The Fermi surface is the surface of constant energy in k space. The Fermi surface separates the unfilled orbitals from the filled orbitals, at absolute zero. The electrical properties of the metal are determined by the shape of the Fermi surface, because the current is due to changes in the occupancy of states near Fermi surface. The free electron Fermi surfaces were developed from spheres of radius kF determined by the valence electron concentration.
The free electron Fermi surface for the an arbitrary electron concentration is shown in Fig.1.
These are Brillouin zones of a square lattice in two dimensions. The blue circle shown is a surface of constant energy for free electrons; it will be the Fermi surface for some particular value of the electron concentration.
It is inconvenient to have sections of the Fermi surface that belong to the same Brillouin zone appear detached one from another. The detachment can be repaired by a transformation to the first Brillouin zone. The procedure is known as mapping the Fermi surface in the reduced zone scheme.
There is also another way to represent the Fermi surface in the reduced and periodic zone scheme. Fermi surfaces for free electrons are constructed by a procedure credited to Harrison, Fig.2.
Thus, in Fig.7,