Modified pressure of relativistic electrons in a superhigh magnetic field

Dong Ai-Jun; Gao Zhi-Fu; Yang Xiao-Feng; Wang Na; Liu Chang; Peng Qiu-He

doi:10.7498/aps.72.20220092

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:

PACS:

4340
4630C
4630L

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).

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References

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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

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

DownLoad: Full-Size Img PowerPoint

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

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

DownLoad: Full-Size Img PowerPoint

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

Figure 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.

DownLoad: Full-Size Img PowerPoint

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)

Figure 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

Figure 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

Figure 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

DownLoad: CSV

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

DownLoad: CSV

[1]	Duncan R C, Thompson C 1992 Astrophys. J. 392 L9 Google Scholar
[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
[16]	Liu J J, Gu W M 2016 Astrophys. J. Suppl. Ser. 224 29 Google Scholar
[17]	Liu J J, Liu D M 2020 Astron. Nachr. 341 291 Google Scholar
[18]	Liu J J, Liu D M 2018 Res. Astron. Astrophys. 18 8 Google Scholar
[19]	Liu J J, Liu D M 2021 Publ. Astron. Soc. Pac. 133 4201 Google Scholar
[20]	Gao Z F, Wang N, Peng Q H, Li X D, Du Y J 2013 Mod. Phys. Lett A 28 1350138 Google Scholar
[21]	Zhu C, Gao Z F, Li X D, Wang N, Yuan J P, Peng Q H 2016 Mod. Phys. Lett A 31 1650070 Google Scholar
[22]	Kubo R 1965 Statistical Mechanics (Amsterdam: North-Holland Publ. Co.) pp278–280
[23]	Peng Q H, Zhang J, Chou C K 2016 EPJ Web. Conf. 10 907003 Google Scholar
[24]	Li X H, Gao Z F, Li X D, Xu Y, Wang P, WangN, Peng Q H 2016 Int. J. Mod. Phys. D 25 165000 Google Scholar
[25]	Lai D, Shapiro S L 1991 Astrophys. J. Lett. 383 745 Google Scholar
[26]	Haensel P, Potekhin A Y, Yakovlev D G 2007 Neutron Stars 1: Equation of state and structure (Berlin: Springer) p326
[27]	Das U, Mukhopadhyay B 2012 Phys. Rev. D 86 042001 Google Scholar
[28]	Chatterjee D, Fantina A F, Chamel N, Novak J, Oertel M 2017 Mon. Not. R. Astron. Soc. 469 95 Google Scholar
[29]	Nandi R, Bandyopadhyay D 2013 J. Phy. Conf. Ser. 420 012144 Google Scholar
[30]	Das U, Mukhopadhyay B 2015 J. Cosmol. Astropart. Phys. 05 045 Google Scholar
[31]	Bera P, Bhattacharya D 2014 Mon. Not. R. Astron. Soc. 445 3951 Google Scholar
[32]	Dong J M, Lombardo U, Zhang H F, Zuo W 2016 Astrophys. J. 817 6 Google Scholar
[33]	Dong J M, Shang X L 2020 Phys. Rev. C 101 014305 Google Scholar
[34]	Dong J M 2021 Mon. Not. R. Astron. Soc. 500 1505 Google Scholar
[35]	Bordbar G H, Karami, M K 2022 Eur. Phys. J. C 82 74 Google Scholar
[36]	Herrera L 2020 Phys. Rev. D 101 104024 Google Scholar
[37]	Shulman G A 1991 Sov. Phys. Astron. 35 50
[38]	Mandal S, Chakrabarty S 2002 arXiv: astro-ph/0209015
[39]	Huang Z P, Yan Z, Shen Z Q, Tong H, Lin L, Yuan J P, Liu J, Zhao R S, Ge M Y, Wang R, 2021 Mon. Not. R. Astron. Soc. 505 1311 Google Scholar
[40]	高志福 2007 硕士学位论文 (乌鲁木齐: 新疆大学) Gao Z F 2007 M. S. Thesis (Urumqi: Xinjiang University) (in Chinese)
[41]	王兆军, 吕国梁, 朱春花, 张军 2011 物理学报 60 049702 Google Scholar Wang Z J, Lü G L, Zhu C H, Zhang J 2011 Acta Phys. Sin. 60 049702 Google Scholar
[42]	Geng L, Toki H, Meng J 2005 Prog. Theor. Phys. 113 785 Google Scholar
[43]	Singh D, Saxena G 2012 Int. J. Mod. Phys. E 21 1250076 Google Scholar
[44]	赵诗艺, 刘承志, 黄修林, 王夷博, 许妍 2021 物理学报 70 222601 Google Scholar Zhao S Y, Liu C Z, Huang X L, Wang Y B, Xu Y 2021 Acta Phys. Sin. 70 222601 Google Scholar
[45]	Gao Z F, Shan H, Wang W, Wang N 2017 Astron. Nachr. 338 1066 Google Scholar
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