# Fermi Energy

1. Oct 17, 2012

### roam

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

The density of gold (atomic mass 197) is 19.32 g/cm3. Assuming each gold atom contributes one electron to the free electron Fermi gas, calculate the Fermi energy in eV.

2. Relevant equations

Fermi energy is given by:

$E_F=\frac{h^2}{8m_e} \left( \frac{3N}{\pi V} \right)^{2/3}$

The total number of electrons is

$N= \int^{\infty}_0 n(E) dE = \int^{E_F}_0 (8 \pi V/h^3) (2m_e^3)^{1/2} E^{1/2} dE = \frac{16 \pi V(2m_e^3)^{1/2}}{3h^3} E_F^{3/2}$

3. The attempt at a solution

To find the Fermi energy I want to use the first equation but I need to know the number of electrons N (which is equal to the number of atoms), and the volume V. I found the volume but I'm not sure how to find N:

$V=\frac{m}{\rho} = \frac{197 \times (1.66 \times 10^{-27}) \ kg}{19.32 \times (10^{-3}/10^{-6}) \ kg/m^3}$

So how can I find the number of electrons? How can I use the second equation to find N without knowing EF?

Any help is greatly appreciated.

Last edited: Oct 17, 2012
2. Oct 17, 2012

### Dickfore

Do you really need the number or something else for EF?

Edit:
BTW, what volume did you actually calculate?

3. Oct 17, 2012

### roam

Yes, I believe I need the actual numerical value of EF to be able to use that equation.

I calculated the total volume of the Fermi gas under consideration.

4. Oct 17, 2012

### Dickfore

Wrong, you are asked to find the Fermi energy by using that equation.

Wrong, the mass that you used is not the mass of the whole sample, but something else.

5. Oct 17, 2012

### roam

Sorry. Yes, that's the volume based on the mass of a single atom.

How can we solve that equation for EF without knowing N?

We don't know how many atoms are in the sample...

6. Oct 17, 2012

### Dickfore

So, how many atoms does a sample with a mass equal to one atom contain? You are given the number of electrons each atom contributes. This should suffice to obtain N.

7. Oct 17, 2012

### roam

So, do you mean we can then just use N=1?