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**Thomas–Fermi screening** is a theoretical approach to calculate the effects of electric field screening by electrons in a solid.^{ [1] } It is a special case of the more general Lindhard theory; in particular, Thomas–Fermi screening is the limit of the Lindhard formula when the wavevector (the reciprocal of the length-scale of interest) is much smaller than the fermi wavevector, i.e. the long-distance limit.^{ [1] } It is named after Llewellyn Thomas and Enrico Fermi.

**Lindhard theory**, named after Danish professor Jens Lindhard, is a method of calculating the effects of electric field screening by electrons in a solid. It is based on quantum mechanics and the random phase approximation.

**Llewellyn Hilleth Thomas** was a British physicist and applied mathematician. He is best known for his contributions to atomic physics and solid-state physics, in particular:

**Enrico Fermi** was an Italian and naturalized-American physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and the "architect of the atomic bomb". He was one of very few physicists to excel in both theoretical physics and experimental physics. Fermi held several patents related to the use of nuclear power, and was awarded the 1938 Nobel Prize in Physics for his work on induced radioactivity by neutron bombardment and for the discovery of transuranium elements. He made significant contributions to the development of statistical mechanics, quantum theory, and nuclear and particle physics.

- Derivation
- Relation between electron density and internal chemical potential
- Local approximation
- Electrons in equilibrium, nonlinear equation
- Linearization, dielectric function
- Example: A point charge
- Fermi gas at arbitrary temperature
- See also
- References

The Thomas–Fermi wavevector (in Gaussian-cgs units) is^{ [1] }

**Gaussian units** constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the **Gaussian unit system**, **Gaussian-cgs units**, or often just **cgs units**. The term "cgs units" is ambiguous and therefore to be avoided if possible: cgs contains within it several conflicting sets of electromagnetism units, not just Gaussian units, as described below.

- ,

where *μ* is the chemical potential (fermi level), *n* is the electron concentration and *e* is the elementary charge.

In thermodynamics, **chemical potential** of a species is energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species in a mixture is defined as the rate of change of free energy of a thermodynamic system with respect to the change in the number of atoms or molecules of the species that are added to the system. Thus, it is the partial derivative of the free energy with respect to the amount of the species, all other species' concentrations in the mixture remaining constant. The molar chemical potential is also known as **partial molar free energy**. When both temperature and pressure are held constant, chemical potential is the partial molar Gibbs free energy. At chemical equilibrium or in phase equilibrium the total sum of the product of chemical potentials and stoichiometric coefficients is zero, as the free energy is at a minimum.

The **Fermi level** of a solid-state body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by *µ* or *E*_{F} for brevity. The Fermi level does not include the work required to remove the electron from wherever it came from. A precise understanding of the Fermi level—how it relates to electronic band structure in determining electronic properties, how it relates to the voltage and flow of charge in an electronic circuit—is essential to an understanding of solid-state physics.

The **elementary charge**, usually denoted by `e` or sometimes `q`_{e}, is the electric charge carried by a single proton or, equivalently, the magnitude of the electric charge carried by a single electron, which has charge −1 `e`. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, *e* is sometimes called the **elementary positive charge**.

Under many circumstances, including semiconductors that are not too heavily doped, *n*∝*e*^{μ/kBT}, where *k*_{B} is Boltzmann constant and *T* is temperature. In this case,

- ,

i.e. 1/*k*_{0} is given by the familiar formula for Debye length. In the opposite extreme, in the low-temperature limit *T=0*, electrons behave as quantum particles (fermions). Such an approximation is valid for metals at room temperature, and the Thomas–Fermi screening wavevector *k*_{TF} given in atomic units is

In plasmas and electrolytes, the **Debye length**, named after Peter Debye, is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists. A **Debye sphere** is a volume whose radius is the Debye length. With each Debye length, charges are increasingly electrically screened. Every Debye‐length , the electric potential will decrease in magnitude by 1/e. Debye length is an important parameter in plasma physics, electrolytes, and colloids. The corresponding Debye screening wave vector for particles of density , charge at a temperature is given by in Gaussian units. Expressions in MKS units will be given below. The analogous quantities at very low temperatures are known as the Thomas–Fermi length and the Thomas–Fermi wave vector. They are of interest in describing the behaviour of electrons in metals at room temperature.

