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The density of states related to volume V and N countable energy levels is defined as: = = (()). Because the smallest allowed change of momentum for a particle in a box of dimension and length is () = (/), the volume-related density of states for continuous energy levels is obtained in the limit as ():= (()), Here, is the spatial dimension of the considered system and the wave vector.
A result is the Fermi–Dirac distribution of particles over these states where no two particles can occupy the same state, which has a considerable effect on the properties of the system. Fermi–Dirac statistics is most commonly applied to electrons , a type of fermion with spin 1/2 .
When considering energy level transitions between two discrete states, Fermi's golden rule is written as = | | ′ | | (), where () is the density of photon states at a given energy, is the photon energy, and is the angular frequency. This alternative expression relies on the fact that there is a continuum of final (photon) states, i.e. the ...
For such a power-law density of states, the grand potential integral evaluates exactly to: [12] (,,) = + (), where () is the complete Fermi–Dirac integral (related to the polylogarithm). From this grand potential and its derivatives, all thermodynamic quantities of interest can be recovered.
The name "density of states effective mass" is used since the above expression for N C is derived via the density of states for a parabolic band. In practice, the effective mass extracted in this way is not quite constant in temperature ( N C does not exactly vary as T 3/2 ).
The following derivation follows the more ... Finding the partition function is also equivalent to performing a Laplace transform of the density of states ...
The density of states which appears in the Fermi's Golden Rule expression is then the joint density of states, which is the number of electronic states in the conduction and valence bands that are separated by a given photon energy.
In other words, the configuration of particle A in state 1 and particle B in state 2 is different from the case in which particle B is in state 1 and particle A is in state 2. This assumption leads to the proper (Boltzmann) statistics of particles in the energy states, but yields non-physical results for the entropy, as embodied in the Gibbs ...