Lawrence Livermore National Laboratory

A Microscopic picture of scission

DRAFT Version 1

March 15, 2010

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Security, LLC, Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

Walid Younes

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Outline

1. Context for a microscopic theory of fission2. Approaching scission3. The nucleus near scission

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Overview of LLNL program

Goal: predict fission-fragments properties (energies, shapes, yields) as a function of incident energy

Two complementary approaches

• Common to both: what is the microscopic picture of scission? Crucial to understanding the entire fission process Crucial to the extraction of realistic fragments properties

Many-bodytheory

Fragmentproperties

Fission-neutronspectrum

Fission chainyields

Fully-mic = HFB+TDGCM (more predictive, less acc)

Mic+Stat. mech (more acc, less predictive)

Informs/guides

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Fully microscopic approach to fission: The Big Picture

Physical Sciences Directorate - N Division

HFB

TDGCM

+ qpd.o.f.

Finite-rangeeff. interaction

Constraints

StaticsPESFrag propsScission id

dynamics

Non-adiabatic

Higher E

Time-evolvingwave packet

Fission times

Fission yields

Yield-avg’edfrag props

Fully microscopic, quantum-mechanical, dynamic approachEffective interaction is the only phenomenological input

Collective Hamiltonian

Coll-intr couplingBased on highlysuccessful BIIIprogram

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Past successes

Predicts/explains cold & hot fission

Predicts realisticFission times

Predicts 238U(,f) TKE to 6%

Reproduces yields for 238U(,f)

Goutte et al., PRC 71 , 024316 (2005)

Berger et al.NPA 502, 85 (1989)CPC 63, 365 (1991)

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Approaching scission

What are the relevant degrees of freedom near scission Discontinuities along the path to scission?

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Fission and the role of collective coordinates: Q20 and Q30

240Pu Most probable path

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Scission configurations in the Q20-Q30 plane: 240Pu hot fission

• Criterion: sudden dropIn neck size• Complex scission lineshape

Younes & Gogny, PRC 80, 054313 (2009)

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A more detailed view: the Q40 collective coordinate

Q20-Q40 map for Q30 = 0 b3/2

• well-defined troughs• barrier between valleys

Focus on symmetric fission

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A more detailed view: barrier between fusion & fission in Q20-Q40

Physical Sciences Directorate - N Division

240Pu, symmetric fission

• ~ 5.6 MeV barrier at Q20 = 320 b, disappear gradually• exit near 300 b (cold), 580 b (hot) or anywhere in between

• Berger et al., NPA 428, 23 (1984)

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Caveat: the Q30 = 30 b3/2 case

• barrier low, with gaps• dynamics can exit early

Barrier from Q20-Q40 map

Q40 analysis exit points fragment properties

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Controlling the approach to scission: the QN coordinate

240Pu, most prob. Q30, hot fission

7.6-MeV discontinuity

Calc at discontinuity with QN

• discontinuity large error in fragment properties• QN ~ neck size controlled approach to scission

Younes & Gogny, PRC 80, 054313 (2009)

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expˆN

N

azz

NQ

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Identifying scission

How do we identify scission microscopically? How do we identify the pre-fragments? What are the fission-fragment properties?

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The path to refining our microscopic picture of scission

Geometric criterion(e.g., neck size)

distinguishes pre and post configurations

doesn’t pinpoint scission

Interaction-energycriterion

pinpoints scission adiabatic treatment of

scission

Molecular-like picture( variation of interaction energy)

Microscopic, non-adiabatic treatment

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Energy-based criterion for identifying the scission configuration

Idea: scission occurs as soon as there is enough energy in system to overcome attractive interaction between fragments

• Use neck size (QN) as constraint to approach scission

• Identify s.p. wavefunctions for left and right fragments tot 1 + 2 + 212, with 12 0 at large separation

• Calculate Eint = EHFB-EHFB(L)-EHFB(R)-Ecoul

• Work in representation that minimizes fragment tails

E

• Scission occurs as soon as Eint = E• Scission can occur with QN 0• Caveats:

• simplified 1D picture• multi-dim fission smaller E

available (Berger et al., NPA428, 23 (1984))

