Solid State Physics - Free electron model

[tigger88]

Homework Statement

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.

Homework 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

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!

 

[Jhero]

Thank you for sharing your thoughts and progress on this problem. I am always excited to see others engaging in scientific inquiry and problem-solving. I would be happy to offer some guidance and suggestions for your question.

Firstly, it seems that you have already solved part (a) of the question and have arrived at a correct answer. Well done! This shows that you have a good understanding of the Debye model and how to apply it. Now, for part (b), let's break down the problem and see which equations are relevant.

We are asked to compare our experimental value of gamma (gamma(expt)) with the prediction of the free electron model. In the free electron model, the specific heat capacity is given by:

Cv = (pi^2)*n*(kb)^2*T / [3Ef]

where T is the temperature and Ef is the Fermi energy.

Now, let's consider the equation you have listed:

gamma = ((pi)^2)*n*(kb)^2 / [2Ef]

This equation is slightly different from the one in the free electron model. In this equation, we are looking at the electronic specific heat constant (gamma) instead of the specific heat capacity (Cv). Additionally, the denominator has a factor of 2 instead of 3. This means that the equations are not directly comparable.

To make them comparable, we can rearrange the equation for Cv in the free electron model to solve for gamma:

gamma = (pi^2)*n*(kb)^2 / [3*kb^2*T]

Now, we can plug in the values for n and T in this equation. For n, we can use the same equation you listed:

n = (rho)*z*Na / A

Where z is the number of free electrons per atom. In this case, we are told that tin has four free electrons per atom. So z = 4.

For the density, we are given the density of tin (rho = 7300 kg/m^3). For the atomic weight (A), we are given the atomic weight of tin (A = 118.7 g/mol). However, we need to convert this to kg/mol, so A = 0.1187 kg/mol.

Now, we can plug in these values and solve for gamma:

gamma = (pi^2)*(7300 kg/m^3)*(4)*(6.02*10^