Consider a paramagnet, which is a
material with \(n\) spins per unit volume each of which may each be
either “up” or “down”. The spins have energy \(\pm mB\) where
\(m\) is the magnetic dipole moment of a single spin, and there is no
interaction between spins. The magnetization \(M\) is defined as the
total magnetic moment divided by the total volume. Hint: each
individual spin may be treated as a two-state system, which you have
already worked with above.
Find the Helmholtz free energy of a paramagnetic system (assume
\(N\) total spins) and show that \(\frac{F}{NkT}\) is a function of
only the ratio \(x\equiv \frac{mB}{kT}\).
Use the canonical ensemble (i.e. partition function and
probabilities) to find an exact expression for the total
magentization \(M\) (which is the total dipole moment per unit
volume) and the susceptibility \begin{align}
\chi\equiv\left(\frac{\partial M}{\partial
B}\right)_T
\end{align} as a function of temperature and magnetic field for the
model system of magnetic moments in a magnetic field. The result for
the magnetization is \begin{align}
M=nm\tanh\left(\frac{mB}{kT}\right)
\end{align} where \(n\) is the number of spins per unit volume. The figure shows what this magnetization looks like.
Show that the susceptibility is \(\chi=\frac{nm^2}{kT}\) in the
limit \(mB\ll kT\).