Theoretical studies of the fission product yields
.
V.A. Rubchenya
Department of Physics, University of Jyväskylä, Finland;
V.G. Khlopin Radium Institute, S-Petersburg, Russia;
Department of Nuclear Physics, S-Petersburg State University, Russia
ERINDA Meeting, January 9-11, 2013, Jyväskylä
 Introduction
 Model
1. Pre-compound processes
2. Shells and fission modes
3. Nuclear friction
3. Isovector quantal fluctuations
 Results
1. Themal neutron induced fission.
2. Neutron and proton induced fission at intermediate energies.
242
Cm sp. decay
10-1
4
He(*10-4)
10-2
10
10-4
SAF?
Be
C
-5
10
10-6
-4
Pu(*10 )
Ternary Fission
-3
yield / fission
238
Binary fission
O
Ne Mg
10-7
10-8
Si
10-9
cluster decay (?)
-10
10
0
20
40
60
80
100
120
140
160
180
fragment mass number
200
220
240
Fission fr.
U FISSION TARGET
EURopean Isotope-Separation-On-Line RIB facility
SF, p n d …
5·10-22
MODEL
Pre-compound stage
Pre-saddle evaporation
10-19
Saddle point
Descent from saddle
1.1·10-19
Scission
Post-scission evaporation
10-8
Fission products
Y(A,Z), YLCP(A,Z), W(ΘF,Ekin), Mn(En,Θn), Mγ(Eγ,Θγ)
Time/ s
Fission product formation cross section in reaction
Ap
Z p ( E p ) + At Z t − > x + Ac Z c −> Z L AL + Z H AH + xx
σ FP ( A, Z , At , Zt , Ap , Z p , E p ) =
∑ ∫ dEc ⋅ Yind ( A, Z , Ac , Z c , Ec )
dσ c ( Ac , Z c , Ec , At , Zt , Ap , Z p , E p )
dEc
Ac ,Z c
Independent yield:
Yind ( A, Z , Ac , Z c , Ec ) = ∑ Ypre ( A + n, Ac , Z c , Ec ) ⋅ PA+n ( Z ) ⋅ Pn ( A + n, Z )
n
Ypre ( A + n, Ac , Z c , Ec )
PA+n (Z )
Pn ( A + n, Z )
is a pre-neutron emission yield
of the fragment isobaric chain
is a charge distribution of the primary isobaric chain
is a prompt neutron multiplicity distribution
The main dynamical effects
1. Pre-compound particle emission: Mnpre-eq, Mppre-eq, Epre-eqelapsed
2. Role of the nuclear friction in the fission:
-
delay time for formation of fission degree of freedom
-
modification of the fission width
-
overdamped collective motion on the descent from saddle
-
Mnpre-sc, Mppre-sc, Eelapsedpre-sc
3. Charge polarization during the descent from sadlle to scission: charge
distribution for isobaric chains: Y(Z/A)
4. Competion between different fission modes as function of composition
and exitation energy of compound nuclei: Y(ACN, ZCN, E*CN)
5. Distribution of excitation energy between fragments: Mn(A, Z, E*scp)
6. Shell structure for very deformed nuclei: shell corrections, fission
barriers, mass parameters, fission modes, level density
Pre-compound stage
We use the two-component exciton model for our objective:
an adequate description of the initial excitation energy distribution and composition
of the compound nuclei in the neutron and proton induced fission at the incident
energy from 10 to 100 MeV.
( pπ , hπ , pν , hν )
The lifetime of exciton state
τ ( pπ , hπ , pν , hν ) = [λπ + λν + λπν + λνπ + λπ + λν
+
+
0
0
em
]
em −1
The proton pre-compound single spectra from given exciton state
dWπs ( pπ , hπ , pν , hν , Eπ ) 2 µπ ρ ( pπ − 1, hπ , pν , hν , Ex − Bπ − Eπ ) inv
= 2 3
σ π ( Eπ ) Eπ
dEπ
π 
ρ ( pπ , hπ , pν , hν , Ex )
The neutron pre-compound single spectra from given exciton state
dWν s ( pπ , hπ , pν , hν , Eν ) 2 µν ρ ( pπ , hπ , pν − 1, hν , Ex − Bν − Eν ) inv
= 2 3
σ ν ( Eν ) Eν
dEν
π 
ρ ( pπ , hπ , pν , hν , Ex )
The two-component exciton model was used for our objectives: an adequate
description of the initial excitation energy distribution and composition of the
compound nuclei in the neutron and proton induced fission. For description of
exciton evolution process the following transitions are taken into account:
- proton particle-hole pair creation;
- neutron particle-hole pair creation;
- conversion of a proton particle-hole pair into a neutron particle-hole pair;
- conversion of a neutron particle-hole pair into a proton particle-hole pair;
- the proton emission;
- the neutron emission;
-after particle emission the exciton evolution process may develop further
until reaching the criteria for transition to the compound stage of the reaction.
The exciton transition cascade is ruptured at reaching one of conditions:
1. exciton number reaches limited value n ≥ nmax;
2. total life time of the exciton stage exceeds limiting value Texc ≥ Tmax which
corresponds to statistical width decay;
3. the number of emitted particles exceeds the limited value
Mn ≥ Mnmax or Mp ≥ Mp max .
Nuclear friction at pre-sadlle stage and transition through fission barrier.
Modification of the statistical Bohr-Wheeler fission width


