Untitled

L,A 1 REPORT 24
LECTURE SERTEG ON NUCBAR PHYSICS
First Serie$: TERkINOLOGY; Lecturer: 'EIMII MCMXLLAgb
LECTURE I: The Atom ahd the NuBkeias
" 11: htehtidn of LIght with Matter
It %If! Forces Holding the Nuclei Together
{' IV: Reactions Betweeh Nuclei
V: RadioactJvity
VI: I) 6 -Decay; 11) Neutron Reactions
t' VII: N~~cI.ear Fission
Bhge 1
1
8
17
24
'
I,;;;
t~.%
32a
g
42
Second Series : RADIOACTIVITY; Lecturers : E. ,Seere
E, Teller
F. Bloch
LECTURE VIII:
IX:
X:
The Radioactive Decay Law
48
48
I, The Radioactive Decay LLPN (Cont'd).
11. Pluctuations. 111. Passflpe Through
Matter 55
-Particles and Their Interactions
with Matter 62
XI:
XII:
XI'II:
XIV:
xv:
XVI :
XVI 1 :
XVIII:
The Theory of Stopping Power 66
The Theory of Stopping Power (Cont'd) 72
Interaction of Charged Particles with
Matter 80
Interaction of Radiations with Matter 83
The Properties of Nuclei
Survey of Stable Kucleus 92
Theory of Alpha Disintegration. Theory
of Beta Particles . 100
I. Positron Emission and K capture.
11. Gamma Radiation 105
a7
Copyrighted

.
~ORRECTIONS
, pI 241 - Equation 48 should read
, ..,
p. 254 - Missing only because of an error in numbering and not
because of censorship.
i?-k
'2%
p. 276 - Equationiat top of page should read
. .

Third Series! NEUTRON P€lS?SICS; Lecturer: J.H. Williams; Page 108
LECTURE XIX: Discovery of the Neutron
108
II
It
XX: Neutron Sources (cont'd)
Typical Reacttons f& Production of ~OUt
irons
114
XXI2 Properties of Slow Neutrons 119
'' XXII: Properties of Slow Neutrons (Cont'd) 125
XXIII:
XXIV:
'Fast Neutrons
Properties of Fast Nqutrons
C r 0 s s Se c ti on Mea suremen t s
135
146
Fourth Series: TWO BODY PRORLEN; Lecturer: C,L.Critchfield:
Page 153
LWTURn XXV:
Nuclear Constants
" XXVI! Neutron-Proton Interaction
153
162
I' XXVII: I) Spin-Spin Forces in the Deuteron
TI) Proton-Proton Scattering 168
'I
XXVIII: Three Body Problems
178
I' XXIX and XXX: Interaction of Neutron and Proton
wfth Fields
185
Fifth Series:
THE STATISTICAL TEEORY OF NUCLEAR REACTIONS;
Lecturor: V, F, 4;Weisskopf Page 207
LECTURE XXXI: Introduction 207
'I XXXII: Qualitative Treatment 210
" XXXIII: Cross Sections and Emission Probabilities 218
" XXXIV: The Excited Nucleus 227
?I
XXXV: Resonance Processes
237
'' XXXVI! Non-Resonance Processes
255
Ju
LECTURE XXXVII:
Diffusion Equations and Point Source
Solutions 287
XXXVIIIt Point Source Problems and Albedo 292

LECTURE XXXIX! The Slowing Down of Neutrons Page 304
11 XL: The Slowing Down of Neutrons in
Water 315
2 XLI: Tho Boltzrnann Equation: Carrections
to Diffusion Theory 323

LA 24 (1)
Sentembar 14, 1943.
1
LECTURE SERIES OH NUCLEAR F'HYSICS
FXRST SERIES: TEFblIINOLOOY LECTURER: E, M. MCMILLN .
LECTURE I:
We consider a piece of matter, for instance copper,
If we magntfy it, we first sea that it consists of crystallites.
They in turn are regularly arrmged arrays of 82;oms. In the
special. case of copper, the eeometrical arrangement i s the same
as a close packed arrangement of spheres, the atoms in one plane
making the following pattern:
I
At the center of each atom there fa a nuoleus, Of coursel the
picture is not really like the ne drawn., 'The elsotrons fill
the whole space, there are no botmdariea, the atoms actually
overlap, thelr limits are fuzzy. In a metal like copper it is
not even clear which electron belongs to which atom, In an insulator,
f'or instance the noble gas Argon, each atam holds on to
its own electrons and each electron belongs to 8 definite atom.
.-
In a chemio-al compound, f'or instance carbon dioxide, tfhe e 8
aped by the atams belonging to one molecule, but eaahmoleas
its own electronsr All the chemical properties of' the
are determined by khe outer electrons,
The question arises what arfect the nucleus has on the
behavior af the atom, The effect af the nucleus is due to its
This charge is an integral multiple of the

positive charge which is of the same magnitude as the charge of
2
the electron,
Atoms must be neutral because a positive charge
would result in the attraction of more electrons, whereas a
negative charga would cause the atws to lose electrons. Each
atom has enough (negatively charged) ezectrons to c&npensate the
nuclear charge.
This number of electrons 2s aZso called the atan-
ic number and dot;eminas the chemical properties of tho atom.
Another significant property of the atomic nucleus is
the nucleus mass,
This mass is much greater than the mass Of the
electrons, Even ths lightest nucleus, namely of hydrogen, is 1840
times heavier than that of an electron,
Thus the mans of matter
is essentially conoentrated in the nuclei,
The mass of the
nucleus depends in first approximation on the number of neutrons
and protons within the nucleus,
The neutrons and protons have
approximately equal mass hut this equality does not hold as
exactly as tho equality of positive and negative charges within
an atm. The protons carry one unit of positive charge, The
neutrons are uncharged,
Tho charge of tho nucleus, that is the
atomfc number, is equal to the numb@r of protons Sn ths nucleus,
The total. weipht of the nucleus on tho other hand, is roughly
eW&l to the weight of a proton (or a neutron) multiplied by the
totah nwnbor OS neutrons and protons within the nuclcus, This
number is callcd the mags number,
'Jhus if two nuolei have thcj
same charge, it docs not moan that they haw the same mass number
because a frivcn nwnbep of protons might be associatcd with a dif-
$'@rent nwnber of noutrons in different nucl-ei,
If two nuclei have
tho same nwnbor of protons thay will belong to the same chemical

dlf'fcront nwnbor of neutrons they will bo different isotopcs,
For instance hydrogen has different ktnds af isotapes,' &very
hydpo$on nucleus has one proton but it may havo 0, 1 or 2 noutr~ns.
Therc cx$s.t: wany more nuclaar spcclie$ than atomic
ordor to designata the nucloar spsoiea WQ writa the atomic symbol,
for instanco H in thc cas0 of hydrogon,
This alroady implias
what tho nuclear charpa. is and thercforo shows how many protons
aro found in thc nucleus,
by an uppor index,
Tho mags nwnbor of tho nuolcvs is shown
The following tablc pivcs the numbor of n6uc
trons and prOtons Tor tho known hydrogcn and helium isotopes:
numbar of protons numbcr of new-
3. trons
B k 0
H2 E 1
H3 1 2
3
B
r,
2 4
Thc? approximate S%Z?- of anatqm is IOe8 cm,
DEfferent
abomic spocics havo dlfforent size but the total variation in size
is not mare than a factor of ton times.
One may visuallza the
size of' an atom by expanding our scale of rncasuromcnts in such a
way that 1 om, will appear 8s ,000 milos, Then an atom will bc
about 1 cm, in radiuls.
Tho 8Sze of the atomic nuclous is about
crr~.~ that is, about lo5 times smaller than the atom,
distancc between the sun a.nd earth is about 100 timas the radius
of the sun,
paint of viow of
tho earth,
The
Thus the yluclous is relatively much smaller from the
its outer electrons than thE: sun as viowed from

4
It was stated that tho lightcst nucleus is about 2,000
timos haavior than an olsctron. /Ire know that in 1 gm, of hydropcn
thoro arc 6 x atoms, thoroforo tho mass of one hydrogsn
nucleus is J1/6 x
grms,
Bofore turning. to tho dimension of the cbctron, we
shall d$.scuss the peculiar dirficulty whfch arises in connection
with visualizing tho electronis within the atoms, It is ofton
stated that an atom is likc a solar system. ThG atom of' iron for
Instance has 26 alcctpons. Can we rnOasuro these as in motion in
approximatcly clliptical orbits around thc nuclcus? If we should
take a microtjcopc of vcry high powou, and rcsolution and take a
photograph of thc inside of an atom at a ,r;rivcn instance, me would
rind a disorderly arranprnwnt of elcotrons within tho atom. One
might then try to tako another photograph a vemj short time later
in order to find thc motion of' thc eloctrons within thcir orbits.
If such a photograph is taken no relation is discovercd botwocn
the positions in tho two consccutive pictures, hc tually if tho
resolving powcr OS tho microscopo would have bccn infinity, the
second picture would have boon 'blank, Thc reason fop this situstion
is that the light which ia uacd in taking tho pictures carries
momcntum and one cannot thcrofore takc a photograph without
,giving a kick to the clcctrons, If good resolving power is
wantod, you must uac light of vary short wavo longth and such
lipht carpios particularly hiph rnomsntwn, Gnc might try to '
cornpromiso in following tho orbit of tho okctrons by using a
lowcr rcsalving power and haping that in this way thc: vcloc.ity
of the clcctrons will not bo too much disturbed in taking a

5
J)
0
photograph but the quant 1.t 8 t ivc e itua t ion is such that tho
nccossary unc c r t ain t ie s in position and momentum aro just
to mako it impossible to definc orbits within atoms,
This conclqsion is a consequence of quantum mechanics,
Quantum mochanics is thc mechanics that appliss to all bodies,
for instance to a tennis ball, but in the case of a large body
like a tennis ball tho laws of quantum mochanics approach the laws
of classical: mechanics so closoly that the two bscome practically
undisbinguishablc, Thc word quantum means amount and bccum in
the designation quantum moohanics bocauso many of thc characteristic
laws of quantum mccharzios am as sociatcd with the fact that
ccrta$n quantitioa appear in amounts that arc multiples of a given
cbaracteristlc amwnt;, Tho most basic of these quantitlcs is the
s o-cal Led "quantum of action'' which is dosienatod This
quantity has tho dimansion of a dlstancc times tho momontm,
h Is the measure a$ the limitation on moasuremont, If the position
o$ a particlc (4.e. its distancc from a rcfcrcncs system)
is well knawn, then little can ba known about its momentum and
vice vcrsa, If tho momcntum is known tho position must bo unknown.
The product of the uncortaintics in momonturn and positian is of
the magnitude h, For a big mass this uncertainty docs not moan
much, This 18 so bccause thb momcntum is a product of mass and
velocity and if tho mass is big, the uncm-tainty in tho velocity
bccomos correspondingly small, but for tho lightest partlcl'os, the
electrons, $he uncortainty of momentum causcs such an uncartainty
in velocity that to dcfins anything likc an orbit for an electron
bocomos impossible4 The only statcrnonts that wc can mako about

ht for instance tako many dnstantanoaus photographs of a
4 certain kind of atom in tho way'doscribod abov~, onc might then
suporimpose those photographs and WQ thon find that tho olcctron
position all1 distribute itself mor0 in cortain rogions (noar tho
nucleus) than in ather rogions.
thc probable donsity of tho c'loctrona inside tho atom, ire, the
probability with which the c3cctrons can bc Found in tho various
volwno elemopts insido tho atom.
Such a plcturo gives an idea of
Tho doscription just givon might induce onc to say that
an olcctron is asbip 8s the wholc atom.
6
HOWOVOF, in an individual
moasurcmcnt the position of an electron may be more sharply de-
ffned and in fact electrons do havo thoir own radii which a r much ~
smaller than thc atomic radius.
Tho conccpt of the electron
radius can be undcrstood by cansidaring the clcctric field of an
olectron.
This clectric flold La nccQrding to tho Coulomb Law
inversely proportional to tho squnrc of tho distance from tho
olectron,
If tho olootron were a point, the ffald mar the
electron would bo infinite, It is not belicvcd that such ** in-
finite fields can oxist and it is asswad thnt at somo givcn
distance from tho c ctron, thc Cqulamb Law CORSCS to hold, This
0
distance has bocn estimated
That is the cloc-
tron radius4and tho nuclear radii a m of comparablc size.
COUPSC,
about 30-13 cm.
tho position of tho nuclci IS not quito so hard to dsf'ino
as tha position of olcctrons duc to tho fast that the nuclei hclvc
much bigger mass and accordin@; to that reasoning unccrtaintics
in their momenta, will not cause very great unccrtaintics in their
volocitics and will not wash out thoir orbits to the seme axtent
Of

VIQ
7
In some of the applications of quantum mechanics which
shall encounter XatBr, the wave Eennths associated with par-
ticles will play a fundamental role,
a given particle is h/mv
Gne lonpth associated with
(h = quantum of action; m : mas8 of par-
ticle; v = velocity of particle; mv = momentum of particle),
%'his
length is actually equal to the uncsrbainty a% position associated
with an uncertninty in momentum of the magnitude mv. De Broglie
was the first to postulate that a wave process is associated with
every particle and it is shown that the wevo lencth is h/mv, This
&$sociation between paptlcles and wave proceaaos is surprising but
it has been actually demonstrated by interference experiments in
which electrons or light atams (for instance hydrogen) have been
reflected from crystal surfaces.
The laws of' this reflection pro-
cess are of the same kind as the laws of X-ray r'eflection and have
,
been described In terns of wave intcrferonca,
It is apparant from
the above statements that the de Sroglie wave lcnpth is closely
related to the uncertainty of electron position and also to the
fuzziness of atomic boundaries,

8
LA 24 (2)
September 265 1943
LZCTURE SE~?XYS ON NCTCL’2AR PHYSICS
, -
FIRST S7RI”S : TTR!‘l INQLGGY LECTtJlBR: ?7 .I$ e MCIJ ILLAN
LECTURE 11: ,
1WT’~RACTICN GF LIGHT TflfITK MATTER
%
The list of simple particles mentioned in the last Xec-
turo comprised electrons, protons and neutrons.
$‘fa may add to this
1Lst light.
It is known that,light consists of‘ eloctromagnctic
waves, but if, as has boon mentioned, the wave process is 8580-
ciated with electrons, n.eutrons or protons, which we ordinarily
consider as particles, then it will, not be too suryrising to find
that light, which ordinarily can bo thought Of‘ a8 a wRVO process,
has certain particle properties, iillhon light bohavss like a particle,
we talk about a Xipht quantum.
There is an important dffforonce between light and
proper particles, for instance tho electrons, Tho electrons have
a strong tcmdencp to persist. fivht quanta do not. They pat absorbed
and omitted, It 3.3 not oasy to count them and for this
reasoh it is impracticable to consider them as constituents of
matter, Gn tho other hand, whonever 1icht does got- absorbed or
emitted, a chunk of anergy of a definite size disappears or is
produced in each such process as though light did consist of pap-
ticlcs cBrpying il certain amount of energy.
This enerpy of a
light quantum is hq whore zfis the frequency of the 1Pgh.t considcred
and h 3.8 the smc quantum constant which appears iur the
dc Broglio rolation conncctinp the momentum of a particle with the

i
1
I
wave longth associated tp it.
,
If the light gGantwn posscsses a definite cncrgy hv, on$
will also expect it to bkrry a momentum. According to the theory
1
. of FgJ,B$ivity this momontmn is hz/Jc, where c is the light v~lo-
city.
bt will be noticog that thc ratio of onereg to rnomc;ntum is
I
'' not thc samG hs ih Nowto)-iian mobhanics. If we write for tho kino&
1
tic onergy mv2/2 (m is mass of gartiolo; v, its velocity) and for
the momentum mv, thcn the ratfa of the 2 quantities is V/2,
that ds
one half of the velocityi of the particle rathar than the velocity
itsclf.
I
1
But vcwtonian mbchanics nppXics only as long as the velo-
city
v is small compared to tho velocity of light c.
hv/c of the momentum of a light quantum can be easily soen to Ggreo
with tho de Broglie rclaItion.
Jcngth of light and
I
I
I
In fact, c/z/is equal to
The value
tho wave
WX l momontum
I
is the samo equation whilch also holds for electrons, noutrons and
protons.
i
I
I
In discussinp llight quanta, it was ncccssary to introduco
1
relativity, ulJhile this lbranch of physics, as wall 8s quantum
mechanics mentioned last1 tims, have an elaborate mathematical
foundation, we nced t9 b(c concorncd horc only with a f'ow s;lmp,lo
consoquenccs of thesc tdoories,
1
I
Nuclear physics too will bc
troatod hero on a simildrly simple dcscriptivo plane, in which tho
clcmontary particles, cl/cctrons, protons and ncutrons, aro intro-
I
duced without doscribind in detail tho evidence that thcy arc
I
olementary particles ani withaut discussing the difficulties to
:I)
whlch this concept leads.
~ C I may ndd hcrc to tho list of tho pro-
e
I
1

10
@
per particles, the positron, whoso existonce was prodictod by thr:
combined application' of quantum mechanics and rslativity, Tha
positron has the smb proportios as an electron, exccpt that it
rSes a positive rather than a ncgative chargc, Tho only rcason
why its discovery occurred much latcr than that 0% tho olcctron is
that the positron is not stable in the prosonce of olcctrons. Tho
positron is attrnctcd by an olcctron to which it happons to comc
elosc., The chargcs of the two particles componsato cach other and
their energy is radiatod out in thc fom of Light, Thus an cloctron-positron
pair is annihilated. It is possiblc to produce posi-
trons but in tho prcscnco of clcctrons which arc found in all OPdinary
mattcr thcy can oxist only a wry short tirnc. Actually the
oduction of positrons i s the rcvcrse proccss of tho annihilation;
in cach production procoss, an elactron-positron paiP appears,
thcrcforc, whcn a productioncannihflation cycle is ondcd, we arc
left with the original nmbcr of clcctrons,
In addition to all thcsc particles, wo shall cncountor
two mom half-known k5nds of pmtZclo8, thc moso- and thc
neqtrinos, Tlno mcsotrons RPO known cxperimcntnllg, Thcy are
creatod in spccial praccsscs obscrvcd in cosmic-ray phJTsics but
oy have not bocn obscrvod up to now in ordinary nuclotlr roactions
Like the positrons thcy diaappcar shortly after thcir creation.
They arc of importancc in the theary of nuclonr forccs which at
a
prcsont is widely acccptod.
Tho ncutrinos haw not bacn obscrvod.
Thcy may bo Qonaiderod. at prcscnt RS a pu~o invcntion introduccd
to satisfy thc conservation laws of cncrgy of momcntum in ccrtain
wcll known nucloar roactions fn which thaso consorvation laws arc

11
agparontly violated,
Cnc important conscqucncc of' the thcozy of relativity is
the conncction bctwaon maas and cnergy, A movinp particlc is heavw
fer than the same partielc at rcst. differonce in mass is propartional
to tha kinetic encrgy, Potcntlal cnergy has a similar
irifluwcc on mass. Tho proportion constant batwocn oncrgy and mass
is the square of tho lQht volocity
%is rclatiorz fa difficult to demonstrate under ordinary conditions
it$ which thc kinotic or patantial onurgy pcr gram of matcrial is
rathor small.
In thcsc cascs, duc to thz high value of thc light
volociity, th& encrgy cha.ngas duo to thht; kinetic or patcntfal oncrgy
are very small cornparad to the orlglnal cncrgy of' tho maasca in-
lvcd, but in nuclcar physlca much grcatcr cncrgics are asso-
ciatea with a pivon mass, and hcro thc chnngos in maas bcoomo de-
toctable.
J ~ O may conslidor thc uncrgy relations in a nuclcar roc
aptfon in which two Doutorons (H2
(ao"
uclci) unit0 t o fomn a-particlo
Tho roaction has aCtualSy not bccn observed,' Its occur-
rbncc is vcrg improbable, bccausc conscrvatton of rnomcntun provcnts
qmrt9cla formed from carrying away tho cncrgy rcloasad by
action as kfnctic onorgy,
Thorcforc, tho cncrgy. r?cloasod
crn$tLod in Lhc form of light and all rcactions involving
n or hbsorption of light arc impraba
c as cornparad to thc
rbactions in which only hoavy particles, noutrons, protons or other
nRclc2, arc involtvcd.
*Zt may bc of intcrcst that a collision of
rot; on, Deuteron and W-particle,
dd'itfon to their more systemic

two Deuterons actually Fives H3 + €I1 or He3 + n (y1 stands for 8
neutron) with about aqua1 probability; These reactions are much
m4re probable than the raactlon mentioned above,
I
I mooretically, 2H24 ~e~ .+ light, lis reaction can tse
used as 8 very simple example for the @neFgy-mass equivalence.
The inaas of H
2 is 2,015 maas units while that of He4 is 4,004 mass
uhits, Therafore, the product nucleus He4 is ,026 mass units
j
lightor than the two original €I2
12
nuclei, This d&Fferonce of mass
corresponds to a difference of energy and since in the reaction
thc nuclear mass and enargy have dsoroasod, the differonce in enew
bbcomos available (in tho present case in tho fom of light) as
1
t&e energy of tho roaction. Tho quantitative connection betwean
crkergy and mags works out in such a way that ,001 mass unit OT
a
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
V$lts (mors accuratcly ,935 M!l[eV),
i? an cnorpy unit 1,6 x
orgs,
Tho alectron Volt (sV) in turn
It is the encqy an electron
aequircs if it falls through a pot0ntlaL difference of 1 Volt.
The relation between inass and energy is freqvcntly used to find
the oxact value of a nuclotir mass,
r
n~.pl~ar reaction tho ~ R S S of all but one particlpants 8ro known and
id tho snorpy of tho roactian is mcasurod;
This can bo dono if in 8
In practical nucloar physics intoraction of light quarita
withmattor plays an important roJ.~.
i$ photo ionizatiok
In thc figure
Cno way of such intoraction

ative ion, namely tho electron ftself, is also fomed
photo ionization, or the praduction of ions by 138
Ad1 actual obsorvatlon in nuclea.?? physics dcpcncls on ionization,
A typical arranpment for such obsorvationa is sketchod in the f o l m
I
chamber
1
i
galvanomcbor
I
t
battr;ry
The air in tho chamber is ordinarily an insulator and so no CUPrent
can be sont by thhc bnttory thraugh the galvanomcter, but if
ionization occurs in thc cbambcr, ions now moving up to the
t
clbctrodcs will carry a cortaln small amount of curront which can
I
bo; road ius. the galvanomctor. Thus by thc curront one can moasuro
ionization,
1) Ionization can bo produced by charged partiales as wall
as: by lfpht, Actu g thc ionization by f movine electrons,

protons, Deuterons and cb-particlcs, is 8. practical basis of
cxperimcntzl nuclcar physics,
exert an electric ficld and if their velocity is high enough, pro-
ducs ianization,
14
These particles movinp past atoms
Such particlos may. enter a chamber like tha one
sfiown in tho above figwro through a thin Toil
I
They aro stoppod by
a 'thicker foil, Onc of' thc: simplo basic axperimcnts Of' nuoloar
,
physio8 is to find out what is the foil thiclcncss that will stop
a curtain kind of particlc carrying a plvcn onorgy,
Thcro arc other interactions bctwcon light and matt@r,
I
Cqc of thcm 1s the Cormpton cffcct,
This is tho scattering of a
light quantum on a frcc clcctron, that is, one not bound in an
atom.
If'
tho clcctron is not froc but its binding energy is nogli-
gtblc compared to the cncrgy of thc light quantum, WQ may st111
talk of a Gompton effect,
In this offcct;, ths behavior of light
can be satisfactorily dcscribcd by a particle piotzlro and vw thore-
fare draw in the following fipurc thc light quantum as a particle,
impinpin? lipfit quantum
0
scattcrod light quentum
Tho momcntwn and oncrg:y impartcd to thc clcctron by the light;
f
qgnn'cum may bc calculated by laws of conservation of cncrgy and
mbrncntum in rclativistic mechanics. Thc shortor arrow on iho scat-
r
tOrcd lipht qLuonturn indicated that thc light quzntwn has pivon part
of its enorgy and rnornE;ntum to tho elcctron,
This is true if thc
clcctron was originalby at rcst, which in przcticc may be assumed
f

15
almost without sxception, Tho offect was discovered by A, E. Comp-
ton,
EIe observsd that most of the X-ray scattered throuph differ-
e& angles has guffered an enorpy loss which dQpend8d on the anvle.
The enarpy losa agrees quantitativsly with the on0 derived from tho
energy-m.orflontum conservation Zaws, The Compton effect' ea the princri'pnl
means by which "-rays (that; is, high frequency lfpht hd#$
roduced in the Compton effect may be detected by thair
ionizinp power,
'
Thors exfats a third kind of interaction betwocn lipht
anh matter, this is the pair production. In this process one
1P:Fht quantum disappears and an electron-positron pair i s created,
T135.s cffoct can occur only for Ini7.h enor,qy 1iFht quanta, natn~fy,
the enarc$y of tho lipht quanta must su.pply at least tho mass of
the electron and the positron, It may in addition supply an arbi-
I
&aq amount of kinetic encrcy to thcse particlos, G3uantftatLvoly
the light quantum must carry approxiaately at least 1 MeV (nom
accurately, at least 1,024 MeV), Tho proccss can not occur in
empty spaco. It requires tho prcsence of mattor, Tho reason for
is is not tha6 the process requires any other raw material than
the onerqy carried by the lipht quantum but pair production in
empty space would violate tho law al? conssrvatfon of momontun.
pracoss can happcn in tho nciqhborhood of 0. nuclrus. Tho
nqcloua having a preat mass can taka up almost arbitrary amounts of
mobontun, It thcmfarc acts like r4 pla'clarm in the n8ighborhoad of
I
which clcctric flolds can act to p~oducc R pair and transfer any
oxtra nomentum to the nuclousr Tho proccas occurs with proatost
prpbability noar to stronyly ohmgad nvclei In whose neighborhood
i
i

16
hiph electrostatic fields axist,
The highost energy lipht quanta
formed in cosmic rays ~ O S O their energy almost completely by the
pair production PFOCOBS,
The hichor onergyy -rays which occur in
ntwlear proccsses lose cncrgy primarily by the Compton effect end
a r absorbed ~ only to a small cxtcnt; by pair production ~ ~ Q C O S I S ~
1-s tho energy of thoT-.rays pots ~OWOP, photo ionization plays an
increasingly important role in tho energy loss in addition to tho
I

17
U
Septambor 21, 1943
LA 24 (3)
LECTURE SERIES ON - NUCLEAR PmSGCS
.,,
st Scrios: Terminology Lecturer: 5, M, McMillan
LECTVFG 111: FGRCES HGLPIlTC ,THE NUCLTET TOGETEER
As tho bcst known kind of forco acting betweon particles,
wo might first considor elcctrpa~tatic forcos.
live know that par-
tilcles of‘ likc charga rcpol eaoh othor and particles ol unlike
attract each Other, tho f OPCO be in8
to the square 05 the distanco,
oqncept of
from 2nf ini ty
from each other.
It is a
anergy ins toad Of that
inversoly proportionax
simplification
of f OFCO.
to
th@
Thc poten-
thc amount of work naedod to bring the particlos
whore thoy do not interact
and potential anorgy is positive,
dofini tc dis tancc
If thorn is repulsion, work will nood to be done
In casc of attraction less than
no work has to bs donc: and tho potcntial energy is ncgativc,
patcntial snergy is the intagral of the forcc,
For thc invorso
law of clectrostatics, tho potential is proportional to on0
Tho
ovcr thc distance,
Thc fipurc shows such an attractiva potantgal,
e
t h distanoa bctwecn two particles boing applfcd a8 abscissa wid
ttp potential as ordinate,
Tharo is no olcctrostatic
B neutron and a
proton since the formcr does not carry a charge, Actually thore
a6poass t9 bo no stzoablc forco acting botwcon thesc two particlos
unless they apprmch to a distance comparable to
-13
tboir rad%$ which is ab0u.t: 10 ern, Thero a
larpc attrac”c;ve forco suddcnly appsars as shown

18
iguro which shows ths poto
a1 encrgy acting between these
twy9 particles*
The dctailcd dspsndonco OF this pot;ontial on tho
distance 28 npt known but the
B u,
A potential af this typ
and protans, but a190 botwcon two neutrons,
a
ding to tho 1
tanGOS, but
atc shape appears to bo that
IdGrablc, narncly
not; only hotweon neutrons
Two protons will repol
cctrostatics at proat dis-
distancos the same kind of
attraction ~ppsaps 88 bstwocn ncutrons and pro-
tons,. "ha goten 1 bctwcon two protons is 21-
lustrated in th.0 graph.
Gne can sea in this
graph tho dSsCinctton botwson long rapgo forces
as thc clcctrostatic forces wh
such as those due to bhc pot
h act at a pwatcr distancc,
whoso effect 9s noticeable only if tho particles am closc,
cowon cxamplo that might rnPLke this
rcpulsivo forc:: occurring whoiz ripid bodice arc broueht
into contaat and the more slowly va
1
A
If wc now want build from sovcral ncud
pratons, it is ncccssary at lcast thc r;d,gias of those
should touch 80 that tho short rnnpQ nttrnctivc forces
o upcratlvc,
mi8 roquircrnont i s not neccsaary in the
c olcctron Tn thc @.toms, which am held to,vather by thG
e long rangc olcctrostatic attraction pnd in which theref'orc tho

l
19
avcrago distance bctwcca particles is much grcatos, than tho radii
of thcso pnrtiolos? Anothur reason why the noutrons and protons
c o:ns tittat ing the nuclei
hcavy part iclc s mq7
aan
easily
so close togothor is the fact that
have srnc.llcr wavo length,
Accord-
ing to quantum mchmicsI it is not possible to confino 8. particle
within a Sl?s. 11 spacc without effoctivcly pivinp it a similarly
sTall wavc lonpth which according to thc de 3, ogl i D relation moans
c? ,hiph momentum,
Heavy particles may possess such R high momentum
without thoraby obtaining an exccssivc kinctic energy which would
surpass tho nuclmr binding( forcos and disrupt thc nuc1Cus.
Sinco
clcctrons confined to nuclei would have such a high kinotic cncrgy
WCJ muat not expect to encountor than as nuclear constitucnts,
othor words, whcn an clcctron is attractcd to 9 nuclous, thc at-
traction will incroasc its kinetic cnergy and mcrncntum and thoro-
by will dccroasc, its wave lcn,yth, but whcn the clcctron has ap-
pxioachcd to within thc distance of thc nucleer radius its wave
Longth is still not as short as the nuclaar radius and thcroforo
one cnn not think of it as bcing confincd to thc nuclQusa++
nqturc operating
cbctrons do not
rangc forcos may
that such short
hccvicr psrticlos,
mcrc is no cssentifll diffcrcnce bctwccn tho laws of
Sn atoms and nuolci but in the ntorns thct light
as I! rulc comG: cloec enough so that thc short
becomo cffcctivc,
It is not at all improbable
rnngc forces 8C t bctwccn clcctrons as bc twecn tho
Only cosmic ray cloctrons possessing vcry high
may bc squcczcd Into a smnllcr spacc than.
thcy usually occupy in atoms, "'his hEppcns in donsc stars,. The
cncr,q which is necessary to pivc c;loctrons high rnomcntum associad
with thcirmoro close confincrncnt Is dcrivod In this caso from
g F avi t t! ti on.
In
!

energios may come clo~o enough to othcr particles and iy1 those
cascs it is rcasonablc to look for short rRngo potmtinls,
Thc: conclusion is that a nuclous tcnds to bc a more or
I
Ic(ss closcly packed assembly of ncutrbns and protons. Such an as-
@
scmbly is shown schomatically,
prossnt protons, the crirclos noutrons,
T ~ G croaacs re-
ThG close
armngemcnt is due to tho short rango attractions.
On tho othcr hand, tbr~ is also a long 'ringc re-
pu'lsion between tho protons which tends to pull tho nuclaua apart,
Jf'wc: consider ncutrons and protons inixcd in a oLrtain proportion
and incrcasG tho total nutnbcr of particles, wc will cxpock that
I
thjc long rclngo mpulsion will cvcntual-17 pravail and dostroy thc
I
ndclcus,
I
Indocd, tlm short rangc forces may bc considcrod tho
same kind af forcc as cohcsfion.
a surfacc which is txs
mattor Rnd with it tho nuclcar rndius (R),
Thoy will tend to give thc nuclaw
small as poxsibfc, irrcrcnsing tho amount of
Tho stabilLzLng tcn-
I S
d&xy of this forcc will bc proportional to tha surfacc! and thcmfore
to R
2 On thc othcr hand, thc total chargo in such a nuclous
is proporttonal to thc valumc, thRt is to R3 and thc disz.uptivc
. Thc:
forcc is thc squp.rc of this chnrgc, dividcd by thc squarc of tho
drlstmcc bctvcon protons which again is R,
Thus, tho dis-
PUptivc f'QrCo8 Would inCPCLlSO aEl R3 X 5 - ~ 4 '?.nd tl?cy will bo..
R%
comc prodominnnt with increasing R,
, a
Tho structuro as dcscribod abovc in a qunlitativc mnnnor
myst not bo tckcn too literally. ,Thurc is n lot of kinctic cncrgy
iG thc nuclcus 9,nd tho phrtichcs must not bc considcrcd as local-
I
izcd, protonso due to thoir ;nutunJ. rcpulsion, hpvC a tcndoncy

* This
21
to'bc pushed out toward thc surfacc, but &,so thc numbcr of neutrons
and protons pcr un%t volwno tend not to dlffcir too @rcatl$.
will bc particularly notlccd in light nuolci most of which have an
approx9matcly equal number of tlautrons and Qrotons, Thc hcnvicst
nublei hnvc almost twice as many ncutrons AS protons but thcrc: is
mason to bclicvc thet q. nucleus consisting of ncutr-ons alonc (or
clsc protons alonc) would not hold togothGP at all, For B bcttcr
undcrstanding of the situation wc shall havc to explain thc operation
Of' the ss+calXod cxclusion~ nri.clp,la, In order to apply this
prfnciplc in pr'abticd. cndols, wc c?lsb hbvc to msntion thc ,snin of
particlcs,
Mcutrons, protons and also olcctP0,ns carry an angular
rnqmcntwn or spin which in quantum units has thc sfzo of 8, One
rmg think of such c pnrticlc as rotating around nn axis which might
I
'ab oriontcd in spncc in differcnt ways. For E completc description
04 such e particlc, it is not sufffcicnt to say where it is or in
which orbit it movcs, 'nut it is also nccGssary to plvc thc oricntntion
of' its rotntioncl @xis. It is a conscqucncc of tho npplfcntion
quantum mcchanfcs thnt for 8 pnrticlc with thc intrinsic
3
aigular momentum z8 the directional angular momonturn may be descTibed
by simply stating that the spin points upward or downward,
antization of direction is closely related to the quantiaai
tion of? angular monentm, It also should be remarked that, where-
l
a$ the angular momentum of elementary particles can be dbserved,
e
there is no direct way to investi~ate their angular velocity*
There is, however, a further indication or rotation of such partic-
X~EI in their magnetic moiaent.
1
Sven neutrons, whose net charge is

The spin assumes its greatest importance in connection
wi,th the exclusion principle,
I
particles rnuat not be in the sane state.'
This principle states that two like
To show how this prin-
le operates and how it is connected with the apin, let us first
sldetr an imapinary spinless electrdn.
One may ~lassify the
electron orbits around the helium nucleus by their energy,
el+ctron will ocoupy the orbit 0% least energy.
One
Then tha second
can not be assipbed the sms orbit bacausa this would
vibrate the exclusion principle.
The second electrm would have
I
tormove in a lsss stronply bound orbit and the holiw atom would
no% have its particular stability.
If, however, wo now remember
that electrons have a epfn which is capable of pointing in two
d!.,CTarant diwctlans, then it is poissibZe to put two electrons
g in their spin dire
tom, and a very stable arrmgament is obtained,
The third
alectran wotzZd have to agree in it@ spin direction with one of the
twp ele'ctrons and tharofovle the exclusion principle provents it
t

23
I
of protons, Tho simplest nucleus of this
, consisting of two ne-utr
tqo poasible spin orleneations in neutrons and protons, the exalu-
I
inoiple permits aJ.1 partiolss in tho Be4 nucleus to occu
e (namely lowest) orbit, IT now, one mope neutron is added,
I
ip musrt go into a no orbit of higher energy, The resulting
ntkleus would be He 5 but the energy of thtxt next orbit is SO high
that it is not hcZd in at all by its conneation with the other
PQpkicla5 and He5 does not exist, If two neutrons a m added,
ldading to 6 the attraction bctwoen tho two additional noutrons
i

J
thb numbor of neutrons
1
1
I
I
i
-nurnbsr
AI1 known nucZci thah fall in a rathor narrow band which
1
stbrts up at 85O oorrosponding to
trbns and protons,
Later the band
thb inaroassd proportion fn nsutro
oughly equal numbor of nou*
This discussdon makos it rrlora understandablo
number of partfclcs in nuclai is Iimitcd,
!
webe prclsant in a nucleus, the a1acbFostati.c forcss would disrupt
I
thb nucleus; if on tho othor
If' tao many protons
adding protons and
incrcaso the numbcu, of' neutr s, then tho ~xclusbon princlp1.s will-
,
recjuf~g tho plarticlos to po intor such high orbits thab tho short;
rahgo attract,ions aro no longor sufftalcnt; t o hold tho nuclcua
1
1
1
September 23, 1943
LA 24 (4)
1
I
I
I
r '
Fgrat Sorics: T~m~nology *N, McMbllan;
LRCTURZ IV:
RZACTIONS BYT'IZTN T€E f\rtTCLEI
I
n+lof,
edch other,
,
In order to be ablo to diactxss the roactions
we hevc to invodtipato the forces with whaoh
These forcss may bc obtainod from their
I
I

which i s shown in tho figure.,
In: thia figure, tho potential cnorgy is glottcd as a Function of
1
th; distanco of tho nuclei ru
static repulsion bctwccn ths nuclci is tho only forco acting,
this, thc patsntlal incrcaacs as thc invcrsc first powor of thc
e
At prcatcr distancss, >tha olcctro,
At smaller distancos, thc jfstickincss" of thc nuclei
cokcs into play; that is, as soon as thc nuclci touch, thcy attrack
cach othor.
This rasults in a drop 9n potential cnorgy which wo
have reprcscnted as a potcntfal wcll.
tdis potcntlal well is so dcep as to ovor-ccmpcnsato thc clootro-
stktic rcpulsion, and ih thirs ~asc tho potcntlal onorgy bccomcs
1
negative, as has boon actually shown in thc fPgurc.
For not too hoavy nuclei,
Tho strcneth of rcpulsion bctwc;cn nuclci and thcrcfcrc
tUs height to which thc; potential barricr riscs, dopcnda on thc
I
prioduct OS nucloi charpcs, This product is srnR1Tcst for W 2 atoms,
I
I
bu!t cvcn in this caso thc potential. barricr af a fcw hundrcd kV 5s
to/ be cxpcctocri,
This potcnt5al valua is of practical irnportancojh
th;s well-known reaction botwocn two bcavy H nuclQP, which is to bc
In
cd below,
For all othcr pairs of nuclcl, tho potantfa1 bar
ovcn highor: that is, of thc ordcr of a fow million volts,
I
or! for two hsavy nuclei ovon considorably ~OPC*
I
I
Tho stpong rcpulslion bctwoon nucloi cxplaine why nuclcar
a raeictlons arc uncommon under ordinary circumstances, Indocd, tho
I
kipctic cncrgy b$ particlos due t o thcmal motion at room tomperae
!

