Consider a solid containing N atoms per unit volume, each atom having a magnetic dipole moment...

Consider a solid containing N atoms per unit volume, each atom having a magnetic dipole moment  Suppose the direction of  can be only parallel or antiparallel to an externally applied magnetic field  (this will be the case it  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 is the temperature and k is Boltzmann\"s constant. Thus, because energy U is, the fraction of atoms whose dipole moment is parallel to B is proportional to e^µB/kT and the fraction of atoms whose dipole moment is antiparallel to  is proportional to e^-µB/kT. (a) Show that the magnitude of the magnetization of this solid is M = Nµ tanh (µB/kT). Here tanh is the hyperbolic tangent function: tanh (x) = (e^x-e^-x)/(e^x +e^-x). (b) Show that the result given in (a) reduces to M = Nµ²B/kT for µB« kT. (c) Show that the result of (a) reduces to M = Nµ²B/kT for µB » kT. (d) Show that both (b) and (c) agree qualitatively with Fig. 32-14

Fig. 32-14 A magnetization curve for potassium chromium sulfate, a paramagnetic salt. The ratio of magnetization M of the salt to the maximum possible magnetization M_max is plotted versus the ratio of the applied magnetic field magnitude B_ext, to the temperature T. Curie\"s law fits the data at the left; quantum theory fits all the data. After W. E. Henry