在超强磁场中修正的相对论电子压强

董爱军; 高志福; 杨晓峰; 王娜; 刘畅; 彭秋和

doi:10.7498/aps.72.20220092

摘要

当前脉冲星领域一个重要的研究热点是磁星. 本文在朱翠等(Zhu C, Gao Z F, Li X D, Wang N, Yuan J P, Peng Q H 2016 Mod. Phys. Lett. A 31 1650070)工作的基础上, 重新研究了磁星超强磁场下($B\gg B$_cr, B_cr是电子的量子临界磁场)电子朗道能级的稳定性及其对电子压强的影响. 首先, 对弱磁场极限下($B\ll B$_cr) 中子星内部电子压强进行必要的回顾; 然后, 通过引入电子朗道能级稳定性系数g_ν和Dirac-δ函数, 推导出在超强磁场下修正的相对论电子压强P_e的表达式, 给出表达式适用条件: 物质密度ρ ≥ 10⁷ g·cm^–3和B_cr ≤ B < 10¹⁷ G (1 G = 10^–4 T). 超强磁场通过修正相对论电子的相空间, 提高了电子数密度n_e, 而n_e的增加意味着P_e的增加. 利用修正的电子压强表达式, 讨论了超强磁场下费米子自旋极化现象、电子磁化现象以及超强磁场对物态方程的修正. 最后, 本文的结果与其他类似工作进行对比, 并对未来的工作进行展望. 本文的研究将为磁星以及强磁化白矮星的物态方程和热演化的探索提供极有价值的参考, 将为普通射电脉冲星等离子磁层数值模拟、高磁场脉冲星辐射机制等相关研究提供有用的信息.

关键词:

超强磁场 /
朗道能级 /
磁星 /
磁化

Abstract

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 2016 Mod. Phys. Lett. A 31 1650070), we reinvestigate the Landau-level stability of electrons in a superhigh magnetic field (SMF), $B\gg B_{\rm cr}$(B_cr is a quantum critical magnetic field with a value of 4.414×10¹³ 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 ($ B\ll B $_cr). Then, we introduce an electron Landau level stability coefficient g_ν and a Dirac-δ function to deduce a modified pressure formula for the degenerate and relativistic electrons in an SMF in an application range of matter density ρ ≥ 10⁷ g·cm^–3 and B_cr $ \ll $B < 10¹⁷ G. By modifying the phase space of relativistic electrons, the SMF can enhance the electron number density n_e, and reduce the maximum of electron Landau level number ν_max, which results in a redistribution of electrons. As B increases, more and more electrons will occupy higher Landau levels, and the electron Landau level stability coefficient g_ν will decrease with the augment of Landau energy-level number ν. By modifying the phase space of relativistic electrons, the electron number density n_e increases with the MF strength increasing, leading the electron pressure P_e 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 B, electron number density n_e and electron Fermi energy $E_{{\rm{F}}}^{{\rm{e}}}$, and the latter two are complex functions containing B; 3) with n_e and $E_{{\rm{F}}}^{{\rm{e}}}$ fixed, P_e increases with B rising; 4) as B 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 ν-level or the transition of electrons from the ν to the (ν+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-B pulsar, magnetar and strongly magnetized white dwarf.

Keywords:

作者及机构信息

1.
贵州师范大学物理与电子科学学院, 贵阳　550001

2.
中国科学院新疆天文台, 乌鲁木齐　830011

3.
南京大学天文与空间科学学院, 南京　210000

4.
贵州省射电天文数据处理重点实验室, 贵阳　550001

通信作者: 高志福, zhifugao@xao.ac.cn

基金项目: 国家自然科学基金 (批准号: 12041304, U1831120)、新疆维吾尔自治区自然科学基金 (批准号: 2022D01A155)、贵州省科技计划(批准号: [2019]1241, KY(2020)003)和中科院高层次人才计划择优支持项目(批准号: [2019]085)资助的课题.

