Study on Nuclear fission at Tokyo Tech.

Chikako ISHIZUKATokyo Institute of Technology, 152-8550 Tokyo, Japan

2

Contents

1. Background of today’s topics2. Research in nuclear fission study by Tokyo Tech.

Implementation of 4-D Langevin equation : • correlated transitions in mass and TKE distributions

of fission fragments• Fragment shape and neutron multiplicity • Fission in SHN region

3. Summary

3

Background: Nuclear fission and SHE

1. Due to complexity of the process as a large-amplitude collective motion, nuclear fission still offers a field of big challenges to nuclear physics, especially, the process from compound nucleus to scission is still a mysterious process

2. Many observables arise as a result of fission, e.g., fission fragment yield, TKE, population of prompt neutrons and gammas which is followed by a series of β-decay: their correlations and distributions must be comprehended in a consistent manner, which is still a difficult subject

3. We have been treating the process before scission by several theories, such as Langevin equation, Antisymmetrized Molecular Dynamics （AMD) and (Time-Dependent) Hartree-Fock（(TD)HF)-BCS, and their outcomes are connected to statistical decay model and theory of β-decay.

4. My talk today shows our recent results with 4D Langevin model.

Systematics of average peak position of light (L) and heavy (H) fragments

<AH>～138

<AL>

A=258-260

258Fm

234U

<AH><AL>

Some kind of transition in fission mechanisms is taking place between them

4

55

Systematics and anomaly in the average Total Kinetic Energy of Fission Fragments

How can we understand these systematical and anomalous trends simultaneously?

(～Coulomb energy of fissioning nuclei)

Nuclear fission treated by Langevin equation

These 2 different d.o.f have different time scales:

Brownian motion

• nucleon motion : 1 to 10 fm/c• shape motion : ～>10,000fm/c

Nuclear shape evolution is driven by random kicks by nucleons in thermal equilibrium (microscopic d.o.f.) given to the nuclear surface (macroscopic d.o.f) inside the surface

6

A tiny particle from the pollen grainsof flowers

Brownian motionis driven byrandom kicks bywater molecules

7

Two-center model(Connected Nilsson model) (Maruhn and Greiner, Z. Phys. 251(1972) 431)

Shape parametrization

: Radius of compound nucleus

Mass asymmetry

Elongation

3( ) , 1, 22

i ii

i i

a b ia b

δ −= =

+

●

●

●

RzZZ 0

0 =

21

21

AAAA

+−

=α

R

35.0=ε neck parameter : fixed●

● volume conservation condition is applied

Rneck

Collective coordinates （4 dynamical variables)

{ } },,,{ 2104 αδδZZq D =

fragmentleft theof mass : fragmentright theof mass :

2

1

AA

3/12.1 CNA=

No parameter adjustment

Deformation of outer tip of each fragment

30×30×30×40=1,080,000 mesh points

{ } { }( ) { } { }( ) { }( )pqqpqP δδ 0, −=We assume no initial distribution

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4D Langevin model

qi : Nuclear shape coordinatepi : Momentum conjugate to qi

F : Helmholtz' free energy 𝐹𝐹 = 𝑈𝑈 − 𝑇𝑇𝑇𝑇mij : Inertia tensor : Werner-Wheeler model or Linear Response Theory

γij : friction tensor: Wall and Window or Linear Response Theory

：Fluctuation dissipation theorem

𝐸𝐸∗：Total excitation energy of the system

Drift term𝑑𝑑𝑝𝑝𝑖𝑖𝑑𝑑𝑑𝑑

= − 𝜕𝜕𝐹𝐹𝜕𝜕𝑞𝑞𝑖𝑖

− 12

𝜕𝜕𝜕𝜕𝑞𝑞𝑖𝑖

𝑚𝑚−1𝑗𝑗𝑗𝑗𝑝𝑝𝑗𝑗𝑝𝑝𝑗𝑗 − 𝛾𝛾𝑖𝑖𝑗𝑗 𝑚𝑚−1

𝑗𝑗𝑗𝑗𝑝𝑝𝑗𝑗 + 𝑔𝑔𝑖𝑖𝑗𝑗𝑅𝑅𝑗𝑗 𝑡𝑡

C.Ishizuka et al., PRC 96, 064616 (2017).

