A perfect crystal has N lattice sites and M interstitial sites (sites between lattice sites). An energy ∆ is required to remove an atom from a lattice site and place it in an interstitial site. Assume that the number of displaced atoms, n, is much smaller than N and M. (a) Find the number of ways to remove n atoms from N lattice sites. (b) Find the number of ways to place n atoms on M interstitial sites. (c) Use the microcanonical ensemble to calculate the entropy as a function of total energy E, and define the temperature. (d) Show that the average number of displaced atoms n at temperature T is given by n2 (N − n)(M − n) = e−∆/kT . Express n for ∆ ≫ kT , and ∆ ≪ kT . (e) Use this model for defects in a solid. Set N = M, and ∆ = 1 eV. Find the defect concentration at T = 1000 K and 300 K. 2