Probability of n position occupied by atoms

[hansbahia]

Homework Statement

If you have atoms that are normally located at the normal lattice positions or at an interstitial position where energy >ε, how can I find the probability that n interstitial positions are occupied by atoms?
If we were to use large N how can I find the fraction of interstitial sites that are occupied as a function of the temperature?
How do I find the microstate?

Homework Equations

NCn
n=E/ε

The Attempt at a Solution

Assuming that there are N normal locations and N interstitial locations and the energy of an atom in a normal lattice position can be set to 0
I know that # of ways to arrange the intersticial atoms is N!/n!(N-n)!
from there I don't know how to calculate de microstate nor the probability
I attempted at a solution of random variables therefore
P(n)=1/2ⁿ(N!/n!(N-n)!

 

[clamtrox]

The way I understand the question is that for each atom, you can either be at the lattice position with energy ε_L or at an interstitial position with energy ε_I, ε_L<ε_I. Is that right? If it is, then each microstate is just of the form L,L,L,I,L,..., telling the position of each atom. The corresponding probability you get from the usual canonical ensemble consideration,
$$p = \frac{1}{Z} e^{-\beta E}$$
and a probability for each macrostate you get using combinatorics, as you already guessed.