- .

If we restore the electron mass and the Planck constant , the screening wavevector in Gaussian units is .

The **electron rest mass** is the mass of a stationary electron, also known as the invariant mass of the electron. It is one of the fundamental constants of physics. It has a value of about 9.109×10^{−31} kilograms or about 5.486×10^{−4} atomic mass units, equivalent to an energy of about 8.187×10^{−14} joules or about 0.5110 MeV.

The **Planck constant**, or **Planck's constant**, denoted is a physical constant that is the quantum of electromagnetic action, which relates the energy carried by a photon to its frequency. A photon's energy is equal to its frequency multiplied by the Planck constant. The Planck constant is of fundamental importance in quantum mechanics, and in metrology it is the basis for the definition of the kilogram.

For more details and discussion, including the one-dimensional and two-dimensional cases, see the article on Lindhard theory.

The internal chemical potential (closely related to Fermi level, see below) of a system of electrons describes how much energy is required to put an extra electron into the system, neglecting electrical potential energy. As the number of electrons in the system increases (with fixed temperature and volume), the internal chemical potential increases. This consequence is largely because electrons satisfy the Pauli exclusion principle: only one electron may occupy an energy level and lower-energy electron states are already full, so the new electrons must occupy higher and higher energy states.

The relationship is described by the electron number density as a function of *μ*, the internal chemical potential. The exact functional form depends on the system. For example, for a three-dimensional Fermi gas, a noninteracting electron gas, at absolute zero temperature, the relation is .

Proof: Including spin degeneracy,

(in this context—i.e., absolute zero—the internal chemical potential is more commonly called the Fermi energy).

As another example, for an n-type semiconductor at low to moderate electron concentration, .

The main assumption in the Thomas–Fermi model is that there is an internal chemical potential at each point **r** that depends *only* on the electron concentration at the same point **r**. This behaviour cannot be exactly true because of the Heisenberg uncertainty principle. No electron can exist at a single point; each is spread out into a wavepacket of size ≈ 1 / *k*_{F}, where *k*_{F} is the Fermi wavenumber, i.e. a typical wavenumber for the states at the Fermi surface. Therefore it cannot be possible to define a chemical potential at a single point, independent of the electron density at nearby points.

Nevertheless, the Thomas–Fermi model is likely to be a reasonably accurate approximation as long as the potential does not vary much over lengths comparable or smaller than 1 / *k*_{F}. This length usually corresponds to a few atoms in metals.

Finally, the Thomas–Fermi model assumes that the electrons are in equilibrium, meaning that the total chemical potential is the same at all points. (In electrochemistry terminology, "the electrochemical potential of electrons is the same at all points". In semiconductor physics terminology, "the Fermi level is flat".) This balance requires that the variations in internal chemical potential are matched by equal and opposite variations in the electric potential energy. This gives rise to the "basic equation of nonlinear Thomas–Fermi theory":^{ [1] }

where *n*(*μ*) is the function discussed above (electron density as a function of internal chemical potential), *e* is the elementary charge, **r** is the position, and is the induced charge at **r**. The electric potential is defined in such a way that at the points where the material is charge-neutral (the number of electrons is exactly equal to the number of ions), and similarly *μ*_{0} is defined as the internal chemical potential at the points where the material is charge-neutral.

If the chemical potential does not vary too much, the above equation can be linearized:

where is evaluated at *μ*_{0} and treated as a constant.

This relation can be converted into a wavevector-dependent dielectric function:^{ [1] }

- (cgs-Gaussian)

where

At long distances (*q*→0), the dielectric constant approaches infinity, reflecting the fact that charges get closer and closer to perfectly screened as you observe them from further away.

