Role of octupole deformed shell effects on the fission of nuclei - - PowerPoint PPT Presentation

April 16-18, 2019

Role of octupole deformed shell effects

• n the fission of nuclei

from actinides to mercury region 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 IAEA Vienna , Vol II, 20 (1974) Data from D. A. Brown et al., Endf/b-viii.0, Nucl. Data Sheets 148, 1 (2018), (spontaneous and thermal neutron-capture).

Motivation

How can we understand this behavior ? Interplay between structure and reaction ?

Mean-field theory with pairing

TDHF

Independent particle Initialisation : ˆ hMF |φi = ǫi |φi Evolution : i dρ

dt = [hMF, ρ]

TDHFB

Pairing correlation Quasi-particles : |ωα = Uα

Vα

• Evolution :

i d|ωα

dt

=

h ∆ −∆∗ −h∗

• |ωα

TDHF+BCS

Based on TDHFB with the approximation : ∆ij = δij∆i Evolution : i dφi

dt = (ˆ

hMF − ǫi)φi i dni

dt = ∆∗ i κi − ∆iκ∗ i

i dκi

dt = κi(ǫi − ǫi) + ∆i(2ni − 1)

New systematic study

First : CHF+BCS

Example : 240Pu

• 1835
• 1830
• 1825
• 1820
• 1815
• 1810
• 1805
• 1800
• 1795
• 1790

E [MeV] 20 40 60 80 100 120 Q20 [b]

Symmetric Asymmetric

Second : TDHF+BCS Details of the calculation

Skyrme functionnal Sly4d Surface pairing interaction ∆x = 0.8 fm Lattice : Lx × Ly × 2Lz = 40 × 19.2 × 19.2 fm3

New systematic study

First : CHF+BCS

Example : 240Pu

• 1835
• 1830
• 1825
• 1820
• 1815
• 1810
• 1805
• 1800
• 1795
• 1790

E [MeV] 20 40 60 80 100 120 Q20 [b]

Symmetric Asymmetric

Second : TDHF+BCS Details of the calculation

Skyrme functionnal Sly4d Surface pairing interaction ∆x = 0.8 fm Lattice : Lx × Ly × 2Lz = 40 × 19.2 × 19.2 fm3

New systematic study

First : CHF+BCS

Example : 240Pu

• 1835
• 1830
• 1825
• 1820
• 1815
• 1810
• 1805
• 1800
• 1795
• 1790

E [MeV] 20 40 60 80 100 120 Q20 [b]

Symmetric Asymmetric

Second : TDHF+BCS Details of the calculation

Skyrme functionnal Sly4d Surface pairing interaction ∆x = 0.8 fm Lattice : 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

Yield (arbitrary units)

230Th 234U 236U 240Pu 246Cm 250Cf

30 35 40 45 50 55 60 Proton 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

Yield (arbitrary units)

230Th 234U 236U 240Pu 246Cm 250Cf

30 35 40 45 50 55 60 Proton 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 − j2 qσ

ρqστ TF

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 with the same q and σ very low. Cqσ(r) = 0.5 Limit of uniform-density Fermi gas. Mask function : → Cqσ(r)ρqσ ρmax

qσ

Nucleon localization function

Fermion localization function

Cqσ(r) =

• 1 +
• τqσρqσ − 1

4 |∇ρqσ|2 − j2 qσ

ρqστ TF

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 with the same q and σ very low. Cqσ(r) = 0.5 Limit of uniform-density Fermi gas. Mask function : → Cqσ(r)ρqσ ρmax

qσ

Example :

• P. Jerabek, B. Schuetrumpf, P.

Schwerdtfeger, and W. Nazarewicz,

• Phys. Rev. Lett. 120, 053001 (2018).