~
tube is about 1/40 of an electron volt, while energies oncounterod
in chemical reactions arc a few clootron volts.
arb far too littlo to allow nuclei to got closc onough to each 0 t h
so, that thoy may touch and rsaction mey startr
i
These energies
arb known to proceed howcver in the intorior of the sun, where
I
tokporaturcs of the order of' many million degrees arc found,
thb laboratory, it is necossary to impart: to thc nuclei high kinc-
bib cncrgios, in order that they should got close anough t o PBBC~;,
28
Nuclear roactione
In order to accclcrato particles sufficiently, dcvfcos cf
two esspntially different kinds have boon used.
tioally shown in tho figure:
In
Tho Q ~ Q is schbma-
c : . , -
A'
* '
+''*.*,
[4'-&-g#Bw
I
--
c
pabt high volocity to nuolci is tlzo USC of thc cyclotron,
dcirico, tho cnarpy is Fivon to tho nucloi in a sorics of small
Xn that
pu'shcst
The nuclci aro hcld in circular paths by a strong magnotic
fi'old. Tho accelerating clectric ficld is actually an oscillating
f'i'cld which exerts ita accclcrating action, always when the nuclei
rcach a cortafn part of tho cfrclo. Ahi2,c tho particlas @et; ELCccloratod,
they mom in incroasing circlos and arc fSnally ojocted
.as fast particles,
Thc tyoe of interaction betwoon nuclei doscribod in tho

*
I
bc@.nning
of this lccturc Sa quite different from th@ interaction
bo,twccn nucloi and ncutrons,
ThZs intoraction potential i s shown
in bha figure as a
Thcro is af cour80 no cloctrostatic repulsion bctwocn thc neutron$
and 'tie nuclous, This is thc roason why noutrons, though thcy arc
quite common particlcs in thc nuclei, have rcrnainad unknown so
lohe]. A ncutron can approach any nuclcus f'xlccly, can rcact with
that nuclcus (for instanc~, it can be capturod) and so it will
stop being a frcc neutron%, Thus in tho laboratory, a. neutron
may bo sccn just "on the fly", Gnc can obtain them by bombarding
a isuitablc nuclous by a suitablo particlo, for inatancc, a proton.
Thc partfclcs which hnvc bccn actually used as bombard-
*
ing particlcs in nuclear reactions are either uncharged or lightly
I
I
charged particlcs,
i
hikh an cncrgy to ovcroornc tho potcntfal barrier, rind approach
anothor nucleus sufficic,ntly close.
flPat colzmn gfvcs tho namc of particles used in bombarding nuclci,
If that particlo is an atomic nuclcus, its natation appcars in
brhckofs,
I
A heavily charged pnrtiolc: rcquirca much too
f'roquontly used particles arc dosignatod,
In tho following tablc, the
Thc sccond column gives spccinl symbols by whicrh thoso
------- .**---------I---
C_f_Y._.
Whcrc i s anothor purely thcorotfcal rca8on why froc noutrcns aro
dif'flcult to obscrf-c,d '??:ley ere supposed to bc unstablc, md arc
cxpcctcd to transioFm by radioactivc oction into protons ornittIng
antclcctron at tho sarni! tho, but A long thc bcforc this radioactive
trnnsfomation could take placc, a noutron is usually capturod;

* dcutcron
I 28
'r 1
1
proton
d
t%
ncutrcm
! radiation, oP' 7 -pays
I
cloctran
:this Xist, thc nuchi El3
P
n
1'
rr3
and He3 might bc addod, but th@y are
i
pr@ctically unavailable and so thq havo not bcon used In our ox-
pcbbcnts of this khd.
If OMC we.nta to use ncukrons ~8 bombard-
ink particles, onc has to dorivo thorn from nuclcar reactions.
If tkc bombarding particlo in a nuclcar ranction tzi'
chargod, one wlll cxpoct no nuclcar rcaction to occur unlcas tho
r
partiolc has suf'ficiont kinctic oizcrgy to DVCPC~C tho
haryior bctwoGn the two particles, Tbt13~ if tho yfold
of: such a rcaction is plottcd against thc cncrg:,- CY. t?nc bombard-.
in& pnrtfc?!~, ono cx~octs tb.c yield to remain zor? v;,3
cngrgy V mr2 then riscj in somo manner,
tho f?.gvi~ by thc; full ~PVC;,
Yl".C?(1 I
quant
1
I r+
! I
l

.,
29
bombarding particle extends exactly up to thb maximum of the
potential barrier. Due to the wave nature of particles, a ceptain
fuzziness of the orbit is unavoidable and even though the energy
does not suffice to carry the partscle right to the top, there
retnains a small probability that the particle may ri leak through"
the barrier.
limited. extent;,
Therefore, if the particle is piven less and less
energy and is therefore turned back according to classical rnecha-
nlcs at preater and greater distances from the top of the poten?
tial barrier, the probRbiLity that it can "leak through" the bar-
rier decrerses rapidly.
But the fuzziness in quantum mechanics is only of a
Tl?a actual yield to be expacted accoI'dp
ink to qwntwn mechanics is shown as a dotted line,
At high ener-
gies, where the particle can get easily over the potential barrier,
the yield mwy approach a constant value or may even decrease again,
The gield curve may show I;, sharp maximum, This is il-
lustrated by the figure.
I
yie Id
T'
-+ enorgy
To understand this phonoinenon, we must consider
ing and
bohbarded particle aftar they have touched end ha,ve formed a so+
*
cazl-ed compound nucleus. This compound nucleus has quantum levels
an@ A sharp maximum occurs in the yield curve if the energy of the
bornbarding particle is just ripht for the formation of such 8 guan
turn level. Thus the rnaximum is due to pn agreement between the
I

enerpies of tha particlos bof'ore reactioh end tho energy state of
a tho compound nucleus, Such an agreomont is called resonance. The
phenomena due t9 such resonance are analogous to those that nre
encounterod in clasaikal rnocknnics when the frequencies
vibrating strings become equal to each other,
of two
The methemetical
theory of those two situations are similar c?nd the shape of the
yield curve near a resonance maximum is essentially tho same as
,
certzin resonp~nce curvos encountered in classical mechanics,
It may hsppcn thRt in a reaction, the produced particles
have greater intrinsic energy content than the initial particles.
In'this caseJ the reaction is possible only if' this energy def'i-
cicncy is supplied by the kinetic enorgy of the original particles,
men a rtlinimum kinetic cnergy is required for tho reaction to pro-
90
ce@d, and tho yield rcmp.lns zero up to this energy,
The energy at
which tho reaction first becomes possible is called thc threshold
of the reaction,
I
Unlikc tho case of the potential barriers, this
threshold is sharp, and below it tho reectfon cannot proceed even
with B small probabflity,
As CJI
exemplc for 1. sirnplo nuclear reaction, we may con-
aidier the reaction between two E2 nuclei.
There is ono possibility
thflt the two nuclei mag stick to,?ether cnd from an CG-pQrticle.
BuC such an a-particle would have a vary high energy contont,
This enorgy might; be usod for tho omission of radiation, And if
it 'is 80 used, the a-prticlo m2.y stick topether permnnently,
hission of radlction takes$ howcvc~?, too lone 8 time and it is
rl,
more probsblc that before a? m y could bo ernittod, t;heCf;-particlc
r
would Ply apept again, It might fly npart into two deuterons, I.&,
I

one would yct back the oripinal nuclei end tho net effect would bc
rnc~oly a collision in which tho deuterium nuclei have doflccted
cach othep,
Gt'ner reactions c.rc also possible,
tektims consistcd of a noutron and e, proton,
of their colllsion, wo hnva two noutrons and two protons and cithcr
31
Bacli of' tho dou-
Thus, in thc rnomont
one of thcsc four pt-wticles mipht fly off', Thus we might got as a
.
result of' the reaction a proton r.nd F. H3so~ n ncutron and He 3
Both these reactions have been obsorved, Thcty proceed with n con-
$idcrable cvolution of onorgy but thc initial
havc had appreciable kinetic oncrgy, othcrvuiso the reaction bccomcs
quftc improbnblo, sincc tho pmticlcs cannot approach sufficiently
c2ose.
Anothcr possiblc rcaction would be tinc formation of a
nuclcus consisting of two protons rnd of nnothcr nucleus consisting
ofltwo noutrons, Thcsc nucloi arc not stable, Finally, throc or
fo

m
ovsntually by using soparntod Li smplssj in which only Lt7 or
Li6 was prcsont,
Thc pnrticular reaction quotod RbdTFo,
is,/ cndothcmic, approxgmntaly 2llV kinotic cnorgy being nacdbd in
thk prstons for the roaction to proCewd,
in0iabn.t: grotbas over tho t oahold will turn in &qatosh part
ihko kenetie cacrgy of khc houtx"oni
onorg.fr ona might gat neutrons of pivan enorgics,
Tho exccbk onoi?&y of tho
By cbntt-ollfng $he proton
$ ~ . q ronction is
I
actually tho best way to produco f'nfrly monoenergotic noutrons In
th& p~rqa of e, fcw tonths of a MV,
!
'
Tho roaction
I ~09+ ~2- ~30-j- n
a vary copious SOUPCC of ncutrons if deuterons of' 3 MV or rndrO
arb umd.
Such Routorons may easily bo producod i n cyclotrons,
Thb roaction 'is strongly oxotbomnio e.nd givos thcrafaro noutrons
high oncrgiea (around 10 I\!V>, Tho ncutrons so prodwcod
,
do'not CVOP haw a well doffnod Cnorgy, Thc 13' produccd in tho
rohcticln may bo Lcft behind in vmious statos of Qxcitation, and
on#WY Of 8"
pons carry away tho varying mounts of cnargy that havo
used vp in exciting
is omitted In thc f'o
at in tho d+d roaction
VOTY light nuchk obtalncd in this
ntually, *the axcitation
citation occups
tion (fop instan
not; havc exoitad lcvela in thc ncighborhood of 3
y avgilablc in thig rdactl~n~ This one
4 MV, which is
thcrcfora
d off as kinotic onorgy of tho reaction products and one
obtains noutrdns of wolbdcfinod oncrgZcs;

Locturor:
Ii,M,McMillan
I
Historically thc first nuclcar transfomntion obscrvod
wss radioQctivlty, Jt W R ~ naticod that Borne mincrals continunlly
waL idmliificd with €Io* nuclei, tho B -radir.tion with olcctrons,
r.nd ths "y -rndiction with qucnta of olcotrmagnotic radiation.
In a radioaotivc process Q nuclrtua unclorpocs a spsntnncous chnngo,
inlwhich it may cmlt a part of its own substcnco.
1
The v-rndiat;im
ho$cvcr corrcsponds morclg to n scttlinp down of a nuclous from a
,
higher cnor.gT stc,t;c into P.
lowcrI mom stablo stRtcl.
Follows some other rcaction n8d is accompnnicd by xz'o chcrnical
change of thc SubstFnoc.
i
It uszznlly
Whcn an =-ray. is emitted, thc cjcctcd pnrticlo carricts
twb chargos end tlic nuGLous is Icft with two chargcs loss, that is,
it mom8 down by two placos in thc poriodic system,
tlid ~ O Qti C on
, Rfi226w Rag22 + Er04
1
An oxcrmplo is
iniwhidh thc clcmant Radium, which is rinalngous in its proportics
I
@
to Barium, is transfornod into tho noblo gas Radon, This reaction
is balanccd in tho sGmo mnnnor as the artificlel rcaotions mom
tiinad aarlicr but r,n osscntinl dif'rcrcncc is that it OCCUPS spon-
tnrilcously,
FO1l.owing tho abovc rcnction, Radon emits in turn an

tances, thc,stickinosa of thc nuclcus will c,auso tho potential
cqorgy t o bo loworedl md near r = Q wo find a patxntinl wall,
Tklc h0riz;ontnl linc not far from tho top of the wcll rcprosonts
tbic oncrgy lcvcl which tbc a,-partiolc occupics within the nua2aus.
This oncrgy IcvcS in a radioactivo nucbus 3.s highor than tho pow
tcntiaP cnorgy a t vcry high r valucs,
This must bo SO bccnusc WQ
know that at high r vaZuos thc Ot;-particlc may appear carrying 00x1,
s&darablo khnctic cncrgy, which kinotio cnorgy must bc dcrivod
from tho onargy that the eparticlc originally poasossod in tho
nuolous, but in opdor that tho G-particlo should cncapo the nu-
cious (moving .long
tlzc dotted linc In tho figuro) it must cros8
~l rcgion of high potantial oncrgy* According to clnssicnl mocharxtcs
this 1s fmpossiblc,
In qucntum mechnnios thc fuzzinoss Q$ par-
thclcs mnkcs ponotrntion through thc potontin1 barrium? possiblc b\l;
t1sc probability of thc particla lcakSng through is vo:-y
small,
This is true in pnrtfculnr if thc potcnticl barrier is high;
, filnds that the avcragc timc which thc a-pnrticlc fs supposcd to
spcnd in thc nuclcuhs bcforc lcalcing out by tho qumtwn rncchnnical
mcthod is in fair aprccrncfit w ith tho tirncs actur:lly abscrvod,
scpsitivo dcpondcnca of th5-a timc on thc hcipht of Chci potcntial
i
bn:rricr cxplnins why radioactivc decay times vary from a smnll
Ono
The
fsaction of? Q
sccond to scvernl billion ycmse
e
s
i
Tho rnnthomationl law of radioactive docay is basod on
th,o fact that each radisactivu :rti;:;~~~ has a constant probability
of! docmy,
l
It 26 to bc notod t;v+, this probability of dccay is
in,dopondcnt of thc time tBo pwticuI,ar nucleus hns already livod,
It fol,lows thnt in n bunch of radionotfvo nucloi, tho nwnbor of

,
i
nuclei dccaying per sccond
proscnt.
If ono plots tho
35
is proportional to tho numbor of nualci
numbor, of nuclai as a function of tho,
givon by tho formula
N =: N,o
" At
No 53 the origiaal number of neutrons (which were present at t=G)
nqdhis the reciprocnl of the so-called mean life,
s$ood that some nuclei live shorter, 2nd cL few,
I
lqneer, th9.n 'this mem life,
active decay by the hcllE-lffe which is piven by
Pt is under-
considerably
It is more usual to dssoribe the rnc3.w
hclf-life Z 0,6g3/'x
The helf-lifs is the time in which hnlf of the original number of
the nuclei hns decayed.
I
In discussing the ,@*decay we shall consider an artifi-
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
obey the spme laws as nRturnlly radioactive ones; but instead of
being enoountsred in nature, they haw to be prOdUc8d by bombarding
a hnturally occurring nucleus by a light nucJouB,
In this way,
maby 6 -active substances have been produced while artificial
activities are very rwe,
CG-
A well-known RrtfficSaZ owactive sub-
stance is obt8ined by bombarding the on0 stabla 2sotops of Sodium
by' Deuterium
N c L .I. ~ ~ H2 w .Na24 .f. H1
I
!

In: this equation we have written e" instond of 10 in order to show
that negatlve electron is emitted, Tha nucleus produced in ths
reiaction, Magnesium, has an8 more charge than Sodium r.nd thus the
total charge is qQnsBrvedr
It has beop stnted that in R nuclous thoro is no room fo$
electrons, How is it then possible that the nucleus should emit
Rnt oloctran? It must be postulnted that tho oloctron is c3retlCcd
in tho p-disintegration ~POCBBS, Cther prooosscs are indeed known
in which pnrticlos me created,
olsotrbn-positron pnfrs from radiationc
pr:ocess in
-Pctj.vJ,ty is the
a-p++e-
Gno mample is the crontion of
Tho c:!smentary
creation
roacti~n. Tho enorgy dolivorcsd ts tho olectron is partly duo to
tho mass dfffcrencc botwoen the proton and tho neutron end partly
it is drawn from tho onerey store of the @-active nucleua ilyl which
thje noutron was orig:lnnlly formedl
ex,othormic.
The above simpln process is
It is ags~m~d that tho nsukron iesslf is @-activer
This actfvity howcvor te.kOg R longror tirnc than other prosoasos
h ncutrona can discpponr p.nd thcrefore the @ -activity of
e
froo nwtrona has as yot not boon obsorved,
EC t ion
AnoAthcr kind of
p-nctivo nuolous is fomed in tha re-
~14 + ~2 .-+015 + n
The Ol5 nuclous decays according to the aquntion
015-815 +

37
Q
This timc a positivc electron is emitted, This kind of' activity
is'n littl o rarer than tho one discussed above, It does not OCCUP
in tho natixrafly radioactive series. The olemontary reaction in
thfs case is
This timo thc olcmentnry roaction is endothomic,
This monnsI
thdt tho proton is sttlblc nnd a positron can be emitted only if
tha ehargy storo of tho fl-active nucleus can supply an onergy sur-
f'icriont to overcome tho mass difforonoo bctwcon proton and neutron,
Sometimes it hr7.ppens that n nucl'cus instead OF omitting
a positron absorbs one of its own electrons,
tran will bc onc of' an inner layer in. an atom, tb?tz E\
K-clcctron,
~'?ij s proccss requires R
As Q rule this elac-
so-called
samcwhat smc.1 Icr cnergy than
tho emission of a positron beccuao af th:: mass (an& corrcsponding
snorgy) of the positron and eloctron.
possible for R nucleus to ccpturo a positron,
In principlo it would bo
But even tho capture
of an electron is nn improbablc procsss in spite of the fact that
tho clactron is neap to the nucleus a11 tho tSmc,
It practically
ncvcr happens that 8 positron on its briof' passage nofir a nuoleus
is absorbed,
ns of the a-decay,
Tho mathematical dcscription of the fl -dccay is tho smne
sccond to hundreds of ycarsr
Tho hnlf-livos mngo Tram R
fraction of a
Slownoss of thcfi3-procoss however
ccnnot bc explained by th:: prcsoncc of a potential barrier slnco
tho clcctron has n long cnough wave lcngth to loak thraugh nuclcar
in exceedingly short
long lives of B -pro-
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
v neutrons P M protons ~ to transform into each other, We must acm
capt the Pact that the elementary creation processes mentioned
above havo a low probability and the correspondine reactions pro-
coed at a slow rate.
This also explains the fact why bombardment
04 nuclei by electrons does not produce typical buclear reactfons;
The only action of electrons on nuclerl which has been observed is
due to the electric Field of the e2ectr.on which ,$ives rise to sirni-
lar excltation vrocessos as absorption of
-rays.
Btransf omations are frequently represented in. nuclear
38
The ordinnto gives the number of neutrons, the abscissa the number
of ppotons. Dots in the diagram rcpresent nuclei, diagonal arrow8
reprosant @ wtransf'omations. Tho Q -transformations actually
limit the repion in this dia pan in which stable nuclei can be
.$ound.
If a nixclous contalns too many protons, it will ernit one
or mom positrons and the nucl.eus ~1.11 thereby return to the belt
of stable nuclei,
sfons will accomplish the same tklinp,
in singlo steps,
If too many neutrons are present, electron omis
These eixissions must happen
Simultaneous emission of two electrons or two
positrons is exceedinqly imprababJ.e, as will be seen from the fact
that the probability of sucli a procsss must be the square of the

39
e
probability of a sinple emission process which we hRve stated is
of itself improbable, No such double process haa ever been obsonred,.
It, can happen howevar t1ia.t; a nucleus nay emit oither 4n
electron or a positron, This occur$ when the nucleus can decrease
its cnorpy efthor by tran.sfoming a neutron or by transforming a
u
proton into a neutron,
Charts of the kind shown above aro useful in foliowing
any kind of quclear reaction. In tho part of a chart shown in tho
figure below, two reactions RPC? shown, In the first a proton is
added; a@--activo nucleus is obtainod and a positron is emitted
subs eq.uet?t ly
-*
-
LA 24 (6)
September 30, 1943
Lr CTURT VI: Z) &DECAY XT, ) NETJTHCN RSAC TI OMS
1) In a nuclear process in wh.lich a,@Hparticlo (oloctron)
if3 emitted, the pt?rticular nuclaua goes T'rom one deflnitc enorgy
state to another dcffnfto enorgy state. Hence ano would cmpcct
l)i
the omitted B-particlo to possess a defdnite kinetic B M B P ~ ~ .
Cxperimentally, howevep, Et is found that the ,@-particles are not
homogcyloous in enorgy
.:
but havo a distribution 8s shown In the
f bllowing f ipro N
Energy

angular rnornentw of L/2 quantum unit,
This oxtraordlnary particlo

ducod by noutrond
88
(1) simplc? cnptura, usually fbilowed by $ha
ma88 @hit Sa~gsr than the: rsrigShR3. hbeXou~3; (2) ine$&tia cob
hielDni the ink~rnal mwgy of tho ~UC~QUS 2s changod,
adbi0h mag bc regarded a8 an absorption of f?
nnd a sztbsoqumt ro-omission of a nsut;ron,
a noutron 5s absolrbod nnd
proton is emitted..
This re-
xzoutron by a nuclcus
(3) !kc; (n,p) reaction
Thi8 raaction pro-
coods w$th fast neutrons, although n slow neutron reaction is known
ogan,
Hqro tha product nuclous has an ntornic number one
smallor t1zr.n tho bombarded nuc1cU-8..
axanplos of this aro
f -?z!c?YH
ni
- L 1 6 -k. n.-
-. , &;a+
(4 1 The b;r:,$ cy} reaction:
Thoso reactions proceed with slaw neutrons, For henvier nuclei,
tho c-particPC must overcome a potontial bhrrisr; honce fast
neutrons are requircd,
(5) The (n, 2n) rcaction in which two
noutrbns are Gmittod requires fast neutrons, (6) For the sake of
complotonoss, elastic collisions are inaludcd,
enorgy of the imadiatod nucleus is unohnngod.
LA 24 (7)
Hero the intarnal
WmJor 5, 1943
ZC'IPURE SERIES 0,N TJUCLEAR PHYSICEJ
ries: Tmninology Lecturer: E,M, McMillan
The word fission mean8 splbtking or brenking apart,
We
am concerned here with a proccse in which a nuclous splits into
two more or 159s oqual parts, Tho product nuclei are Fairly heavy.
h fission processes arc known In which tho initial nuclous
is one af the hoaviost nuclci,

43
The reason f0r the finsS-on procoss is the olectroatatio
botwoen protons,.
In highly charged nuclei the potential
*his repulsion ip SQ gpeat that by breaking apart the
nuclaua can P berate energy,
Tg LJlustrate this, WQ plot the ma88
defect as a function of the atomic numbor far stab
I
mass
defect
atomic number
-*
Fig. 3.
dof'oct ia a rnaasuro for the potential enorgy per particle
MUCZOUS, Ono obtalns it by subtracting from Ch.o nuc1cq~
o nearest intogar (which is the nm.bor of neutron
the nucleus cnd has been cnllod tho mass x1
0
dividing by mass nwnbc3r).
By going from Uranium to the SissSon pro-
ducts, this patontial enargy per unit particle has considera
sed and in the fiasion procees an oncrgy of tha ordor of
be liberated,
for a long t b o but it has been co
i%iZ;y aP auch onorgy liberation ha b~en known
sfon Qccurs when Uranium 28 bornbar d by neutrons, f
ducts aro radboactivc.
idorod4 unlikelyr, Actup~lly fig-
ThesQ nctivitios have been observed, but for
D. oonsldorablc t ims thcy were not Intorpretod RS due to fission pro-
ducts bocaUsc of thc prcconcoived idea that the occu6renco of tho
,

I
fissiori process is practically impossible.
44
Finally Hahn observed
that on@ of' the activities is definitely due to Barium.
nucleba has only little more than half the mass of Uranium, it be-
came evident, thgt tho Uranium nucleus did break into large frqmcn2;s
Physicisbs the8 started to look for highly ionizfng particlast
Since this
GoULd be predicted thai fission di.J.1 give rise to sudh particles,
because the great amounts of energy llberated in the process must
evantually be dissipated in the form of ionization -processes+ The
highly ionizing particles were easily found by letting fission prom
ceed in an ionization chamber in which the ions formed are drawn out
by an electric field. and collected on the electrodes.
Et
The pulses
and corresponding ionizations obtained w0re m m times ~ stl?onger
than thoso produced by =-particles
most strongly ionizing particles,
which up to that time were the
A picture explaining the fission process is based on the
droplet model ,of heavy nuclei.
posed to be held togsther by surface tension,
like an elastic skin.
Liko droplets, heavy nuclei are sup-
Surface tension acts
It can be explained by the tendency of the
constituent particles of the droplet to get in as close touch with
each cther as possible,
many contacts as they would in the interior.
On the surf'ac'e the particles do not make as
Thus a surface means
ve potential energy, ancl there is consequently a tcndono-y
to Form as small surfaces as possible.
The charge af the nuclei acts in the opposite manner from
surface tension in trying to gull the nucleus apart,
As long an
the droplet is spherical (as shown in Fig, 2) the charge, being
symmetrical, will not be able to produce motion.
If, however, the
nucleus assumes an ellipsoidical shape, tho charges will accumulate

the droplqt
sion can be
charged nuclei
.tron capture,
larger tho charge, the smaller elongation w ill cause
to break up4
This pi tum explains therefor0 why fis-
+
expected wlth greatest probability in the most highly
In UranSum and Thorium, fission has been induced by nsu-
If the neutron is captured, its binding energy be-
, come@ available, This energy is a8 a rub eventually. emitted in
the f'om of ?-radiation, For that diatlon, however, consider-
able tbe i s needed.
The first effect oi the binding snergy is to
set the particles within the nucleus into motion,
may sot in and may pivs to the nuclmm, temporarily, an elliptical
shape which 2s well known to
Then fission may re~ult~
234
isotope u
Uranim has two pri ipal isotopGs,
is ve~y rare).
An oscillation
CUP in the oscillation of droplets.
and'U235 (a third
The u238 nucleus ia the parent of the
a
Radiuln series,
nuclod undergo fisiion,
u235 is that of th@ Actinium ser2~s. Both these
In u238 the fission has a threshold of a
littlo mope than 1 M.RcV~ The PPssSon cross-section,
rf, varies
with the enorgy of bombarding neutrons as shown in tb?e following

figure .
I
r
4
46
Fig* 4
The threshold energy Ls the amount of enorgy needed, beyond tho
binding energy of neutron, to carry the nucleus beyond thc stability
point, that is, to give to tho nucleus a suFFiciently ellip-
soidfcal. 8hape.
Since this energy I s only needed to overcome a
potential barrier, tho questton arises why by tho leaking through
process fission does Mot occur spontaneously.
Tho answer is two-
fold:
first, the barrier I s high and broad; End socand, spontan-
cous fissSon does occur, but at an cxcocdingly S!.OW
Orr c
There 18 no threshold 3.n tho fission u":'""',
rato,
mhis nucleus
obtains by nautron cqturc an ovon number of neutrons and sincre
nuclei with an even number of nwtrons have a stronger binding
energy, mor0 enor8y is liberated in this noutron capture procmss
than in the capture of U238.
Therefore, enough energy is available
to 8tmt the fission procsss ev~n if' the neutron brings no kinetic
energy along.
Fission in Thorium bohaves in a way which is similar to
fission in u 238 ,
The ,@-activity of the fission products may bo explainod
a
by considering the plot of tho stable nuclei,

oqual fraponts, the corresponding points indicated by tho solid
line are considerably above the roglon of stable nuclei. The fiesion
products will then get back ta that region by a series of
a -decays, indicated in the figure by small arr3ws- Many chains
of this kind have been found, Most of those activftics were not
known previously. Rowever, soma oF them near tho end of the chafns
and 1yIng close to the region of stability are known artificial
activitfos, These activities have holpod to identify tho mass
numbors of these radioactive chains, The atomic numbers could, of
course, be found by invostigating tho chemical propertios.
While thc product nuclei in fission haw approximakclg
equal weight, there is a marked tendency for a slight dissymmetry,
Tn tho most probably f'lssion process, tho ratio of massc8 of' tho
flssion products is distinctly diffcren'c from unity. In spit0 of
many atitempta, no explanation has bsen found which is quite simple

49
v\rh$gh the nunbsr of atoms has decreased to 1/2 of .their original value. 1% is seen
from the radioaotiae decay formula that ecm l/~ Also one has
from which it follows tha%
AT z h 2
= ,6931 .
The designation mean life os
is justified by the fact that it is actually the
average of the time an atom lives before decaying.
In fact the number of' atoms
whidh have lived for a time between .t; and t 4- dt is J!J(t) Ad%, Multiplying by
t, integrating from 0 to a, and dividing by the original number of %toms, one
gets for the mean life
a
L/?T(O) tN(t)A.dt l/n
0
It; is easy to verify tihat this expression is 1/A
3
which by definitios is equal to
The number of atoms decaying per unit time is defined a8 . activity.
...i... .- 1%
is equal %a. N(t>a , Ro-t;ivities are mensure'd in units of curies. k curie corres-
ponds to 3.7 x lolo disintegrations per second.
The reason for the choice of this
numbor was that it was supposed to be tho ntxnbor of disintegrations pcr socond
occurring in one gram of' radium.
Re0en.t measurements seem to show that the number
of disintegrations In radium is actually somowha% lowor (the latest valtw is
3r46 x 1O1O dfs/sco).
But it is better to retain tihe number 3.7 x 1O1O dis/sec as
a fhod unit.
In modicino a different meaning is often adop-bed for the expression
"radium equiva1ont.l'
Thus a milligram radium equivalent may mean an amount of!
substance whirth gives rise bohind a 10 m load shteld to the same density of
ionization as would be caused by 1, mg of radium.
Tho reason fur such a dsfinit-ion
e pOwer,
Tho change of
the amount of radioactive substanco with time may be quite
is that the biological effects of radioactivo su.bstances aro dm to their ionizing
complicated if tihe radioaotiva substanco is itself a product of radioaotiva decay

.
~r'
I
Own moxe particularly if tihe radioactive, stibstance is a member of a long radio-
'How complicated such chains may bocomo can be il.lustratod by tho
example of bhs Urabium family* The following tablo is esseylfially takon from tho
hemistry and Yhysios, 27Lb Ed$tion, page 315,
Tho da6a given in *he handboak are %he results or' an intort&ianol
agree-
QS+
available information in 1950, In the meani;ime, ohangas aro
hese chatlges have been incorporw6sd in tho following tabla.
These are
-P-"f -
----
Uranium
changes neodod,
JN"W.TIQiL
-- --CrUCC-..ICII.*rY.-"---
TABLE OF TEE RADIOACTIVF, ELRIIRRTS AND THRXR CONSTAPITS
-".-*I -., +- -.-.-
J$arne
Series
--I-
Symbol
R ~ l f period --.-.-- '$ Radiation Iso-bops -"-
Uranium I 01. 4.56 x lo9 yrsI a, v
Uranium X1 u41 24.6 dGyS
49 Th
Uranium X2
u-xz 1,15 mine 49 PR
Uranium IT (Brovium)
UII
2,68 x lO5 yrs. QL, U
Ionium
Io 6,9 x 104 yrs. U Th
Radium Ra 1690 yrsr ad Ra
Radon (Rmlium omanat%un,Niton) RQ 3.85 aaya Rn
Radium A ' RWA 3.0 min. a Po
Radium B Ra-B 26.8 min, 43 Pb
' Radium C Ra-C "35 min 99.97% 4 B$.
Radium C' Ra-C' ~4.4 10-4 s80 a Po
Radium D (Radiolead) Ba-D 16.5 yoars 4 Pb
Radium E Ra-R 5,O days 49 Bi
Radiurn F (Polonium RWF 136 days Cic Po
Radium fi 7 Ran' I O + V e * V 8 * I
qe)e.eae Pb
Pb206
The ohango in tho number of' atoms N1
Ra-C .*Cia.***. ;03'$u* Bi
Ra-C" 1-4 mine 8 T1
- c
Ra SZ" 4 e e s *#e;**** Pb
1
of tho firs+ mornbor of p. radio-
acrtivo sorios oboya tho s'implo differonti41 decay law monkionod in tho boginning
af tho lecture
1)
wharo A1
in 112,
dIV1 = - XlNldt,
is tho docay constant of the first moinbar of tho sorbs, Tho chdnga
bho numbw of atoms of thn second member of tho series is
dflz $Hid% - &32dt

51
@rw of the right hand side i s the inoreaae in +he number of atoms N2
due $0 the deaay gf tho firs% member of tha
wrih for the khird member
tions hold for furtiher mombare of the series.
The system of equations
81 4 &I-,, e"
a33
,-h3t
ents all, "21, a319 etc*, may be dstermirned
the sxpreesions for N1, N2, N3!
into %he dif'feren-bial equa.t;ions,
ha time rata of ohango of these quantities, and second, writing tho
for NlI NZ, N3# @to,, for t= 0, and equating ttheso quahkities 00 tho
a1 amounts of the radionotive subdtancas. In this way a nectoneary and
suf'f%oleM'b numbor of squations is obtained to find all tho ooefflic
'
far
%im@ dependoms of Rl, N2, @toI, ikl 6ome of tho oases of p
tanoe in %he Ra family are tabulated in the book of 1bej. Curie.
conseqwnce Qf'
brim. If'., eeg,, a uranium solut
the equations discussed is the
is IAft standine; 90r a tilw long oompared $0
of tho fir&
daugh.0
wry skate w411 establish itself in which the smo numbe
and are formed por unS time.
This is t
nl, or if the atzio of the nutnbor of uranium atoms t o tho
1 atom,
same as tho rabio of the period of tho uranium aOom 'bo
This simplo concop% of radioaotive equilibrium is basod on
ho origivlal mothcr substanca U1 has tag exceedingly long xifa and i ta
e noglootad during tho growth of the daughter subs'WwQe
If U-XL
is precipitatad out oP o. tiraniurn solution, all tho U-XI will bo fgrmed in the

preoipieate and none in the solution,
52
The solution, however, retains all of the
~ * b substanoe r
U1C
a
deony in the precipitate.
figure,
Subsequently, U-X~ will grow in the solu+ion, wliile it will
This growth and decay are illustrated in the following
U-Xl activfty inprecfpltate
J
U-X1
activity in solution
The decay of U-Xl
-t
in ths precipi6ate follows the simple exponential radioactive
decay law, The grmrth of' U-XI in the solution can be obtacliimcl from the fact?
that ohcirjcai $cy+Ir2-*

e
53
method is used for determining tho half-life of radium. In this case, however,
the oldest radium samples (about 40 pars old) begin to show an adtivlty slightly
smaller than they had originally,
Periods in the range betweon a few seclonds and a year may bo cohven-
ielntly dstormfned by following Srho change of actifvity with time,
Most; of tho
known artificial activities fall into this rango,
This is duo to tho fat% 'chat
activities with very short or very long lives are hard to dctoc'c,
Rvon where
the simplest determination of life tima is feasible, a prooiso dotormination of
tho llfa timo is difficulfj.
Thus tho bost known poriod ol" a naturally activo
olcmonk, that of rad.on, is known to a precision of *05$, wliilo .the poriod oi' tho
woll knovm artificially activo nuclcus P32 is known to .2$,
The usual accuracy
of half-livos is less than 1 or 2$,
This cxporimontal unocrkainty may givo rho
to cons5.d.erablc orrors I% amounts of a radioactivo substanoo aro to bo calculatod
from tho docay formula from moasurcrncnts taken long aft-cr th'ni: timo in which onc
i s intoros+odr
Special method6 ar,i nccdod to moasuro poriods *-diioh arc shortor than a
second.
One motlzod axpplicablo for radioactivo gasos is to let tho gas with a
known vclooity strcam 5 , o. ~ pipc, past a sorios of collecting clcc-brodos, as
shown in %lis figure.
collecting olcctrodcs
Tho ourront collcctcd by t-hoso oloctrwclcs is proportional to -the ionization
caused in their neighborhaod by the radioactive gas and this in turn is proportional
to the number of radioaotive atoms remaining in tho stream.
From this
measurement and the known volocity of %he gas, the dQCQY period may be calculated.

A similar method has been used for known gaseous substances.
But
54
instead of the velocity of the gas stream, tho recoil v9locity of the radioactive
atoms was utilized, tho recoil being due to the a-emission by which the short
lived subs'cance was formed,
This method is not reljable bacauso instead of
single atoms, groups of atoms or crystallities may recoil, thus making elre
recoil velocity uncertain,
A more suooessful method utilizes ooincidence counters.
13-coun
counter
coi L;+----l
cedeno6 circuit
Let us suppose that a parent substace emits 6-rays and tlioreby transF6rms into
ana-active substanoe of vary short periodo The substanoct i s placed nexti to an
and 60 a &-.counter and -the counters are connocted to a coincidence oircuit
which will give counts only if the two counters aro activated within a time t,
apart, , Than .tho number of coinoidanco counts to be expcctod i s
"&
Here Cmax 9s the number of counts to bo expeotod if tic is chosen as a very long
-At
time. This quaritity is multipliod by tho intogral of tho activi.ty Aa Of
unit substance owr %he period t,, Thus C gives the number of counts obbainod
ff only tlzoso &-disintegrations are sffective which occur aikhin B time tc
aftor t;ho @ -disintegrations.
Rapoating this cxparimont for various times tC
e
one can find from .tho variation of C. This oxporimcnt has bean onrried out
for the Ra-C -1 Ra-C' pair of radiosotivn SLibstanccs.
It is claimod that; thSs
mothod is capablo of mcasuring half-lives ~C~WOOM 10" and low7 scoon'&s.

$. 24 (9)
55
Lsoturer:
E* Segre
One applicaC5.m of kke radioactrive decay law is tho de'aarrninatioa of
anium oresI If
originally no lead was prosont in the o m and if tl?coro has remairzod undia%urbed
shoe its form&ion, the lead formed by rildioaotivs decay will give a meamre of
has passed since tho famation of the ore, Another similar method
s the helium formed by raclioac;tjive dacay and remaining in *he ore From
.
(I
P
t e age of tho old@$% oras, th.e preaumablo
Values of tho oraor of 109 years ~i.avo bQoM obtained,
of the earth can be d&@rmlned*
ivo moasurcmen'cs aro usually made by countj.ng disintsgratione.
ooourring in those caun+bs in'woduao errors into bho moasuramontst
nvasfiigato tho influanoo of tho
Tho simplost case is if tho eubstafice undor investigation doos not
rociably during the fimo of .the mcastro
shall QOY18idOF suoh a
uma khat tho aoti
ha probab511by o
w corresponds t
in a second Inahad of the
o find pn wo subdivibc tho scoond in&o
ivision fino anough 10 th& tho probability of find'sng
-.-IC---,
I -.u"I
aioulaLlon o f those probabilikios

*
oaunb irr on0 subdivision 2s nogliglblo. To find OW oounti in such a sub-
div5sSon has a probabili'by m/k, To ind n p~rtiolas ilz kho first n divisions
nono Zn %ho f6Jlowfng k-n qubdivisloii has tho probabili-by
(m/k)n (1 -
56
counts distributcd In any manmr Rmong tha
e
k intervals, wc must
k
is last oxprossion by 2rlio binomla1 caoffiaicht (n> i*c., tho number of
ways in which n intQrvals m n bo choscn among k. Thus wo obtain For pn
I
bcaomo s
, gn z (k) (m/k)n (1 - m/k)lrc'
ot' k go t o infinity, t;hc first facOor in %ho abovo oxprossion
(E) 1
wharoas tho last wil
-m
k(k - L),*.(lc * ~1 1) -,kg
nr
k-+ ns
c^---r-..ir*-il-rr*I .-..u--.
--
tho formula simplifios to
ynn
Pn : - Chm
11 s
lcnwm as Poisson's formula, Xn thu fallowhg figure R plot of p~
8 dofinod only for in.togra3
tcr p$otting thosc in"egrn1 valuas ano can Snterpola%o tho
ro@t of' Iiho ourvo,
It ia usual in dc+orm%nmtions of half-livos to ploG tho
occurronco pn of a glven numbor of counts.por socand n aga:Lns%
n and to cornparo this plot with Poisson's formula, lf n Can bo
Sit, one lias khoroby obtained tho doshwd

value of the counts par second and has at %ha same time oheoked thu propr
57
1)
sta-bistical functioning of tho coun*arr
If m is vary large, the plot of the Poisson formula 1.001~s as IndSoa-bed
in Glza follawing figurer
pn
1
m
The ourvo has now a rathor symmetrical sharp maximum around the value n * m
and goes quickly to zero when the dovia-bion of n trom rn baooms appreciablev
In Chis case, Polssonts formula may be replaced by the Gauss approxhcition
1 . 9 *-(n - m>2/2m
-i&Gz
Pn a *
This formula can be clerZved by assuming %hat
M .. m is small compared t o m,
using the Stirling agproxima$ion for %he fao%.t-orial, baking %he logarithm of Pn
and expanding into a Taylor series near "Clie, maximum n me
The wi4th of the curve in tho above figure oan be defined by choosing a
value w. in such a way that one half the area under %he ourve lies in the reg$on
whara In - rnl is smallor than w. This means that +he deviation O f m from n
has the same probabili-ty of being greater or of being less than wI
ah
-
be shown to be
w 0.6745
Ahothsr hteresbing quantiity is the average value of (D=m)2.
The value
Thia 5s
*
denoting n
gPn (n - d2-
m by x and %he average value of its equars by x2 and, roplaoing
by an inttngral (which is psrmtssiblo for %he Large rn values oonsidered hare)
II.

a
we obtain
a time
t, we have an even chanoo that by repeating the measurement many Cfmes
the kverage result will be betwen m - 0.6745Elfl;;l and. m+0.6745 4- m or outside
that interval. 0,6745 m is called the probable error and m the standard
4- 4-
doviation. Tho standard doviation increases with m, the relative standard
deviation F / m = l/llfir decreases with m.
If wo count 100 event6 the relativo
standard deviation of our measurements 5.s lO$, if WB count 1000 events the r o b
tive standard dsvia%ion is 3.1$,
cCc.
It is sometimes important to know the probability of an error cqual to a
multiple+ times tiha standard devia.t.ione
Theso probabilities are tabulaked, ocg,
in the Handbook of Physics, 27th odi-tion, page 199.
The quostion. sometimes arises what conolusSons can be drawn about tho
docay constant if one finas no counts during tho timo of observation.
I+ foLlows
from Poisson's formula that the probability of finding no counts in an interval
ia which m
counts should hava bocyl oxpocted is
po = e-m
A closoly connoctod quostion is that of the ooincidenca corrections to
be appltsd to obscrved counting rates.
Lek us assme that a counter takes the
time z to rocovmo This means khat af.t;er a count, +ha oouvftar remains insen-
.'*
sitivt: €or a period z If' we observe a counting rate mexp , the true oounting
the exponential factor representing the correctiotn due to the :?act thab counts
are missed if they follou eaoh oCher at intervals smaller than z e Assuming
that the time z is short;, so khat the probabili%y of missing any count due to
the above qffeot is small, one finds that the correcJbd counting rate mcory
2
is

expresaed 'In terms of the experimental countitng rate moxp by
59
Mora involved prob~erns arise .if the substance decays appreciably while
the experiment is performed,
One praatical problem is to find which pariods of
measurement are best suited for tihe d8tf3rmfnatiOn of the half-life of the sub-
stanoe.
One might guess that it will be besf to let the subsl;ance daoay for man&'
half-lives and to take advantage of the great change in counting rate which has
accllrred in the meantime. IIowever, if one waitis too long, there is too grsat a
loss of accuraoy due JGo the reduoed counting rate and tho corresponding greater
influenoo of
fluctuations.
In the following figure the logarithm of the oounking
rate is plotftd against the the,
t
+-
-t I
.c.
Acoording t o the radioac-bive deoay lavr this plot should be a stira%@t l%w,
Zf
one detiemines ifs value at points 1 and 3 one might expect a graater acourag.
in the slopo.
But this 5s true only if the moasuremont at 3 doos not bocoma too
inaccurate as hac been ipdicatod in the figuro by the vor'cical line.
It is found
actually $hat i-b is bsst Go choose tho %vo points 1 and 2 at which caunting rates
are-dstorminad one, or one and a half, half-lives apart.
The fluotuationa whioh we have discusfwd am a consequence of tho radio-
actim docay law,
It is thcreforo of fundamontal intarest to sco whoChor those
fluctuations cmform to tho predictions, This was found to be tihs aase whoncmr
0 maasuramonts vmro porf ormod with tho nooassary prooautions. Oyto mistake whioh
frequently has led $0 spurious doviations from tho prodictud law OS fluotuations

ia to neglock Ghe coincidence correction of the counter,
60
I) C-.'.---W^-----
IIIa PASSAGE OF ?ARTICLES TFIROUGH PUTTER .--u
--- -part iGle s
IBasuring deflections of gE-particles in eleatric and magnetic fields
RuOhorford had established that a-particles are helium nuclei,
The charac.teristic property of the G-rays in which we are hsre inter-
ested, is their defihite range.
This mealzs thaCa-par%icles of a givon velooity
(for instance&-particles emitted by a thin foil of polonium) w ill traverse a
given distance in air before ba'lng stoppod.
This can bo domonstrated in the
7Nilson cloud chamber,
The Wilson cloud chamber is a chamber containing satwated vapor.
Sudden
expansion of the chamber causes superr$atura+ion in consequence of adiabatic
doolingc The droplets ivill then condense around ions formed by the passage of
the &.-partiole khus marking the track of %he &-rays.
The .tiracks seen in the
Vilson chamber are of qukbo uniform lengthe
The stopping 0% C;c=-particl~:s is due to collisions with elac-brotzs.
Since
the mass of the electron ja about 7,400 .times smaller 6han that of the
&-parliiclo,
tl?~ cz; -parti.clo is not deflected by such a oollision appreciably
and its path remains visi.bly dxaigllGe Xowever, the enorw lost in such
co$llsions accv.inll?ates.
1% occurs somctimes tha% an c\; =-particle ooll.ides with a
nucleus and suffers a sti-ong deflection.
Vi1 s on chamb r
Thcso events can be easily seen in the
The i'ollen~ing flgum shows the density of' ionization j along the pa%h
abssi .ssa x
I
is the clj.stanoe
i ,'

travelled in air, The above curve is called the Bragg curve.
It s last and best
detemination 9s due to Livingston and Bollovmyz).
Close to the
end of the rang
the Brag curve a%tains its mximum, making about 6,000 ion paths
pes mm of
air.
At tho end of the range tho ionization declines sharply iio zoroI
Ian$zati.on in air 5s accompanied by loss of energy.
Tach ion path oorres-
ponds to an energy loss of about 32 eV, This figure is fairly indoponden% of
%ha vel.oci2;y ai' thecikpar'tiolo.
Thus the Bragg curve gives no% only a measure
1
I of %ha ionization density but also of tbe onsrey lost per mm of path,
1 It is of groat intorost %O fihd out the energy loss ofc&-purtichs in
other w"cerials khaii air.
Thus iZ; is important to knw %he energy lost by the
a f s vthen eutering a chambor or oounter through a thin foil.
The onerig loss or
stopping power i12 various substwoas is host measured by giving the vaJ.uo of the

*
Octobor 14, 1943
LA 24 (10)
62
LECTURR SERIES ON NUCLBAR PHYSICS
__I-- rr.-nrri..-.rr2.-"., .e. .*IIyu-II-.I- ....--
t
$ocond Scries t Radioactivity Lecturer! E, Scgre
LECTURZ X:
n --.&--.-&"--------
~*PARTICJ.,ES AIfD TIiEIR Ii'JTRRACTIOM WIT73 liIATTER
-.*-I -*--I*..-.-.--*------
In cloud chambcr photographs of ana-omitting source, it is ovidant that
tho tracks oxhibit a, more or loss charactoristic longth, known as Gba range
(v. Rusctti, p. 303).
In tlic main, the onorgy of thc &-particlo is dissipatod
in collisions with olactrons: tho path of liho Ctppartic3.o is not dcf'lcotQd by
them many cncountcrs bacause o f t-ho rclatij.vely groat mass of tho a-partiolc
comparod t o that of thc clectron.
TJpon closo inspection of thc trsxcks, Q ~ C may
soc tho faint- tracks of tho projoctod olcc-ixons, all having their origins in tho
hoavy track of the drpartic1.c
Occasiondly ancX-traclc shpws a f ork-likc
structuro; hero t-hc a:-par%iclc has suPfcrcd a nuclcar collision nrzd my bo
approciably doflswhcd, .tho pa'i;h of tho struck particle bcivig tho other branch of
thu Pork,
A monsu.rcmonti of tho anglcs and rangos (onorq) of tho various
colliding bodics slwvrs that- tho oonsi:rva-i;ion laws of momcn-bum and cncrgy aro
sattsfiad.
Sometimes, one finds that kinetic enora i s apparsni;lg not conserved.
In thoso cases an inelastic collision or nuclear reaction has taken placOr
The many collisions of ana-particle wi%h electrons result in ion production
along its path# The number of ions per u~I.t lcng-bh ts oalled the specific
8ragg curve) of specific ionization VS r angs is shown in
the follovring figure ,
*
range

63
The speolfic ionization inoraases with decreasing velocity of theaqartiole,
1)
attains a maximum near the end of the path and then drops to zeroI (See for 8
, quanfitafive graph, Livings-ton and RQlloway, Phys, Rev, P
54 .- 18, 1938,)
%he
tlze
5 10
range
lL
20
As an example,CZ-par-ticlesfroln polonium ham an energy of 5,298
EBV and a range
of 3,842 cmr
-- .. *-.-
-
Stragglinci If the ranges of' an lni%ially homogeneous beap of a-parfiolos are
measured, one finds valuos dis-tributod about a certain value, the moan range R,
with a deviation of
one or two percent.
A plot of the number, n(x), whioh have
a range greator tliaiz
x
is shown in tho following figure.
n(x>
.L
Tho value of
x at whioh n(x) 1 1/2 n(0) is oqual to Rm The int;orsoc%ion with
the x-axis of tho .tangent to tho curve at 3 is tho extrapolated rangob
31) Tho distribtuion of ranges (tho phenomenon of straggliw) may bo
oxplainad by the statistical fluctuation of tha onergy loss per ooll$sion, and

e
64
s% the ohmge of’ theahpartic16 varies several thousand kimss, 1.8.)
He’, He, along its path. Those changes, however, ocour almost
en%irely tn the last few mil[limsts~s of rangec
\5
$he stopping power of a substance 2fthe spaco rate of enorgy loss of the
implngihg partioles as they travc;rse the substBnoe, io@,, if F = stopping power,
i
T = kinetic energy of particle, x
P
-dT/dx,
space aoordina~s,
Consequently the range R ia givon by
R 0 TO
R $ ilx = - jOdT/F =
0
The ralattvo stopping power of a su,bstanoe is dofined as
approximately indopondonti of onorgyo
stiopping power divided by the density of the substanco.
mnp -I.-mCrC- in air
rmgo in subtibst;ame’
Finally, ?;ho mass stopping power is tho
The mss s2;opping pmor
s’d 7

7
e.
65
scatterer at ona fogus. The ohange from tho iniiifal direckion of approach to *he
final dirowbion of the par”cio1a is %he so& ing angle 8, The hypo.tjh&ioal,
distance of closest approach if the parOicJe were not deF’lected is tho so-callad
impact parameter, b or tiha above figure,

*
LRCTURE
OoBober 26, 1943
SERIES OlT NUCLEAR PIIYSIGS
L-c- ., *- cu-
- L
cond Series: Rgdioaotivity Leoturerr F. Bloch
LECTURE XLI TIT? TEEORY OF STOPPING POVP3R
-Cw-.,+.”UlrUICUII
_..I. I. --a-
~ircrs*’hw.-urc-.cI
We shall calculake the energy which oharged particles lase when pas
tter.
This onerm 106s detormines JGhs range of chargod particles (ergr,
es or protons),
Tho onergy loss
s also olosely &nneo%od wSQh ionlza-
d by the charged particlo in %ha substanoa through which it p1as0sC We
shall first discuss the Cheo of st~pp!ing powor as givon by Bohr. Hi
only tho sirnpkst comcep+s of c2a
ioal maohagioso
WB consider B particlo with oharge Zo and volooity Ya 1% ixzteraots
wiOh an atom.
llore particularly we shall examine tho interaction with ona
shckron. ot chicgo e and mas8 M wrt.l-hin that a-koni, Tho e1oc”cron shall ’bo
an aguil!,brium position with olaetic forces and shall haw tha vfibra-
tionax Troqucncy &$..
The olootron was originally at rost.
1’ro shall, aaloulato
the amount of cnci’gy AT that %he s2;ron ob$ains durfng the passage of the
lo* Tho charged partiole % thon have lost 2;ho same amount of onorgy,
it has changod i%a onergy by -AT.
In tho ffguro, tho pasit-ion of tho clcctron la show by tho point
I

Tho rnin5,rp.m
6 '7
value of %his distance, that is, tho dis-banca of closest appfroach, is
equal to b, JGhlB distance is called %he lision parameter. The angle
inGluded by the path of the oharged partiale and by
r is oalled $, Wti shall
system whoso x direrottoll ooilzoides with the path of the
a-partioxti and. whose y diroabion is perpendicular to it-. At ths poiht of
olosast approach of the charged particle t o the electran wo set x s 0,
We
furthermore assme -khat for this point also t = 0.
Then at all other times we
have
x m vfi,
Ifire shall solve %ha problsm of tho energy %;ransfor to the elea%ron In ati
approximate but very shpljeway.
The approximation to be used requires; 1) that
the eLectron be considered 88s al.ly as free during the time of collision;
a'b the displacement of the electron during %he collision shall be amall
to the distanoe of cLoseslt approach b.
In ordar to soo wlmther the first assumption is valid lrm shall consider
$he y-compcwn!; of %he force af interaction
F =
"&
r2
of this component i s zero at t E -mand t tcX) e
It has a WXimUm at
B width of %his maximum (%ha* is %he %imo during which ElJr is more than
s maxArnum valuo) WB oar1 tlzs collision .time
The condition that
the elecr-bron may be considered as fr
is fulfilled if Z is small compar
*he, perrod of vibrat
Q
l/a
Indeed if' tl7i.s is tho case the electron mows
collision through a dis%anco which is small compared to 1Gs vibra-
Gional ampli%u&, i.o., the oleotron remains close, to its equ$liW.um positioh,
librim position tlza elastio for~os are small, *hay may be
elQGtTQn may b9 aonaidorod a3
TO shall diSOUS9 *he

*
the
68
validity of our second assumption later,
Tn order to caloulat-e AT we shall first compute the momentum given t-o
eleciwon by the charged particle. According to our second assumption, %he
alegtron has moved a small distance compared to
b
and the force will be prao-
e same a6 though the eleotron hqs not moved at all,
Under these con-
dition@ one sees that the net change in the x-cbmponen*b of the momebtum of tho
electron, dp, i s equal to zero, Indeed tho momentum in +tihe x direction that
the electron has obtained while the charged particle was approaching, will be
cancelled by a momentwn change in the opposite direc.t;ion thb$ occurs while the
charged particle recedec. Thus the only net momentum change is in the y-dire&ion
and after %he passage of %he charged particle *he electron is left vibrating in
that direction.
This momentum change
pY
is the time integral OS the y-oompanent
Sub s t iky0 tlvlg
one finds
1
sin2 $
dt* kx ---
r = b/sin $j>~d v
Ap, ' = ;+-Ze2 5
Irp
sin $ d $ =
2Ze2
0
The corresponding &ange in kinetic enorgy Is
1 2 ~ 2 ~ 4
AT ='z;;; IPx2 + ApY2) 2 -
lnb2v2
\ I
La holds as long &s b,inc: bg bmBx* If' b
s mado above are not valid.
is ou*side this region, *he
We shall calculate bow the anrgy lost by tho particlo when passing
J
through a distance &x, Lot u.s assme that them are N cleo%rons per conti-
motor oubed. In a cylindrical shell of length ax, inner and outor radii b and

e
69
b + db $here will be 23Fbdbd XN e9leotrons. Tho energy transhrred to these
AS = Rhx
7 --c-
Bfl'zZOB
db c Ndx c- 4 z%4 log % bmax $ .
'ET7 b ' m VZ rnin
bmin
(i
From th5s squal;ion tlm energy loss per aentiiineter &T/Ax
may be obtaifiad. as
soon as $he values of bmin and bmax arb dsbmnined.
DiscussSon of brain
--*-
For b - 0 our approxbdto formulae would give AT a00 which is
Actually tha maximum volnoity a Fxee slootron may obtain in a colli-
v and therefore tho maxiinurn energy loss i s
22
I
r
G
I
ose b in suah a manner tlmt %he calaulafed onarc3 loss shall bo equal
imal energy loss, %hen $his value ~f b
will mark the limit 'beyand
whiah the approximate value of the energy loss can no longor be used.
At %his
1
value of b,
the assumption that tho electron has been displaced during %he
through a distance small oompared to b is no longer valid, This
b we shall choose as bmtn we obtain
2 z v
3
m
(2v)2 = v-
i 9 m b,in2v2 2
(&vr =
I-.
ze
a-
,* binin mv 2
, A treatmen% which is rigorous for small values of b u88s the
d farmula for the soatitwing of the charged partiole by %he ehotron.