Younes & Gogny, AIP proceedings 1175, 3 (2009)/arXiv:0910.1804v1

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Identifying the pre-fragments: choice of representation

QN = 0.01 • HFB solution defined up to unitary trans• Free to choose representation• Arbitrary rep can lead to large frag tails• With microscopic def of fragments, we see

the tails• Tail-minimizing rep (via orthogonal

transformation of s.p. wave funcs)• produces reasonable Eint

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Choosing a representation that minimizes tails

Define localization parameter

• For a pair of states: Identify pairs of states (i,j) and angle such that

• Gives

2

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zdz

zdzzdz

i

z i

z

i

iN

N

1 for completely localized qp

0 for completely unlocalized qp

22jiij

j

i

j

i

cossin

sincos

ijij

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Tail reduction at scission: Q20 = 365 b, Q30 = 60 b3/2, QN = 1.55

Before wave function localization After wave function localization

Operation does not affect total energy, but allows• identication of left and right pre-fragments• definition of a separation distance• calculation of interaction energy

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Results: comparison with observables for 239Pu(nth,f)

Total kinetic energy(expt data have ~ 10 MeV)

Average neutron multiplicity

Remarkable results for a parameterless calculation!

Younes & Gogny, AIP proceedings 1175, 3 (2009)/arXiv:0910.1804v1

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The molecular-like picture of scission

Competition between attractive nuclear and repulsive Coulomb forces creates scission valley and barrier

Requires non-adiabatic calculation, otherwise no scission barrier:• stop HFB calcs at config where

there is almost no nuclear interaction between pre-fragments

• “freeze” pre-fragment configs• separate by translationÞ sudden approximationW. Nörenberg, IAEA-SM-122/30, 51 (1969).

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Non-adiabatic separation of fragments

Start from HFB for 240Pu with Q20 = 350 b, QN = 2

Apply unitary transform to localize those sp wave functions that extend into the complementary fragment (Younes & Gogny, arXiv:0910.1804v)

Translate pre-fragment densities (Younes & Gogny, PRC

80, 054313) Calculate the energy

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The molecular-like microscopic picture of scission

Sharp drop at Q20 = 370 b for adiabatic calc (hot fission)

Non-adiabatic calcs for different starting Q20, QN

• Scission barrier decreases with Q20 and QN

• For hot fission, scission barrier disappears between QN = 1 and 2

This is still a static picture TDGCM dynamics

• Pre-scission energy available to overcome scission barrier

• Some of that energy may be taken up by collective transverse d.o.f. (Berger et al., NPA428, 23 (1984)) and, possibly, intrinsic excitations

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Application: microscopic Wilkins model for mass yields

Based on Wilkins et al., PRC 14, 1832 (1976). (See also S. Heinrich thesis) Static microscopic calcs of fragments at many deformations

Calculate energy of two-fragment system as a function of separation d Identify distance d at scission such that

Boltzmann factor gives probability distributions: exp(-Etot/Tcoll)

ZL,AL,b,TintZH,AH,b,Tint

d

Tcoll

0int

d

E

d

Etot

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The semi-microscopic approach: Mass yields

236U(nth,f) using LDM(Wilkins et al., 1976)

239Pu(nth,f) using microscopic theory(Our work, in progress…)

Already better than LDM. Should improve with:• proper treatment of anti-symmetrization• more fragments included• intrinsic temperature• revisit Pauli blocking in odd-A and odd-odd

nuclei

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Conclusions

Quantitative, microscopic picture of scission is essential for a predictive theory of fission

Near scission, new collective d.o.f. become relevant (QN, d) Molecular-like picture of fission provides solid framework to understand

scission• Requires the identification of left and right pre-fragments and their

interaction energy• Microscopic definition of scission

• Sudden approximation at scission• Non-adiabatic separation of the fragments

0int

Ed

F

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Application: interaction energy for 240Pu symmetric fission

Densities at large and small QN Interaction energies as function of QN

This is nonsense!

Eint between well-separated fragments should be small, not > 3 GeV

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Caveat: topology of the PES and the need for fission dynamics

Q20-Q40 map for Q30 = 60 b3/2

• valleys well separated again• exit near Q20 = 370 b