Nucleon localization function

Fermion localization function

Cqσ(r) =

• 1 +
• τqσρqσ − 1

4 |∇ρqσ|2 − j2 qσ

ρqστ TF

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 with the same q and σ very low. Cqσ(r) = 0.5 Limit of uniform-density Fermi gas. Mask function : → Cqσ(r)ρqσ ρmax

qσ

Schematic system

Example of 240Pu

240Pu

Example of 240Pu

240Pu

Example of 240Pu

240Pu

Example of 240Pu

240Pu

Example of 240Pu

Hypothesis

The octupole shell effects are important in the fission fragment

Other systems

Other systems

Other systems

Why the fragments have octupole deformation ?

Similar effect on fusion reaction

40Ca+40Ca, E3− = 3.7 MeV 56Ni+56Ni, E3− = 7.5 MeV

• C. Simenel, M. Dasgupta, D. J. Hinde, and E. Williams, Phys. Rev. C 88, 064604 (2013).

Why the fragments have octupole deformation ?

Similar effect on fusion reaction

40Ca+40Ca, E3− = 3.7 MeV 56Ni+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. and Particle Physics, 2015

Results from systematic calculation

In both calculations, the region Z ≃ 54 , N ≃ 88 is favorable for octupole deformation .

Experimental results

144Ba is found to be octupole in its groud state. Bucher et al. PRL 116 (2016).

Constraint HF+BCS octupole deformation with Sly4d

Result from constraint calculation of the heavy fragment

The 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

ǫn[MeV] 0.0 0.5 1.0 1.5 2.0 2.5 3 4 5 6 2 3 4 5

• 17
• 16
• 15
• 14
• 13
• 12
• 11
• 10
• 9
• 8
• 7

ǫp[MeV] 0.0 0.5 1.0 1.5 2.0 2.5 3 4 5 6 2 3 4 5 52

Conclusion

Mechanism

The Nucleus-Nucleus interaction at the scission configuration favors the octupole shapes Shell structure favors octupole shape in the region Z ≃ 52-56, N ≃ 84-88 Actinide fission fragments are driven in the region Z ≃ 54, N ≃ 86

• G. Scamps, C. Simenel, Nature 564, 382 (2018).

Similar effect for other systems ?

• P. A. Butler,

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

• 10
• 5

5 10 y [fm] 10 5

• 5
• 10
• 15
• 20

z [fm]

15 20

0.0 0.1 0.2 0.3 0.4 0.5 Cn

1

R u

8

K r

Comparison with experimental data

• 10
• 5

5 10 y [fm] 10 5

• 5
• 10
• 15
• 20

z [fm] 15 20 0.0 0.1 0.2 0.3 0.4 0.5 Cn

1

R u

8

K r

Experimental article : A. N. Andreyev, et al. Phys. Rev. Lett. 105, 252502 (2010).

Single particle energies

Structure of 100Ru (Z=44 and N=56)

50 44 50 52 56

β2 β3

• 14
• 13
• 12
• 11
• 10
• 9
• 8
• 7
• 6
• 5

ǫn[MeV]

0.2 0.3 0.4 0.5

• 12
• 10
• 8
• 6
• 4
• 2

ǫp[MeV]

0.0 0.1 0.2 0.3 0.4 34

Structure of 80Kr (Z=36 and N=44)

28 36 38 50 36 44

• 14
• 13
• 12
• 11
• 10
• 9
• 8
• 7
• 6
• 5

ǫn[MeV]

• 14
• 13
• 12
• 11
• 10
• 9
• 8
• 7
• 6
• 5

ǫp[MeV]

0.0 0.2 0.4 0.6

β2

0.0 0.1 0.2 0.3 0.4

β3

32

β3 = 0 β2 = 0.75

178Pt 180Hg 190Hg 198Hg

Z=28 Z=50 N=50 Z=36 N=82 N=56 N=52

Neutrons Protons

60 80 100 120

A

60 80 100 120

A

50 75 100 125 150

A

50 75 100 125 150

A

• G. Scamps and C. Simenel, arXiv :1904.01275

CHF+BCS calculations : Hg isotopic chain

178Hg 182Hg 184Hg 186Hg 188Hg

Z=34 N=42 Z=36 N=46 Z=36 N=47 Z=37 N=49 Z=35 N=46

180Hg

Z=36 N=44

188Hg

Sym. Z=44 N=57 Z=44 N=56 Z=43 N=57 Z=46 N=56 Z=45 N=62 Z=44 N=56 Z=40 N=54 0.0 0.1 0.2 0.3 0.4 0.5 Cn

178Hg 180Hg 182Hg 184Hg 186Hg 188Hg

• 1390
• 1385
• 1380
• 1375

E[MeV] Symmetric Asymmetric

• 1410
• 1405
• 1400

E[MeV]

