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 protonneutron quasiparticle randomphase approximation (pnQRPA) and carry out statebystate 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
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 [
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
In the early stage of collapse (for densities lower than a few
The first calculations of stellar electron capture rates for iron group nuclei have been performed by employing the independent particle model (IPM) [
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 [
Electrons of energy
The nuclear calculations for the cross sections of reaction (
From the nuclear theory point of view, the main task is to calculate the cross sections of reaction (
In the present work, in (
Parameters for the renormalization of the interaction of proton pairs,
Nucleus 







^{66}Zn  1.0059  0.9271  1.7715  1.7716  1.2815  1.2814 
As it is well known, the pairing parameters,
The experimental separation energies in MeV for protons and neutrons of the target (
Nucleus 







^{66}Zn  7.979  11.059  7.052  7.454  8.924  5.269 
Subsequently, the excited states
The solution of (
For the renormalization of the residual 2body interaction (Bonn CD potential), the strength parameters, for the particleparticle
Strength parameters for the particleparticle
Positive parity states  Negative parity states  












0.827  0.547  0.686  0.854  1.300  0.994  0.200  0.486  0.622 

0.336  0.200  1.079  0.235  0.200  1.200  0.200  1.200  1.200 
At this point, it is worth mentioning that, for measuring the excitation energies of the daughter nucleus
The shift (in MeV) applied on the spectrum (seperately of each multipole set of states) of ^{66}Cu isotope, daughter nucleus of the electron capture on ^{66}Zn.
Positive parity states  Negative parity states  


0.90 

5.00 

2.50 

6.80 

2.55 

3.85 

2.50 

2.60 

1.75 

3.55 

0.55 

3.00 
Comparison of the theoretical excitation spectrum (resulting from the solution of the QRPA eigenvalue problem) with the lowlying (up to about 3 MeV) experimental one for
We must also mention that, usually, in nuclear structure calculations we test a nuclear method in two phases: first through the construction of the excitation spectrum as discussed before and second through the calculations of electron scattering cross sections or muon capture rates. Following the above steps, we test the reproducibility of the relevant experimental data for many nuclear models employed in nuclear applications (nuclear structure and nuclear reactions) and in nuclear astrophysics [
In this work we perform detailed cross section calculations for the electron capture on
At this point of the present work and in order to increase the confidence level of our method, we perform total muon capture rates calculations [
Despite the fact that the muon capture on nuclei does not play a crucial role in stellarnucleosynthesis, it is, however, important to start our study from this process since the nuclear matrix elements required for an accurate description of the
The calculations of the muon capture rates are performed in three steps. In the first step we carry out realistic statebystate calculations of exclusive ordinary muon capture (OMC) rates in
Due to the fact that there are no available data in the literature for exclusive muon capture rates, the test of our method is realized by comparing partial and total muon capture rates with experimental data and other theoretical results [
The percentage of each multipolarity into the total muon capture rate evaluated with our pnQRPA method.
Positive parity transitions  Negative parity transitions  


Portions 

Portions 

8.22 

7.94 

21.29 

44.21 

2.85 

13.32 

1.58 

0.34 

0.01 

0.23 
In the last step of testing our method, we evaluate total muon capture rates for the
For the sake of comparison, the abovementioned
Individual contribution of polarvector, axialvector, and overlap parts into the total muoncapture rate. The total muon capture rates, obtained by using the pnQRPA with the quenched value of
Total muon capture rates 


Present pnQRPA calculations  Experiment  Other theoretical methods  
Nucleus 







^{66}Zn  1.651  4.487  −0.204  5.934  5.809  4.976  5.809 
After acquiring a high confidence level for our nuclear method, we proceed with the main goal of the present study which concerns the calculations of the electron capture cross sections. As mentioned before, this includes original (see Section
The original cross sections for the electron capture process in the
From the energy conservation in the reaction (
It is worth mentioning that for low momentum transfer, various authors use the approximation
While performing detailed calculations for the original electron capture cross sections on
Original total cross sections of electron capture on the
From the study of the original electron capture cross sections we conclude that the total cross sections can be well approximated with the GamowTeller transitions only in the region of low energies [
As it is well known, electron capture process plays a crucial role in late stages of evolution of a massive star, in presupernova and in supernova phases [
In an independent particle picture, the GamowTeller transitions (which is the most important in the electron capture cross section calculations) are forbidden for these nuclei [
For astrophysical environment, where the finite temperature and the matter density effects cannot be ignored (the initial nucleus is at finite temperature), in general, the initial nuclear state needs to be a weighted sum over an appropriate energy distribution. Then, assuming MaxwellBoltzmann distribution of the initial state
The results coming out of the study of electron capture cross sections under stellar conditions are shown in Figure
Electron capture cross sections for the
The percentage contributions of various multipolarities (with
Total
Positive parity transitions  Negative parity transitions  



Portions 


Portions 

31.164  25.96 

5.288  4.41 

52.779  43.98 

13.409  11.14 

6.921  5.77 

3.262  2.72 

5.499  4.58 

0.905  0.75 

0.244  0.20 

0.299  0.25 

0.208  0.17 

0.042  0.04 
In performing statebystate calculations for the electron capture cross sections, our code has the possibility to provide separately the contribution of the polarvector, the axialvector, and the overlap parts induced by the corresponding components of the electron capture operators. In Figure
Individual contributions of the polarvector, (
As mentioned before, our code gives separately the partial
Before closing, it should be mentioned that the
Furthermore, in core collapse simulations one defines the reaction rate of electron capture on nuclei given by
Furthermore, the average neutrino energy,
The electron capture on nuclei plays crucial role during the presupernova and collapse phase (in the late stage
In this work, by using our numerical approach based on a refinement of the pnQRPA that describes reliably all the semileptonic weak interaction processes in nuclei, we studied in detail the electron capture process on
Our future plans are to extend the application of this method and make similar calculations for other interesting nuclei [
The eight different tensor multipole operators entering the above equations (see Section
These multipole operators contain polarvector and axialvector parts and are written in terms of seven independent basic multipole operators as follows:
These multipole operators, due to the Conserved Vector Current (CVC) theory, are reduced to seven new basic operators expressed in terms of spherical Bessel functions, spherical harmonics, and vector spherical harmonics (see [
In the context of the pnQRPA, the required reduced nuclear matrix elements between the initial
These matrix elements enter the description of various semileptonic weak interaction processes in the presence of nuclei [
In (
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research has been cofinanced by the European Union (European Social FundESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: Heracleitus II, investing in knowledge society through the European Social Fund.