Consider an idealized crystal with N atoms. Each atom can occupy its own lattice point or a neighboring interstitial position, representing a defect in the lattice. Assume that each atom has access to one unique interstitial position (i.e., there's no competition for sites). Let E be the energy necessary to move an atom from a lattice site to an interstitial position, and let n be the number of atoms in interstitial positions. Assume the system is in equilibrium at temperature . (a) If N and n are fixed and known, how many possible states are there? (b) Use physical reasoning and perhaps some simple math to explain what the limiting values of the entropy must be for 0 and ? The "best" way to think about this takes no calculation at all. (c) What are the partition function Z and entropy for the system in terms of , N , and E? (d) How many defects are present in the system at temperature ? In one or more places you may find useful the binomial expansion