In the oase
,><
thab
mv
70
the second JGerm Liz the denominator oan bs neglsc
and the integral reduces to
the simpler form which has been used above. At small valuer$ of b the Inbgral
goes to zero.
This eives a qualitative justification for using %he simpler
integral
b
for outting off the irrbegration at
P 282
bmih
mv
D ~ S C L I S ~ of' ~ bmax O ~
I*^.. -..-,"
c- .%-,.c** .
-L.
For too large values of the collision parameter
onger be considered as free. In fa&, coll'lsl,on time is appro
and this t5me will become great
l/&J when b becomes suf.Piciently large,
vibrational. period of the
For large values of b, that
is, when b/v))l/W
the variation of the Toroe of interaction botwesn charged
and eleotron is slow compared to .t;
tion between tihe charged pa
18 cnd %he eloctroq may
treated by Che so-called adiabatic-, approxi.mat ion,
Ih this approximatLon, ths
is displaced whsn %he chargod particle approaches but whan 4.5,
Ipeocsdes
ron is eased baok Into its original position without- iti taking at$
i
reford ju.stifio
ur integral to the poilit where
and to neglect oontri greater b valuos, Thus we have
b,in
into tho enercy loss formula
obtained Bbove, one fhds
* been
c-dT ;1N
dx
sd out this formula fs un proximste. The o
carried ou-b by Bohr rigorously and he obtains
2. "N.---
4 e $e4
log
1,123 mvz
--.r-.*C L
dx mv2 Ze2U

0
Wcro n donotw tho nhboq af
YIO
may us0 tho samo oxprossion
73
atoms por aubio cantimotor, For tho uppar limit
as vms dcrivod in crlassioal fhcary, namoly,
hmvovor, must bo kangod, This classical. lowor limit has tho‘ folloying s2gnif9-
onaidor tho kinetic onorgy ‘of tho cloo2;ron in t hs framo of
i
%ho ohmgad partiolo ie at rest, thon bbin Is the distance at
he pokential energy between the eleotron and 6he charged parflcle becomes
o tha-t; kfnetiio energy,
electron and charged
Now in quant;um neohanics a closer approach
article than the de Broglie vmv~ length is meaning-
WQ have to use therefore @or the lover 1imi.t;
i
bmin (quankum) = %/m.
’ Thi indeed so if tha lowar limit- in guanhn theory ha? a highohor value than
the previously derived classlical lower 1j.m32;, becauss a5 “.‘1$ approaoh closer
%han %he quantum lwier4.irni.t defraction phenomena will OCCW~ and make an effeo-
tdve approaoh impossible.
If hotmvor,
%he olaaaiaal 10~1er-12rn.S;t; should %urn 0u.t;
to ba greater, then down to %hat lower
limit, classical theory 3s applioablo and
be UBOdr
Thus, %he QlaSSiCal or quantum
sd according to afhethher tho ratio
r small compared to uni%y. Assuming Ze2/1;% = 1, one ob%

*
which factor does 110%
dotailed calculation.
74
follon from o w argwzhnt but must bo obtained by a more
Tho stopping power just g h n is valid If tho following throe condi-
a
tions are fulfilled:
(4 viua>> ro
(b) Z@2/*V{{ 1 * 3.
(4"2))flOj.
Condition (a) means that at the upper limit where wo have made use of the f'rc-
quoncy of the oloctron, tho distanoe of' tho charged parkiclo of tho atom should
'be great compared to the atomic radius
ro+
ThSs condiAiion i s nscegsary in
ic field of tlze charged particle should be homogeneous
the atom whenever %he oscillakos 'urea-tmont of the elaotrou has been made use ofr
Unless this condition is satisfied, the analogy to t5e interaction with ligh%
rea-bmeylt by virtual oscillators breaks downo
Co;iclition (b) mem~ thati
%he quantum Sower-Xfrnlt i.s pater than %ha classical 1o:mr-lj.mit
and that
therefore Clm quan-bum 1m*mr-lImi7b should. be usedr Fina!.ly, condieion (0)
expressss the requirement ti22a-b the upper limi.1; of
b
must be considerably
greater bhan the lower limtt of b aud that therefore there shall bo a region
in which our aBproximationi are valid*
ConditIQna (a) and (c) bofh mean n
ically khat *he veloai.hy of the
harp3 particle must be great compared to the valo
0 f -bhe
with whlch the Gharged particle inferacts.
For condikian (a) this Is
fly because
row is equal to ~ ~ 1 In , condition (c) one may replace
'h@l(that
is, tho sxci'bation energy of the electron) by twice %ha kinetic
energ$ of the electron which is a guanrt,i.(;y of the same order of magnitude,
Then
(e) roduces to

which again means that vQ)vel.
Condition (b) i s also satisfied if “))vel#
IV
ele ptrons +
If v and
vel become comparable, our treatment it3 no longer valid
and an exact treatment for this case has no% yet; been carried out,
The fhoa-
retical difficulties are paralleled by a complicated behavior of tho charged
particles a% relatively slow vslooitles,
If‘ their vsloci-by bocomes equal to *he
veloci-by of %he outermost electroiis in the sluwing dovrn medium, then the oharged
par”cclss. my pull ouli electrons from the slovdng down modlwn, attaoh them .Do
khernseXves, and thus reduce their affootlve charges,
In lator collis5ons “cess
charges may again bo lost.
Nhilo tho chargad particle had its charge roduoed,
itra ionizing powor and anergy Ioss will also bo smaller.
Since various charged
pa.rti.31~5 will spend variaus times in tho staiie of reduced oharge, tlio actual
” r range
of %he individual particles will dif%er, This dSff ;renco of range is
aalled straggling,
If an t&-particle has EVO. enorgy of 72 lcV, then its valoci%y is tho
same ad %hit of an eleokron with 10 eV.
This is %he avei*aqs of the ou-l;ermox+i
electrons.
Thus, vc must expec2; ‘chat our formula of s-bopping pwmr for
&-particles becomes Snapplioable under 72 kV1 ActualLy, sexious deviations
from the formula start at the somawha% higher energy of 200 kV.
‘I/he correspond-
ing limit for deuterons i s at 100 kV and for protons a+ 50 kV,
From the formula of the slxpping povmr an estimate of the range of the
particles can be obtained, For e, we replace the energy 10,s~ dT/dx
hange in kin,e%io energy of
iole -dBlc,n/dx.
The formula for the

*
This
Forge;el%-Sng about the logarithmic factor, one ubtains
PQIX~U~& sliws that the quantiky Elcinx ohanges at a constant rate along %he
16
I
path of %lis par%icle and that therefore %ha range should be proportional to %he
square of the original kinetic onergy.
Actually, the range varies vti=bh a &omem
what lower power of the original elnerpj.
is bhe stopping power formula.
A quantitative evaluatiion of the sDopp2~g povrer formula requires knnwl-
edge of all absorption frequenoies and all oscillator s%rongths,
These quan2;iWs
iven in an analytic form for %he hydrogofi a%om and in this case the
requirad calaula+iions have besn carrisd ouG,
Far heavy atoms oha my ude a orudoly simpLffiec1 sbatis-bical model.
Thia model oonsiders tlze elec%rons as a gas, %he IC
o energy or %he e1sotro)ls
being balanced IIT the avorag9 e1eotirosta”ciu potent
find the oscSlLat ion frequency of th5.s gas by hydrodyix!ni-r; ca’J.oulat
uso +ha% froquenoy instead of
CJi and use trq Ju.,uriZ nuliber of aleow
an atom Zo
instead of tlzs oscillator s%reng-b
The vibrational frequency in tho statiskical
the ‘tsound veloc.i--by’’ divided by the a,tomlc radius,
Tho 1),l.t;ruF(+ sacli.us can be shovn
”.
to vary as 2, 1/3*1 Tho sound velooity is of the saylle ordor of magialtude 8s the
obtain
locity of tho electrons and % 8 praporOiona1 t o Z,L/3, Thus li
dT 4.fFe4z2
L-rl
ckC 7
lag
a
ula, R Is the so-oalle
Rydberg frequency, that is fqR is tho
cw-
*%rhhoaviar &%oms ha smaller atomic dii which %s in contradiohe
experience about- atomic rad.ii as cus d, IIovevar, in
the customary definit-ion the atomic radius is essentia
the orbS.%s of‘
the few ou-kermosl; electrons vrhereas in our presen+ disc:
average distaizcs of electrans from tho nucleus, the av
eleotrons,

ionizlation energy of %ha hydrogen ationi, and
ating it, one may 'aajust 5t in suo
17
is a numerical factor, Inatead
way as t o give agreement bet;aaan
the formula and %he oxparisnce for on0 haavy akom, f instanca for gold. Then
la describes tho dependence of "cho st'ogping pawror on the a'ciomic number
sfaotory vmy.
The formula for tho stopping power of eleoGrons i s tho same a8 the for-
mula for other charged particles rriOh the tilxooption that under
faetor '2 cloos ndc appear :
2 mv
uo to %hc fact that i mining the lowr limit of b t
mw leng.t;h musC 530 uscd vrhich corresponds to kho vclooity of' $he atomic elaotron
e of rsfbronoa wkro tho conbo
chargsd particlo is at rest.
of the onooming partiole, as
ravity of tlint ol@cCran and *ho
olboity is practically equal to tha
that oncoming par.l;ic;le i s heavy,
leu$ If the part le i. s an electron its
s one half the velocity o
&lparticlss and of
oharged oosnSc-ray particles the motion OS the
ticle must be treated aclciording to rtslativis meohmlcs. This l-fill hnve %m
) the region in whieh the elea$ria field aots is oontracfed by
Ths Dim@' during which the
same factor; 2) %he strength of %he electrio f is j*nQrQasQd by
Ore, the momentun ah6 energ *trans-
ron during the c
airr the same as in unrelativis$ic
ver, l3he Limits for t parameter b chakp, The vpor
limit I s horeased by tho reciprocal of
tion, we oar], go ta groat
os tho period of the atornic electron.
In %he lower
a~8 Length must agdnba used vhich is given by

LA-24 (19)
80
LRCTURl3 $ERIE
c-"L-lrr..l*..Ic
Sooond Series : Radioaotivity Lcoturer; R, 'Pallor
---
CIlARGED PARTfCLES GilITI-1 rTATTER
*-"..v-.yL)-..i.-uyCr^w.m.--C~~
Tn the previous leoture it vras sh n that, apart fro
of aner~ loss of a moving hwpd partiole (
nu-orsoly proporiiional to % voloci%y Y of the
is faotor arises beonusa thc momenCum transferred lio an alaotron is
proportional to the "kime of collision'' which In turn is inversely proportional
to .the particle volocity.
Ilgnca n par*iclo
nga should therefom bo wEZO Expcrimntally, hot-ravcr,
WE3/'. The d epanoy ma37 bo removed by a more
tho onergy loss oquc,,tion.
Jt will '00 raoallod that
incldont :7nrtiolo nil1 transfor oi~orgy to tho atomic nloctrrotzs $2 i t s voloo
or than the oloc%rron volooitios,
Approximhly, tho numbor of olootrons
om vhich hwo volooj.-tics swllor than v,
and oar! thorci'oro roooivo
onorgf from tho particle, is simply proporkfonal t o v (provided v i s in tho
oloctron velocities). CDYW~VW
*he ra-b of onorgy loss is n m
go iq proportio
0 to noto %ha% tho,ol)or@ loss per
he, speed and charge of the 9noidelif pnrtiKle,
Fnrticles of' the saw '
ere at n rate proportional *Q square of -their rsspeoCSrre Zfs,
so
at n 4 1vN &?partide (2 = 2) has the same range as o. 11fV p
The enerar vrliiah is talcen from the inaiden'c parbicle my be used to
~JI sleotron from atom (ioniza$fon), 'or rnv only produce electronic exoitation
with subsequent om sion of a 1%

e
81
in detection of CC-particles),
Andhher possdble prooesb, and of considerable
impardanocs in contlsoCion Ivi2rh bl.dag5dn2 erfL"ec6s 02 radiation) is molecular
n with subsequent dissooiation into atom or ibnsi Takiwrg inti0 account
these various processes together with seoondary effects, one finds that the
average energy per ion pair i s roughly 30 eV.
An approximate measure of the
biologtoal effeots of radiatiaiz on a part of the body is the amount of enerfg
absorbed by the particular region.
There are, of course, qunli%ntive di:?farences,
becnuso the density of logs is qukh different along pcH~s of varioua j.i!;ipin@.ng
pnrticles, so thnt recombin?.tion protlesses m%y be of another clznracter in the
cam of, shy, a heavily ionized 4L-fraok fis oompared to a veak fi-track,
Ianiz
, --.*-
by heavily charged partfolesb
.---.-..-- - -I-
We have seen thnt the Bra@ cur~@ of nn a-partiale, o. measure of the ~
specSf5c ionization along the pnth, increases slovrly to ~1. mnximuin and. *hen drops
s1znrpl.y to zero+ rhe corresponding QLITYQ for fis5sion par-3.cles slions only a
_ -
in speolfic ionization along its pn-bh. ::hat :is the exp1:in:ci;ion
the
di.i.??erencol
The ini%ial velocity of A fission pnr2;icJ.o 5.s
apprcxi.lmt@XY
ce
One ccn estiinzte it.s initial ohargs because tho fisflion fragmeDt
vi11 8hod all the electrons having veloci2;ies smaller than its ovm,
As it pro-
ceeds dong its pa th, %he particle vd11 pick up electrons, RS 2, result the
speoific ionization will tend .t;o deorease,
At npproxinctoly 1 MY, +he parthle
becomes neueral; its pnth a6 R ohnrged pnrticla is nhou'b five to ten times
an the distiawo it trnvels as n neutral pc.rticle,
The range is ti'bou$
2 am, in air,
of Elec-t;rons.
*.
ft is difficult,to asoribe a range to eleckrons traveling through
OQ~UW of the excossive straggling.
In contrnst to the es$enfilnlly
il)
strni&t; pc.th of an a-particle, an electron my be considerc2bly dcfleoted in
n oolllsion, so that it w ill have a more or loss rnndon pnth.
Secondly, %he

*
energy spaatrum of #-pa clee emitted from a nucleus is not monochromatic (as
in the aase OS W s ) but s a considerabls spread, The combined effect gives a
range distribution which imitates an exponential absorption curv
There are seyeral prpasslsss by wliioh the enorgy nf a fast moving elea-
tron may be dissipabed. When ran electron passos -through the electric field of a
nucleus, it is QGCQh3ra%0d,
This ameleration gives riae to radiative energy
lasses (bremsstjrahlung).
They are appreciable for elecCron energies of several
Mev.
The electron-electvon radiative lossas are quite small inasmuch as there
can be no dipolo, but oiily quadripole, radiaCion.
The eLeotron suffers energy loases in Its collisions with eleatrons.
An approxima'ce formula has been given for this rate of
energy loss, and we have
aeon it depends on the squam of the atomio numbor of thc nuclei,
In .these
oollislons, all of the atomic electrons par%iaipate and ionization in the trctv-
ersed maDerial i s produced,
Tho ratio of ionizstion loss to radiative loss is
givah roughly by
Rato of ionization loss I ZLmev)
R a . t ; e T ? m L % X -
--.I
8 O r
where Z is the atomic numbor of kl10 rnnt;orial, E is +he kinetic e
million elgoZ;ron volts.
To give an idoa of how much shielding is nocessary to stop
one may give an axample.
4-
particbps,
For aluminum 100 mg/om2 will stop 100 kv betas, and
1 g/cm2 will be adequate for 3EW betas.
1% is approximakely %rue that %he
stopping power per unit muss is the 8aine for all tho olomonts.
. .r-...u.--.--ruCu..-
Reflectian of fi pa_r2;iales.
.._I_
Coefficients of reflec.Gion bstwen 1/10 and 1/2 are, qui%e usual,
Heavy
nuclei by virtue of %heir charge are more Eiffoctive in producing ,largo angle
soattaring than 'light nuclei,
Honce pb reflects fi particles much butter than,
I
say,, paraffin6

e
84
Sn particular, tho first roaction is irnportmn-b booousc it givos information con-
corning tho bindi.l?,g cnorgics cnd forces betwoon clomoqtary pnrticl-os,
Gross-
soctions for this 6ypc of' roactions arc rolakivoly small, being of tho ordor of
m2 or smallor.
Tho two ronatlons clitiod nbovo roquirc ralativoly low
3. -cncrgios, i.c. about 2 IJcv,
Elost nuclcar photoolfccts requiro 5-8 Flov
%t-rcys
The interac6ion OF ?-rays with rnaLter are in the maiD interactians
with electrons.
The phenomena may be described as the photo-electric effect,
Compton scatCaring and pair production.
We shall gee %hat the Parims inter-
ac.S;ions depend difforontly on nuolear charge and the energy regions in which each
effect becomes important is more or lesa well defined.
Photoel@otric effect,
.-A. ---".
__I
A bound electron absorbs energy and leaves the atom.
.IC-
Vost of the
energ? of the %-quantum is given to thu elec.l-rrm, most of the mom@ii%uin is given
to Che residual positive ion,
Fach shell o:? electrons wound a nucleus will con-
erilsute whose onera is smaller thavl the eneru of the ? quantum.
Among them
the shells whose ionization energy is closes% to the energy of the ?-ray
makes
the greatest GOYLtribLi'GiOn,
pray energies are usually greator tlian the ioniea-
Lion energy of the IC electrons so that the innermost; shells have the larpst
effebt,
For sufYicientl*y energetic ?-rays,
tho photoelectric absorptioki
ooefficjant goes 8s Z5/E 7/2 where Z is tho atomic number of the
5
and E the ?-ray enera* This formula indicates a very great differ-
9
ance b0.knreen the respective coefficients for Pb and C,
Them i's actually a
great difference in absorp6ion coefficient in the region whore photoelecbric
effect $8 the main source of absarp'cion,
This is the X-ray regtoi?,.
Compkon Sffect
_LII
magnitude of %he scaWiering of a phokon by an elecGron can be

88
uiicer.t;ainty of 30,000 e1eoi;ron. f~ol-bs, This aoouracy oomparcs favorably with bhat
of good optioal spectrographs,
The method of mass deterclinafion utilizes comparisons bs%ween ions of
marly qual rnwses suak as1
Differanoos of -these inasses are than measured. For acaurate measuraments, 5.C is
best if the ion ooncen6ratioas in the discharge are so matched that %heir intensities
are appraxim,%ely equal.
The scale OS measuring masses used in physids is based on the iaotope
0l6.
Tho mass of' this isollope 3.s S Q ~ equal to accurately, 16 unl%fl, then one
i;l&sS unit oorrosponds to 930 Ibv,
This quantity might be compared
with the mas8
of the olectron whioh aorresponds to ,til Pietr,
In ohemistry, it is usual to set @he atomic weight of the natural iso-
topic oxygan equal t o 16,
+,00031 x low2* grams,
In %his c&s@, the unit of nuclear masses Is 1,66035
In discussing a$omic masses, the mass defect is of'
interest.
1% is defined as 34 - A, whore I! is -the 1x83s rneacursd Ill %he phystoal
scale, and A is the integer number closest to 11 aL?d is called tho mas8 numbor,
A further qc'n-bi-ky frequently used is the packing frac,tiol?, which is defined as
F'I - A
In the following figure thb packing fraction is plot.t;od &gains% the
nuclear chargo,
s

The be& figure of Chis kind is found in %he ar6iole of EEahn and Mattauoh,
a9
Physikalische Zeitschrift, 1440, The packing fraction has a minimum a6 about
&other
method of determininG WSSRS is by balancing reaction energies.
For instance, the mass of the iieu.t;ron bas bean deOarmined in thi5 manner,
mass, of cau.rse, co~ild not be determii?ed in the mass spectrograph because neu-
This
trans caiinot be deflected Sn the electromagnetic fields of tha% apparatus,
the determina+ion of the neutron mass the reactfoil
For
Hi2 t hv -Hll *). 11~ 0
is used, Knowing the mass speotrographic values of the mass for M12 aiid lill
which are 2.0147 aad 1,00812 and ~isi.ng tho minlmwn snargy 2,17
Nev at which Gha
light quantum can cause disiiitegratio:1, one fbds for the mass of the neutron
1.00812.
In many othor CBSBS,
cross-ohecks are possible by using mass spectra-
grapliic values of %he masses and data from nuclear reactions.
Size of Nucleus
e. _”I ..“
-I-*---
.,-
One may de-bermiiie the size of’ atomic nuclei from various reaotions,
some of these, hoviever, like reactioim of capture of c low neutrons, givs errat;ia
values. l’he aatual size of tlza nucleus can be definod 8s the radius to vdiich
guolear forces exteml, and is usually ob6ained by inv6stigating the
deviatfohls from the Rutherfwd scatt-wing or by investigating the potential
barrier effects in c&-Clecays
and ot.her nuclear reaotiotis involving chwged par-
ticles. OKW fj.:icis that nuclear radii arc? given approximately by 1.4 x 10-13~1/3.
This maiis %hat %lie nuclear volume is proportionate to the iiuclear mass.
-..-
l!Tuolear Donsl-by
.I-.
different nuclei.
It follow that nuclear densities are approximately +lie same For

first disoovered in a structure of spsctiral lires which 1s CalledAhyperfine
90
s%ructure.
These structures were first observed at the end of the last. centwy,
but their explanation by Pauli and their more dehiled study goes baolc only two
deoades. The explanation of the hyperfine structure is as followsr The orbital
motion of the eieotrons produce8 maglietic fields,
magnetic moment associated with %he nuclear spins,
These interact with the
The interaotion elzero
rnod5fio s the sprectral lines in a differen% way, aocording to $he orientation
of the nuclear spin in the magnotio field of the electronic mokion; thus tho
lines are split,
From the limber of the Compomnks, the spin may be determined,
while from tho sizo of the splitting, "Fie nuclear magnetic moment may be oaScu
1 at e d
Daiia on magne0ic spins and rnRgnetio moments have been summarized up
to 1939 by llorsching (Zeitschrift fur PhysFk),
A now method for the determination of the spin and magne.l;ic, moments
has been davolopad by Xtarn and Wabi.
This method USOS atomic or molecular
beams passing through static and oscillating magnotio fields,
more direct than the om involviiig the hyperfine sZiructure.
have dotermined by this method, the magnotic moment of the noutrons which, of
course could not be dstermined by the uso of hyperfine structure.
The mothod is
Bloch and Alvarez
The following
table suznmarizos the results for %he neut;ron, the proton and the deukeran:
spin -- I
magnetic moment
.(I.-. --_11 -1
n 1/2 ? "1.93
HJ, 1/2 2b78
I1 2 1 0,865
Tlne spin of %ha iioutron has been assumed as 1/2.
This value is probablo, but
*
not quite oer.l-aj.n, The magnotic momon.t.s are measured in units of
eh
m
is tho mass of %he proton,
The minus sign i n the mp;notic momoiit of

positive aharge.
Vuo le ar s-bat 5. st i as
t; obays Dho %om
he premnt piattxre $hat nucloi with svon mass
,

92
. T11us one.muld have Qo believe by at nuolei oontain posi6rons 1
in addl%$on to sle&ironsr
A rather 8Z;rong argument has been given which shwrs tha% neither elao-
011s c~ii exht in polel.
If' a partiole is danffnsd to' a region
lirmir d$mensiorr$ &q then the momeiihm has an uncerb;ain%y */Aq, tha% is,
$he manreliburn may have any value from zero up 00 abou-b this latter value. l''70vr in
%he nuoleus Aq is a few timas om. From tho GO e spoading momentum uiicsr
tiainty one may aaloulats t
sprea4 in klnetlo energy by the formula
P2
-.I
2m
where p is tha momoiitum, and m fs the 8s of the partxiole in question. I%
.tihe pareicle is 8 proton or a neutmn, $hen tho uncertainty in kfrieOio energy' 1s
of the order of the nuclear bindi
OY~QF~IBS.
If, hovrsvor, the pni..ti,ole v m
Icinetig euerpg mould bs much bigger due to the
small value of $he mass 0% the electron. Aotually, i k i -t;l?

*
92;
The table shovss that nuclei vri.t;h an odd nusnbor of protons ar an add numbor of
mutrans and an odd number of pxotakis WQ particularly few. Thora &re only four
that; these nu.oloi form a simple sories. 3;
bo r@marked -bkx&t; not o'2&r
arc3
there fovmr kinds of nuclei with either an odd numbor of noukrons or an odd
number of protons, bvA also tlicat such i:uclei aro loss abundant;.
In connection wit& tho fa& %ha% nuoloi with an odd number of nsu%rons
or an odd. number of protons are rolativoly rare it may b? montionod that lluclei
vdth avr odd value of 2, i.0.
with an odd nunibor of pi*ck~h-~ ;iovor have more than
tno isotoopos, IZth the exception of tho ljghtas2; Q13i?E':'.n, thoso isotopos dikfor
by two L~QLX'G~OIM.
Similarly, if onc oansidars nuclol with a given odd number of
'?
sh QY?S

95
One might imagine the figure made more complete by cadding the energy of
the nuoleus, that is, the enera needed to pull apart all the constituent nw-
tronl; and protons,
One may plot this energy in the third dimension perpondia-
ularly ta *he plane of Flgufe 2.
The enargy surface obtained in this mannor will
have o valley along the dotted line near which tho s%able nuclei clustcr,
Curve as shown
CuZ;%,Zng the enorgy gurfaco along an isobario line one obtains &n onergy
o o ns t a nt
The straight livlos shown in the f'ig~r~ am energies of ehe various isobario
nuclei,
One of thaso energies will lzavo the smallest value and a corrospondihg
nuoleus will be the only stable isobar. The oorrosponding stablo nualwj is
indioatwd by a dot vshilo the uylstablo ones 4.136 shown as crossesb
shown which indioato the dkoction in which nuclear
Arravs are also
-transformations taka plaoo (I
Tho situation depicted in Figwe 3 holds only for nuclei vrith an odd mss numbor.
Isobars of such nuolsi must either contain an won nwnbor of noutrons and an odd
number of protons os olss ~n ovon numbcr of protons and an odd number of hautrons.
Thare Ss no systoma-tic deviahion bOtWQQn the onorgies of them two kinds of
ass number is won, than tho nuclai my eithor contain an moM
rons atzd an won number of protons, or else, an o4d numbor QS
P
neutrons and an odd number of protons.
Tho formcr kind of nucloi have system-
atioally 8, lawor cncrgy than thc lattor kind,
Thus .tho nuclear onergias will
lie on two roughly parallol curves, tho highor on0 oorrosponding to tho odd-odd
QBSC, the 1mrcr on0 Go tho avon-ovon CRSO. This is shown in Figure 4.

96
e
E
2 odd N odd
,Z even N even
A
constant
Fig* 4
It viE1 be noticzed that In this case more than one stable nuolaus may exisf.
In
the figure,
such nuclei w e shown.
The one on the right h
han the one to the left, But the nuoleus with the higher egergy
oould ljransforrn into %he one of tho loiver energy only by simultaneous emission
or two charges which is an exceedingly i d3&tble process
Emission of one
electron or positron) would carry us t o the upper ourvs and tvauld require
snere rather than produce it. It will also be notiGed from
on the highsr ourve may transform into G nuoleus of ~ Q W energy ~ F &%her
king to the right or the left, The
that suoh a ~iucleus emits positrons and elec%ron$.
This has actually been
ob served for several nuchi *
From the sysbernatics of the nuclei, aonolusiona about nualear forces
r nutlet. It is %Q be explained by
wtrone and protons are similar.
The explanation also uses the Pauli prtlnaiple, which asser.t;s that in each state
9
only one parkicle oan exist8 that is, in a gtvsn orbit within the nuoleus one
may have not more than two neutrons (or 80% more thm two pra.t;ons) differing 9u
the direcfion of their spips. From th quality of the foro s acting 012 mutrons

*
and proGons it- follows that up to a given energy %her@
of orbits available far neutrovls and protons.
97
will be a similar number
PF one then filla up the lowes$
orbits available up to a given epargy, one mi13 have to use B roughly eqLtal
number of neutrons and probons .,
This is t o be explained with the help of ths Coulomb law,
It is
easier 60 add a neutron than a proton Do a nucleus whioh is already heavily
charged,
In lighter nuclei, the Cablomb ~O$?O@E are considarably smaller than
%he nuclear forges,
In heavier nuclei Stho Coulamb energies aS the several pro-
tons due to the longer range of 6liase forces add up and beaoma oveiitually
comparable to the nuclsar energio s
Mass defeob and volwne proportional to mass number,
--I .---- -crlucI. .ua"-
These facts ir?dicaCe forces having a saturation ahsracter, +ha% is, a
situaWan 5.11 which one particle oon'xibutos offectivaly $0 the to'l;aX energy by
interaq+iihg with a llhited number of &hQr particles.
The aame situation is
encoun-borad in C;rystalS
Bj.nding onergies of DeuBaron and a -partiole
-I-.-+.--.*--~ .I* * r *nrrUu.lr- -.-.-
Tho fact +that tlia &-particle has a more khan LO times greater binding
enera than the Douteron sliav~s that tho saturation has not been roached at the
Zatter przrWcle.
In the &-par.t;icle
the saturation i s actually reaohed.
Thls
a% R particle, for instimice a nou
nuoloua with another neutron and %wo otihar protona.
an, can effcc+iv~ly intafact in a
Tho usus1 intorprotation is
that a par%icle oan intQracat with all oChor parfioles which aro parrnit-bod by the
InciplQ t o move in tho same orbiti and %hat tho intorao6ion does no%
dopond yory strongly on tho differcnk spin directions vJ'hIch tho particles my
*
possess in this orbit,
The dc Dislhtogration.
LI*-.--IUl*.-IUUC.
The &-particI.as
omitted by a nuc2ous have a well dcfivlsd onargy and

m
range. A vory small fraction of tlia a;-partido has somotimos a considerably
longor rnngo. Tho main faot of tho uniform onorEy of tho a ' s may bo o::plaiulcd
by assuming that the &particlo has boen srni-blied by a nuoleus of R givon enorjg
and that arnothor nuclous of a givon miergy is left bohind after OhoQCI-dooay,
The anorgy difforonoo is thon oarriod away by tho C;C-part.iolo,
Rnat liar important law ~f tho &wadj.oaativity is tho eonnoctioii bc%?voon
tho range of tho par'ciolos and tho docay aons%mt,
Thts is tho Geiger-Nuttal
relation,
It statos thab the logarithm 0% tho docay oonstrrlit is a linoar funm
tion of
tha railgo,
Actually one finds that tho relation botiwocn tho logarithm
of tho dacay constant and tha rango is sligJitly diffcron% for tho radium,
aot initun, ntid trhoriuh famillo s,
Tlzc throo straight liiio s -&pro sonting the
relations for thoso thrm familios arc liovmvor parallel and their diatalioos from
oach othor ara small,
If oil0 had plottod %ho logarithm of tho doaay oonstmt
as a fuiiotion 0% thc onorEy ratilicr than as o, function of tho rango, tho plot
would again bc in good approximation a straight linc.
Thc oxplana%ion of tho Goigar-1Vuth.l relation is closoly oonnoctod
with the shape of fiho potential anergy aoting botwem tlia d-parbicle and the
frwtion of the nucleus which +:le tr;-parbiole has Just left, At radii bigger
than the nuclear radius ro chmm in the 'figure, the interaction is a Coulomb
repulsion as shoTim in +he same Tigura.
At distances closer than ro some kind
of an at0raction must @xist, otliemvise %hers would bo no rea on why the origMal
hualeus that ha3 emitted the &-particle
had stayed together any length of' timer
In the potential valley at distances smaller than
rQ
the &-partiole
had in tihe
original nucleus an en~~cy.r level at; +lie anergy value El

99
I
Figure 5,
After thea-particle has moved out t$o suffioiently greati r
values BO that all
Znteraations including the Coulomb interaction have become negligible, the
Cjpartiole will PcSsseS; that energy E in %he form of kinetic enera.
If' now *lie product nucltsus is bombarded by&*particLes of energy E,
pure Rutherford scattering is observed, iiidicatinG that at the radtus router to
which the =-particle
can penetrate aooording to classical mcrhanics, the unmodi-
fied Coulomb force still holds,
There arises the quostion:
in what way can the
dr-partiole OY'OSS the potential barrier bebveen rinser and router (see
figure 5 ).
This quss2;ion has been answered by Gurne$r and Condon, and by Garnow.
In qu4ntwn mechanios leakage through a potenthal barrier is possiblec though %he
probability of suoh a process becomes small when the barrier becomes high or
broad, The time of decay may be estimated by imagining the &-partiole oscillating
wi%hin the potential hole, by calculating the number of times the &==partiole
bumps into the barrler per second, sand Pi)mlly, by multiplying with %he
probability that the amparticle penetrates .the barrier in one collision.
The facC that the a-particle can penetrate through tho barrier is
*
conneoted with the &partiole's wave nakuro. An illustration from optPas may
be g5.ven. If light impingos on tho intarface between glass and air from the aide
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
exponeriCla1ly 4eoaying field 3n air and par+ of' the 13gh-b will be 6rammfttad to

*
104
nuolei are known whioh emit in turn, an alpha, a beba, and then another alpha.
The 4x0 aLphas have quite definite energhs, yet the beta partiele does no$,
Some years ago,' attempts were made to meagure the energy of the betas by using
a very thick Pb w alhd oalorimeter,
ft was hoped at the time tihat all the energy
of the emitted particles would be measured.
However, an amount, aorrespottdfng I
o thw ama under the distribution ourve, vas found.
Bohr made the suggestion th& perhaps energy and mornenWm vms not con..
served ina-4$sintegration;
seoond partiole .IC
still valid,
A spsuial hypathesia made by Pauli was thak a
the neutrine -- was emitted, 80 that aanserva'binn laws were
Attempts ham been made to oorrelate the maximum enera E in a begs
speofirwrn wifih the disintegration aonstantA I as was dona in the uase of
&,-partiales (Geiger-Nutall laws), "he Sargent ourves, a plot 02 log h YS E,
we an attempt in +his direction.
From absorption measurements in aluminum and using a standardized
method of extrapolating tho rangeeenorgy ourve, on0 obtains
R 0.543 E 0.16
,
where E is in million electron volts and R in cmb (Fsathor rolation)r
A satisfaotory theory of beta-disintegration must explain the oon-
tinuous spotrum and give a oorroot rahgowenorgy rolation.
Formi has dewlopod
a thoosy with bho neutrino hypothosisr
In tho nuolaus a neu~ron is aonvar%ed to
a proton 'oogother w9.tjh tho omission of an olootron* Tho rolo of Z;b olao..t;ron 5.8
similar to that of' photon in radiation thoory. Tho probability of omissZon OS
tron with onor#
in tho interval El E # AE for lfght nuole3. is gSvon by
P{E)dB g H%7 t{U$Vmdy12 (Eo - Edl3
whore E is in unl2;s of C the voloaity of light, E, %hc maximum

The noutirop is expe&ed to be
astive with Q mean life of several
hours. Experimental evide'noe 9s hckin oaqse the neutron renot6 relatSvely
quiakly k th any nu'clsus that may be pr
tr Hwm@r, experimefits hhvo boon
proposed to mocasurc3 this disintogratlon oonstaht.
Novombor 18, 1943
LA-24 (18)
Lectmor: E, SO~PO
UCTURE XVXfXt
ON AND K CAPTURE
I
it pr>si%rons, giving riw to 1Jsobars with one
nuclei.
This roaakian OOOU~S if tho avail-
able enorgy is at loast 600 kv, tho she~gy oquivnlsnt of tho gosl.ttronz, that is,
$1 tho mass of $he parent nuolous sxcoods thn2; of tha daugh$er by at lwst tho
mass of tho posttron.
If $he onwgy nvtrilnblo is lees, then tho nuolous will
0
capturo an oloo.t;ran fram the innormost shell and subsaquently emit an X-ray

106
quantum,
This so-called K (electron)-oapturs by the nucleus also ooours at $he
higher energies but la less possible,
2) , Gamma ,Radiat;ion
w. , ,
I
The eleotromgnetio radiations emitted by radioactiva wbstanoes are
oalled ?-rays.
The Y-rays are aharaaterized by %heir frequenoy, or 8s is more
usual by the energy of a quaRtwn expressed in electron volts OF in unsts of
2 The range in energy of 9 -rays from rzsturally radioaotive subs.t.anos.s is
approximately 60,000 eV to about 3 Mev (million tslootiron volts),
By bombarding
Li with pro4jons more energstio ?-,rays ara obtained, values up to 17 MeV have
been faundr
The highest; energios In tho lwb have boen renohed with tho XersG
betalwon,
The theory of ?-ray omission is similar to that of the emission of
light from atoms and moleoulad, oxoept that tho magnitudes of aomo of tho quan-
tities involvad are muoh different,
The probability of a transition from one
onore fitate say m, to another lover enorgy etato n, and tho omission of Q
light quantum is &ven by
whore 4.d is 2fi times %ha frequonoy of the omtfkod radiation, C tho volooity of
light.
P i s tho so-oallod matrix olomont of tho transition and is tho ol0a.Dria
momont avcrhgod over tho oigonfunctians of tlio two statas m and p* Equation
(1) roprosonts tho first torm of an expcnsiqn in powars of co/x whom ro 3s
tho nuclear ra@ius and A tho wavolongth of %he omittbd quantum,
PhysicalLy,
tho flrst torm in tho oxpansbon corrosponds to tho dipole radiation, %ho aoomd
ta %ha guadrupolc radiation, otor In atornib prooossos, is 60 largo compnrod 1
to tho radius of tho atom, SO that the soaond term is vory muohismallor than tho
*
first,
For nuoloar procdsms, huwovor, )r and ro do not diffor as grocxtly; con-
sequantly tho second term mczkos an approciable contribution,
Tho magnitudosr of
the matrix ~ lomont~ vary over o. oansidorably w2da rnngo corrosponding to wido

is ejeoted, then
'kin
Enuolsus a Eg
This prooess is known as internal cot~version, Photographs onfl -speotrograph
show tha2; the ihternal ootlversion electrons, sa is to be expected, form very
homogeheous groups in contradistinotion %o disintegration -alectro~ls which have
a oont$nuous distrj;but%on in energyr
Usually a group of lines of varying inten-
sity is observed cmrresponding to ejoation of electrons from different onergy
levels of the atomv
Oacasionally3. -radiation is acreompanted by fl-disintegration and the
questSon arises, as to whether the radiation precedes or follows %he disintegration,
If' inkernalfy converted eleatrons are presen4i and are analyeed to deter*
mine the energy differences beheen the various groups, a11 anwer o m be given.
For if the spacing corresponds to an Xnray diagram of the paront nuolsus, thovl
%he 5 -radiation precedes %he disintegration, whereas if tho spac32ng is that of
the dawghter nuclous, the cmnvarge is truo.
The ratio of the number of oonvorsion sleotrons to the number of
3, -quan.l;wn is lcnovm as the convorsion coeffiaient,

108
reLatlvely very smdl 80 %hat the me
Zy large, %he exaifed level $8
r8'
life Pf %ha% leveb S,
be mefastable,. Le% us
are in the ground state, %hey
underg -disintegra+ion wi-tih a half- 18mr If on Lhw othw hmd,
nuclei are in the first sxaited level, wbiah has a half-3iie for @)I
L
-eminsfan
en the observed halfmlife of t
sinae the rate defermining fact
proof of suah "isarnsri.o!' nucl
sequentfl remission &a ,found ts be
the Blower ra2;e of
hat they oan be produeed by exaita-
s or e?.tzo-brons, first; t o a higher level, from wbioh tihey reaoh the
by 3 -@mhsS$Ot1 of
gh, ra%e r simple ohemical te thni
rt J.
H+ Williams
POD Was found by Both@ .
(1930), (Naturwiss -#39 753 (1931)). They
published the results of tmme experiments i n whbh the 1igh.t elsmen%s L1, Be,

