Free Electron Gas

The total energy of a molecule with three degrees of freedom is:

The specific heat is:

This aligns with the degrees of freedom for rotational and vibrational contributions.

In 3 dimensions, each dimension contributes from kinetic energy and from potential energy, giving per dimension. Thus:

Since , this becomes:

The total kinetic energy of free electrons in a 3D gas at 0 K is:

where is the Fermi energy. This result comes from integrating the energy density over the occupied states in the Fermi sphere.

The energy gained by an electron in an electric field over a distance is:

Given , , and :

The drift velocity is calculated using:

Using resistivity, can be derived from the relation:

Given , , , , and :

Substituting into :

The energy of an electron confined in a box is given by:

where , , and .
Substituting values:

The frequency of the emitted X-ray is:

Given :

The energy of the emitted photon is:

Using , where and :