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Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 1 / 40
Center for Computational Sciences
December 10-12, 2018
CCS workshop
Impact of pear-shaped fission fragmentson mass-asymmetric fission
Guillaume SCAMPS
Collaboration : C. Simenel
Motivation : understand the shell effects on fission
Empirical behavior of actinide nuclei
J.P. Unik, J.E. Gindler, J.E.Glendenin et al. : Proc. Phys.and Chem. of Fission IAEAVienna , Vol II, 20 (1974)
Data from D. A. Brown et al., Endf/b-viii.0, Nucl. DataSheets 148, 1 (2018), (spontaneous and thermal
neutron-capture).
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 2 / 40
Systematic comparison for actinide
Empirical behavior of actinide nuclei
K.-H. Schmidt et al. Nuclear Physics A665 (2000)
MotivationHow can we understand this behavior ? Interplay between structure and reaction ?
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 3 / 40
Mean-field theory with pairing
TDHFIndependent particleInitialisation : hMF |φi〉 = εi |φi〉Evolution :i~ dρ
dt = [hMF , ρ]
TDHFBPairing correlationQuasi-particles : |ωα〉 =
(UαVα
)Evolution :i~ d|ωα〉
dt =( h ∆−∆∗ −h∗
)|ωα〉
TDHF+BCSBased on TDHFB with the approximation : ∆ij = δij ∆i
Evolution : i~ dφidt = (hMF − εi )φi
i~ dnidt = ∆∗i κi −∆iκ
∗i
i~ dκidt = κi (εi − εi ) + ∆i (2ni − 1)
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 4 / 40
Why does we need pairing ?
Fission barrier : 258Fm
-1930
-1920
-1910
-1900
E[M
eV]
5 6 7 8 9 10 11 12 13 14 15
R[fm]
TDHF
TDHF+BCS
G. Scamps, C. Simenel, D. Lacroix, PRC 92, 011602(R) (2015).
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 5 / 40
Why does we need pairing ?
Fission barrier : 258Fm
-1930
-1920
-1910
-1900
E[M
eV]
5 6 7 8 9 10 11 12 13 14 15
R[fm]
TDHF
TDHF+BCS
G. Scamps, C. Simenel, D. Lacroix, PRC 92, 011602(R) (2015).
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 5 / 40
Why does we need pairing ?
Fission barrier : 258Fm
-1930
-1920
-1910
-1900
E[M
eV]
5 6 7 8 9 10 11 12 13 14 15
R[fm]
TDHF
TDHF+BCS
G. Scamps, C. Simenel, D. Lacroix, PRC 92, 011602(R) (2015).
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 5 / 40
Influence of pairing on fission process
0
50
100
150
200
250
300
350
0 10 20 30 40 50 600.0
0.2
0.4
0.6
0.8
1.0
-15 -10 -5 0 5
E
t[10−22 s]
ni
ǫi
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 6 / 40
Influence of pairing on fission process
0
50
100
150
200
250
300
350
0 10 20 30 40 50 600.0
0.2
0.4
0.6
0.8
1.0
-15 -10 -5 0 5
E
t[10−22 s]
ni
ǫi
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 6 / 40
Influence of pairing on fission process
0
50
100
150
200
250
300
350
0 10 20 30 40 50 600.0
0.2
0.4
0.6
0.8
1.0
-15 -10 -5 0 5
E
t[10−22 s]
ni
ǫi
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 6 / 40
Influence of pairing on fission process
0
50
100
150
200
250
300
350
0 10 20 30 40 50 600.0
0.2
0.4
0.6
0.8
1.0
-15 -10 -5 0 5
E
t[10−22 s]
ni
ǫi
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 6 / 40
Influence of pairing on fission process
0
50
100
150
200
250
300
350
0 10 20 30 40 50 600.0
0.2
0.4
0.6
0.8
1.0
-15 -10 -5 0 5
E
t[10−22 s]
ni
ǫi
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 6 / 40
