Example: two-level paramagnet (noninteracting spins)

Canonical solution for N independent two-level magnetic moments in a field: partition function, magnetization, Curie law, energy, and heat capacity.

Example: two-level paramagnet (noninteracting spins)

Consider $N$ independent spins (or magnetic moments) in a uniform magnetic field $B$ . Each spin has two energy levels

E_\pm = \mp \mu B,

so the system is a canonical-ensemble model (see canonical ensemble ) with a simple discrete spectrum.

Partition function

For inverse temperature $\beta=1/(k_B T)$ (with $T$ the temperature ), the single-spin partition function is

z(\beta,B)=e^{\beta\mu B}+e^{-\beta\mu B}=2\cosh(\beta\mu B),

Z(\beta,B)=z(\beta,B)^N=\big(2\cosh(\beta\mu B)\big)^N.

F(T,B)=-Nk_B T\log\!\big(2\cosh(\beta\mu B)\big).

The magnetization is the field derivative of $F$ :

M = -\left(\frac{\partial F}{\partial B}\right)_T = N\mu\,\tanh(\beta\mu B).

The susceptibility is

\chi=\left(\frac{\partial M}{\partial B}\right)_T = N\beta\mu^2\,\mathrm{sech}^2(\beta\mu B),

and for high temperature (small $\beta\mu B$ ) this gives Curie’s law

\chi \approx \frac{N\mu^2}{k_B T}.

U = -\frac{\partial}{\partial\beta}\log Z = -N\mu B\,\tanh(\beta\mu B).

Differentiating in $T$ gives the heat capacity at fixed $B$ :

C = \left(\frac{\partial U}{\partial T}\right)_B = N k_B(\beta\mu B)^2\,\mathrm{sech}^2(\beta\mu B).

The entropy can be written as

S = -\left(\frac{\partial F}{\partial T}\right)_B = N k_B\Big[\log\!\big(2\cosh x\big)-x\tanh x\Big], \qquad x=\beta\mu B.