2 1/ 2


t
(
)
(
)
1
Γf (t ) = Γf 1 − exp −
+
−
γ
γ

τ d 


BW
LDM
B
10

f
τ d = ln
 T
ω gs
and
ωb

γ
 1

+


 2γωgs ω gs 
β
γ=
2ωb
are collective frequences at equilibrium and saddle shapes
β denotes the reduced dissipation coefficient
*
*
<
0
,
at
E
E

th
,
β (E* ) = 
2
*
*
β 0 (d 0 + d1T + d 2T ), at E ≥ Eth
Competition between particle evaporation and fission channels defines the fission chances.
Saddle-to-scission descent stage
• saddle-to-scission
time is altered by the nuclear dissipation
((
) )
τ ssc = τ ssc (γ = 0) 1 + γ 2 + γ .
• Saddle
and bifurcation points and valleys on the potential-energy
surface of fissioning nucleus determine the properties of fission modes
Smoothed primary mass distribution is formed by 5 fission modes:
~
Y pre ( A) = C SY YSY ( A) + C SI YSI ( A) + C SII YSII ( A) +
C SA1YSA1 ( A) + C SA2YSA2 ( A)
Each asymmetric component is influenced by corresponding nuclear shells:
YSI is defined by 132Sn shells : Z = 50 , N = 82
YSII is defined by deformed shelll : N = 86 – 90
Superasymmetric YSA1 YSA2 modes are defined by splitted
78
Ni shells : Z = 28 , N = 50
Fission mode weights as function of the excitation energy
for 239U compound nucleus
102
239
CSY
101
CSII
Cm, %
100
U
CSI
10-1
CSAI
10-2
10-3
CSAII
10-4
10-5
0
20
40
60
80
excitation energy, MeV
100
Primary isobaric charge distribution parametriztion
~
Ppre ( Z / A) = Ppre ( Z / A) Foe ( Z )
~
Ppre ( Z / A) =
 ( Z − Z ( A)) 2 
1
exp−

2
2σ Z ( A) 
σ Z ( A) 2π

Foe ( Z ) ∝ exp((Π ZH + Π ZL )δ Z ( Ac , Z c , Ec ))
δ Z ( Ac , Z c , Ec ) =
δ Z ( Ac , Z c ,0)
, Π H ( L ) = + 1

Z
1 + exp(( Ec − 10) / 2)
−1

if
if
1 − 0.1( Z c2 / Ac − 35.22) 2 , Z c2 / Ac > 35.22

δ Z ( Ac , Z c ,0) = 1,
Z c2 / Ac ≤ 35.22
0,
δ Z < 0.

Z
Z
is even
is odd
Frozen quantal fluctuations in the charge equilibration mode
Time evolution of isobaric width in the harmonic approximation
V ( Z ) = V ( Z ) + 12 C ZZ ( Z − Z ) 2
0,6
σ2(Z/A)
freeze-out point
0,4
adiabatic
solution
0,2
scission time
drneck/dt =2 fm/(10-21s)
0,0
0
25
50
75
time, 10-23 s
100
125
M ZZ
Ac2 Lneck + 2rneck
16 3
= r0 m
2
Zc Nc
rneck
9
||