New systematic study
First : CHF+BCSExample : 240Pu
-1835
-1830
-1825
-1820
-1815
-1810
-1805
-1800
-1795
-1790
E[M
eV]
0 20 40 60 80 100 120Q20 [b]
SymmetricAsymmetric Second : TDHF+BCS
Details of the calculationSkyrme functionnal Sly4dSurface pairing interaction∆x = 0.8 fmLattice : Lx × Ly × 2Lz = 40 × 19.2 × 19.2 fm3
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 7 / 40
New systematic study
First : CHF+BCSExample : 240Pu
-1835
-1830
-1825
-1820
-1815
-1810
-1805
-1800
-1795
-1790
E[M
eV]
0 20 40 60 80 100 120Q20 [b]
SymmetricAsymmetric
Second : TDHF+BCS
Details of the calculationSkyrme functionnal Sly4dSurface pairing interaction∆x = 0.8 fmLattice : Lx × Ly × 2Lz = 40 × 19.2 × 19.2 fm3
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 7 / 40
New systematic study
First : CHF+BCSExample : 240Pu
-1835
-1830
-1825
-1820
-1815
-1810
-1805
-1800
-1795
-1790
E[M
eV]
0 20 40 60 80 100 120Q20 [b]
SymmetricAsymmetric
Second : TDHF+BCS
Details of the calculationSkyrme functionnal Sly4dSurface pairing interaction∆x = 0.8 fmLattice : Lx × Ly × 2Lz = 40 × 19.2 × 19.2 fm3
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 7 / 40
TDHF+BCS systematics results
TDHF+BCS
50
52
54
56
ZH
230Th 234U 236U 240Pu 246Cm 250Cf
A
82
84
86
88
90
NH
Comparison with experimental data
Yie
ld(a
rbitra
ryunits)
230Th
234U
236U
240Pu
246Cm
250Cf
30 35 40 45 50 55 60Proton number Z
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 8 / 40
TDHF+BCS systematics results
TDHF+BCS
50
52
54
56
ZH
230Th 234U 236U 240Pu 246Cm 250Cf
A
82
84
86
88
90
NH
Comparison with experimental data
Yie
ld(a
rbitra
ryunits)
230Th
234U
236U
240Pu
246Cm
250Cf
30 35 40 45 50 55 60Proton number Z
Conclusion :The TDHF+BCS calculation reproduces well the Z=54 behavior. But why ?
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 8 / 40
Nucleon localization function
Fermion localization function
Cqσ(r) =
[1 +(τqσρqσ − 1
4 |∇ρqσ |2 − j2qσρqστTF
qσ
)2]−1
A. D. Becke and K. E. Edgecombe, J. Chem. Phys.92, 5397 (1990).
Physical meaning :C ∈ [0 : 1]Cqσ(r) = 1 Probability to find another particle withthe same q and σ very low.Cqσ(r) = 0.5 Limit of uniform-density Fermi gas.
Mask function :
→Cqσ(r)ρqσ
ρmaxqσ
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 9 / 40
Nucleon localization function
Fermion localization function
Cqσ(r) =
[1 +(τqσρqσ − 1
4 |∇ρqσ |2 − j2qσρqστTF
qσ
)2]−1
A. D. Becke and K. E. Edgecombe, J. Chem. Phys.92, 5397 (1990).
Physical meaning :C ∈ [0 : 1]Cqσ(r) = 1 Probability to find another particle withthe same q and σ very low.Cqσ(r) = 0.5 Limit of uniform-density Fermi gas.
Mask function :
→Cqσ(r)ρqσ
ρmaxqσ
Example :
P. Jerabek, B. Schuetrumpf, P.Schwerdtfeger, and W. Nazarewicz,Phys. Rev. Lett. 120, 053001 (2018).
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 9 / 40
Nucleon localization function
Fermion localization function
Cqσ(r) =
[1 +(τqσρqσ − 1
4 |∇ρqσ |2 − j2qσρqστTF
qσ
)2]−1
A. D. Becke and K. E. Edgecombe, J. Chem. Phys.92, 5397 (1990).
Physical meaning :C ∈ [0 : 1]Cqσ(r) = 1 Probability to find another particle withthe same q and σ very low.Cqσ(r) = 0.5 Limit of uniform-density Fermi gas.
