# Presentation: 265588

## The LLNL microscopic fission theory program

**W. Younes**

**This work performed under the auspices of the U.S. department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344**

## The program in a nutshell

- Goal
- Calculate energy partition among products (fragments, neutrons, gammas) for induced fission
- Team
- W. Younes, D. Gogny (20%)
- Methodology
- Statics: constrained HFB with finite-range interaction
- Dynamics: time-dependent GCM built on HFB solutions, with Gaussian overlap approximation
- History
- Methodology developed by BIII (France) since early 80’s
- In development at LLNL since fall 2006

**Fully microscopic/quantum mechanical/self-consistent, and dynamical**
****** ****Effective interaction is the only input**

## Past successes

**Predicts/explains cold & hot fission**

**Predicts realistic**
**Fission times**
**(microscopic theory**
**of “dissipation”)**

**Predicts ****238****U(********,f) TKE to 6%**

**Reproduces yields for ****238****U(********,f)**

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

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

## The microscopic method: statics

- Based on highly successful work at BIII
- Main tool: finite-range, constrained Hartree-Fock-Bogoliubov
- nucleus is built up from individual protons and neutrons
- only phenomenological input is effective inter-nucleon interaction (Gogny, D1S)
- finite-range interaction  mean field and pairing treated on same footing
- constraints introduce external fields to “mold” nucleus into desired “shape”
- choose set of “collective” coordinates (e.g., Q2, Q3, Q4)
- LLNL implementation:
- _F_inite _RAN_ge  _C_onstrained _H_artree-Fock-_B_ogoliubov with _R_apid _I_teration _E_xecution (FRANCHBRIE)

## The HFB code FRANCHBRIE for fission statics

- Can do HF, HF+BCS, HFB with Gogny effective interaction
- Uses separation method for 2-body matrix elements
- Assumes axial & time-reversal symm., but not reflection symm.
- Deformed H.O. basis, 2 parameters , z can be optimized
- Multiple constraints:
- <N>, <Z>, Q1, Q2, Q3, Q4, dfrag, <Nleft>, <Zleft>, <Nright>, <Zright>
- Numerically stable for large bases (e.g., 26 shells in z direction)
- Coulomb can be included in pairing field
- Coulomb exchange can be exact or Slater approx
- 2-body center-of-mass correction term can be included
- Implemented in both serial and parallel versions

## The challenge of fission calculations in a one-center basis

- Code allows different number of h.o. shells in radial and z dirs (after Warda et al. PRC 66, 014310, 2002)
- The idea: put more shells along z dir
- Graph shows problem size for full (black line) and Nz  Nr bases, when parity is broken

**Reduced (****N****z**** ******** N****r****) basis is**
**essential to keep problem**
**size manageable**

## Application: 240Pu energy versus elongation & asymmetry

- Q2 (b)

- Q3 (b3/2)

- EHFB (MeV)

**240****Pu**

**Densities along most likely path**

- Reflection symmetry is spontaneously broken
-  asymmetric fission (Q3  0)

**Notes:**
Here is an example of the static calculations we have performed for 240Pu.  The energy surface was calculated with constraints on the quadrupole mom (elong) and the octupole mom (mass asymm).  Note the valley that corresponds to the  most probable path to scission.
This path implies a spontaneous breaking of symmetry that leads to asymmetric fission as the  most likely fission mode, which is known to be the case experimentally for low-E fission.
The animation on the right shows the sequence of nuclear densities sampled as the nucleus travels along the valley.
I will show results later in the talk that correspond to fission at the end of this valley.
However, note that the valley is not that deep, this means that in the dynamical treatment, the wave packet built via the GCM will be spread out around the valley, and you will end up with a distribution of fragment masses, as is seen experimentally.
As a side note, to emphasize the richness of the technique, note the formation of clusters in the densities, it might be interesting to integrate out the number of particles in the clusters, identify, try to understand what they are...

## The manifold picture of scission

- Q20-Q40 map shows two HFB solutions
- Fission valley: unbroken
- Fusion valley: broken
- Barrier separates valleys
- ~ 5 MeV for low Q20
- Disappears at large Q20
- Describes cold/hot fission
- Cold  fragments with no Ex
- Hot  fragments with high Ex
- + everything in between

**Berger et al., NPA 428, 23 (1984)**

**240****Pu: calculation for most likely Q****30**

**Additional coll. var. Q****40**** leads to new physics!**
**Observed experimentally: Signarbieux et al., **
**JPL 42, 437 (1981)**

## Application: calculation of “tepid” fission for 240Pu

**Integrate densities**
**separately:**
** ****135****I/****105****Nb**
** ****d = 16.9 fm**
** ****TKE = 185.5 MeV**
** ****TXE = 18.9 MeV**
** ****E****x****(****135****I) = 10.2 MeV**
** ******** ****<********>(****135****I) ****≈**** 1.04**
** ****E****x****(****105****Nb) = 8.7 MeV**
** ******** ****<********>****(****105****Nb) ****≈**** 1.00**

**HFB calcs with 6**
**constraints!**

**Repeat on (Q****20****,Q****30****,Q****40****) mesh ******** huge parallel job**
**For further study: is Q****40**** the ideal collective var?**

## Mapping the 240Pu hot-scission line: method

- Take small steps in Q20 and Q30, seek drop into fusion valley
- When fission-fusion barrier disappears convergence can be tricky
- Example of converged pre-scission (unbroken) solution:

## Mapping the 240Pu hot-scission line: method

**Mapping the ****240****Pu hot-scission line: method**

- Take small steps in Q20 and Q30, seek drop into fusion valley
- When fission-fusion barrier disappears convergence can be tricky
- Example of converged pre-scission (unbroken) solution:

