# Presentation: 122338

## Compound Nucleus Contributions to the Optical Potential

- Ian Thompson,
- Jutta Escher and Frank Dietrich
- Nuclear Theory and Modeling Group,
- Lawrence Livermore National Laboratory

- UCRL-PRES-235658

- This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344, and under SciDAC Contract DE-FC02-07ER41457

- JUSTIPEN

- Jan 2008

## Nonelastic Channels

- Optical Potential for n+A Elastic Scattering:
- monopole folding potential,
- + _dynamic polarisation potential_ from all non-elastic reactions.
- Direct Reactions,
- Examples: collective inelastic states or pickup
- All remove flux from the elastic channel
- Effect on elastic scattering is an imaginary contribution to the optical potential, giving
- Reaction Cross Section
- Full Calculation: DPP has Real & Imaginary Components
- Energy dependence of these related by dispersion integrals.

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- Jan 2008

## Compound Nucleus States

- CN States are Long-lived Resonances
- narrow peaks in an incident-energy spectrum.
- Remove Flux from the Elastic Channel, which Flux is emitted some long time later:
- either back to the elastic channel,
- or by -ray or particle emissions.
- After a long time,
- No remaining information about the incident beam direction,
- Decays are isotropic (subject to conserved quantum numbers)

- JUSTIPEN

- Jan 2008

## The Optical Potential

- Defined to include the effects of all ‘fast’ absorption from the elastic channel
- when averaged over some interval *I* »*D*, where *D* is the level spacing
- So CN states give optical-model absorption
- This is to treat separately:
- Shape Elastic
- From the optical potential
- Compound Elastic:
- What only much later feeds back to the elastic channel.

- JUSTIPEN

- Jan 2008

## Average Widths

- To calculate the optical potential, need information about (average) CN resonances.
- The ratio of the average width of the resonances <> to D gives the reaction cross section loss in the elastic channel :
- 1 – |*S*opt|2 =  2 <>/*D*
- (This is the ratio needed for Hauser-Feshbach calculations)
- BUT: to calculate the <>/*D* ratio, 	microscopic details needed,	either _statistical_, or _schematic_.

- JUSTIPEN

- Jan 2008

## Schemes for finding <>/D

- <>/*D* is the fraction, total-width/spacing.
- Consider doorway states
- (those reached from first particle-hole step)
- These will be ‘fractioned’ into all the final CN states,
- BUT:
- Initial doorways and final CN states have similar <>/*D*
- SO:
- try to model the doorway states so they have correct average physical widths <> and spacings *D*

- JUSTIPEN

- Jan 2008

## Coupled Channels Models

- Try to explicit couple elastic to CN states
- Too many to do all of these, so:
- Just focus on the _particle-hole Doorway States_
- Do coupled-channels calculations:
- _Either_: pure particle-hole excitations in mean field,
- _Or_: Correlated p-h states from Random Phase Approximation (RPA) model of excitations (so include some residual interactions in target)
- (Starting to) Unify Direct Reaction and Statistical Methods

- JUSTIPEN

- Jan 2008

## Steps in OM calculation

- Nucleus AZ: here 90Zr.
- Hartree-Fock gs + RPA excitations
- Transition densities gs  E*(f)
- Folding with effective Vnn  Vf0(r;)
- Large Coupled-channels calculations
- Extract S-matrix elements *S*'
- Hence:
- Reaction cross sections R(L) = (2L+1) [1–|*S*|2]/*k*2
- Elastic 
- Use partial reaction cross sections R(L) in HF models
- (If desired) fit to find elastic optical potential
- 	 [An optical potential = convenient way of generating R(L; E) ]

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- Jan 2008

## Particle-hole & RPA levels

- Spherical HF calculations from Marc Dupuis
- Using Gogny's D1S’ force (Vso=–115 MeV)
- Harmonic oscillator basis, 14   where   = 13.70 MeV minimises the 90Zr gs
- RPA calculation of spectrum
- (removing spurious 1– state that is cm motion)
- Extract super-positions of particle-hole amplitudes for each state.

- Note: this only asmall fraction of all the levels!

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- Jan 2008

## Folding with effective Vnn to get transition gs  E*(f) 

- Use Love’s effective Vnn derived from M3Y
- (fit with Gaussians)
- direct + approximate (ZR) exchange
- Folded with RPA transition densities using Fourier method
- Derived transition potentials Vf0(*r*;) from gs to each excited state, of multipole 

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- Jan 2008

## Coupled channels n+A*

- Add Woods-Saxon real monopole V0(r)
- NO imaginary part in any input
- Fresco Coupled inelastic channels at Elab(n)=40 MeV
- E* < 10, 20 or 30 MeV, with ph and RPA spectra.
- Maximum 1277 partial waves.
- RPA moves 1– strength (to GDR), and removes c.m. motion and enhances collective 2+, 3–

- RPA:

- PH:

- n+90Zr
- at 40 MeV

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- Jan 2008

## Predicted Cross sections

- Calculate reaction cross section R(L) for each incoming wave L
- Guidance: compare with R(L) from fitted optical potential such as Becchetti-Greenlees (black lines)
- _Result_: with RPA and all 30 MeV of spectrum, we obtain about HALF of ‘observed’ reaction cross section.
- Optical Potentials can be obtained by fitting to elastic *S*L or el()

- PH

- RPA

- n+90Zr
- at 40 MeV

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- Jan 2008

## Damping of Doorway States

- Doorway States couple to _further_ ph states: the 2p2h states
- (giving 3p2h, including incident nucleon)
- So: Doorways damped just like the incident 1p state!
- Try using _observed_ 1p damping for _each_ of the doorway states?
- (ignoring escape widths of the RPA/1p1h states)

- NOT a large effect in this approx.
- Unless excited states damped More

- JUSTIPEN

- Jan 2008

## Improving the Accuracy

- RPA model has Low-Lying Collective + Giant Resonance  States.
- Is this structure Accurate?
- We should couple Between RPA states
- Known to have big effect in breakup reactions
- Pickup reactions in second order: (n,d)(d,n)
- Re-examine Effective Interaction Vnn
- Especially its Density-Dependence

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- Jan 2008

## Resonance Averaging

- At lower energy these CC calculations will give resonances, from closed inelastic channels.
- Must Average theoretical curves over resonances
- _Or_ use Complex Energy. For interval I:
- <*S *(E)>  = *S *(E + *i* I)
- Note: CC calculations with only doorway states have only SMALL level densities:
- Much Smaller than Observed CN-resonance level density.

- JUSTIPEN

- Jan 2008

## Conclusions

- We can now Begin to:
- Use Structure Models for Doorway States, to
- Give Transition Densities, to
- Find Transition Potentials, to
- Do large Coupled Channels Calculations, to
- Extract Reaction Cross Sections & Optical Potentials
- Still Need:
- More systematic calculation of Doorway Widths
- Higher Level-Densities of resonance, & their Averaging.
- (Starting to) Unify Direct Reaction and Statistical Methods

- JUSTIPEN

- Jan 2008

## www.kernz.org

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- Jan 2008