Bloch's theorem: an unbound electron moves in a periodic potential as a free electron in vacuum, except for the electron mass me becoming an effective mass m* which may deviate considerably from me (one can even use negative effective mass to describe conduction by electron holes). Effective masses can be derived from band structure computations that were not originally taken into account in the free electron model. From the Drude model Many physical properties follow directly from the Drude model, as some equations do not depend on the statistical distribution of the particles. Taking the classical velocity distribution of an ideal gas or the velocity distribution of a Fermi gas only changes the results related to the speed of the electrons. Mainly, the free electron model and the Drude model predict the same DC electrical conductivity σ for Ohm's law, that is with where is the current density, is the external electric field, is the electronic density (number of electrons/volume), is the mean free time and is the electron electric charge. Other quantities that remain the same under the free electron model as under Drude's are the AC susceptibility, the plasma frequency, the magnetoresistance, and the Hall coefficient related to the Hall effect. Properties of an electron gas Many properties of the free electron model follow directly from equations related to the Fermi gas, as the independent electron approximation leads to an ensemble of non-interacting electrons. For a three-dimensional electron gas we can define the Fermi energy as where is the reduced Planck constant. The Fermi energy defines the energy of the highest energy electron at zero temperature. For metals the Fermi energy is in the order of units of electronvolts above the free electron band minimum energy. Density of states The 3D density of states (number of energy states, per energy per volume) of a non-interacting electron gas is given by: where is the energy of a given electron. This formula takes into account the spin degeneracy but does not consider a possible energy shift due to the bottom of the conduction band. For 2D the density of states is constant and for 1D is inversely proportional to the square root of the electron energy. Fermi level The chemical potential of electrons in a solid is also known as the Fermi level and, like the related Fermi energy, often denoted . The Sommerfeld expansion can be used to calculate