Homework Set # 5 - Due September 29

1. Consider a system that can be in any of the following 5 states: 0, +, -, ++, and - - with energies E(0)=0, E(+)=E(-)=ε, E(++)=E(- -)=2ε. If the energy E of the system is such that ε-δ<E<ε+δ, (with δ<<ε), find the probabilities P(0), P(+), P(-), P(++) and P(- -)

   a) in the microcanonical ensemble;

   b) in the canonical ensemble.

   In both cases plot the probability P(ε) vs ε. Hint: you need to provide the actual numerical value of each probability and check that they are normalized as expected.

2. Problem 7.1

3. Problem 7.3

4. Problem 7.7

5. 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) intersticial, slightly displaced from the normal site. Assume that the crystal contains M normal sites and M intersticial sites. The atoms in the normal sites have energy e_0 while the atoms in the intersticial 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 intersticial 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 intersticial 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 intersticial sites for the crystal formed by M cells in the limit e_1-e_0>>kT.

6. Problem 7.8

7. Problem 7.9