109
and B were bombarded by polonium alpha particles. A wry penetrating “radiatiopf1
was emSItted which vas more diffioule to absorb by lead than tho shorbst wav~
- lowlgth 1(cY)ays Ohon known, thosa of I‘hC’’, Their orlginal explanation was tha%
the radiation was light of a vary short wavo length.
Today wc know that ths
inoreased absorption of wry short WWo*lQng‘ah @)f -rays by the prooass OS pair
areaLima would alimina$e this explanation.
They war0 lasd to *he aasurnption %ha%
tihs Itradiation” was pho.t;ons on other grouurda, bosSdos tihat of trbsorption measure-
mentis, on0 being that an traoks were obsorued in a cloud chamber,
Carrying their hypothosis of hard to its logical conolusion on
the basis of the Klein-Nishina absorption curve as a f(E >, thoso early
”r
axporimehters gave an enorgy for this “radia+ion” of from 7 to 55 Move Their
wbsorpgion moasurcmants were mado wi%h Gobgar Fountor dotootors,
In a nuoloar
squation they pos%ula%od that the reaotlop involved was
Supposo for an exoroiso wo oheolc tho onorgy balanco of Ghis equation
by subeti,tuting masses in Z;ho oqua2;ion
qBe9 9,01504
$Io4 - 4.00389
-1ZTiRZ
c33 I 13,00761
-‘TUTEZ mass uni+is
Ono mass unit
931 Mov
Enorgy availablo EG-b10,5 Mov,
The problom was next attaohad by Curia and Joliot (Comptos Rondus 194
273, 708 (1932) )., Tho important difforonco was that in Chcso resoarohos, thin
window ionization ohambers instoad of Goigor gountors wore usod as dotootors of
n.lnbul’
tho radiation.
Thoy found %hat if matorial aontainlng €1 were into.rposod

a
botwoora. tho &on Ba SOU~CO and tho ionization Qhambor, tx
'ianixa$ion was obswved.
obsorvod,
2arg~ incroaso in
110
For absorbors of othor kinds only small docroasos 'vmo
Thay showod that this fvloraaso in iolziza~ion was duo %o the prescnco
of H roooils leaving; thc "absorbcw" and ontoring thoir ionization ohambor.
Cloud ohambar studias revaalad tho prosonce of those proton roooils as ~013. RS
Ho racails in n Ha gas.
By moasuring tho lengths of thoso roooil tracks it tlrns
possfl~1a to de'cormine tho onora of the roooils,
Soma of thoso rocoil protons
had enorgias up 00 5 MOT,
No traoks wro absckod for tho primary ltradSationtl
which thoy assmod to bo? -rays.
Curio and Jo1io.t; oxplninod thoso obsorva.Dions
by saying that the onorgy lirnnsfor ocourrad in a Compton pro~~sa~ Tho? "ray
enarm otilovlated in this way turned out to bo 50 Mov.
Unfortunately, tho oar-
oulations for othor roooil pnrticlos on Z;ho basis of this %hoary $avo diffaronf
?-ray enorglrss,
They WO~O highor for hanvior rocoil nuolai,
Chadwiok (Prac, Roy. SOC. A* 136, 693, (1932),) solved %hJs d ilom sa
c-
aonvincingly tha% ho reooived tho Nobol prize, A sohomatio diagram of the
apparatus usod follows.
FFg '
1
-**
7-
I
I
7
VQCUUm paraffin ionization ahambas
l
amp3iFiQr
Tho measured rocoil onorgios of H' and
nualai vmro 5.7 M w nnd 1.2
Mov
cospoctivoly.
On the basis of tho phofon hypotiho sis tho oalculatad ltradlatioatl
onoF6;y would bo 55 MQV and 90 Mov rospocti-valy.
Thus 'Gho photon bypathosis was inconsistant with tho obsarvntionsr
Either tho laws of aonservation of onorgy and momantum or the hypothosis as to
*
the naturo of tho pherzomnon was 'Go bo rojeotod,
Chadwick thorefam assumed
that tho "radiation" was a new pnr%iolo of mas8Nmss of proton arzd having Zero

mutual forces between nuclei are fhe subdoof of Mr.
Artifioial Souroos of Neutrons
3.12
Critohfieldrs leoturesr
Not a source of monoonorgotic neutrons,
This mode of disintegration withdj.particlas as tho bombardikg particle
1
which results in neutron produotioh has beon obsorvod for sovcwal light oloments,
Li7, 1311, NI4,
FL9, Na23, Mg24, A1Z7, but is moa6 intenso from Be, then 3 and Li.'
Those souroes af neuJcrons am assentially v&y woak, ralativo Do
sources of whioh we shall speak lator,
Tt was impoosiblo up bo 1942 to get
sourco8 in OhSa manner using Po as tho G-sourco which give mor0 than ,405 r/goo
or oxpressod in anathor way tho yield is 60 n/lOs stoppod, On the o6hor
hand, if tho &,$s &re obtainod from R, which is intimately mixed with Bo, the
numbor of noutsrons em$tt.l-od per mor par mq. of Rn is of tho ordor of 104/soa.
With a curb of Ra on0 thwofora has an appreciably notivt7 sourcoo
Unfortunalioly
' 3( -rays are prasont hero whore %boy aro abscn9 in tho Po &~ourcos, except
tw 1 h ?.',ha
A complete disousaion of @Len roac%Sons, %ha enorgies of tho neutrons
and of .Oxlo Z X S S O G ~ ?*rays ~ ~ Q ~
is given in Livingstion and Bcthe, Rev', Mod, Phya',
9, pagee 303 to 308, 1937,
*-
Photoe1oct;rio Produa@ion of Nautrons,
With two of the li&% elaman4x H12 and Bo noutrons oan bo produoed
4 '
by tho 7 nmys of' ThC" or aqy ?trays whose onergy OXoQeds fjhg bi'nding enorgy
of tha neutron in tha primary nuolOuar
The probability of $his process is low
and as a oonsequonco tilio yield of those SOUYGOS
-
Examplos - H2+ hW-+H1 .t n'+
Q
is oorrospondingly' lawc
Chadwfak and Goldhabar, {Procr Royv SOC, 151, 479 (1435),) usad tho ihCtt +L*rays
of anergy 2,62
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.
113
Sinoo E, E ~8 oonsaquonoo of their having approximately cqual
P
0 masses, tho olnergy equation moms
The analysis of %he data gav0 Q "-2.2
MCIV
This moans tho naukm is bound fn*tho H2 nudous wi%h an sncrgy of
212 MWVI
Knowledgo of this binding anorgy is of groat thoorotical importance
sinoo it allows us to dskormine the maso of tho nautwon,
Mass of Nou6ron
.-...I
RowrlOing the pho.i;ordisintagrat;ion equatlon
Mn a 1,00893 f?OOOO5
A tnblo in Livingston and Botho, page 353, swnnarizos tho avidmoo on
?-araac%;iQns up to 39370
A modem usoful SOUFOO of noukroyls usiylg ?-rays on light elemonts
illustrchtas tho advanoos in toohniquo mada in nuolsnr physics in rooenf yomsu
As we vrS11 dlsou~ls IcL~Q~, It is possiblo to produoo an isotopo of
by dou4xwon bombarbont of Br vfhioh has a l/Z Iifo of 80 days nnd omits a "')+-ray
of suffid.en6 anorg to dlssoclake Bo9 and givo nautrons of 16Q-k~ onergyt
This souras hnq boon extromoly usoful RS Q SQUTOO of poln%ively monoonorgo'eio
Y
e

134
December
2, 1.943
Third Series: Neutron Physics Lecturer : J. H. Williams
Neutron Sources (Continued).
Neutrons may he produced by
the bombardment af certain nuclei with protons, deuterons or alpha
particles, that is, a nuclear faaction takes place and @
emitted.
neutron is
To initiate the re~ctim, the bombarding particle mu8t
penetrate the potential barrlar surrotmding the target nucleus
In other words9 .the projectiles must have a sufficiently high
kinetic energy
We may distinguish two methods used to e.ccelerate such
particles, namely direct and resonance acceleratcrs,
Tne first
is long vacuum tube one end of which is at a very
high potential and contains a sourca OS ionsS and the other i s
usually at ground potential and contains the target.
The ions
traverse the vacuum tube acquiring kinetic energy came sponding to
the drop in potential,
The resonance acceleratorg or cyclotron, consists cf a
flat, hollow cylinder
resembling the letter D. An
slepara ted into two sections each
oscilla tine voltage difference
applied between the ~ W O “doesft OS the order of.
and &t a Frequency of several rnapacycles.
a hundred kilovolts
Along the axis, a
‘magnetic field of several thousand. gauss is applied,
The ion
i

*
115
source is on the axis, Under the Froper conditions, the ions
travel in circular orbits of increasing radii and Increase tketir
kinetic energy during each passage from one af the dees to the
ather. The sucees~ of the cyclotron is due to the fact that the
angular frequency of the motion is independent of the energy (ieeo
of the radius of curvature of the orbit). When the voltage frequency
is equal to the angular frequency, resonance conditions
obtain. 1t'l.s possible by such resonance means to attain energies
of the order of several million electron voltso
In this type, the Incomlne; particle is a proton and the emitted
one a neutron. Written in detail:
ZA + HI-+ (Z + I)* + nt
where A is the mass number and 2 the atomic number, Q is the
energy of the reaction, It turns out that Q in th

116
and 1.86 MEV. Thus this method is very convenient for producing
mono-enerpetic neutrons simply by controlling the bombarding voltage.
It is clear from consideration of momentum and energy conservation
that Tor a given bombarding voltage, the energy of the emitted,
neutron depends on the direction of emission, so that neutron6
I
observed in the backward direction of the beam will have less energy
than those in the forward direction, SQ that sne may obtain a range
of neutron energies as a funct$on of angle. Practically, there is
an effective spread of bombarding energies because of the fact that
some of the neutrons will be slowed down in the finite thickness of
the tarpet before they initiate a nuclear reaction. This variation
can be made arbitrarily small- but is usually of the order of 5 to
50 kv for intensity reasons,
I_di,_n_) reaction: Here the typo is
ZA .c ~ 2 - (Z I 1W 4 n1 I Q
The so-called (d,d) reaction is an example:
H2 4 H2 .crlt He3 4 nt + 3.2 MEV
This reaction provides a copioits source of neutrons. Yields of the
order 09 IO8 neutrons/sec are possible with bombarding currents of
the order of lOrA impinging on heavy ice targets. The efficiencies
are of the order of lnh05 deuteronso A desirable feature is that
the reaction has an appreciable cross-section at low bombarding
energies. One difficulty is the preparation of thin tarpets. Often
gas targets are used but the yields are correspondingly smaller.
Another example is
c612 + HI2
~*lb* N13 + n' + Q

117
e
with Q t -,25 lvaEV9 which means that the reaction is endoergic.
The energy spectrum of the emitted neutrons is not simple because
of the C13(d,n)N14 reaction.
ut_eRra_tions muted by neut= &a) re&ctiongA
An example of this type is
n' + N14-He
4
+ B 11
One can obtain cloud chamber photoeraphs of this process to
establish its identity beyond question, but it is a somewhat
tedtous procedure since many photographs must be taken before one
finds evidence for the reaction. Another well known reaction is
and also
nf C Li6- H3 9 He4
330th of these reactions have relatiy?ely very large cross-sections
for thermal neutrons and are very useful for the detection of
neutrons by detecting the ionization produced by the emitted
cparticles. A common procedure 3.3 to line the walls of the
ionization chamber with a layer of boron. Often proportional
counters are used which contain gasious boron trifluoride and the
individual particle pulsas are detected.
&D) _r_q&tian: Symbolically this reaction is
*
This reaction is usually endoergic, the case ZA c,~Nl4 being an
exception.
This type is the simple capture
process in which a neutron is absorbed and a gamma quantvm

I
118
e
omitted.
Process was first observed by Fermi in the production of
artificially radioactive isotopes by slow neutron irradiation,
&.$I
r3Gion:
ZA + nt ...lp ~A-1-t 2n
It 9s clear that this react.ion requires at least an amount of
energy corresponding to the binding energy of a neutron in the
nucleus. Hence fast neutrons are required to make reaction go.
This process ban only be reliably identified if the resulting
nucleus is radioactive,

*
LA-24
December 7,
1843
(21)
119
Third Series: Neutron Physics Lecturer:: J, H, Williams
Slow neutrons are produced by allowing fast neutrons to
pass through paraffin, The collisions of the neutrons with the
protons in paraffin result in the initial energy of the neutron
being divided up between the neutron and the proton. Thus In
successive collisions the neutron loses energy until it finally
winds up with thermal energies I)
The process of slowing down of the neutrons depends on
the collision cross-section between neutrons and protons, We shall
discuss the concept of a collision cross-section, as it has been
introduced in the kinetic theory OF gases.
Let us assume that two gas atoms collWe if they come
closer to each other than the distance s, Assume further that the
relative velocity of the two particles is V. Then, %he particle
under consideration wllZ sweep out per unit time a. volume Ss*V and
the particle will coll%de with all collision particles within this
volume, Therefore the number of collisions par second is $T.s2Vn
where n is the number of collision partners per unit volume.
Assumln'g that a11 of the relative velocity Y is due to the motion
of the incident particle, the distance through which this particle
0 travels between colllsions, that is the mean free path, is given by

1)
I
,&=l/fls%. (in the kinetic energy of gases the incident particle
and its collision partners move with the same average veloqity
which introduces into the farmula or the mean free path a factor
1/42.)
distance x
The probability that a particle should move through a
without making a csllision is given by e -x/k re;
e -xm% Here the colZislon cross-section o t.f$s' has
been introduced,
This cross-section i s in gas kinetic theory the
geometric area within which collisions take place.
the above exponentSal formula may be used to meaaure the cross-
--
.L
La
Experimentally,
section, For this purpose a $'lab of the material made aut of the
collision particles and having thickness x
is placed into a beam
of incident neutrons with intensity Io, A smaller intensity I: will
emerge from the slab since the neutrons which made the collision
are missing from the beam, Then the ratio of the two intensities
Ss given by the formula I
=cox
,/ro
X
I
0
Nuclear cross-sections have usually the order of
magnitude 10 -24,,2
This quantity has been introduced as a unit
of nuclear cross-sections. It is called the barn.

3.21
e
The collislon cross-section and mean free path o f neutrons
in paraffin vary a great deal. with tha energy of the neutrons.
crawwmction Is a few barn9 above a My whereas it i o fifty barna
for thermal energies,
few cm, while thermal neutrons have a mean free path af a few mm.
The
At the hSgh energies the maan free path is a
ThB ~nergy loss d h natttybn in a collision with a probdti
depends on the angle of deflecti6M. rf the ifiitial energy of the
neutrons is E,
collisions i s Eo/en.
then the average energy of the neutrons after
to.reduce fast neutrons to thermal energy.
Thus apnroximately 20 collisions are necessary
The number of slow neutrons that can be obtained in a
given volurqe of paraffin depends on the rate of production of sluw
neutrons and on the rate of absorption of slow neutrons by the
hydrogen in the paraffin,
H' + n'4H2 + htb
This reaction is only one example of the very
common process in which a slow neutron is absorbed by a nucleus and
a )t
active.
-ray is emitted,
This absorption is due to the process
The resulting nucleus Is frequently rad5o-
(This is not the case when neutrons are absorbed by HI
The production of radioactive nuclei provides us with an easy means
of measuring the capture cro9s-section9 iqes9 the part of the calli-.
sion cross-section which is duo to the capture process.
n

*
122
To perform the measurement we use a thin sheat containing n absorb-
ing atoms per om2,
and serr~nd.
We let N nuclei impinpapan the sheet per cm2
c ,
Then the number of r%dio-ac$ivs nuclsP fo~r~lad, per cm2
aqd second. is given by nNo,,
Thus the capture croasmxtlon may be
measured by determining the radio-activity formed b
Some nuclei have very high thermar capture orosstsections.
For instance, the capture cross-sectlan of cadmium is 3,000 barns,
While this cross-section I s very muoh greater than the geometric
cross-gection of the nucleus, tha scattering cross-8ection at car-
responding energies i s only a fev barns, that is, comparable to the
geometric cross-section,
A further peculiarity of the capture cross-sections at law
energies was found by Moon and Tillman (Proc. Roy. Soc,
A =_.467
(1936) ) They observed that the capture cross-section changes
rapidly with the neutron energy showing maxima as can be ~een in
the followinp figure e
The maxima are separated bp distances of the order of 10 electron
0 . vults on the enerpy scale.

123
The crude explanatfan of this phenomenon is resonance.
The word resonknee suggests the analogy with the great energy trans-
Pers which become possible
quencies are equal to each other,
between vibrating systems If the fre-
If the rest energy o f the initial
nucleus, plus the rest energy of the neutron, plus the kinetic
enemy carried in by the netttron, happens to agree with the energy
of a compound nucleus made up from the Initial nucleus and the
neutron, then the rearbIan becomes more probable
For instance I ths
above reaction of neutron cap turs by hydrogen may be written in
form HI .I?' n? ++H$-r-)-H 2 $I hz/ where the carnpound nucleus H2* is
an excited state of the nucleus H2,
Not only capture but a .1~0
scatter5hg may be describd with the help of compound nuclei,
iristance, the scattering of neutrons by protons may be written in
the fallowing form H1 4- h'+H 2 jc -.-E1 .f. nf
For
The various means by
*
which the compaund nucleus may disappear, that is, emission of a
quantum or emission of a neutron, or possibly emission of another
particle, compete with each other, At low naut;'ron energies emission
of a 9 -quantum and consequent neutron capture predominates for
most nuclei.
The first theory of resonmce sarture of neutrons was
given by Breit and Wignsr (Phys. Rev. s...T19 (1936) >; Rather then
discuss the complete theory we shall describe the behavior of the
neutrons
A) if their energy is very close to a resonance
B) if their enorgy is very low.
In this way, we shalL understand the peneral appearance of the
curve for at which has been given above, that is, the general rise
I

1)
of with demeasine energy (8) and the superposed resonances (A).
If condition [A) is PulfllLed, the collision cross-section
is deteranlned by the sinple neiphboring resonance with energy E,.
Then, according to Breit and Wlgner a neutron with energy E .
possesses a collision cross-section
'
Here k is the de Broglie wave length divided by 2'R of the neutron
of energy E and, fir is the same quantity for the case that the
neutron possesses the resonance energy E,, f is the width ~f the
resonance and Fn is that part of the width which is due to the
neutron emission, A datafled explanation of these quantities will
be given in a later lecture.
If E - E,
is great compared to r the co?-.iision cross-
section w ill fall off inve r s e ly as the square
of the resonance.
In exact resollitnc~ only the
sz both sides
the denominatar and at this point (or rather cl-ose tc this point)
the cross-section attains its maximum.
7"-r
2 t~m remains in
IF on the cther hand, the energy osf the neutron.is small
compared to all resonance energies (assump K!.[J~ B?, then the energy
dependence of the cross-section is determined by the variation of
5 with energy. In this region, the cross-section varies as 1
over the velocity of the neutron.
on the spacing of the resonance levels.
level is reached the l/v law breaks down.
The size of this region depends
When the first resonance

December 9,
1943
LA-24
Third Series: Neutron Physics
Lecturer:
Lf the width of the lowest level
is greater than the
energy E, of the lowest level, then the Z/V Law will break clown when
the neutron energy becomes comparable t o the width r
The above statements concerning the region of
the l/v law can be proved from the energy dap-endence of
cross-se~tion as given by the Breit-Wkgner formula (see
lecture).
This energy dependence has the form
validity of
the capture
end of last
One sees that Cr,
d/dE
will behave as
-1/2 ))
log E
l/v if the condition
r *}
is
satisfied, This in turn is proved if either
ar

126
One very important substance in slow neutron physics is
wdmiun. Its absorption crossIsection as a function of energy is
roughly represented by Fig. 1,
Its strong absorption for thermal neutrons is due to a resonance
lying very close to thermal energies, A few tenths of a millimeter
of Cd shielding is sufficient to remove from a distribution of
neutrons the thermal neutrons.
The first experlments, by which information on the Cd
absorption and on the distribution of thermal neutrons was obtafned,
were performed at Columbia, The experimental arrangement is shown
in Fig. 2.
'B
Ra+ Be
Cd

*
e
127
On the left side of' this figure a paraffin I1howitzerlr 28 shown,
which is a block of paraffin shaped as shoyn in the figure and containing
a radium-beryllium scmrce at the indicated positjon, The
paraffin blook emits S ~ Q W neutrons from a11 of its surfaces hut the
arrows indicate a particularly stronp and somewhat better directed
beam of slow neutrons ppoduced by the peculiar shape of the howitzal;,
This neutron beam impinges upon a velocity selector which is showp
in the figure to the right of the howitzer.' This velocity SeZectar
consists of two rotating discs mounted on an axis at a distance D
from each other and two stationary discs located before the first
and behind the second rotating disc, On each of the Tour discs, Cd
segments are mounted in such a way that there ar0 on every disc 50
Cd segments and 50 segments free of Cd, This arrangement is shown
schematically,below the velocity selector in the figure, To the far
right of the figure a boron trifluoride counter is shown which
detects the slow neutrons that have passed through the velocity
selector.
The velocity selector is rotated with a velocity of n
revolutions per second, The Cd strips of the rotating and the
adjaxent stationary discs wSll at a given time owrlap, so that the
,
neutrons can pet through the Cd-free parts of the disc. Shortly
thereafter, the Cd segments on the rotating and stationary discs
will alternate so that no neutrons can get through. IT the velocity
of the neutrons is such that they will find either the flrst or the
second pair of rotating and stationary discs closed, then a minimum
of transmitted neutron intensity will he observed, Assuming that
all neutrons have the same velooity v, this minimum should occur
I

when the relation
r*l
v E lOOn/D
is satisfied, It has been found that the neutrons omitted by the
howitzer have an approximate velocity of 2,500 meters per second,
This corresponds to thermal velocities,
A somewhat more detailed
1
analysis showed thgt the neutron velocities have approximately a
Maxwelltan distribution shown in the figure.
In this figure, the number of neutrons having a given anergy E is
plotted against that energy, The maximum of that curve shifts to
higher energies when tho temperature is raised,
An InvestigatSon of the resonances occurring at a few eV
was first carried out by a somewhat indirect method, This method
takes advantage of the fact that the reaction
has a cross soctian obeying the l/v law up to quite high energies.
The experimental arrangement is shown in Fig. 4.
7°F"
13 absorber
Source *
Cd box
Detector fotl

129
On the left hand side is indicated a neutron source (this might be
a howitzor or another arrangement)# The neutrons emitted by this
source pass through a boron absorber of thickness x,
are carried out with various thicknesses X.
Experiments
On the right hand side
of the figure is seen 8 detector foil which i s made of a substance
mes radioactive under neutron bombardment.
It is in thisz
substance that we want to study tha neutron capture r@sonances.
Tho detoctor foil is enclosed in a Cd box.
. ment are made!
Four kinds of measure-
I. no boron absorber present, no aadmim present,
2. no boron absorber present, cadmium present,
3. boron absorber present, no cadmium present,
4. boron absorber presant, cadmium present.
Comparing measurements 5 and 3rl one Finds what fraction
of the thermal and higher energy neutrona have been absorbed by the
boron,
Comparing measurements 2 and 4, one finds what fraction
*
higher energy neutrons have beon absorbed by tho boron (the ,
thermal neutrons are stopped by the Cd box),
Comparing tho d$fforenco of 1. and 2 with tho difference of
3 and 4, one finds what Traction of the thermal ne'utrons haw been

130
absorbed by the boron,
We plot in Flg. 5 the quantity log Ax, that is, the
logarithm of thc intensity observed in the detector foil as a
of' the thickness of the boron absorber, The lower one of
the two stralght lines in the figure
"j A
l - ~ . ,
-7t
is abtained by taking the ratio of msasurements 4 and 2.
curve, therofcrre, refers to high energy neutronso The higher lino
is obtained by. dividing (3 - r> by (1 - 2),
thoreEdre to thermal noutrons,
The intensity va~5'itions with boron
absorber thickness may be represented by formulae of tho type
-Kx -noox
A, s AOo =IO
ThSa
This curve refers
Tho slope of the curws 6ivl;s the absorption co9fficiont X, The
coef%lc?.cct j . boron ~ is proportion41 to -&ho-1/2 powor
rgy of t3:e :ioii$r0111s, Therofore, one obtains for the
ratio of rssonanco oncrgy to thermal onorgy tho expression
o m can calculate the resonance energy,
b

One limgta an of the mothod $uaL
cribed f s that It;
assumes that neutron
the neutron beam only by
absorption in .boron a ing in boron. This 3.8 a valid
asmmption for small neutron energies where tho capture cross-
Bection in boFon is la~g6, For em ies above a kw, soattaring by
boron disturbs the me8$ur@mentd
1 In this manner, the relsonance captures fop v&rious
have bosn explored, For medium and heavy
found that redonances spacod Irrsgularly at 8, dista
of 10 volts from each other are present and that these! POS
have wzdths of the order of 1 volt. In light elements no such
found, Theso elements shau show m'uoln mom broa
i
ea which are much more widely spaced, if In fact they do not
ovexz
ft i s more difficult ta study tho resonances o f an isatape
which doas not; becoma radioactfve when It captures a neutron but
'
s Into anothor stable isotope,
A rather inadequate mothod
which has been applied in such cases is, based on the comparison of

*
The neutrons originating from tho SOUPGO
and then after having flown a ~ Q C Q distcznce ~ D
132
fall on n noutron absorber
dstoctod in a
nautmn Gountoq for instnnke in BF chamber, Tb souma is not
3
g noutsons a13 tho Limo bot'bnaY dtirimg shopt Sntsrvals of
'clrn~~ let us say of 10 micromconds duration, sepamacd by longer
time interval$, of far instance 100 microsecond$,
The neutron
pulees arnitted by the SQU~CQ are shown in tho uppar half of Fig. 7.
*-* . I C - P U l B @ Of SOUPCe
Tho lower half of this flguro shows the time intorvnls during whYch
tha rroutron detector $s sensitive, It is seen that the ddmtar
will not roact to neutrons unless* they arrive t after they have
been-emittad, Thus only neutrons with 8 voloclty
e
are observed. By varying tho thickness of tho absorber, ona can
find the abssrpt;,ion coefficient for those partlcular neutrons in
the material of the absorber, By varying the time delay A t om
can investigate the depondenco of Po upon neutron anergy,

133
sensitive time of tho datcctor which CQ~QS
L period Jator,
This is
called recycling,
Thls difficulty can bo minZmieod by choosSng a
long rctpeat period for the emission pulscs and the corresponding
sansitive intervals
By tho mothad descrtbed bbovo it WBS possible to explore
the propedies of' $IuW kc&aibns dj? ho abduk $0 cv,
We shall no summarize briefly tha intoraction of neuktons
with some element@. This will bo done in connection with a table
which in part has bson taken from A, Both@, Rev, Mod, Phys. 29

I) a
$>
1
H1
ETa s tic soatt ering 20 1.7 Efficient neutron
slow-up &orL
of thermal
rleutrons
Elastic scattering 4
1.7
3x6
Neutron absorbed
alcpha emitted
goo
310
tl
C= Elastic Scattering
3000
H3-4 Elastic scattesing 10
and neutron absopbed
proton emitted
st? Elastic scattering 12
and radiative capture
n
5
6000 Produces active
ZSQtOpe

December 24, 1943
~~-24 (23)
UTURE $E 1 ;HY IC
Third Series: Neutron Physics Lecturer f J ,H, Willlams
Generally speaking, neutrons are produced as fast; neutrons,
There ape three principal react2ons in which fast neutrons are
produced, In the first one an alpha particle enters the nucleus
and a neutron is emitted, The reaction is symbolized by (a&
This reac.tion has mostly historical interest since it was the first
reaction in which the production of neutrons was obsemved.
A second type of reaction in which neutrons are produced
is the reaction in which a deuteron impinges on the nucleus and a
nrsutroi? is emitted, The symbol for this reaction is (d,n>, Twa
reactions of this type have become of great practical importance.
One is the Be(d,n) reactton in which'the Be is the bombarded
ntlcLeus and which gives a very copious source of neutrons. Another
is the D(d,n) in which the deuteron hits a stationary deuteron in
the target. In this reaction rnonoenergeliz neutyons can bs produced.
The energy of the neutrons depends ?yi the energy of the
impinging dm'teron, Nsutronv between 2 1vle!v .?,:i2 7 Mev may be pro-
The third important source of neutrons is. the (p,n) reactions
and in particular the Li(p,n). In this reaction an Li
nucleus is bombarded by protons and neutrons emerge leaving the Be

*
nucleus behind.
This reaction i a endoenergetic, that is, the
protons musk have a certain mlnimm energy in order that the
136
r~ectlpn ahouXd proceed at a18. Monoene~ket3.c neutrons between 10
kflsvolts and 2 million volts may ba produced by this reaction for
the similar range of incident particle energies, $.eoo up to 3.5
Mev.
The Li(p,n) reactlon may be written in somewhat mors
detail ixl the following form
**
~17
where the energy of reaction Q
suspect that instead of the above rea
takes place
where Se7',9
+ HI n1+ ~ e+ 7 Q
is a negative quantity, one might
Li7 H1- n I). Be7'4- Q1
&n excited form of the ;Be nucleus and Q1 Is 8 greater
negative reaction energy, If this second reaction proceeds, the
neutron produced would bQ no Langer monoenergetic.
this possibiliey has been found,
ion the fdllowing rsaction
No evidence of
It is true that the Li7 nucleus
(which i s quite analcgms to the Be nucleus having as many neutY'ons
as the Bo7 'ia:3 ~)cckons 3iiC RS many protons as ths Be7 has neutrons)
has an excitaii atnm of 485 kilovolt
hat the Be'? h
There is some reason to
a similar excited state and the presenoe of
such a state might giv rise to S~OWQT nautrons whenever the Li(p,n)
reaction is used to produce neutrohs ab e 3.1'2
.Measurest of. Fast Nqutrqu
milaion volts,
Ths first method of measuring the energy of fast neutron8
made use of the Wilson Cloud chamber,
The arrangement by which this
be perfosmed is shown In the following f igura i

137
,
On the left hand. side the position of the neutron source i s indi-
cated,
shown.
Qn the right hand side, a Wilson Chamber 3s schematically
The chamber $8 fillad with a gas of small atomic weight
such as hydrogen or helium, The path of the neutron is also ihdicated
in'the f$gure. The heavy line in the chamber is the track of
a recoiling particle, If this track makes the angle B with the
continuation with the path of the neutron then from this angle and
from the energy of the recoiling parlicle, the original energy of
ran may be calculated, In partlcular, if the recoiling
particle is a proton then the energy of the neutron is given by the
2
E, t E, cos Q
1)
Where Er is the energy of the recoiling particle and En is the
energg of the neutron, it is usual to count only the tracks
which go in the forward direction and make a small angle 0 (for
smaller than 10') wi$h the direction of the incident
he factor cos2 Q in the above formula may be
and the energy of' the recoiling proton is aqua1 to the
energy of tha inoident neutron,
One danger of this pmcedure is that the neutron might
have bean scattered In the wall of the chamber and while the protan
seems to go farward, it actually does no$ lie in the continuation

2.38
sf the neut;son which hag hit it,
I
Another difficulty is
cks of the recoiling particfe are rather lo~g and they
d d;n the ch er. The anergy 04 such recoils cannot be
chamber measur
A fwkhor diff icu
tn eva2mtin.g results of cloud
fits as walZ as of other meksuremants of fast
noutrona i s tha* a pr0vious bowledge of the dependende of the
callision cross section on tho anergy f the incLdent neutron is,
A typtoal depsndende of thi8 c
olLowirtg figure t
SS sectlaxi 6 (wh
mtia cross selStion) on the neutrqn energy E,
.li_
E,
Umfortwately sometimes, as far instance, in the case of helltun
ndence is mors involvrad EL^
shown in the figurb:
is shown
e
resonance is nd 2 MeV, ThSs resonance
to d great:
between J and 2 rn112l.m
of this resonance them? many tracks were e
8hQWing the presonce o
Recently pho
ha incident neutron has
, Previous t o the disc
cluud ohamber-s In measuring the energy aP neutrons,
the phQtographic plates eon$cd.ns hydrogen,
ous3.y interpreted as
and 2 MeV,
n USQ~
instead of
cause fani;zat$on and leave @evelopable grains in the waka
The emulslon of
The rocoil protons

,
I
Wilson Chamber only those tracks are $0 be counted which make a
g2e with the direction of the neutron,
The photographic
technique has tho Pollawing advantages a$ aomparod $0 tho Wilson
chamber*
First9 the stopping power of the omuJ-sion is g?eatar9 the
lsnbth of the tr~cks is only a few thoasandths of an Inch and the
.danger that ths tsack IoaVtss the emulsion bafore ending is smaller',
S@c?ond, in tho photo$ra$hi@ plate %Hers heod be much less material
than there is in a Wilson chcmbar; consequently the danger of a
scattered neutron causing Q recoil. i s decreased.
Thirdly, the
photographic plate is continuously sonsitive while the sensftivity
of a cloud chambcr I s rostr$ctcd to the moments of expansson, The
photographic technique has howavar the disndvantage that it becomes
Jnaccurabe in couqting slowor neutrons,
The recoils produosd by
utrans make few grains and since the number of the grains
produced is used in determining the neutron energy, fluctuations
in the number of grains mako such anergy doterminations aifl$icult,
The photographla mothod must be calibrated by using R reaction of
known enorgy
A less straightforward, but perhaps mme trlxstworthy
mothod of rncaszxring nautron enargios makes uso of the ionization
chambers,
Such a chamber is shown 6chemnticnlZy In the following

filled with hydrogen.
ians produced by tho rocolling protons.
140
"fie field wlthln the chamber collects the
0111.the sight hand side the
eonsloction to tho ampltfying grid is shown in which the charge pro-
duced by the ;loris is amplified SQ as to give a measurable effect,
The following figuro shows these amplified current pulsesp each of
which corrosponds t o one recoiling particle according to the angle
which tho recoiling proton makos with tho neutron.
The rocoiling
particle will carry more or Xess energy and the size of the pulse
will vary come rspondingly e
Amp 1. iff e, P
output
I
9
lJlilLnii
l-cI3L"..b
\
O m may plot tho number of pulsos which have more cne'rgy than a
given value E agninst srmplifior output
b?mber
of
pu 3.8 e s
pulse size or energy,
Very small puZsos cannot be counted bacause they arc? obscured by
%ha background or noise oE tho nmplifiey,
By dffforontiating this
curve9 one obtains the number of r(ccoil!-i?g particles of a glven
encrgy.
This I s shown in the Tollawing grnpht
I
Number of recoils
If. tho gas in the ionization chamber was hydrogen then a second
dlfforentiation gives tho number of primary neutrons as plotted
against their energy.

*
Ember of privariea
The figure$ given above refer to the case of monoenergetic incident
neutrons.
If a distribution of neutrons with wrious energias
impinges OR the ionization chamber, figures of the following kind
may be obtained:
Number of pulsos
above a given energy
I
Tn w
Ember of primary
neutrons
I
Inswoad of recording the pulses of various hoig,,t,
one mighu use an
anplificr wfth a bias such as to cut out all pulses las~
than a
certein energy.
The sonsitAvity of such a counter is shown below:
S0nsdtivity
-E
Dslow the bias the
amplifier-dotactor will not respond at all. At

142
higher ener(gies the sensitivity will rise to a plateau and than
docrease.
Instend of using an ionization chamber fillod with hydrogen
gas, one mSght use a cbambor in which the side facing the *
neutron SQU~CO is coated with a thick layer of paraffin.
Such chambers are then fillod with argon,
sufficiently heavy so that argon recoils carry rolatlvely little
energy and tho bias may be tzd3ustod in such il way as to count only
the proton rocoils originating in the paraffin,
such a chamber is shown in tho following figure:
Sen s i t ivity
L..=L
---- - %2
The argon nuclei are
The sensitivity of
Th..ere 2s this time an inzraasing ,rise o f sensitivity above the bias
value,
This is duo to tho fact that hfghor e~iergy neutrons produce
protons which hcive R longer range llsn the paraf?'.k ad. have therefore
'

14 3
0
a
The coun$er Is filleti with hydrogen,
Instead of a ~oll&ting plate
it has a collecting wire near which Parge electric fields are
present,
In this field electrons are aocehrat~d towards the wire
causing further ionization and in thts way greater- pulse result,
With such proportional cQunter8, pulses as srilall as 10 kv may be
detected whllle the ionization chamber described above is insensitive
up to 100 kv or higher.
Ionization chambers also may be filled with BF,,
chambers the neutron reacts with the Bra
products of total kinetic energy about 3 Mav.
pulses are produced.
3
In such
nucleus giving rise to end
Thus sufficiently big
Not only the measurement of the energy of fast neutrons
but also the measurement of their number is a diSficuXt question,
In one method due to Fermi and Amaldi
the neutrons are slowed down
in it water bath of sufficiently big volume surrounding the neutron
source on all sides.
After. beinp slowed down, the neutrons am
detected by rhodium or indium foils distributed in the bath.
From
the known cross sections of thewe foils for slow rleutrons, one may
determine the original number of fanSt neutron?, In order to do that
one has to integrate over the
the bath,
the detectors in
This last operation may be avoided if instead of the
foils the slow neutrons are detected by manganese dissolved in the
bath, BY stir Ping this solution and taking sample afterwards,
one may find the integral of neutron absorption in the bath and from
this quantity the original number of fast neutrons may be calculated.
A theoretically very effective way of counting fast
neutrons and measuring their energy is shown in the following *sketah:

..
On the left hand side, an incident neutron beam is shown.
This
ges on an Iankmtion chamber which an the side of the"
neutron beam has a thin paraffin film. This film is folhwed by a
collimator which" is a plate transverged by naxvowe1: channels point-
' ing in the original direction of the neutron, These channeLs will
A
let through only such protons whose direction colncidss with the
direction of the neutrons, From the number of protons in
the thin paraffin layer, from the colllsion cross section between
protons and neutrons and from the number of pulses of various siaes
izat3.m chamber the number of neutrons with various
._
-
not work, The reason
-c--_m
is that there is too much hydrdgen in other places than in the thin
rotons corne from
We shall discuss the following processes ich occur when
ring, ilzelastic
8, (nJn> process,
processes, These
rmation abaut these
een the neutron

14 5
0
$bt$xbs and thk neutran detector.
if the scatterer is smdll enough
and if thb neub!dn has beed ikfiected in the scatterfhg pyoeess by a
I
big enough angle9 then this neutron in the originlra bbam will miss
the detector and no neutron outside the original beam will have an
appreciable chance to be scattered into the detector.
In this wag
the major p’orlion of the elastically scattered neutrons w ill be
missing from the beam,
neutrons vdll be only partially observed,
Similarly the fnalastically scattered
In the case of the
ally scattered neutrons an argument may be given which
t these neutrons are distributed spherically after the
z
scattering process. In fact the scattering process may be described
by the following equation:
- - Q1+ ZA z(A3.1)’ zA*+ n1
This equalton shows that the first act in an anelastic scattering
probess is the formation of a compound nucleus with the atomic
number A 4-1,
neutron will have forgotim about its original direction of ink-
dence,
to be scattered
When this compound nualeus emits the neutron, the
Thus there is no reason for inelastically scattered neutrons
part Icularly small angles and the scattering
geometry described above will cause most inelastically scattered
neutrons to miss the de te c t OT o
(with It
The ne u t r oqs which are captured
gamma emission) or whrich undergo a (n,&dg or (n,p) reaction
be naturally missing from the original beam.
In the
process the neutrons emitted are again spherically distributed and
will therefore in all probability miss the detector.
Thus practic-
ally wha t eve r reactfon a neutron hag made with the scattering
material the result will be that the neutron w ill not appear in the

I C
14 6
detector
rblated to
The intensity I transmitted by the bcatterer will be
the intensity Xd
df the 6rdgiba3. beam by the formula:
Here small
scatterer,
4n%d
I/J, 0 8
n is the numbep of nuclei
x is the thickness of the
per cubic
scatterer
aJ1 C~QSS sections or the total cross section.
centimeter In the
and d is the sw of
December 16, 1943
LA+24 (24)
Third Series Neutron Physics
--.- LECTURE SJRIES OcN_ELEAR, PHYSIC$
LECTURE XXIVt PROPF,RTIE$ OF'
CROSS SECTION
The total. cross section of a substance for fast neutrons
can be measured by a direct transmission experiment In which the
source,. scatterer, and detector are situated in what is known as a
"good geometry". This good geometry is so defined that the solid
angle subtended by the scatterer at elther the position of ,the
source or of the detector is small.

*
14 7
In
scatterer at
by a neutron
removal from
the detector,
this case the neutrons from the source strike the
practically normal incidence and any process undergone.
in its passage through the scatterer will result in its
the-beam and its consequent failwe to be recorded by
The neutrons counted by the detector will consis$
almost entirely of those which have had no interaction with the
atoms of the scatterer.
A very small percentage of the detected
neutrons wfll consist- of those scattered (elastically or inslasti-
@ally) through very small angles, but in a really good geometry
they will constttute a negligible fraction.
A transmission experi-
ment in such a good geometry wi.11 therefore measure the total cross
section for all processes.
If the incident neutrons are normal to
the scc7.ttf;ererS which may be supposed to have 8 thickness Ax and a
number t~1 atoms per cm3, the ratio of the detected intensities with
and without scatterer will be given by
- 14,
-nom
0
where 6“ 2s the total CTOSS section.
In contrast to this measurement with good geometry, trans-
Mission experiments in poor geometry are performed in order to find
the elastic scattering cross section.
In this case the solid angle
subtended by the scatterer at the positfon of the detector is very

14 8
Q
large,
This makes It possible for neutrons scattered through large
angles to be scattered into the detector, thus compensating
.----
Sauroo
De t e c t ox-
*-.-.- _._""
Scat t oyer
PGGR GCGMSTR'SI
for those scattered neutrons which miss the detector.
I
Since this
compensation is almost exact, scattering processes alone would have
'
j
if the detector is biased so as not to record neutrons below a
cartarn energy, the neutrons which are &elasticallyl scattered will
still contribute to the measured reductions in beam intensity along
with all the other processes with the exception of elastic ssattering
This experiment with poor geometry and biased detector
therefore measures the sum of all cross sections with the exception
of. tke cross section for elastic scattering, To find the elastic
scattering cross section alone, w0 therefore subtract the cross
section as measured in poor geometry with a blased detector From
that measured in good g'eometry,
Capture cross sections are easily measured if the nucleus
resuZt$np from the addition'of the neutron 1s radioactive, The
*
C~OSS section can than be obtained from a measurement of the
!I inttuced activity. When the resulting nuofaus is not radioactive

149
there is na easy means for measurinp the capture CPOSS section, X t
sometimes happens that one isotope of a given element will become
radioaofive upon capture of a neutron while 30me other isotope of
the same element will not, However9 it is found empirically that
the capture cross sections of isotopes of a given element are of the
same order of magnitude, Therefore one can sometimes estimate the
captum cross section of an isotope which does not gfve a radioactive
product.
Dividing the energy range Snto roughly three regions, the
resonance Pegion from 0 to 10 kv, the medium fast region from 10 kv
to 1/2 Mev, and the region of fast neutrons from 1/2 Mev on up, one
gets a behavior of the capture cross section with energy which looks
like this:
crc
0 10 kv
In the resonance repian there are sharp resonance peaks in the cross
sockLon superposed on a l/v background. Tn the medium fast region
the resonances are much broader and are discrete only for( the
lighter elementsp say up to oxygen, For the heavier elements the
broad levels overlap and give a smooth behavior something like l/v,
In the fast region above 1/2 Mev the behavior is more like l/E.
For the very fast neutrons the cross section is determined principally
by the area of the nucleus and the competition with other

e
possible processes like inelastic scattering which becomes more
probsbla
the energy of the neutron$ increases.
The cross section for an (n,& or an(n,p) reaction can be
measured by putting a sample of the dement in question inside an
ionization chamber which w ill count the alpha particles or the
pr~tom directly, when the chamber is placed in a neutron flux.
cross section for an (n,2n) reaction is extr)emely difficult to
pt in the case that the residual. nucleus Os radioactive.
Inelastic Scattering of a neutron is characterized by a
reduction in the energy of the neutron as a result of the scatterXng
process,
In light elements liko hydrogen or helium or carbon this
reduction in energy comes from the no
a1 classical collision of
two particles of comparable maw and. such collisions should really
The
be called elastic, However for the heavier elements lika lead OT
gdd, the neutron enters the nucleus and emerges with reduced
enorgy leaving tho residual nucleus in an axcited state,
excited n~clous then falls down into the ground state with the
emissdan of a @pray, Thus a nuclenr inelastic scattering is always
nied by one or more 'prays,
The
The probability for inaxastic
scattering I~CPQ~S~S
with increasing atomic weight sinco the
nu~loi, havlng many constit nt particles, have a large
variety of closely spaced energy 1
els into which the nucleus can
be raised by the addition af the energy of the incident neutron,
Qualitative fnformation about Inelastic scattering can be
obtained by measuring the activity induced by the inelastically
*
scattered neutrons in various detectors whlch respond to neutrons in
different energy ranges. For example, siLver or rhodium can be used

as radioactive detectors of slow neutrons and aluminum ox silicon
ns detoctars of the f3st neutrons.
By measuring the activities in
thcse Botectors with and without the scatterer one can get some idea
of how the SneLastically scattered neutkons have to be dogrnded in
cnergy. Experiments have also been performed in which theq-ray
intonsdty which accompanies any inelastic scattor$ng is rnm6urad.
The magnitude OS this intensity gives an indication of the cross
section for the inelastic scattering. However the experiment is
complicated by the possible presenco of 3. -rays in the source, Not
much Information about the energy of the inelastically scattered
neutrons can be obtained from the energy of the? -rays since the
excited nucleus will in general emft a variety of ?-rays of
different energy corresponding to the many states into which the
nucleus can fall before reaching the ground state.
Complete infor-
mation would be given by n measurement 00 the spectrm of the
inelastically scattered neutrons but this is F?
very difficult thing
to do.
@ & m C T S RADIATION PROTECTION
Electromagnetic radiotion like x-rays or 3. -rays damage
tissue by praducing fast electrons (through the photoelectric
effect or the Coripton effect) which cause lonization and disruptfon
of the molecular structure of' the tissue, These electrons have R
relatively small ionization per unit; path length and ix correspondingly
long range. Neutrons will in many cases give rtse to
energetic heavy chargod particxes like protons or bclrparticles which
will pave R vory high specific ionimtion but R short range. Fast
protons cuuld be produced by collision of a fast neutron wSLh a

162
r,
hydrogen atom in one of the molecules. Slow neutrons may bo
m@urod nnd give rise to q&tys or in some cases, like those of
Agatnst 3/ -rays, lead Or cancre te walls of suff iciene
thickness give good protection. Lead is of no value against
neutrons, however I Materials aontalning hydrogen, like water,
paraffin, or c~ncrete, are ganerally used bec&use of their effoativeness
in slowing the noutrons down to 8 point where they can be

I L A 24 625) 153
, \
'.. .:, *
December 21, 1943
LEC'IWE SERIZS ON NUCLEAR PHYSIC$
. . , . . . ,
Fourth Series : Tm-Bod.y Problem Lecturer: C. L. Critchfield
..). c -. LXC'MJRE XXV: . NUCLRAR CObJSThflTS . '
,. ,
11.1(,~
, ' %,
,->
I:+
. . . ,
There aye two kind; qf nuclear quanti t3e.s:
'5
Quantize& and
., not qvantized. The quantized' numbers are: mass number,- charge,
',* L '
',, spin, statistics and parity,
The qtla'ntiths that are not quan-
.Fized are the nuclear radius, .the nuclear mass, the magnetic mom-
\
eht and the electrical quadruple moment.
.~
It is believed. that eventually there will exl st a thevy
of ithe quantized nuclear numbers correl.ating these to each ' ather.
SUCH, n theory shall link the varjous numbers t.2 each other and
:, ' posslbly to other 'physical qiiantltl.es. IJp ti nw there has been
: I ).
. ,.d
.,
(,'.*'W.'
. .
I $,'.
, G I
'\i
' \,, . .
( '
, ,
' i
I
mly one sipnificant attempt in thl.s direction,
'.
PaiJ-li tried to
cwrehte spin and sttitfstics by statlng that particles wTth a
half ,integral spin have Fermi-Qirac statist1 cs while particles
vj th an i-Qteger spin have Pose-Einstein statistics.
\, The nani.ng of Fermi-Dirac statist! cs mag he explained
with $he help tj? the example ~f the H2 molecule.
'that the slsins
tho Value 1/25 are Rarallel (trlplet state).
StatLstics, which ap&.es
Let us assu.me
"?,the two protons of^ this mqlecu.lo (which have
Then Fermi-Dirac
to the protons, postulates that an:
'intprchange of the posl$ions of th,e, protons will came the wave
. .
\, \\\ " .
fundtions tq change sign, 1 , -

