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
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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