A system of magnetic spins is made of $N$ localized domains at temperature $T$. The spins are associated with the energy
$N$
$H = -J \sum_{i=1,3,5...N-1} \sigma_i \sigma_{i+1} - \mu_o H \sum_{i=1}^N \sigma_i$
where the parameters $J$, $\mu_o$, and $H$ are positive, and $\sigma_i = \pm 1$ for all sites $i$.
Assume that $N$ is an even number, and note that the first sum is over odd integers.
a) Obtain the canonical partition function for this system.
b) Calculate the internal energy per spin, $u = u(T, H)$. Make a plot of
$u(T, H = 0)$ vs. $T$.
c) Calculate the entropy per spin, $s = s(T, H)$. Make a plot of $s(T, H = 0)$ vs
$T$.
d) Obtain expressions for the magnetization per particle,
$1$
$N$
$m = m(T, H) = \frac{1}{N} \mu_o \sum_{i=1}^N \sigma_i$
and for the magnetic susceptibility,
$\frac{\partial m}{\partial H}|_T$
