Microscopic phase-space
exploration of nuclear fission
arXiv:1609.0142 [nucl-th] (2016)

Yusuke Tanimura (GSI)
Collaborators:
Denis Lacroix (Institut de Physique Nucléaire Orsay)
Sakir Ayik (Tennessee Technological Univ.)“Neutron star mergers: From gravitational waves to nucleosynthesis
International Workshop XLV on Gross Properties of Nuclei and Nuclear Excitations”
Hirschegg, Kleinwalsertal, Austria, January 15-21, 2017

Outline• Introduction• Overview of theories for fission• Stochastic mean-field (SMF) theory• Spontaneous fission of 258Fm• SummaryNuclear fission• Importance– Astrophysical process– Energy production– Synthesis of super heavy elements– Production of radioactive isotopes
fission with TDHF

• Theoretical challenges– Large-amplitude process of quantum many-body system– Quantum tunneling of many-degrees of freedom– Coupling between collective and internal (single-particle)
degrees of freedom

Theories for fission• Conventional strategy
1.
2.
3.

select a set of relevant collective degrees of freedom (DOF) Q
ex. elongation, mass asymmetry, etc.
construct potential energy V(Q) and collective inertia parameterssolve the equations of motion of Q to get fission observables
lifetime, fragment mass/charge distribution, etc.

• Approaches based on
the macroscopic model + shell correction
have been successful• Microscopic models are still under development• With microscopic approaches– energy density functional (EDF) theory is employed– less phenomenological assumptions

Theories for fission• Conventional strategy
1.
2.
3.

select a set of relevant collective degrees of freedom (DOF) Q
ex. elongation, mass asymmetry, etc.
construct potential energy V(Q) and collective inertia parameterssolve the equations of motion of Q to get fission observables
lifetime, fragment mass/charge distribution, etc.

• Approaches based on
the macroscopic model + shell correction
have been successful• Microscopic models are still under development• With microscopic approaches– energy density functional (EDF) theory is employed– less phenomenological assumptions

We aim to establish a microscopic theory fordescription of the fission process

Microscopic models
1. Static approach  talk by S.A. Giuliani• With energy density functional
theory (Skyrme, Gogny, RMF)
• Fission paths on the potential
energy surface given by constrained
mean-field calculations• Relevant DOF? Mass parameter?
• Dynamics is not fully treated2. Dynamical approaches
TDHF (Time-dependent
Hartree-Fock)TDGCM (Time-dependent
generator coordinate method)• No need to select collective DOFs
and compute mass parameters• Fully non-adiabatic• Collective motions are nearlyclassical• No spontaneous symmetry
breaking• Quantum treatment of collective
degrees of freedom• Relevant DOF? Mass?
• Numerical cost rises rapidly withnumber of collective DOFs

Negele et al, PRC17, 1098 (1978),
Umar and Simenel, PRC89, 031601 (2014).
YT, Lacroix, and Scamps, PRC92, 034601 (2015).

Goutte et al PRC71, 024316 (2005),
Regnier et al, PRC93, 054611 (2016).

EEQQ

2. Dynamical approaches
TDHF (Time-dependent
Hartree-Fock)TDGCM (Time-dependentgenerator coordinate method)• No need to select collective DOFs
and compute mass parameters• Fully non-adiabatic• Collective motions are nearlyclassical• No spontaneous symmetry
breaking• Quantum treatment of collective
degrees of freedom• Relevant DOF? Mass?
• Numerical cost rises rapidly withnumber of collective DOFs

Negele et al, PRC17, 1098 (1978),
Umar and Simenel, PRC89, 031601 (2014).
YT, Lacroix, and Scamps, PRC92, 034601 (2015).

Goutte et al PRC71, 024316 (2005),
Regnier et al, PRC93, 054611 (2016).

EE

Q

Q

2. Dynamical approaches
TDHF (Time-dependent
Hartree-Fock)

PQ• No need to select collective DOFs
and compute mass parameters• Fully non-adiabatic• Collective motions are nearlyclassical• No spontaneous symmetry
breaking

Collective phase spaceA1
TDHF

t>0

A
t=0
Q

e.g. fragment mass distribution

yieldEQ

A2

fragment mass

Our method

S. Ayik, PLB658, 174 (2008)

TDHF  Stochastic mean field (SMF) theory• No need to select collective DOFs
• Fully non-adiabatic• Quantum fluctuations by initialstate sampling• Symmetry breaking

PQ

Collective phase space

t=0

SMF
Q

 microscopic and dynamical
description of fission
yield

e.g. fragment mass distribution

fragment mass

Stochastic mean-field theory
S. Ayik, PLB658, 174 (2008)

Stochastic mean-field theory

If the initial many-body state is a Slater determinant:

1h110p• Configuration-mixing effect• Symmetries breaking

00Stochastic mean-field theory
An application to an exactly solvable model

εV+-Lacroix, Ayik, and Yilmaz, PRC85, 041602(R) (2012)

Quasi-spin of Lipkin model with SMF
TDHF: nothing happensApplication to spontaneous fission of 258Fm• Interaction: SLy4d + pairing• δρij is truncated within awindow Δε around εF

1h=110p00Total kinetic energyand fragment mass
•Width of distribution isreasonably reproduced•Peak position is shiftedfrom TDHF value•Does not reproduce theasymmetric shape of TKE
distribution

Scission
point model

SMF

expt. 257Fm

Scission
point model

Wilkins et al., PRC14, 1832 (1976)

Deformation of fragments“Projection” of motion of wavepacket
onto certain collective subspace• Quadrupole deformation isrelaxed after scission• Octupole deformation
oscillates after scission

Summary and perspectives• Aim: Fully microscopic and dynamical description for fission• We applied the SMF theory to take into accountthe quantum fluctuations missing in TDHF
– fluctuation of ρij is introduced at t = 0 by random sampling– spontaneous fission of 258Fm: possible to obtain realistic TKE andfragment-mass distributions• Induced fission• Initial fluctuation determined from GCM calculation• Systematic calculations

With and without
dynamical pairing

TDHF + BCS and TDHF FOABCSFOA

Evaporation of particles

Application to spontaneous fission of 258Fm• Interaction: SLy4d + pairing• δρij is truncated within awindow around εF1h=110p00

Application to spontaneous fission of 258Fm• Interaction: SLy4d + pairing• δρij is truncated within awindow around εF

1h=110p00