: @(u) =fi 0<br />
\oof(x) sin ux dx<br />
30<br />
Applying this to our problem:<br />
i'<br />
By the definition of the delta function this becomes<br />
Hence<br />
dividing by<br />
r94"C;<br />
e<br />
y i e Ids'*
311<br />
To evaluate the first integral let<br />
Then;<br />
The first term on the right yd6lds zero while the second becomes:<br />
Similarly<br />
Thus :<br />
To find C we remember that:<br />
Lo1<br />
qdvol=Q<br />
Let u<br />
I-~/~~C' then:
*<br />
Hence :<br />
Qe<br />
3" (4mp<br />
That this result is what we should expect is easily seen from the<br />
3 12<br />
following argument, Since at r s 2fi q has reduced to l/e of its<br />
value at the origin, we see that<br />
is a measure of the width of<br />
curve representing q. If were very small our curve should be<br />
very narrow (a) .<br />
*<br />
This is ph..~s?c~l?~y plausible, since small<br />
in dicates fast<br />
neutrons and ther;rl we would expect to be grouped near the source.<br />
On the other hand large means slower neutrons. These<br />
would be expected to be more uniformly distributed than the faster<br />
ones, That Is just what our formula leads us to conclude.
I<br />
0 -1”<br />
Plane Source Representations,<br />
If instead of a point<br />
source the solution would be of<br />
+<br />
-z2/42<br />
q = Q e<br />
pflT<br />
source we have an infinite plane<br />
the form:<br />
Q = neuts/cm<br />
2<br />
sec from<br />
source<br />
Let us suppose now that we have a medium finite and of<br />
width (a) in both x and y directions and infinite in the - and - z<br />
directlons.<br />
e quat ion :<br />
Assume a point source at (0, 0, 0). A solution of the<br />
which satisfies the boundary conditions at x<br />
- a/3, y o L a/2 is:<br />
..-<br />
where A? and n are positive integers, C(k) is a constant depending<br />
on k, and
3 14<br />
p2 s (d2 +, m2j,, q2+ k2,<br />
2<br />
a<br />
(k is any real number)<br />
The general solution i s then found by adding up all possible solutions.<br />
This means swnming over m and A? -and integrating over k.<br />
Thus<br />
2<br />
C(k) >: A cos& cos& cos kz e **-P z,<br />
m,k 1,m a 8<br />
Applying the orthogonality principle, integrating, and using the<br />
fact that a point source may be represented by a delta function we<br />
gat:<br />
We note that for >) a (i,e, for the lower energy neutron) the<br />
higher harmonics die out. Hence, with this zipproximation our<br />
expression becomes:
315<br />
LA-24 (40)<br />
February 29th, 1944<br />
LECTmE SERIES ON NUCLEAR PHYSICS<br />
Sixth Series: Diffusion Theory<br />
Lecturer: R, F. Christy<br />
LECTURE Vz<br />
THE SLOWING DOWN OF NEUTRONS IN WAX%<br />
In this lecture we shall consider the f,ollowing question,<br />
What happens when a Ra-Be source of neutrons is placed in a large<br />
water tank? What is the distribution of neutrons in space and<br />
energy?<br />
Suppose we were to meazure the distribution by means of a<br />
cadmium covered indium foil. Such a counter records the number of<br />
neutrons of the indium resonance energy (slightly above 1 ev).<br />
Plotting<br />
fi(r2A) against r (where r is the distance from the<br />
source and A is the activity measured) we obtain results such as<br />
those in the diagram,<br />
.<br />
\ L.<br />
\<br />
\<br />
\ d-- Gauss i a n<br />
\
cally straight,<br />
We note that at large values of<br />
This implies that in that region:<br />
/<br />
316<br />
r the curve is practi-<br />
$I<br />
We also note that the curve of measured neutron density differs<br />
considePBbly at large r from what it would look like if it<br />
followed a Gaussian distrfbutlon (dotted line),<br />
The above distributlon of neutrons is readily recognized<br />
as being the same as that of neutrons from a point source which have<br />
not yet had any collisions. That this is so is seen from the fact<br />
that the probability that a neutron will go a distance<br />
a collision is<br />
r without<br />
2<br />
to<br />
e -vi and that we must have a factor Qf 1/4‘Kr<br />
satisfy geometrical considerations,<br />
Now it is easy to demonstrate using classfcal mechanics<br />