1
1
E
GDR
σ Z2 ( A) =
=
2 M ZZ CZZ 2 CZZ
drneck
≈ 2 fm −21
10 s
dt
Approximation of the freeze-out point configuration
Δtip
p
p
n
n
AL, ZL, {εL}
AH, ZH, {εH}
L
H
V pot = VCoul + Vnucl + Edef
({ε L }) + Edef
({ε H })
LDM
E def = E def
+ δU + δE pair
Vpot for 248Cm -> 78Ni +170Er
0,7
0,6
0,4
-9,000
-6,000
-3,000
0,3
0
3,000
0,2
6,000
9,000
78
Ni deformation
0,5
12,00
0,1
15,00
18,00
0,0
0,0
21,00
0,1
0,2
0,3
0,4
0,5
0,6
heavy fragment deformation
0,7
fragment charge number
70
3.500 -- 4.000
3.000 -- 3.500
2.500 -- 3.000
2.000 -- 2.500
1.500 -- 2.000
1.000 -- 1.500
0.5000 -- 1.000
0 -- 0.5000
-0.5000 -- 0
-1.000 -- -0.5000
-1.500 -- -1.000
-2.000 -- -1.500
-2.500 -- -2.000
-3.000 -- -2.500
-3.500 -- -3.000
-4.000 -- -3.500
-4.500 -- -4.000
-5.000 -- -4.500
-5.500 -- -5.000
-6.000 -- -5.500
-6.500 -- -6.000
-7.000 -- -6.500
-7.500 -- -7.000
-8.000 -- -7.500
shell corrections at scission point
60
239
Np
50
40
30
20
60
80
100
120
140
fragment mass number
160
180
10
~ min
∂ V pot ( A; Z )
CZZ =
∂2Z
AL=70
2
236
U
Am
252
Cf
243
Vsc.p.min (MeV)
0
-10
-20
24
26
28
ZL
30
32
Z =Z
V ( Z ) = V ( Z ) + 12 CZZ ( Z − Z ) 2
1
2
M ZZ


Z −Z
1
 H n 
Ψn = 
n
 σ ZZ
 σ ZZ π 2 n! 
E n = (n + 2 )ω Z ,
1
2
~
P ( Z / A) ∝ Ψ0 =
σ Z2 ( A) =
1
σ ZZ
16 3
Ac2 L + 2rneck
= r0 m
2
9
Z c N c rneck
2



−
Z
Z
(
)
 exp−
2 
2σ ZZ 


CZZ
ωZ = 
M ZZ
2


−
(
Z
Z
)
exp−
2 
σ
π

ZZ 

1
2 M ZZ CZZ ( A)
239
0,6
Np
<Z> - ZUCD
0,4
0,2
scission point model:
with shells
LDM
fit of LDM function
0,0
(d2Vpot/dZ2)-1<Z> (MeV-1)
-0,2
0,40
0,35
0,30
0,25
0
10
20
30
40
ACN/2 - AL
50
60
70
The approximation of the liquid drop charge distribution parameters
Deviation from uniform distribution:
2
2
2
 A