If a point charge *Q* is placed at **r**=0 in a solid, what field will it produce, taking electron screening into account?

One seeks a self-consistent solution to two equations:

- The Thomas–Fermi screening formula gives the charge density at each point
**r**as a function of the potential at that point. - The Poisson equation (derived from Gauss's law) relates the second derivative of the potential to the charge density.

For the nonlinear Thomas–Fermi formula, solving these simultaneously can be difficult, and usually there is no analytical solution. However, the linearized formula has a simple solution:

- (cgs-Gaussian)

With *k*_{0}=0 (no screening), this becomes the familiar Coulomb's law.

Note that there may be dielectric permittivity *in addition to* the screening discussed here; for example due to the polarization of immobile core electrons. In that case, replace *Q* by *Q*/ε, where ε is the relative permittivity due to these other contributions.

For a three-dimensional Fermi gas (noninteracting electron gas), the screening wavevector can be expressed as a function of both temperature and Fermi energy . The first step is calculating the internal chemical potential , which involves the inverse of a Fermi–Dirac integral,

- .

We can express in terms of an effective temperature : , or . The general result for is

.

In the classical limit , we find , while in the degenerate limit we find

- .

A simple approximate form that recovers both limits correctly is

- ,

for any power . A value that gives decent agreement with the exact result for all is , which has a maximum relative error of < 2.3%.

**Paramagnetism** is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, diamagnetic materials are repelled by magnetic fields and form induced magnetic fields in the direction opposite to that of the applied magnetic field. Paramagnetic materials include most chemical elements and some compounds; they have a relative magnetic permeability slightly greater than 1 and hence are attracted to magnetic fields. The magnetic moment induced by the applied field is linear in the field strength and rather weak. It typically requires a sensitive analytical balance to detect the effect and modern measurements on paramagnetic materials are often conducted with a SQUID magnetometer.

In solid state physics, a particle's **effective mass** is the mass that it *seems* to have when responding to forces, or the mass that it seems to have when interacting with other identical particles in a thermal distribution. One of the results from the band theory of solids is that the movement of particles in a periodic potential, over long distances larger than the lattice spacing, can be very different from their motion in a vacuum. The effective mass is a quantity that is used to simplify band structures by modeling the behavior of a free particle with that mass. For some purposes and some materials, the effective mass can be considered to be a simple constant of a material. In general, however, the value of effective mass depends on the purpose for which it is used, and can vary depending on a number of factors.

In quantum statistics, a branch of physics, **Fermi–Dirac statistics** describe a distribution of particles over energy states in systems consisting of many identical particles that obey the "Pauli exclusion principle". It is named after Enrico Fermi and Paul Dirac, each of whom discovered the method independently.

An ideal **Fermi gas** is a state of matter which is an ensemble of a large number of non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density, temperature, and the set of available energy states. The model is named after the Italian physicist Enrico Fermi.

In physics, **screening** is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases, electrolytes, and charge carriers in electronic conductors . In a fluid, with a given permittivity *ε*, composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force as

In atomic physics, **hyperfine structure** comprises small shifts and splittings in the energy levels of atoms, molecules, and ions, due to interaction between the state of the nucleus and the state of the electron clouds.

The **magnetic moment** is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include: loops of electric current, permanent magnets, elementary particles, various molecules, and many astronomical objects.

In physics, a **wave vector** is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important. Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

The **Seebeck coefficient** of a material is a measure of the magnitude of an induced thermoelectric voltage in response to a temperature difference across that material, as induced by the Seebeck effect. The SI unit of the Seebeck coefficient is volts per kelvin (V/K), although it is more often given in microvolts per kelvin (μV/K).

In condensed matter physics, the **Fermi surface** is the surface in reciprocal space which separates occupied from unoccupied electron states at zero temperature. The shape of the Fermi surface is derived from the periodicity and symmetry of the crystalline lattice and from the occupation of electronic energy bands. The existence of a Fermi surface is a direct consequence of the Pauli exclusion principle, which allows a maximum of one electron per quantum state.

