saddle point shapes of nuclei - Frankfurt Institute for Advanced Studies
saddle point shapes of nuclei - Frankfurt Institute for Advanced Studies
saddle point shapes of nuclei - Frankfurt Institute for Advanced Studies
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SADDLE POINT SHAPES OF NUCLEI<br />
Poenaru, Gherghescu, W. Greiner,Complex fission phenomena, Nucl. Phys. A 747 (2005) 182.<br />
Poenaru, W. Greiner, De<strong>for</strong>mation energy minima at finite mass asymmetry, EPL 64 (2003) 164<br />
Poenaru, W. Greiner, Nagame, Gherghescu, Nuclear <strong>shapes</strong> in fission phenomena, J. Nucl.<br />
Radiochem. Sci. Japan 3 (2002) 43<br />
Dorin Poenaru
OUTLINE<br />
• Nuclear <strong>shapes</strong> and mass asymmetry<br />
• Equilibrium and scission configurations<br />
• Examples <strong>of</strong> <strong>shapes</strong> (given<br />
parametrization)<br />
• David L. Hill’s method<br />
• Integro-differential equation (no param.)<br />
• Phenomenological shell effects<br />
• Fission into 2, 3, and 4 fragments<br />
• Experimental attempts<br />
• Conclusions<br />
Dorin Poenaru
SCIENTISTS and INSTITUTES<br />
NSTITUTES<br />
Walter GREINER, FIAS<br />
Joseph H. HAMILTON, DPA<br />
Yuichiro NAGAME, ASRC<br />
FIAS = <strong>Frankfurt</strong> <strong>Institute</strong> <strong>for</strong> <strong>Advanced</strong> <strong>Studies</strong>, J. W. Goethe University,<br />
<strong>Frankfurt</strong> am Main, http://www.fias.uni-frankfurt.de<br />
DPA = Department <strong>of</strong> Physics & Astronomy, Vanderbilt<br />
University, Nashville, TN, USA, http://www.vanderbilt.edu<br />
IFIN-HH = H. Hulubei National <strong>Institute</strong> <strong>of</strong> Physics & Nuclear Engineering,<br />
Bucharest, http://www.theory.nipne.ro<br />
ASRC = <strong>Advanced</strong> Science Research Center, Japan Atomic Energy Research<br />
<strong>Institute</strong>, Tokai, http://asr.tokai.jaeri.go.jp<br />
Dorin Poenaru<br />
Akunuri V. RAMAYYA, DPA<br />
Radu A. GHERGHESCU, IFIN-HH IFIN HH<br />
Dorin N. POENARU, IFIN-HH IFIN HH
HYSTORICAL MILE-STONES<br />
MILE STONES<br />
• 1929 Fine structure in α−decay<br />
• 1939 Induced fission<br />
• 1940 Spontaneous fission<br />
• 1946 Ternary (particle-accomp.) fission<br />
• 1980 Prediction <strong>of</strong> cluster Radioactivity<br />
• 1981 Cold binary fission<br />
• 1984 cR experimentally confirmed<br />
• 1998 Cold α and Be accompanied fission<br />
Dorin Poenaru
FISSION FRAGMENT MASS ASYMMETRY<br />
Symmetry <strong>for</strong> some<br />
Fm, Md, No<br />
isotopes when the<br />
most probable<br />
fragments are very<br />
close to the doubly<br />
magic 132 Sn<br />
Dorin Poenaru
EQUILIBRIUM QUILIBRIUM CONFIGURATIONS<br />
Nuclear gs de<strong>for</strong>mation and fission fragment<br />
mass asymetry are not explained within LDM.<br />
One should add shell corrections to calculate<br />
Potential Energy Surfaces, PES eg<br />
Equilibrium: ∂E<br />
∂E<br />
∂q<br />
1<br />
=<br />
∂q<br />
= 0<br />
• Minima (ground state, shape isomer)<br />
• Sadde-<strong>point</strong>s: maxima on fission valley<br />