154
The cmsequence is that a rqtational state mi.th no n0d.e (J = 0)
d?es not occi?r simultaneously.
A31 even Values ~f the rotational
qvantim number J are exclud.ed. Odd J values are alloved and
the lovest state correspmds to 3 .= 1. This state of hydrQgen
is called orthohydrogen.
Let u.s now cmsider the D2 molecule.
Let 'tis again assme
that the spin 0.p the devtrons (which have the value 1) are para-
llel. Then tlie Bose-Einstein statistics whJ,ch applies t? the
deuterons 'pmtijlates that the even'rotatiqnal states OF the rnole-
&le, are alone pTesent while the odd. rqtational states do &?&
xcvr.
d eu. t er on.
This stake of the deuteron molecule is. cn1.led ortho-
There also exists a secsnd stnte fw- both the hyd.rqpen
and. the. deu.term in which tho spins o?' the nuclei are nut parallel
and in nhtch those rdational states are allowed which did nqt
Occur in the ortho states and t.hose rotational states are missing
whjch were present in the ortho-molecil.1-es.
called pnrahydrQFen and. paradeuteriurn.
These states are
The experience ab711.t the
statistics qf nuclei is a strmg.argument j.n favw of the nuclei
heing cmstitu.ted frm neutrons and. protqns rather than from
prqtons and electrons.
Assuming Fermi-Dirac statistics f w all
these elementary par'tj.cles (prqtons, electrqns and neu.trms) ,it
follows that a nucleus cmtajning an qdd number of slvh particles
behaves accwdinp to the Fermi-Di.ra,c statistics whilo a nuclew
cqntaining an even number qf particl4s has Rose-Tinstein statis-
tics.
NTW the deuteron, i-? it vere cmstitu.ted from pr3tons and
electrons, would have to contain two protms and one electron,
that is, an ogd number of particles and. should, therefwe, have

155
Fermi-Dirac statistics, wliereas In reality it, has Bosa-Einstein
es. Xf on the 7 t h ~ hand, prrltqns and neutrons are the
ents of the sru.c$ei, a 4euterqn Is made up of ane pratm
and onemntron, that, of an even number qf particles which
edicts Rose-Finstein statistles,
ong the non-uixmtizec! qii
e radius can hot be determined v
e values have been qbtai
ItJ es characteristie of
accurately.
t an early date in the
nbcl.ear physics by scattering 04' alpha particles.
The pPTncLlple of' this determination I s that when the alpha parr
ticle ap3roaches to within the s vm o
he radii of the scattepinfr
nr: the alpha particle then tho Rutharfwd scatterinp Inw
ceases Ita hold, From thls anomalous catt,er.ing csf alpha particlos
4 2
adfi in tho order of 10 CM have kecm qbtained.
Phe electric quadruple moment of a raucleus for a sharply
deftned qmntity has not Fe~n
determfned up tq now wtth v&y
high
*
Tt has been first dis
d rneaswOd by 1 vest i Pa t Sng
errine strvctvre qf' atgntc spectrum. This hyperfine slrucdire
to the intaractton of' the magnekic moment of the ebecw
e ta the spin
ita1 moment sf' the
e nucleus Differant
cl.ear and eLectTica1 angular morn
wont energies and cavse Q nRrrryc~
s which is as a rille less than one
on descr-1.bed abme gives a
fine strvcturee Small deviations
hcavy nuclei, These
additional term in the ener

expl-anation which depends qn the? sqvare of tho cqsine of the
angle Inclvded between dlroctims of the nuclear ancl el-ectric
956
angvlnr moments, PhysicntLy thl.5 Lima1 term is due to
an olectrlc qiiadruolp moment of tho nucleus whl ch interacts with
the inhqmqgeneity q? thr. electric field prqd,tlced hv t h elcctr-+Is.
~
Thcro is also me rothpr direct detwmj,natjm of the quadruple
moment qf the deilteron which IW
The masses 9"
mns8 spectr9gravhic wwk.
shall Clscuss later.
the nuclei arc knwm qllttp accllr~t~1.y frw~
and megnotic flc3,ds inrhlch focus ions T? a riven !.veJght into a
given pqsi ti qn,
this methqd have been carried qvt, he Ralnbrldgc; fw lipht nticloi
and by Dmqpster for hcrvy nucLoi,
This method uses crqsscd el(3CtTic
Rcccnt acvyate determjnatj 3ns of masses by
BainbridEe was able to obtain
high intensities by f'xx~slnq a beam of groat angular deviations,
In hi)s sot-up the positi9n af the fwvs dopends on the maSs qf
the nucleus vcry accurately as a linear function,
In this v~ny
it was possible to obtain mass determinations Isf high accuracy
by comparing the 'ocus Q? ions OF nzarly the same wei-ght,
Thus In his car1,y 141vk he cwparod the mPss of the hellurn 1qn
1R4.t;h the MBSS qf the triraatqm1.c DHFI.
The actual value deduced
frqm this measurement for the mass of thc dovterm titrncd out to
to oxygen masses g?vcn by Astgn.
carried out vith abixdant deutarium s~urces and evaluntcd. 1NitdZ
the correct helium to qxygen ratio gave very 8ccvratc deutaron
masses.
gics.
More rccent work of Rainbridgo
NL~CIOQT rnaSses aro ~llso used lin balancing reactlon oncr-
A systematic sti4y of" thcsc rmctiqn cnergles lcad Bcthc

n
H
1,60893
1 * 09812 ' He3

158
It is svTp3:ising that by morely d.ovbling thc numbor gf particles
within tho nvC&%s tho bindlng rnergy should be Increased by a
fnotor of 13.
Thc magnotic moment is also knmn vith very hirh accuracy.
The beet detorminations in!
by Rahi nnd his cqllahwfltwq.
e carried out at Cplinmbin University
They ueed an ingenious method.
Tlic meam.vemsnts wwe carriad out on rnolccvlcs like H2 or D2 in
vhich the olqctrona nre paired and the effect ~f tha electron spins
cancel each other.
through 3 magnetic fields,
gmous and d,ePLccts thc: molocular be~m,
genotis.
motion.
h beam of the moloct~le passes c-msequcntly
Thcs first 9.F these fio'tds is inhomo-
Arovnd this field the nuclear spins perform a prwessing
The t!i%rd fScld is Inhoni.1 nous again and brings the
molecular beam back t~ c",
The second field is horn&-
fxus in Tvhich thc molecv1.@s moct inde-
penden$ of tho spin qrientztim Q-' the nuclei, In addftim ts
this arrmgewnt an osci?I.ntj,nr;t mngnetic, fi ~ l is d si1pcrp~secl on
the hamogcnws mngnotic flr57-8. Th$a wcill ating fYo1.d iy.parts
energy to the nucl.ci crlusinp thc nucZear soin tr, jump from ono
stcte into c.nd,he,r.
If this happons thc socmd. inhomogcnavs field
will no lonpar compensate thc crleflnctfm cclv.sct3 by the Fimt in-
11s flcld and any chanpc in winntption cmased by the
oscil-lating f2GI.d Tlrj.11, glvo rlsc to EI diminishwl numbor of mole-
cules arriving in thct "(3c1.t~~ A rhanyc: In tho nuclear spin ~rfcn-
tRtfon w111 occw howevw mly i.p the frcqvency of the mcillating
magnetic field is in resonance with the precession of thc nucl.oi?.r
spin.
This precession depends on the s'tl?cngth of the hr>mogonaus
fiold, on the magnetic moment $.nd on tha value of the nuclear
spin. The last quantity may bo detclrmincd from
for instancc by tho analysis of tho spcctrttm.
other oxpcTiment S,

159
fn this way th.en, tho mngnetic rnomont mag be dc&rminod to a high
accuracy,
Actually *’he procisiqn of tho oxpor2ments is liniitod
only by the 2cngth of tho rcpim of tho hqmogcnous electric fidd
in thEt even the tins of flight of the moleculo in this region is
short.
T ~ P accuracy of sesonmce betwan an, oscillating magnetic
fiold and precession frcqucntly is limited.
Vory accwi?.to values
haye been obtained- by the rs?.ztiva megnitvdes of rnagnct5.c momcnts
proccssitm frcqvcncies of tho H and D nuc1ou.s.
is used.,
A rm1”e complex plctvre l.6 Ilstained I$ the> H2 molecule
Here thE C rotcltjnnal stnte can not bc vsed br.cnuse in
this stc?.tc thc nuclear spins hwe opposite directions and the
magnetic moments cmoonsntce
state (J u 1) is t o be used.
Tkorcfwo thc first rqtatlonal
‘In this state also thc spin of the
, ntzcloi add up to mc and th2To ~ ~ rcsult ~ ~ altqgothcr 1 1 9 mtnt1ona.l
and spin stntcs,
Thc em-rglc3s of them states me nf”ectcd by
tho magnetic Interoeticm hctwwn thc prolma and ’nv tho h13tir)n of
the ~l~ctrons induced by rnoLwvlnr rotatlm.
Retweon the 9 statos
mcntioncd nhovc, two sots of‘ 6 transitions are possihlt? and cor-
respondingly ono dme~ves tvto sets of 6 dips of intensity,
order to oxpl.ain thr. oxnct frcqiii’ncy at iKihJ.ch these dips occur,
assumptjons have to bc! made abmt the intcractiqn hetirvem thc mag-
netic rnommts of the nuc2.ci c7nA t;hc rnngnotic fic??.d pr?dt”.ccd by
In

tho rotntkng molecvla.
16 0
Giving apprqprinte Values to thaso cqndi-
tions the position of. .the 6 dips can be explained.,
Hnvfng deterrnincd the molocvLwr qilantitios that have an
effect on tho nuclear mnkncts thc! position Q?
rnay ba predicted..
in ivhjeh J 3 1 and also thc: spin fs equal to
the dips In D2
The ppodictions
hawever did not cbock wit19 the experimental rasults me agroclroi.nt
could. he obtqincd only bv assuninp an adcljtional term in thc enor-
gy which depands m the sqilare of‘ thc cminc? Inclilc?cd by the nux-
lcar rnqment and the mQlecular crlxis. This fs the kind c>f term that
should be if thr? detttcrm h2.d a quadruple morwnt.
Rabi, Rnrnscy
and. othcrs have indeed beon led to as~rno an electric qvadruple
mmnent for the deutrriurn;
Par this mcslccir3,e /)ne may lnvostipato thc. St2.t~
Tho discovcry or the a1indrupl.e rnowmt
Q?? thr deuterium
csmo as a svrprise bocnusc cal..culntiwls on the htncllng cnorgy
find other properties .of the deu.term were based r>n R
symmetric model of tho deuteron whjclz did not porm!t
of any ailndrvple mnmont.
s.r’.mple
the -xistance
The mt:sciwe sf such t? moment shows
that evon. in the simplest two-hqdv problem r7f nuclcar physics
quadrup1.e f’orces appear,
moments of thc simplest niiclei mmsured in units of nuclear
magnltrons
Thc: follqwinp tahLr? gives thc magnetic
H! 2.785
1
n -1. (73
D2
@
855
Tho negative side attczchcd to the magnetic mmcnt of thc neutron
Is naccssary in qrdor that the pTotqn ann’ the neutron rnagncts
lincd up in a parc?l-lcl way shovld give Dt least npnroximat*:ly
the magnetic momcnt of the deuteron.

L A 24 (26)
162
Fourth Series:! "vc, Body Pmhlem Lecturer: L. C, CritchfiePd
*
The earliest atterpts to bitild a mwe detailed thesrv 9f
orces were cmcerned dith explaining the binding ener-
pies of the lirpht nuclei, particularly of the deuteron.
It was
nded that the bindinp energy of the helium nv.cl.eus was thirteen
ti.ws that ~f the deuterm elthwph tlie nvv?"nr of attractive links
Pmir particles is only six, a t mosts av?pared wjth one
between twg particles,
"ripner minted o3.t that the extraordinar-
ily large bqn.ding in the alpha particle ccxzpared with that of tho
indicated that the kinetic energy of the partlc1.e~ in the
c?evtelt.lon almost compensptes the mutual attractim an@ that thls can
be theGcase if the attractive f'wces have 3. very shwt ranFe. ioerq
the ranpe of Forces is srrlal.ler than the wcive length of the gar-
ticlea.
In the alpha particle, on the Qther hand, the number of
attrinctive bmds is prqportimately preatsr than the increase in
kinetic energy, and a 1mer average energy Is possihle.
EhQrt ranpe fwces between nuclear part4 cles makos possible
many simplifiqatims in the tlieory of the deuteron.
Essentially,
we may say t!vt if ncvtrm and wotm are farther apart than a
distance they d . not ~ inf'ltience each other ap:-reciably, but at
separations less than g strxtFr; attractive ~orces act and there is
.

the expct s%pe of the reeion cannot he important tq the des-
criptisn of the stater Accmd.fnpXg, it my. he nssl?msd that the
patenticll energy is a c?nstc?nt Vas inside the rad.ivs
outside it.
Let r be tho rolative cmrdimte of tlic urqtcm. ?v!th
3rd zero
resmct to the n.evtron, 14 the mass of" &?tm m novtron, znd E
TrTc accordingly let

The solution Is then,
164
their
smoothly
I
I
a8 Vo
Thus the
the
by protons, It i.s tlwn assvmcd thzt i s t1-m

*
been considered,
165
So far, only tho space coomjinates af the particles have
It is well established that; ths ground ntat@ of
the deukepon i s a triplet, i&@J tho spins of neutron and proton
a m parallel,
let state.
Thus the above rough calaW,ation applies to the trip-
Furthermore, the wave fu.nction asawned i s spherical-ly
symmetrical SQ that no quadrupole moment is obtained,
In spite of
these defectsl, however, tha scattering induced by the forces thus
derived will be calculated rand cornparad with experiment.
oomputa
Scattering by short range force$ bS particularly easy to
especially if' the wavewlcngth of th3 colliding particxes 9s
1ong'e.r than tho range of forces,
Under these conditions a flnite
orbital mgular momentum will prohibit the particles from coming
close enough together to be akttracted and there is no scattering
if the particles pass each other at a distance larger than a wave-
length.
Consider a plana wave for the neutron incident upon a pro-
tan. This wave represents a aefinito relative velocity v of
ftcollision" but for all possible values of Z;ho impact parameter and
hsnce all angular rnorqent.
According to the foregoing argument de-
flection will bo expcrdcnced only by those collisions of ZDPO
angu-
lar mmcmtum, ire, in ths spherically symmetric state. h t the in-
cident beam be ropresented by 9.. ikr cos0 &
Tho spherical part
oS'qin absencc of a potential well. can be determined by averaging
over all 0,
yoo = sin kr
-E?--
In tho presence of a field the wave will be strongly refracted fn-
side P = a and, so far as the wavo at 2aqp distavlcos i s con-
Lc
carried, tho effect will be to introduce R phase shift e The
e
general form of tho spherical part wlth potential woll is thon

166
ab
with 0 and 8 to be doteminedl, The wave yv" must represent tho
spherical part of the inoidsnt beam plus a scattered wave, and the
Zatter must have the form Seikr/kr. kence
The number of scattored neutrons per unit volma i s then
2
sin 6
--zET--
k r
and the total number of particles mattered by the proton per second
k
The incidtont cii.n~:=lt eensity postulated iu v klence the effective
cross-seck3.cn :xaaoni;ed for scattorlng 9s
k
Dekormination of .5 can be made approximately a8 follows:
the banbarding energy of the neutron8 is aaawnod to be small campared
with depth of the potential wall, hence the wave function
inside the range of' forces, 3, is substantially the same a8 far
the itable deuteron, To join functions smoothly? therefore
+-

e
d' tho bltidihg energy bf t;ha dezrt6,rop
Thus the binding dnergg of the deuteeon determhed the scattterihg
crosarsection, at least if the callision is suffidiantly slow4 conid
'
parison with experiment, howevsr, showed that tho predictsd CPOSBsection
is a little hi'gh at high enepgy (E>2Mv) and about a factor
10 too small at low ehe~?gyi To explain the discrepancy Wignsr
pointed out this cross-section 19 calbulated kroni what is known
absbt tkie b9ipkek btate of the deutehn Only and that it is posd
il
sible to ohooao a I'biilldlhg'' dnergy, for die sihglat skate ih
such a way as to account for the experimontai $esult$r Since Lhi-oe
out of four collisions between neutron and proton ape triplet colj
lisions and on8 is singlet the result for the complete cross-sac.-
tion is
E is tho energy of collision in ths center of mass system and hence
is onowhalf the bombarding energy if the prokons may bo considered
at mst. Thc value of ET needed to fit experiments is of thhe order
of 0.10 MeV but the sign of it is not determined becauso sin
2
6
is determined from the aquaro of tho logarithmic derivative at a,
By taking into account the effect of the finite kinetic
encrgy inside the distance a the expression for fbecomes, in
next approximation

168
Al%hough the spherically symmetrical solution for the
1)
deuteron must be incomplete because of the existence of the quadru-
pale moment it represqnts the grosg propertias quite well,
Bof'oro
considering refinements of tho theory, therefore, it ia worthwhile
to gtudy, the implicd dependence of tho force between neutron and
proton upon their relative spin orientation, With parallal spins
thsrc is a binding energy, 2.16 Mov and with opposite spins
(singlet state) the binding enorgy is very close t o zero. Thus the
relation between V, and a2 obtained in an approximate way abovo applies
vsry closely for the singlet state, For tho triplet the cxact
relation determining V 5 V1 is, in the units used above,
Thus for 2 = 1 the potential wolf. i s almost twice as doop for the
triplet state as for the singlot. This indicates that the spins
of the nuclear particles play a dominating role in the attraction
between neutron and proton,
LA 24 (27)
December 28, 1943
LECTURE SERXES ON NUCLEAR PHYSICS
Fourth Series: Two Body Problem Leoturer: C,L. Critchfield
LECTUm XXVII:
I) SPIN-SPIN FORCES IN THE DEUTER
ON
PRO,TCN-PROTON SCATTERING
rx)
@*
T.
The theory that has been developed for the deutei-on is
generally satisfactory except for describing the electrical quadrupolo
moment. In order to aocount for the quadrupole moment Sbhwinger
used a spln-dependent interaction similar to that between mag-

netic dipoles,
The wave equation then becomes
169
whore
+ $1
SN is the neutron spin operator (Pauli matrix), 6p tho pro-
ton spin operator andvdetomines the amount of the spin-spin
potential,
;I
Sinco the additional forco introduces components of ? '
In a rion*spherical combination it $s in general impossible t o aasme
spherical symmetry f or the solixtion,
For the singlet staee of' tho
dstiterond however, the dipole-dipole interaction cannot affect the
motion of th.e papticles and its average value is zero BO the same
equation as before ie obtained and the same relation betweon Vo and
a2 holds,
For the triplet state the intoraction does not vanish
but rather tends to change the spin directions of the particles and
hence a380 thoir orbital motion,
The intoraction is invariant to
reflection in thc center of symmetry so the orbital stato into which
the dcuteran might go is limitod to even parity and the only pos83.7
bility l a a D stater Symmetry of the operator In 6j, and bp shows
that tha spins of the neutron and protan remain parallel and tho
solution will bo a mixture of 3S, and 3D, states.
Tha mount of
3D, that i s mixod with the 3S, to obtain the lowost snorgy is detsr-
minod by the zliee of "p which is, in turn, choson to give the Correct
value of the quadrupole rnmont, which is 2.73 x 10 -27 2 om
Thc observed quadrupole mmont is positive and that means
that the charge distribution is "cigar shapedVrs Mathematically the
quadrupole moment is defined as tho average of (1/4#8 2 - r 2 ) over
the charge distribution ob.t.ained from the c~olutlon of tho wave

~
e,
equatlsn.
170
Qualitatively it is readfly demonstrated that when two
parallel dipoles interact the energy will be lowest; when they lie
on sn axis and that suoh a ,coqffguratlon 5s brought about by in*
em6 between D waves and S w8vea. The pwt of the D
wave that varies a8 3 cos2 62
*I 1 will interfere with the S wave
becaqse the spins are tho setme for the two waves,
Along tha axis,
ther8fOm, amplitudes add whereas the parts of the D wave that
are large near 0 z (H/2)(mL>O)
netic quantum numbers and intensities add,
one phase of
do not have identical spin mag-
The result is, with
D wave, a prolate charge distribution about the
spin :axis and with the other phase an oblate charge distribution,
The sign of" is thus chosen to glve bQe prolate distribution and
-13
the valus of y appropriate to a range of force 2,80 x 10 om
(a 1) is Ot775, The corresponding depth of potential well is
v, 8 27 mo2 which is close to that for the depth of the gislglet;
2
W e l l Vor 25 m@
In faot Schwingar points out that if the range
of forces had been taken 2.7 x lo -13 cm the two potentials would
have been the same and the entire dependence upon spin of the
neutron and praton is imparted by the dipole-dipole Interaction,
It may be noted hers that if' this dipole interaction had been due
to the magnetic moments it would be of the opposite sign and its
ed value much smaller than that found above.
3
The amount of D-wave in tho lovest state ia found to be only
4 percent, The charactoristic influence of the dipole forcos on
scattaring, capture and photoelectric disintegration are corroa-
pandih@y mall, mounting in general to 2 percent corrections,
Formulas obtainod in the simple
S wave theoyy are thsrafore

3-71
0
adeqyate for most calcu5ations and fail only 18 the true intcrpretation
of the forces and in predicting the quadrupole moment, The
ble effeot; of :'this evidently vory anal1 adrnlxturo of or'bfion
i>s that gsssntl8lly the same depth of potential well
obtain8 for bgth triplet and singlot state. In other words, the
difference in depth found necessary in the spherical states is
just made up by tho strength of the spin coupling but the '13 state
to which tho 3S is coupled is very pd t;o excite, .
A reasonable thaoq and method of calculation is then available
for the deuteron provided that the energies are not too high.
At high energy of collisian the potential well can no longer be
oonnidered sna1.l compared with a wave length and the shape of the
potential CUTW will matter, This energy is roughly equal to the
depths of the potential wells, say 10 MeV, in the canter of mass
Actually, the noutron-proton cross sectim has been obt
24 Mov (12 Mev in the c tep of mass frame) by Sharp
results are fairly well accounted for by tho simpla theory
g P- waves, Abovo t the effects ?f the shape of
and' of tho response t ions with highor angular
momontum should bo in convinci
\
.II)
TI
The range of forces bbtwepn heutron and proton h8s been taken
to be 2.8 x cm (a l)'* The ape two methods of establishing
this value For the range, The first. to be oonB€dered is the scattering
of protons by protons and detection of deviations from the
Rutherford law.
tho range of foroes fn the deuteron

from khese results assumes that the rangas are the same,
second method
and Ha4
As in
involves
the
the cam Q f
onergies available are
making estlmates
O f the deuteron,
The
172
of the masses of H3, He 3
neutron-proton forces the experimental
capable of f inding onSy the nuclear fQPCeS
exerted between two protons when they collide Zn an S-state,
interaction may therefore again be represented by a "square well"
but calculation of the influence of such a ffw~llt' on the scattorad
distribution I s BQmQWhat more complicated for two charged particles
There are three reasons for the complication : The first is that
the 908. t te ring affect Of the "we 11 i s superposed on that of the
electric fields so that intorference terms appear in addition to
thoss due to nuclear forces alone;
The
the socomd roason is that the
Coulomb farces are long range and modify the inciden2; wave even
at infinite distanco so that the simple cxponentials are not solu-
tions at any distance from the point of collision,
In addi ti on,
the collision of two protons if influenced by the statistics of
the particles and a lsecond klnd of interference is introduced,
The Fermi-Dirac statistics admit collision in a IS stat6 and ox-
3
clude the SI so we are here concerned only with the nuclear forces
between protons Of opposite spin,
is -
The wave equation of the reduced system in the Coulomb field
vIr,e;y)=u . .
M
where v is the relative velocity at infinit@ separation. Let
3
metrical, part of the solutionr Thon the equation for u(r) becomos
dr

have the same asymptotic exponential form RS vi except for a
IE
ohan$Q in sign,
This is the case if
Tho sign and the phase,B are not &yon by tho considorations
buts is a function of velooity only, The canplebe theory shows
The spherical portion of the wavo in a Coulomb field ha3 tho
asymptotic $om
The effect of
tb.e forces between protons will be to intPOdUGe a
constant phaae shift Ko 2n such a way as to leave the incoming part
of u unchanged,
“he tom OP the wave that has been influenced
by tkie nuclear forces is therefore
Uf t ,u3iiK,
The asymptotic form of the perturbed wave then becomes
e
If the particles were not identical tho crogs section would
be gtven by the absolute square *of the curly brackets,
In classi-

caL theory, &he cross section for identical particles is the sum
of the c,ross, seckians a('@) + de+ 0).
is an additianal ponsideration to be mader
175
But in quantum theozy there
If one praton is at ra
and the other at rb the wave function describing the state is
~~('a) *JrG(rblG yl(rb) yz(Pa).
in the triplet,
Therefore (since rate) = r,@+@))
tho saattering
at R-8 which is thesame as at%+B interferes constructively with
that at 8 in the singlet colldsLons, i,e, 1/4 of the'time, and
interferes aastructively in triplet collisions (314). If we call
the quantity in curly brackots f',(0) the cross section for scattering
in unit aslid ang3.e at 8 then becom s
This effect w88 first discussed by Mott,
(g).
+ ' ~ '
The first term is the pure CouZomb scattering# the second the pure
0os(&a,0s2(1/2)0+
cos2(i/z)e
potential scattering and the last the interference tam. TO get
the woss*8ection for scattering between @ and 84-68 multiply by
2flsiln 8dB; and to obtain the cross saction in the laboratory
Ko)
, -
j
1

c
system replace every 0 by 20,
taryf and a$ this angle
176
The largcst effect of the potehtial
At a million volts colllsion energy? is about 1/6 and it becomes
smaller as the energy increasos. The effect of a $ma91 phase
shXft due to nuclear forcos is therefore greatly amplified at
these energies,' Further, due to interference, the sign of KO is
determined .'I
Experiments on proton-proton scattering have been carried out
with good precision by Tuve, Hafstad, Hoydenburg and Herb, Kerst,
Parkinsdn and Plain..
The latter have carried the collisiovl energy
up to 2.4 Mev where the ratio OF cross soctions is 43 and KO is
480* At 1 MeV KO 2 33O,, Both experimental groups accelerated
protons by electrostatic,generators a d scattered them in hydrogen
gas rrlaking counts at several angles to the beam,
Comparison of the result8 obtained wieh those that are expected
frm a square we31 can now be made,: For this purpose,,
however, it i s necessary., in general,, to know both the bounded and
* -
unbounded solutions t o the wave equation in the Coulomb field,..
These solutions are called
F and G respectively,
The phase
shift, KO, has been defined to relate to the asymptotio foms of
Fo and W0 for Sdw'aves which are taken t~ be */2 out of phase at
.-

*
inside
177
large r;. At the boundary of the.well the combination Fo COS
sin KO must fit onto the wave function, Fir that applies
the well
and this determines KO&
Fy = bit G& tan K,
'";L T+ cfo tan K,
If we g.ssume that the Coulomb field has
not affected Fb and a,
greatly we gat tho usual ralation
9.k.Q Mevk
Breit and others have shown that about 0.8 Msv sh6txld be
added as an average effect of the Coulomb repulsion insicto thc WGLL
The more exact calculation gives 11L3 Mev as the dspth of the well
and also shows that good agreement w ith results at all energies is
obtained,
The rough method used would be applicable only at
energies very high compared with the Coulomb rapulsion at
Po = 62/rnC 2 4
Using the exact wave functions a. search for the best value of
Po has been made by Breit et el and the decision roached that
2 2
0 /mc is the most acceptable (a = l), A value of B 0 cy5
gives a noticeably Inferior fit as ala0 does a = 1425, The depth
of welS pertaining to a = 1 is 11,3 Mov in good aereenaent with the
most camful estimates of the depth of tha singlst deuteron well
of the same width, 11,6 MeV;

178
e
December 301 1943
LA 24 (28)
LECTUFG3 SERIES ON NUCLEAR PHYSICS
Fourth Series! Two Body Problem Lecturart C. Lc Critchffeld
pwrum xxvIIr: THREE BODY PROBLEMS
A second method of determining the range of huclear forces is
obtaihed in estimating the binding energy of the nuclei containing
three partic&esBH3 and He3,
The binding energies of these nuclei
are approximately equal, the difference being caused
tho electro
statlc repulsion of the two protons in He 3 +*
Since an attractive
force bet'ween two protons hag been demonstrated in scattoring experiments,
it follows from the equality of bindlng in H3 and Ha3
'chat the attractive force exists between neutrons,, Attractive
forces between all particles in these nuchi and in He4 are
also found necessary in getting agreement between observation and
the estimates,
The method of
c alculn ti ng
binding energies that we shall ap-
PlY
is to
problem"
reduce the 'chreerbody problem to an "equivalent
certain as sump tions and avo raging over tho
two body
third
particlo.
The method was first used in this connection by Feon-'
berg, Consider the nucleus H3 which has a binding energy 16.3 mc2.
Assume that the wave function that describes any two particles in
H3 is the same as for the deuteron* In order to exprcss the douteron
wave function as a single algebraic function tho form of the
two particle function will be assumed to be
whore (2/d)3/2 is essentially the radius of the wave of the deu-

179
tsron,
&rCher asmane that the wave fixnation for a19 three par-
tiales is a symmetric product of throe such deuteron functions in
the three separations r 328 q3 and pZ2;:
PO8
ten
one
hihe
I
%e potential eneygy af' eaah particle ~$32 depend upon the
ition of each of tho other two) but an effective two body PO-
Carrying out tho integral OVBP dT3 (the volma elomont of particle
particles is the sarne as in the deuteron except that the range of'
fomas is a larger.
The total potential is three times that
between any two particlea,
In Q similar calculation the kinetic
sner$y may be averaged and the result is that the fotal kinetic
is three thes that in the d'euterqn,
We shall apply the results obtained above to determining the
equivalent solpkion with a ttsquaro" potential well. Let be
the binding energy of I?; the wave squat$.& far the two body

problem "equivalent" to H3 is then
180
a r 1
-
M
Ar --
-I
mixture of singlet and triplet potentials, The two neutrons is
€I3 are certainly in 8 singlet state and if' the spin of the proton
ia parallel to ons of the neutron spina it forms the triplet with
that neutron and half triplet and half singlet wlth the other.
The expected relatkon between vT and a can be determined from the
deuteron calculations and is a simple avorage of singlet and trip
let relations
a- a
This average is based on the assumption that th8 affect Qf the I3
3
wave in i~l the same bctwoen poirs of particlea in tie triplet
state as in the dmateron.
The other relation between vT and
from the oquivalent two body equation for H 3
a may be obtained
That equation may
bo mad@ formally the sme QS for the triplet deuteron by substi-
tuting
*
wheree is the binding energy of the triplet deuteron. Then the
D
relation between X~V a is the Sam8 as botweon vo and
T

a for the triplet deuteron,,% i . e . q
The approprlato VdUe Of x2 is 0.78 and
This, combinad with the relation obtained from tho deutoron (H2)
has the solution 1
a = le08
for equal d@ptbs of well4 vT, The valus for the range of forcos
thus obtained is hiphcr than *hat; indicated by proton-proton sent-
taring but only by 8 porcsnt,
and the result obtained i s satisfactory,
The method is very rough, of COUPSO,
If a neutron-neutron
force had nob been postulated the agromont would be unacceptable,
There is a dif'feroncso in binding betwaen He3 and H3
about '76 Mov that should bo duo to the CouLomb repulsion of the
ppotons.
DE : i.
We may estimate tho expected rspulsion as follows:
,
oO-(3/2) ( drBf
( Q2/r 1 ( r2dr )
.& , , , ~,, : , , ,
$m-(3/2,(t/re) Q
dr
."(%)
, ,
2 represento the reoiprocal square o f the cxtent of tho wave
1/2
of
o!
f'unctfon and mag be taken approximately aquaa t o MG/fi2
= 0,43 in
units of m 2C,/4..
4
e
This givas Or.65 Mev for the calculated Coulomb
energy in satisfactory apoomont with the oxperimsntal value con-
.sldoring; the approximations mado.,
Similar throe and four body onloulationa have been made

y tho variational method with substantia y the sane result for
102
oarroot accounting Qf the bi 3ng ensrgios or H3, Ha3 and,
There is another ty
tal interest to the thoory of'
i
a-body problem O f fundman-
on, This is the scatter-
neutrons by hydrogen rnoleoufes* The theory bas been worked
out 'by Schwingor and TB Ejr and haa ahown that the scattering cross
of tho molcculs 2s oxtranslg sensitive to whother the sing-
to of tho deuteron i8 real or ''vtrtual".
Experiments on
scRttw4ng of liquid-air notltrons havc been intorproted by tho
thoory to prove concZusivaly that the singlet state is virtual,
ieO*, thePo is bawd SlnglOt Stat@'
Tho groat decSsiv
cause af' an accidental near cnncollation of
tion,
tho exprimant come8 about be-
tams in tho aa1cul.a-
Details of tho cnlculntions will not bo given here but tho
principal feature of them Is adily deacribed. The cnlculnted
aross section for noutron-proton scattering duo to the triplot
-24 2
wo11 aXona is 3*50 x ZQ at low enovlgy whoreas the observed
-24 2
La 53 x 10 ern From Wignor's hy.po$hosis the singlot
soattaring must have a CPO~SS saotion 41,5 x lo2* om2,
If
ia
* orda
tho triplet and q tho singlet CF
0; 11*8 [r;
Scattoring ?Prom the hydrogon molecule will show inter..
etwoon spharical waves coming from tho two nuclei if tho
th of tho neutron Is long compared with tho internuclsar
separatian, At room temporaturs tho wave length i s of the sana
do 8s the scgnratlan. Calcvlation of ths CPOS8

and
mounts to 41 if tho singlet stab i s stablo, ire', bo and
havo tho smo sign, and only to 0,38 if the singlet is unr
ro is thus d otor of ov@r 200 in cross section de-
pending upon tho atabilitg
of tho factor is due to tho
the singlet stnte and thc large size
act that sin6 2s so nearly equal to
0
-3 times sinal* catterinp; fj=* 3 = +J.
5 . 2
The right; side is 53 for tho stablo singlet nnd 20 for an unstGabSe
3 irqb t
Tho oxporimontal proce ro i s to cornpara
Of para-hydrogen with that of orthohydrogen (or rather the USURZ.
1 for llquid air
to the furtho?? oomplicat
ion is subject
tpons may convert
one kind of hydro@n infm the othbr, fn this cam the wavo
function of the final molecule has ono si n if tlzo spin of' one prochanged
and the opposito sign if the spin of the other i s
The interforone 8 then dsstructivo and the CTQSB 880-
tion propoptional to
O"conv rrd
(SW 6, - sin0,) 2

185
09 Qourso the oonvorsion is accompanied by n. change in xwtational
state because of the atatfstics,
%onv
Sclzwirrger an4 Teller have shown
is of the 883110 order of magnitude as
0"
when tho parahydrogen is mostly in the ground state, J
whm tho nautrons have su
state i s *023
00
so that only
0, and
low energy w i l l its CrodS see-
so oxtrornely small for tho cas8 of an unstable singlot
n, %e energy required tb rniso 8 para molecule to ortho
e unablo to oonvort
octron volts, hence liquid air neutrons (,012 ev)
pa to orbho
perimsnt is Chat the acattering cros~ seotion OF ortho-H 2
much larger than that of para hydrogen. This provos conaluaivdly
thak tho singlet deutaron is not stabla,
Exautly the same kind of considerations have boon mado by
Schwingor Tor the possibility that tha noutron has spin f3/2jf\ in-
stead of (L/2;)* *
onergy would then bo a quintet,
%e Icv of the deuteron that lies near zero
In this case however, the fortul-
tous oancekxation of toms goas not OCCUP cnd all ross s~ctfons
in molecular hydrogen am a$' ths same order of' magnitude,
the experiments prove not only that the singlet; state of thc deu-
teron is unstable but ~lso that ths ncuhron spin $8 (1/2)fi a
24 (29 & 30)
Thus
& XXX: INTERAC F NEUTRON AND PROTON WITH
Neutrons and protom interact with the fields that hnve
tablOshQd by classical physies nameiy, tho gravi tati ow~l

1
sB Oravitattonal phenomena am
ar procemes.
zhe
hand, am very 3ngaFtan
captum, emhaion of gnmma
G dissociation are
Trs. addition to thsse we
s. One of thsao fa
its rnagnotlc moment and th&
ts magnetic moment,
s there are
fields that influance tho nut-
e~vatione on th.
tho field of moson8,
o of this thoory ia to m-
e nature of
beta activity is cirawnacrfbcxl to sornc oxtant, Some fundamental ,
sa of the neutran is a PO-
tho deutoran.
Also, RccuraCe

I 180
tea particlea, l/fs is the WBVB function of the ground state
of the deuteron and e, FQP s;tmpLlicity we shaS1 assws
q8 in tho f ~ m
Y?

189
is
1 = c(E2$, H2/8m~) e , E Z / ~ * ~ ~
and the CYOSSI section for photo electrtc disintegration (only one
polarization is effective)
If we had normali2ad the neutron-proton Pmwave function8 to unit
@ndrgY the factor pp would have been unity,
Calculation of M is straightforward. Consider thO die
E
pols moment In the direction, B:: Q,
ME sJy8(@I'/z C O s @ ) ~ d ~
D
neutron and grolj'on and T, oxpressed as a function of E andE
1
becomes
% = /8T/3)(@2/Cio)( fi*/Ite )
(@* 1 -t E/€" )3
For the 3. -rays of Th C ', used by Chadwick and GoSdhaber, the cal-
culated CTOSS soction is 6.7 x lon2' cm2
To the CrOB8 section rE must be added the O~QGS soction
due to elfact of the magnetic v@CtQP,CM, The perturbing enerey
is then p. €i, and the matrix olomont, Mmj contains not only tan

190
s bat a sun over t spina, n o
ts to Et 1s state or poai.t;ive
vel of the doukeron in tho poaj,
is very Large
at low onorgy
relative weakness of t
ron magnetic mamonts to bo

I)
neutron by proton,
198
* (€3, / $B) ( h2 BA/ 2
of photo disintegration ilq simple radiative capture of
Syatam A is composod of one photon and tho
stable deuteron; %he photon is capable a$ two polarizwtLons and tho
n of three, hence eA 9 6,
ono proton cacb of
sect
apturo 18 than
8gstsm B aontains on@ neutron and
Tho mom
3/2 ( 2*d/kc ' (a;.,+ % )
ectxio and magnotic effects have dilfsront dependenciss on
tho onsrgy of the nsutron-proton gygtern.
At htgh energy $he mag-
y low energy,. howover, tho
gnotic offact ppedominatad and
rc: 1,4 x 10 -28 6 E fE?
utron at. 1-00112 tompernture, I/~U c,v,, E 2 X[~O Q~v,.
*24 2
% = .,I8 x 10 Gm
lrYhcn the noutron was diacravared thc way wa8 opened for
ament of a field thoory of beta radiation,
Foml In analogy to a&ocCromagnotie themy,
mi# was
!bo fundamsn-
ptions am:" .
nuc lo on ) .,
ron a,nd proton 8ro two
(2). Energyr spin and stati6tic.s am oansorvod in bsta
111) pRd€Gtlon by tho introdpotio f the Pauli mutrino,
($1 T ~ Q rest maeg OS the nautrino is zero (or nsarly

its spin is +
t with the'
o are radiated when neutron
1
and a neutrino are radi
n.gw to proton an6 8 positron
hangoa to nazltron.
(6) Elslotran and neutrkno sharo tka availab onorgy in
a11
sible pr6partions.
Xn analogy t o the inte
n botween chargad particle8
and light quanta, tha
(26)
numbor of
w = g2
matrix element c
e
The interaction eon8
dypam the amp
at thg YlU(J2OU8, p ep%ikkoa
*iC
and/0,, are tha

194
and, as in tho case of
dou tar on
the photoolcctric disintegration of thc
Tho shapo of the electron spoctrum is thus determinod,,
Exporimen-
tally. tho nwnbor per unit enorgy (or unit momontum) is ma8surods
divided by pE or its equivnlcnt in the Coulomb field and tho square
root of the resulting nmbor plottad against E,
In tho carefully
done work with nuclei that hnvo only ono beta activity tho plot is
a good straight line, tha Intercept of which W, Thc thoorg Sa
thoraforo substantinted by tho exporimcnts,
If the nuclonr charge is lcrgcr than, say 20, or if tho
electron encrgy is particularly low the sine wavc 'Is not a. good
2
approximation for tho clcctron W~VC,. 11 mom oxact value of c
is plus the charge nwnbqr of' tho residual nucleus If a positron
9s emittad and minus the chargo number for cloctrun emission,
Tho probability of omission of an olsctron with energy
botwcon E End E -k dE is

4
195
The total probebi
trino per saoond is gftren by the integral OF P($)dE OV8r @l@ctPoll
'energies
EI ma2 to E = W, . For ~>>mt@ th$s integral 5s
czosaly equal to
+
h z 60Q c fi
w5
Tho decay constant, A, $a proporblonal to the fifth power Of
total, energy relaased by the tranaftion,
Moasuremgnts
the half livas of the light radioaotive aloments makes
a detomination of the ''Fermi conatan
mare'is a class of positron mittera "chat
larlv well suited far deternining gr
JZght radioaotive yzucolol. c
i s p
This clasa oomprisoa the
tha
taining one mope proton than neutron,
ra proton turns into a neutron with the emission of a posI*
tron and, since the nuclear forces am avidontap the sms between
all pairs of paPticlos, -bho wave Punotion. of the neutron in the
final nuoleus should be very neBr3y tho same as the wave funo.t;ion
of tho Initial proton,
In thia
nu~lei, their half lives and cncrglbs roleasad am give
on tho following pago,
nttng two*pOssiblo final statoar
e I.I" 3'
A table of thsse
In some Instances two oncrgiQs are givsn
g the Wansltlon mloasing tho most energy,
The life time Is dotor-
tion goes to tho excited etate ths nucleurj subsequent1
IF tho transir

I
196
MAXIMUM KINETXC
sR$xa
1# I6
93 m , 092, 2.20
52'6.0 8 ;LJ
? b 1
2; 20
%
11.6 I 2m82
1.7, 3,o
3.54
3,. 69
J a b 2 8 fl 3 ,,87
2,4 I 4.33
3188 s 4.43.
e07 s 4.94
We have assmod thkt klucbron and neutrino are emitted
lizLi angular momsntum,
the omitted park5
The rnaxbwn amount OS spin that
8s is, t,horoforo,% . Zn order
the nucleus bs ohmged 1
WOOM rloutro
lsld contain tha
tog,
Rmnl'a oI?t?igi
trnn@tion from o m spin t o anot
locttxon or boCh
to bo am$tted with orbital.
il)
e
93% amp5iLudtm of wavos of highar wngulor momcn-%wn
knallar at'tha ntzozoar radiua than the amplitude of tho

e
s
I
and such transitions are tsmed "forbidden".
197
The half-ligo
of a forbiddon transition is therefore much longer than that of an
allow'ad
appears
transition of tho same cnorgy release,
Certain nuclei produce allowed transitions alkhbugh it
certain that the nuclear apiM changes by$).
This is parti-
cula~3y true of He6 which must have spin 0 but decays at an "al-
lowed'' rate to LIB with spin%,, Xn Fermi's theory such a transim
tion would be forbidden,
To acco
of span k l# Gmow and Teller postulated a spin dependant interac-
tion with tho electron-neutrino fiQld,
t fop transitions with cha.nge
The colculatlons made thus far apply only to allowed
transitions in light nuclei.
To oxtend thorn to heavy nuolQi and
bidden transitions more exaot accouet must be takon of the
wave functions for relativistic motion in a Coulomb fiold and there
are aevcral interesting fcaburas of the solution that are Drnpha-
sized in betadecay theopy.
FJELQ
THEORIES OF NUCLEAR FqRCES
The SUCCBSS of Femfrsr theory of beta decay promptsd an
a't;taqst to explain the forces between nuclear pccrrticles by the
same interaction.
Thus the olectron-neutrino field should play
the same role in t h ~ attraction between nuchar particles as tho
eloctromagnotic field plays in the force8 batween ohargsd particles
At the same time ths anomalous charaoter of the rnagnebtc moments
of neutron and proton might also bo explained by the interaction
with this field (~Viok).
The attempt was not successful, howevc
For two very good reasons, either OF which fs dufficicnt,
the forces calqulated are much too weak t o account for the ob-
First,

sorved strength an6 range of
198
nwlear f'oroes; seaondly, ths theory
gives attraction, In first approximation, only bekwcen neutron and
proton,
The forces betwoen like particles nppaar Zn second appro..
ximsltion and are repulalve,
Two linos of endmtvor have been pwausd in tryZng to con*
struct a field theory of nuclear forces.
One generaliaod the beea
bntoractionr
,
Thia was first don@ by Garnow and Teller, and by
Wentzol, who postulated an intoraction with pure ePoOtran fields,
80 that neutron and praton would be able to croate an 6leotron-
positron pair as well a8 an electron-neutrino pair,
This same
type of "pair emission" forces waa lator extonded to meson pairs
(Mllarshak) on ?;ho asaumption that tho meson has spin
On acw
count of the flexibility in SntroducZng new fields th.ose theories
are able to account for ths nuclear forces,
jsctionablo features of the results, howevor,
There aro aavoral ab-
In the electron*
positron patr tbeory, for exwpl8, special forms of tho states into
which tho cloctrons are arnittod have to be assmod in order to
avoid scattering of slow neutrona by the atomic electrons* In the
meson-pair theory the cnlculatod CPO~S section for scnttaring of
cosmic ray mesons by nuclei is higkor than that observcd excopt
For very weak interaction Eorcos (and a rather long range for the
forms), Them is R general objectton to the mson pair theory
that obsarvatiohs ahow about 30% mare positive mesons at sea SevoZ
than negative onc8,
If they were croatod in pairs tho numbor
might be expected to bo more nearly oqual,
Pending further oxparl-
mental ovidenco of
the spin of tho meson, howeivw, thc possibibity
of a nmson-qpair f

e
Tho othor class of attqnpts to describe nuclear forces
by n field tlzeory is typifiod by more direct ganeralizations of
electramagnotic theory,
sented by Yukawa, in 1935, before: the meson was dlhcavered.
Yukawa found that the range and strength of nuclear forces could.
199
Th@ First theory of this class was pre-
be understood if tho nuclear particles omitted "hsavy quanta" hw-
ing a finito rest ma98 of aboub 200 times the electron rest mass.
This quantum was conoeived as having Bossdstatistics, the sgm~ a8
a light quantum, but for simpl;lcStg the spin was assumed to bo
zerod
hrthermoro tho heavy quantum could have one unit of elac-
tric charas posittve or nogativs and. Yukawa suggested that the
heavy quantum could di sintearate intg elaatron and neutrino glvivlg
beta radiation.
The discovery of tho realmeson, of about tho samo mawt
by Neddormeyor and Anderson stimulated interest in Yukawafs theory,
Then it was proveh that chargod quanta of zwo spin gave repul-
sion between neutron and proton in the 33 deuteron,
To correct
this obvious defect the spin of the quantum was as$umed tic) be
unity,
The free apace wave equation9 of tho meson of spin one are
therefores very sim91ar to those of a light quantum, ike.
MaxwoLZ's
equations, Thus thore will be the vector field quantities and
I H and potentials for tlilose fi@ldsyand &,
The inclusion of a
finite rest mass) however, introduces a characteristic length whioh
we shall denote by l/&where
This Lengthl may be. usad to generalize Maxwell's squat$.ons for froe

