In condensed matter physics, the Fermi surface is the surface in reciprocal space which separates occupied from unoccupied electron states at zero temperature. The shape of the Fermi surface is derived from the periodicity and symmetry of the crystalline lattice and from the occupation of electronic energy bands. The existence of a Fermi surface is a direct consequence of the Pauli exclusion principle, which allows a maximum of one electron per quantum state. The study of the Fermi surfaces of materials is called fermiology. Theory Consider a spin-less ideal Fermi gas of particles. According to Fermi–Dirac statistics, the mean occupation number of a state with energy is given by where, is the mean occupation number of the state is the kinetic energy of the state is the chemical potential (at zero temperature, this is the maximum kinetic energy the particle can have, i.e. Fermi energy ) is the absolute temperature is the Boltzmann constant Suppose we consider the limit . Then we have, By the Pauli exclusion principle, no two fermions can be in the same state. Therefore, in the state of lowest energy, the particles fill up all energy levels below the Fermi energy , which is equivalent to saying that is the energy level below which there are exactly states. In momentum space, these particles fill up a ball of radius , the surface of which is called the Fermi surface. The linear response of a metal to an electric, magnetic, or thermal gradient is determined by the shape of the Fermi surface, because currents are due to changes in the occupancy of states near the Fermi energy. In reciprocal space, the Fermi surface of an ideal Fermi gas is a sphere of radius , determined by the valence electron concentration where is the reduced Planck's constant. A material whose Fermi level falls in a gap between bands is an insulator or semiconductor depending on the size of the bandgap. When a material's Fermi level falls in a bandgap, there is no Fermi surface. Materials with complex crystal structures can have quite intricate Fermi surfaces. The figure 2 illustrates the anisotropic Fermi surface of graphite, which has both electron and hole pockets in its Fermi surface due to multiple bands crossing the Fermi energy along the direction. Often in a metal, the Fermi surface radius is larger than the size of the first Brillouin zone which results in