2<br />
2<br />
(min.)<br />
∂ E ∂ E<br />
∂E<br />
∂q<br />
1<br />
∂E<br />
=<br />
∂q<br />
2<br />
= 0<br />
2<br />
Dorin Poenaru<br />
2<br />
∂q1<br />
2<br />
∂ E<br />
∂q<br />
∂q<br />
2<br />
1<br />
∂q1∂q<br />
2<br />
∂ E<br />
∂q<br />
E =<br />
2<br />
2<br />
2<br />
( 1, 2)<br />
q q E<br />
< 0
G STATE, VALLEYS, SADDLE POINTS<br />
Dorin Poenaru
POTENTIAL ENERGY SURFACE <strong>of</strong> 264Fm<br />
Symmetrical mass<br />
distributions<br />
GS=ground state<br />
SI=shape isomer<br />
SP1=<strong>saddle</strong> <strong>point</strong><br />
SP2=<strong>saddle</strong> <strong>point</strong><br />
— Fission valley<br />
at η = 0<br />
Dorin Poenaru<br />
264 Fm
TWO PATHS FROM SADDLE TO SCISSION<br />
Y. Nagame et al.<br />
J. Radioanalyt. and Nucl.<br />
Chemistry 239 (1999)<br />
97.<br />
Dorin Poenaru
THREE HREE DECAY MODES OF 234 234U LIGHT FRAGM.<br />
α<br />
28 Mg<br />
100 Zr<br />
SHAPES ALONG FISS. PATH<br />
Dorin Poenaru
252 Cf<br />
Cf 146 146Ba Ba + 96 96Sr Sr + 10 10Be Be<br />
Given parametrization and fission path<br />
Dorin Poenaru
GIVEN PARAMETRIZATION<br />
Möller, Madland, Sierk & Iwamoto, Nature 409 (2001)<br />
5 dimensional shape coordinates<br />
Dorin Poenaru
∇v r<br />
DAVID HILL’S HILL METHOD (I)<br />
Thesis (Princeton 1951) – large amplitude motion<br />
<strong>of</strong> an incompressible fluid with irrotational flow<br />
=<br />
0,<br />
∇ × v r<br />
mass density µ under bulk <strong>for</strong>ce<br />
=<br />
0<br />
<strong>of</strong> electrostatic nature and a pressure distr. p 1<br />
r<br />
dv<br />
r<br />
µ = χ − ∇ p<br />
dt<br />
1<br />
H µ<br />
2<br />
1<br />
≅<br />
Dorin Poenaru<br />
−∇<br />
p<br />
r<br />
F = −∇U<br />
∇<br />
2<br />
H = 0<br />
2<br />
= U + p1<br />
+ v p( z ) = U ( z ) + σΚ(<br />
z )<br />
Pseudopressure p, surf. tens. σ, and curvature K
DAVID HILL’S HILL METHOD (II)<br />
D.L. Hill: The Dynamics<br />
<strong>of</strong> Nuclear Fission,<br />
Proc. <strong>of</strong> the 2 nd United<br />
Nations International<br />
Conference on the<br />
Peaceful Uses <strong>of</strong> Atomic<br />
Energy, Vol. 15,Geneva,<br />
1958<br />
Dorin Poenaru
STATIC PATH ON PES<br />
Dorin Poenaru
STATIC PATH ON CONTOUR PLOT<br />
Dorin Poenaru
INTEGRO-DIFFERENTIAL INTEGRO DIFFERENTIAL EQUATION (I)<br />
Κ(<br />
z ) =<br />
1<br />
ℜ<br />
a<br />
+<br />
1<br />
ℜ<br />
b<br />
=<br />
1<br />
ρ(<br />
1 + ρ'<br />
Dorin Poenaru<br />
2<br />
)<br />
1 / 2<br />
−<br />
ρ'<br />
'<br />
2<br />
( 1 + ρ'<br />
ρ = ρ (z) is the surface equation (cylindrical symm.)<br />
minimizing the de<strong>for</strong>mation energy<br />
E def<br />
LDM<br />
s C<br />
( α ) = E + δE<br />
= E + E + δE<br />
with constraints: const. de<strong>for</strong>mation α and<br />
α =<br />
z +<br />
c<br />
L<br />
z<br />
c<br />
R<br />
mass asymmetry η<br />
η<br />
=<br />
A1<br />
− A2<br />
A + A<br />
1<br />
2<br />
)<br />
3 / 2
INTEGRO-DIFFERENTIAL INTEGRO DIFFERENTIAL EQUATION (II)<br />
The pressure at the surface<br />
r r<br />
ρ ( r ) + 2σΚ<br />
( r )<br />
eV<br />
s<br />