Thrw ctlffkwent assumptions about the tma electric
202
charge on the sqson8 are generally
sidered,
These are (3.) Utn-
charged, (2) either positive or neg6ttive (+e),
(3) both charged
and unehapged.
The theories developed on thess assumptions are
referred to as neukra'l, chargad and symmetrical theories raspec-
tiuelg.
Tho requirements on the. 8's and f*s for neutron aad pro;.
ton a$@ somewhat different $'or the different theorlea#
Tt is nscesaary to account for oquality of neutronmneuc
tmn, proton-proton, and neutron roton forGes In the stnglot state;
In the neutral theoryI whiah does not; distinguish in any other way
between neutron and proton W(3 have Ql 2 g2 and fl k f2 for any
nrxcleon.
Consoquently the potential function for a singlat:
q' T2"2 ,3, and the third term v&nighes, Sa
The, potential funotion Po& the triplet state is
vl(P) = (e2 3- 2/3 f2I (@*"/P)
4- dipole-dipale tern,
The ertatlc, g 2
term is repulsive for both states and must be ovw-
come by the =2f2 in the singlet and by the dipOlQAdlpbl@ tsrm in
the triplet,
In the triplet the repulsion is increased by 2/3 f2
Bethe has shown that g m
le6 and triplet binding energle
taken to vanish ana the &in$*
orreot--y ab-
tained by c
ipoke term has been discunssed unrjer the theary of the
deuteron,
In that discussion an attractive atatic potential was

203
e
mcnt with ouscrvations on the deuteron with f = 3,246.
In tho charged meson theory there are. no neutral mesons
postulated and the meson must have either &e4 A neutron can emit
only a negative meson and turn into a proton, and @ proton can
emit only a positive meson turning into a neutronr This has an
important effect on the nrathre of the forces. The neutron and the
proton are Considered to be two states of the same particle, The
property of being one or the'othsr is called the isotopic spin in
analogy to the ordinary spin. A nucleon may have m$= -9 in which
case it is a proton or m ~ & = for a neutron,
analogous to ms * -8 and fig f 8
quantum numbers of the nucleon or eloctron,
This is directly
For the two possible rnapnetic
The significance of
the formulation of isotopic spin is evidenced in constructing
rnanyFperticls wave functlons in aocordance with the Pauli exclub
slon principle,
By thfs principle, and the assumption that neu-
tron and proton reprosent two states of the same particle, the
wave function of the deuteron, for example, must be antisymmetric,
Thus the '3
stat6 of deuteron, which is obviously symmetric in
spin and coordinate space, must be antisymmetric in isotopic spin
spaoe; this means that if' the two particles exchangc charge, as in
the emission and absorption of charged mesons, the complete wave
funotion of' tho 3S changes sign on each exchanp.
hand the complete wave function of the 'S
on the other
will not change sign,
sincc it is already antisymmetric duc,to the spin state and the
isotopic spin function must be symmetric,
The nature of charge exchange is thereforc
the two particle potential function by giving a plus
manifested in
sign to states

204
*
with’even 3 L and a minus slgn to states with 0d.d S +Le The
static Forces are thus attractgve far some states and repulsive lor
others, instead of being alwetys repulsive as in the neutral theory,,
Unfortunately the char$ed kheory gives no forces between like parbecause
of the impossibility of exohange In that case and
cann9.t bs seriously considered 8s a theory of nuclear forces, It
represents the simplest case of QxOhanegs fOPCeB, I’rowever, and we
shall need the, description of such forccsa in the sgrmmetricsal
theoqh
The symnmtrical theory postulates that both charged and
ged mesons are represented by ths field.. Slnce the chargad
part contributes nothing to the forces between like particles the
ohoics Of g, = gp and fY1 * f for the neutral mesons is rulsd out
P
$or, then Che forces between unlike particles and batweon like
Ica in the ’S
would bo equal bQcRusc 0‘2 the neutral rnesoYla
Theladditionnl charged field between unlike particles
wowlD then make them unaqual,
neutron-neutron and nsutron-proton states are
Tho potentials in the singlet
v0 (n,n) = (gn2 - 2fn 2)(e “EZr/p)
= (gngp 2fnfp) + 2(gn
1 2 . cc 2f lf 1) (e-KT/r)
n P
rl, 1 c fn 1 etc,, apply to c d meaoylsl The factor 2 in
ed Bxgression takes acc
The unprirned g and F re
the two kinds of cJhapged
o neutral rn@sms1 as before,
We can equate Vo (n,nS, V, (p,p) and V,(n,p) by taking
= -gp n 3 gpl = @, 2 * gp2
f, = mf rnl rgl fn2 M f 2
0 P P

.I,
205
which has the symmetrical solution: all lg/ are equal and ~ l l
if1 are equal. Only tho phases thesf3 mlmbers diJY@r*
Tho ralative merits of tho neutral and symmetrical
I
theories have been determined by Bathe $0 whom thfs semi-classical,
description of nuclear fields is also due. Bothe Finds that tho
neutral theory with Q = o and f 0 3r24a can account for Che two
body problems satisfactorlky.
Thars is one artifioe neaessary,
however, and this arises becauee of the 8ivergenoe of the dipoledipole
forces at r s 0, Ra the wave f'unction of a particle is
concentrated 1x1 a region of radius r tho rninhum kinetic energy 1s
of the order K2/2mr2.
The averagB dSpole-dipole potontlal, on ths
other hand, is proportional to l/P3*
the radius of the wave function the lower the energy and cohse-
quently there is no finfte state of lowest enwgy,
it is customary tolassme thBt the l/r3 law or attraction cease3
at a certain "cut off" radiuq,
This shows that the smallar
To remedy this
In the neutral theory Bethe finds
it nwessary to cut off at 0,32 the range of force (,32/K) if it
is assumed that the potential remains covlstant at smaller r and
equal to the value at cut off,
The theor$ "can give8 the carreot
upole mmont and also the correct S-states and Slscattering*
The same method applied to the symmotrioa
to a cut off at (1.3) (l&)
heory laada
and to a quadrupole moment of the,
signa Thus tho num"e5cal results, at least for the
D states, are hlghly unsatiafacto
and
Qualitatively tho symmetri-
ea1 theory is to be preferred because the charged meson6 appear
-*
in cosmic rays; they might reaeonably be supposed to disintagpats
into electron and neutrino; and they mako possible, in princip3eI

206
an understanding of Lhs anornolous magnatiQ moments of neutron and
@
proton,
In case the mesons are supposed to emit beta particles
the Fern% constant is a product of the intoraction constants per-
taining to nucleon-meson interaction and meson-beta interactions,
The decay period of a free meson, which has been accurately mea-
sured by Rossl, ’ should depend only on the meson-beta interaction
constant.
There RP~, therefore, thr.ae determinations of the two
constants and they are in violent disagreement,
all tihis the calculated burat producing cross section of ohargsd
mesons in the Rtmosphere is muoh JaPger than the observed oross
see t 1 on,
In addition to
From 8 quantitrtive point of view, the neutral meson
theory is the most promZalng analog of the eloctromagnetio theoq,
Perhaps part; of the reason for this l a that it need not explain
the prop0re%e~ of
the obwrved, chargod mesons4
There i’a some
indication Pram the scattering of High energy neutrons by probons,
Ler, tho cas@ in which the P-wave bocomes important, that the.
nsutral theory predicts 8. too large cross sootion,
The symmetri-
cal theory accounts for tho exparbents on fast neutrons satis-
fczctorily.
s
Zn conclusion it O m on3.y bc said that it is hoped that
further work on scattering CTOSB aectlons at higher energlea,
angular distributions in scattering and photoeloctrlc procsasos,
the naturo, particularly the spin, of the mesons# @to.,
will guide
us in tomulnting n fiold theory of nuclear forces that will ex-
plaSn everything.

LECTURE SE2zLE;S ON NUCEEAR PHY
Lecturer:
V, F,: Weisskopf
The ca>cuI.ation and the theoretic prediction of the properties of
is made difficult chiefly by two reasons: our ignorance as to the
e of $the nuclear
inability to solve the complicated,
bbms which we face in the quantum mechanical treatment of almost
us except the deuteron,
Some attempts have been made, however, to
abt$in Qualitati,ve results from the theory in order to interpret the vast ex-
perhankel material on nucleap reactions which been collected so Tar, In this
,
of nuclali,
account is given of the statistical metbods of describing the behavior
These methods use aa few as possible actual assumptions concerning
ar forces or tha nuclear structure.
marized in the following pointsr
Their main assumptions may be sum-
A) Thb nucleus can be considered as a itcondensed phase" of the neutron-
proton system in the tharaodynainical aense.
"he neutrons and protons form a
*
ich they are dsnsely
ed, so that tihe nuclear matter has definite
t the volume lis proportional to the number of constituent8,
By a@-
aripal form, one can, erefore, introduce a nuclear' radius
-til
where A Is the number of constibuents and ro s -I:*# ko em.
The results from this formula are in fair agroement with values Pram~experbnts
in which the si~s of the nucleus is involved,
Furthermore - and this is the
of assumption A - the characteristic psopsrties of nuclear matter,

208
especially the close proximity of all constituents, are maintained even if the
*
nucleus is excited to energies high enough SQ that St rdght emit one ar more
constituents, This fs valid beclause the bability of emission of the donstituents
is very small, 90 that the nucleus is in a wt3,J.l defhHed state before
having emitted the particle,
It is in a stehe which has essentialky a e s m
properties as .the lower excited states which do not emit partioles,
This is in definite contrast to the situation with excited atmns,, An
e4eotron escapes very rapidky if excited above Sonisa.t,ion snsrgy and no states. of
tb atoms in which the electron is still oloss to the atom could possible be de-
fi#ed in this regiov of excitation,
'Ilia nuclear conditions however, are similar
to that of a liquid drop or a solid body, dnce khe heat energy of such systems
b
may weal be much higher than the work necassary to evaporate one singbe molecub.,
Tho state befare emission of this molccul@ hiis a life tba lQng enough to bo well
dofined,
Emission af a constituent by means of nuclear excitation may then be
divided into two steps:
first the oxcitation to the excited state, wd then the
emission of the particle from the excited state.
The analogous proces# in atoms,
.
however, must be describcd in one single step, since the time interval until the
electron leavus the atom is not long enough to define a regular quantum statc
in which the electron is sti4.k within the atom.
The assumption A is no longer fulfilled for very high excitation
anergies of the nucleus,
as in the condensed phwe
The range of anergy in which the nucleus can be con-
ria from nucleus to nuclaus, but it is a&f@
to assume that the assumption is true f ~ a r rang@ of excitation up to a large
fraction of its total binding energy?
E) According to Ikshn, a nuclear reaotion can be divided into two well
od states.
The first is the formtion by the incident particle of a
compound nucleus in a well defined state; the second is the disintegration of,
the compound system into tha product nucleus and an emitted particle,
The second
I) stagu can be treated as independent of the first atago of the processe 'She basis

209
of this assumption is the classical picture of a nucleus as a system of pasthies
.
with very strong interaction and short range forces,
If the incident partido
comes within the range of the forces, its unsrgy is quickly shared among all
constituents we1L before any r'cemission can ocmr., The state of the compound
nucleus is now no longer dependent on the way it was formed and could hava been
produced by any other particle with corresponding mer& incident on a suitable
nucleus,
Its decay into an emitted particlo and a residual nucleus is thus
independmt of what happened before 4
Bohr's assumption is certainly right for a oase in which them is only
one single quantum stato of: tha compound nucleus which can be formed by tho in-
cident particle conaidering its energy and othdr circumstances @
Then, evidcntly,
the properties of this state are well defined and independent of the way it has
been excited,
It is very probably not a good assumption if tha state of the
cirmpound nucleus which is formed by the incident particles consists of a super-
position of several stationary status,
'fhe course of the process then depends
tgtrongly on thc relativo phases of tho states, and is thmofora not independent
af the initial procoss of excitation.
In the case, howeverl that the density
bf states of the compound nucleus is very high, so that their widths overlap
each otha strongly, a groat many states can bo excited simultaneously by the
incident particle,
This phase relation my be at rdfidom znd the resulting
pPocesscs, themfore, independent of the way of excitation.
Bohrls assumption regains approximate validity for this case,
It is possible that
'his is made more
than probable by the classical picture of strong binding forces used above.
Actually, classical considerations are justified in the region of high level
density wit'n overlapping lcvols where no quantum selectivit, v occur.s4
C) The statistical method assums the existence of averagl: values of
certain mvgnitudes averaged over stctes within R not too wide t3xcitation energy
interval.
energy,
The ?"vexages are supposed to be slowly varying functions of that
The ~gnitudes concerned here are level distances, transition probabili-

tain statas, etc~ The validit
ful applications.*, Xf the energ
210
is assumption m n onZy be proved
rvals of averaging must be taken
LECTURE SERIMS QN
V, %, WGisskopf
Ad
Tbr: garieral chhractar af the energy states of a nucleus om be described
lowing' menner~ Thcre isj 50 far, no definite regularity found estabtho
spacing of tha onorgy Levels, The distance between Lha lowost
of the order of ma.
1 Mdv* Solsttimes they lie Qloser and in
groups. Information ?'bout 1,
ecently measworn@
sible to detarmin B directly, It is interesting
t there is no i
est levels on
dependence of the average dis-
alight tendency to smaller disi
tances for highctr weight,
*
It is 'safe to assw~o that tht, level dclnsity increases with higher
energy as is oxpdCtod of elmost eny mechanical system,
tha stronger the mow particlas the nuolcus consists of.
o sxpox'imental muterial mailable
l'hb inorease
Thdre is,
the slow neutron capture oxporim@nta provide some very soant

evidence on the Xsvsl de&
The states of th nuckeus, W i t
a finite lifetim bacaGsc. of the possibi
231
citation as is explained below,
ception of th ground stat@, have
f radiabivs transitions to lower
states and of cjoction of particles,
(The emission of@rays
aan be naglmtud
hero since its probaki&ity is extremely smalli)
l'he states thus hnva a finite
which is connected with the lifetbe %hof .tho nth skate according t o
the rclakion
the total width
I$! the finite li5etSmo is due to different emissions,
he sm or partial widths# which themselves are propor-
tional to the specific ern$ision
-.. A"
' 2s the radiation width and
litLos* We therefore can write :
he partial width corresponding to
of a particle ci2 whish may be o, proton, a noutron, or anaek-;partiole,
tho emission probabil
per second of a partiole & by the
Thle width of the law lykng levels is purely a radiation width as long
tfon energy is lower thnn the Lowost binding enorgy of a particle,,
The bindbg anorgy Ba of a p&rtieAe 5 i s the energy ncoessar-y to remove it from
the ground state and have the residua9 ru~?leus in its ground stab,,
above I
t&e gram4 state.
The binding energy B,
We there-
ciLation energy Qf Lhs nth love1
of e. proton or a nuutron is found
to be about 8 Mav for nucJ,ei which are not taa near %he lower or upper end of
ic tab&, @hs vor olernents about up t o oxyg do not show .my
I From then OM)- 6 rcge hut thz value my be several
B heavy and of the table (A320Q)
g snergy is smlle
af the weight, of'Uc
arags of 6 MeV for the elemen58
Thsse figures are taken from the total binding onergg Wo of the nucleus;
he eno~gy neoessary to d~ornpoa~~ the nucleus into neutrans and protons,
which is given By tho mss defect* The toba
inding energy 'No is roughLy

212
proporbion&. to *the number A of constitucnka and it is therefare justified to
assume that the 'binding energy BB, OS one particle is W,/A,
which leads to the
figure s rrlant iane,d above &
We know, howcverj from the &-values of p-n reactions that it requires
a8 much as 4 Mav to replaoe n neutron by a proton4
This showe thaC
large flVctuatSans in Sa from the averng,s value must be expected.
With sufficient excitation energy En, a pwticle 8 can be emitted with
different energies, corresponding to the differont states in which the residual
nucleus can bo left behind,
We thsn can subdivide ran in2;o differant partial
widths:
(1)
s the width corresponding to an ornissj-on bf
a with the spatlid
hat the residual nucleus should bs: left in the statead
The widths 4" increase w5th higher excitation cnorgy since, firstly,
icles can be emitted and secondly, mors dfffuront states #&of the
nuclei are possible, so that the number .of t'esms in the bum (1) in*
creases, and finally, it is expected that the probabuity of ekssion of a
partScle increases with its velocity.
At a certain excitation enorgy E
the
width A" becoms larger thsn the level distance and the ldvels overlap,
The nuclear levc?&s in tho region abavc En= 3, can be invostfgatcd by
the bombardment of nuclei with pastLclos
of a given an~x'gyg
If the particle
hits the nucleuy and is c;hsosbod, it forms a compound nuclcus with &m GxCitation
energy Ba-kc J This absorption should be ap ociable only if the energy BzI 44
0, or within thu width of, an excitation level of the wmpound nuclsus,'
monoenergdtic partiole beams, it is possible to investigate, by varying
I
peotrum of the compound nuclsus from 3, upwards,
We expect to
obwrvs ltr@sonanccil absorption when Sa
Resonance levels have been observed wlth
6 aqua1 to an excitation enorgy E,;
tich and protQn bombrzrdmmts on
averago dlstanau of LsvcLs

213
with an excitation energy above Ba becomas vary sntall with increaszng atomic
number so that the energy of the bombarding particle beam cannot be dofined
e
sharply enough to separate them in heavier nuclei. No regonance has been observed
with beams of protons, deuterons, oreparticlcs for elements heavier
than zinc (2 qO),A sr66,
Although it is even more difficult Lo produce monoenergetic neutron
beams of arbitrary energy, there is one energy region in which neutrons can be
used with great success; naniely the rogion of wry low thermtt energies.
aan be slowed down until they rcach thermal or nearly thermal energies,
Nautrons
Their
energy is then very woll defined compared to the poor energy definition of beams
of fast particles.
Their anergy can also be measurad to a high accuracy by means
of absorbers which have, in that region, n known snorgy dependent absorption
(baron) +
Recently methods have been developed to produce neutron bums with very
sharp energy up to 5 eve
Resonance absorption with slow neutrons can, therefore, be expccted even
for heavy nuclei.
Although it is not difficult to produce similar low anergy
beams of chnrged particles, they are useless bceausa they do not ponetretc the
Coulomb bcrrier of the nucleus.
The resonance capture experiments with slow natrons give informtion
as to the shape and width of the sesoniince levels of the compound nucleus and will
be discussed in connoction wfth the quantitative expre~sions. It is found that
thu main contribution to thc width of thus@ lcvels is the radiation width.
This
means that a captured slow neutron will in all probability stay within thc com-
pound nuclcus, sincc the most probably effect is the wdssion of’),-quantwn after
which tho excitation dnolrgy sinks below 3,
and no particle omission is possiblec
Some evidence as to the level density of the compound nucleus cm be
obtained from the number of nuclei which are found to show resonanccj cilpture of
slow neutronsI
This nwnbor indicates the probability that a level lies in c?
re-
gion of a few volts above Be+,
One obtains by that method p, lsvel distance of

2 14
roughly of the order 10 ov at the excitation energy Ba for dements whose atomic
c)
number A is about 100 or higher, It is much larger for lighter elsmcntsj in the
region of Ad60 it is probably of the order of 100 ev.
We now proceed to tho discussion of the cboss swtions of mtud nuclcw
reactions, According to assumption B, we may write for the cram section 6(a,b)
of a nuclear reaction in which a nucl.ous is bombarded wfth a particle g and a
pwtlcle .- b is e jectod:
,
. (2)
Here fa the woss section for the formation of the compmnd nucleus,
and 'Qp the relative probability that the particle b is emitted,
The cross section 6 can bi. written in the following form8
a
ba55i&;l
Here Sa is tho maximum value (g can possibly assum and e 1s tho
a
''sticking probability" of a which necassarily is smaller than unity. For
neutrons whose wave length
is large compared to tho nuclear radius R Sn?$Tx ;B
c*
For charged particles, Sa can bo roughly defined as the cross section for penc-
trstion to the surface of the nucleus and is strongly dependent on the onorgy in
1
the familiar way of the Gamow-penetration of Coulomb barriers,
discussion is found in Section XXXIII*
The quantitative
6 2 3 tlic $robability that a formtion of the compound nucleus takes
plnca, if the particle has reached the nucleus.
As a function of the onergy,e
1
will display the reeonance propartizs, mentioned before.
It w ill be small betwoen
tho resonances and large when B +e is equal or near to an excitation levo1 of the
Q
compound nucleus, From a oertain value 4 on, tho widths of the levels overlap
and 6 becomas a smooth function which, however, must be smaller or oqual to
unity.
Naturally, an average value of f over the resonance is needed if the
e
t
to
incident bdam is not sharp enough in onesgy to separate tho ~JV&,
It is reasonable to assume that e becoms unity if & is small compared
a
the distance between thc? nuclear constituents. Then, classical considorations

215
can be applied, which make it a~em vo y plausible that every ppxticle which comcs
bange of ths nuclear $orccs is incorporatad into a cornpaurld nucleus,
th
is SuAfilled for energies abovs 10 &7t,
I
Thu'second factor in (2
clo is anittad by the compound nucILe
ivc probdbility that a pnrti-
s evidently given by
re p"k i+ the; satin
/
of a particle b, amraged 5ver the compound st~tes
whigh arc exci
atage of the process,
particles c which me ejected,
TLo sun in tho denofibn8.tor shau
c
a
(3 )
ound stabs, for the omission
be extended Over aJJ,
TLG wjig~t&i~n of t k . ~ la furdG2gr:t ;.ob different; from tho oal-
culation of the a&, TLa
fornation of a aompaund nuc3eus and there
are tho probabilities of tho opposite gpoaoss to the
a fundaraefltql relet2on be tw@@n
in
A few qualitative conclusions can be drawn without using the formul2s.
The for neu6t.m emission is always much grsa'csr than the
emission, or the rb for any ChargBd particle, because of the Coulo
barrier, except under unusual condiblons 'chat the bin
is considerably smaller than that om Of bhe neutron, $0 that tho proton has much
more energy. at its disposal; Thus, if a reaction with neubron omissi
geticakxy possible, this is gene
probable than all. the otha
are found only Just abbve the threshold of w
with neutron
when the neutron has very smal energy. me
energy dependent and beconie smaller with larger 2, Nuclear re
s therefore are very Pare far heavr olemcnts,
P
f
Deuteron emissions have never bwn observed,
This, is becauge OS tho
to remove a deuteron from the compound nucleus; it ia equal to the
in4 two constitumts &nus the small. amount of the detu

216
energy* The emission of a neutron alone is therefore by far much more probable,
What can be predicted as to the energy distribution of the emitted
particles? Aftar emission of a particle, the residual nucleus is left in on@ of
its excited states, say with an excitation energy EB or in its ground statat
The energy
where E is the sxcitaticm energy of the compould nucleus which 1s in turn dete?r-
mined by the energy e of the incident particle sr producing the compound
;a
nucle;us: E z Ela.CQaI To every level E of the residual nucleus corresponds
k9
and energy value e Thus tho energy distribution of the particles omitted
bfi
consists of a series of peaks which are a picture of the spectrum of the residual
nucleus, thQ highest energy corrosponding to tho lowest level of the residual
nucleus,
In the special case that the incident particle is the sanie a8 the
emitted one - (n,n) - (p,p) process - we thus obtain
I
B
A - Graund level of initial nucleus
B - Ground level of final nucleus
C - Ground level of compound nucleus
The highest energy corresponds to the elastic scattering, This
b
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,
217
Actually it is impossible to distinguish between the folLowiizg fQur contributions
to the elastic scattering, which give rise to scattered wave-functlohs which &re
superposed coherentlp)
a) Scattqrbg by the Coulomb field if any
t
b) Scattering.jby the &ape of the nucleus in analogy to the
optical diffractAon
c) Penetration to tho surface of the nucleus and reflection
there
d) The forming of a o
emission with squa
und nucleus wit
It is, howover., important to distinguish d) from the other contributions in ordor
which cqrresponds to the fsrmtgon of the compound nucleus can be dcdined by con-
wctring it with

general (apart from individual fluctuations) proportional to the energ;y.Q
218
and
to the penetration probability through the potential barrier in case of oharged
1 a decreasing function of (Eb.
p&rtiCluS. P(cb) is an increasing function,CJ(E
P
We thdrafore obtain cn energy diatribution in the region in which the 10~018 of
tho residual nucleus become undissolved, with a definite maximum which is in the
case or" uncherged pnrticles very similar to a Maxwellinn distribution,
discussed quantitatively in Section XXufVI.
This is
January 20, 194.4
LECTUW SdRIES ON NUCLEAR PHYSICS
Fifth Series: The Statistical l'hcory Lecturer: V. F, Weisskopf
of Nuclear Reactions
UCTUW xXxI11, gross' Seqtions and Emission Probabilities
Tho values of the cross sections of the formation of a compound nucleus
by impact of a particlo with a nucleus, and the probabilities of tho inverse
process, tha emission of a particle by a nucleus, obey certain general rulss,
which are derived from the geome~rical'propertics of the problem,
Let us consider the cross section $('!of
the formation of a compound
nucleus.
It is appropriate to decompose my cross section 6 of a nuclear process
R
with an angular momQntum fi 1 in raspect to the sccttering center:
into tho partial cross sections @ belonging to tho contributions of the particles
)a-,
The angular monieqtwn in respect to the scattering center of a classical particle
_." (6)
*
moving in a straight beam is given by m v d whore d is the nearest distance of a
straight directisn of motion to the center, Quantum theory admits only integer
multiples of % as values for the angylar momentum and it is justifiod to 'say

IJI
219
that, in a straight beam, ell particles nzoving along lines, whose distnnca d from
.ChQ center lies between
$ave an angular morrantwn .kR.
us all particles of angular momentum
asmhn area perpendicular to the beam
of the size ($?gf'i)$%! A nucloar rtlaction which removes particles of a given
angular mornentum,frorn the beam, cannot remove more thsn come in end thus never
A
ss section 6 larger than that area:
I- .-.". (7)
A more exact derivation of this relation is given below,
It is to bs expected that partioles, whose nearest approach on o.
'. ?
straight lina, d, is largljr than the range W of any forces coming from the -scat-
terer, wul not be scattered at all, Thus 6 150 if d >I2
R
$%4
We obtain the maxjmwn possible cross section of a system if we assume tha$ 6"
d"
has its m;sxhm value for .h

*
that every subwmm consists of a superposition of i331 incoming and n.n outgoing
spherical wave of equal intensity (first and second tvrm in tho sqtaaro brccl,ceb){
If a perticle wave of this form impinges upon a nuclcfus, the superr
position is changed in two ways:
2) the intensity of tho outgois spherical. W&VP
is diminished due to absorption of the parkiclos (the latter may or may not be
reemittcd); b) the phase of the remaining outgoing witv~j is changed,
Both effects
give rise to a scattered wave since tho outggolng wave is na longur ablo to inter-
fere with thc. incoming wave in tho sam way as in (y),
thd only way ,j.n which there
is no radiation from the center.
Thus tha aatual wave function ft has not tho
asymptotic form (LO) but
taken place, *q is srnaller than unity, 2.b
waves b
ht us, docompose (loa) into the incoming subwavrts:
.: f'g,e*.~rjs~
and the outgoing ones
$,; 3 *YE '\/2~
% kr
1
is the phase shift of tho outgoing
il @-~(~~+L.N.,J-*(O? .-__
4
--,.---

c wave :
The d
to a plme wave
221
nce betwoen (loa) wd (10) is the wava which one has to add
) $0 get the actual wav9
/f.'i
* I
it id therefore the scattered
The crcm aectian for elastic scattering, for a single d, can be obtained
by calculeting the current of the 4th suiwe.ve. of #-- )I$ .through a large
sphere and dividing'it by the current of the incident wave.
1
' f ci-qe rig) d(&+i) " ..
This cen also be writthn in tho form
This gives
(14)
It is seen froni (12) and (14) that curhain relations hold bctwaen the
Capture cross section is given by (7).
"he elastic cross soctbn, however, has
a higher upper limit, which is reached if r)'sl and a g 2R/#:
--%
This mnxhm can only be reached if 7 S 1 or if 6')s #
c
This can also be understood in tho? following wzy:
. ,
the maximum effect
of elastic sez*ttering is obtained by a change of phase of #' of tho second term
in the square bracket of (10) (tho outgoing subwave) without reducing its
This is identical, with adding to the plana wc.vy just twice the outavd.
Thus the cross section corresponding to this, is 22 times larger
than the one resulting from absorbing thc outgoing subwave, Hence we gat a
maxhurn cross section far scattering Pour times largar than for absorption,
*
given value of@ "I.
ljJXitL3
1'
**f
'9
The expressions (12) and (14) limit the possible values of @ R (4 for a
4
This is indicatod in Pig. 1, whoro thc upper nnd lower
6''. flf" ,.is plottad against It is seen that it is nccessarily
IC) assues its msurimunl value(& A + i ) ~fi F .
djl
We foLlowing theorem is OS interest, since it connccts the total cross
ith the amplitude of the elastic scattering; the total cross section is

222
e which also can be writton in tho farm:
1
q ftd -.-%J.&”pAfl RX%R+I)
wheso An(1-W aigJ] is the value by which ono has to multiply tho outgoing
subwave of the undisturbed plane wave in order to get the scattored subwavo(b3),
i
Fig, 1: Upper and Jowor limit of ths elastiu cross section for given capture
cross section.
If the scsttering object (nctuczlly tho range of the scattering forces)
is large coni9ared , ~ll’wc?vc?s for which kfi(.gcan be absorbed, but the wava
for which 1 - R/n )) 1 vanish for r 5 R and thus arc not influancud and should
a have gicl ;31 0 If R/j( is a large number, it is a goad approximction to
L

the,fact that a
interference effect Pf a ac
kon out by tho sh&daw,
becomes com-
value vnly for vvry few values Q
cind 5t is masthy lower

a
224
nucleus in the state a; b) the umission by tho compound nuclcus, of a particld
a with an engular mornenturn$,!,, and Iswing the residual nucleus in the stato G.
The two processes a) and b) arc invurse. The first procuss is rmasurod by E! cross
section 6aaf and the secmd by Bn emission probability Gak Wo should c?lso
ifiod the direction of khc apin of the emitted particlo in respocL to
some arbitrary mist We omit this and assume in the following that the spin is
given together with tho angular momentum A e
In the definition of CaGAit i s
assumed that the incoming beam is spread over an energy intervd largo enough
to uxcita many component levels, T;*,B is the average value of the emission
probability, averaged over thsse.lavds which can be excited by the first process
within a snmll. anergy region,
In order to siivd indices, we replace the triple index (ad) byeand
writs and T;r/ , wherever it is practicable, The result of the following
considerations will be the relation
whae D is the avcmgo distcnca between the levels, which can be oxcited, Before
c
going into the detailed dcrivntion, it is useful to understand, why a rslation of
this chwacter is expected to exist,
Let us first assurm that no other process but the one charnctcrized by
can follow the c@porption given by
In this case, the level width is
equal to r& It is to be expected that $a is proportional to the ratio p'
(524
of the width to the distance of lwmls, becam this magnitude measures the rela-
tive portions of the spectrum in which absorption is possiblo.
It is plausible
m&ximum absorption is reached, ifr/ is of the order of unity, The
mimwn value of rf is fik '(2ktl)and
d'D
WQ may expoct P, relation of the typQ
where the constant $s of the order unity, The detailed calculation shows that it
II
is 2rt/f,k+ 1).
This considsration also holds true if tha coriipound nucleus
created by the absorption of the particlo a. ccvl decay in suvoral WF.YS.
-
This ef-

228
fect would broaden the lines but would not change the clbsorption avurege ovw
the lines,
9 Let us now oomparo the probabilities of the two invorsb processes,
The probabilities should be equal if they cndud in states of eque-1 weight.
Since
this is not the cnse, they should ba egucQ aftzr dividing thm by the stntjstical
weight af tho ebd stQteS,
Tho pyobability Wl per second of the creation of tho compound nucleus
by tho particle a if it is enclosed in n big sphsro of radius R is giwn by
_. 4" V (15)
W i " ~ f h .. ~ n ~
whore v is the velocity of the particle and g its wave length.
understood in tho following wey:
This can bo
if 1IJZgCc'aaumos its maxiiiiw valw (2k+f)fl
the particle with an angulzr niorientum )iRis sur@ to be captured on its wcy to
the contw of the big sphere,
The averago probability par unit kbm Is bhen
v/2R shoe it nlay be fowxl anywhere along (z diamter, {Its angular momentum
buing fixed, it mves on lines which havc a given distance ,(,% from thc centor,
The length of these lines is na&y 2R,
In C~SQ @&is
invarsu ~TOCQSSOIS.
6
since R i s assumed very large: a >)R&
snxillsr than its ~.ililximrUn value this probebility 5,s reducod in tho
hg ratio and (15)results.
\Ye now compme the statisticr,l wights ,of tht! end status of the two
If the anergy of tho incident particles lies botween Q and
4
the statistical woight of the end state is -Uc(E) dE, hhoro U,(E) is the
level density of the compound nucleus at tho excitation energy E which will. be
sclcction rules,)
(Only thess Iuvcls are couqtod, which can be excited owing to angular
r/,
Tho probability of the inverso process is ~;t
and the statistical
weighb of its end state is thr! stetistical waigbt of a fro@ particle with the
L
*
cnorgy between 6 and
We thus get
+dIE,and the angulcr nornentm 6 in a sphere of a radius
.- -
-

By expressing cross sootion ca by Q dimensionloss mgnitudc .Q s
which never can be larger than unity, an
Ita,
y introducing the a.vcrngo love1 dis-
momentum and with the rasidual nucleus left in a &finit@ state a, is thus bound
by an upper limit (Q 4 L) :
distance D.
6 (16b)
A few remarks are necessary concorning the significance of the lovol
nuoleus, frorn which the particlc can be omitted.
the levels which con bs exciked by the particle .-
Q wi$h tho angular riionentwn
(and given spin), hitting a nuclous in the state Crj, On the other hand, Lis
tho pnrtial width for the snlissfon of a,
the lovols descrihd cbove (whose nverngo distcnco is D) within a cortdn sml1
energy ragion,
D is defined as the distcncs bQtwccn tho levels of the coritpound
Zt is tho diatcww betwoen
with the SCZI:IC? properties awragod ovar
Expression (160) is not bound to this definition of Db
Far
m%r@le, D could bG dafined as tha tweragu distance between
levels of tho
coiuponont nucleus in this energy region; is then understood 11s the averngu
over all thGse levels, (for ciost of whiohra So), tho rolation (ab&) is then
d, In Cha following we will, howuver understend by D End ra
average distance and width of the levels which man emit tho particle 8 with
tho properties specified above.

*
227
LECTURE 3E3S1ES ON NUCLEAH PHYSICS
s: The Statistical Theory Lecturer; V, F. W&iskopf
gf Nuclear Reaction8
LECTUM XXXZVe Th~Clted NQCZ~US
st us consider a vluc1tzus in an excited state n in which it i s able t o
icles.
The st&.Ce tz has a finite Llfebsule and the probability of finding
rn is the width of
eus in $his state decays accordinG to the law e cf”t e
the state, In general bhe state n can emit different particles B and these
y be emitted with different anguLar momenta Asand the residual
be left behind in different states&.
!Therefore tho width Tncan
be subddvided
where
%a,
3
the following way:
rn, $ (18)
B ‘the partial width for the n of a with an angular rnomentw
fng the resfdual nucleus in the state ‘@L,
The value of
is determined by intranuclear and extranuclear
and it is useful t o separate them into the following ones: the probabil-
of emission of a particle a is proportional to the fol-
1) me final velocity which it acquires after having left the
ftdr having penetrated the p entia1 barrier if there i3 any;
&) Q$ bhe barrier which fs defined as the
f intensity of an outgoing spherical wave, corresponding to
face through the potential barrier to the
t is useful tQ consider as PO
barxier, not only the Coulombfield
ged particles, but also the potential of the Csntrlfugal force
i
;*

e
$'A2&
A.
228
(4 *I)&-# the case of an emission of a particle with an angular momentum
This force is described by a repulsive potential, inversely proportional to
the square of the distance, and acts on a particle in the same way as the Coulomb
repulsion. T,fi&)is a function of the charge and of the angular niomentum of a
and of the energy

e
229
being emittod by the original nucleus as, for example, a statu in which tho
particle has a very high angular momentum in respect to the residual nuclous. In
order to describe the excited status of the nuclexs zbove $which aro defined by
.
our assumption A,
%
it is necessary to introduce a device to oxcludc thu solutions
of (2Q) which correspond to a residual nucleus plus a fmdy moving indqwniient
particle which is not created by an emission of the compound nucleus.
This can be
done by mans of boundary conditions for
which should express tho Pact Lhct,
in case the state n includes frec particles outside the nucleus, thoy must be
represented by outgoing spherical 'CIICVI?~. With these conditions the equation (20)
no longer has a continuous spectrum above E. It then hcs discrete Gigenvalues
which belong to quantum states of finite lifctime, becouso of their c;bility to
onli t particles ,
Let us introduce a radius 8, the nuclaar radius, which is the short-est
distance from the nucioar center at which prnctieslly no nuclear force acts upon
vn
a nuclear particle.
(The particle might, however, at that distance bo under
influence of, say, the Coulomb force, ) Tha uigenfunctions yn haw the following
form for ra>R:
Here X are the eigenfunctions of the states 4: of the residual nuclaus whioh is
a#
-
left behind, if tho pnrticlc a is remowd from tho nucleus; pn Is the wave
function of the particle R outside the nucleus, 4; (Ea) is a solution of tha
following one body equation of the pcrticle a in tho space outside of the nucleus:
A

where YQ is e lcrge distilncc! zt which Ghl: kinotic onorgy of c".lJ,'.snrticlos mitt4
is inuck lt?rger than tho potontieil crwrgy Vpw[rF.) cnd thu cuntrifugnl forcl:
. Tho swlu can b;: expresslud by boundcry conditions:
-c 1 =O
n
24) k,, is givw by (a)
Per Pa =% (24)
n
wiiicti 1 ~ 0 ~ G A
part of kmis positive,,)
aid thct sign of thc squcrc, root must bo
Thew Bonditions and, of courset compctiblu with thu ~YEV~ oquation (20)
Tho equation (20) with thc (camplux) boundary conditions (23) or (24) has soh-
tions fop which '& is nucossarily comp2c.x:
wn =E,- irya ' (25)
oigenvc?.luo rizcan~ thrlt tht? st&b decays dxponentially e~oordi& to
Ft ,
,

so that rn is the rociprocal half life OS the width of the st&e n.
23 1
(Actually kn c..lso is coinplox because of thc fact that (21) cdntakis the
complex VJn. It ~xprossus the fact 'chat the outgoing wave (23) changes in intensity
with re, It increases due to the fact, that the nuclear stato decays expanan+
that the parts farther away aro emitted in an earlier stago. This
change of iintansity czlong thd outgoing wave, howvor, is only a very sn~ll effect
na long a? thc width r" of the levo1 is srimu cornpcr'$d ta tht, 3 t?nyy of thu
eniittod pcrticla, llnc ratio bQtwoen rwal and ink7ghary parts of k, is equal
to tho distance trcmllsd by the emit ed pwLic3.e within tho lifotinie of thc level
ed in wavdungthsX'(k *)-' . This ratio will be wry lar o in all
intarost and we ncgltlct?frorn no n the imginnry part of ka A completely
axcopt, of courss the C~SB TIn < Ba { itntion energy less than boundary
anergy) in which
rn,s
is purely hginary nnd r".: 0,)
ih now express tho lwd wtdth r A in term of't.heQq, n . The veluo of
the rcciprocal life time of the level n and tlwreforc equal to the numbor
of palkticlcs emittQd by the nucleus per second.
following way:
We first subdivide rnin the
rn = E r;&, (26)
a,a,a
whero &is thc: partial width for, tho emgssian of 2 with an angular momentum 1 ;
leaving the nucleus in the state a. Ue surround the nucleus by a sphere wikh (he
radius ro and dotermine the number of pcrticlos 2 wilt% tho above proporties crossn,
ing that sphurv por second, which in turn is equal toGL& Thus WI.? obtain (the
index a d is repincod bycx:)
undor the condition :
9n thz limits 1 e' 1 < r.
Q
(27)
owr all vnricblos
*) If pn is 2, slowly deccying stctc3, so thnt thc
particle dsnsiJdy 'wtxwn'thc boundary R of thu nucleus and ro is wry srrtall com-
pemd to the ddnsik,y hsidt! of tho, nuclwa, wo c m also norrmlizo
in tugral
by the
n
In the oxpression (27) the v;?luc. of *-,n is still dt.pcndont on tha field outside
of thc nuc!-a~s. Since thG latter is wvll known it is usoful to show its influence
1 2
is the vdocit of tho pnYklcle at ro~~~(?~,l
is, in the
usud htiro, x-3 t$m~s tho probability of finding a particlo Q in
om volwnu unit at ro.
'

explicitly in the expression for tha width, ?% thoroforc express n (ro)l
232
by tha value of the s m function at the nuclusr radius Re Vh introducit thd
magnitude Tasg(€)which
is tho ratAo by which the "(intensity x r2)1f diminishes
in an outgoin; wave of a particle 2 with an onwgy 6 and an angular momntum 16,
if it pehotratos thQ field around the nucleus from R to roe ro had boon chosen
outside of dl fictlds of forcrs so that the 1l(intsnsity x r 2 )It is coxistant outside
r,. T,g&)can
be cnllcd the transmission coefficient of thc potuntial barrior.
It cnn be calculntod from tho wave cquation of i? parliclt: with an onorgy and
&n angular momntum .46 in the fiold outsi.de of the nuclcus, whew solution wc
will call Fa
0.
r(r) for this purposu:
[& ."
If we choosu .the p&rticular solution which
has tho asymptotic form
wc get
T ddpend9 on the naturi: of
F-e
F

233
far xcc 1
with x k,R. e v&lue of Ta kcG> for chargod particles is diffic
put* sxactly, ~GC~USQ of tho wry SLQW ca
enoe of the expressions for t
enfunction of the Coulomb tiel 0 region considerad horc. 2t 2s
enorgy E.
e8 stronglr with
An approximate cxprwersion, c
char chargo 2 and with fa2S2ng
cd by using tho N.X.B.
method, 5s
(28)
1
is the charge of the pnrticla
is the anergy barrier height for
The valuas of c
mwgy E of the

*
Expression
254
The Pactw G depends on the value of the aigenfunction C& at the boundary af the
nucleus and thus reprosents tho influenee at the intranuclear structum.
(29) soparntes the offects of tho forcos outside of tho
could therefore ba called partial wgdth witho-ut 'outsidc potential barrier,
This
separation of the barrim effect and eho nuclear effwt is by no mucins completx!,
n
It is to be expected that the value ~f..~~~~(R)&I.so depaqds on the €arm of the
WEV~ outside of Rk sincc tho Inside and the outside p ~ of ~ the t function must
join smoothly,
Under certain conditions, howovw, thore is good reason to expect that
the valuds cP0lf\(@do not dopwd on tho bahwior outsidc of the nucleus, 130-
cause of tho strong forces inside of the nucleus, the averagw momentum pi of a
particle inside ,of the nucleus is very high and corresponds to an unwgy of 20 to
30 Lev.
The average monientum is an indication in what space intcrvnl the wave
functSon chenges ita veluc considorably. :!e dofinr: an c2Verc?gu wL?v'c length
xi =
i
and wo can say that bhc? W~VB function insidc the nuclous chcngus its
value by an cmount of the ordm of its avcrco.gc size, if tho coordimtc, of one
portich is chnngvd by h+
1'
much smallm than pi:
If the momontum po outside of the nuclom surfacu is
tho wave function insidw the nucluus must be joined at t h nucltmr ~ surface to
e, tvaw function, whoso villuo chnngos wry slowly cornpard to conditions insido,
ft 1s thOn a good approxhtian in thu duturmination of tho TGVQ function inside
to mswrs: that
t
for all pmtiolos. The vcluGs for
aa.4
arc not very difforunt from thc: truu on@*
guts with this boundary condi4ion
Sinco this condition does not contain

any reforanoe to the properties outaide, tho genwal behavior of 4,
(R) is to
oc" i" 235
I
I
be oxpsdtod the sane for chargod and uncharged particles and also the szmc for
ciiffsranC Rt s,
Particularly, the aqprago vnlucs
neighboring luvols
should bo practically indopndcnt of tho energy 6. a of tho outgoing particlu.
It
will depend only on the properties OS tho excited nuclous, whosl: excitation
enmgy is Ba+f4, an enorgy
- c.
valud 1 aaLof
validity aP (31) end docs not
the pnrtizl
which is much lcrgcr then
chcngo ap2wclablgr in thct region.
in the region of
'l'hi. avcrcgi:
width OVM neighboring 1C;v@ls, is thcn Given by
whcre C(3) is a funation of tho total excitation energy i= Ijk 4- ea only, It
dQcs not vary strongly in an onorgy intaval which is sm11 compnrod to 38a .
Furthergoru , At is c.,pproximRtuly uqu&Q for tho different pcrticles which can bd
arnittcd by Chc! conpound nucleus,
and for t hG diffsront vnLuos of o.ngular monicntum
of thew pwtic1c.s.
The uaofulnoss of this approximation con bc illustratod in n few
oxahplea:
thc herap ixirtial width for the emission of a neutron with
is gj.von by:
F = c(EC}dq (33)
which is ~ssuntidly proportional to the square root of thc outgoing noutron
mwgy in the rGgion in which (31) is valid,
Tm ?.wrap cross scction over neighboring lcvols for the capture of s-
n~utrons can be obtained from (16):
%= ---- D
This relation ShQw thu wll lcnotvn
slow s-natrons ( 1x0).
(34)
1 l ( a '
'/u law for thc ccvpturo cross sdction of
Thc particl width for thG emission of neutrons with 1 # O is givcn by

236
It is thus proportional ta e itg for mill cmrgies. The emission
prababllitics of chargud pcrticlcs haw tho trmsnzission cosffiaicnt TaL(G)
as thG dominwring factor.
z'hu factor C appccring in tha ncutron widths cannot bo datmniiiud but an
.c...
uppor limit can bo sat: WG know from (17) that % 5 ?&T
Since (33) should bc valid up to cbout
1 bkv wo obtain
for noutrons &to).
Thc: exprimentally d~tr:r&od vnluos (see Scotion 5)$ indiontc that C is not much
smeller then D ea, if the energy units arc chooson in 1ievtss
/
So fc?r tho radiation of the nuclear statos has beon neglectod. It intron
, which we call the rodiation wid?&, !he
duccls another additional width
n
tatcl width A of the lcvol is thm giwn by
rn
Tho radtction width consists of a sum af partial. widthsr
transition8 to spacial hvels f9 below n:
fP
corresponding to
u; is thc! matrix dcmont of the from n t o . P
Tho spoctrwn of khc nuclous can be dividd a) into 2" tlstc?bJ.a" region
from 0 to
to rndiatiori; b) in E 'lroson~.nco~J rvgjon abovc 3 whwe tho stntas arc able 'ca
(36 1
where the lvvcls omit light quanta only and the width is moroly duo
Qmit ptzrticlt\s, but still aro smrt from each otlicr by mor3 then their width so
th2.t thoy form c? discrot

c
January 27, 1944
LA 24 (35)
237
UCTUiU 3El.&3 ON NUCUAd PHYSIC3
Fifth Series: The Statistical Thecrry Leeturcr: V. F, Ykisskopf
of Nuclear Reactions
A) The Brcit-WiPntlr Formu2
The cross sections of nucloar rmctions, which wuro cnlculated from the
emission probabilitios in Section WIII, woro definQd as avorag“ cross sections
over 8 region oI enurgy of the incaming particle, which includas mny resonance
levels.
This suction is dovotod to tho study of t h crosq section BS a function
of energy, if the anergy is skierply definod.
phcnomenc. a3 deskxibed qualitativcby in Zection %XII.
In this casu vro oxpuct resonance
:‘io first investigP.tc tho onmgy relations: if eoCia tho energy of the
incoming particle, the onurgyeb of an emittdd particle is givon by:
% = Ea+D,--Bb+E,-Eg (38 1
whcre E
aru the excitation enorgios of the initial and final nucluus
cb and %3
respcctiwly, an4 Ba and i$, we tho binding energius of a to the initial
nucleus or b b0 the final nuclcus rospoctivafy.
The cnorgyca is thus dependent
on €a d i s in general c.l- not equal to thr: onergye,$
of a particlo b umittcd by
any of the excited statm n of tho compound nucleu$, excqt if is just in tho
middlo of n rosclnancu;
This is tho rxsa if Fa+*Ba is just equal to tho
oxcitation energy of thc compound stat3 n,
The expression of ths cross section of a nuclear reaction in the
od by meit and ?fignor” and later by Both@ and
1) Broit and iligner, Phys, Rw. 519 (1936)
I

e
238
Placeek2), by Peierls and Kr.pur3) and: by Sicgwt4), In tho following it is not
attempted to give an exact dcrivation of it, The anelogy to the rosonancc of
dmped oscillators is uscd, to mkc tho main foaturus of tho formula as plcusible
as possibluc
!le first introduce thr? concept of effective wid6hs. According to (29)
22
tho partial widths depend on the energy of the outgoing particlo by ntonns of
n w
the fsctor kd Tb (Cpg Since the oncrgy E
nuclezr reaction is not nocdsserily equcl.1 t oe
@
fin
of thd outgoing parkicls in a
, it is usoful to introduce the
uffectivo width') (cfj) which is a function of It is obtainod from ths
fd
B'
actuel width by tcking thc energy dcpundant factor at th+ value cfi instead of
This rolntion is not quitit exact
I
bucczuao of tha slight energy depundsnco of the
other factors in (281, namLy
I .
(F) l2 ; however, the rosonanc~ rugion dous
not uxtand furthm than the rugion of validity of (31), in which I
not cnergy dopcndcnt,
fflfl(R)/:s
ht us assume that tho energy af an incoming particle a lies within an
intcrvcil d which i s much s m l l G r thnn thv width of a rosonnnco lcv~l of the
compound nucleug. Furthwnorc we ossumo thct thu wwgy is nmr to the onwgy
En at which thc particle would bo at thd mmir;ium of a rosonmcG5), and compara-
tively far 2.~2.~ from othor rusonenoes so that wc do not wed to consider the
influonco of othm lwds. The cross wction c(E)as 2. function of thd wiergyt$,
B
has than thd cheractwistic L*Gscxw.iicd dqwndonco :
(1937 1
3) Poiorls and K~pur, Boy. Soc, Proc. 166, 2'7'7 (1938),
4) SiegGrt, Phys, f2cv, s, 750 (2.939).
.
5) en should, in accordanco with Suction XXXIV, bo writton a," , cnd
shouLd
haw an indox Ea. In this section we omit this indcx in tho cnljrgy of the
incident particlo and the cnGrgy ab which it is at rusonancG with thG nth
IWCZ.

