Impacts of Strongly Magnetized Degenerate Plasma on the Electron-capture Rates

. The strongly magnetized degenerate astrophysical plasma is investigated. A relativistic Hartree self-consistent ﬁeld method is applied to calculate the screening potential. A proﬁle from a 15 M ⊙ core collapsing supernova (SN) progenitor is applied to evaluate the electron-capture rates of 54 Fe. It is found that the screening potential at high ﬁeld is enhanced compared with the previous study. If the ﬁeld is high enough and only the lowest Landau level is allowed, two orders of magnitude reduction of the capture rates are found in the high-density region. Such deviations of the electron capture rates are essential since the rates determine the neutron richness of the progenitor model as well as the iron core mass, which are crucial for MHD-Jet SNe explosion calculation.


Introduction
The weak interactions determine the nucleosynthetic path in the supernovae (SNe) and also its explosion models [1][2][3][4]. For the MHD-Jet SNe, it is believed that the collapsing and bouncing phases have relativistic degenerate electron gas in a strong magnetic field of the order of 10 14−16 G [5] in the inner region. The thermodynamics of the electrons (positrons) in such an environment is different from the field-free case and could further deviate the nuclear weak interactions significantly. Prompted by this fact, this work focuses on the microscopic effects of the relativistic, magnetized degenerate plasmas on the weak interactions. The general Hartree self-consistent field method of the relativistic degenerate electron gas [6,7] is applied to calculate the strong screening potential in the background magnetic field. The impacts on the weak interaction rates from the highly magnetized degenerate plasma are also investigated. We apply a profile from the 15 M ⊙ progenitor model to calculate the electroncapture rates of iron group nuclei. The results are essential to determine the iron core mass before the explosion and the electron fraction Y e during SN's collapsing phase.

Impacts of Strongly Magnetized Degenerate Plasma
The Electron capture rate of a nucleus (Z, A) is contributed by individual capture rates λ i f from one of the initial states i to all possible final states f [9][10][11][12]. Considering the background magnetic field, the integrand in phase space of the capture rate reads where the electron momentum transverse to the field direction is quantized into separate Landau levels, the maximum occupied level n max is determined by the Fermi energy E F of the electrons (positrons). The electron energy ω B is given by ω 2 B = p 2 + m 2 e + 2eB. µ B denotes the chemical potential within the background magnetic field. The integration limits of Eq. 1 are set by the reaction threshold ω ′ l . Q i f is the electron capture transition energy, determined from the nuclear masses. F(Z, ω) is the Fermi function which corrects the distortion of the electron wave function [13]. Inside the realistic astrophysical plasma, the interaction rates are varied from the vacuum case due to the many-body effect inside the plasma such as the Coulomb screening correction. The screening could mainly change: (1). the threshold energy for the capture is varied by the amount ∆Q c = µ c (Z − 1) − µ c (Z) [14], where µ c is the Coulomb chemical potential [15]; (2). The energy of the captured electron ω is reduced by the screening potential ∆V(B), which is obtained under the relativistic degenerate condition [6,7] (see Ref. [8] for more details).
In Fig. 1, we compare f i j under different density (ρY e ) and field strength. For an extremely strong magnetic field, E F decreases monotonically as a function of B, only the Lowest Landau Level (LLL) is occupied, and the value of f i j shows the same trend since the Fermi energy determines the maximum energy of electrons. With decreasing magnetic field strength, the new Landau level pops up to cancel the extra energy carried by the magnetic field. Therefore, f i j decreases firstly, and then increases since the cancelling effect decreases monotonically with decrease of the magnetic field. Until the next Landau level pops up, such a pattern will be repeated. The left panel of Fig. 1 shows the ratio between screened and unscreened f i j : the screening could suppress f i j for about 20% under high density and strong magnetic field case (the right bottom part), while for a weak magnetic field, screening could reduce f i j only for few percent. This is consistent with the previous study [13] where only the field-free screening is taken into account. On the right panel, we compare the ratio of f i j between the scenarios with and without magnetic field. For every (ρY e , B) combination that corresponds to a popped-up new Landau level, f i j value could be enlarged for 1.5 times due to the larger value of E F . Once the field strength becomes lower than 10 14 G, it becomes the same value as the B = 0 scenario. Since λ i j ∝ f i j , such impact on the phase space should leave the imprints in the final electron capture rates.

Electron Capture Rates of Iron Group Nuclei
The importance of the shell-model rates in the pre-SN stage has been investigated previously [2,3]. It was shown that weak interaction rates significantly affect the central electron fraction Y e at the onset of the core-collapse, which further affect the nucleosynthetic path. The detailed analysis of Y e change after Oxygen burning also find that the critical flow during the final stages of stellar evolution is much closer to the valley of stability, and several essential electron capturing nuclei (e.g., 54,56,58 Fe, 55 Mn, 53 Cr) are identified [2,3]. Inspired by the previous studies, we first focus on the rate of electron capture on 54 Fe. We calculate the reaction rates in different sets of stellar conditions (i.e., the combination of density, temperature and Y e ). Those combinations are appropriate for the inner core of a collapsing 15 M ⊙ star, the profile of which is taken from Ref. [13]. Fig. 2 shows the reaction rate of 54 Fe(e − , ν e ) 54 Mn as functions of ρY e . All the rates are converged to the same value for electron chemical potentials µ e larger than about 25 MeV, while at lower µ e values, the capture rates are more sensitive to the magnetic field strengths. The screening could suppress the electron-capture rates as expected, and the suppression is stronger for the lower µ e value. It is also worth mentioning that, due to the magnetic field, µ e is also a function of B, as discussed. Therefore, for the same combination of (ρ, Y e ), µ e decreased as the increasing field strength, the electron capture rate could be reduced by almost two orders of magnitude for the low density and high field region. In high-density region, the field strength that allowed only the LLL occupation is also high, therefore B < B crit makes the indistinguishable trend of the reaction rates in high-density region. The electron capture process could reduce the electron number density and the neutrino emission from electron capture could carry away the energy and entropy of the core. Although the dominant composition during the collapse phase of the core is protons and neutrons, it has been shown that the heavy nuclei play an equally important role for the Y e evolution during the collapse [16]. Therefore, it is expected that such an influence of the magnetic field on the iron group nuclei could significantly impact the collapse and bouncing phase. It is worth extending to the sophisticated MHD-Jet SNe calculation.

Conclusion
In this work, the electron capture rates of iron group nuclei are re-evaluated for the strongly magnetized degenerate plasma. The degeneracy of the plasma makes the Thomas-Fermi approximation invalid. A relativistic generalization of the Hartree self-consistent field method is applied to calculate the strong screening potential. The screening potential could reduce the electron capture rate more significantly than the field-free situation. Moreover, the strong field strength only allows the lowest Landau level occupation. Therefore, the weak interaction rate is dramatically suppressed. A profile from a 15 M ⊙ core collapsing SN progenitor model is applied to calculate the electron-capture rates of 54 Fe. It is noticed that two orders of magnitude reduction is found between the rates for plasma with density ρ ∼ 10 9 ∼ 10 11 g cm −3 and magnetic field B > 10 14 G . These impacts have potential significance since electron capture on iron group nuclei is the leading reaction that reduces Y e , which finally could affect the r-process path and also SNe explosion models. Then, such results should be applied to the collapse and bouncing phases of MHD-Jet SNe calculation.