Partition function and heat capacity

[mprovod]

2.1 Consider an array of N localised spin-1/2 paramagnetic atoms. In the presence of a magnetic field, B, the two degenerate spin states split by ±μB, where μB is the Bohr magneton.

 

(i) Derive the single particle partition function for the system.

(ii) Show that the heat capacity C can be written as

C = dU/dT = NkB((D/T)^2)exp(D/T)/(exp(D/T)+1)^2

(here, Kb is Boltzmann constant)

 

and find the value of the constant A. Show that C has a peak at a temperature Tpeak = AμBB/kB where A is a numerical constant. Determine A.

 

 

 

I think I found the answer to part (i), which I think is 2cosh(beta*μB*B), where beta is just the greek symbol. However, I can't figure out the second part, especially how to get the relation for C. Once I have that it should be fine. Thanks for help.

[timo]

- I don't understand your answer to question 1.

- For the 2nd question, I would straightforwardly express the energy via the simple canonic sum, take the derivative wrt. to T of this expression and hope to get the desired result. Did you try that and what did you get?

- There is no constant A to be seen in the expression for C. Did you mean D?

 

Sidenote: You can use TeX on sfn by enclosing the code with [ math] TeX-code [ /math] (without the blank). It makes expressions a bit more readable.

[mprovod]

- I don't understand your answer to question 1.

- For the 2nd question, I would straightforwardly express the energy via the simple canonic sum, take the derivative wrt. to T of this expression and hope to get the desired result. Did you try that and what did you get?

- There is no constant A to be seen in the expression for C. Did you mean D?

 

Sidenote: You can use TeX on sfn by enclosing the code with [ math] TeX-code [ /math] (without the blank). It makes expressions a bit more readable.

I've figured it out. Thanks a lot.