1. Review Paramagnets
2. Langevin function
3. Internal energy and heat capacity
4. Why low temperature?
5. Quantization, Brillouin functions
6. Ferro, Anti-ferro magnetism

• Work out typical value of T

The question what is a paramagnet gave answers in terms of models, as well as observations. M is proportional to H/T at low fields is the answer. Plot inverse susceptibility versus temperature is a good indication to check. Class discussion given the plot M(H/T) create the plot M(T/H) took some time. Need to look at low temperature behavior, or large x limit, where the 1/x is the dominant deviation from 1. I did not catch this in class, but this is an example where we do not see the correct low temperature behavior! Classical models don't work there. Homework for next time is to calculate Brillouin functions. We looked at the values of the Bohr magneton and Boltzmann factor, both similar in ev/T and eV/K, so 1 Tesla is about 1K. Need for low temperatures in Cp measurements, also because we have lattice specific heat. We see AF phase transitions, hence we need the total magnetic field, including interactions. We had only ten minutes to discuss effective fields, I hope that was useful. I did it in the context of Langevin.

1. Review based on dU=… Done effects of TdS term, added HdM, now mu dN.
2. Quantum states with variable numbers of aprticles, V determines epsilon
3. New Lagrangian, do not like book aproach, extra term - lambda2 ( sum nPn - N )
4. Gibbs factor, Grand partition function
5. Grand energy, relate to entropy, first law

• Work out Lagrangian
• derive Omega = -pV based on extensive variable. Similar to G = mu N.
• simple example e_s = s hbar omega, n_s = s (gives unphysical results, only one quantum number)

It took the groups about 15 minutes to construct the new Lagrangian, with good questions being asked. The basic idea is there, but some details are not. Next time ask question about the meaning of variables, they are all averages. Need practice constructing grand partition function, next time.

1. Relate grand pf to regular pf. Remark on analytical properties.
2. Calculate for independent particles Z= 1/N! Z1^N, no singularities.
3. Simple examples.

• State (s) : e = s hbar omega , n = s Too simple!
• State (s,t) : e = s hbar omega1 + t hbar omega2 , n = s + t
• State (s,t) : e = s hbar omega1 + t hbar omega2 + s t e_mix , n = s + t

Rearranging the summation to sum over n first is always a bit of a problem to understand. It seems like a math trick. Make sure that everybody understands that all terms are present. Showing next that independent particles have no phase transition is fun. Class worked on example 1, which took a while. One question was the difference between N and n_s, average versus actual. The quantities in thermo equations are always averages. Problem 2 was assigned for homework.