Strongly screened electron capture rates of chromium isotopes in presupernova evolution
1 Introduction
At the presupernova stage, beta decay and electron capture (EC) on some neutronrich nuclei may play important roles in determining the hydrostatic core structure of massive presupernova stars, thereby affecting their subsequent evolution during the gravitational collapse and supernova explosion phases (e.g., Dean et al. 1998 ; Caurier et al. 1999 ; Juodagalvis et al. 2010 ; Liu 2013 , 2014 ; Liu et al. 2016 ; Liu & Gu 2016 ; Liu et al. 2017 ). For example, beta decay (EC) strongly influences the time rate of change of the lepton fraction (e.g., the time rate of change of electron fraction Ẏ _{e}) by increasing (decreasing) the number of electrons. Some isotopes of iron, chromium and copper can also make a substantial contribution to the overall changes in lepton fraction (e.g., Ẏ _{e}), electron degeneracy pressure and entropy of the stellar core during its very late stage of evolution. Many of these nuclei can be appropriately tracked in the reaction network of stellar evolution calculations. The lepton fraction (e.g., Ẏ _{e}) is bound to lead to an unstoppable process of gravitational collapse and supernova explosion.
Some research has shown that EC in iron group nuclei (e.g., iron and chromium isotopes) is a very important and dominant process in supernova explosions (e.g., Aufderheide et al. 1990 , 1994 ; Dean et al. 1998 ; Heger et al. 2001 ). In the process of presupernova evolution, chromium isotopes are a very important and crucial radionuclide. Aufderheide et al. ( 1994 ) investigated EC and beta decay for these nuclei in detail in presupernova evolution. They found that the EC rates of these chromium isotopes can be of significant astrophysical importance by controlling the electronic abundance. Heger et al. ( 2001 ) also discussed weakinteraction rates for some iron group nuclei by employing shellmodel calculations in presupernova evolution. They found that EC rates on iron group nuclei would be crucial for decreasing the electronic abundance (Y _{e}) in stellar matter.
On the other hand, in the process of presupernova evolution in massive stars, the GamowTeller (GT) transitions of isotopes of chromium play a consequential role. Some studies have shown that βdecay and EC rates of chromium isotopes significantly affect the time rate of change of lepton fraction (Ẏ _{e}). For example, Nabi et al. ( 2016 ) detailed the GT strength distributions, Ẏ _{e} and neutrino energy loss rates for chromium isotopes due to weak interactions in stellar matter.
However, their works did not discuss the problem of how strong electron screening (SES) would effect EC. What role does EC play in stellar evolution? How does SES influence the EC reaction at high density and temperature? In order to accurately calculate the EC rates and screening correction for numerical simulations of supernova explosions, in this paper we will discuss this problem in detail.
Based on the linear response theory model (LRTM) and random phase approximation (RPA), we study strongly screened EC rates of chromium isotopes in astrophysical environments by using the ShellModel Monte Carlo (SMMC) method. In the next section, we discuss the methods used for EC in stellar interiors in the cases with and without SES. Section
2 EC rates in the process of stellar core collapse
For nucleus (Z, A), we calculate the stellar EC rates, which are given by a sum over the initial parent states i and the final daughter states f at temperature T. This expression is written as (e.g., Fuller et al. 1980 , 1982 )
Based on the theory of RPA, the EC rates are closely related to cross section σ _{ec}, and can written as (e.g., see detailed discussions in Dean et al. 1998 ; Caurier et al. 1999 ; Juodagalvis et al. 2010 )
The electron chemical potential is obtained by
The total cross section in the process of EC reaction is given by (e.g., Dean et al. 1998 ; Caurier et al. 1999 ; Juodagalvis et al. 2010 )
The total amount of GT strength is S
_{GT}
^{+}, which is calculated by summing over a complete set from an initial state to final states. The response function R_{A}
(τ) of an operator
The strength distribution is related to R_{A}
(τ) by a Laplace transform
For degenerate relativistic electron gas, the EC rates in the case without SES are given by (e.g., Dean et al. 1998 ; Caurier et al. 1999 ; Juodagalvis et al. 2010 )
In 2002, based on the LRTM for relativistic degenerate electrons, Itoh et al. ( 2002 ) discussed the effect of screening potential on EC. The electron is strongly degenerate in our considered regime of densitytemperature. This condition is expressed as
