LANL FIESTA: Fission School & Workshop SEPTEMBER 8–12...
Transcript of LANL FIESTA: Fission School & Workshop SEPTEMBER 8–12...
LANL FIESTA: Fission School & WorkshopSEPTEMBER 8–12, 2014, ELDORADO HOTEL,
SANTA FE, NM, USA
Fission Barriers and Fission-Fragment Yieldsin the Region 170 < A ≤ 330
Peter MollerTeoretical Divison, Los Alamos National Laboratory
Collaborators on this and other projects:W. D. Myers, J. Randrup(LBL), H. Sagawa (Aizu), S. Yoshida(Hosei), T. Ichikawa(YITP), A. J. Sierk(LANL), A. Iwamoto (JAEA),S. Aberg (Lund), R. Bengtsson (Lund), S. Gupta (IIT, Ropar),and many experimental groups (e. g. K.-L. Kratz (Mainz), H.Schatz (MSU), A. Andreyev (York), K. Nishio (JAEA) . . . ).
More details about fission (PRC 79 064304, PRC 84 034613,PRC 88 064606), other projects (beta-decay,masses), associ-ated ASCII data files, interactive access to data (type in Z, Aand get specific data, contour maps) and figures are at
http://t2.lanl.gov/nis/molleretal/
Potential Energy of Deformation
We use the macroscopic-microscopic method introduced by
Swiatecki and Strutinsky:Epot(shape) = Ema r(shape) +Emi r(shape) (1)
The macroscopic term is calculated in a liquid-drop type
model (for a specific deformed shape).
The microscopic correction is determined in the following
steps
1. A shape is prescribed
2. A single-particle potential with this shape is generated.
A spin-orbit term is included.
3. The Schrodinger equation is solved for this deformed
potential and single-particle levels and wave-functions
are obtained
4. The shell correction is calculated by use of Strutinsky’s
method.
5. The pairing correction is calculated in the BCS or
Lipkin-Nogami method.
Shape Parameterizations
For small distortions we use multipole expansions, for
example the � parameterization:
r(�; �) = R0(1 + 1Xl=1 lXm=�l�lmY ml )
For large deformations near the outer saddle in the actinide
region or beyond we use the three-quadratic-surface
parameterization:
�(z)2 =8>>>>>>><>>>>>>>:
a12 � a12 12 (z � l1)2 ; l1 � 1 � z � z1a22 � a22 22 (z � l2)2 ; z2 � z � l2 + 2a32 � a32 32 (z � l3)2 ; z1 � z � z2
36 72Kr36
Scale 0.20 (MeV)
0.00 0.10 0.20 0.30 0.40 0
20
40
60
3
4
4
5
5
5A
xial
Asy
mm
etry
γ
Spheroidal Deformation ε2
Discrepancy (Exp. − Calc.)
Calculated
Experimental
σth = 0.559 MeV
FRDM (2012)
0 20 40 60 80 100 120 140 160 Neutron Number N
− 10
0
10
0
10
0
10
Mic
rosc
opic
Ene
rgy
(MeV
)
E(def?)? E(def?)?
In fission, what are the shapes and related energiesinvolved in the transition from a single ground-stateshape to two separated fission fragments?
234U
234U : Asymmetric valley at Q2 = 76
εf1 = − 0.1000 εf2 = 0.2500 MH/ML = 135.7/98.3
234U : Symmetric valley at Q2 = 76
εf1 = 0.1500 εf2 = 0.1000 MH/ML = 119.3/114.7
136I 98Y
117Pd 117Pd
From: Nucl. Phys A469 (1987) 1
0.75 1.00 1.25 1.50 1.750.50
0.75
1.00
Distance between Mass Centers r (Units of R0)
Fra
gmen
t Elo
ngat
ion
σ (U
nits
of R
0)
Family of shapes considered
From: Journ. Phys. G: Nucl. Part. Phys. 20 (1994) 1681
0.75 1.00 1.25 1.50 1.75 2.000.50
0.75
1.00
1.25
Distance between Mass Centers r (Units of R0)
Fra
gmen
t Elo
ngat
ion
σ (U
nits
of R
0)
− 4 − 2 0 2 4 6
Potential energy for 258Fm
E (MeV)
Q2
45 Q2 ~ Elongation (fission direction)
15 εf1
~ Left fragment deformation
εf1
εf2
15 εf2
~ Right fragment deformation
15
⊗
⊗
⊗
⊗d ~ Neck
d
35 αg ~ (M1-M2)/(M1+M2) Mass asymmetry
Five Essential Fission Shape Coordinates
M1 M2
⇒ 5 315 625 grid points − 306 300 unphysical points⇒ 5 009 325 physical grid points
Graphics by P
eter Möller
Fission Barrier and Associated Shapes for 232Th
Separating ridge Symmetric mode Asymmetric mode
0 2 4 6 8 10 Nuclear Deformation (Q2 / b)(1/2)
− 5
0
5
10
Fis
sion
-Bar
rier
Hei
ght (
MeV
)
Experimental
Calculated (FRLDM (2002))
rms = 1.00 MeV
Discrepancy (Exp. − Calc.)