that in an elastic collision with a proton B neutron can never be<br />
scstt,c:wd at an angle greater than go0. Hence the average scattering<br />
angle will be between 0’ and 90’. live see, therefore, that on<br />
coLlic?ing with protons the directton of the neutrons is not much<br />
changed and so we would expect the neutron density to follow a law<br />
similar to the above (i.e, we expect A 0’ (l/r2)enr/LX where -)-L<br />
is actually somewhat greater than the true mean (free path), A<br />
further reason why we should expect such a distribution is seen when<br />
we remember that the scattering cross-section for hydrogen decreases<br />
rapidly with increasing neutron energy, Thus the mean frea path is<br />
largest at higher energies and so most of the distance traveled by<br />
the neutrons is covered in the first few collisions. But in the<br />
first few collisions the (l/r*)e-”klaw is followed, IVe conclude
317<br />
m<br />
from this argument also that the last is the law we should expect to<br />
find for the neutron density for large r.<br />
A concept which we shall find useful in considering our<br />
problem is that of the mean square distance r2 traveled by a<br />
neutron.<br />
This quantity is defined by:<br />
where f(r) is the kxnctional dependence 0.f the neutron density on<br />
the distance (r) from the source.<br />
__L<br />
As an example let us calculate r2 for the case where<br />
our neutron density follows the (l/r2>e-r’h<br />
law to the first<br />
collision and then follows an age diffusion law (i.e, proportional<br />
to e -r2/47=). Let us denote the first portion by the subscript (1)<br />
an?. the second by the subscript (2).<br />
Then:<br />
.r 2 =:
1<br />
3J8<br />
Substituting<br />
Hence :<br />
pre-age<br />
dif fusioq<br />
we<br />
diffusion<br />
In order to calaulate the distribution of ne<br />
k<br />
consider the following!<br />
From the poipt source the neutrons go to a ppi$t p<br />
following the (l/r?)ae-”.h<br />
I , ] ’ , 1,<br />
law. From rl they go a<br />
I<br />
2<br />
age dZTTusion (i,e, following e*r /4z). We can 4<br />
6<br />
pofnt rl as a source for the age diffusion which pr<br />
‘1 < ~<br />
neutron reaohes r TnCagrating per the whole volwe2<br />
I<br />
4<br />
contribution of egch su h source we get:<br />
I<br />
” k b i 1
319<br />
Substituting<br />
we get<br />
c03 B<br />
?-
320<br />
*<br />
This integration with respect to rl cannot, unfortunately,<br />
be done analytically,<br />
To get an approximate analytic solution we<br />
can do the following I Approximate the distribution of (l/r2}e*r/)L<br />
by (l/~>e-~/~. In this form we can do the integral. L should be<br />
2<br />
chosen so that the mean square distance traveled (T- ) comes out the<br />
2<br />
same. Originally Pr2 was 2x so we must have:<br />
We note paFt$cul&dj? hbw analogous L2 is to "r: in the slowing<br />
down equation. There we found<br />
-2 I<br />
3:<br />
carried quite far,<br />
slowing down equation by a diff'usion equation.<br />
- 6T, This analogy can be<br />
approximation is good only over a small energy range and for<br />
not too large compared to<br />
Under certain conditions we can even replace a<br />
However9 such an<br />
fiT.<br />
In our water tank problem it is fairly obvious that a<br />
certain percentage of the neutrons emitted by the source per second<br />
(Q) will escape from the tank without ever undergoing any collisions<br />
and so will not be adequately described in terms of a diffusion<br />
equation.<br />
To correct for this we can replace OUT<br />
strength Q by one of strength QJ- such that:<br />
source of<br />
r
*<br />
We can *en<br />
321<br />
find the distribution by assuming It to be of the form$!<br />
- 4 L -(2R - rP/L<br />
c - e<br />
4nrL2 e<br />
C is de‘term’ined fmm the ?elation:<br />
s”,<br />
2<br />
C - (,@*’/L c ,e-(2R - r)/l) 4”lr dr s Q‘<br />
4XrL2<br />
6till another, though mor8 foTrna1, way of treating the<br />
problem ‘3% as follows;<br />
First consider slowing down and then thermal<br />
dif ftlsi0t1.r<br />
The standard age equlati6n la:<br />
With spherical symmetry, this reduces tbO<br />
where R*= ra4ius of tank.<br />
