Density of states (DOS) is a property in statistical and condensed matter physics that quantifies how closely packed energy levels are in some physical system. It is often expressed as a function g(E) of the internal energy E, or a function g(k) of the wavevector k. It is usually used with electronic energy levels in a solid. In 3 dimensions, for example, the density of states in reciprocal space (k-space) is , where V is the volume of the solid. Statistical physics, one of the fundamental theories of physics, uses methods of statistics in solving physical problems. ... Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... An energy level is a quantified stable energy, which a physical system can have; the term is most commonly used in reference to the electron configuration of electrons, in atoms or molecules. ... The internal energy of a system (abbreviated E or U) is the total kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the total potential energy associated with the vibrational and electric energy of atoms within molecules or crystals. ... A wave vector is a vector representation of a wave. ... Properties The electron is a fundamental subatomic particle which carries a negative electric charge. ... In jewelry, a solid gold piece is the alternative to gold-filled or gold-plated jewelry. ... Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ... In crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that for all lattice point position vectors R. The reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice. ... In crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that for all lattice point position vectors R. The reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice. ...

A more precise definition is as follows: g(E) dE is the number of allowed energy levels per unit volume of the material, within the energy range E to E + dE (and equivalently for k).

Calculation of the Density of States

To calculate the density of energy states of a particle we first calculate the density of states in reciprocal space (momentum- or k-space). The separation between states is fixed by the boundary conditions. For free electrons and photons within a box of size L, and for electrons inside a crystal lattice with lattice constant L, periodic (Born-von Karman) boundary conditions can be applied. Using the free particle wavefunction of a plane wave this implies In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ... In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ... In quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued function ψ defined over a portion of space and normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared of the wavefunction |ψ(x)|2 is the... In the physics of wave propagation (especially electromagnetic waves), a plane wave (also spelled planewave) is a constant-frequency wave whose wavefronts (surfaces of constant amplitude and phase) are infinite parallel planes normal to the propagation direction. ...

where n is any integer and dk the separation of different k states.

The total number of k-states available to a particle is the volume of k-space accessible to it divided by the volume of a single k-state. The volume accessible is simply the integral from k = 0 to k = k, and the volume of a k-state is .

In 1D this is

and in 3D

These expressions can then be differentiated with respect to k to give the density of states at a given k value:

To find the density of energy states, the relation between energy and momentum for a particular particle is used, to express k and dk in g(k)dk in terms of E and dE. For example for a free electron: , In physics, the free electron model is a possible model for the behaviour of electrons in a crystal structure. ...

This gives a density of states at energy E per unit volume,

Density of States of the Free Electron Gas in One Dimension

The use of the term gas here means that we do not allow the electrons to interact with one another. We assume that we have N electrons constrained to exist in a circular geometry, with radius L. This has the effect of imposing periodic boundary conditions; this is physically acceptable for a system that is very long. In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ...

The eigenfunctions are then indexed by the label n, which takes on all integer values: The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

This eigenfunction has energy eigenvalue

We now remember that we have a gas of electrons, not just a single one, and we start to fill the states.