Authors and contacts

1.
School of Physics and Electronic Science, Guizhou Normal University, Guiyang 550001, China

2.
Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi 830011, China

3.
School of Astronomy & Space Science, Nanjing University, Nanjing 210000, China

4.
Guizhou Provincial Key Laboratory of Radio Data Processing, Guiyang 550001, China

Corresponding author: Gao Zhi-Fu, zhifugao@xao.ac.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12041304, U1831120), the Natural Science Foundation of Xinjiang Uygur Autonomous Region, China (Grant No. 2022D01A155), the Natural Science Foundation of Guizhou, China (Grant Nos. [2019]1241, KY(2020)003), and the High Level Talent Program support project of Chinese Academy of Sciences, China (Grant No. [2019]085).

文章全文 : translate this paragraph

参考文献

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施引文献

图 1 中子星内部弱磁场极限下相对论电子压强P_e随电子数密度n_e的变化

Fig. 1. Relativistic electron pressure P_e with electron number density n_e in the limit of weak magnetic field inside a neutron star.

下载: 全尺寸图片幻灯片

图 2 不同磁场下中子星内部电子压强P_e随物质密度ρ的变化

Fig. 2. Relation between electron pressure P_e and matter density ρ in neutron stars with different magnetic fields.

下载: 全尺寸图片幻灯片

图 3 本文与其他强磁场中电子数密度和电子压强研究的对比　(a)强磁化白矮星中电子压强P_e随ρ变化关系; (b)中子星壳层电子数密度n_e随ρ变化关系; (c)磁化白矮星中(最大电子费米能量E_Fmax = 20m_ec²)电子压强P_e随ρ变化关系; (d) 两种不同的理论模型下白矮星中电子压强P_e随ρ变化关系

Fig. 3. Study of electron number density and electron pressure in strong magnetic fields by other authors and their comparison with this work: (a) Relationship between electron pressure P_e and ρ in a strongly magnetized white dwarf (WD); (b) relationship between the electron number density n_e and ρ in the crust of a neutron star; (c) electron pressure P_e as a function of ρ in a magnetized WD with maximum electron Fermi energy E_Fmax = 20m_ec²; (d) electron pressure P_e as a function of ρ in a magnetized WD under two different theoretical models.

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图 4 中子星内部费米子完全极化场景下饱和磁场强度B_s随粒子数密度n的变化关系　(a) 质子/电子完全极化下B_s vs. n_e/n_p; (b) 中子完全极化下B_s vs. n_B (n_B为重子数密度)

Fig. 4. Relationship between the saturated magnetic field strength B_s and the particle number density n in a fully polarized neutron star fermion matter: (a) B_s vs. n_e/n_p in a fully polarized scenario for proton/electron matter system; (b) B_s vs. n_B in a fully polarized scenario for the neutron matter system (n_B is the baryon number density).

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图 5 不同磁场下中子星内部相对论电子的磁化率χ与电子数密度n_e的变化关系

Fig. 5. Relation between the magnetic susceptibility χ and number density of relativistic electrons n_e in neutron stars with different magnetic field strengths.

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图 6 中子星内部磁场B随物质密度ρ的变化关系

Fig. 6. Relation of the magnetic field B and matter density ρ in a neutron star.

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表 1 在相对论平均场TMA参数模型下n_N, $ E_{\text{F} }^{\text{e} } $, P_e, P和M的部分计算值

Table 1. Partial calculations of n_N, $ E_{\text{F} }^{\text{e} } $, P_e, P and M in a relativistic mean field model with the TMA parameter set.