The coefficients (Ah) can be determine& as follows:<br />
T= 0 we know the distribution is<br />
For<br />
Hence;
ll:<br />
..*L-<br />
--.<br />
322<br />
To find the number of thermal neutrons escaping from the<br />
tank we consider the ordinary diffusion equation and assume each<br />
mode, which we shall denote by the subscript n, satisfies the<br />
equation. Thus:<br />
As a solution assume:<br />
Substituting in our equation we get:<br />
Hence :<br />
Naw for the number of neutrons captured per second is:<br />
s<br />
’V f.<br />
-0ct vol= 1 .+ $pfl”Ln<br />
7<br />
B7j.l *hhg g d vol is equal to the number of thermal<br />
-. Svnl L3 Pa vos Sdvol<br />
I vol<br />
neutrons prDC!i(:ed ps second.<br />
just equal to the riumber produced minus the number absorbed is:<br />
Hence, the number escaping Which I S
323<br />
LA-24 (41)<br />
March 2, 1944<br />
LECTURE SERIES ON NUCLEAR PHYSICS<br />
Sixth Series: Diffusion Theory<br />
Lecturer: R, F. Christy<br />
_LECTURE VI: THE BOLTZMANN EXJUATJON:<br />
- CORRECTIONS TOXFFUSION THEORY<br />
Boltzmann Equation_<br />
In this lecture we shall consider some corrections to the<br />
elementary diffusion theory previously treated,<br />
More exact methods<br />
depend on either the Boltzmann equation or an integral equation,<br />
shall make our approach through the former.<br />
To derive the Boltzmann equation for diffusing neutrons,<br />
consider the following:<br />
Let f(x9 y, z9 vx, vy9 va> be the density<br />
of neutrons at the point x,y,z having velocity components between:<br />
vx and vx+ dv,, v Y and and vz and vd + dv,. It is<br />
readily seen that:<br />
c<br />
\<br />
&e 7.0 '3 ity<br />
f d vel<br />
ordinary neutron denkllty<br />
We<br />
Now the tot,3:L rate o:? change of<br />
f with time, which we denote by
324<br />
*<br />
But this total rate of change must be equal to the number of neutrons<br />
entering the given velocity range per second minus the number lost<br />
by collisions, Let 4T= ea+ c$where sc$ is the absorption cross<br />
section per unit volume and cK3 is the scattering cross section. In<br />
terms of these symbols the number of neutrons lost per second is:<br />
vof, The number entering the velocity range is found by averaging<br />
vo f over all angles. Thus,<br />
where<br />
due element of solid angler<br />
When the above are substituted, the Boltzmann equation becomes:<br />
where w0 substitute<br />
*<br />
In passing it is interesting to observe that this<br />
equation is considerably simpler than the ordinary Boltzmann<br />
equation of kinetic theory. This is due to our neglecting collisions<br />
between neutrons and the fact that we have considered the<br />
objects hit by the neutrons to remain stationary. The great simplification<br />
is that the ordinary Bol'czmann equation Is non-linear<br />
while our above derived equation is linear and so is considerably<br />
more tractable<br />
Let us consider the Boltzmann equation in the Steady
325<br />
e<br />
state, Then it reduces to:<br />
7 * ef =-vrf<br />
- Assume further that we have an infinite medium, that f e kx and is<br />
independent of y and z. (This is one of the few instances in<br />
which the Boltzmann equation may be solved.)<br />
Our equation becomes:<br />
vx kf = -vsf -1- VGT<br />
Averaging over all angles and substituting .s g vx/v z Goa 6, we have:<br />
Hence<br />
This equation serves to determine p(<br />
The first approximation for<br />
small x/r and<br />
diffusion theory result L2<br />
Tdrgives K2 = 35% which is the same as the<br />
l/3%~ , However, If is not much<br />
less than 05 the BoZtzmann equation shows that we must correct our<br />
value for 6/<br />
A rather good second order approximation is:<br />
2<br />
e Hence the value of L<br />
that should be used for more accurate work is:
326<br />
3 i-7n 2h<br />
x*<br />
Another important instance in which we can correct<br />
ordinary diffusion theory through use of the Boltzmann equation is<br />
in the matter of neutron flux.<br />
Until now we have used -( )~v/3)Vn<br />
for the flux (io@. we have said the diffusion constant D, was<br />
+Xv/3).<br />
A more accurate D can be obtained as follows:<br />
