### 2.2 Effects of magnetic fields

Typical pulsars have surface magnetic fields . Magnetars have much higher magnetic fields,
. The properties of the outer envelope of neutron stars can be drastically modified by a
sufficiently strong magnetic field . It is convenient to introduce the “atomic” magnetic field
It is the value of for which the electron cyclotron energy is equal to ( is the
Bohr radius). Putting it differently, at the characteristic magnetic length
equals the Bohr radius. For typical pulsars and magnetars the surface magnetic field is significantly stronger
than . As a result, the atomic structure at low pressure is expected to be strongly modified. The
motion of electrons perpendicular to is quantized into Landau orbitals. Assuming that ,
the electron energy levels are given by , where is the Landau
quantum number and the z-component of the electron momentum. The ground state Landau level
is nondegenerate with respect to the spin (the spin is antiparallel to , with spin quantum
number ), while the higher levels are doubly degenerate (). The
cyclotron frequency for electrons is ; it is 1836 times larger than for protons.
The Coulomb binding of electrons by the atomic nucleus is significantly less effective along
, while in the plane perpendicular to the electron motion is confined to the
Landau level. Therefore atoms get a cylindrical shape and can form linear chains along . 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 [287]). The density of the condensed
phase at zero pressure (i.e. at the stellar surface) and zero temperature is
where . For each element, there is a critical temperature at given , below which a
magnetically condensed phase exists at . The values of at several for ^{56}Fe, ^{12}C, and ^{4}He
are given in Table 1.
We now briefly consider the effects of the magnetic field on plasma properties at finite pressure .
The magnetic field strongly quantizes the motion of electrons, if it confines most of them to
the ground Landau state . Parameters relevant to a strong quantization regime are

and
The field is strongly quantizing if and . On the contrary, a magnetic field is called
weakly quantizing if many Landau levels are occupied, but still . Finally, is nonquantizing if
. The temperature and density are shown in Figure 2.
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).