At zero temperature, N atoms fully occupy the sites of a simple cubic crystal lattice. At...

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At zero temperature, N atoms fully occupy the sites of a simple cubic crystal lattice. At a finite temperature T, M of these atoms are displaced from their lattice sites to occupy the interstices of the lattice (i.e., points that lie centrally between lattice sites). You can assume that the number of interstices is equal to the number of lattice sites, and that the atoms are displaced into the interstices in a way completely independent of the atoms' original lattice sites. (a) What is the number of ways of taking Matoms from the lattice sites and putting them on interstices? (b) Assuming that both N and M are very large numbers derive an approximate expression for the entropy of the system. (c) The energy required to displace an atom from a lattice site into an interstice is E, and the total internal energy is therefore U = ME. By considering how the temperature T is related to entropy, derive an expression for the fraction of occupied interstices, M/N, in terms of E and T. At zero temperature, N atoms fully occupy the sites of a simple cubic crystal lattice. At a finite temperature T, M of these atoms are displaced from their lattice sites to occupy the interstices of the lattice (i.e., points that lie centrally between lattice sites). You can assume that the number of interstices is equal to the number of lattice sites, and that the atoms are displaced into the interstices in a way completely independent of the atoms' original lattice sites. (a) What is the number of ways of taking Matoms from the lattice sites and putting them on interstices? (b) Assuming that both N and M are very large numbers derive an approximate expression for the entropy of the system. (c) The energy required to displace an atom from a lattice site into an interstice is E, and the total internal energy is therefore U = ME. By considering how the temperature T is related to entropy, derive an expression for the fraction of occupied interstices, M/N, in terms of E and T.