**Wrong! Looks can be deceiving!**

## Mapping the 240Pu hot-scission line: method

- Take small steps in Q20 and Q30, seek drop into fusion valley
- When fission-fusion barrier disappears convergence can be tricky
- Example of converged **post**-scission (**broken**) solution:

**Iterating further shows the solution is in the fusion valley **

## Mapping the 240Pu hot-scission line: first results

**Mapping the ****240****Pu hot-scission line: first results**

**Scission line**

**Separation between fragments**

**Fragment masses**

**Radical change near symmetric limit!**

## Basis size and optimization

**Basis size and optimization**

- Currently using 0 = 8 MeV, q  / z = exp[3/2/(2 +1)] (_[www.phynu-cea.fr](http://www.phynu-cea.fr/)_)
- Crucial to optimize 0 and q
- In progress via parallel mapping of EHFB(0,q) at each point (10-cpu years)
- Will probably affect post-scission most
- Currently using Nr = 13, Nz = 22
- Crucial to explore basis-size effects
- Next: Nr = 15, Nz = 25
- Smaller steps in Q20, Q30 needed

**Energy along scission line: before optimization**

## Conclusion

- LLNL program based on successful BIII fission program
- Finite-range HFB code written and in use
- Currently extracting static HFB solutions
- Mapping hot-scission line
- Investigating basis-size and optimization effects
- Next: extract fission-fragment properties (Ex, TKE, shape)
- Along hot-scission line
- Everywhere else ( Q20, Q30, Q40 constrained)
- Next: dynamic treatment
- Fragment yields
- Fission times

## The microscopic method: dynamics

- Generator-coordinate method + Gaussian-overlap approximation
- Solve HFB for values of collective coordinates on a mesh
- Construct wave packet as linear superposition of HFB solutions
- weights are given by variational procedure that minimizes energy
- Simplification: assume overlap of HFB solutions is Gaussian function of difference in collective-coordinates  favor similar “shapes”
- Wave packet is spread over all nuclear configurations
- Wave packet is allowed to evolve naturally out to scission
- Yields _Collective Hamiltonian_ built from single-particle d.o.f.!

- Fully quantum-mechanical approach

**Notes:**
Once you have constructed the static solutions of interest, there is a well-defined, self-consistent procedure to put the nucleus in motion and treat fission dynamically.
The procedure is the GCM...

## Hot vs. cold fission: dynamics

**The cold-fission mechanism:**
** *****Q*****20****-*****Q*****40**** coupling from two sources**** **

**Notes:**
Q: how does the system get past the barrier between the two valleys in cold fission?
A: via the coupling of the collective d.o.f. Q2 and Q4, as shown by Berger et al.
The coordinates are clearly coupled in both kinetic and potential terms in the collective Ham. (by the  way, to re-iterate, Hcoll falls right out of the formalism, nothing is added a posteriori here)

## Hot vs. cold fission: dynamics

**The cold-fission mechanism:**
** *****Q*****20****-*****Q*****40**** coupling from two sources**** **
** ****Transverse motion**
** ****excitation above barrier**
** ****slowing-down along longitudinal dir.**
** ****effect mimicked by ad-hoc dissipation term in semiclassical models**

**Naturally built into the microscopic**
**approach**

**Notes:**
The animation is a schematic representation of what happens.
The nucleus is moving along, then via the coupling of Q2 and Q4, energy is taken out of the longitudinal motion, and pumped into the transverse motion in V1.  As a result, the nucleus is excited to higher collective states in V1, like climbing up a ladder.
Eventually, the system climbs above the barrier, and runs away into V2.
This mechanism is much more likely than tunneling
This effect, which occurs naturally in the microscopic method usually has to be simulated in semiclassical models of fission by adding dissipation terms.
Instead this coupling between collective dof leads to calculated fission times that are quite realistic.
This was one of the triumphs of the technique obtained by Berger et al. in the late 80’s.

## Project timeline

**Low-E fission**
**(statics)**

**Low-E fission**
**(dynamics)**

**Inter-E fission**
**(dynamics)**

**High-E fission**
**(statics)**

**High-E fission**
**(dynamics)**

**FY08**

**FY09**

**FY10**

**FY11**

**FY12-13**

**Statics ******** frag. properties (E****x****, TKE, structure)**** ******** emission spectra**
**Dynamics**** ******** ****fission yields, fission times**

## Adiabatic approximation for low-energy fission

**E****x**** < 1.8 MeV ******** no coupling to intrinsic excitations**
**1.8 MeV < E****x**** < 3.6 MeV ******** no 0-to-2 qp coupling from HFB, weak coupling from GCM via ********Q**
**E****x**** > 3.6 MeV (e.g., 5 MeV) ******** strong coupling possible ******** not adiabatic**

**E level at 1****st**** barrier**
**(lowered by triaxiality)**

**2-qp energies remain**
**around 1.8 MeV**

**E****x**

## FRANCHBRIE execution times

- Typically 40 iters/calc (varies a lot, could be reduced by 2 or more)
- Coulomb in pairing  1.28
- Exact Coulomb exchange  1.52
- Coulomb in pairing & exact exchange  1.81

|  | Nr = 13, Nz = 19 | Nr = 15, Nz = 22 | Nr = 17, Nz = 25 |
| --- | --- | --- | --- |
| Q2 | 153 s/iter | 532.5 s/iter | 745 s/iter |
| Q2, Q3 | 166 s/iter | 576 s/iter | 808.5 s/iter |
| Q2, Q3, Q4 | 179 s/iter | 623 s/iter | 871.5 s/iter |