# Schottky's defect

1. Feb 12, 2012

### kudoushinichi88

1. The problem statement, all variables and given/known data
A ‘lattice gas’ consists of a lattice of N sites. Each of these sites can be empty, in which
case its energy is zero, or occupied by one particle, in which case its energy is e. Each particle
has a magnetic moment of magnitude μ which in the presence of an applied magnetic field
B, can adopt two orientations (parallel or anti-parallel to the field). Evaluate the mean energy
and mean magnetic moment of the system assuming that the particles are not interacting with
each other.

2. Relevant equations
$$Z=\sum_r e^{-\beta E_r}$$
$$p_r=\frac{1}{Z}e^{-\beta E_r}$$

For a system of n defects in a system of N sites,
$$\frac{n}{N}=\frac{1}{e^{\beta \epsilon}+1}$$
where ε is the energy associated with the defect

Mean energy,
$$\bar{E}=\frac{\partial \ln{Z}}{\partial\beta}$$

3. The attempt at a solution
My problem is that I'm not sure whether there are three separate states for each site, or are there only 2 energy states and the schottky's defect must be considered separately.

If I consider that there are 3 possible states, then the possible energies are
$$\epsilon+\mu\beta,0\ \textrm{and}\ \epsilon-\mu\beta$$

But if this is not right, then I'm not sure how to go about on this question.