Question

If n is the number of conduction electrons per unit volume and m the electron mass then show that the Fermi energy is given by the expression

EF = h2/8m (3n/π)2/3

Answer 1

For the conduction electrons, the number of states per unit volume with energy in the range E and E + dE, can be written as n(E)dE where n(E) is the density of states.

Now, for a free electron gas

Let P(E) be the probability function which gives the probability of the state at the energy E to be occupied. At T = 0 all states below a certain energy are filled (P = 1) and all states above that energy are vacant (P = 0). The highest occupied state under the given conditions is called the Fermi energy.

The product of the density n(E) of available states and the probability P(E) that those states are occupied, gives the density of occupied states n0(E); that is

n0(E) = n(E)P(E)

The total number of occupied states per unit volume is given by