In most paramagnetic materials, the individual magnetic particles have more than two independent states (orientations). The number of independent states depends on the particle's angular momentum "quantum number" j, which must be a multiple of 1/2. For j=1/2 there are just two independent states. More generally, the allowed values of the z component of a particle's magnetic moment are mu_z=-j delta_ mu,(-j+1) delta_ mu, ldots,(j-1) delta_ mu, j delta_ mu, where delta_ mu is a constant, equal to the difference in mu_z between one state and the next. (When the particle's angular momentum comes entirely from electron spins, delta_ mu equals twice the Bohr magneton. When orbital angular momentum also contributes, delta_ mu is somewhat different but comparable in magnitude. For an atomic nucleus, delta_ mu is roughly a thousand times smaller.) Thus the number of states is 2 j+1. In the presence of a magnetic field B pointing in the z direction, the particle's magnetic energy (neglecting interactions between dipoles) is - mu_z B. (a) Prove the following identity for the sum of a finite geometric series: 1+x+x²+ cdots+xⁿ= frac1-xⁿ+11-x. (b) Show that the partition function of a single magnetic particle is Z= frac sinh left[b left(j+ frac12 right) right] sinh fracb2, where b= beta delta_ mu B. (c) Show that the total magnetization of a system of N such particles is M=N delta_ mu left[ left(j+ frac12 right) operatornamecoth left[b left(j+ frac12 right) right]- frac12 operatornamecoth fracb2 right], where coth x is the hyperbolic cotangent, equal to cosh x / sinh x. Plot the quantity M / N delta_ mu vs. b, for a few different values of j. (d) Show that the magnetization has the expected behavior as T rightarrow 0. (e) Show that the magnetization is proportional to 1/T (Curie's law) in the limit T rightarrow infty. (f) Show that for j=1/2, the result of part (c) reduces to the formula derived for a two-state paramagnet.