is in equilibrium. V s is the electrostatic potential<br />
ρ e is the charge density<br />
2<br />
ρρ'<br />
' −ρ' −[<br />
λ + λ z + ℑ(<br />
ρ,<br />
z )] ρ(<br />
1 + ρ'<br />
1<br />
2<br />
2<br />
)<br />
3 / 2<br />
ℑ = 10 XVs<br />
( z , ρ)<br />
Lagrange multipliers λ1,2 −1<br />
= 0<br />
X = fissility V s is expressed by a double integral<br />
Conditions<br />
ρ ( z1<br />
) = ρ(<br />
z 2 ) = 0<br />
dρ( z1<br />
, 2 )<br />
= ±∞<br />
Dorin Poenaru dz
INTEGRO-DIFFERENTIAL INTEGRO DIFFERENTIAL EQUATION (III)<br />
z 1,2 are the intercepts with z axis at the 2 tips.<br />
Symmetry: z 2 = z p = - z 1 . New function<br />
Vsd s<br />
2 2<br />
u( v)<br />
= Λ ρ [ z(<br />
v)];<br />
( v)<br />
= z −v<br />
/ Λ<br />
Dorin Poenaru<br />
z p<br />
1 2<br />
u'' = 2 + [ u'<br />
+ ( v − d + Vsd<br />
)( 4u<br />
+ u'<br />
u<br />
5X<br />
= V<br />
2Λ<br />
− a −vb;<br />
2<br />
)<br />
3/<br />
2<br />
u ( 0)<br />
= u'(<br />
vpn)<br />
= 0;<br />
u '( 0)<br />
= 1/<br />
d;<br />
d and n are input parameters related to elongation<br />
& number <strong>of</strong> necks. Runge-Kutta numerical meth.<br />
]
REFLECTION ASYMMETRICAL SHAPES<br />
Equations <strong>for</strong> left hand side (L) and right h.s. (R).<br />
Matching ρ L(0) = ρ R(0)<br />
2π<br />
M L = ( 1+<br />
η)<br />
= πΛ<br />
3<br />
−3<br />
L<br />
And similarly <strong>for</strong> right hand side.<br />
∫<br />
vp<br />
0<br />
n ≠<br />
u<br />
L<br />
,<br />
Alternatively: and continuity <strong>of</strong> ρ’’(0)<br />
L<br />
1/<br />
2<br />
L<br />
n<br />
R<br />
1/<br />
2<br />
L<br />
Dorin Poenaru<br />
u ( v ) / Λ = u ( v ) / Λ<br />
( v)<br />
dv;<br />
( d −v<br />
) u ( v ) = ( d − v<br />
L<br />
pL<br />
pL<br />
R<br />
p<br />
L<br />
1/<br />
2<br />
R<br />
⎧3<br />
Λ 0 = ⎨ ∫ 2 0 ⎩<br />
p v<br />
L u<br />
pR<br />
) u<br />
1/<br />
2<br />
R<br />
( v<br />
pR<br />
)<br />
L<br />
p<br />
R<br />
⎫<br />
( v)<br />
dv⎬<br />
⎭<br />
1/<br />
3
DEFORMATION ENERGY<br />
Liquid Drop Model (LDM)<br />
E<br />
LDM<br />
X =<br />
0<br />
= E − E<br />
0<br />
0<br />
= Es<br />
( Bs<br />
−1)<br />
+ EC<br />
( BC<br />
0 0<br />
E /( 2E<br />
) - fissility<br />
C<br />
s<br />
De<strong>for</strong>mation dependent surface energy<br />
y(x) = surface equation with (–1,+1) intercepts on x axis<br />
Dorin Poenaru<br />
−1)<br />
0<br />
2 2/<br />
3<br />
0 2 −1/<br />
3<br />
Es = as(<br />
1−κI<br />
) A ; I = ( N − Z)<br />
/ A;<br />
EC<br />
= aCZ<br />
A<br />
B s<br />
1/<br />
2<br />
2 + 1<br />
2<br />
2<br />
d ⎡<br />
2 1 ⎛ dy ⎞ ⎤ z2<br />
− z<br />
= ∫ ⎢y<br />
+ ⎥ dx;<br />
d =<br />
2 4 ⎜<br />
dx ⎟<br />
2R<br />
−1⎢⎣<br />
⎝ ⎠ ⎥⎦<br />
0<br />
1
B c<br />
COULOMB ENERGY<br />
Assume uni<strong>for</strong>m charge density ρ0e = ρ1e = ρ2e 5 + 1 + 1<br />
5d<br />
= ∫dx∫dx' F(<br />
x,<br />
x')<br />
;<br />
F(<br />
x,<br />
x')<br />
= { yy1[(<br />
K − 2D)<br />
/ 3]<br />
•<br />
8π<br />
−1<br />
−1<br />
2 2<br />
⎡<br />
⎤<br />
2 2<br />
2 3 ⎛ dy ⎞<br />
1 dy<br />
⎢2(<br />
y + y − x − x + x − x ⎜ − ⎟<br />
1 ) ( ')<br />
( ')<br />
⎥ +<br />
⎣<br />