0 20 40 60 80 100 120 140 160 Neutron Number N
− 10
0
10/0
10
20
30
40
50/0
10
20
30
40
50 F
issi
on B
arrie
r H
eigt
(M
eV)
Calculated Fission-Barrier Height
1 2 3 4 5 6 7 8 Bf(Z,N) (MeV)
130 140 150 160 170 180 190 200 210 220 230 Neutron Number N
80
90
100
110
120
130
Pro
ton
Num
ber
Z
Calculated Fission-Barrier Height
1 2 3 4 5 6 7 8
Bf(Z,N) (MeV)
SF Observed (t1/2<30d)
140 150 160 170 180 Neutron Number N
90
100
110
120
Pro
ton
Num
ber
Z
JR: NPF 2010
Brownian shape mo.on
Nuclear deforma5on energy: Edef(i,j,k,l,m)
Q2
45 Q2 ~ Elongation (fission direction)
35 !g ~ (M1-M2)/(M1+M2) Mass asymmetry
15 "f1
~ Left fragment deformation
"f1
"f2
15 "f2
~ Right fragment deformation
15#
#
#
#
d ~ Neck
d
Five Essential Fission Shape Coordinates
M1 M2
$ 5 315 625 grid points % 306 300 unphysical points
$ 5 009 325 physical grid points
Bias poten5al: Vbias(i) = V0 (Q0/Q2)2
Scission: Cri5cal neck radius c0 ≈ 2.5 fm
Level density parameter: aA = A/(8 MeV)
Temperature T: E*‐ Edef = aAT2
Metropolis walk:
P. Möller et al, Nature 409 (2001) 785
i
j
k
l
m
=> V(χ) = Edef + Vbias
Change shape: χ -> χ’ ? V(χ’) < V(χ): move with P = 1
V(χ’) > V(χ): move with P = exp(‐ΔV/T)
χ
N. Metropolis et al, J Chem Phys 26 (1953) 1087
a b
c d
Exp. 233U(n,f) Calc. (6.54 MeV) 234U
30 40 50 60
0
5
10
15
20
25
Exp. 239Pu(n,f) Calc. (6.84 MeV) 240Pu
0
5
10
15
20
25
Yie
ld Y
(Zf)
(%)
Exp. 235U(n,f) Calc. (6.54 MeV) 236U
Exp. 234U(γ,f) Calc. (11.0 MeV) 234U
30 40 50 60 Fragment Charge Number Zf
205At120 E* = 17.41 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
206At121 E* = 18.63 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
204Rn118 E* = 14.10 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
205Rn119 E* = 15.20 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
206Rn120 E* = 16.20 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
207Rn121 E* = 17.14 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
208Rn122 E* = 17.93 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
209Rn123 E* = 18.75 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
206Fr119 E* = 13.98 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
207Fr120 E* = 14.85 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
208Fr121 E* = 15.71 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
209Fr122 E* = 16.60 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
210Fr123 E* = 17.40 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
211Fr124 E* = 18.02 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
212Fr125 E* = 18.72 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
217Fr130 E* = 14.74 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
218Fr131 E* = 14.40 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
209Ra121 E* = 14.54 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
210Ra122 E* = 15.26 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
211Ra123 E* = 16.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
212Ra124 E* = 16.40 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
213Ra125 E* = 17.11 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
214Ra126 E* = 16.94 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
215Ra127 E* = 15.93 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
216Ra128 E* = 15.01 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
217Ra129 E* = 13.80 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
218Ra130 E* = 13.25 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
219Ra131 E* = 12.81 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
212Ac123 E* = 14.65 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
213Ac124 E* = 14.99 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
214Ac125 E* = 15.52 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
215Ac126 E* = 15.36 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
216Ac127 E* = 14.19 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
217Ac128 E* = 13.08 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
218Ac129 E* = 12.40 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
219Ac130 E* = 12.03 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
220Ac131 E* = 11.71 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
221Ac132 E* = 11.34 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
222Ac133 E* = 11.13 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
