Consider solid containing N atoms per unit volume, each atom having a magnetic dipole moment suppose the direction of , can be only parallel

Consider solid containing N atoms per unit volume, each atom having a magnetic dipole moment μ suppose the direction of μ, can be only parallel or antiparallel to an externally applied magnetic field B (this will be the case if μ is due to the spin of a single electron). According to statistical mechanics, the probability of an atom being in a state with energy U is proportional to e -U/kT, where T the temperature and k is Boltzmann's constant. Thus, because energy U is –μ ∙ B, the fraction of atoms whose dipole moment is parallel to B is proportional to e-μ B/kT and the fraction of atoms whose dipole moment is antiparallel to B is proportional to e-μ B/kT.

(a) Show that the magnitude of the magnetization of this solid is M = N μ tan h (μ B/kT). Here tanh is the hyperbolic tangent function: tanh (x) = (ex - e-x) / (ex + e-x).

(b) Show that the result given in (a) reduces to M = Nμ2B/ kT for μ B < kT.

(c) Show that the result of (a) reduces to M = N μ for μ B > kT.

(d) Show that both (b) and (c) agree qualitatively with Figure