2 ⎝ dx'<br />
dx ⎠⎦<br />
2<br />
2<br />
⎧ 2 2 ⎡ 2 x − x'<br />
dy ⎤⎡<br />
2 x − x'<br />
dy ⎤⎫<br />
1 −1<br />
K − K'<br />
K⎨y<br />
y1<br />
/ 3+<br />
⎢y<br />
− ⎥⎢y<br />
1 − ⎥⎬}<br />
aρ<br />
; D = 2<br />
⎩ ⎣ 2 dx ⎦⎣<br />
2 dx'<br />
⎦⎭<br />
k<br />
K, K’ are complete elliptic integrals <strong>of</strong> 1st and 2nd<br />
kind. After scission E =<br />
2<br />
e Z Z / R<br />
Dorin Poenaru<br />
C<br />
∑<br />
i≠<br />
j<br />
i<br />
j<br />
ij
PHENOMENOLOGICAL SHELL CORRECTIONS<br />
Adapted after Myers & Swiatecki. Number <strong>of</strong> nucleons<br />
Ζ , N proportional to the volume.<br />
= ∑ ∑<br />
δE<br />
( δE<br />
+ δEni)<br />
= C [ s(<br />
Zi<br />
) + s(<br />
i)];<br />
i pi N<br />
i<br />
Ni are magic numbers. C=6.2 MeV, c=0.2 from fit to<br />
exp. masses and def. Variation with de<strong>for</strong>mation:<br />
C ⎧<br />
L ⎫<br />
= ⎨∑<br />
i<br />
δE<br />
[ s(<br />
Ni<br />
) + s(<br />
Zi<br />
)] ⎬<br />
2 ⎩ i<br />
Ri<br />
⎭<br />
For magic nb. δE is minimum.<br />
L i are the lengths <strong>of</strong> fragments.<br />
Dorin Poenaru<br />
s −<br />
2/<br />
3<br />
1/<br />
3<br />
( Z)<br />
= Z F(<br />
Z)<br />
cZ<br />
−<br />
5/<br />
3 5/<br />
3<br />
3 ⎡ Ni<br />
− Ni<br />
5/<br />
3 5/<br />
3⎤<br />
−1 F( n)<br />
= ⎢ ( n − Ni−1)<br />
− n + Ni−1<br />
⎥;<br />
n∈<br />
( Ni−1,<br />
Ni<br />
) is Z or N<br />
5 ⎣ Ni<br />
− Ni−1<br />
⎦
CALCULATED SHAPES <strong>for</strong> X = 0.60<br />
X = 0.60 (e.g. 170 Yb) η = 0 n L = n R = 2<br />
d L = d R = 1.40, 1.50, 1.70, 1.91(SP) input parameters<br />
α/R 0 = 1.31, 1.64, 2.10, 2.30 de<strong>for</strong>mation<br />
Dorin Poenaru
CALCULATED SHAPES <strong>for</strong> X = 0.70<br />
X = 0.70 (e.g. 204 Pb) η = 0 n L = n R = 2<br />
d L = d R = 1.40, 1.70, 2.00 , 1.55(SP) input parameters<br />
(α/R 0 ) SP = 1.82 SP de<strong>for</strong>mation<br />
Dorin Poenaru
DEFORMATION and ENERGY vs input parameter<br />
X = 0.60 (e.g. 170 Yb)<br />
and 0.70 (e.g. 204 Pb)<br />
η = 0<br />
n L = n R = 2<br />
Maximum energy at<br />
the Saddle Point<br />
Dorin Poenaru
CALCULATED SADDLE POINT SHAPES<br />
η= 0 n L = n R = 2 d L = d R = d<br />
X = 0.60, e.g. 170 Yb<br />
d=1.91 α/R 0 = 2.30<br />
X = 0.70, e.g. 204 Pb<br />
d=1.55 α/R 0 = 1.82<br />
X = 0.82, e.g. 252 Cf<br />
d=1.38 α/R 0 = 1.16<br />
Dorin Poenaru
η<br />
≠<br />
0<br />
ASYMMETRICAL SHAPES (I)<br />
Two ways to obtain: (1) n L = n R = n<br />
(1) n=2<br />
X = 0.77<br />
e.g. 238 U<br />
d R =1.4<br />
d L =1.4,<br />
1.45,<br />
1.5, 1.6<br />
n ≠<br />
(2) L R<br />
L L<br />
Dorin Poenaru<br />
n<br />
d ≠<br />
L<br />
d ≠<br />
d<br />
d<br />
L
η<br />
≠<br />
0<br />
ASYMMETRICAL SHAPES (II)<br />
(2) n L = 4, n R = 2, d L = 2.47, 2.44, 2.39<br />
X = 0.60, e.g. 170 Yb d R = 1.46, 1.43, 1.40<br />
α/R 0 = 2.06, 1.98, 1.84<br />
Dorin Poenaru
MASS ASYMMETRY AT THE 2 nd SADDLE POINT<br />
The 2 nd barrier height is<br />
lower at some finite<br />
mass asymmetry.<br />
Dorin Poenaru
SP MINIMUM DUE TO SHELL EFFECTS (I)<br />
Binary cold fission. Saddle <strong>point</strong> energy vs mass<br />
