Statistical Physics: Problem 9 - A Spin Model

[broegger]

Hi.

I'm having trouble with this statistical physics thing again. I am given this exercise:

Problem 9 – A spin model

In a solid at temperature T the atoms have spin 1 so that the m quantum number takes on the values m = 0, ±1. Due to an interaction with the electrostatic field in the crystal, the states with m = ±1 have an energy which is higher by ε than the state with m = 0.

1. Find the average energy E(T) of the nuclei and the associated heat capacity C_V(T). Sketch both functions.

I'm really confused about some basic things here. The Boltzmann distribution states that:

p_r = \frac1{Z}\exp{(-E_r/kT)}.​

This is the probability that the system will be in a particular state r with energy E_r, right? I don't know the number of states of the system or their energies - I know only the individual energies of the atoms.

I have nevertheless tried to determine the mean energy as a function of the temperature in various ways and I always end up with a function that approaches some constant value asymptotically at large temperatures. This would correspond to the heat capacity approaching zero as T tends to infinity (the heat capacity, C_V(T), being the derivative of E(T)), which I don't think makes sense. I don't know if I'm making myself clear...

Any help would be appreciated.

 

[Physics Monkey]

Hi broegger,

In your case the spins don't interact with each other, so the energy states of the whole system can be labeled by the individual spin states. Because the spins are independent, the whole partition function factors into a piece for each spin, and it's just a few lines to obtain the full partition function. Also, the energy should approach an asymptotic value in the limit of high temperature. This becomes obvious when you think about the fact that the energy of the system is bounded from above.

hope this helps.

 

[broegger]

Yea, thanks. So if there is N atoms we would have Z_{tot} = Z^N?

I think what was confusing me was the fact that I used to think of the heat capacity intuitively as "the amount of energy needed to raise the temperature by 1 degree", and so, the heat capacity approaching 0 seems to imply that you could raise the temperature indefinitely without supplying any energy. What does in fact happen if you keep supplying heat even though the system can't absorb any - there seems to be many possibilities?

Thanks for helping out.