Problem 1
Consider a solid of volume V containing N sites (N>>1). Each site is predisposed and susceptible to trap
only one particle. A trapped particle can take two energies: -$\epsilon_1$ or -$\epsilon_2$, with $\epsilon_2 > \epsilon_1 > 0$. We neglect the
interactions between particles trapped by two different sites. The solid is in contact with a thermostat
set at temperature T and a reservoir of particles of chemical potential, ?.
a) Calculate the grand canonical ensemble of the solid.
b) Calculate the average number of trapped particles <n>, and the average energy of the solid as a
function of T and ?. (<..> denotes average quantity)
c) Calculate <$n_1$> and <$n_2$> the average numbers of trapped particles in the energy levels -$\epsilon_1$ and -$\epsilon_2$,
respectively, as a function of T and ?.
d) Calculate the average number, <$n_0$>, of empty sites (no trapped particles).
e) Let us apply the above results to the case of electrons of magnetic moment, m, which are subjected
to a magnetic field, B. The energy levels 1 and 2 are thus: -$\epsilon_1 = \epsilon_0 + mB$ and -$\epsilon_2 = \epsilon_0 - mB$ where $\epsilon_0 > 0$.
Level 1 is the level occupied by an electron of spin opposite to the magnetic field (Zeeman energy +mB)
whereas level 2 is the level occupied by an electron of spin parallel to the magnetic field.
Using the results above, calculate the average magnetic moment <M> of the occupied traps (sites)
by the electrons. Determine the high and low temperature limits of the average moment. Comment??
