[NOTES/SM-04005] Applications of Canonical Ensemble to Paramanetism

The paramagnetism of a substance is a result of magnetic moment associated with spin. Assume each atoms has spin $\frac{1}{2}$ magnetic moment and that the atoms interact weakly with each other and with the other degrees of freedom of the substance.

\noindent Probability distribution is a product of single particle probability distributions.
$$P(E,\vec{r}_1,\cdots,\vec{r}_N)=P(\vec{\epsilon}_1,\vec{r}_1),\cdots
P(\epsilon_n,\vec{r}_N) $$
The magnetic interaction energy is $=-\vec{\mu}\cdot\vec{B}$. Each atom has two energy levels.

\begin{align*}
\text{Two states}\qquad\qquad\begin{cases}E_-=-\mu H &\text{parallel}\\
E_+=+\mu H&\text{antiparallel}\end{cases}
\end{align*}
The probabilities, \(P_\pm\) of the two states for a system in equilibrium with a heat reservoir at temperature are given by
\begin{align*}
P_+=&C\exp(-\beta\epsilon_+)
=C\exp(+\beta\mu HJ), \qquad\qquad \beta=\frac{1}{kT}\\
P_-=&C\exp(-\beta\mu HJ).
\end{align*}
The average magnetic moment per atom,
\(\bar{\mu}_H\), is given by

\begin{align*}
\bar{\mu}_H=&\frac{\mu P_++(-\mu)P_-}{P_++P+_-} = \mu
\frac{\exp(\mu H\beta)-\exp(-\beta\mu H)} {\exp(\beta\mu
H)+\exp(-\beta\mu H)}\\
\bar{\mu}_H =&\mu\cdot\tanh\left(\frac{\mu H}{kT}\right).
\end{align*}
The dipole moment per unit volume, also called {\tt magnetization}, is
$$\bar{M}_0 = N_0\bar{\mu}_H$$
For high temperatures, \(\mu H/kT \ll 1~ \) and
\begin{align*}
\tan\frac{\mu H}{kT} \approx \frac{\mu
H}{kT}\,,\quad \bar{\mu}=\frac{\mu^2H}{kT}
\end{align*}
In the low temperature limit, \(\mu H/kT\gg 1~\)
\begin{align*}
\tan\frac{\mu H}{kT}\approx 1.\quad \bar{\mu}=\mu
\end{align*}
The magnetic susceptibility, in the two limits, is given by
\begin{align*}
\chi = \bar{M}/H = \begin{cases} N_0\mu^2/kT & \text{high temp}\\ N_0\mu & \text{low temp}\end{cases}
\end{align*}
\Flagged[Plot of susceptibility against temperature to be included]

At low $\beta$, {\it i.e.} high temperature, we have
$$\chi \rightarrow \frac{N_0\mu^2}{kT} \qquad M\propto H$$
and we recover Curie Weiss law \[\chi \ \propto \ \frac{1}{T}.\] For large $\beta$, $\chi$ independent of $T$, and $M$ is independent of $H$.