Mask function :
→Cqσ(r)ρqσ
ρmaxqσ
Schematic system
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 9 / 40
Example of 240Pu
240Pu
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 10 / 40
Example of 240Pu
240Pu
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 10 / 40
Example of 240Pu
240Pu
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 10 / 40
Example of 240Pu
240Pu
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 10 / 40
Example of 240Pu
240Pu
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 10 / 40
Example of 240Pu
240Pu
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 10 / 40
Example of 240Pu
HypothesisThe octupole shell effects are important in the fission fragment
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 11 / 40
Other systems
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 12 / 40
Other systems
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 12 / 40
Other systems
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 12 / 40
Other systems
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 12 / 40
Why the fragments have octupole deformation ?
Similar effect on fusion reaction
40Ca+40Ca, E3− = 3.7 MeV56Ni+56Ni, E3− = 7.5 MeV
C. Simenel, M. Dasgupta, D. J. Hinde, and E. Williams, Phys. Rev. C 88, 064604 (2013).
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 13 / 40
Why the fragments have octupole deformation ?
Similar effect on fusion reaction
40Ca+40Ca, E3− = 3.7 MeV56Ni+56Ni, E3− = 7.5 MeV
C. Simenel, M. Dasgupta, D. J. Hinde, and E. Williams, Phys. Rev. C 88, 064604 (2013).
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 13 / 40
Octupole deformation systematics
Skyrme Skm*.
S. Ebata, and T. Nakatsukasa, Phys. Scr.92 (2017) 064005
Gogny D1S
LM Robledo - J. phys. G : Nucl. andParticle Physics, 2015
Results from systematic calculationIn both calculations, the region Z ' 54 , N ' 88 is favorable for octupole deformation .
Experimental results144Ba is found to be octupole in its groud state. Burcher et al. PRL 116 (2016).
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 14 / 40
Constraint HF+BCS octupole deformation with Sly4d
Result from constraint calculation of the heavy fragmentThe gain in energy due to the octupole softness drives the fission to the Z'54
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 15 / 40
Structure, 144Ba, Z=56, N=88
Q2 - Q3 potential energy surface
Single particle energy
50
56
82 84
88
Q20 [b]
Q20 [b]
Q30 [b3/2]
Q30 [b3/2]
-10
-9
-8
-7
-6
-5
-4
-3
-2
ǫ n[M
eV]
0.0 0.5 1.0 1.5 2.0 2.5
0 0
3 4 5 6
0 2 3 4 5
-17
-16
-15-14
-13-12
-11
-10
-9
-8-7
ǫ p[M
eV]
0.0 0.5 1.0 1.5 2.0 2.5
0 0
3 4 5 6
0 2 3 4 5
52
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 16 / 40
Structure
Single particle energies
50
56
82 84
88
Q20 [b]
Q20 [b]
Q30 [b3/2]
Q30 [b3/2]
-10
-9
-8
-7
-6
-5
-4
-3
-2
ǫ n[M
eV]
0.0 0.5 1.0 1.5 2.0 2.5
0 0
3 4 5 6
0 2 3 4 5
-17
-16
-15-14
-13-12
-11
-10
-9
-8-7
ǫ p[M
eV]
0.0 0.5 1.0 1.5 2.0 2.5
0 0
3 4 5 6
0 2 3 4 5
52
Experimental results
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 17 / 40
Deformation energy at the scission. Simple scission point model
E(N,Z) = Eβ3=0.35(N,Z) + Eβ2=0.8(Ntot − N,Ztot − Z) + e2 Z(Ztot − Z)Dsc
(1)
With Dsc=17 fm. On the map, E(N,Z)− Emin is shown.For 240Pu, Ntot=146 and Ztot=94
Octupole deformation
30
35
40
45
50
55
60
65
Z
60 65 70 75 80 85 90 95 100 105
N
Spherical heavy fragment
30
35
40
45
50
55
60
65
Z
60 65 70 75 80 85 90 95 100 105
N
The energies have been calculated with the CHF+BCS theory Sly4d
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 18 / 40
Deformation energy at the scission. Simple scission point model
E(N,Z) = Eβ3=0.35(N,Z) + Eβ2=0.8(Ntot − N,Ztot − Z) + e2 Z(Ztot − Z)Dsc
(1)
With Dsc=17 fm. On the map, E(N,Z)− Emin is shown.For 240Pu, Ntot=146 and Ztot=94
Octupole deformation
30
35
40
45
50
55
60
65
Z
60 65 70 75 80 85 90 95 100 105
N
Experimental data
0.1
1
10
Yield
45
50
55
60
65
Z
70 75 80 85 90 95 100N
240Pu
The energies have been calculated with the CHF+BCS theory Sly4d
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 18 / 40
Identification method with the nucleon localisation function
This method assumes that the pre-fragments have reflexion symmetry.J. Sadhukhan, C. Zhang, W. Nazarewicz, and N. Schunck, PRC 96, 061301(R) (2017).
