The total partition function of N identical, independent particles is $Z^N/N!$ where Z is the partition function of a single particle. To find out the correct magnetization due to N identical atoms of magnetic moment $\mu=-g \mu_B J$, the partition function is taken to be $Z^N$. See page 4 of this note. http://physics.unl.edu/~cbinek/Paramagnetism%20unit%2022.pptx Why is the factor $1/N!$ missing?
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Firstlly, magnetization does not care about the $N!$. It's thermodynamic quantities such as specific heat that need the $N!$ Secondly it is not clear whether the powerpoint slides are talking about a paramagnetic gas or a crystal. In the crystal the atoms are, in effect, distinguishable because they can be specified by their positions: if they were free to move and exchange places with one another, then we would have to keep the $N!$, As long as they don't move there is no thermodynamic consequence to their identity. The crystal microstates are therefore counted as if the atoms were distinguishable.
answered Nov 10, 2020 at 13:16