The Sack–Schamel equation describes the nonlinear evolution of the cold ion fluid in a two-component plasma under the influence of a self-organized electric field. It is a partial differential equation of second order in time and space formulated in Lagrangian coordinates. The dynamics described by the equation take place on an ionic time scale, which allows electrons to be treated as if they were in equilibrium and described by an isothermal Boltzmann distribution. Supplemented by suitable boundary conditions, it describes the entire configuration space of possible events the ion fluid is capable of, both globally and locally.
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In this Dossier
- 👉 Boltzmann distribution in the context of Sack-Schamel equation
- Boltzmann distribution in the context of Thermal fluctuations
- Boltzmann distribution in the context of Frenkel pair
Boltzmann distribution in the context of Thermal fluctuations
In statistical mechanics, thermal fluctuations are random deviations of an atomic system from its average state, that occur in a system at equilibrium. All thermal fluctuations become larger and more frequent as the temperature increases, and likewise they decrease as temperature approaches absolute zero.
Thermal fluctuations are a basic manifestation of the temperature of systems: A system at nonzero temperature does not stay in its equilibrium microscopic state, but instead randomly samples all possible states, with probabilities given by the Boltzmann distribution.
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In crystallography, a Frenkel defect is a type of point defect in crystalline solids, named after its discoverer Yakov Frenkel. The defect forms when an atom or smaller ion (usually cation) leaves its place in the structure, creating a vacancy and becomes an interstitial by lodging in a nearby location. In elemental systems, they are primarily generated during particle irradiation, as their formation enthalpy is typically much higher than for other point defects, such as vacancies, and thus their equilibrium concentration according to the Boltzmann distribution is below the detection limit. In ionic crystals, which usually possess low coordination number or a considerable disparity in the sizes of the ions, this defect can be generated also spontaneously, where the smaller ion (usually the cation) is dislocated. Similar to a Schottky defect the Frenkel defect is a stoichiometric defect (does not change the overall stoichiometry of the compound). In ionic compounds, the vacancy and interstitial defect involved are oppositely charged and one might expect them to be located close to each other due to electrostatic attraction. However, this is not likely the case in real material due to smaller entropy of such a coupled defect, or because the two defects might collapse into each other. Also, because such coupled complex defects are stoichiometric, their concentration will be independent of chemical conditions.
View the full Wikipedia page for Frenkel pair