• 1430
• 1425
• 1420

E[MeV]

• 1450
• 1445
• 1440
• 1435

E[MeV]

• 1470
• 1465
• 1460
• 1455

E[MeV]

• 1485
• 1480
• 1475

E[MeV]

20 40 60 80 100 120

Q20[b]

Preliminary results for 188Pb with Sly4d and BCS

188Pb

AH=94 AH=96 AH=100 AH=102 5 10 15 20 25 30 35 40 45 50

Q30[b]

20 40 60 80 100 120 140 160

Q20[b3/2]

188Pb

• 1475
• 1470
• 1465
• 1460

E[MeV]

20 40 60 80 100 120 140

Q20[]

Preliminary results for 188Pb with Sly4 and Constrained Hartree-Fock + BCS

188Pb

AH=94 AH=103 AH=107 5 10 15 20 25 30 35 40 45 50

Q30[b3/2]

20 40 60 80 100 120 140 160

Q20[b]

188Pb

• 1475
• 1470
• 1465
• 1460
• 1455
• 1450
• 1445

E[MeV]

20 40 60 80 100 120 140

Q20[b]

Thank you

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 95 100

N

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=0.42 Red contour line : density of a fissioning 258Fm (asymmetric mode)

Identification with nucleon localisation function

Top : NLF of a 144Ba with a constraint β3=0.42 Bottom : 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)

Preliminary results on 188Pb with Sly4 and BCS

Z=28 Z=50 N=50 Z=36 N=82 N=56 N=52

Neutrons Protons

Table

A NH ZH

• def. H.

NL ZL

• def. L.

198 63.1 43.4 Elong. 54.9 36.6 Comp. 196 62 43.4 Elong. 54 36.6 Comp. 194 61.7 43.5 Elong. 52.3 36.5 Comp. 65 45.5 Elong. 49 34.5 Elong. 192 60.9 43.5 Elong. 51.1 36.5 Comp. 64.5 45.6 Elong. 47.5 34.4 Elong. 190 58.7 42.7 Elong. 51.3 37.3 Comp. 64 46 Elong. 46 34 Elong. 188 62 45.4 Elong. 46 34.6 Elong. 186 57.3 43.2 Comp. 48.7 36.8 Elong. 184 56.9 43.5 Comp. 47.1 36.5 Elong. 182 56.1 44 Comp. 45.9 36 Elong. 180 55.8 44.5 Comp. 44.2 35.5 Elong. 178 56.1 46.1 Comp. 41.9 33.9 Elong. 176 55.6 46.4 Comp. 40.4 33.6 Elong. N Z NH ZH

• def. H.

NL ZL

• def. L.

96 82 56 47.5 Comp. 40 34.5 Elong. 98 80 56.1 46.1 Comp. 41.9 33.9 Elong. 100 78 55.7 43.5 Comp. 44.3 34.5 Elong. 102 76 56.5 42 Comp. 45.5 34 Elong. 104 74 58 40.8 Comp. 46 33.2 Elong. 106 72 58 39 Comp. 48 33 Elong. 108 70 54 35 Comp. 54 35 Comp. 112 68 56 34 Comp. 56 34 Comp.

Deformation energy

2=0.8 3=0.25 2=0.25 3=0.25

• Edef. [MeV]
• Edef. [MeV]

25 30 35 40 45 50 55 60

Z

25 30 35 40 45 50 55 60 65 70 75

N

• 5

5 10 15 20 25 30 35 25 30 35 40 45 50 55 60

Z

• 5

5 10 15