impactofpear-shapedﬁssionfragments onmass ......guillaumescamps...

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).


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 ?


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)


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).


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


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


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


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


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 ?


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




Cqσ(r) =



qσ

)2]−1



Mask function :

→Cqσ(r)ρqσ

ρmaxqσ

Example :

P. Jerabek, B. Schuetrumpf, P.Schwerdtfeger, and W. Nazarewicz,Phys. Rev. Lett. 120, 053001 (2018).




Cqσ(r) =



qσ

)2]−1



Mask function :

→Cqσ(r)ρqσ

ρmaxqσ

Schematic system


Example of 240Pu

240Pu


Example of 240Pu

HypothesisThe octupole shell effects are important in the fission fragment


Other systems


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).


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).


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


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


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


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


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


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).


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)


Identification with nucleon localisation function

Top : NLF of a 144Ba with a constraint β3=0.42Bottom : NLF of a fissioning 258Fm (asymmetric

mode)


Identification with nucleon localisation function


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)


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)


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).


Similar effect for other systems ?


Experimental data of 180Hg


A. N. Andreyev, et al., PRL 105, 252502 (2010)

Experimental data of 178Pt


I. Tsekhanovich, ArXiv :1804.01832

Similar effect of the octupole deformation ?


CHF+BCS calculation

-1415

-1410

-1405

-1400

-1395

E[M

eV]

0 20 40 60 80 100 120

Q20[b]

SymmetricAsymmetric


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]


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


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


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


Conclusion

The fission process magnify the octupole shell structure


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)


Outlook : Fission recycling

Z=28

N=50

Z=34

N=56

Region where BCS approximation is forbidden (due to continuum)


Thank you


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


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


impactofpear-shapedﬁssionfragments onmass ......guillaumescamps...

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