223Ac134 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
224Ac135 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
225Ac136 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
226Ac137 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
217Th127 E* = 13.02 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
218Th128 E* = 11.99 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
219Th129 E* = 11.17 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
220Th130 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
221Th131 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
222Th132 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
223Th133 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
224Th134 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
225Th135 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
226Th136 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
227Th137 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
228Th138 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
229Th139 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
224Pa133 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
225Pa134 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
226Pa135 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
227Pa136 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
228Pa137 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
229Pa138 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
230Pa139 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
231Pa140 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
232Pa141 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
230U138 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
231U139 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
232U140 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
233U141 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
234U142 E* = 11.00 (MeV)
20 30 40 50 60 70 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
172W E* = 28.05 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
173W E* = 29.14 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
174W E* = 29.86 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
175W E* = 30.88 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
176W E* = 31.60 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
177W E* = 32.75 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
178W E* = 33.56 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
179W E* = 34.62 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
180W E* = 35.32 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
181W E* = 36.21 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
182W E* = 36.42 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
183W E* = 36.92 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
184W E* = 36.70 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
185W E* = 37.13 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
186W E* = 37.37 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
187W E* = 38.18 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
188W E* = 38.74 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
189W E* = 38.24 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
190W E* = 38.29 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
191W E* = 38.47 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
192W E* = 38.98 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
193W E* = 39.22 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
194W E* = 39.91 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
195W E* = 40.78 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
196W E* = 41.51 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
175Hg E* = 11.77 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
176Hg E* = 11.62 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
177Hg E* = 11.41 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
178Hg E* = 11.32 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
179Hg E* = 11.68 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
180Hg E* = 11.81 (MeV)
50 60 70 80 90 100 110 120 130 140 150 Fragment Mass Number Af
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Yie
ld Y
(Af)
(%)
4 8 12 16 20 24 28 32
Fission-Barrier Height (MeV)
90 100 110 120 130 140 150Neutron Number N
