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.
Read more about this topic: Density Of States
Famous quotes containing the words states, distribution and/or functions:
“Courage, then, for the end draws near! A few more years of persistent, faithful work and the women of the United States will be recognized as the legal equals of men.”
—Mary A. Livermore (1821–1905)
“My topic for Army reunions ... this summer: How to prepare for war in time of peace. Not by fortifications, by navies, or by standing armies. But by policies which will add to the happiness and the comfort of all our people and which will tend to the distribution of intelligence [and] wealth equally among all. Our strength is a contented and intelligent community.”
—Rutherford Birchard Hayes (1822–1893)
“Those things which now most engage the attention of men, as politics and the daily routine, are, it is true, vital functions of human society, but should be unconsciously performed, like the corresponding functions of the physical body.”
—Henry David Thoreau (1817–1862)