asymmetry with and without shell effects.<br />
Dorin Poenaru
SP MINIMUM DUE TO SHELL EFFECTS (II)<br />
Binary cold fission. Saddle <strong>point</strong> energy vs mass<br />
asymmetry <strong>for</strong> Th isotopes<br />
Dorin Poenaru
SP MINIMUM DUE TO SHELL EFFECTS (III)<br />
Binary cold fission. Saddle <strong>point</strong> energy vs mass<br />
asymmetry <strong>for</strong> U isotopes<br />
Dorin Poenaru
CONTRIBUTION OF SHELL EFFECTS<br />
Dorin Poenaru
MASS NUMBER OF THE HEAVY FRAGMENT<br />
Mass number <strong>of</strong> the heavy fragment corresponding<br />
to the minimum <strong>of</strong> the <strong>saddle</strong> <strong>point</strong> energy<br />
Nuclei X d L –d R η A 1<br />
238 U 0.769 0.04 .04988 124.93<br />
232 Th 0.754 0.06 .07517 124.72<br />
228 Th 0.758 0.08 .09512 124.84<br />
Dorin Poenaru
TERNARY FISSION (I)<br />
Binary fissility X = 0.60 e.g. 170 Yb n L = n R = 3<br />
d L = d R = 2.25, 2.80, 7.00 α/R 0 = 1.65, 2.31, 2.73<br />
Dorin Poenaru
TERNARY FISSION (II)<br />
Config. d = 7, E/E s 0 = 0.134 is not far from a<br />
true ternary fission 170 Yb 56 V + 56 V + 58 Cr,<br />
Q=83.64 MeV, (E t -Q)/ E s 0 = 0.239 <strong>for</strong> spherical<br />
<strong>shapes</strong> in touch<br />
For 170 Yb 10 Be + 80 As + 80 As<br />
Q=70.86 MeV, (E t -Q)/ E s 0 = 0.147<br />
For 170 Yb 4 He + 83 Se + 83 Se<br />
Q=87.48 MeV, (E t -Q)/ E s 0 = 0.103 — higher Q-value<br />
Dorin Poenaru<br />
and lower fission barrier
QUATERNARY FISSION<br />
Binary fissility X = 0.60 e.g. 170 Yb n L = n R = 4<br />
d L = d R = 2.3, 2.7, 4.00 α/R 0 = 2.14, 3.14, 3.23<br />
Config. d = 4, E/E s 0 = 0.214 is not far from a<br />
true quaternary fission<br />
Dorin Poenaru<br />
170Yb 42Cl+ 42Cl+ 43Ar+ 43Ar Q=53.17 MeV, (Et-Q)/ E 0<br />
s = 0.324 <strong>for</strong> touching<br />
spheres
EXPERIMENTAL ATTEMPTS<br />
• Theory: R. D. Present (Phys. Rev. 59 (1941) 466): Uranium<br />
tripartition would release about 20 MeV more energy than binary<br />
• Exp. Rosen & Hudson (1950): 235U + nth . Triple gas filled<br />
ionization chamber. Yield <strong>of</strong> 6.7 per 106 binary fission acts<br />
• Iyer & Coble (1968): 235U + He ions <strong>of</strong> intermediate energy.<br />
Also optimistic.<br />
• Schall, Heeg, Mutterer & Theobald (1987): spontaneous<br />
fission <strong>of</strong> 252Cf. Triple coincidences with detectors at 120o Pessimistic: yield under 10-8 per binary.<br />
Dorin Poenaru
CONCLUSIONS<br />
• One can obtain <strong>saddle</strong>-<strong>point</strong> <strong>shapes</strong> by solving<br />
an integro-differential equation<br />
• There is no limitation imposed by the space <strong>of</strong><br />
de<strong>for</strong>mation coordinates (no parametrization)<br />
• Fission barriers <strong>for</strong> true ternary and<br />
quaternary fission are lower than <strong>for</strong> aligned<br />
spherical fragments in touch<br />
• Mass-asymmetry in binary fission is explained<br />
by adding shell corrections to the LDM energy<br />
• Experimental confirmation <strong>of</strong> true ternary<br />
fission still needed<br />
Dorin Poenaru