Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 3.19.
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 can now get the temperature from
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.