Lawrence Livermore National Laboratory

1PLS Directorate, Physics Division – LLNL, Livermore, CA2CEA, DAM, DIF, Arpajon, France

3University of North Carolina, Chapel Hill, NC

Accurate Reaction Cross-section Predictions

for Nucleon-Induced Reactions06/24/2010

G. P. A. Nobre1,*, I. J. Thompson1, J. E. Escher1, F. S. Dietrich1, M. Dupuis2, J. Terasaki3 and J. Engel3

Prepared by LLNL under Contract DE-AC52-07NA27344

Performance Measures x.x, x.x, and x.x

*nobre1@llnl.gov

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PLS Directorate - Physics Division - NPS

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1: UNEDF project: a national 5-year SciDAC collaboration

TargetA = (N,Z)

UNEDF:VNN, VNNN…

Veff forscattering

Structure ModelsMethods: HF, DFT,

RPA, CI, CC, …

TransitionDensity [Nobre]

Ground state Excited states

Continuum states

Folding[Escher, Nobre]

Transition Densities

KEY:UNEDF Ab-initio InputUser Inputs/Outputs

Exchanged DataRelated research

Eprojectile

Transition Potentials

Coupled Channels

[Thompson, Summers]

Optical Potentials[Arbanas]

Preequilibriumemission

PartialFusionTheory

[Thompson]

Hauser-Feshbach

decay chains[Ormand]

Compoundemission

Residues (N’,Z’)

ElasticS-matrixelements

Inelasticproduction

Voptical

Global opticalpotentials

Deliverables

UNEDF Reaction Work

ResonanceAveraging[Arbanas]

Neutron escape[Summers, Thompson]

or

Two-stepOpticalPotential

PLS Directorate - Physics Division - NPS

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Nuclear Excited States from Mean-field Models

Mean-field HFB calculations using SLy4 Skryme functional Use (Q)RPA to find all levels E*, with transition densities from the g.s.

Uncorrelatedparticle-hole states

Correlated p-h states in HO basis

Correlated p-h states in 15 fm box

Neutron separation energy is 9.5 MeV.Above this we have discretized continuum.

Collaboration withChapel Hill: Engel& Terasaki

PLS Directorate - Physics Division - NPS

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Diagonal Density

PLS Directorate - Physics Division - NPS

Example of diagonalDensity for 90Zr

Example of diagonalDensity for 90Zr

RPA

Folding of densities with n-n interaction Transition potentials

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Transition densities to Transition potentials

Diagonal folded potential Off-diagonal couplings

Natural parity states only: no spin-flip, so no spin-orbit forces generated.No energy or density dependence. Exchange contributions included implicitly.

All potentials real-valuedAll potentials real-valued

(So far)PLS Directorate - Physics Division - NPS

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PLS Directorate - Physics Division - NPS

Reaction Cross Sections with Inelastic Couplings

(Q)RPA Structure Calculations for n,p + 40,48Ca, 58Ni, 90Zr and 144 Sm Couple to all excited states, E* < 10, 20, 30, 40 MeV Find what fraction of σR corresponds to inelastic couplings

Not Converged yet! E* < 50, 60, 70, …?Not Converged yet! E* < 50, 60, 70, …?

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Inelastic Convergence

Coupling to more states gives larger effect

Convergence appears when all open channels are coupled

PLS Directorate - Physics Division - NPS

For reactions with protons as projectile, inelastic convergence is achieved with less couplings due to the Coulomb barrier

Protons as projectile

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Coupling Between Excited States

PLS Directorate - Physics Division - NPS

g.s.

At not too low energies:

Individual cross-sections change very little, except for some few states: up to 20%

Overall sum of reaction over states remains the same

Supports the concept of “doorway states”

At not too low energies:

Individual cross-sections change very little, except for some few states: up to 20%

Overall sum of reaction over states remains the same

Supports the concept of “doorway states”

Details in paper being prepared

for submission to PRC

Details in paper being prepared

for submission to PRC

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Pick-up Channel: Deuteron Formation

PLS Directorate - Physics Division - NPS

N. Keeley and R. S. Mackintosh* showed the importance of

including pick-up channels in coupled reaction channel (CRC)

calculations.

N. Keeley and R. S. Mackintosh* showed the importance of

including pick-up channels in coupled reaction channel (CRC)

calculations.