70
80
90
Pro
ton
Num
ber
Z
ε2=-0.130 ε4= 0.010 ε6= 0.010
T1/2= 1.74 (s) Folded-Yukawa potential λn=34.88 MeV
λp=32.50 MeV a= 0.80 fm
∆n=0.99 MeV ∆p=0.51 MeV
(L-N) Ti Hg+e+ 81 180 180 80
B f
QEC
0 5 10 15 Excitation Energy (MeV)
0.0
2.5
5.0
Gam
ow-T
elle
r S
tren
gth
Asymmetric Symmetric0.2 0.4 0.6 0.8
Fission-Yield Valley-to-Peak Ratio
90 100 110 120 130 140 150Neutron Number N
70
80
90
Pro
ton
Num
ber
Z
10 15 20 25 30 35 40 45
Fission-Fragment Mass Division (MH-ML)
90 100 110 120 130 140 150Neutron Number N
70
80
90
Pro
ton
Num
ber
Z
Contrasting Fission Potential-Energy Surfaces Hg↔U
Ichikawa et al., PRC 86 024610 (2012)
180Hg 236U
0
2
4
6
8
1012
14
Quadrupole Moment q2
−0.3−0.2−0.1
0.0
0.1
0.2
0.3
MassAsymmetryα g
−50
510
1520
PotentialEnergy(MeV)
180Hg
sample text sample text sample text sample text
0
2
4
6
8
Quadrupole Moment q2
−0.3−0.2−0.1
0.0
0.1
0.2
0.3
MassAsymmetryα g
05
10
PotentialEnergy(MeV)
236U
c
<Eex(a)> = 7.86 (n,f), E*=6.84 234U
30 40 50 60 Fragment Charge Number Zf
0
5
10
15
20
Cha
rge
Yie
ld Y
(Zf)
(%)
b
<Eex(a)> = 11.60
(γ,f), E*=11 234U
0
5
10
15
20
Cha
rge
Yie
ld Y
(Zf)
(%)
a
<Eex(a)> = 11.32 (n,f), E*=6.54 240Pu
0
5
10
15
20
Cha
rge
Yie
ld Y
(Zf)
(%)
(nth,f): Eex = Etot − U(gs) = 6.54 MeV Eex = Etot − U(10) = 9.73 MeV
Eex
240Pu
0 2 4 6 8 10 Nuclear Deformation (Q2 / b)(1/2)
− 5
0
5
10
Ene
rgy
(MeV
)
234U
(nth,f): Eex = Etot − U(gs) = 6.84 MeV
Eex = Etot − U(10) = 6.22 MeV
(γ,f): Eex = Etot − U(gs) = 11 MeV
Eex = Etot − U(10) = 11.4 MeV
Eex
− 5
0
5
10
c
<Eex(a)> = 7.86 (n,f), E*=6.84 234U
30 40 50 60 Fragment Charge Number Zf
0
5
10
15
20
Cha
rge
Yie
ld Y
(Zf)
(%)
b
<Eex(a)> = 11.60
(γ,f), E*=11 234U
0
5
10
15
20
Cha
rge
Yie
ld Y
(Zf)
(%)
a
<Eex(a)> = 11.32 (n,f), E*=6.54 240Pu
0
5
10
15
20
Cha
rge
Yie
ld Y
(Zf)
(%)
exp. th.,no odd
(n,f), E*=6.54
236U
30 40 50 60 Fragment Charge Number Zf
0
5
10
15
20
25 C
harg
e Y
ield
Y(Z
f) (%
)
READ(LU,’(2f10.3)’) r,rw
idiv =(N+1)/2if(N+1 .eq. 2*idiv) r = r + (rw-1 +0.01)*2*1.0
E(I,J,K,L,N) = r
exp. th.,odd
∆ = 1.0 (n,f), E*=6.54
236U
30 40 50 60 Fragment Charge Number Zf
0
5
10
15
20
25 C
harg
e Y
ield
Y(Z
f) (%
)
82
92
124 120
104 106
108
110
86
114
128
Hexadecapole Deformation ε4 0.00 0.00 0.00 0.08 0.08
272110 λp = 34.80, ap = 0.80 fm
− 0.4 − 0.2 0.0 0.2 0.4− 8
− 6
− 4
− 2
0
2
Spheroidal Deformation ε2
Sin
gle-
Pro
ton
Ene
rgy
(MeV
)
exp. th.,1p1qp
∆ = 1.0 (n,f), E*=6.54
236U
30 40 50 60 Fragment Charge Number Zf
0
5
10
15
20
25 C
harg
e Y
ield
Y(Z
f) (%
)
exp. th.,odd
∆ = 1.0 (n,f), E*=6.54
236U
30 40 50 60 Fragment Charge Number Zf
0
5
10
15
20
25 C
harg
e Y
ield
Y(Z
f) (%
)
exp. th.,1p1qp
∆ = 1.0 (n,f), E*=6.84
234U
30 40 50 60 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
exp. th.,odd
∆ = 1.0 (n,f), E*=6.84
234U
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
exp. th., no odd
(n,f), E*=6.84
234U
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
∆ = 1.0
exp. th.,1p2qp
(γ,f), E*=11
234U
30 40 50 60 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
∆ = 1.0
exp. th.,odd
(γ,f), E*=11
234U
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
exp. th., no odd
(γ,f), E*=11
234U
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
exp. th.,1p1qp
∆ = 1.0 (n,f), E*=6.54
236U
30 40 50 60 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
exp. th.,odd
∆ = 1.0 (n,f), E*=6.54
236U
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
exp. th.,no odd
∆ = 1.0 (n,f), E*=6.54
236U
0
5
10
15
20
25 C
harg
e Y
ield
Y(Z
f) (%
)
exp. th.,1p1qp
∆ = 1.0 (n,f), E*=6.54
240Pu
30 40 50 60 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
exp. th.,odd
∆ = 1.0 (n,f), E*=6.54
240Pu
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
exp. th.,no odd
(n,f), E*=6.54
240Pu
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
a b
c d
Exp. 233U(n,f) Calc. (6.54 MeV) 234U
30 40 50 60
0
5
10
15
20
25
Exp. 239Pu(n,f) Calc. (6.84 MeV) 240Pu
0
5
10
15
20
25
Yie
ld Y
(Zf)
(%)
Exp. 235U(n,f) Calc. (6.54 MeV) 236U
Exp. 234U(γ,f) Calc. (11.0 MeV) 234U
30 40 50 60 Fragment Charge Number Zf
exp. th.,1p1qp
∆ = 1.0 (n,f), E*=6.84
234U
30 40 50 60 Fragment Charge Number Zf
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
exp. th.,1p1qp
∆ = 1.0 (n,f), E*=6.54
240Pu
0
5
10
15
20
25
Cha
rge
Yie
ld Y
(Zf)
(%)
exp. th.,1p1qp
∆ = 1.0 (n,f), E*=6.54
236U
∆ = 1.0
exp. th.,1p2qp
(γ,f), E*=11
234U
30 40 50 60 Fragment Charge Number Zf