T0

oo ne 2n2am X.

KbT v 1 + 2bn and use the approximation (Potekhin, 1996a)

where £ = ln{1+exp[x - xo(y)]}, xo(y) = 1/(1 + 0.623y1'603), c(y) = 0.9422 y1'7262, a(£) = Vn/2 + (0.103 +0.043£2) y/£,

"(£) = 1 + 0.0802 y/£ + 0.2944£ + 0.043 £3.

This approximation reproduces correct asymptotes at small and large x and y, and remains accurate within 0.6% at any x and y. The chemical potential i at a given density ne can be found by the numerical inversion of Eq. (4.14) with the use of the fit (4.23).

The x-derivative of the right-hand side of Eq. (4.23) reproduces the exact derivative dF(x, y)/dx with a maximum relative error of 2% . Using this derivative in Eq. (4.19) and replacing d/(0)/dc — -5(e - ¡) we get the electron screening wave number

Integrating Eq. (4.15) by parts, we obtain

Nb (c) /(0) dc = Pr 9-(1 + 2"n)1/4/i/2(X-,T-),

where Pr = mec2/XC3 is the relativistic unit of pressure introduced in § 2.3.1. The Fermi-Dirac integral I\/2(x,T) is readily evaluated using Eqs. (2.54)-(2.56).

Let us comment, in passing, that the kinetic pressure of an electron gas, calculated as the quantum-mechanical average ne (pava), is anisotropic in quantizing magnetic fields. For instance, the kinetic pressure in the transverse direction, P± = ne{pxvx), is much smaller than P = ne {pzvz), if the field is strongly quantizing. However, the kinetic pressure is only one part of the total pressure in the magnetized plasma. As proven by Blandford & Hernquist (1982), a deficit of the kinetic pressure in the transverse direction is exactly balanced by the pressure excess caused by magnetization currents. Thus, the total actually thermodynamic pressure is isotropic at any field strength, and Eq. (4.25) is always valid.

Strongly quantizing magnetic field. Let pF0 = hkF0, eF0, and TF0 denote, respectively, the non-magnetic Fermi momentum, energy, and temperature at a given density (§ 2.1.2). We reserve the notations pF = hkF, eF, and TF for the same quantities in a magnetic field. We keep the parameters xr, jr, and 3r expressed through pF0, as in Chapter 2. For instance, xr = pF0/mec is a convenient measure of the density regardless the magnetic field strength.

At T < Tf, one can replace (—df(0)/3e) in Eq. (4.16) by the delta function 5(e — £f):

J me c2

= ¿2 b H gn (! + 2bn) [x^ 1 + x2n — ln(xra + ^1+ x2n)], (4.27)

where xn = cpn(eF)/en(0), and Pr is the same as in Eq. (2.66). The Fermi energy eF at a given ne is found by the inversion of Eq. (4.26).

A magnetic field is called strongly quantizing, if it confines most of the electrons to the ground Landau level. This occurs at sufficiently low temperatures and densities. In this case, from Eq. (4.26) one obtains eF = mec2 \J 1 + x2B, where v „ ^ 30.2 Z A

while A' and (Z) are the mean effective atomic mass and charge numbers, respectively (see § 2.1.1); xr and p6 are introduced in § 2.1.2. With increasing density at a fixed B, the electron number density ne reaches some critical value nB, at which eF = ei(0) and degenerate electrons start to populate the first excited Landau level. From Eq. (4.28) we see that nB = 1/(n2\[2 am). Hence the strongly quantizing regime occurs at T ^ Tcyci and p < pB, where

Comparing Eqs. (4.28) and (2.3), we see that kF = (4/3)1/3(p/pB)2/3 kF0 in this regime. Therefore, TF is strongly reduced at p ^ pB, compared to its non-magnetic value TF0:

The nondegenerate electron gas obeys classical statistics. According to the Bohr-van Leeuwen theorem (see footnote 1 on page 54), the magnetic field in this case does not affect the EOS. On the contrary, the EOS is changed drastically, if the electron gas is strongly degenerate and the magnetic field is strongly quantizing. In that case only the n = 0 term survives in Eq. (4.27), and the EOS can be presented in the form

P =777^2 [XB YB - ln(xB + YB)] = 7T2^ « 1 , (4.32)

(2n)2 2n2 Yad BYad 1

where XB and YB are given by Eq. (4.31). In Eq. (4.32) we have introduced a quasi-adiabatic index Yad which, in general, depends on xb, but takes on the constant values 3 and 2 in the non-relativistic (xb ^ 1) and ultrarelativistic (xb » 1) limits, respectively. Compared with the non-magnetic case, Eq. (2.70), Yad is higher (the density dependence of P is steeper), but the numerical value of P is lower everywhere except in the vicinity of the first Landau threshold. This means that a strongly quantizing magnetic field softens the EOS of degenerate electrons.

