Advances in High Energy Physics

Volume 2015 (2015), Article ID 398796, 11 pages

http://dx.doi.org/10.1155/2015/398796

## Electron Capture Cross Sections for Stellar Nucleosynthesis

Division of Theoretical Physics, University of Ioannina, 45100 Ioannina, Greece

Received 11 July 2014; Accepted 8 October 2014

Academic Editor: Athanasios Hatzikoutelis

Copyright © 2015 P. G. Giannaka and T. S. Kosmas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

In the first stage of this work, we perform detailed calculations for the cross sections of the electron capture on nuclei under laboratory conditions. Towards this aim we exploit the advantages of a refined version of the proton-neutron quasiparticle random-phase approximation (pn-QRPA) and carry out state-by-state evaluations of the rates of exclusive processes that lead to any of the accessible transitions within the chosen model space. In the second stage of our present study, we translate the abovementioned -capture cross sections to the stellar environment ones by inserting the temperature dependence through a Maxwell-Boltzmann distribution describing the stellar electron gas. As a concrete nuclear target we use the ^{66}Zn isotope, which belongs to the iron group nuclei and plays prominent role in stellar nucleosynthesis at core collapse supernovae environment.

#### 1. Introduction

Weak interaction processes occurring in the presence of nuclei under stellar conditions play crucial role in the late stages of the evolution of massive stars and in the presupernova stellar collapse [1–6]. As it is known, the core of a massive star, at the end of its hydrostatic burning, is stabilized by electron degeneracy pressure as long as its mass does not exceed an appropriate mass (the Chandrasekhar mass limit, ) [6–10]. When the core mass exceeds , electron degeneracy pressure cannot longer stabilize the center of the star and the collapse starts. In the early stage of collapse electrons are captured by nuclei in the iron group region [6, 10].

During the presupernova evolution of core collapse supernova, the Fermi energy (or equivalently the chemical potential) of the degenerate electron gas is sufficiently large to overcome the threshold energy ( is given by negative values of the reactions involved in the interior of the stars) [11] and the nuclear matter in the stellar core is neutronized. This high Fermi energy of the degenerate electron gas leads to enormous -capture on nuclei and reduces the electron to baryon ratio [12, 13]. In this way, the electron pressure is reduced and the energy as well as the entropy drop. One of the important characteristics of the early pre-explosion evolution is the fact that electron capture on nuclei (specifically on nuclei of the pf shell) plays a key role [14, 15].

In the early stage of collapse (for densities lower than a few g cm^{−3}), the electron chemical potential is of the same order of magnitude as the nuclear value, and the -capture cross sections are sensitive to the details of GT strength distributions in daughter nuclei. For this reason, some authors restrict the calculations only to the GT strength and evaluate -capture rates on the basis of the GT transitions (at these densities, electrons are captured mostly on nuclei with mass number ) [9–12, 15, 16]. Various methods, used for calculating -capture on nuclei during the collapse phase, have shown that this process produces neutrinos with rather low energies in contrast to the inelastic neutrino-nucleus reactions occurring in supernova [17–20]. These neutrinos escape the star carrying away energy and entropy from the core which is an effective cooling mechanism of the exploding massive star [21]. For higher densities and temperatures, capture occurs on heavier nuclei [8, 10, 13–15]. As a consequence, the nuclear composition is shifted to more neutron-rich and heavier nuclei (including those with ) which dominate the matter composition for densities larger than about g cm^{−3} [1, 15, 21, 22].

The first calculations of stellar electron capture rates for iron group nuclei have been performed by employing the independent particle model (IPM) [2–4]. Recently, similar studies have been addressed by using continuum RPA (CRPA) [25], large scale shell model [26, 27], RPA [11], and so forth [28]. In the present work -capture cross sections are obtained within a refined version of the quasiparticle random phase approximation (QRPA) which is reliable for constructing all the accessible final (excited) states of the daughter nuclei in the iron group region of the periodic table [29–38]. For the description of the required correlated nuclear ground states we determine single-particle occupation numbers calculated within the BCS theory [29, 31, 32]. Our nuclear method is tested through the reproducibility of experimental muon capture rates relying on detailed calculations of exclusive, partial, and total muon capture rates [23, 24, 39–42]. The agreement with experimental data provided us with high confidence level of our method and we continued with the calculations of electron capture cross sections in supernova conditions (where the densities and temperatures are high) using the pn-QRPA method. In this paper, we performed calculations for isotope (it belongs to the iron group nuclei) that plays prominent role in core collapse supernovae stellar nucleosynthesis [18, 19, 43].

Our strategy in this work is, at first, to perform extensive calculations of the transition rates for all the abovementioned nuclear processes, assuming laboratory conditions, and then to translate these rates to the corresponding quantities within stellar environment through the use of an appropriate convolution procedure [9, 14, 15, 21, 26]. To this purpose, we assume that leptons under such conditions follow Maxwell-Boltzmann energy distribution [9, 26].

#### 2. Construction of Nuclear Ground and Excited States

Electrons of energy are captured by nuclei interacting weakly with them via boson exchange as follows:The outgoing neutrino carries energy while the daughter nucleus absorbs a part of the incident electron energy given by the difference between the initial and the final nuclear energies as .

The nuclear calculations for the cross sections of reaction (1) start by writing down the weak interaction Hamiltonian which is given as a product of the leptonic, , and the hadronic, , currents (current-current interaction Hamiltonian) as follows:where with and being the well-known weak interaction coupling constant and the Cabibbo angle, respectively [31, 32, 44].

From the nuclear theory point of view, the main task is to calculate the cross sections of reaction (1) which are based on the evaluation of the nuclear transition matrix elements between the initial and a final nuclear states of the formThe quantity stands for the leptonic matrix element written in coordinate space with being the 3-momentum transfer. For the calculation of these transition matrix elements one may take advantage of the Donnelly-Walecka multipole decomposition which leads to a set of eight independent irreducible tensor multipole operators containing polar-vector and axial-vector components [44] (see Appendix A).

In the present work, in (3) the ground state of the parent nucleus is computed by solving the relevant BCS equations which give us the quasiparticle energies and the amplitudes and that determine the probability for each single particle level to be occupied or unoccupied, respectively [31]. Towards this aim, at first, we consider a Coulomb corrected Woods-Saxon potential with a spin orbit part as a mean field for the description of the strong nuclear field [45, 46]. For the latter potential we adopt the parameterization of IOWA group [47]. Then, we use as pairing interaction the monopole part of the Bonn C-D one meson exchange potential. The renormalization of this interaction, to fit in the isotope, is achieved through the two pairing parameters , for proton (neutron) pairs, and the values of which are tabulated in Table 1.