Zc
LDM
−4 Z c
 c − A  +

( A) = 0.1297 − 0.0069
+ 1.001 × 10 
δZ

Ac
Ac   2





2
3
2
2




Z
Z
A
 × 10 −7  c − A 
 − 144.63 + 6.8186 c
− 0.08558 c

Ac
Ac  

 2




The reverse stiffness parameter:
2
2
2




Z
Z
1 LDM =  − 1.161 + 0.07409 c
+
− 0.000896862 c

Ac
Ac  

C ZZ



2
2
2
2






Z
 × 10 −5  Ac − A 
 − 7.102 + 0.27956 Z c
− 0.00336 c


Ac 
Ac  
2







Excitation energy dependence
δZ ( A, Ec ) = δZ
CZZ = C
LDM
ZZ
LDM
+ (δZ ( A,0) − δZ
+ (CZZ ( A,0) − C
LDM
ZZ
LDM
1 − exp(− βEc )
( A) )
β Ec
1 − exp(− βEc )
( A) )
β Ec
Neutron emission:
•Pre-equilibrium stage
•Pre-barrier evaporation from compound nuclei
•Emission at the descent form barrier to scission
•Post-scission evaporation from the primary fission fragments
Results
Computer code FIPRODY (FIssion PRODuct Yields) based on the theoretical
model was developed to calculate the characteristics of the light particles
induced fission; photo fission and spontaneous fission. The model parameters
were determined by comparison between calculated and experimental
characteristics of the proton and neutron induced fission
of the heavy actinides from Th to Cf.
100
242
Pu(p, f), Ep=55 MeV
-1
10
d2Mn / dΩndEn, ster-1 MeV-1
10-2
10-3
10-4
10-5
Θn=107o
10-6
exp. data
pre-eq.
pre-sc.
post-sc.
total
-1
10
10-2
10-3
10-4
10-5
10-6
Θn=18o
0
10
20
30
En, MeV
40
50
65
235
U (nth, f)
fragment charge number
60
55
Z=50
50
45
40
A=78
A=132
35
30
25
60
Z=28
80
100
120
140
fragment mass number
160
1.000E1
3.266E0
1.067E0
3.485E-1
1.138E-1
3.718E-2
1.214E-2
3.966E-3
1.296E-3
4.232E-4
1.382E-4
4.515E-5
1.475E-5
4.817E-6
1.573E-6
5.139E-7
1.678E-7
5.482E-8
1.791E-8
5.849E-9
1.910E-9
6.240E-10
2.038E-10
6.658E-11
2.175E-11
7.103E-12
2.320E-12
101
100
10-1
yield (%)
10-2
10-3
235
U(nth, f)
theory
LOHENGRIN
ENDF/B6
10-4
10-5
10-6
10-7
10-8
60
80
100
120
140
product mass (amu)
160
180
Comparison theoretical calculations (red points) with Lohenrgin’s results for
fission product isobaric chains A = 80 – 106
in the thermal neutron induced fission.
235
U(nth, f)
5
10
4
10
103
A=81
A=80
A=82
2
10
1
10
100
104
3
10
A=83
A=84
A=85
2
10
cross section, mb
101
100
4
10
103
A=88
A=86
A=87
A=89
A=90
A=91
A=92
A=93
A=94
102
101
100
104
103
2
10
1
10
0
10
104
103
2
10
101
100
30 32 34 36 38 40 42 30 32 34 36 38 40 42 30 32 34 36 38 40 42 44
fission product charge number
235
U(nth, f)
105
104
103
A=95
A=96
A=97
2
10
101
100
cross section, mb
104
103
A=98
A=99
A=100
A=101
A=102
A=103
A=104
A=105
102
101
100
104
103
102
101
0
10
104
103
A=106
102
101
100
30 32 34 36 38 40 42 30 32 34 36 38 40 42
30 32 34 36 38 40 42 44
fission product charge number
Comparison between theortical calculations and LOHENGRIN data ( PRC 70, 2004)
245
Cm(nth, f)
-3
10
Z29
Z28
-4
10
243
Cm(nth, f)
10-3
10-4
10-5
10
10-6
10-6
Z=28
Z=29
Z=30
Z=31
Z=32
Z=33
Z=34
Z=35
-5
Z31
Z30
-2
10
10-2
-3
10
-3
10
10-5
10-4
YIELD ( % )
10-6
Z33
-1
10
Z32
-2
10
10-3
-4
10
-5
10
Z35
Z34
0
10
YIELD ( % )
10-4
10-5
10-1
10-2
10-3
-4
10
-1
10-1
10-2
10-2
10-3
10-3
10
-4
10
65
70
75
80
85
90
95
Z37
100
Z36
100
10-4
Z=36
100
10-1
Z=37
-1
10
-2
10
10-2
-3
10
10-3
-4
10
65
70
75
80
85
90
95
70
75
80
85
PRODUCT MASS NUMBER
90
95
100
10-4
65
70
75
80
85
90
95
70
75
80
85
PRODUCT MASS NUMBER
90
95 100
U(p, f) mass distributions
0.1
0.1
0.01
0.01
yield, per fission
yield, per fission
238
238
U(p(20 MeV), f)
theory
exp.-PSAC/1998
1E-3
1E-4
1E-5
80
100
120
140
0.1
100
120
140
160
0.1
0.01
yield, per fission
