Roles of nuclear weak rates on the evolution of degenerate cores in stars

Electron-capture and β -decay rates in stellar environments are evaluated with the use of new shell-model Hamiltonians for sd -shell and pf -shell nuclei as well as for nuclei belonging to the island of inversion. Important role of the nuclear weak rates on the final evolution of stellar degenerate cores is presented. The weak interaction rates for sd -shell nuclei are calculated to study nuclear Urca processes in O-Ne-Mg cores of stars with 8-10 M⊙ (solar mass) and their effects on the final fate of the stars. Nucleosynthesis of iron-group elements in Type Ia supernova explosions are studied with the weak rates for pf -shell nuclei. The problem of the neutron-rich iron-group isotope over-production compared to the solar abundances is shown to be nearly solved with the use of the new rates and explosion model of slow defraglation with delayed detonation. Evaluation of the weak rates is extended to the island of inversion and the region of neutron-rich nuclei near 78 Ni, where two major shells contribute to their configurations.


Introduction
Roles of the rate of nuclear processes driven by electron-capture and β-decays on the final evolution of electron-degenerate cores of stars are investigated.Standard electron-capture and β-decay rates of nuclei presented in a recent literature [1] are those for (1) sd-shell nuclei by [2], (2) pf -shell of LMP [3], (3) pf − g-shell of LMSH [4], and (4) Fuller-Fowler-Newman (FFN) rates [5].Besides them, an approximate two-parameter formula for the rates is obtained [6] to be used for nuclei, which have no available data tables.
Here, we study nuclear Urca processes in stars with O-Ne-Mg cores by employing a set of new weak rates computed by the USDB [7] for sd-shell nuclei.Nucleosynthesis of iron-group nuclei in Type Ia supernova explosions (SNe) (see [8] for a recent review on nucleosynthesis in Type Ia SNe and hydrodynamical explosion models) is discussed by using a new Hamiltonian GXPF1J [9] for pfshell nuclei.We also extend our study to the weak rates for nuclei in the island of inversion, where 3 Electron-capture rates in p f -shell nuclei and nucleosynthesis in Type Ia SNe New shell-model Hamiltonians in pf -shell, GXPF1 [14] and GXPF1J [9], can describe spin properties of pf -shell nuclei quite well.The Gamow-Teller (GT) strengths in Ni isotopes and M1 strengths in various pf -shell nuclei are also well described [15].In particular, for 56 Ni, the GT strength distribution is nicely reproduced [15,16].In Type-Ia supernova explosions, a plenty amount of 56 Ni is produced.Production yields of neutron-rich nuclei and the lepton-to-baryon ratio (or proton fraction) Y e depend sensitively on the rate of e-capture on 56 Ni.The ratio of the production yields of 58 Ni over 56 Ni can be reduced nearly by half for GXPF1J compared to KB3G [17].
The problem of over-production of 58 Ni, 54 Cr and 54 Fe compared to the solar abundance was discussed [18] with the use of the capture rates of FFN [5].A possible solution of the problem with slower e-capture rates was discussed in [3,19].The problem could be solved by using slower e-capture rates of GXPF1J.Here, we use the W7 and WDD2 model [18] as explosion models of Type-Ia supernova explosions starting from C-O white dwarf with a mass 1.38 M ⊙ .The W7 model proceeds by a fast deflagration while the WDD2 model proceeds by a slow deflagration with delayed detonation.Electron-capture rates of GXPF1J are used for pf -shell nuclei with 21≤Z ≤32 and those of KBF [3] otherwise.At high densities above a few 10 8 gcm −3 , slower e-capture processes on Ni and Fe isotopes lead to less production of neutron-rich nuclei (see [20] for the details).The results of final element abundances are shown in Ref. [20].In the WDD2 (W7) model, over-production of neutron-rich Cr, Fe and Ni isotopes compared with the solar system abundances is suppressed within a factor of 2 (2-3).This is much smaller than the case of FFN weak rates, where the over-production factor becomes as large as up to 4-5.

Weak rates for nuclei in the island of inversion
Urca processes for nuclear pairs in the island of inversion [21] such as 31 Mg-31 Al and 33 Mg-33 Al pairs have been pointed out to be important for the cooling of neutron star crusts [22] ) are sucessfully reproduced by SDPF-M Hamiltonian [23] with sd − pf -shell configurations.The important contributions from 2p-2h components are found in 30 Ne and 32 Mg.We discuss the weak rates for 31 Mg-31 Al pair.The SDPF-M fails to reproduce the energy levels of 31 Mg, that is, 7/2 − state becomes the ground state while the experimental g.s. is 1/2 + .The Urca density can not be clearly assigned for the weak rates for SDPF-M.This shortcoming can be improved for the effective interaction obtained by EKK (extended Kuo-Krenciglowa) method [24], which can properly treat Q-box calculations in two-major shells without divergence problems [25].Energy levels in 31 Mg are well reproduced, and more important roles of p-h excitations are noticed compared with the SDPF-M case.The 4p-4h components are found to be as much as the 2p-2h components in 32 Mg.The weak rates are evaluated with the EKK method and they prove to be close to those obtained by taking into account the available experimental data as shown in Fig. 1.For the EKK approach, the Urca density can be assigned to be log 10 ρY e = 10.14.

Summary
New weak rates for sd-shell and pf -shell nuclei are obtained with USDB and GXPF1J Hamiltonians, respectively.The rates for sd-shell are used to study evolutions of the O-Ne-Mg cores of stars with 8-10 M ⊙ .Nuclear Urca processes for A = 23 and 25 nuclear pairs are found to be important for the cooling of the cores and determination of the fate of the stars whether they end up with ECSNe or CCSNe.The rates for pf -shell are used to study nucleosynthesis of iron-group elements in Type Ia SNe.Relatively smaller e-capture rates for GXPF1J compared with KB3G, KBF and FFN are found to considerably suppress the over-production of neutron-rich iron-group isotopes, especially in the WDD2 explosion model with detonation.The weak rates for GXPF1J are tabulated for further study of astrophysical processes [26].
[21]rge sd − pf shell in neutron-rich Ne, Na and Mg isotopes near N=20.Lowering of 2 + 1 states and enhancement of E2 transition strengths show evidence for the breaking of the magicity at N =20[21].