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 19 / 40
Identification with density
Technique of : M. Warda, A. Staszczak, and W. Nazarewicz, PRC 86, 024601 (2012).
Green contour line : density of a 144Ba with a constraint β3=0.42Red contour line : density of a fissioning 258Fm (asymmetric mode)
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 20 / 40
Identification with nucleon localisation function
Top : NLF of a 144Ba with a constraint β3=0.42Bottom : NLF of a fissioning 258Fm (asymmetric
mode)
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 21 / 40
Identification with nucleon localisation function
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 22 / 40
Identification method with octupole degree of freedom
Identification of the fragments as a function of time for the fission of 258Fm
All of the systems are identified as 144Ba with different β3 values (resp. 0.14, 0.39, 0.39 and 0.42)
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 23 / 40
Identification method with octupole degree of freedom
Identification of the fragments at the scission for the different elements.
All systems are identified as 144Ba with different β3 values (resp. 0.28, 0.28, 0.27 and 0.44)
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 24 / 40
Conclusion
MechanismThe Nucleus-Nucleus interaction at the scission configuration favors the octupoleshapesShell structure favors octupole shape in the region Z ' 52-56, N ' 84-88Actinide fission fragments are driven in the region Z ' 54, N ' 86
G. Scamps, C. Simenel, arXiv :1804.03337 (2018).
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 25 / 40
Similar effect for other systems ?
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 26 / 40
Experimental data of 180Hg
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 27 / 40
A. N. Andreyev, et al., PRL 105, 252502 (2010)
Experimental data of 178Pt
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 28 / 40
I. Tsekhanovich, ArXiv :1804.01832
Similar effect of the octupole deformation ?
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 29 / 40
CHF+BCS calculation
-1415
-1410
-1405
-1400
-1395
E[M
eV]
0 20 40 60 80 100 120
Q20[b]
SymmetricAsymmetric
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 30 / 40
CHF+BCS calculations : Hg isotopic chain
178Hg
182Hg
184Hg
186Hg
188Hg
188HgSym.
Z=44N=57
Z=44N=56
Z=45N=59
Z=45N=56
Z=46N=62
Z=40N=54
Z=35N=42
Z=36N=46
Z=36N=47
Z=35N=47
Z=34N=46
-1390
-1385
-1380
-1375
E[M
eV]
SymmetricAsymmetric
-1412
-1410
-1408
-1406
-1404
-1402
-1400
E[M
eV]
-1430
-1428
-1426
-1424
-1422
-1420
-1418
E[M
eV]
-1448
-1446
-1444
-1442
-1440
-1438
-1436
E[M
eV]
-1482
-1480
-1478
-1476
-1474
-1472
-1470
E[M
eV]
0 20 40 60 80 100 120
Q20[b]
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 31 / 40
180Hg
190Hg
198Hg
Z=28
Z=50
N=50
Z=34
Z=56
N=56
60 80 100 120
A
50 75 100 125 150
A
50 75 100 125 150
A
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 32 / 40
178Pt
180Hg
190Hg
198Hg
Z=28
Z=50
N=50
Z=34
Z=56
N=56
60 80 100 120
A
60 80 100 120
A
50 75 100 125 150
A
50 75 100 125 150
A
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 32 / 40
Deformation energy of the fragments
2=0.8
3=0.25
2=0.25
3=0.25
25
30
35
40
45
50
55
60Z
25 30 35 40 45 50 55 60 65 70 75
N
-5
0
5
10
15
20
25
30
35
25
30
35
40
45
50
55
60
Z
25 30 35 40 45 50 55 60 65 70 75
N
-5
0
5
10
15
0. 0. 0.