Friction term White noise

1*2 ij i jE m p p

Ta

−=

𝑔𝑔𝑖𝑖𝑗𝑗𝑔𝑔𝑖𝑖𝑗𝑗 = 𝛾𝛾𝑖𝑖𝑗𝑗𝑇𝑇

: 2-center Woods-Saxon+Strutinsky+BCS

Transport coefficients

A coupled stochastic differential equation

𝑑𝑑𝑞𝑞𝑖𝑖𝑑𝑑𝑑𝑑

= 𝑚𝑚−1𝑖𝑖𝑗𝑗𝑝𝑝𝑗𝑗 , (𝑖𝑖, 𝑗𝑗 = 1, … , 4)

{ } 0 1 2{ , , , }q ZZ δ δ α=

( ) (0) (0) ( )LD shellF T F F Tδ= + Φ Φ(𝑇𝑇): Ivanyuk, CI, Usang, Chiba, PRC97, 054331(2018)

Pashkevich, NPA169, 275(1971)

9

Multidimensional Langevin modelfor nuclear fission=combination of Hamiltonian dynamics + thermodynamics

:conserved:increasing

ES

Space of collective coordinates {qi}: U, S, T, VF=U-TS

Heat Bath: TMicroscopic d.o.f.

Energy of each subsystem is not conserved but only their sum is Entropy of each subsystem is not necessarily increasing but their sum is

FrictionRandom force

i ii

dU TdS PdV K dq= − −∑i i

i

ii i

i i i ii i

ii

dU TdS PdV K dq

U SK Tq q

F U TSdF SdT PdV K q K dq

FKq

= − − ⋅

∂ ∂= − +

∂ ∂= −

= − − = − ⋅

∂= −

∂

∑

∑ ∑As we assume Langvin equation, we implicitly assume that system is in quasi-equilibrium so that the temperature should not give the driving force of the system (In the reality, the nucleus as a heat bath is so small that T may change but should not be driving the system)

microcanonical

canonical

K: general force(drift term)

10

Free energy surface F at T=0

(236U, direct calculation)F.A.Ivanyuk, C.I., M.D.Usang and S. Chiba, Phys. Rev. C 97, 054331 (2018)

11

Example of Langevin trajectories (236U, 20MeV)

(MeV)F

RzZZ 0

0 =

Rneck=0

12

234U 238Np

240Pu242Am

Predictions for mass distributions (Ex=20MeV)

236U

Result for LDP only

13

Anomaly in averaged peak positions

14

Peak structure of FF in Fm region

No parameter adjustment is required

Reason of such a sharp change is to be explained later in this presentation (if time allows)

236U

Superlong

Supershort

Q-value

Mass-TKE correlation and its decompositionClear transition of fission mechanisms, symmetric mode begin superlong for 236U, while it is supershort for 258Fm

258Fm

15

16

ResultsM. Usang et al., Scientific Reports 9, 1525 (2019)

Transition of symmetric mode from super-long to super-shortat Es-254

Transition of the main modefrom asym. to sym.at Fm-258

Transition of the main modefrom sym. to asym.at No-256

17

Results of mass-TKE correlations(ε=0.35)

Usang, Ishizuka, Ivanyuk and SC "Correlated transitions in TKE and mass distributions of fission fragments described by 4-D Langevin equation", Scientific Reports 9, 1525(2019).

Unik & Violasystematicscorrespond to the standard mode

18

Fragment shape just after scission

H LL H

,

,,{ }

,

( ) 1L H

L HL HV

L H

AV

ρ ⊂= rr

density with sharp-cut approx.

, , 2 2 2 320

, , 3 2 2 330

5 ( )(2 ) ,16

7 ( )[2 3 ( )]16

L H L H

L H L H

Q z x y d r

Q z z x y d r

ρπ

ρπ

= − −

= − +

∫

∫

r

rQuadrupole moment

Octapole moment

1919

Direct evidence that Light fragments are much more elongated compared withcorresponding heavy fragments:Strong influence of magicity of 132Sn or A=144 (to be explained later)

Direct evidence of superlong mode in the symmetric component→Low TKE

20 20( ) ( )Q L Q H>>

Q20(A) exhibits a saw-tooth structure

20

Q30 of fission fragments from 240Pu

• Saw-tooth structure is seen also for Q30

• Q30(L) ≤ Q30(H) while Q20(L) >> Q20(H)

• Large Q30 around A～144 having appreciable yield

21

Effect of deformed shell magicity around A=144 plays very important roles in interpreting fission observables

22

Q20 of 256Fm (left) and 258Fm (right)

22Magicity of A=132 & 144

Magicity of 132Sn

Magicity of A=144

Importance of dynamical treatment256Fm

258Fm

0.51.0

1.52.0

2.5

-40

-30

-20

-10

0

10

-0.6-0.4 -0.2 0.0 0.2 0.4 0.6

E def (M

eV)

α

z0 / R

0

258Fm

0.51.0

1.52.0

2.5

-40

-30

-20

-10

0

10

-0.6-0.4 -0.2 0.0 0.2 0.4 0.6

256FmE def (M

eV)

α

z0 / R

0

23

24

Itkis et al., Nucl. Phys. A944(2015)204-237

Peak position at broader region of nuclei

AH～140

AL～132

25

26

Mass distributions and TKE for superheavy nuclei (Ex=7MeV)

• There seems to be really multimodal fission (we can see at least 3 dominant modes): A=208, 144 and ?