B$\ll $B^*					B > B^*
n_N/fm^–3	$ E_{\text{F}}^{\text{e}} $/MeV	P_e/(MeV·fm^–3)	P/(MeV·fm^–3)	M/M_⊙	$ E_{\text{F}}^{\text{e}} $/MeV	P_e/(MeV·fm^–3)	P/(MeV·fm^–3)	M/M_⊙
0.0013	2.924	4.9×10^–10	3.78×10^–6	0.0289	3.351	8.41×10^–10	3.79×10^–6	0.0311
0.0211	23.49	2.03×10^–6	6.79×10^–5	0.0593	27.62	2.88×10^–6	7.36×10^–5	0.0613
0.0772	68.58	1.47×10^–4	0.0021	0.0517	81.06	2.87×10^–4	0.00258	0.0543
0.1332	107.89	9.04×10^–4	0.0143	0.2904	128.65	0.00182	0.0179	0.2932
0.1554	120.90	0.0014	0.0229	0.4201	145.13	0.00295	0.0725	0.4241
0.2003	143.58	0.0028	0.0475	0.6884	175.48	0.00632	0.0861	0.6965
0.2338	158.31	0.0042	0.0724	0.8808	183.72	0.00762	0.0965	0.8912
0.3206	190.04	0.0087	0.1624	1.2945	251.49	0.0267	0.2105	1.3062
0.3556	200.78	0.0108	0.2092	1.4236	273.35	0.0372	0.2761	1.4327
0.4186	218.29	0.0151	0.3065	1.6071	312.72	0.0637	0.4211	1.6223
0.4746	231.98	0.0193	0.4068	1.7263	347.67	0.0974	0.5816	1.7412
0.5446	247.31	0.0249	0.5479	1.8312	391.12	0.1561	0.8278	1.8522
0.6076	259.75	0.0304	0.6880	1.8947	429.70	0.2264	1.0272	1.9132
0.6846	273.65	0.0374	0.8737	1.9444	456.80	0.2905	1.3092	1.9675
0.7266	280.73	0.0414	0.9809	1.9621	480.23	0.3551	1.4782	1.9853
0.8396	298.23	0.0528	1.2845	1.9830	526.73	0.5135	1.9521	2.0061
0.9156	318.40	0.0655	1.5925	1.9916	586.65	0.7478	2.5316	2.0342

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表 2 相对论平均场模型下n_N, ρ, B, n_e, |MB|, ΔP和P_//的部分计算值, 这里选择TMA参数组和密度依赖的中子星强磁场模型

Table 2. Partial calculations of n_N, ρ, B, n_e, |MB|, ΔP和P_//in a relativistic mean field model. TMA parameter set and a density-dependent magnetic field model for a neutron star are selected.

