Density of States and Distribution Functions
The density of states plays an important role in the kinetic theory of solids. The product of the density of states and the probability distribution function is the number of occupied states per unit volume at a given energy for a system in thermal equilibrium. This value is widely used to investigate various physical properties of matter. The following are examples, using two common distribution functions, of how applying a distribution function to the density of states can give rise to physical properties.
Fermi-Dirac statistics: The Fermi-Dirac probability distribution function, Fig. 4, is used to find the probability that a fermion occupies a specific quantum state in a system at thermal equilibrium. Fermions are particles which obey the Pauli Exclusion Principle (e.g. electrons, protons, neutrons). The distribution function can be written as
is the chemical potential (also denoted as EF and called the Fermi level), is the Boltzmann constant, and is temperature. Fig. 4 illustrates how the product of the Fermi-Dirac distribution function and the three dimensional density of states for a semiconductor can give insight to physical properties such as carrier concentration and Energy band gaps.
Bose-Einstein statistics: The Bose-Einstein probability distribution function is used to find the probability that a boson occupies a specific quantum state in a system at thermal equilibrium. Bosons are particles which do not obey the Pauli Exclusion Principle (e.g. phonons and photons). The distribution function can be written as
From these two distributions it is possible to calculate properties such as the internal energy, the density of particles, specific heat capacity, and thermal conductivity . The relationships between these properties and the product of the density of states and the probability distribution, denoting the density of states by instead of, are given by
is dimensionality, is sound velocity and is mean free path.
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