# Fermi Energy

## Homework Statement

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.

## Homework 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

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

## 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.

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

Edit:
BTW, what volume did you actually calculate?

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

Edit:
BTW, what volume did you actually calculate?
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.

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.

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.
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...

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...
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.

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