23 9
1)
a(+ F(0 (43 1
wherc d(E)is thc effective totc21 width of the resonance, which $6 thu sum of
all effuctivc: partial widths:
6&1 E E? (43d
B&
is the energy (given by (38) ) with which tho particle is omitted. n e sum
c,,
is taken over a11 possible index triples b41'
F(6) is 3. slowly varying
function ofe, which we determine later, If is in resonance, 6 ran thc
\ rB
energies E arz tho sarie as they would be if th3 particles wcm cmlttod by the
resonance level n indopondently. Thus 6(gn) =An,
Exprcssion (43) cnn bo understood by analogy Lo the caw of forced
vibrations of 3. dcrriiped oscillator with cz proper frequoncy V
~nce of a pirriodic forcc F cos trt with a frequency v
to bt: due to a fruquoncy dcpendunt friction forcc -mb(v)X
undur the influ-
The damping is asawd
(x is thu dis-
placcmnent of the oscillator),
On0 obtains for thc: displacommt xt
And f.or the avorage energy cbsorbud by the oscillator per unit thw:
In this oxpression, tarms smller than of the order (%-Pv(q
me negluct-
cd (mcond term in (44s)). The timo duoendance of the frw vibration of this
oscillator would bc: x hl COS % ; its intonsity in this case
docays wpponontinlly with thu dscny constcnt o(t/,).
From this wc: cm undmstcnd dxprljssion (43) for thd cross sGct$on, by
*
rdpllacing in (Skb) fraqucncics by cnorgios. d(t/)goi!s ov~r into thc: &fm$ivu
total width o(c)of thc: lmd, whoso v2.1~~: for &= krn is rlso (aftvr division
by 'h ,) the actual dr3wy conatcnt of thc: 1 ~ ~ just 1 , as &(%) is thc: dl;cay

constant of the intensity of the oscillator.
The function F(€) in (43) is an
240
analogue to a frequency dependent 'force F, acting on the oscillator.
with its inverse:
We deterdne the function F( ) in (43) by comparing this cross section
particle with the energy
the deoay of the compound nucleus with an emission of a
within the small interval de . 'ilne probability of
this emission P(e)dG must be proportional to the same resonance factor, which is
contained in c(€), but should give, after integration over the total width of the
level, the total emission probability Tn (the partial width for the emission of
a,
the particle a>
c, is an energy interval large compared to the width. This is fulfilled by
since
6@)/2?T
?an(e) &py (bp +
)e
be. (45 1
This relation is right if "'&n(e)and 6" (&)are sufficiently slowly variable
over the resonance region.
It &so would be fulfilled by &tin;S the aotual partial width Fz
instead of the effective partial width ?*?&)as the first factor in (45). It
is evident, however, from the derivation of the energy dependence of 9 "(e) in
4x4
(42) that the emission of a particle with an energy e is proportional to?n(C)
rather than to
:;?e now apply the expression of detailed balance (16) to calcuiate the
4%
* where
function F(4) in (43) from (45). Since the end state of $he'capture process has
the weight unity in this case, dG(E)dE must be replaced in (16) by unity,
obtain instead of (16):
is the wave length corresponding to 6 , From this we get by using (45):
We

U
(6) &"(E) 24 1
4 = ?cs2 (@)P.+ [-Y6&]2 (46)
This is the cross section for capture of the particle
a regardless
what reactiok should follow, Since the bn in the numberator of (461 was writ-
ten in (k3a) as a sum of the partial widths belonging to all processes which can
be initiated by the capture of
a, it is self evident that the cross section for
a special nuclear reaction 5 4 6 is given byz
e q$*?ye)Z(kJ-
-Q(aLlfl) =fix (c-gn)a*l*n(G&gJF (47)
If the resonance en~rgies f" of several levels are near enough to e ,
the different contributions add up coherently;
tensitiea must be sunmd,
be understood from the analogy with a forced oscillation.
to (h4a) is proportional to the complex Factor 1
pears in the form ,/?+-E" +ib% ) '
The cross sectioli of a6 f b3,a:, 1
the wplitudes and not the in-
One then obtains an expression, whose form again can
Tha amplitude according
+ix which here
,.@%P,k)reaction with the initial particle
l
a incident with an energy +Z. and a given angular mornentum1-h upon the initial
nucleus in the state Q,, resl~lting in an emission of b with an angular momentum
1' and with the final nucleus in 10 , is given by:
*
Here f;, are the resonance values:-&n = z,yI--*Ba
Ea , i4n is $he
f the excited state n OS the compound nucleus acting as intermediate
state, &% is the energy of the statem of the initial nucleus;
/d.crs
phase difference of the initial and the final sta,to and it vanishes if cX;.""fl
and 3.' t,jp
is the
are the effective partial widths of n, corresponding to the
emission of a and b respectively, with the residual nucleus left in a, or fl .
f(€) is a slowly varying function of
ea which is small compared to the r@son-
ance contributions of the first terra, . It is introduced to replace tho contribu-
tions of the second term in the amplitude (4ka) and to inc;lude the contributions

states n and
z
(%md n
k, and kb are the wave numbers correspo
8 a or b respectively, Hare the
abed in the *119. E' (6) is the @me
aftm division bY the factors
The numeratar of the ~8sonance terms consists here only of nuclear
s independent of the cnargies af the particles,
In formias (48) and (49) dJ ,stata;s 6)
6)
n of tho nuclei are con*
sidered aa of weight unity. Zf song$ of 'thew stabw are degenerated it is so01'3-
re useful to suni over all substates, Tho tots1 cross section is the sum
of all crow sections from every single qubstate of $% to every one of c In
cwe 03 angular rmmentum degenaracy these summations can be performd and Bethe:
te A.
Tho initial and final states are no denoted with capitallotters
stead of the greek letters (C~fl> in order to indicate that they cow
prisa all magnetic substates. The compound state is called C, I/ 1' andjlj/
are the arbitaL and tba total angular molllenturn of the incident or emitted parti-
ectively;
J is tho total angular momentum of' incoining particle and
cleus and therefore also the total spin of the coirrpound state C,
is the, amplibuds
of the compound state C in regpect tc) the emiesion
tide a, with orbital and total angular momentum L and 3 and with the
residual nucleus fn A,
The sumnation sign 1'
C with the spin J,
c
It is evident that @&A
=o wess/j++j)l j-i(
Indicates that the sum should be taken only over states
The sumr.;stion over J 1s outside dF the absolute square signs
'

which 'chat thd cantr cPlCtefjng with Egerent total
moment
In most cases in which (50) Ls appl%ed, the contributi
4
of B
with a
e exfendsd t
utie glso rgdiative phenGmena sutsh
ar reaction in wkii&i
ia particle
ent upon a nucleus in the srta
and a light quantum 2s wdtted
ending in the stla,

m B)
A fm general conclusions cats be drawn from the ono-Zeva1 formula (47);
If there is only ano einission procoss poadible from tha corrlpound
nucleus, which thi!
can put oh
cwsaPi2.y is the reamissios of the absorbed particle, we
and we get; Par the only poslsibls process, the elastic scat-
elastic scattering.
It thura i.3 bne eorty3uCing ~FQCQSS, say the ofixl-ssion of b, we obtain
which i s off reBo
but, in resonance, weakm

consideration:
246
if the proton would be emitted with highor energy than tho one of
1)
thb ndutron, the energy of the residual nucleus would bG lowsr than thc ona of
the initial nucleus, and the two nuclei would be isobars diffGring by onv unit of
charger nus the initial nucleus would bop-Unstable, which is impossible:. If
tho emitted proton had equal or ldss energy than thc nautron it wwld not be able
to pcnetrat,: the potential barrier,
1% first assumo that thdm is only one level sufficiently near to tho
onurgy to which the com2ound nuclas can be excrited by thd neutron. \'TU then apply
(5b) and uso (33) for thc: partial. widthTaA of the neutron, F'urtherrnoru,
' 1 1
JaIkE since tho noutron can contribute only its intrinsic spin, 932, and
The cross
is thcn:
(55)
which is obtaincd by putting ?( s C&" also according to (33) .!
60
The cross sdction for emissioii of a light quantum or anothm particle
This cross section shows the characteristic €-* dapndmae on the neutron
t
ondrgg (tho so-called l/v law), If thhi? wicith bc is wry Xarg6, tho e'g-factor
is the doterraining factor for tho energy doyndonce. This is the casu obgf,) in
thu boron reaction:
where thc total width is datmnjuld minly by th3 ~Z-entission width,
Tho &-pBr-
ticbs are mittod with an urisrgy of about 2.7 MeV, and am able to pass above
the barrier, The corrwsponding width 72 is therefom vory large and was found
to bt: sbout 200 Kay.
Thus the boron raaction follows tho

of a light quantum, (as in triost Dams of he
246
nuclei) the capture OPOQ~ section
C) Wdrimctnt-
A number of resananc; phonomma haw bwn obsorvod with bombardribnts of
oi,
Staub and Tatc17) haw obasrvGd a strong resonance in th\: elastic
.sf neutrons by helium, Thus rQmni&nce could be assign9d to a doubbt
(54).
itud by p-neutrons and was ana
Hcrb and collaborators kn Wiscons
ed by mdans of a formula sinxLlar to
and Tuve et aJ9) havc found row-
nanco~ by inasuring ' thQ emitted ?*r&y intunsZt7 as a function of thu enorgy of
tho bomlwrdGd protons on Li, F, a. Tho? prays @re cithur crnittod by tho coni-
pound nucl~~ il;scLf or by th\: oxcitud fSna.1 nuclwus which is loft over afta
omission of an &-particlo (thu lattur pos$$blo occurs in F). The obsurve
9 dray intensity shows sharp maxhti if the enorgy of thG bombarding protons is
such that B love1 is a citod in thu COLipQUnd nucl@us. Ttiua tho resonances giva
a picburu of tha syct/rurrri of
leus abovc! the proton binding
nd which, bQoawc! af ssldction rubs,
emit GL-particlc:s are so broad
8) Herb, Kurst, McKibbtm, Phys, itev
&irnot, Hwb, Parkinson, Phys, 1%
Plain, Hwb, Hudson, Ijlarro

.-
247
resonanw. The most striking rcsult of a comparhon batweon thc thrw o;tc.mcnts
is the grcatdr leu4 density in the ha.vi+r nuclei. Tho avoraec love1 distanco
is about 500 Kciv in Li, 200 Kcv in F, and 30 Kev in Al.
thc levds arc? a sum of particle: and radiation widths.
The obsurvad widths of
Thuy ari: 11 Jim in Li,
about 8 Kcv in F and of tho order of on3 or two K m in Al,
Tho luwl density in A1 is so high that ono would not expoct to bo able
to find resalvabld rJsonancas in dumcnts much havier than Al with chargGd
partiall;: beam.
Hdsonanws in huaviw nuchi w~ro found with slow neutrons only,
The formula (56) for thu capture of slow nwtrons can thon be tcsted, and tho
diffmant menitudes: ec +ec and C; can be dotm&wdr
nQutron width C& is usually much smalldr than thd radiation width; b;r, thurc-
ford put d"W3i.c
and thG forriiula ($6) rGduces thcn to:
In this caw, tho
In thc swmd hcpression thc dnurgy f is exprdmUd in GV.
c
In this formula tho Dopplor cffdct has bdzn nugl0ctt.d. &causu of tho
thmnal motion of thc: nucllzi, tho rtilative energy batwcwi neutron and nuclleus is
not uxactly cqugl to thc! noutron cnwgy.
around ths nuutron unargy E witih r?. Gaussian distribution: $,(/2x e*pE(E-!$v&@
whum the Doppld width K is giwn by
K=2(mekT/M) A
The rolatiw wmgy E. is distributed
and m/N is thu ratio of thu msscs af tht, nJutron and tho nucleusI
Sinw (56a) givvs th3 dopondoncr: of d(e)on thu relative enurgy, thc3
cross wction as a function of thct actual. c3nmgy 3 i s givdn by:
c
If thd Dopplm width K is Si&U corAipnrr;rd to thc: natural. line width 6
Doppler 4fmt is nGgligiblt: and (568) can bu us& with as thf: actual nGwtron
(57)
the

c
endrgyr If K i s Larp compared to 6 the ahapo of the absorption
rely by th3 Dopplvr dffwt and WF: gat:
24 8
lino nuar
Howover if the anorgy
is sufficiently far away from rosonznce, G-G>> K
c ,C End d of a reswmnce levul:
1) Thd diruct wiwthod of mnquring bh+ absorption of, or bhG radioactivity
0
produced by, a mor?ochrorm,tic nautron beam with vnriablr: unurgy.
fixpurhmts with
rilonoenergljtic ndutrans hi;w been first performed by AlvarGz by mums of a i,iodu-
lated nwtron SQU~C~ and by a suitably aodulatud amplifier, which registers only
nvutrons which havti arrived at the counter jXr s1 spucificd thzu intvrval sftur
uction,
I3zche-r and Baker have jpiprovod this mthod and wcru abll; to
thc nuutrons,
rang@ of encrghs invdstigstod Pram 'two to ebout 5 cv.
This rwthod gives $&,turi3
diroctly as a functAon of thc anlilrgy e of
Thc quzntity mcaaurdd in an absorption r-iuasuretmnt is actually tho
SWJ of ccepturL. and the scathring cross suction,
For low enorgius and n~ar
2) T~L: boron absorption mthQd, Thu boron absorption 1;idLhod is used for
dupendonct: of the cross soction of th\: (na$ proccss in baron, which is
bld foY thu absorption of ndutrons boronu ThG PrinciPlQ of this
cuthod consists of thd ~follov~ing: n ~,lc?,terial is r.ctivnt;rd by zbsorption of
nc;utrons from p. bJ.m contcining a continuous spcrctrw of neutron ohcr@Gs.
Thd
of thd activation is r&nsurod afto covding the matd%d.
with boron.
ction is pToportionc1 to thc absorption In boron and thdr$:foru pro-
to @;' if f?,c is thu wmgy of thu nwtrons at msonanca. As shbwn
bufom,, JVUY imteriab CapturUs to i! curtain atmt very slow n3utm-M of thwwl

important, and tha intensity is given by:
&lation (58) is! corrdct if C!&n$))W , which ;,i&ns that tho andrgy rc:gions
sffi;ct\zd by thc: Dopplvr uffdot $.r$ coriiplcrtuly e.bsorwd.
4) Thd activity mthodq Tho activity producud by a nuutron barn of
given spoctrurn is riiaasurod, aftJr shiolding off tho 'thzrmal nuutrans.
ndutron spoctrurii is usually obtainud by slowing dovfn high unorgy noutrons in
light mc.terial as carbon or pp.raffin,
thm proportional to
y&
4
This Method givds c mI?surc,
sy
of
de= mr*+~ AC
inddpcndcint of thr; Dapplor df&%,
Tho
TI~G intmsity p~r' ondrgy intdrval de is
5) Tho mcsur,JrtiGnt of thG thdmo.1 ?.bsorption cross scction. 'he thurml
absorption ccn bd usGd for thc: dwtGrrlninntion of f+:vc?l constcat8 only if tho
oncrgy of all highdr bwls is lnrg\: cofiipcrod to thu lowost onQ, and if no luvd
*
bdow zero unurgy is of cny influmct..
Tht: lowst 3 ~ vnust d b6 ~ OQJ onough so
that thu absorption at thmml cinurgids is taking placu in the I'wingN of the
rus0nrZncc: ti&"
In cava the one-levo1 formnula is applicable, WJ obtain:

261
I
-11)
from niethod 2): ec .
frois niuthod 3); Q"~ if ac >I> K or from thick Iqws,
Thus thc! throd unknown qupntitius E. C I &, c cen bd dotwrmindd, JnconsiEstcs
bwxusu of tht, contribution of othJr hvds.
1Lthods 3) 2nd 4) may giw too
largd vnlues bduause thay includc: thc: uffuct of higher 3.~vds4
A tab3.u of thu charactGristic vc+lucis far 6011id lGw3.s of diffurcnt ulu-
iwnts is givsn below*
Only dmonts arc: includcd whurc: the ttviduncc soms to bu
not too contradictory.
Thu nuinburs in peronthcsis e.t tho r&.wncos
indicek thu
muthods usud, as list4 pruviously,
C~SOS w h v tho spin of thu nvutron is addud to, or subtmctud f.rorsi, thu spin oft
tho initial nucleus,
Uopplor width in this cnso is dominmt,
C
Thu value of A in As cannot bw doturriinod bocausc thu
The radiation width, rF which, in thwu cases, is practiocl2.y Gqual to
C
thQ total width A , is vury nd?.rly Qque.1 in all c8.swsc It sham uriusunlly
little fluctuction.
?%is r~letivt! constancy is connuctod with thG fkct thct
rtzprdsants a sum of wqy partial widths corrcsponding to r-zdictive transitions
tQ n gr6F.t nuGibor of lowor stctos.
Tho fhctuntions of single trnnsition proba-
*
bilitias wc nv3ragJd Out, Thd valws of C show somewhat gruahr fluctuations,
"h~y rc(prt.amt e singlir transition probability pad my dlspund mor0 critic,n.lly on
thG propwtius of thu individud ldvobr
Thu vnlu~s C' m-wIi to fulfill the, candition (36) fairly vrLLT,
Tho vc?J.ul:

Bombarded Life time ~,Ccvl Ref . Tr Ref. Enx10 * CW* Ref.
Element of product
AS76 27 h 40 1(2,3 1 32 5 1I3)
Aul'* 2.7 days
d -
669 (1937) '
Phys. Rev. 53, 18 (19401
Phys. Rev, 59; 154 (1.941)
-
* *

*
2k (341
255
UCTWl38 8FiRI;E;S ON NUCLEAR PHYSICS
ies: lzse Statistical Theory Lecturer: V. F. VJeFsskopf
of Nuclear React ions
L!ECTU.dE XXXVI:,
Non Reaonance Processes
AI
If the energy of the incfdent particle is higher than a certain limit,
resontanbes do no longer occur. This can
anergy cannot be made mOnQChrOmatic enough to resolve the levels; it can be due
to the Doppler effect; the D~Qpler width reaches, for example, 50 ev for an
inddent particle energy of 250,000 ev, if the nucleus is 100 th3s heavier than
41s; 50 ev is of the order of or
sonames $or heavy nuclei. Furthermare, at energies of several Mav the
actual width of the level reache8 the order of level distances, This can be seen
in the following way: according to (17) the partial width for the endssSon af 8.
fker leaving the nuoleus in a given state, reaches the value (2k+l)D/211;
&dual nucLeus can be le&
in several excited statas, the corresponding
tance D,
d up and gLve rise to a total width, equal or larger than the level dis-
Thus,, even under ideal conditions of' a monochromatic beam and zero
m, rescmances dbappear at atr energy of several &vts.
In order to compute the cross qrsctions of non-resonance processes, we
use Elohr's assumption 3 (see introductiofi). The CPQSS section c3n then be ex*
* where
pressed by (2)
t~ (a,
b) +2= ra *tlb (2)
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:
256
Here
is the partial width for the emission of b irrespective of in what
- -
state the residua;l nucleus is left: r averaged over the levels
Ak bB/t
of the compound states excited by a, A- is the averaged total width.
-
The partial widths 7~ (we use again the abbreviated index&; see
page '7, part XXXIII) are connected with the cross section Caof the inverse
process: the formation of a compound nucleus by the particle a of angular
momentum 1 and a nucleus in the state 05. This relation is given by the
expression (16):
The cross section Sa appearing in (2) is Gal with 6 being the ground
state of the initial nucleus. Thus we get for (2):
(2a)
where the summation over 1 and 1' refers to the angular momentum index in the
abbreviated indexa, or,@
respectively.
The consistency of the expressions (2), (3), (16), with the resonarlce
formula (48) is shown by averaging (48) over an energy regionbe, which contain9
many resonances.
If the level distances are large compared to' the widths, this
integration is a sum over the contributions of the energy regions around every
single level.
One then obtains:
The sum is extended over all levels within the energy interval Ae .
This relation is right if the term f(g) in (4.8) is sufficiently small compared to
the resonance contributions.
After introducing the average width 2; over neigh-
boring levels, we obtain:
-1
where D= [c+ (E,)] . is the average level distance. After sumriiing over k! and 1'

*
We
e57
and / j we obtain the sane expression as (2a). This shows that expression (2) for
a non-resonance reaction is in agreement with the resonance formula in the region
in which the latter one is valid (D )) )#
now proceed to the quantitative discussion of the magnitudes occur-
.
ring in (2). We make use of the intimate connection between partial widths r&
and cross sections which is given in (16a).
In some energy regions the cross
sections are better known, in others the widths; either magnitude can be used to
determine the other,
absorbed) particle,
We distinguish three energy regions for the emitted (or
The characteristic magnitude to distinguish the three regionp
is Kawhich is the wave number of the particle near the nuclear surface and which
is defined as follows:
M a
kt,
2 2
11, LOW energies: /Kat .C h, . This region includes all energies
near to the one for which it just passes over the barrier or, if there is no bar-
2
rier, all energies smaller than% k:/zm . It includes also energies which are
2
not sufficient t~ pass over the barrier. In this case ff a is negative .
2 2
111. Penetration region: K, < - k, . This includes the cases of
penetration of barriers with energies much lower than the barrier height.
In the first region, claasical considerations are justified since the
wave length of the particles is small compared to the nucleus and of the order or
*
Smallw of the distance between constituents. 1Ve may therefore assume that every
particle that comes to the surface is absorbed.
nearly equal to its maximum value and get:
Hence we put 8&,,eaqual or

258
The second expression follows from (16a). is a pure number smaller than
unity which goes to unity for high energies and is introduced to take care of the
transition regions,
The second energy region is the same region in which condition (31) in
Section XXXIV is valid.
The value of fia can therefore be obtained from the
expression (28) in a simple way.
the form (32):
It was shown there that (28) can be written in
where C(E) is a function of the excitation energy E of the compound nucleus,
which is independent of the nature (and angular momentum) of the emitted particle,
In case of an s-neutron (1 = 0, T = 1) this expression should join (59a) at ka=k,
and it is therefore tempting to put
where
cCFJ=t(D&t)(P1+1)(1 / k 0 )
may be some function of E only. Since, in the extent of the Region 11,
E does not vary by much, we may assume that the value of is approximately
constant in this region and of the order unity as in Region I. We then get:
Region SI (593)
The second equation comes from (16a).
In the third region (high barrier), the wave length x, will be always
ao small that the potential energy outside the nucleus does not change appreciably
over a distance %a. Thus the wmcalculation is . applicable and it yields

259
Here Pa is the penetration of the barrier, which is, in WKB approximation, given
Here rl and r2 are the radii between which the potential energy V is larger khan
the total energy E.
we obtain Pa= (ka/&)
By comparison of this with the VJKB expression (29) for Ta
Ta.
Thus the regions I1 and I11 join smoothly.
It is interesting to investigate the Region 11 by means of the VKB
method of approximation.
'Illis region is defined by the condition that the wave
length ,&of tho particle outside of the nucleus is long cornpared to the distance
between tho nuclear constituents.
If we assume that 5 is still small enough to
assume the potential fields outside the nuclear surface (centrifugal or Coulomb
force) as slowly varying, it is allowed to apply tho TiKB approximation for the
wave function from the outside up to the
nuclear surface, There however, conditions
change abruptly in distances small compared
to 'f; . It seems to be appropriate to as-
sume that, once a particle has panetrated
this surface, it will not return without
change in energy; in other words, it will
It
v
have formed a compound nucleus,
Thus the
penetration process is equivalent to one
dimensional problem with a potential energy
of the type indicated in Fig. 2.
Here a
particle comes from the left side over a
smooth potential barrier.
At x=O, the
potential abruptly drops to a value which
would give to the particle a wave length%
comparable to the wave length inside of the
nucleus, which is of the order of the Fig. 2

260
distance butween nuclear particles , (Vie call the corresponding wave number: EG)
*
This region oxtends to tho infinite, so that the particle does not return, If a
wave of particles with an energy E (A particles per 'second) comes from the left,
it is partially reflected (even if there is no barrier, because the potential
energy changes abruptly at, x = 0).
The current which is not reflected but pene-
trates to the right (B
particles per second) can be calculated easily by 'vbKI3
where XI
is the point at which the potcwtial energy V becomes larger than the
total energy E of the particle, and Vo is the value of' V at x=O. B/A is tho
ratio of the penetrating particles to thc incident ones,
According to this ex-
pression, the cross section for the formation of the compound nucleus is given by
the maximum possible cross section, multiplied with B/A:
sinc*$

Region I:
The energy is more than 1 or 2 Mev above the barrier,
261
ier .
Region 11:
Region XII:
The energy is less than 1 or 2 Mev above or below the bar-
The energy is more than 1 or 2 Mev below the barrier.
We now apply these considerations to the calculation of neutron cross
sections, The total cross section cn for the formation of a compound nucleus by
neutrons, is given by
momentum 6 1
Cn=
where 6" is the contribution from the angular
f 9, L-
We first consider neutrons of low energy E {

(N
Fi9.3 Neutron CI-GSS Section vs. Erlerjy

as functions of the energy e.
k K
B is given by 0 - 2 ( 21
1 263
The energy is given in units of (kR)&* The curve
, This sum converges rapidly for the values
L=O
of E far which it is plotted so that the upper limit of the sum does not enter.
B is proportional to C,V, and A is proportional to the cross section itself.
It is seen, that, with the plausible value for koR ~ ' 3 , the cross saction(Jln goes
smoothly over into its asymptotic value RR 2
.
For a heavy nucleus of Afi.200, the radius R is near to 9 x SO'13
cm,
so that (kR)2 is a measure of the energy in units of 250 Kev,
Hence we must ex-
pect cn to follow a l/v law for low energies up to about 130 Kev and to assume a
value near flR2S2 3 x low% above that energy,
The cross section 0"for charged particles can be computed as follows:
it is given by (60) as long as their energy is appreciably lower than the barrier.
The IVKB method can be applied to calculate the penetration factor, if the barrier
extends over a region which is large enough, so that the potential energy does not
change appreciably over one wavelungth of the particle,
This is the case for
higher nucloar charges (2 } 20).
If the energy is well abovc the barrier, the
expressions (59a) give the cross sections directly, since there is then no pene-
tration factor to calculate,
sions (59b) should be used.
For energies near the barrier, however, the expres-
The VfKB method in this region is no longer reliable,
since the remaining barrier is necessarily very low and extends over a narrow
region,' In the following comptttations, this region has been left out completely.
The validity of the WKB solution, applied at low energies, has boon sxtonded up
to energies equal to the barrier, where it was joined with the solution valid at
energies much higher than the barrier.
These two solutions join smoothly at that
energy since the WKB expression for the penetrability P of the barrier becomes
unity at the barrier.
The justification df this procedure lies essuntially in the
fact that, the barrier heights for different angular moments of the incident
particle differ by amounts of the order of 1 Mev, so that, for a given energy of
the incident particle, only one or two angular moments lead to a case of a Ftegion
,

.:c) 264
11, Most of the partial cro8s sections gare . thus computed correctly .
The cross section for the ponetration through the barrier is a very
sensitive function of the nuclear radius.
The following tables give tho values
for two sets of nuclear radii, one givon by d * 1.5 x A 113 , the other by R=L3
x A l/3
It seems that the second set with its steeper excitation functions, fits
better to the erpx-imental material available.
-
the rb for dl particles b emitted by the compound nucleus.
the sum f" =x
I3
The calculation of q of expression (2) involves the computation of
A? kP1
They are given by
each term of which can be computed from (59a), (59b),
(60) with reasmatle accuracy, apart from the factor 4 which does not differ much
from unity.
L.? thl final nilclew can be left in many oxcitdd statesfl, the sum
can be exgresscd PS Pri inkegred and we obtain from (16a):

.4
05
06
.7
-8
09
1.0
1.2
124
1.6 1;8
2.0
1
40
i Ca
2s55
6,37
12.3
18.3
26.5
35.1
40,3
50,o
56,2
60.4
64,O
66,2
B 1 6.4'7 I 10.06 15.60
120
a
0
0
,0239
.
298
1.72
6.13
15-7
27.6
53.6
71.7
83.4
93,o
i02,
8-72
6.0
7.1 74,6
106 145
I 18.731 9.70 116.24
d
r
L
LI
.84E
3.42
8.15
15.1
23.2
33.7
4613
54.8
63.8
70.2
73.7
78,3
64
Zn
30
8,28 x5&74 -
I
90
Zr
- - I
TT
Gc/ I P
0
0
. 00468
.099E
. 871
3.90
11,2
20,6
44.7
61.9
71-4
€33.3
86.9
14 2
Ed
60
,00433
32;o 7 i.85
.02
I1
19
12,67 I 121.45
"--
73c
3,04
8,23
16.1
26.7
38,8
49.3
69. 9
79.3
86,9
3.03.
110
7,17
-
----
_13
* 0499
, 531
2.88
8.79
17,3
38, I
88.8
91-6
.15
.37
.41
50
10- 98
.04
.14
I
26.5 44;6
173
v
1 70"
5L
27,O
79.4 67.5
96.9 94.4
I
I
4
0
0
0
0223
,410
2.87
11.7
I
1)

266
where the sua in the denominator is extended over all particles c wh$ch ccqn be
emitted in this reaction. It is evident that the integrand in (64)
TABLE: I1
The cross sections 6 and Ca @re given for 8 typical elements as
P
function of the energy :
cp 3 @ = n(x) '10-26 CWl*
n is the number given in the tables, x is a masure of the energy in units of the
barrier hekht B given in each colwnn: e== x.B . The columns headed by I corres-
pond to a nuclear radius R = ro All3 with ro = 1.3 x lOe13 cm, the colwnns
headed by XI, correspond to ro = 1.5 x
cm,
The cross section q(6~) for the penetration of the deuteron is equal
to the one of the proton for en element with a charge s'mller by a factor (2) 1 413
and with an energy smaller by (h) h
is the probability of emission of b with an energy between
and e +dk. It
describes the energy distribution for the outgoing particle as a function of its

energy e . (This is identical with expression (4) ),
267
*
It is interesting to discuss the connection between the distribution
(67) and the Maxwell distribution of particles evaporating from a hot sphere, as
indicated in the introduction.
The distribution I(E, ) ch be written
According to statistical mechanics, $(E) is the entropy of the residual nucleus
with the energy E.
By writing
and by using the well-kncwmstatistical expression
we obtain
=i/T
Mere T is the temperature at which, according to (67), the residual nucleus has
the average energy E, -. If Cb is a slowly varying function of the energy,
as it is in the case of the emission of neutrons, the emitted particles have a
bxwellian energy distribution, just as particles would have, that are evaporated
by a material with a temperature T.
This toniperature is, however, not the
ntemperature of the compound nucleus before emission11 , but the ftternprature of
a nucleus after emissionfl. Here we understand by Ittemperature of a nucleus in a
state of energy Elf the temperature at which it has an average excitation energy E.
This comes from the fact that the Itevaporation” of one particle constitutes a
relatively large loss of energy which reduces the temperature considerably.
The
*
Maxwell distribution of the emitted particles is determined by the temperature
-
after the emission.
If 6b is a strongly varying function of the energy, as it is in case of
charged particles emitted, the Maxwell distribdtion is deformed by the factor a“b
in (68). Since cT~ is usually a strongly increasing function for charged parti-

e
268
cles, the maxhwn of the distribution will be shifted to higher enqrgies..
The limitations of tho validity of the Maxsv 11 distribution in ni clear
emissions are obvious from the method of derivation: it is assumed (a) that the
level density of the residual nucleus in the region of possible excitation is
high enough to permit statistical treatment, (b) that e (( Eb m, which means
that the energy of the emitted neutron is small compared to its mcurimum available
energy,
Since the level density of nuclei is unknown to a great extent, it is
hard to predict the limits of validity in terns of Mevs,
3t )
In recent experiments
it seems that the level
-
density of the lower lyhg levels is rather small, viz.,
about one or two levels per MeV in the first or second Idlev of epitation energy.
The maximum energy Eb p robably must be at luast 5 - 7 MeV in order to expect
a shape similar to a Mawhian distribution for the emitted neutrons by elements
in the middle or at the heavy end of the periodic system.
The values of tho temperature T, which are expected in nuclear
reactions, can be estimated in the following way:
the average energy E of a
system is a monotonically increasing function of the temperature T,
If the temp-
erature is very high, so that all degrees of freedom are excited, E is propor-
tional to T,
This is certainly not the case in a nucleus, when it is excited by
a normal nuclear reaction, If one considers the nucleus very roughly as n gas of
A particles concentrated in a voluma (G R3), one finds that this ttgasll is de-
genersted if the excitation energy is of the order or less
.3As Meve Heavy nuclei should therefore be considered as highly de-
J
generated tlgasoslt
The stroilg interaction between the particle does not affect
r)
this cor,c!-usion which also holds for an electron gas in metals in spite of their
--u_
-L-.-,uru-I-uI-.c-
WFlkins, T, R, and Kuerti, G., Phys, ibv, z, 1134 (1939).
Wilkina, T. R., Phys, Rev..@, 365 (1941).
Dicke, R, H. and Nhrshall, J., Phys. Rev. Q, 86 (1943).
Dunlsp, H, F, and Little, R, N.,
Phys. Rev, 60, 693 (194l)*

interackion,
269
The function E(T) ha8 ta vanishing derivative$ ( &E/ 4 T)T=o = 0 for
c T= 0. (This means that the specifio heat is aero for T*O.aocording to the third
law of thermodynamics,) The expansion of E(T) near T*O etar‘ts therefore with a
quadratic term, If a system is highly degenerate, we may write
E = bT2 (70 1
and neglect the higher powers of T, which are small in this case, From %his we
and finally for the level density: 7
pE
W(ES =C (711
One can try to adjust the constants a and C, 80 that this equation
reproduces our rather scant knowledge on level densities. A fair choice for A } 60
is e*g,,
cad = [;Z/(WOjJ (MeV)”’, Ceven = 9 Codd? a = 3 35 (A-40)* (MeV )”’
if thB eXCitatiOn energies E are measured in Mev,
Codd and Ceven refers to
nuclei with odd’ or even A respectively.
The expression (71) is adjusted to our knowledge of level densities for
very low excitation energies and for excitations in the region of capture of slow
1
neutrons. In the latter case only levels, whose spin differs by kzfi
1
from
the initial nucleus Eire observed. The level density in the expresslon (711,
however, should include all levels which can emit or absorb the particles b with
the energy Eb Our knowledge of level densities and of the distribution of
angular momenta among the levels is insufficient to make any distinction of that
kind at the present %be,
From (70) an estimate can be given of the nuclear temperature which
Ir
appears In expression (69) and which determines the Maxwell distribution of *he
outgoing neutrons. If the maaimum possible energy of the outgoing neutrons is .
Eo, the temperatures are given in the following table in Mev’s according to

I-
60 150
-1
5 1w-15 76
10 1.63 1.07
15
2.00 le31
I-
230
-
66
.93
1.14
These temperatures can be obtained by bombarding the
nucleus with neutrons of an energy Eo, or by any process leading
to the same excitation energy of the compound nucleus.
270
Expression
(69) shows that the maximum intensity is emitted with an energy
2kT, if'ris a slowly varying fuiiction of E
to be in case of neutron ernissfon.
:i/
which serve to compute the factor 'Y\baccording to
are shown in Fig. 4.
107
c-
LO6
102
J Y. -/-
as it is expected
With the expresston (71) for the level density It is
possible to calculate the functions f ( ebmax defined in (65)
(65). Examples
101 '
1
.rl)
/I I I I
0 2 4 6 8 10 12 14
Fig.4. The functions fn,fp and fvare given as functions of energy
for the compound nucloi Cu, Zr and Sn, These values must be multiplied
by 2 or 0.5 if the residual nuclous is odd-bdd or even-even,
.
respectively. Expression ('71) for the level density Is used in the
c ornput a ti on

271
*
The reaction types, discussed in this section, are ordered aocordtng
to the nature of the incoming particle,
Resonance reactions are excluded, since
they were discussed in Section V,
1. Neutron Reactions
The cross section %(e) for the formation of a compound nucleus for
energies e above the resonance region is given by (63a) and (63b), and is represented
in Fig. 2,
in ,n ) Reactions, This is not a real nuclear reaction. eince the bom-
barded nucleus remains unchanged, It may however be left in an excited state
after reemission of the neutron, in which oase we speak of inelastic scattering
of neutrons by the nucleus,
Once the compount nucleus is formed, every other
particle except the neutron has to penetrate a potential barrier to get out.
The
neutron width, therefore, is much larger than any other particle width, and also
larger than the radiation or fission width if E is sufficiently high. Under these
conditions. 7n N t and the cross section for the (n,n) reaction is given by OB.,
its elf.
There is a difficulty in interpreting the observed (n,n) cross section
by means of the expression (2), due to the presence of elastic scattering.
This
is the part of the n,n reaction in which the outcoming neutron has the same energy
a8 the incornbg one.
As mentioned in Section XXXI, page 8, part of the elastic
scattering is due to effects in which the neutron is deflected by the nuclear
potentials, and thus does not penetrate and form a compound nucleus.
of the scattering is evidently not ineluded in the expression (2).
This part
The observed
cross aection of an (n,n) reaction therefore contains a part which has nothing to
do with the formation of a compound nucleus,
The waves of the scattered neutrons
due to external effects (in which no compound nucleus is formed) combine coheront-
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,
2 72
These two effects
II,
together give the elastic scattering.
The observed cross section of an (n,n) reaction therefore contains a
part which has nothing to do with the formation of a compound nucleus. The ob-
served C(n,n) can be much larger than Cr,, especially at low energies, when the
elastic scattering is a relatively large part of the total scattering. It is, of
course, possible to separate tho elastic from the inelastic scattering by dis-
kinguishing between the scattered neutrons of different energy.
The cross section
qn for inelastic scattering, which Is due to the neutrons which have left the
nucleus in an excited state, is certainly lower than CJI,, because it does not
contain the contribution from the elastic scattering, which corresponds to a real
penek-sfion into the nucleus and a suhsequent re-erni$sion by the compound nucleus
wtLh kko sarw emrgj as it came in.
The pax% cf the elastic scattering coming from the ren1,formation of the
csnpovnd ste

15 rn
2 73
There is only very
little experimental material available on the spQctrum
of the scattered neutrons by heavier elements, Dunlap and Little*) measured the
spactrum of neutrons scattered by Pb with an initial energy of 2.5 MeV and found an
inelastic scattering which indicates several excitation levels of Pb.
The h.2n)-reaction occurs if the energy of the incident neutron is sufficiently
high, the residual nucleus after the re-emission of the incident neutron,
. may still be excited highly enough to emit a second neutron.
This is possible only
if en- Gib is larger than tho binding energy E, of a neutron to the bombarded
nucleus where Gn is the energy of the incident neutron and et,, is the energy of
the first emitted neutron. The probability of (n,2n) reaction is therefore given
by:
where I(€ )de
given by (67).
is the probability of emission of a neutron with the energy f. ,
For energies en which make a (n,2n) reaction possible, the energy
distribution can be well approximated by 8 Maxwell distribution (69) with a temperr
ature T, with which one obtains (with. q, =c rCg2 .):
If A
AE-Gf) - B,
-0 the energy surplus over tho threshold energy of the reaction I- is large
compared to tho temperature T, the (n,2n) reaction should be the dominant reaction
.
and its cross section is then nearly equal to 6,. The (n,n) reaction cross section
should become correspondingly smallerr This should occur for A,E 9 2
or 3 Mev for
e
elements in the middle of the periodic systems. For very high primary enorgies -0
e > 2B
n
(n,3n) reactions will occur.
qunlap and 'Little, Phys. Rev.

