Modified pressure of relativistic electrons in a superhigh magnetic field
Authors: Dong Ai-Jun , Gao Zhi-Fu , Yang Xiao-Feng , Wang Na , Liu Chang , Peng Qiu-He ,   School of Physics and Electronic Science, Guizhou Normal University, Guiyang 550001, China,   Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi 830011, China,   School of Astronomy & Space Science, Nanjing University, Nanjing 210000, China,   Guizhou Provincial Key Laboratory of Radio Data Processing, Guiyang 550001, China
Published: November 2022

Magnetar is a kind of pulsar powered by magnetic field energy. The study of magnetars is an important hotspot in the field of pulsars. In this paper, according to the work of Zhu Cui, et al. (Zhu C, Gao Z F, Li X D, Wang N, Yuan J P, Peng Q H <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/doi.10.1142/S021773231650070X">2016 <i>Mod. Phys. Lett. A</i> <b>31</b> 1650070</ext-link>), we reinvestigate the Landau-level stability of electrons in a superhigh magnetic field (SMF), <inline-formula><tex-math id="Z-20230117140609">\begin{document}$B\gg B_{\rm cr}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20220092_Z-20230117140609.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20220092_Z-20230117140609.png"/></alternatives></inline-formula>(<i>B</i><sub>cr</sub> is a quantum critical magnetic field with a value of 4.414×10<sup>13</sup> G), and its influence on the pressure of electrons in magnetar. First, we briefly review the pressure of electrons in neutron star (NS) with a weak-magnetic field limit (<inline-formula><tex-math id="Z-20230117140625">\begin{document}$ B\ll B $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20220092_Z-20230117140625.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20220092_Z-20230117140625.png"/></alternatives></inline-formula><sub>cr</sub>). Then, we introduce an electron Landau level stability coefficient <i>g</i><sub><i>ν</i></sub> and a Dirac-<i>δ</i> function to deduce a modified pressure formula for the degenerate and relativistic electrons in an SMF in an application range of matter density <i>ρ</i> ≥ 10<sup>7</sup> g·cm<sup>–3</sup> and <i>B</i><sub>cr</sub> <i><inline-formula><tex-math id="Z-20230117140650">\begin{document}$ \ll $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20220092_Z-20230117140650.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20220092_Z-20230117140650.png"/></alternatives></inline-formula>B</i> < 10<sup>17</sup> G. By modifying the phase space of relativistic electrons, the SMF can enhance the electron number density <i>n</i><sub>e</sub>, and reduce the maximum of electron Landau level number<i> ν</i><sub>max</sub>, which results in a redistribution of electrons. As <i>B</i> increases, more and more electrons will occupy higher Landau levels, and the electron Landau level stability coefficient <i>g</i><sub><i>ν</i></sub> will decrease with the augment of Landau energy-level number <i>ν</i>. By modifying the phase space of relativistic electrons, the electron number density <i>n</i><sub>e</sub> increases with the MF strength increasing, leading the electron pressure <i>P</i><sub>e</sub> to increase. Utilizing the modified expression of electron pressure, we discuss the phenomena of Fermion spin polarization and electron magnetization in the SMF, and the modification of the equation of state by the SMF. We calculate the baryon number density, magnetization pressure, and the difference between pressures in the direction parallel to and perpendicular to the magnetic field in the frame of the relativistic mean field model. Moreover, we find that the pressure anisotropy due to the strong magnetic field is very small and can be ignored in the present model. We compare our results with the results from other similar studies, and examine their similarities and dissimilarities. The similarities include 1) the abnormal magnetic moments of electrons and the interaction between them are ignored; 2) the electron pressure relate to magnetic field intensity <i>B</i>, electron number density <i>n</i><sub>e</sub> and electron Fermi energy <inline-formula><tex-math id="M1">\begin{document}$E_{{\rm{F}}}^{{\rm{e}}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20220092_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20220092_M1.png"/></alternatives></inline-formula>, and the latter two are complex functions containing <i>B</i>; 3) with <i>n</i><sub>e</sub> and <inline-formula><tex-math id="M2">\begin{document}$E_{{\rm{F}}}^{{\rm{e}}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20220092_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20220092_M2.png"/></alternatives></inline-formula> fixed, <i>P</i><sub>e</sub> increases with <i>B</i> rising; 4) as <i>B</i> increases, the pressure-density curves fitted by the results from other similar studies have irregular protrusions or fluctuations, which are caused by the transformation of electron energy state from partial filling to complete filling at the<i> ν</i>-level or the transition of electrons from the<i> ν</i> to the (<i>ν</i>+1)-level. This phenomenon is believed to relate to the behavior of electrons near the Fermi surface in a strong magnetic field, which essentially reflects the Landau level instability. Finally, the future research direction is prospected. The present results provide a reference for future studies of the equation of state and emission mechanism of high-<i>B</i> pulsar, magnetar and strongly magnetized white dwarf.