yield, per fission
80
pre-neutron fragment mass number
pre-neutron fragment mass number
238
U(p(50 MeV), f)
theory
exp.-PSAC/1998
1E-3
1E-4
1E-5
U(p(35 MeV), f)
theory
exp.-PSAC/1998
1E-4
1E-5
160
238
1E-3
80
100
120
140
pre-neutron fragment mass number
160
0.01
238
U(p(20 MeV), f)
theory
exp.-PSAC/1998
1E-3
1E-4
1E-5
80
100
120
140
pre-neutron fragment mass number
160
Calculated fission product yelds
102
Sr
Pd
Sn
Xe
238
U(n, f)
En=50 MeV
1
10
Cross Section (mb)
100
10-1
Ni
10-2
-3
Tb
Fe
10
10-4
10-5
10-6
10-7
40
50
60
70
80
Neutron Number
90
100
Calculated fission product yields
238
60
U(n, f)
En=50 MeV
mb
50
40
mb
30
1E2
1E1
N
Z
60 E = 5 MeV
n
1
0.1
0.01
50
1E-3
1E-4
1E-5
40
1E-6
1E-7
30
1E-8
40
50
60
70
80
90
N
http://www.gsi.de/eurisol-t11/database
100
65
232
Th(p, f), Ep=25 MeV
60 fission product cross sections (mb)
1.000E-10
3.981E-10
1.585E-9
6.310E-9
2.512E-8
1.000E-7
3.981E-7
1.585E-6
6.310E-6
2.512E-5
1.000E-4
3.981E-4
0.001585
0.006310
0.02512
0.1000
0.3981
1.585
6.310
25.12
100.0
55
Z
50
132
Sn
45
40
35
78
Ni
30
40
50
60
70
N
80
90
100
Comparison with mass spectrometer experimental data in 238U(p, f) for Rb
102
102
Rb
Rb
101
100
cross section, mb
cross section, mb
101
238
U(p, f), Ep/MeV
50 Tracy
50 theory
10-1
10-2
85
90
95
100
238
U(p, f), Ep/MeV
60 Tracy
60 theory
10-1
10-2
85
90
mass number
mass number
102
102
Rb
Rb
101
100
238
10-2
85
U(p, f), Ep/MeV
70 Belyaev
70 theory
90
mass number
95
cross section, mb
cross section, mb
101
10-1
95
100
238
10-1
10-2
85
U(p, f), Ep/MeV
90 Belyaev
90 theory
90
mass number
95
Comparison with mass spectrometer experimental data in 238U(p, f) for Cs
102
102
Cs
Cs
101
100
238
U(p,f), Ep/MeV
40 Tracy
40 theory
10-1
10-2
cross section, mb
cross section, mb
101
130
135
140
100
238
U(p,f), Ep/MeV
50 Tracy
50 theory
10-1
10-2
145
130
135
mass number
2
10
10
Cs
Cs
101
100
cross section, mb
101
cross section, mb
145
mass number
2
238
U(p,f), Ep/MeV
60 Tracy
60 theory
10-1
10-2
140
130
135
140
mass number
145
100
238
U(p,f), Ep/MeV
90 Belyaev
90 theory
10-1
10-2
130
135
140
mass number
145
Comparison with IGISOL_TRAP data in 238U(p, f) at Ep=25 MeV
102
102
cross section, mb
100
cross section, mb
101
Zr
101
10-1
10-2
10-3
10-4
10-5
10-1
10-2
10-3
10-4
10-5
10-6
10-6
10-7
10-7
10-8
10-8
90
95
100
105
Mo
100
110
95
100
102
115
102
1
10
101
Sn
0
10
cross section, mb
cross section, mb
110
mass number
mass number
10-1
-2
10
-3
10
-4
10
-5
10
-6
10
Cs
100
10-1
10-2
10-3
10-4
10-5
10-6
-7
10
10-7
-8
10
105
120
125
130
mass number
135
10-8
130
135
140
mass number
145
150
Sn - neutron rich isotopes yields
132
2
10
101
cross section, mb
100
10-1
Sn50
Sn in 238U(n(γ), f)
En=10 MeV
Eγ=15 MeV
10-2
10-3
10-4
10-5
10-6
10-7
10-8
115
120
125
130
A
135
140
Computer code FIPRODY can be used for calculation folloiwing output in the
proton and neutron induced reactions at energy up to 100 MeV:
• Evaporation residues cross sections.
• Pre-compound neutron and proton emission spectra.
• Pre-compound proton and neutron multiplicity.
• Pre-scission light charged particles (p, d, α) emission multiplicities.
• Pre-scission neutron spectrum.
• Post-scission light charged particle multiplicities.
• Post-scission neutron spectrum.
• Fragment kinetic energies.
• Pre-neutron emission fragment mass yields.
• Fission product mass yields.
• Fission product yields for isobaric chains.
• Fission product yields for elemental chains.
Thank you for your attention!