In the equations of § 4.1.2, the summation over discrete Landau numbers n reflects consecutive population of new Landau levels with growing density, which leads to magnetic quantum oscillations of thermodynamic and kinetic functions (see, e.g., Lifshitz & Pitaevskii 1980). When the field is weakly quantizing, these quantities oscillate, as a rule, around their values obtained neglecting the magnetic quantization. For first-order (bulk) thermodynamic quantities (P, U, n), the oscillations are relatively weak, whereas for second-order quantities (Cv , CP, kTF) they are more pronounced. For example, the oscillations of the density exponent xP defined by Eq. (2.38), will be shown in Fig. 4.11 on p. 204. The oscillations are smoothed by the thermal broadening of the Fermi distribution function and by the quantum broadening of the Landau levels (particularly, owing to electron collisions; see Yakovlev & Kaminker 1994, for references).
For example, we can mention the well known de Haas-van Alphen effect — oscillations of magnetic susceptibility (e.g., Landau & Lifshitz 1993). In quantizing magnetic fields, these oscillations can be large, resulting in high magnetization. Canuto & Chiu (1971) suggested that this effect could lead to a spontaneous magnetization of the electron gas in neutron star envelopes, the so called Landau orbital ferromagnetism - LOFER. They found that the oscillations of the kinetic pressure due to the Landau quantization of degenerate electron gas may result in a state of permanent quasistable macroscopic magnetism, which is stronger for higher pressure. Neglecting the broadening of the Landau levels, the authors obtained a density dependence of the maximum LOFER field strength, which implied that in the outer neutron star crust (at P < 10115 g cm-3) LOFER could produce B up to — 1012 G (and still larger B in the inner crust). However, the thermal and quantum broadening of oscillations prevents the spontaneous magnetization. For instance, Schmid-Burgk (1973) showed that LOFER is smeared out by the thermal broadening in the outer crust of a neutron star at T > 104 K.
The de Haas-van Alphen effect may also violate the condition for thermo-dynamic equilibrium which states that the field strength should increase with the growth of the magnetic induction. This instability leads to the formation of domains with alternating magnetization (Lifshitz & Pitaevskii, 1980). Bland-ford et al. (1983) showed that in a neutron star envelope with B = 1012 G this instability may develop at T up to 107 K. However, since the magnetization is weak (a few percent of the field strength), this effect can hardly have any observable consequences.
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