For a relativistically degenerate electron liquid, Jancovici ( 1962 ) studied the static longitudinal dielectric function. Taking into account the effect of strong screening, the electron potential energy is written as
The screening potential for relativistic degenerate electrons from linear response theory is written as (Itoh et al. 2002 )
The screening energy is sufficiently high that we cannot neglect its influence at high density when electrons are strongly screened. The electron screening will make electron energy decrease from ε to ε′ = ε –D in the process of EC. Meanwhile, the screening relatively increases threshold energy from ε _{0} to ε _{s} = ε _{0} + D for EC. So, the EC rates in SES are given by (e.g., Juodagalvis et al. 2010 ; Liu 2014 )
The nuclear binding energy will increase due to interactions with the dense electron gas in the plasma. The effective nuclear Qvalue (Q_{if} ) will change at high density due to the influence of the charge dependence on this binding. When we take the effect of SES into account, the EC Qvalue will increase by (Fuller et al. 1982 )
Therefore, the Qvalue of EC increases from Q_{if}
to
We define the screening enhancement factor C to enable a comparison of the results as follows
3 Numerical Calculations Of EC Rates and Discussion
The influences of SES on EC rates for these chromium isotopes at some typical astrophysical conditions are shown in Figure
According to our calculations, the GT transition EC reaction may not be the dominant process at lower temperature. On the other hand, the higher the temperature is, the larger the electron energy, the larger the density and the higher the electron Fermi energy become. Therefore, a lot of electrons join in the EC reaction and the GT transition would be very active and be the dominant contribution to the total EC rates. Figure
The GT strength distributions play a significant role in supernova evolution. However, the GT^{+} transitions are addressed only qualitatively in presupernova simulations because of insufficient experimental information. The general rule is that the energy for the daughter ground state is parameterized phenomenologically by assuming the GT^{+} strength resides in a single resonance. Charge exchange reactions (n, p) and (p, n) would, if obtainable, supply us with plenty of experimental information. However, any available experimental GT^{+} strength distributions for these nuclei cannot be obtained except for theoretical calculations.
Table


Based on pnQRPA theory, NKK analyzed the nuclear excitation energy distribution by taking into consideration particle emission processes. They calculated a stronger GT strength distribution from these excited states compared to those assumed using Brink’s hypothesis. However, in their works, they only discussed the low angular momentum states. By using the method of SMMC, the GT intensity distribution is discussed in detail and an average value of the distribution is actually adopted in our paper.
Values for the screening factor C are plotted as a function of ρ
_{7} in Figure
Table


Synthesizing the above analysis, the effects of charge screening on nuclear physics (e.g., EC and beta decay) come at least from the following factors. First, the screening potential will change the electron Coulomb wave function in nuclear reactions. Second, the electron screening potential decreases the energy of incident electrons joining in the capture reactions. Third, the electron screening increases the energy of atomic nuclei (i.e., increases the single particle energy) in nuclear reactions. Finally, the electron screening effectively decreases the number of higherenergy electrons, whose energy is more than the threshold of the capture reaction. Therefore, screening relatively increases the threshold needed for capture reactions and decreases the capture rates.
4 Concluding remarks
In this paper, based on the theory of RPA and LRTM and by using the method of SMMC, we investigate the EC rates in SES. The EC rates increase greatly by more than six orders of magnitude as the density increases. On the other hand, by taking into account the influence of SES on the energy of incident electrons and threshold energy of EC, the EC rates decrease by ∼ 40.43%.
ECs play an important role in the dynamic process of the collapsing core of a massive star. It is a main parameter for supernova explosion and stellar collapse. SES strongly influences EC and may influence the cooling rate and evolutionary timescale of stellar evolution. Thus, the conclusions we obtained may have a significant influence on further research of supernova explosions and related numerical simulations.
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