• Due to the fact that the light fragment of A=208 component hasa broad distribution in Q20, the TKE distribution for that component is also broadly distributed

27

A=208

Results

3 peaks 4 peaks

Exp. data in Itkis

A=132

28

M. Albertsson et al., arXiv:1910.06030

29

Mass distributions and fragment Q20

for superheavy nuclei (Ex=7MeV)

Tilted Q20: Rare earth: A=144 and above

Totally different deformation for L and H fragmentsSpherical heavy fragment + light fragments sort against Q moment

30

deepinel.scat.

deepinel.scat.

QF

Itkis+2002Itkis+2002

Present

Fission fragment m

ass yield

Present

Exp. data by Itkis et al.

A=132

OurLangevincalculation

31

Z. Matheson et al., Phys. Rev. C 99, 041304(R) (2019)Microscopic study on fission of 294Og, using DFT

Of Heavy Fragments

Comparison with recent microscopic study

Bf of 240Pu

292Fl

294Og

Our Langevincalculation

32

Systematics

Z2/A1/3 M. Usang et al., Scientific Reports 9, 1525 (2019)

33

Conclusions

1. We have calculated the deformation of fission fragments in terms of Q20 and Q30 to show the fission fragment shape more appropriately than the δ-parameter in the 2CSM parametrization, which shows shape of fragments only at the outer tip

2. It was found that there is a clear saw-tooth structures in the distributions of Q20 and Q30 , and those values for heavy and light nuclei are very different in the region where there are 2 peaks in the mass distribution of fission fragments: Q20 (L) >> Q20 (H), Q30

(L) ≤ Q30 (H) and Q20 (symmetric ) > Q20 (L,H)3. Dependence of distributions of Q20 and Q30 on initial excitation

energy were also investigated4. In 258Fm, where the mass distribution consists of a sharp single

peak with wings, the main part has a very small Q20, while the 2 wings have very different values of Q20 : Q20 (L) >> Q20 (H)

5. Similar analysis in the SHN is ongoing, and we already found interesting results

34

35

36

37

Present(Ivanyuk)Present(Ignatuk)Experiment (Nishio+)

Mass distribution of 180Hg

mass-resolution convolutedσ=3

No smearing

Ex=41.2MeV

Ex=33.1MeV

Ex=30.2MeV

Ex=23.9MeV

Ex=21.0MeV

Effects of multichance fission

Ex=21.0MeV

38

How we can find fusion-fission components in experimental data of SHE fission-like events

Itkis+2002

Free energy surface (FES) for 302120

Finite temperature correction to the free energy

int 2

( ) ( ),(0) (0) (0) (0)

shell

LDM shell

E aTF E TS F F TF E V V E

δδ

== − = ∞ +

= = = +

• Calculation single particle energy 𝜖𝜖𝑗𝑗 based on 2-center Woods-Saxon model

• Then, we calculate shell correction at finite temperature

Computation of shell correction

( )/ ( )/

( ) ( ) ( ), ( ) 2 , ( ) ( ) ,

1 1, 2 ,1 1k

T Tshell k k e

k

T T Tk k eT T

k

E T E T E T E T n E T g n d

n n N ne eε µ ε µ

δ ε ε ε ε∞

−∞

− −

= − = =

= = =+ +

∑ ∫

∑2 '( ) ,

ig f ε εε

γ γ −

=

∑2

0 20,2,...

( ) ( ), 1,2

x Mn

n n nn

aef x a H x a anπ

−

+=

−= = =

+∑( ) Teg n d Nε ε

∞

−∞=∫

• We do the same procedure for pairing correction

( ) ( ) ( )

( ) ( ) ( )

( ) 2 log (1 ) log(1 )

( ) ( ) log (1 ) log(1 )

shell shell shell

shell

T T T Tk k k k

k

T T T Te e e e

F T E T T S T

S T S T S T

S T n n n n

S T d g n n n n

δ δ δ

δ

ε ε∞

−∞

= −

= −

= − + − −

= − + − −

∑

∫

42

Multidimensional Langevin modelfor nuclear fission=combination of Hamiltonian dynamics + thermodynamics

:conserved, :increasingE S

Space of collective coordinates {qi}: U, S, T, VF=U-TS

Heat Bath: TMicroscopic d.o.f.

•Energy of each subsystem is not conserved but only their sum is

•Entropy of each subsystem is not necessarily increasing but their sum is

Friction γRandom force gR

•As we assume Langvin equation, we implicitly assume that system is in quasi-equilibrium so that the temperature should not give the driving force to the system

•In the reality, the nucleus as a heat bath is so small that T may change but should not be driving the system

ii

FKq∂

= −∂

i i i ii i

dF SdT PdV K q K dq= − − = − ⋅∑ ∑

ii

FKq∂

= −∂