Answer: Consider a solid containing N atoms per unit

Chapter , Problem 45

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Consider a solid containing N atoms per unit volume, each atom having a magnetic dipole moment 71. Suppose the direction of 71 can be only parallel or antiparallel to an externally applied magnetic field B (this will be the case if /1 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- UlkT, where T is the temperature and k is Boltzmann's constant. Thus, because energy U is - /1- B, the fraction of atoms whose dipole moment is parallel to B is proportional to ellBlkT and the fraction of atoms whose dipole moment is antiparallel to B is proportional to e-IlBlkT. (a) Show that the magnitude of the magnetization of this solid is M = NfL tanh(fLBlkT). Here tanh is the hyperbolic tangent function: tanh(x) = (eX - e-X)/(e' + e-X). (b) Show that the result given in (a) reduces to M = NfL2BlkTfor fLB ~ kT. (c) Show that the result of (a) reduces to M = NfL for fLB ~ kT. (d) Show that both (b) and (c) agree qualitatively with Fig. 32-14.