Thermodynamic functions of degenerate magnetized electron gas
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, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS
Thermodynamic Functions of Degenerate Magnetized Electron Gas V. V. Skobelev Moscow State Industrial University, Moscow, 115280 Russia email: [email protected] Received April 13, 2011
Abstract—The Fermi energy, pressure, internal energy, entropy, and heat capacity of completely degenerate relativistic electron gas are calculated by numerical methods. It is shown that the maximum admissible mag netic field on the order of 109 G in white dwarfs increases the pressure by a factor of 1.06 in the central region, where the electron concentration is ~1033 cm–3, while the equilibrium radius increases by approximately a factor of 1.03, which obviously cannot be observed experimentally. A magnetic field of ~108 G or lower has no effect on the pressure and other thermodynamic functions. It is also shown that the contribution of degen erate electron gas to the total pressure in neutron stars is negligible compared to that of neutron gas even in magnetic fields with a maximum induction ~1017 G possible in neutron stars. The neutron betadecay forbid deness conditions in a superstrong magnetic field are formulated. It is assumed that small neutron stars have such magnetic fields and that pulsars with small periods are the most probable objects that can have super strong magnetic fields. DOI: 10.1134/S1063776111130115
1. INTRODUCTION It is known [1] that pressure in white dwarfs is mainly determined by degenerate relativistic electron gas. On the other hand, these astrophysical objects can have quite strong magnetic fields (up to 109 G [2]). Therefore, it is of interest to study the influence of magnetic field on the pressure P, chemical potential μ (or the Fermi energy EF = μ T = 0 ), volume energy density U, entropy S, and heat capacity C per unit vol ume. It is important in what follows that the wavefunc tion of the Dirac equation for an electron in a mag netic field [3] contains the exponential factor (the z axis is directed along the field) p 2 1 exp i ( p 3 z – p 2 y ) – ⎛ x γ – 2⎞ . 2⎝ γ⎠
(1)
Here, γ = eB , p3 is the momentum along the field, p2 is the quasimomentum determining, according to (1), the position of the center of the wave packet on the first axis, X = p 2 /γ,
(2)
so that ∞
∫ dp
–∞
In the absence of magnetic field, the thermody namic functions are described by the expressions [4] ∞
4π f p 2 dp, n e = 2 3 e ( 2π ) 0
∫
∞
=
∫ γ dX = γL , 1
2
4π 1 f p 2 dp, P = 2 e p 3 ( 2π ) 3 0 ε
∫
(4b)
∞
4π f εp 2 dp, U = 2 3 e ( 2π ) 0
∫
where ε =
2
(4c)
2
p + m , ne is the concentration, and – μ⎞ + 1 f e = exp ⎛ ε ⎝ T ⎠
–1
(5)
is the distribution function. In the case of fully degenerate electron gas (T = 0), integration gives the expressions [5]
L1 2
(4a)
EF = (3)
0
where L1 is the normalization length along the first axis (L2, 3 are these lengths along the second and third axes). 791
2
2
2
1/3
m + pF ,
(6a)
p F = ( 3π n e ) , 4
m f ( z ), P = 2 24π
p z
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