In solid-state physics, the **free electron model** is a simple model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with quantum mechanical Fermi–Dirac statistics and hence it is also known as the **Drude–Sommerfeld model**.

In statistical mechanics, a **grand canonical ensemble** is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium with a reservoir. The system is said to be open in the sense that the system can exchange energy and particles with a reservoir, so that various possible states of the system can differ in both their total energy and total number of particles. The system's volume, shape, and other external coordinates are kept the same in all possible states of the system.

In physics the **Einstein relation** is a previously unexpected connection revealed independently by William Sutherland in 1904, Albert Einstein in 1905, and by Marian Smoluchowski in 1906 in their papers on Brownian motion. The more general form of the equation is

A **quasi Fermi level** is a term used in quantum mechanics and especially in solid state physics for the Fermi level that describes the population of electrons separately in the conduction band and valence band, when their populations are displaced from equilibrium. This displacement could be caused by the application of an external voltage, or by exposure to light of energy , which alter the populations of electrons in the conduction band and valence band. Since recombination rate tends to be much slower than the energy relaxation rate within each band, the conduction band and valence band can each have an individual population that is internally in equilibrium, even though the bands are not in equilibrium with respect to exchange of electrons. The displacement from equilibrium is such that the carrier populations can no longer be described by a single Fermi level, however it is possible to describe using concept of separate quasi-Fermi levels for each band.

**Surface-extended X-ray absorption fine structure** (SEXAFS) is the surface-sensitive equivalent of the EXAFS technique. This technique involves the illumination of the sample by high-intensity X-ray beams from a synchrotron and monitoring their photoabsorption by detecting in the intensity of Auger electrons as a function of the incident photon energy. Surface sensitivity is achieved by the interpretation of data depending on the intensity of the Auger electrons instead of looking at the relative absorption of the X-rays as in the parent method, EXAFS.

The **Thomas–Fermi** (**TF**) **model**, named after Llewellyn Thomas and Enrico Fermi, is a quantum mechanical theory for the electronic structure of many-body systems developed semiclassically shortly after the introduction of the Schrödinger equation. It stands separate from wave function theory as being formulated in terms of the electronic density alone and as such is viewed as a precursor to modern density functional theory. The TF model is correct only in the limit of an infinite nuclear charge. Using the approximation for realistic systems yields poor quantitative predictions, even failing to reproduce some general features of the density such as shell structure in atoms and Friedel oscillations in solids. It has, however, found modern applications in many fields through the ability to extract qualitative trends analytically and with the ease at which the model can be solved. The kinetic energy expression of Thomas–Fermi theory is also used as a component in more sophisticated density approximation to the kinetic energy within modern orbital-free density functional theory.

**Static force fields** are fields, such as a simple electric, magnetic or gravitational fields, that exist without excitations. The most common approximation method that physicists use for scattering calculations can be interpreted as static forces arising from the interactions between two bodies mediated by **virtual particles**, particles that exist for only a short time determined by the uncertainty principle. The virtual particles, also known as force carriers, are bosons, with different bosons associated with each force.

In quantum mechanics, **orbital magnetization**, **M**_{orb}, refers to the magnetization induced by orbital motion of charged particles, usually electrons in solids. The term "orbital" distinguishes it from the contribution of spin degrees of freedom, **M**_{spin}, to the total magnetization. A nonzero orbital magnetization requires broken time-reversal symmetry, which can occur spontaneously in ferromagnetic and ferrimagnetic materials, or can be induced in a non-magnetic material by an applied magnetic field.

In solid state physics the **electron heat capacity** or **electronic specific heat** describes the contribution of electrons to the heat capacity. Heat is transported by phonons and by free electrons in solids. For pure metals, however, the electronic contributions dominate in the thermal conductivity. In impure metals, the electron mean free path is reduced by collisions with impurities, and the phonon contribution may be comparable with the electronic contribution.

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