W A RICHTER

ITHEMBA LABS

UNIVERSITY OF THE WESTERN CAPE

B A BROWN and C WREDE

NSCL, MICHIGAN STATE UNIVERSITY

Determination of the important 30P(p,γ)31Sastrophysical rapid-proton capture reaction

rate

Introduction

OUTLINE

Brief review of interactions USDA and USDB

The WBP interaction and negative parity calculations

Matching experiment and theory

(p,gamma) reaction rates

INC interactions USDA-cdpn and USDB-cdpn

Conclusions

The thermonuclear rate of the reaction 30P(p,γ)31S reaction is important for interpreting nova nucleosynthesis in the A ≥ 30 region.

We present results for the first time of calculations in a full 1ћω model space for several negative parity states. We use USDB-cdpn for positive parity.

Spectroscopic factors and proton-decay widths are calculated for input into the reaction rate. Available experimental data is used in conjunction with the calculations to obtain an estimate for the reaction rate.

INTRODUCTION

1983: Hobson Wildenthal obtains USDinteraction by fitting SPE and TBME (3+63=66) to ~ 450 energies in sd shell

2006: Alex Brown and W R did model-independent fits including neutron-rich nuclei and previously omitted nuclei – a total of 608 levels in 77 nuclei

Brief review of Interactions USDA and USDB

For USDA, 30 well-determined Linear Combinations (170 keV rms)

For USDB, 56 well-determined LC’s (130 keV rms)

Generally good agreement with experiment for all sd-shell observables calculated with the effective interactions USDA and USDB [ Richter, Mkhize, Brown, Phys. Rev. C 78, 064302 (2008) ] Optimal g factors and effective charges were determined from least-square fits to 48 magnetic moments, 26 quadrupole moments, 111 M1 transitions and 144 E2 transitions.

For level energies USDB provided a superior agreement (130 keV rms fit deviations). USD overbinds both the n-rich F and O isotopes. Both USDB and USDA gave improved binding energies for neutron-rich nuclei compared to USD

In 1989 W E Ormand and B A Brown (NP A 491, 1) reproduced 42 b coefficients with an rms deviation of 27 keV and 26 c coefficients with an rms dev of 9 keV using a charge-dependent Hamiltonian for A=18-22 and A=34-39 (IMME: B = a + bTz + cTz

2).We use USDA/B for the charge-independent part. The full composite Hamiltonians are referred to as USDA-cdpn and USBD-cdpn

cd refers to charge-dependent and pn to the proton-neutron formalism used in Nushellx

The INC interactions USDA-cdpn and USDB-cdpn

The WBP interaction

The WBP Hamiltonian (Warburton and Brown 1992) was designed to reproduce energies of 1ћω states for A = 10 - 20.

It also contains the sd-pf Hamiltonian (Warburton, Becker and Brown 1990) designed to reproduce energies of 1ћω states in nuclei with A = 35 – 43.

WBP has not been applied before to the middle of the sd shell due to the large dimensions involved.

Gamma widths for positive parity states from USDB-cdpn with effective charges/g-factors shown earlier (PRC 78, 064302).

For neg parity states a const gamma width of 9.1 meV was used (corresponding to an ave exp half-life of ~50 fs for mirror states in 31P). Reason: B(E1) small due to cancellation between s.p. compts of low-lying configs and those for the collective GDR (~ 20 MeV) – these weak E1 should be checked against exp.

Our calculated rates are shown in Fig. 1, based on the resonance energies in Table I.

Halflives: P-30 ~ 2.5 min. S-31: 2.55 sThus direct reaction expts generally not feasible, so use stable targets and light-ion transfer or charge exchange.

S-32(d,t)S-31 Irvine MLL Munich 2013 (p,d) Ma ORNL-HRIBF 2006 P-31(He-3,t)S-31 Wrede Yale-WNSL 2001 Parikh MLL Munich 2011

Si-28(α,n)S-31 Doherty Gammasphere ANL-ATLAS 2012

Experiments to determine the reaction rate for P-30(p,gamma)S-31.

P-30(p,γ)S-31 in inverse kinematics Kankainen NSCL

Populating L = 0 P-30(p, γ)S-31 resonances with Cl-31 decay Bennett NSCL 2014, Saastamoinen TAMU 2011Kankainen IGISOL 2006

C-12(Ne-20,nγ)S-31 Jenkins Gammasphere 2006

P-30(d,nγ)S-31 Kankainen GRETINA & S800 NSCL

CONCLUSIONS

• We have considered the major aspects leading to uncertainties in calculating the reaction rate.

• Calculations were done for the first time in this mass region in a full 1ћω model spaceto assess the important contributions of negative parity states.

• Ambiguities in the experimental assignments make it difficult to achieve a one to one correspondence with theory.

Hopefully our results will help to guide the next generation of experiments.

A special workshop was convened in July 2014 in Debrecen, Hungary as part of the Classical Novae in the Cosmos to discuss the P-30(p,gamma)S-31 reaction rate problems.

Thank you !

THANK YOU !

PRC 89, 062801 (2014) Comments re Table I

The USDB-cdpn energies have been shifted down by 240 keV in order to align theory and experiment for the well-established 3/2+ T=3/2 level at 6280 keV. Implication is a systematic defect of the USD Hamiltonians.

The energies of the 1ћω states also shifted down (by 354 keV) to align theory and exp for the 11/2- level at 6833 keV.

The 1ћω states of possible importance for the rate that are not matched with exp are given at the bottom of the table.

All levels up to 6.8 MeV can be matched with theory except for the exp levels at 6160, 6357, 6420 and 6796 keV, but the latter two are of questionable existence.

Above 6.8 MeV two 1/2+ levels at 6961 and 6975MeV have a good association with theory (latter a candidate for the T=3/2 level).

Between 6.8 and 7 MeV there are 4 unmatched exp levels and 7 theor levels, indicating levels not yet observed.

B A Brown, W A Richter, C WredePhys. Rev. C 89, 062801 (2014)

C Wrede AIP Advances 4, 041004 (2014)

Shell-model interactions are constructed in the cross-shell model space connecting the 0p and ls0d shells.The interactions have three distinctive 0p-shell, cross-shell, and 1s0d-shell parts. The latter is taken to be the previously determined W interaction. The 0p-shell interaction is represented by two-bodymatrix elements and the cross-shell by either a potential or by two-body matrix elements. The interactions are determined by least-squares fits to 51 0p-shell and 165 cross-shell binding energies.It is found that the addition of monopole terms to a potential that is otherwise similar to that of Milliner-Kurath leads to a great improvement.

Why measure excitation energies ?

Coulombpenetrability

Boltzmann e-E/kT

+_ 100 KeV

Shellmodel(OXBASH)

32Cl(p,g)33Ar Reaction rate for T=0.4 x 109 KExample: contribution from one individual resonance (5/2+)

mole s

cmeMeV][)(1054.1

39T

MeV][rE605.11

2/39

11

ATvN A1

2 1

(2 1)(2 1)pr

T

J

J J