# Solid State Physics - Free electron model

1. Nov 6, 2009

### tigger88

1. The problem statement, all variables and given/known data

The following constant volume heat capacity data, Cv, were obtained for a 0.05kg sample of tin at low temperature. (The sample was maintained in the non-superconducting state by the application of a magnetic field). Assuming that tin obeys the Debye model of lattice thermal capacity,
a) deduce the experimental electronic specific heat constant (gamma).
b) Compare your answer with the prediction of the free electron model, assuming tin has four free electrons per atom.

Atomic wt of tin: 118.7 density of tin 7300kg/m^3

I have already solved (a), but after far too much time spent on (b), I have been unable to get the correct answer.
In the original question there is a table of data given to calculate the experimental value, but as I have already found this (and it is correct), I have left it out.

2. Relevant equations

Part of my problem is that I don't know which equations are relevant. Here are some that might be:

gamma = ((pi)^2)*n*(kb)^2 / [2Ef]
where kb = Boltzmann's constant; Ef = Fermi energy

n = (rho)*z*Na / A
where Na = Avogadro's number; A = atomic weight; rho = density

Ef = [(hbar)^2 /(2m)](3(pi^2)n)^(2/3)
where m = mass of electron

3. The attempt at a solution

The answer to (a) was found to be gamma(expt) = 110 J/(m^3 K^2).

I've tried plugging in the values, but I get an answer off by a factor of 10. I have been given the numerical solution (1.28), but I get 12.9.
Clearly I'm not using the correct equations, or something, but this was supposed to be a trivial question, and for some reason, afer spending hours trying to get it right (and number-crunching multiple times to ensure no miscalculation) I've had no success!

Thanks very much!