Reference: Daniel V. Schroeder, *An Introduction to Thermal Physics*, (Addison-Wesley, 2000) – Problem 3.19.

We’ve looked at a two-state paramagnet and generated some curves involving entropy, energy and temperature using numerical methods. Here we’ll derive the analytic solution.

This is a system of magnetic dipoles which, when placed in a magnetic field , align themselves so that their magnetic moment points either parallel or antiparallel to the field. The energy of a dipole that is aligned with the field is lower, and we’ll call it , so that the antiparallel dipole has energy , and the total energy of the system is

where an up arrow indicates parallel alignment and a down arrow antiparallel.

The net magnetization is then

The multiplicity of states is the same as a set of coins with heads, so

We begin the analytic solution by looking at the entropy directly, and using Stirling’s approximation.

We can now get the temperature from

Using the chain rule, 2 and 1 we get

Exponentiating both sides gives

which can be solved for :

The magnetism is, from 3

Finally, the heat capacity at constant magnetic field is (using the derivative of the tanh function: ):

These analytic formulas give the curves we saw earlier using the numerical solution.

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