2 74
The fn.p) reactions naturally are loss probable than the (n,n) reactions,
c
because of the potential barrier which prevents the proton from leaving the
nucleus as easily as a neutron.
The cross section is given by:
where the fts are given by (65). This cross section assumes measurable values
only if the energy of the incident neutron is several Mev above the reaction
threshold, beoause the outgoing proton needs that much energy to penetrate the
barrier,
The rather steep dependence of C(n,p) on the energy gives rise to an
apparent threshold which is much higher than the actual threshold.
If the nucleus which islcrsated by the (n,p) reaction is a negative elec-
tron emitter, the final product of the,@-dewy is identical with tho initially
bombardeq nucleus.
In this case the reaction
energy Q = Cp - en of tho (n,p)
reaction can be calculated by the energy law.
The energy balance in the nuclear
react ion
n+X=Y+p
is given by
En+ EX = Ey+ Ep
where
and Ey are the binding energies of the nuclei X and Y in their ground
states, The ,&-decay has the following energy balance:
%= %---mn+m c e-+ ht/
P
where my, and mp arc the mass energies of the proton and the neutron and
Eo in-
cludes the mass energy mc2 and the kinetic energy CGin of the electron emiteed;
h U is. the energy of a 7 -ray which may accompany the ,@ =decay.
Be thus get for
the +value:
Q =%- %=mn-mp- e-- hn/ = 0.7 Mev - hiV - G& (72 )
In,&) reactions have an extremely small cross section in heavy nuclei .
]I,
because of the potential barrier, which makes it very improbable that the
&-particle leaves the nucleus, If the neutron energias are extremely high,

is
however, (nt&) reactions may occur in heavy nuclei.
275
There will be more chance
to observe these reactions at very heavy elements, where the binding energy of %he
f -particle is sqallcr ,
The (n,’))
reaction is usually called radiative neutron capture and was
disausscd extensively in the r~~onanc~ region in Section XXXV.
energy region the average value over many resonances is observed.
a.-(n 99 = q (T!,
In the higher
The processes which compete in heavy elements with the radiation, are the re-
emission of the neutron and, in some elements, the nuclear fission.
the latter from these considerat5ons.
of a neutron with an angular inomentum .,df~ is given by
. .
We exclude
The cross section for the radiative capture
r-
c
I
is the average radiation width and -1 r,
the neutron width for the compound
states, which are excited by a neutron with an angular momentm . 1% . If the
primary neutron energy is not too high (less than 1 Mev), the probability of in-
elaotic scattering is relatively low, so that the re-omission of the neutron is
R
essentially tho process inverse to the capture”). stands than in the relation
(16) to 5(”), and me get for the total capture cross section:
This can be calculated with the help of (59b). For small energies (
A!= 0 contributes and we obtain fkom (73) and (59b) by assuming
>> rn
of the incoming neutron,
>> R) only
(Ql
a value of 3*corresponding to 2 MeV, one obtains
-----.---
$E the angular momentum of the bombarded nucleus is difforent from zero, the
neutron can bo re-omitted with an angular momontum different from th
angular monenbum of thE Cap%U&Qd one, ’ The psrfcty rule cal~~ows only ld’-/’I =
2, 4 etc., and a closer investigation shows that the probability of /--/’#O is
;1)1 rather small.
(74 1
With

cT"
* if
6 (n
P) =t$"dz--) v
io+ (&< r&
E is measured in ev. is & magnitude which should be not far from unity:)
An example of the energy dependence of cr(n,y) is given in Fig, 5, which
276
shows expression (73a) with the following choice of constants:
R = 9 x
k$= 3, D/2fl= 45F )

high.
278
Then the advantage of the neutron emission, which comes from the absence
of a barrier is made up by the high energy of the competing proton re-emission,
e
In this case qr, can be smaller than 1 and is determined by 'qn Z= fn/(fn+fp).
The f's are functions of the maximum energy Emax of the respective particles as
described previously. If E, m a f,,
P
particularly if the nuclear charge Z is not too high, (See Fig. 4.) A case of
this sort is found in the Ni62 (p,n) Cu6' reaction which has a threshold of 4.7
%- 1
MeV ,
In general, however, the cross section of a (p,n)-reaction
should be
very closely equal to 3 in Table I1 for different nuclei. Comparison with experiments")
have shown that a nuclear radius R = ro A u3 with ro N 1.3 x lom8 cm
fits better for elements in the middle of the periodic system between Ca and Ag,
The energy distribution of the emitted neutrons, is similar to the
distribution in an (n,n)-reaction.
The highest possible energy occurs when the
final nucleus is left in its ground state, and the emitted spectrum corresponds
to the excitation spectrum of the final nucleus in the way that lower energies in
the spectrum differ fram,the highest by an amount equal. to an excitation energy
of the nucleus.
For high initial. proton energies, this spectrum becomes s ~ l a r
to a i'ulaxwell distribution.
If the mrni;nwn energy E,
of the neutrons of a (p,n) reaction is
larger than the binding energy of a neutron to the final nucleus, the latter can
be left in an excited state high enough to emit a second neutron,
We then obtain
a b,2n)-react=. The probability of this event is given by a srimilar expression
to (71a); rn has to be replaced by WP. The energy distribution I(€) of
the emitted neutrons can be represented here also by a illaxwell distribution and
* 'TWaisskopf
aid Euving, P~Ys. Bev. d, 472, (1940).

&e is the excess energy of the proton over the (p,2n) threshold.
279
The (pa?) reaction is the most important resction in case the proton
energy is below the (p,n) threshold.
The only competing process is then Che re-
emission of the proton r(p,p) reaction] , which is not probable since the
-
proton has to penetrate the barrier again. The cross section is given by:
n
and is vanishingly small above the (p,n) threshold because fn rises rapidly and
becomes larger than all other fts, No resonance effects should be expected in
the (p,?)
reactions since in heavy elements, in the resonance region, the energy
of theof the proton is much too small to penetrate the barrier to an observable
extent,
The (p,p) reaction can be observed if the proton energy is not sufficient
to emit a neutron with appreciable probability,
Since the Butherford scattering
by the Goulomb field is always present and usualiy much stronger than any nuclear
reaction, a (p,p) reaction can only by observed either by registering scattered
+t 1
protons with less energy than the incoming ones or by finding the bombarded
dH:)
nucleus in an exLited state
order to be observed.
which must be a metastable (isomeric) state in
The cross section of a (p,p) reaction is given by:
fl- (p, p) = "r -*r+--$--li.K
s
It rises steeper than cp with increasing energy ep of the incoming proton, as
long as fp < f3, since f
P
also is a strongly rising function of ep* qP,P)
is then essentially proportional to the square of the penetration of the barrier.
3. GC-Particle Reactions
These reactions are very similar &o the proton reactions.
The (CC,n)
II)
3t) R. Wilkins, Phys, Rev. 60, 365 (1941).
R. Dicke and J,. IvIarshall, Phys. Eev, Q, 86 (1943).
3Ht) S, Barnes, Phys, Rev, $6, 414 (1939).

B O
reaction is the most probable one and its cross section is given by an expression
like (75) with Crcr, instead of 5,
threshold energy.
after replacing
(77).
T~ is almost unity except below or near the
The (G,2n) reaction is given by the same expression as (76)
by Q, and so is the (a??) reaction cross section given by
The latter one is unobssrvably small sime, below thd (cX,n) threshold,
the a-particle hardly can ponetrate the barrier.
The (CL$p) reaction has a cross section of the form
- tions especially with low Z and E,.p > en -,
It is hard to estimate the threshold energies for the different
and is always smaller than the (CL,n) r&ction but can have observable cross sac-
a-reactions, because of tho fact that the binding enerm of tP.e a-particle to
the compound nucleus is not wall known, but is neaded to determine the excitation
energy of the compound nucleus.
The binding energy Badcan bo estimated as fol-
lows:
four nuclear particles are added, which have had in the a-particle a
binding energy of 28 Mev.
Thus the .binding energy of the =-particle
should be
BG= 4B - 28 MeV, where B is the average binding energy of the two protons and
two neutrons added. B is not well enough known to make any guess about BG. Recent
rssulis on (CL,Zn)
have shown that, with 15 &lev&-particles,
(G2n) reactions were not observable. This shows that, at least in the investi-
Bnl + %2 - B -2 15 Mev, where B
gated elements:
are the binding energies
"1,2
of the first and second neutron of the (dX;,2n) reaction,
4* Deuteron Reactions
The d-reactions follow in general the same laws as the proton raactions.
They'distinguish themselves from all other reactions because of the very high ex-
* +b)
citation of the cornpcmnd nucleus, The excitation energy is equal to the kinetic
energy of the deuteron plus the binding energies of two particles reduced by the
R. N, Smith, Dissertation, Purdue University, unpublished

0
281
relatively small internal binding energy of the deuteron of 2.15 Niv. It amounts
to about 14-16 MeV plus the kinetic energy, This is one of the reasons why
deuteron-reactions have a relatively large yield.
The (d,n) and (d,2n) reactions obey the same laws as (75) and (76).
bd is naturally smaller than 5 at ths same energy. The ~iiaximum energy €n -
of the outgoing neutron is usually much higher than the deuteron energy because
of the high excitation of the compound nucleus.
Thus, there is no positive
threshold for the (d,n) reaction, and Ykn is always nearly unity.
The threshold of the (d,2n) reaction can be calculated from the
.&'energy
of the produced radioactive nucleus if it is a positron emitter, Ac-
cording to the same principles as in (72), we find
Here ED L 2.15
(d,2n) threshold = €+ $. h t/ + m ,-'InpfED
-. 4.
= %n+ hu+2.85 ~ i v
Ivkv is the internal binding energy of the deuteron.
The (d,p) reaction should be much less probable than the (d,n) reaction.
Actually, however, (d,p) reactions are observed with equal yield as (d,n)
reactions.
This has been explained by Oppenhcimer and Phillipz) and is due to
tho following process: the distance between the proton and the neutron within
the deuteron is relatively large, narnely -3.5 x cp.1 and its binding force
60 of 2 bv is un*mxxily small,
This is why the deuteron becomes strongly
"polarizodlt when it approaches a nucleus; the proton goes to the farther end and
.the neutron to the nearer end,
If the neutron arrives at the nucloar surface,
the proton is skill some distance away and has still a potential barrier to penu-
%rate, The nuclear forces, which then act upon the neutron are much larger than
ED, and the deuteron is likely to break up, when the proton still is relatively
far away from the surface. After breaking up, the proton is only under the infbuence
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
into its parts by the Coulomb field of the nuclear charge.
The apparent effect,
however, is a (d,p) reaction.
The cross section of the Oppenheimer-Phillips process is much larger
than 6d3 since the latter is given by the probability that the charge reaches
the nuclear surface, whereas, in the Oppenkeimer-Phillips process, it reaches a
point some distance outside of the surface.
(d,/) ) processes do not occur with measurable yield since the compound
nucleuB always is able to emit a neutron with considerable energy,
5 Radiative Processes
Nuclear rwctions can be initiated by the absorption of a ?-ray.
It
is evident that the energy ht., must be larger than the lowest binding energy of
a particle.
The compound state is, in this case, a highly excited state of the
initial nucleus.
The levels are then RO close that no resonance can be expected.
The cross section for the cliffwent processes is given by
o-(?,b) = (T+(fb/t
a
where b is the emitted particle and the sum is extended over a11 competitors.
The most cofmon reaction of this type is the (T,n) reaction.
art3 possible, but LL’F/
pete with either FJ:’Ct:J’*&
mcii venker.
3: :i?utron emission.
fa
(9,~) reactions
Any other particle WGU~! not be able to com-
Exceptj-or.:: ta tAis are found in
very heavy elements where photo-fission occurs.
The excited state is then un-
stable against sp3..tttting into two fragments
The VL.?.W
of G;, Is connected with the transitim probabilities between
nuclear states, and before describing its properties, it is necessary to discuss
other radiative nuclear processes.
*
Radiative trnnsitions between low-lying excited states have been ob-
served in great number in the ? -ray emissions accompanyingB-decay andGdecay,
In analyzing term systems, it was found out, that, transitions occur between

283
states which differ by two units of angular momenta, Aj =. 2 and not only between
I)
states differing by one or no unit, as it is the case with atomic spoctra. It is
known that the change of two units corresponds to a qundrupole radiation which
2
should be smaller in the ratio (R/%) compared to the dipole radiation emitted
in the transitions A j=l,O.
This ratio is about 1000 in tho observed region of
hU-5 x 105 ev whereas the observed quadrupole transitions were about equally
probable as the dipole trcnsitions,
This has been explained by the fact that,
dipole radiation only occurs if the center of charge is moving relative to the
center of mass, whereas quadrupole and higher pole radiation is emitted also in
motions where the two centers stay coincidente In a nucleus, the charge is nearly
evenly distributed over the mass, since protons and neutrons are not distinguished
in their nuclaar motion. Thus the center of mass and charge stay practically
together and this strongly seduces the dipole radiation. According to
the observations, thds reduction is strong enough to reduce it to tho strength
of the quadrugole rp? ie t icn ,
We themfoYY exile^::, that the absorption of 3/ -rays by nuclei, also
follows the law nf quidrdy&3 3r dipole absorption,
The ratio of the transition
probabilities in qur-tGr.:po3.e and dipole radiation is pr o?ot*tJi.oml to 2 1 ', and we
expect therefore ~.,zhl-y qmL*ip.de transitions for the a!uxption of T-rays of
both types of radisiio:i. vjre cqually probable for lejs 3v.a 3
bkv.
Quadrupole rric'.itlt,ior, corresponds in the partick ;-ict,ure of light, to
the emission or ~~JT',~:.LoI: ol' a photon with an orbital arigller momentum of one
unit, whereas in tk- d;.r;~:~e ?%diation, the photon only posses3es an intrinsic
tfsphll of one unit and no orbital momentum.
sponds to the absorption of a p-garticle,
Thus quadrupole absorption corre-
The cross section Tv ,md its energy dependence can then bo estimated.
We use the fact that tho wavelength 'x , even for energies as high as 30 k v is
still larger than nuclear dimensions.
The absorption probability should then be

284
just proportional to the intensity E2 of tho electric field 6 at the nucleus.
*
(The same principle, namely that the probability should be proportional to the
neutron density at tho nucleus, lt3&
to the l/v law for neutron capture,)
intensity for a photon is, firstly, proportional to tho frequency, since E2fi~ 11%'.
Secondly, because of the fact that it is a "p-photon" with en angular momentum
one, its intensity at thc nucleus is proportional to (R/?i2
) (cf, page 7, Section
XWrIV)
This
Thus the probo"bi1ity of absorption is proportional to U3. The cross
section 67 is l/v times the probability which gives here, because of v = c:
0;z = C (hGy
.3k)
where C is a constant
e
This relation is found to be well fulfilled according to (y,n) experi-
ments made by Bethe 2nd @ntner'"*)
who bombardad a number of elements withy-rays
of 17 iv1ev aid of 12 lev.
The constcnt C shows surprisingly little change ovor
the psriodic system and is C = 0.65 x cm2/(Mev)3, This value can also be
brought into connection with the observations on T-rays between low-lying ele-
ments: by cpplying expression (161, the width for the emission of a ?-ray can
D is the level distance in the region of the upper level.
If this is applied to
low-lying lovels of h t/ FJ 1 lvlev with a level distcnce of 0 ~ 0 . 5 hv we obtain
rpw10-3 ev,
This is in good agreement with the known transition probabilities
among low-lying nuclear levels 4 ,
The r?.dini;ion width
.---
++) Weisskopf, Phys. Rev, 3, 318, (1941).
3Mt) Bethe and Gentner, Z. fur Phys. 112, 45, (1939).
of levels of the compound nucleus created by
) H. A, &the, Rav. Nod, Fhys. 2, 229 (1937), see also experiments bylialdmn,
Phys. Rev.

285
neutron cbsorption is a composite hzr-gnitude and canQot be compared directly with
rv . rr represents the sum of all transitions from the level of the energy E
to lower lying levels, whose number is very high.
Formula (78) gives the radia-
tion widtb for a transition from an excited state to the ground stat2 only.
one applies (78) &so to transitions to excited states, the radiation width 'F
If
is givcn by:
PE/*
where r7 is the width 2s a function of the emitted rcdiation and Uis the level
density and E tho excitation energy of the level, whose r; is calculated. If E
is assumed 9 Mev and the expressions (71) are used for the level densities (A-1
100) one obtajae from (79): f, *uO.3 Nev which is rathar close to the observed
values in Table I, uut somewhat large,
The applicetion of (78) to transitions to excited stF.tes is question-
is slow~y *;nriable where D is the leva1 distance, if one varies either the upper
level or the lower levo1 of the transition.
In this case one would get instead
whereW(0) is the level density necr the ground state, which is of the order of
3.5
One then gets for Em9 MEW:
r;?= 2.0 x loa4 D' where D' is the
level distance at the upper level, which is of the order of 40 ev. This gives
definitely too srmll vclues.
The truth lies probably soucwhere batween (79) and
(80)
The dependence of r;l on the excitation energy E is given by (80) as
proportional to E6 and by (79) as very slowly indroasing function of E, Hence it
*
is very difficult to predict the dependence of rr on the excitation energy,
Tho expected spectrum of the emitted radiation fron a state with the
excitation energy E depends nlso on the assumptions made. The intensity I(z/)dv

* energies;
286
as a function of frequency is given by the integrand of (79) or (80). If (80)
is valid, thc distribution is proportional tdd and has a maximum at high
3
the transitions to the lowest stntas nre the most probable ones. If
(79) is valid, the intensity has a maximum soniewhere at intermediate oncrgies; it
is proportional tov5 and also to thG level density at the end state, which de-
creases sharply for the transitions of larger energyr

287
February 15, 1944
LECTURE SERIES ON NUCLEAR PHYSICS
Sixth Series: Diffusion Theory Lecturer: R,F. Christy
LECTURE 37:
DIFFUSION EQUATION AND POINT SOURCE
SOLUTIONS
In the first lectures we will discuss the diffusion of
neutrons which have come into energy equilibrium with their surroundings
(thermal neutrons). Later we will discuss the slowing
down and diffusion of fast neutrons and corrections to the diffusion
theory of thermal neutrons,
The Diffusion Equation
The flux of thermal neutrons is --Dan where n is the
thermal neutron density and D the diffusion constant,
D can be
obtained from simple kinetic theory arguments and equals xv/3
where xis the mean free path and v the neutron velocity,
"standard" velocity of themal neutrons 1s 2.2 h/sec.
related to the total cross-sectiorrU-by
where N is the nwnber of nuclei per cc,
sec, =
is
The
the relation 1: 1/N<
Tho diffusion equation can then be written
div (D grad n) 4- productfon/cc sec. - absorption/cc
$n/ At
The production/cc sec.
position,
is usually denoted by q, a function of
If O-, is the absorption cross-section, the absorption/
cc sec. is equal to Nd'nv; a or, writingh= 1/N Ta G the mean
free path for absorption, the absorption/cc sec equals (v/A)n=dT
where
is the mean life of the neutrouis,

This 1ead.s t o the diffusion equation in the following
288
forms*
If we consider the time independent diffusion equation
- xv CJ2n
3
D# n - t q = o
- A nfq = O
2
where L
=AM3 = DT*
Plane Source in an Infinite Medium
L is called the diffusion length,
Let the source be of strength Qn/cm2 and occupy tho
plane 2; = 0, With q = 0 everywhere except 2; = 0, the equation
I
reads
-L.- d2n n =
dz 2
'zz
which has solutions e and Since the neutron density
must approach zero for z =
n=Ae
fa the correct solution must be
-..
In order to relate A to Q, we note that the flow out from the
--
SOUPC8 per cm2 is
2 xv
r &.v A sQ
3 g)z P O - 3 Ti;
so A = SV
and n = 3LQ 0 - 1
ZG
This equation ,can be used to determine L experimentally

289
r)
for such materials as paraffin or water whero LX3 cm is small
compared to convcnient transverse dimensions of about 1 mater,
A source or” thermal neutrons is not directly available but can be
obtained as follows. A fast neutron source is placed below the
plane z = 0 in some slowing medium. At Z = 0 a removable sheet
of Cd (which selectively absorbs thermal neutrons) is placcd,
Above z = 0, data on n is obtained for the Cd in place and with
no Cd, The difference represents the effect of the Cd which is a
thermal source or sink and follows the above equation.
If the diffusing medium cannot be considered infinite in
the transverse direction and if the source can be considered to
have the transverse distribution cos *3c;/a cos Ty/a where a is
the transverse dimension, then the solution is
Substitution in the diffusion equation
- d2n 2 2
i---
d n d n - n
dx 2 2+---2 ;2 0
dY dz
leads to
r e.-
g+.-
2
a
1
- 1
a b 7
= o
which determines b,
This relation 9s frequently used for the
determination of L in media whore L is relatively long, of ordor
Point Source in an Infinite Medium
Here we write tho diffusion equation with g = 0 in

290
*
spherical coordinatos, considering only spherically symmetric solu*
tions.
if u = rn then u satisfies the
d2u -
7
2
dr
e qu a ti on
= o
The condition that n be finite at r -&gives
and
,=A 8 -r/L
r
The source strength
Doprassion of Neutron Density Dear a Fofl
d simple cxprcssion for the depression 6f the neutron
density near a foil can be obtained only if we assume the foil to
have thc form of a spherical shell, Let its thickness b? t, ab-
sorption cross-section Fa, and its nuclear density be N.
The diffusion equation is
Let n = qT A
I2
- e? -r/L which satisfios the equation and has in
r
general the right form. The number of neutrons absorbed per
second by a foil of radius ro is

291
0
The fractional depression in neutron density at the foil is
i
-
A
e
-r/L
r
0 -
1
qT
14- +Po+ VL
3NCr t
a
which can be considerably larger than the fraction of neutrons
absorbed in a single traversal of the foil which is N C t r
a
a
r

292
February 17, 1944
In the idsf; I.eci;;;:e
iwo considered the effect; of a detector,
such as an indrivm .Y9il7 or? tho neutrcm density in a medium in which
theilmal neut.ron..; we~cj c‘,irCi:!sing. It was shown how, as a first
ap~rzximation, t h i‘a?.l ~ coiild. be replacod by a sphei’r? and fairly
simple roslG.ts obt3t:md.
ta the prohi em.
Let us now consider a mo~e exact approach
S.c,-gpose we ccjimider the element of area (dA) of the fail
surface as a paint absaorber, If p is the coordinate of the
element r3A and r ta the cccrdinate of the point at which we want to
know the neutron densj.tyy, then the effect of a point sinh such RS dA
is given by:
To evaluate G
- L
Ce
- Ir -‘TI
we constdor the following:
The number of netltrnns absorbed per second may be set equal to the
number passing thrmgh the surface of a very small sphere surround-
ing our point sinh, fiToa;r the number of neutrons absorbed per second
equals Ntqnv6A wbers :
N : number of atoms/cs in foil

c cfa
t thickness of foil
= absorption cross-section of foil material
v~ : neutron velocity
The number of neutrons passing into a very, very small sphere is
given by:
293
Therefore :
1_3 4ffXv C = -Nt%n
3
v dA
The effect of the whole foil is then the sum of the effects of each
element. On adding these effects and passing to the limit as the
number of subdivisions is increased we get the total effect of the
foil at any point r to be
If we assume our source to be such that in the absence of the foil
the neutron density in the region in which we are interested is
everywhere constant and equal to one, then the neutron density at
any point is merely given by the sum of the contributions of the
foil and the source without the foil
Thus
*
The more pFeclse approach to our problem therefore is seen to lead

2 94
*
to an integral equation, Since this equation is difficult, if not
impossible, to solve exactly the usual procedure is to have recourse
to approximating the foil by a sphere,
Point Source in a Finitesphere
As another example of the way in which the diffusian
equation may be appliQd Let us consider the problem of a point source
in a finite spherical medium. The general solution in such a case
is:
r r
where A and B are arbitrary constants to be determined, B may be
found ira terns rf A by ems of the relation; n(R) z 0 where R is
the outpi’ ~ai!?-us of’ ’the sp1iere.
4R/L
R t’ -A e
anl! the solution becomes:
The cmstkirt A may then bs determined by .equating tha
* As
Up until now the solutions to the diffusion equation that
we have found have involved only a few terms, In general such representations
will not satisfy the boundary conditions, Instead we
must have recourse to aore complex methods, (e.g. the use of Fourier
Series). an illustration of suoh a problem we can take the case
of a point source embedded in a rectangular medium of length a
in

295
0
both x and y directions but infinite in extent in the and - 3
directions, We take axes such that the source which is at the center
of the medium is at the point (0, 0, 0). The general solution of the
equation c7 2 n- 0 is:
3
To determine bmn we can use the fact that each term separately must
satisfy the differential equation. Hence b, is given by the rela-
tion:
To find the coefficients A, we consider the fact that we have a
point source,
Such a smrce can be represented as 6 (x,y) where 6
stands for the Dirac delta function, Were we are assuming a source
of slow neutrons, though in practice a fast neutron source is used.
Let us expand 6(xpy) in a Fourier series.
We determine the Bmn’S as follows:

296
e
To relate the A's and Bls we must use the fact that the number of
neutrons flowing out of the plane z = 0 per second in a glven mode
must equal the number produced per second in that mode. The latter
is equal to Bmn
2xv JK,,, 2 xu m% nKy
Number flowing out ==T A, COS 7 cos
Hence our solution is:
If the so'u:~ce stxngth were Q instead of 1 the above solution must
be multfplied by a factor of Q. We note that B, decreases rapidly
with increasing m and n and so at great distances the only significant
term in our series will be the first. Thus at great distances
from our source:
This presents a very convenient method of measuring L since the
decrease of n with z enables us to findbll and from bll we can
readily compute L,
solution of the Diffusion Equation in an Iqfinite Medium with a as a
Function of Position.
As another type problem suppose we consider the solution

*
of the diffusion equation with a term due to the production of
neutrons (9) by a rather special method. The equation involved is:
297
If g P 0, the solution is (assuming an infinite medium spherical
symmetry)
-'IL
n=Ae where A
r
is some. constant.
As shown in a previous lecture A can be related to the source
strength Q by:
A .P
4.R Av
Hence :
- 3Q e
-r/L
n-- -
r
4T xv
Suppose now q f (x, y, 2).
We can solve the problem as follows: The solution for a1 point source
has the form given above. Our f'unction q can then be treated as an
infinite number of point sources continuously distributed throughout
the medium. The total effect will then be the sum of the effects
due to each point. If ;r is the point at which we want to how the
neutpon density and rf the point where our source is located we
have :
c
Sometimes the above integral can be readily evaluated and
under such conditions represents a very great simpliEication of the
problem. In utilizing this method it is very important to take
advantage of whatever symmetry the problem offers. Thus if plane

298
e
symmetry were exhibited the appropriate plane solution of the diffusion
equation, should be chosen. If cylindrical symmetry were the
case it would be well to find the solution for such conditions and
then apply the method here followed of integrating to gat the contributions
of the many point sources making up the q function.
A quantity of considerable utility in neutron diffusion is
the albedo or reflection coefficient,
If a finite medium with a
point source were left in empty space, the neutrons reaching the
bounding surface would diffuse out and be lost, Now, if another
medium were surrounding the first one, some of the neutrons reaching
the boundary would be reflected back in, thus increasing the neutron
density in our original medium,
The magnitude of the reflection
effect is measured by the albedo? of the, in this case, outside
medium.
Actually the albedo of a medium will depend, not only on
the medium, but also on the angular distribution of neutrons falling
on the medium.
Albedo of Various Media'
\fire w ill neglect this effect.
For this purpose we must consider the angular distribution
of neutrons at a point. We let be the cosine of the angle
between the velocity of the neutron and the radius vector and write
I
'
nv(rp) = + LAW +p~(rg
rl
*

299
\ ,and nv(r,p) = $A(r) -phA'(r)
Tho total flow in the positive radial direction is called the flow
out
flow out = P nv(r,p),pdr = [Aw - ij x A ~ (rj
The total flow in
the negative &dial direction is called the flow in.
')=flow
in z
Now the albedo of
the inside surface R of the medium is
1 -r/L
We can take n = c. e
r
do - - dn z - 1 e
dr r
($ +
The larger is R the better the albedo and for R
the albedo
approaches that of a plane
2 1
= I -5
l + Z &
3 L

0
If in addition we make the medium non-absorbing, L =a,

in diffusion theory nl n2 at the boundary
Ap dnl Ag
and - ,-, =
3 dr
dn2
3 dr
Instead of one af these we can use the ratio
- -
at the boundary
301
NOW this ratio for a medium can be written in terns of the albedo so
is another possible form of the boundary condition,
Extrapolated End Point
Let us examine the neutron densltg. in a plane semi-infinit6
non-absorbing slab, "e will show that if the neutron density
inside the slab is continued to the edge and beyond it would vanish
at a distance
3
Xbeyond the ed&.
n
/
/
/
/
- /
x-
We have, as before

*
302
The solution deep in the slab w i l l be nv = A(x) = a(l 4- t) where L
is the extrapolated end point. Shce A(x) = 0 for x = -L
At(x) z $
nv(x,r.) = 2
The total flow in the +x direction is
J
whichmust vavlibh at x = 0 since there are no neutrons entering the
slab.
An exact solution of this problem leads to L z .7104x actually.
Angular Distribution of neutrons Emerging from a Surface
As above we may take nv(x) = a(l -k )
I
We may consider these neutrons uniformly distri-
buted in angle and calculate the probability of
escape from the surface at angle p (per unit
range of
nv (p)
).“I 00
= 1, kk+$ a) e-’pwx
= i{y+gy2)
So then the flux from the surface -cos 0 4 cos2 8
z
X

303
Actual Distribution Near a Plane Boundary of a Non-absorbing Medium
Actual angular distribution in n -p +@y* , 0 0 m ;
Tho extrapolation of' the asynptotic solution vanishes at
,7104Xfrom the surface,
Very close (in a mean free path) to the
surface the neutron density decreases and has a singulasitg in
slope at the surface, The value of n at the surface is
of tihe value of the straight line solution at the su~face.
-
.*n
a

304
February 24, 1944
LECTURE SERIES ON NUCLEAR PHYSICS
Sixth Series: Diffusion Theory Lecturer: R, F. Christy
SECTURE 39: THE SLOWING DOWN 0.F NEUTRONS
Fermi’s ARe Eauation
The preceding lectures have all dealt with the diffusion
of thermal neutronsn Lot us now consider how neutrons coming from
a fast neutron soui“c”o :ce s!-med 5~vm- In the discussion the
quantity q will kij *JG::I’~ cox~ve~;.1.eni:, :-I,
1s defined as follows:
It fs obvious that the -i;c-ta_? EL,.
must equal the noo produced per second,
of
q we must have:
o+’ neuti-cnr: c:rs.;sing energy E
Hence from our definition
\
* the
where Q is the total number of neutrons produced per second.
The mechanism by which the neutrons are slowed down is
that of collision with other particles. The amount of kinetic
energy lost by a neutron in a given encounter may be computed by the
methoas of classical mechanics as applied to elastic collisions. In
general, if a neutron hits a particle of mass M the ratio of its
opaiginal en@l”gy (B) to its final energy (E’) is a fWLction both of
mass M and the angle between the orfginal path of the neutron

305
1)
and its path after the collision.
angles we can conclude that
By appropriate averaging over all
where C is a constant which depends only on M.
usually written as:
This relation is
where is the average of .
It can be shown that:
where the approximation is better for heavier nuclei (i.e. for
larger MI* A few sample values are:
for hydrogen: 5 P 1
for deuterium: 5 = 0,725
for carbon: 5 = 0.158
for oxygen: 5 0 = 0.120
Using the above we may find the mean number of collisions
suffered by a neutron in slowing down, Suppose the neutrons start
off with energies of 2 Mev and end up at thermal energies (about
1/30 ev), Then the mean number of collisions:

*
The
treatment now to be presented for slowing down is
somewhat limited in its application - even more so than is ordinary
diffusion theory. In the latter case our only important approximation
was that grad n be small, In slowing down theory it will
be demanded that the number of collisions be large and also that the
mean free path (A) not vary much between collisions, These
assumptions are necessary if we are to be able to replace the essentially
discontinuous process of energy loss by collision by a continuous
process, (It may be noted that the most common slowing dom
substances, such as paraffin and water, fail to satisfy the conditions
very well and so any treatment of paraffin, water, and
similar materials by the methods given here may result in considerable
error.)
306
The starting point in treating slowing down phenomena is
the so-called Fermi Age Equation. We shall. give here a non-rigorous
.justification rather than a formal derivation of the equation. Our
standard diffusion equation was:
By analogy we write:
*
If we consider t in this last equation to be a variable depending
on the past history of the neutrons we can relate this to the neu-
tron energy. Thus: - AhE per collision h number of
3
dt
collisions per second.

307
*
But Adn E per collision =k
V
and number of collisions per second =x
Hence :
Then our original equation becomes:
We introduce the quantity 2: (the ttAgeft) by the ,relation:
On substituting we get:
(Fermi's Age Equation)
We note that the tlAgefl has the dimensions of length squared. It
plays very much the same role in slowing down as the quantity L
2
did
in thermal neutron diffusion,
is a function of the energy (E) to
which the neutrons have been slowed and may be found from the
*
where Eois the energy with which the neutrons leave the aource.
Point Source Solution
As our first example of the use of the Age Equation let us

308
*
calculate the distribution of neutrons In space and energy coming
from a point source. We shall consider an infinite medium with
spherical symmetry. Our equations are:
The equation involving the Laplacian reduces to:
We shall solve this equation by the standard method of separation of
variables :
Let
Then
rq = f(r) 0 (TI
s=fd2
f d%/
fl dT
Since the left hand side is a function of r alone and the right of
7 alone we must have:
Thus we get the two equations:
d'f
ti> +-'k2f = o

309
*
The solution of (2) is: 121 = e -k2”C
The solution of (I) Is:: f = A sin kr+ B cos kr
t Solution is: q D, 5
m e
(A sin kr J- 3 cos kr) e
r
-k2y
where A and B are arbitrary constants depending on k.
stay flnite at r = 0, we must have B z 0.
Since q must
Hence:
q z & sin kr e
r
-k2”r: ,
Since we have no boundary conditions to be satisfied, any real value
of k will satisfy our equations.
The general solution, which is the
sum of‘ all possible solutions, is then found by by multiplying the
above expression by dk and integrating from - fl to -COO
Using the fact that we have a point source we can determine the
‘ function A(k). We know that;
q( T = 0) =: c 6(r>
where C is a constant to be determined later whiczh depends only on
the source strength &. b(r) is a delta function such that:
b(r) =: 0, r # 0 and
c
30
r2 h (r)dr t 1
Substituting in the above we get:
500
r Ch(r) P A(k) sin kr dk
-00
As the integrand is symmetric (as will be seen from later results)
written as:
I

3 10
rCb(r)
a 2 5
cx;,
A(k)sin kr dk
0
Multiplying by
- 1 yields
Now the Fourier Sine Transform tells us that
fla
If f(x) =fi pI (u) sin xu du
then: @(u) =fi 0
\oof(x) sin ux dx
30
Applying this to our problem:
i'
By the definition of the delta function this becomes
Hence
dividing by
r94"C;
e
y i e Ids'*

311
To evaluate the first integral let
Then;
The first term on the right yd6lds zero while the second becomes:
Similarly
Thus :
To find C we remember that:
Lo1
qdvol=Q
Let u
I-~/~~C' then:

*
Hence :
Qe
3" (4mp
That this result is what we should expect is easily seen from the
3 12
following argument, Since at r s 2fi q has reduced to l/e of its
value at the origin, we see that
is a measure of the width of
curve representing q. If were very small our curve should be
very narrow (a) .
*
This is ph..~s?c~l?~y plausible, since small
in dicates fast
neutrons and ther;rl we would expect to be grouped near the source.
On the other hand large means slower neutrons. These
would be expected to be more uniformly distributed than the faster
ones, That Is just what our formula leads us to conclude.

I
0 -1”
Plane Source Representations,
If instead of a point
source the solution would be of
+
-z2/42
q = Q e
pflT
source we have an infinite plane
the form:
Q = neuts/cm
2
sec from
source
Let us suppose now that we have a medium finite and of
width (a) in both x and y directions and infinite in the - and - z
directlons.
e quat ion :
Assume a point source at (0, 0, 0). A solution of the
which satisfies the boundary conditions at x
- a/3, y o L a/2 is:
..-
where A? and n are positive integers, C(k) is a constant depending
on k, and

3 14
p2 s (d2 +, m2j,, q2+ k2,
2
a
(k is any real number)
The general solution i s then found by adding up all possible solutions.
This means swnming over m and A? -and integrating over k.
Thus
2
C(k) >: A cos& cos& cos kz e **-P z,
m,k 1,m a 8
Applying the orthogonality principle, integrating, and using the
fact that a point source may be represented by a delta function we
gat:
We note that for >) a (i,e, for the lower energy neutron) the
higher harmonics die out. Hence, with this zipproximation our
expression becomes:

315
LA-24 (40)
February 29th, 1944
LECTmE SERIES ON NUCLEAR PHYSICS
Sixth Series: Diffusion Theory
Lecturer: R, F. Christy
LECTURE Vz
THE SLOWING DOWN OF NEUTRONS IN WAX%
In this lecture we shall consider the f,ollowing question,
What happens when a Ra-Be source of neutrons is placed in a large
water tank? What is the distribution of neutrons in space and
energy?
Suppose we were to meazure the distribution by means of a
cadmium covered indium foil. Such a counter records the number of
neutrons of the indium resonance energy (slightly above 1 ev).
Plotting
fi(r2A) against r (where r is the distance from the
source and A is the activity measured) we obtain results such as
those in the diagram,
.
\ L.
\
\
\ d-- Gauss i a n
\

cally straight,
We note that at large values of
This implies that in that region:
/
316
r the curve is practi-
$I
We also note that the curve of measured neutron density differs
considePBbly at large r from what it would look like if it
followed a Gaussian distrfbutlon (dotted line),
The above distributlon of neutrons is readily recognized
as being the same as that of neutrons from a point source which have
not yet had any collisions. That this is so is seen from the fact
that the probability that a neutron will go a distance
a collision is
r without
2
to
e -vi and that we must have a factor Qf 1/4‘Kr
satisfy geometrical considerations,
Now it is easy to demonstrate using classfcal mechanics
that in an elastic collision with a proton B neutron can never be
scstt,c:wd at an angle greater than go0. Hence the average scattering
angle will be between 0’ and 90’. live see, therefore, that on
coLlic?ing with protons the directton of the neutrons is not much
changed and so we would expect the neutron density to follow a law
similar to the above (i.e, we expect A 0’ (l/r2)enr/LX where -)-L
is actually somewhat greater than the true mean (free path), A
further reason why we should expect such a distribution is seen when
we remember that the scattering cross-section for hydrogen decreases
rapidly with increasing neutron energy, Thus the mean frea path is
largest at higher energies and so most of the distance traveled by
the neutrons is covered in the first few collisions. But in the
first few collisions the (l/r*)e-”klaw is followed, IVe conclude

317
m
from this argument also that the last is the law we should expect to
find for the neutron density for large r.
A concept which we shall find useful in considering our
problem is that of the mean square distance r2 traveled by a
neutron.
This quantity is defined by:
where f(r) is the kxnctional dependence 0.f the neutron density on
the distance (r) from the source.
__L
As an example let us calculate r2 for the case where
our neutron density follows the (l/r2>e-r’h
law to the first
collision and then follows an age diffusion law (i.e, proportional
to e -r2/47=). Let us denote the first portion by the subscript (1)
an?. the second by the subscript (2).
Then:
.r 2 =:

1
3J8
Substituting
Hence :
pre-age
dif fusioq
we
diffusion
In order to calaulate the distribution of ne
k
consider the following!
From the poipt source the neutrons go to a ppi$t p
following the (l/r?)ae-”.h
I , ] ’ , 1,
law. From rl they go a
I
2
age dZTTusion (i,e, following e*r /4z). We can 4
6
pofnt rl as a source for the age diffusion which pr
‘1 < ~
neutron reaohes r TnCagrating per the whole volwe2
I
4
contribution of egch su h source we get:
I
” k b i 1

319
Substituting
we get
c03 B
?-

320
*
This integration with respect to rl cannot, unfortunately,
be done analytically,
To get an approximate analytic solution we
can do the following I Approximate the distribution of (l/r2}e*r/)L
by (l/~>e-~/~. In this form we can do the integral. L should be
2
chosen so that the mean square distance traveled (T- ) comes out the
2
same. Originally Pr2 was 2x so we must have:
We note paFt$cul&dj? hbw analogous L2 is to "r: in the slowing
down equation. There we found
-2 I
3:
carried quite far,
slowing down equation by a diff'usion equation.
- 6T, This analogy can be
approximation is good only over a small energy range and for
not too large compared to
Under certain conditions we can even replace a
However9 such an
fiT.
In our water tank problem it is fairly obvious that a
certain percentage of the neutrons emitted by the source per second
(Q) will escape from the tank without ever undergoing any collisions
and so will not be adequately described in terms of a diffusion
equation.
To correct for this we can replace OUT
strength Q by one of strength QJ- such that:
source of
r

*
We can *en
321
find the distribution by assuming It to be of the form$!
- 4 L -(2R - rP/L
c - e
4nrL2 e
C is de‘term’ined fmm the ?elation:
s”,
2
C - (,@*’/L c ,e-(2R - r)/l) 4”lr dr s Q‘
4XrL2
6till another, though mor8 foTrna1, way of treating the
problem ‘3% as follows;
First consider slowing down and then thermal
dif ftlsi0t1.r
The standard age equlati6n la:
With spherical symmetry, this reduces tbO
where R*= ra4ius of tank.
The coefficients (Ah) can be determine& as follows:
T= 0 we know the distribution is
For
Hence;

ll:
..*L-
--.
322
To find the number of thermal neutrons escaping from the
tank we consider the ordinary diffusion equation and assume each
mode, which we shall denote by the subscript n, satisfies the
equation. Thus:
As a solution assume:
Substituting in our equation we get:
Hence :
Naw for the number of neutrons captured per second is:
s
’V f.
-0ct vol= 1 .+ $pfl”Ln
7
B7j.l *hhg g d vol is equal to the number of thermal
-. Svnl L3 Pa vos Sdvol
I vol
neutrons prDC!i(:ed ps second.
just equal to the riumber produced minus the number absorbed is:
Hence, the number escaping Which I S

323
LA-24 (41)
March 2, 1944
LECTURE SERIES ON NUCLEAR PHYSICS
Sixth Series: Diffusion Theory
Lecturer: R, F. Christy
_LECTURE VI: THE BOLTZMANN EXJUATJON:
- CORRECTIONS TOXFFUSION THEORY
Boltzmann Equation_
In this lecture we shall consider some corrections to the
elementary diffusion theory previously treated,
More exact methods
depend on either the Boltzmann equation or an integral equation,
shall make our approach through the former.
To derive the Boltzmann equation for diffusing neutrons,
consider the following:
Let f(x9 y, z9 vx, vy9 va> be the density
of neutrons at the point x,y,z having velocity components between:
vx and vx+ dv,, v Y and and vz and vd + dv,. It is
readily seen that:
c
\
&e 7.0 '3 ity
f d vel
ordinary neutron denkllty
We
Now the tot,3:L rate o:? change of
f with time, which we denote by

324
*
But this total rate of change must be equal to the number of neutrons
entering the given velocity range per second minus the number lost
by collisions, Let 4T= ea+ c$where sc$ is the absorption cross
section per unit volume and cK3 is the scattering cross section. In
terms of these symbols the number of neutrons lost per second is:
vof, The number entering the velocity range is found by averaging
vo f over all angles. Thus,
where
due element of solid angler
When the above are substituted, the Boltzmann equation becomes:
where w0 substitute
*
In passing it is interesting to observe that this
equation is considerably simpler than the ordinary Boltzmann
equation of kinetic theory. This is due to our neglecting collisions
between neutrons and the fact that we have considered the
objects hit by the neutrons to remain stationary. The great simplification
is that the ordinary Bol'czmann equation Is non-linear
while our above derived equation is linear and so is considerably
more tractable
Let us consider the Boltzmann equation in the Steady

325
e
state, Then it reduces to:
7 * ef =-vrf
- Assume further that we have an infinite medium, that f e kx and is
independent of y and z. (This is one of the few instances in
which the Boltzmann equation may be solved.)
Our equation becomes:
vx kf = -vsf -1- VGT
Averaging over all angles and substituting .s g vx/v z Goa 6, we have:
Hence
This equation serves to determine p(
The first approximation for
small x/r and
diffusion theory result L2
Tdrgives K2 = 35% which is the same as the
l/3%~ , However, If is not much
less than 05 the BoZtzmann equation shows that we must correct our
value for 6/
A rather good second order approximation is:
2
e Hence the value of L
that should be used for more accurate work is:

326
3 i-7n 2h
x*
Another important instance in which we can correct
ordinary diffusion theory through use of the Boltzmann equation is
in the matter of neutron flux.
Until now we have used -( )~v/3)Vn
for the flux (io@. we have said the diffusion constant D, was
+Xv/3).
A more accurate D can be obtained as follows:
Let us find
7
vxfo This is:
From this we see that a more accurate diffusion constant is:
Thus, while
the exact relat2on is:
-
In all, the above work when averaging over angles to find
9 we tacftly assumed to be independent of the angle of scatter-
,-I
* -

~
327
-41
tfig (f.e. Isotropic scattering). If le aid floe have sphsyidal
symmetry we could have expanded SS in a power series in cos 8,
Thus
The effect would be to give us a dlighkly different value for ? and
would serve to complicate the problem.
Assuming such an expansion neatjrd#ktr& i.f; hbrns ofit that
still an additional correction for & is necessqryi de get
I /
Let US Introduce a quptity called the trandbdvt 6Ybrtd lagCkib$S
defining it by the relation:
o&, .= b(F-cos e)
where a is the average of cos 0 over all collisions.
*ly we obtain the equation:
Then for
where M
is the mass of the scattering nucleus, Then
We see that for light elements, such as hydrogen, the correction is
quite important, On the ather hand, for heavy elements the correction
is negligible.
StlZl another way in which the Boltzmann equation can be

328
4-
of use to us is in helping u8 to derive the Age equation, We shall
not carry this derivation through here, but will merely indicate the
method to be followed,
First assume a certain energy change per scattering.
follows that the number of neutrons entering the given velocity
It
range is -f(vl), where vr represents the neutron velocity at a
somewhat higher energy level. Then we can represent f(v’ by a
Taylor series and break off the series after the first two terms.
Thus:
f(v’) =f(v) +(v’--v) $$
This is then substituted in the Boltzmam equation I
and the same
general procedure is followed as previously, We obtain a relation
between and a’I;/dv The quantity ‘T is then related to V2q and
and 8/Av to &q/&?;
Application to Water:
BfndinP: Corrections:
To illustrate the application of the formulas we have
developed, let us consider the apparent; paradox that arises in com-
puting the diffusion length for water.
Since the cross sections
for hydrogen are much larger than those for oxygen, we need only
take hydrogen cross sections into account. for hydrogen when
measured turns out to depend on V. At thermal velocities it is
about 40 barns. Hence : 40/3. for hydrogen is 0.33. When
*
we substitute in our Formulas for
differs greatly from the value L2
L2 for water we get a value that
2
8.3 cm which has been measured.
Here it seems that our formulas are wrong and so must be thrown
away.
Fairly good agreement, though, can be reached by using the

329
following argument:
I
In the calculations it was assumed that the mass of the
hydrogen atom is 1. Actually it is effectively greater than this,
since the hydrogens are bound to a greater or lesser extent to the
heavy oxygen atoms. Actually it is found that for'snergies greater
than 1 ev the hydrogen atom is practically free and so has an effective
mass of about 1, For energies less than this, the effective
mass is somewhat more than 1,
For epithermal energies (Le. energies slightly over
thermal) Cr, for hydrogen is approximately 20,. For hydrogen eom-
pletely bound (this corresponds to 0 energy) theory leads us to
2
g 80, For thermal energies we should
expect CS to be 20 (w)
then expect fl5 to be somewhere between 20 and 80, The observed
cross section is the geometric meanp 40. It corresponds to a
hydrogen binding that is about halfway complete.
By analogy we expact the 1 - cos 0
term also to be same-
where between its two extrerrleei of l/3 and I, If we again take the
Using this value we get rather good agreement in the measured and
computed values for I,'.