0. 0. 0.
28
34
50 52
56
100Ru
78Se
β2
β2
β3
β3
-14
-13
-12
-11
-10
-9
-8
-7
-6
-5
ǫ n[M
eV]
0.0 0.1 0.2 0.3
0.2 0.3 0.4
-14
-13
-12
-11
-10
-9
-8
-7
-6
-5
ǫ p[M
eV]
0.0 0.1 0.2 0.3
0.2 0.3 0.4 0.5
30
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 33 / 40
Conclusion
The fission process magnify the octupole shell structure
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 34 / 40
Preliminary Gogny-TDHFB calculation
x and y direction :Harmonic oscillatorbasis nx + ny ≤ Nshell
z direction : Cartesianmesh nz= 55
Nbase ' 3000full cartesian meshabout 100 000 degreesof freedom
Y. HashimotoGuillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 35 / 40
Preliminary test on 230Th
TDHFB(GognyD1S)
TDHF+BCS(Sly4d)
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 36 / 40
Preliminary test on 230Th
TDHFB(GognyD1S)
TDHF+BCS(Sly4d)
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 36 / 40
Preliminary test on 230Th
TDHFB(GognyD1S)
TDHF+BCS(Sly4d)
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 36 / 40
Preliminary test on 230Th
TDHFB(GognyD1S)
TDHF+BCS(Sly4d)
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 36 / 40
Preliminary test on 230Th
TDHFB(GognyD1S)
TDHF+BCS(Sly4d)
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 36 / 40
Outlook : Fission recycling
Z=28
N=50
Z=34
N=56
Region where BCS approximation is forbidden (due to continuum)
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 37 / 40
Thank you
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 38 / 40
Comparison TDHFB - TDHF+BCS
TDHF+BCSBased on TDHFB with the approximation : ∆ij = δij ∆i
Evolution : i~ dϕidt = (hMF − εi )ϕi
i~ dnidt = ∆∗i κi −∆iκ
∗i
i~ dκidt = κi (εi − εi ) + ∆i (2ni − 1)
Theoretical differenceNumerical cost : TDHFB requires 1000 times more numerical resourcesTreatment of continuum states : BCS gas problemContinuity equationNumber of pairing degrees of freedom ( HFB ∆(r), BCS : ∆i i )Spatial dependence of the pairing correlation
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 39 / 40
Comparison
A. Bulgac, P. Magierski, K. J. Roche, and I. Stetcu, PRL 116, 122504 (2016).
Table – TDHF+BCS results for 240Pu
# Q0 [b] E∗0 [MeV] Tfis [fm/c] ZL NL TKE [MeV]1 45.4 1.46 6480 40.21 60.77 171.52 46.7 0.8 4830 40.83 62.68 181.83 50.5 -1.16 26970 42.2 64.83 181.84 53.0 -2.13 6750 41.39 63.05 177.95 56.8 -3.5 4800 40.99 62.85 177.26 59.3 -4.3 5400 40.45 62.17 178.47 63.1 -5.31 6630 39.55 59.58 162.78 71.9 -7.8 1020 41.8 63.28 179.9
Guillaume SCAMPS Impact of pear-shaped fission fragments on mass-asymmetric fission December 10-12, 2018 40 / 40