The modified pressure of relativistic electrons in a superhigh magnetic field

Magnetars are a kind of pulsars powered by magnetic field energy. The study of magnetars is an important hotspot in the field of pulsars. In this paper, based on the work of Zhu Cui et al. (2016), we reinvestigate the Landau-level stability of electrons in a superhigh magnetic field (SMF), <em>B</em>>> <em>B</em><sub>cr</sub> (<em>B</em><sub>cr</sub> =4.414×10<sup>13</sup>G is the quantum critical magnetic field) and their influences on the pressure of electrons in magnetars. Firstly, we briefly review the pressure of electrons in neutron stars (NSs) with a weak-magnetic field limit (<em>B<< B</em><sub>cr</sub>). By introducing the electron Landau level stability coefficient g<sub>n</sub> and the Dirac -δ function, then we deduce a modified formula for pressure of degenerate and relativistic electrons in a SMF with an application range of <em>ρ</em> ≥10<sup>7</sup>g cm<sup>-3</sup>and <em>B</em><sub>cr</sub><em><<B<</em>10<sup>17</sup>G. By modifying the phase space of relativistic electrons, a SMF can enhance the electron number density <em>n</em><sub>e</sub>, and decrease the maximum of electron Landau level number<em> ν<sub>max</sub></em>, which results in a redistribution of electrons. As <em>B</em> increases, more and more electrons will occupy higher Landau levels, the electron Landau level stability coefficient g<sub>ν</sub> will decrease with the increase in Landau energy-level number ν. By modifying the phase space of relativistic electrons, the electron number density <em>n<sub>e</sub></em> increases with the MF strength, leading to an increase in the electron pressure <em>P<sub>e</sub></em>. Utilizing this modified expression of electron pressure, we discuss the phenomenon of Fermion spin polarization and electron magnetization in SMFs, and the modification of the equation of state by the SMFs. We calculate the baryon number density, magnetization pressure, and pressure differences parallel to and perpendicular to the magnetic field under the relativistic mean field model and find that the pressure anisotropy due to the strong magnetic field is very small and can be ignored under the present model. We have compared our results with other similar works, and found that there are similarities and differences between our work and other similar work. The similarities include: (1) the abnormal magnetic moment of electrons and the interaction between them are ignored; (2) The electron pressure is related to magnetic field intensity <em>B</em>, electron number density <em>n<sub>e</sub></em> and electron Fermi energy <em>E</em><sub>F</sub>(e) and the latter two are complex functions containing <em>B</em>. (3) Given <em>n<sub>e</sub></em> and <em>E</em><sub>F</sub>(e), <em>P</em><sub>e</sub> increases with <em>B</em>; (4) As <em>B </em>increases, the pressure-density curve fitted by other similar work will have irregular protrusions or fluctuations, which is caused by the transformation of electron energy state from partial filling to complete filling at the<em> ν</em> -level or the transition of electrons from the<em> ν</em> to the (<em>ν</em>+1)-level. This phenomenon is believed to be related to the behavior of electrons near the Fermi surface in a strong magnetic field, which essentially reflects the Landau level instability. Finally, we look forward to the prospects of the future research. The results provide a reference for future studies on the equation of state and emission mechanism of high-<em>B</em> pulsars, magnetars and strongly magnetized white dwarfs.