n_N/fm^–3	ρ/(g·cm^–3)	B/G	n_e/cm^–3	\|M\|/G	\|MB\|/(dyn·cm^–2)	ΔP/(dyn·cm^–2)	P_///(dyn·cm^–2)
0.0013	2.535×10¹²	1.000×10¹⁴	1.051×10³²	4.277×10¹¹	4.277×10²⁵	8.385×10²⁶	1.196×10³⁰
0.0211	3.992×10¹³	1.003×10¹⁴	5.689×10³⁴	2.841×10¹³	2.845×10²⁷	3.641×10²⁷	3.324×10³¹
0.0722	1.014×10¹⁴	1.011×10¹⁴	1.418×10³⁶	2.428×10¹⁴	2.485×10²⁸	2.567×10²⁸	8.147×10³¹
0.1332	2.521×10¹⁴	1.073×10¹⁴	5.520×10³⁶	6.049×10¹⁴	6.964×10²⁸	7.055×10²⁸	5.651×10³¹
0.1554	2.940×10¹⁴	1.116×10¹⁴	7.781×10³⁶	7.638×10¹⁴	9.508×10²⁸	9.607×10²⁸	2.509×10³⁴
0.2003	3.789×10¹⁴	1.247×10¹⁴	1.301×10³⁷	1.089×10¹⁵	1.796×10²⁹	1.708×10²⁹	2.719×10³⁴
0.2338	4.423×10¹⁴	1.393×10¹⁴	1.744×10³⁷	1.868×10¹⁵	2.604×10²⁹	2.619×10²⁹	3.049×10³⁴
0.3206	6.065×10¹⁴	2.011×10¹⁴	3.017×10³⁷	5.406×10¹⁵	8.143×10²⁹	8.175×10²⁹	6.645×10³⁴
0.3556	6.727×10¹⁴	2.377×10¹⁴	3.563×10³⁷	5.406×10¹⁵	1.285×10³⁰	1.291×10³⁰	8.716×10³⁴
0.4185	7.917×10¹⁴	3.237×10¹⁴	4.572×10³⁷	7.676×10¹⁵	2.485×10³⁰	2.493×10³⁰	1.329×10³⁵
0.4746	8.978×10¹⁴	4.244×10¹⁴	4.580×10³⁷	1.237×10¹⁶	5.203×10³⁰	5.217×10³⁰	1.836×10³⁵
0.5447	1.031×10¹⁵	5.860×10¹⁴	6.647×10³⁷	2.249×10¹⁶	1.318×10³¹	1.321×10³¹	2.613×10³⁵
0.6076	1.145×10¹⁵	7.685×10¹⁴	7.704×10³⁷	3.353×10¹⁶	2.577×10³¹	2.583×10³¹	3.243×10³⁵
0.6846	1.295×10¹⁵	1.042×10¹⁵	9.012×10³⁷	5.225×10¹⁶	5.445×10³¹	5.453×10³¹	4.133×10³⁵
0.7265	1.375×10¹⁵	1.215×10¹⁵	9.725×10³⁷	6.533×10¹⁶	7.932×10³¹	7.944×10³¹	4.675×10³⁵
0.8386	1.586×10¹⁵	1.763×10¹⁵	1.160×10³⁸	1.109×10¹⁷	1.955×10³²	1.958×10³²	6.165×10³⁵
0.9156	1.774×10¹⁵	2.347×10¹⁵	1.342×10³⁸	1.678×10¹⁷	3.938×10³²	3.943×10³²	7.996×10³⁵

下载: 导出CSV

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[2]	Gao Z F, Li X D, Wang N, Yuan J P, Wang P, Peng Q H, Du Y J 2016 Mon. Not. R. Astron. Soc. 456 55 Google Scholar
[3]	Gao Z F, Wang N, Shan H, Li, X D, Wang W 2017 Astrophys. J. 849 19 Google Scholar
[4]	Kaspi V M, Beloborodov A M 2017 Ann. Rev. Astron. Astrophys. 55 261 Google Scholar
[5]	Shen J, Wang Y, Zhou T, Ji H 2017 Astrophys. J. 835 43 Google Scholar
[6]	Shen J, Ji H, Su Y 2022 Res. Astron. Astrophys. 22 015019 Google Scholar
[7]	Mereghetti S, Pons J A, Melatos A 2015 Space Sci. Rev. 191 315 Google Scholar
[8]	Zhao X F 2019 Int. J. Theor. Phys. 58 1060 Google Scholar
[9]	Zhao X F 2019 Astrophys. Space Sci. 364 38 Google Scholar
[10]	Zhao X F 2020 Chin. J. Phys. 3 240 Google Scholar
[11]	Rabhi A, Pérez-García M A, Providéncia C, Vidaña I 2015 Phys. Rev. C 91 045803 Google Scholar
[12]	Chatterjee D, Elghozi T, Novak J, Oertel M 2015 Mon. Not. R. Astron. Soc. 447 3785 Google Scholar
[13]	Shen J, Zhou T, Ji H, Wiegelmann T, Inhester B, Feng L 2014 Astrophys. J. 791 83 Google Scholar
[14]	Farooq F, Nabi J U, Shehzadi R 2021 Astrophys. Space Sci. 366 86 Google Scholar
[15]	Liu J J, Liu D M 2018 Eur. Phys. J. C 78 84 Google Scholar
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