40Ca(d,d) elastic scattering

*Physical Review C 76, 024601 (2007) Physical Review C 77, 054603 (2008)

d

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Contribution of Transfer Channels

€

0hω

€

1hω

€

2hω

€

3hω

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0s€

0p€

0d

€

1s

€

0 f

€

1p€

0g

€

0s1/2€

0p3/2

€

0p1/2

€

0d5/2

€

1s1/2

€

0d3/2

€

0 f7/2

€

1p3/2

€

0 f5/2

€

1p1/2

€

0g9/2

€

0g7 / 2

€

1d5/2

€

(2)€

(4)

€

(2)

€

(6)

€

(2)

€

(4)

€

(8)

€

(4)

€

(6)

€

(2)

€

(10)

€

Nhω

€

nL

€

nL j

€

2 j +1

€

2 j +1∑

€

2€

6

€

8

€

14

€

16

€

20

€

28

€

32

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38

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40

€

50

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2

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50

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20

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8

HarmonicOscillator

FiniteWellN=2n+L

With spin-orbit force nocc(j) Closedshells

Sum

Neutrons

Protons

There are many nucleons in the targetthat can be picked out to make a deuteron.

Effect depends on binding energy and size of bound state wave functions: Given by the mean-field model

Large contribution to σR: closer to OM!

Significant non-orthogonality effects

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Non-Orthogonality and Fraction of σR

PLS Directorate - Physics Division - NPS

Behaviour of non-orthogonality is sensitive to changes of the deuteron potential:

Coupling to 90Zr(n,d,n) channel gives a large increment, approaching to the optical model calculation.

Non-Orthogonality has an additional effect.

Better definition needed!

Using Johnson-Soper* prescription:

Vd(R)=Vn(r)+Vp(R) Vd(R)=Vn(r)+Vp(R)

αCC < αCC+CRC and αCC+CRC+NOαCC < αCC+CRC and αCC+CRC+NO

Using the Daehnick et al.§ potential for the deuteron.

*Physical Review C 1, 976 (1970)§Physical Review C 21, 2253 (1980)

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Comparison with Experimental Data

PLS Directorate - Physics Division - NPS

Good description of experimental data!

Good description of experimental data!

There is still possibility for improvements.

Inelastic convergence when coupling up to all

open channels

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Comparison with Experimental Data

PLS Directorate - Physics Division - NPS

Good description of experimental data!

Good description of experimental data!

Inelastic and pick-up channels account for all reaction cross sections

arXiv:1006.0267Submitted to PRLarXiv:1006.0267

Submitted to PRL

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Summary of Results at Elab = 30 MeV

PLS Directorate - Physics Division - NPS

Inelastic + Transfer with non-orthogonality

Inelastic couplings only

Inelastic + Transfer

Phenomenological Optical Model

Targets

40Ca, 48Ca, 58Ni, 90Zr, 144Sm

With all couplings, calculations agree with experimental data

arXiv:1006.0267 - Submitted to PRLarXiv:1006.0267 - Submitted to PRL

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Elastic Angular Distributions

PLS Directorate - Physics Division - NPS

• Provide complementary information on reaction mechanisms• Are sensitive to the effective interaction used

Density-dependent effective interaction:

•Resulting coupling potentials improve large-angle behavior, still need improvements for small angles.•Work in progress to treat and then test UNEDF Skyrme functionals.

Our approach predicts a variety of reaction observables.

Data provides constraints on the ingredients.

Our approach predicts a variety of reaction observables.

Data provides constraints on the ingredients.

Results will be shown in paper being prepared for

submission to PRC

Results will be shown in paper being prepared for

submission to PRC

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Conclusions

Inelastic (Q)RPA couplings account for a fraction of reaction cross-section

To achieve convergence, couplings to (at least) all open channels is necessary

Coupling to pick-up channel is very important• Deuteron potential• Non-orthogonality

Concept of “doorway states” is a good approximation: simplifies the problem, saves computational time

Coupling to inelastic and deuteron channels accounts for (almost) all reaction cross sections

Angular distributions are sensitive to the effective interaction

PLS Directorate - Physics Division - NPS

arXiv:1006.0267 - Submitted to PRLarXiv:1006.0267 - Submitted to PRL

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PLS Directorate - Physics Division - NPS

Future Work - Next Steps

Incorporate UNEDF functionals into folding potentials and examine effects, in particular density-dependence (Jutta Escher)

Use densities from deformed QRPA code (Terasaki & Engel, Chapel Hill, NC)*

Analyze reactions on a range of nuclei, using spherical and deformed QRPA transition densities and functionals from UNEDF*

Two-step approach (Ian Thompson)

Couple to even higher states to achieve convergence*

Solve consistency issues about deuteron potential, break-up, triton coupling

Draw conclusions about the most important ingredients for a predictive reaction calculation.

*Computational Challenges

Future Publications:

• Detailed paper to be submitted to PRC• Proceeding for INPC2010, Vancouver