paramagnetism in many paramagnetic materials individual spin particles

Question

Paramagnetism. In many paramagnetic materials, individual spin particles have more than two spin states. The allowed values for the z-component of a particle's magnetic moment are: µz = -j8, (-j+1)8,..(j-1) 8µ, j8, where 8 = a constant equal to the difference in u, between adjacent states. The result is that the allowed energies for a single particle are: a. Show that the single-particle partition function Z = {sinh [b(j+1/2)]}/sinh(b/2) where b = 8,B/t. Hint:You can use the identity 1 + x + x² +...+ x¯ (1-x¹¹)/(1-x) . In 2-3 sentences, define what magnetization M means. Then derive the total magnetization of a system of N such particles and show that M = N8 [(j+1/2)coth[b(j+1/2)] - (1/2)coth(b/2)]where coth x = (cosh x)/sinh x c. In 2-3 sentences, explain what Curie's Law means. Show that in the high-temperature limit, that your magnetization is proportional to 1/T. Hint: First, show that coth x = 1/x + x/3 when x <<1.