Let us find<br />
7<br />
vxfo This is:<br />
From this we see that a more accurate diffusion constant is:<br />
Thus, while<br />
the exact relat2on is:<br />
-<br />
In all, the above work when averaging over angles to find<br />
9 we tacftly assumed to be independent of the angle of scatter-<br />
,-I<br />
* -
~<br />
327<br />
-41<br />
tfig (f.e. Isotropic scattering). If le aid floe have sphsyidal<br />
symmetry we could have expanded SS in a power series in cos 8,<br />
Thus<br />
The effect would be to give us a dlighkly different value for ? and<br />
would serve to complicate the problem.<br />
Assuming such an expansion neatjrd#ktr& i.f; hbrns ofit that<br />
still an additional correction for & is necessqryi de get<br />
I /<br />
Let US Introduce a quptity called the trandbdvt 6Ybrtd lagCkib$S<br />
defining it by the relation:<br />
o&, .= b(F-cos e)<br />
where a is the average of cos 0 over all collisions.<br />
*ly we obtain the equation:<br />
Then for<br />
where M<br />
is the mass of the scattering nucleus, Then<br />
We see that for light elements, such as hydrogen, the correction is<br />
quite important, On the ather hand, for heavy elements the correction<br />
is negligible.<br />
StlZl another way in which the Boltzmann equation can be
328<br />
4-<br />
of use to us is in helping u8 to derive the Age equation, We shall<br />
not carry this derivation through here, but will merely indicate the<br />
method to be followed,<br />
First assume a certain energy change per scattering.<br />
follows that the number of neutrons entering the given velocity<br />
It<br />
range is -f(vl), where vr represents the neutron velocity at a<br />
somewhat higher energy level. Then we can represent f(v’ by a<br />
Taylor series and break off the series after the first two terms.<br />
Thus:<br />
f(v’) =f(v) +(v’--v) $$<br />
This is then substituted in the Boltzmam equation I<br />
and the same<br />
general procedure is followed as previously, We obtain a relation<br />
between and a’I;/dv The quantity ‘T is then related to V2q and<br />
and 8/Av to &q/&?;<br />
Application to Water:<br />
BfndinP: Corrections:<br />
To illustrate the application of the formulas we have<br />
developed, let us consider the apparent; paradox that arises in com-<br />
puting the diffusion length for water.<br />
Since the cross sections<br />
for hydrogen are much larger than those for oxygen, we need only<br />
take hydrogen cross sections into account. for hydrogen when<br />
measured turns out to depend on V. At thermal velocities it is<br />
about 40 barns. Hence : 40/3. for hydrogen is 0.33. When<br />
*<br />
we substitute in our Formulas for<br />
differs greatly from the value L2<br />
L2 for water we get a value that<br />
2<br />
8.3 cm which has been measured.<br />
Here it seems that our formulas are wrong and so must be thrown<br />
away.<br />
Fairly good agreement, though, can be reached by using the
329<br />
following argument:<br />
I<br />
In the calculations it was assumed that the mass of the<br />
hydrogen atom is 1. Actually it is effectively greater than this,<br />
since the hydrogens are bound to a greater or lesser extent to the<br />
heavy oxygen atoms. Actually it is found that for'snergies greater<br />
than 1 ev the hydrogen atom is practically free and so has an effective<br />
mass of about 1, For energies less than this, the effective<br />
mass is somewhat more than 1,<br />
For epithermal energies (Le. energies slightly over<br />
thermal) Cr, for hydrogen is approximately 20,. For hydrogen eom-<br />
pletely bound (this corresponds to 0 energy) theory leads us to<br />
2<br />
g 80, For thermal energies we should<br />
expect CS to be 20 (w)<br />
then expect fl5 to be somewhere between 20 and 80, The observed<br />
cross section is the geometric meanp 40. It corresponds to a<br />
hydrogen binding that is about halfway complete.<br />
By analogy we expact the 1 - cos 0<br />
term also to be same-<br />
where between its two extrerrleei of l/3 and I, If we again take the<br />
Using this value we get rather good agreement in the measured and<br />
computed values for I,'.