Non-quantizing magnetic field. If the temperature or density is high enough, the electron distribution is smeared over many Landau levels, and one can replace NB(e) by N0(e). Then the field is non-quantizing. This happens either at p » pB or at T » TB, where

(in this chapter we assume that T ^ Tr). In the relativistic regime at p > pB, Tb is smaller than Tcycl, because the distance between excited Landau levels

Figure 4.1. Characteristic parameter domains in the p —T plane for iron matter at B = 1012 G. Solid lines show TF and Tpi; the dot-dashed line presents Tm (T = 175). The dotted line display Tf at B = 0 (Fig. 2.2). Long-dashed lines show Tb and pB and separate the regions of strong (the lower left sector) and weak (the lower right sector) magnetic quantization, and the domain of the non-quantizing field (T > TB).

Figure 4.1. Characteristic parameter domains in the p —T plane for iron matter at B = 1012 G. Solid lines show TF and Tpi; the dot-dashed line presents Tm (T = 175). The dotted line display Tf at B = 0 (Fig. 2.2). Long-dashed lines show Tb and pB and separate the regions of strong (the lower left sector) and weak (the lower right sector) magnetic quantization, and the domain of the non-quantizing field (T > TB).

for the electrons with c — CF is ^ frug < hwc, where wg(e) = eBc/c is the electron gyrofrequency.

In the non-quantizing magnetic field, many Landau levels contribute to sums over n in Eqs. (4.16) and (4.25). In this case, the summation can be approximately replaced by the integration. Then, integrating by parts, we can reduce Eqs. (4.16) and (4.25) to Eqs. (2.51) and (2.50), respectively.

If p > pB and T < TB, the Landau quantization can remain important for a phenomenon under study. In this case the field is called weakly quantizing. Usually it happens if only a few Landau levels are populated. Higher-order thermodynamic quantities (such as the electron heat capacity, entropy, magnetization) are much stronger affected by magnetic fields in this regime than the bulk quantities (for instance, the electron energy density, chemical potential, pressure).

A density-temperature diagram. Characteristic p -T domains for the outer neutron-star envelope composed of iron are shown in Fig. 4.1 for B = 0 and 1012 G. Partial ionization is taken into account in the mean-ion approximation. The electrons are degenerate below TF; the ions are classical above Tpi. The mean-ion charge number Zeff has been evaluated assuming that the pres sure created by free electrons and by free ions with this Zeff equals the pressure given by the finite-temperature Thomas-Fermi model of Thorolfsson et al. (1998) (discussed below in § 4.3). For comparison, the dotted line reproduces non-magnetic TF from Fig. 2.2. Finally, the long-dashed lines separate three regions, where the magnetic field is strongly quantizing (to the left of pB and considerably below TB), weakly quantizing (to the right of pB at T < TB), or classical (above TB).

Non-relativistic limit. Thermodynamic functions of the ideal electron gas in a magnetic field simplify in the non-relativistic limit (pF ^ mec, T ^ Tr). In this case the electron pressure and number density are given by k T 1

Here, Ae is the electron thermal wavelength given by Eq. (2.27). In the nonde-generate regime (T » TF), one has Iv(x) ~ ex T(v +1), were r(v + 1) is the gamma-function. Therefore, Eq. (4.34) yields Pe = nekBT and

where

This provides an explicit analytical form of the Helmholtz free energy F^ = (X0 — 1) NekBT (in this chapter we do not include the rest energy me c2 into the free energy). In the non-quantizing field (Ze ^ 1), the last two terms in Eq. (4.35) cancel out and the classical non-magnetic result is recovered, Fj[je) = NekBT [ln(neA^/2) — 1]. In the strongly quantizing, nondegenerate regime (p < pB and TF ^ T ^ Tcyci), the last term of Eq. (4.35) vanishes, which yields

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