Homework Set # 7 - Due October 23

1. Consider a system of N non-interacting particles. Each particle can be in one of the following 3 energy states: E_0=0, E_1=ε, and E_2=2ε. If the number of particles N in the system is such that 3-δ<N<3+δ.

   a) How many states the system has and what are the energies of those states?

   b) If we know that the energy E of the system is in the range 2ε-δ<E< 2ε+δ provide the probability P(ε) in the microcanonical ensemble and make a plot showing P(ε) vs ε.

   c) Now provide the probability P(ε) in the canonical ensemble and make a plot showing P(ε) vs ε. Hint: Write the partition function Z and find β.

   d) Now provide the probability ℙ(ε) in the grand-canonical ensemble and make a plot showing ℙ(ε,N=3) vs ε and ℙ(E=2ε,N) vs N. Hint: Write the grand-partition function and find μ; once ℙ(E,N_m)<0.001 you can assume that ℙ(E,N_m)=0.

2. Consider a monoatomic crystal with M atoms that can be placed in two different kind of positions: 1) normal, i.e., at the sites of a squared lattice and 2) interstitial, slightly displaced from the normal site. Assume that the crystal contains M normal sites and M interstitial sites. The atoms in the normal sites have energy e_0 while the atoms in the interstitial sites have energy e_1 with e_1>e_0. Assume that each site can be occupied by either 0 or 1 atoms.

   a) Write the grand-canonical partition function for one single cell of this system, i.e., a cell composed by one normal and one interstitial site.

   b) Now assume that the average number of atoms per cell is <N>=1 and obtain the chemical potential.

   c) Now find the average population <n> of the interstitial site for one cell. Consider the limits e_1-e_0>>kT and e_1-e_0<<kT.

   d) Assuming weak interactions find the average population <n> for the interstitial sites for the crystal formed by M cells in the limit e_1-e_0>>kT.

3. Problem 4.1

4. Problem 4.4

5. Problem 4.8