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2.2 Effects of magnetic fields

Typical pulsars have surface magnetic fields B ∼ 1012 G. Magnetars have much higher magnetic fields, B ∼ 1014– 1015 G. The properties of the outer envelope of neutron stars can be drastically modified by a sufficiently strong magnetic field BBB. It is convenient to introduce the “atomic” magnetic field B0
2 3 B0 = m-ee-c= 2.35 × 109 G . (17 ) ℏ3
It is the value of B for which the electron cyclotron energy is equal to e2∕a0 = 2 × 13.6 eV (a0 is the Bohr radius). Putting it differently, at B = B 0 the characteristic magnetic length a = (ℏc∕eB )1∕2 m equals the Bohr radius. For typical pulsars and magnetars the surface magnetic field is significantly stronger than B0. As a result, the atomic structure at low pressure is expected to be strongly modified. The motion of electrons perpendicular to BBB is quantized into Landau orbitals. Assuming that BBB = [0,0, B], the electron energy levels are given by εn(pz) = c(m2ec2 + 2ℏ ωcemen + p2z)1∕2, where n is the Landau quantum number and p z the z-component of the electron momentum. The ground state Landau level n = 0 is nondegenerate with respect to the spin (the spin is antiparallel to BBB, with spin quantum number s = − 1), while the higher levels n > 0 are doubly degenerate (s = ±1). The cyclotron frequency for electrons is ωce = eB ∕mec; it is 1836 times larger than for protons. The Coulomb binding of electrons by the atomic nucleus is significantly less effective along B BB, while in the plane perpendicular to B BB the electron motion is confined to the n = 0 Landau level. Therefore atoms get a cylindrical shape and can form linear chains along BBB. The attraction between these chains can lead to a phase transition into a “magnetically condensed” phase (for a recent review on this topic, see Medin & Lai [287Jump To The Next Citation Point]). The density of the condensed phase at zero pressure P = 0 (i.e. at the stellar surface) and zero temperature T = 0 is
A ( Z ) −3∕5 ρs ≃ 4.4 × 103 --- --- B61∕25g cm −3 , (18 ) 56 26
where B = B ∕1012 G 12. For each element, there is a critical temperature at given B, below which a magnetically condensed phase exists at P = 0. The values of Tcrit at several B for 56Fe, 12C, and 4He are given in Table 1.

Table 1: Critical temperature (in K) below which a condensed phase exists at P = 0, for several magnetic field strength B12 = B / 1012 G and matter composition. From [287].
B12 = 10 100 1000
56Fe × 105 × 106 × 107
12C × 105 × 106 × 107
4He × 105 × 106 × 107

We now briefly consider the effects of the magnetic field on plasma properties at finite pressure P > 0. The magnetic field strongly quantizes the motion of electrons, if it confines most of them to the ground Landau state n = 0. Parameters relevant to a strong quantization regime are

ℏ-ωce 8 3A-′ 3∕2 −3 Tce = kB ≈ 1.343 × 10 B12 K , ρB = 7.045 × 10 Z B 12 g cm , (19 )
∘ ------- TB = Tce if ρ < ρB , or TB = Tce∕ 1 + x2r if ρ > ρB . (20 )
The field BBB is strongly quantizing if ρ < ρ B and T ≪ T ce. On the contrary, a magnetic field is called weakly quantizing if many Landau levels are occupied, but still T ≪ TB. Finally, BBB is nonquantizing if T ≫ TB. The temperature TB and density ρB are shown in Figure 2View Image.

The magnetic field can strongly modify transport properties (Section 9.5) and neutrino emission (Section 11.7). Its effect on the equation of state is significant only if it is strongly quantizing (see Section 6.4).

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