Gamow-Teller strength distributions and electron capture rates for 55Co and 56Ni

The Gamow-Teller strength (GT) distributions and electron capture rates on 55Co and 56Ni have been calculated using the proton-neutron quasiparticle random phase approximation theory. We calculate these weak interaction mediated rates over a wide temperature (0.01x109 - 30x109 K) and density (10 - 1011 g cm-3) domain. Electron capture process is one of the essential ingredients involved in the complex dynamics of supernova explosion. Our calculations of electron capture rates show differences with the reported shell model diagonalization approach calculations and are comparatively enhanced at presupernova temperatures. We note that the GT strength is fragmented over many final states.

Indeed, the efforts to simulate the explosion numerically are found to make a substantial difference in the ultimate outcome, depending upon the progenitor models. Because the final outcome of the explosion depends so sensitively on a variety of physical inputs at the beginning of each stage of the entire process (i.e., collapse, shock formation, and shock propagation), it is desirable to calculate the presupernova stellar structure with the best possible physical data and inputs currently available. The energy budget would be balanced in favor of an explosion by a smaller precollapse iron core mass.
The evolution of the massive stars and the concomitant nucleosynthesis has been the subject of much computation [1]. During the later part of their burning cycles, these stars develop an iron core and lack further nuclear fuels (any transformation of the strongly-bound iron nuclei is endothermic). The core steadily becomes unstable and implodes as result of freeelectron captures and iron photodisintegration.
The collapse is very sensitive to the entropy and to the number of leptons per baryon, Y e [2].
These two quantities are mainly determined by weak interaction processes, namely electron capture and β decay. The simulation of the core collapse is very much dependent on the electron capture of heavy nuclides [3]. In the early stage of the collapse Y e is reduced as electrons are captured by Fe peak nuclei. The late evolution stages of massive stars are strongly influenced by weak interactions which act to determine the core entropy and electron to baryon ratio, Y e , of the presupernova star, and hence its Chandrasekhar mass which is proportional to Y e 2 [4]. Electron capture reduces the number of electrons available for pressure support, while beta decay acts in the opposite direction. Both processes produce neutrinos which, for densities ρ ≤ 10 11 g cm -3 , escape the star carrying away energy and entropy from the core. Electron The FFN rates were then updated taking into account quenching of GT strength by an overall factor of two by Aufderheide and collaborators [11]. They also compiled a list of important nuclides which affect Y e via the electron capture processes. They ranked 55 Co and 56 Ni the most important nuclei with respect to their importance 3 for the electron capture process for the early presupernova collapse.
We account here the microscopic calculation of electron capture rates in the stellar matter for the nuclei 55 Co and 56 Ni using the proton-neutron quasiparticle random phase approximation (pn-QRPA) theory.
The pn-QRPA theory [12][13][14] has been shown to be a good microscopic theory for the calculation of beta decay half lives far from stability [14,15]. The pn-QRPA theory was also successfully employed in the calculation of β + /electron capture half lives and again satisfactory comparison with the experimental half-lives were reported [16]. The pn-QRPA theory was then extended to treat transitions from nuclear excited states [17]. In view of success of the pn-QRPA theory in calculating terrestrial decay rates, Nabi and Klapdor used this theory to calculate weak interaction mediated rates and energy losses in stellar environment for sd- [18] and fp/fpg-shell nuclides [19]. Reliability of the calculated rates was also discussed in detail in [19]. There the authors compared the measured data of thousands of nuclides with the pn-QRPA calculations and got good comparison (See also [20]). Here we use this extended model to calculate the electron capture rates in stellar matter for 55 Co and 56 Ni pertaining to presupernova and supernova conditions. The main advantage of using the pn-QRPA theory is that we can handle large configuration spaces, by is diagonalized in three consecutive steps. Single particle energies and wave functions are calculated in the Nilsson model [21], which takes into account nuclear deformations. Pairing is treated in the BCS approximation. The protonneutron residual interactions occur in two different forms, namely as particle-hole and particle-particle interaction. The interactions are given separable form and are characterized by two interaction constants χ and κ, respectively.
The selections of these two constants are done in an optimal fashion. Details of the model parameters can be seen in [16,22]. In this work, we took χ = 0.  is an integral over total energy and for electron capture it is given by In the above equation, w is the total energy of the electron including its rest mass, and l w is the total capture threshold energy (rest + kinetic) for electron capture.  In our calculations, we summed the partial rates over 200 initial and as many final states (to ensure satisfactory convergence) to get the total capture rate. For details we refer to [19].
Realizing the pivotal role played by 55 Co and 56 Ni for the core collapse, Langanke and Martinez-Pinedo also calculated these electron capture rates separately [24]. They used the shell   [24].
Our electron capture rates for 55 Co and 56 Ni are shown in Figs. 3 and 4, respectively. The temperature scale T 9 measures the temperature in 10 9 K and the density shown in the legend has units of g cm -3 . We calculate these rates for densities in the range 10 to 10 11 g/cm 3 . Fig. 3 and How do our rates compare with those of [24]?
The comparison is shown in Fig. 5 and Fig. 6 for 55 Co and 56 Ni, respectively. Here the right panel shows the rate of [24]. Our rates are depicted in the left panel. These calculations were performed for the same temperature and density scale as done by [24]. ρ 7 implies density in units of 10 7 g cm -3 and T 9 measures temperature in 10 9 K.
For 55 Co, our rates are much stronger and differ by almost two orders of magnitude at low temperatures as compared to those of [24]. At higher temperatures our rates are still a factor of two more than those of [24]. Our calculation certainly points to a much more enhanced capture rates as compared to those given in [24]. The electron capture rates reported here can have a significant astrophysical impact.
According to the authors in [11], e   (rate of 7 change of lepton-to-baryon ratio) changes by about 50% due to electron capture on 55 Co (and about 25% for the case of 56 Ni). It will be very interesting to see if these rates are in favor of a prompt collapse of the core. We also note that authors in [3] do point towards the fact that the spherically symmetric core collapse simulations, taking into consideration electron capture rates on heavy nuclides, still do not explode because of the reduced electron capture in the outer layers slowing the collapse and resulting in a shock radius of slightly larger magnitude. We are in a process of finding the affect of inclusion of our rates in stellar evolution codes and hope to soon report our results.  [24]. E i (E j ) represents parent (daughter) states. For comparison the calculated GT strength by [24] is shown in the lower panel. Here the energy scale refers to excitation energies in the daughter nucleus.