We start with the kinetic energy of free electrons, here in a representation which involves the momentum p.
De Broglie first speculated on the possibility to assign wavelike properties to particles. He proposed a relation between momentum p and the inverse of the wavelength λ with Planck's constant h as proportionality factor. Put this into the equation for the free electron energy and take the wavenumber k= 2π/λ for convenience.
At first glance this does not look much different. However, there is a fundamental change in the underlying interpretation because the movement of the electron is described by a travelling wave.
The electron is described by a wavenumber k and an angular frequency ω. We can no longer give an exact position but only a probability for presence or absence. Many fundamental experimental findings like, for example, electron interference at double slits supported the validity of this description.
The next step is to extend the description into three dimensions, although we will sometimes revert to the one dimensional representation for illustration. The wavenumber k in the equation above turns into a wavevector with three components kx, ky, and kz and k is now the length. These are related by common vector operations.
For the description of a material with finite dimensions we should consider a certain shape and require the wavefunctions to have zero amplitudes (nodes) at the surface, outside the material the wavefunctions should be zero because we do not normally observe free electrons.
The mathematical description, however, becomes much easier if we use a simple geometry and less restrictive boundaries. We will consider a cube with Volume V=L3 and periodic boundaries, also known as Born-v. Karmann conditions. The periodicity imposes restrictions on the choice of possible k-values.
We end up with many allowed k vectors. They point from the origin to individual points, and all of these points are separated by 2ω/L. We may consider this as a cubic lattice, and in future we will call it k-space. An electron with a given energy E may travel into any direction but the energy defines the "length" for its k-vector with respect to the origin.
In the next step we will look at more than one electron. Let us make a very brutal assumption here; we suppose that the electrons do not interfere with each other except for the Pauli exclusion principle that states that a maximum of two electrons with opposite spin can go into the same state. A state, by the way, is a set of coordinates in k-space. We put the first electron into the state with the lowest energy (k=(0,0,0)) and try to fill all the other electrons into states with the same or the next higher energy. For a large number of electrons we obtain a sphere, the so called Fermi-sphere in k-space. All lattice points inside the sphere are occupied and all the states on the surface will have the k-value which is often sloppily referred to as Fermi-momentum kF even though it is not a momentum but a a wave vector.
We can relate the radius of the Fermi sphere to the number of electron N. Each state occupies the volume of the primitive cell (2π/L)3 and is filled by two electrons. Thus, we can simply count count how many times we can fit the volume of the primitive cell into the Fermi shpere. This number is then equal to N/2:
4/3 π kF3 / (2π/L)3 = N/2
Thus:
Finally we can express the radius of the Fermi sphere kF with the Fermi energy EF by using the relation between wave vector and energy which has been given further up further up on this page.
