the problem is about statistical thermodynamics:

The magnetization M of a system is the magnetic moment density - that is, the magnetic moment per unit volume, ! H M= oi = BV' This means the expected value of magnetization is related to the expected value of energy via (M) = BV The magnetic susceptibility x is defined as the measure of how much our paramagnet will "become magnetized" when exposed to an external magnetic field B, (M) X= OB V Suppose our system has a density of spins n = N/V. (d) Extra Part (Not for Credit) Show the following results for the magnetization (M) and susceptibility x of our paramagnet as functions of temperature, (M) = nutanh MB nu2 KBT X = kBT cosh? HB KBT (e) Consider some reference magnetic field scale Bo. First, demonstrate or argue that parameters b = B/Bo, T = KBT/MBo, m = (M)u, and x = xBou are non-dimensional and intensive (i.e. independent of the size of the system). Graph m-vs-T and x-VS-T at fixed B = Bo and comment on what happens in the T - 0 and T - co limits at fixed B? Then graph m-vs-b and x-vs-b at fixed T = MBo/KB and comment on what happens in the b - 0 and b - too limits. [Note: Only a brief discussion of the graphs is expected but think about why it makes sense for the magnetization graphs to look the way they do and what high/low temperature and field mean for how the spins in the paramagnet behave.](f) Use the partition function Z to determine the Helmholtz free energy F. How can we use our knowledge of F to determine the entropy, 5'? How do we expect F and S to relate to (E)? [Supplementary Part (Not for Credit): Now do it! Solve for S and explicitly Show that your relation between F, S, and (E) is satised]