Modified Fermi Energy of Electrons in a Superhigh Magnetic Field

In this paper, we investigate the electron Landau-level stability and its influence on the electron Fermi energy, $E_{\rm F}(e)$, in the circumstance of magnetars, which are powered by magnetic field energy. In a magnetar, the Landau levels of degenerate and relativistic electrons are strongly quantized. A new quantity $g_{n}$, the electron Landau-level stability coefficient is introduced. According to the requirement that $g_{n}$ decreases with increasing the magnetic field intensity $B$, the magnetic-field index $\beta$ in the expression of $E_{\rm F}(e)$ must be positive. By introducing the Dirac$-\delta$ function, we deduce a general formulae for the Fermi energy of degenerate and relativistic electrons, and obtain a particular solution to $E_{\rm F}(e)$ in a superhigh magnetic field (SMF). This solution has a low magnetic-field index of $\beta=1/6$, compared with the previous one, and works when $\rho\geq 10^{7}$~g cm$^{-3}$ and $B_{\rm cr}\ll B\leq 10^{17}$~Gauss. By modifying the phase space of relativistic electrons, a SMF can enhance the electron number density $n_e$, and decrease the maximum of electron Landau level number, which results in a redistribution of electrons. According to Pauli exclusion principle, the degenerate electrons will fill quantum states from the lowest Landau level to the highest Landau level. As $B$ increases, more and more electrons will occupy higher Landau levels, though $g_{n}$ decreases with the Landau level number $n$. The enhanced $n_{e}$ in a SMF means an increase in the electron Fermi energy and an increase in the electron degeneracy pressure. The results are expected to facilitate the study of the weak-interaction processes inside neutron stars and the magnetic-thermal evolution mechanism for megnetars.