**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.

- 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] }

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

For the example of semiconductors that are not too heavily doped, the charge density *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

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

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.

Given a Fermi gas of density , the highest occupied momentum state (at zero temperature) is known as the Fermi momentum, .

Then the required 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] } (in cgs-Gaussian units)

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 (in cgs-Gaussian units):

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 ,^{ [2] } which has a maximum relative error of < 2.3%.

In the effective temperature given above, the temperature is used to construct an effective classical model. However, this form of the effective temperature does not correctly recover the specific heat and most other properties of the finite- electron fluid even for the non-interacting electron gas. It does not of course attempt to include electron-electron interaction effects. A simple form for an effective temperature which correctly recovers all the density-functional properties of even the *interacting* electron gas, including the pair-distribution functions at finite , has been given using the classical map hyper-netted-chain (CHNC) model of the electron fluid. That is

where the quantum temperature is defined as:

where *a* = 1.594, *b* = −0.3160, *c* = 0.0240. Here is the Wigner–Seitz radius corresponding to a sphere in atomic units containing one electron. That is, if is the number of electrons in a unit volume using atomic units where the unit of length is the Bohr, viz., 5.29177×10^{−9} cm, then

For a dense electron gas, e.g., with or less, electron-electron interactions become negligible compared to the Fermi energy, then, using a value of close to unity, we see that the CHNC effective temperature at approximates towards the form . Other mappings for the 3D case,^{ [3] } and similar formulae for the effective temperature have been given for the classical map of the 2-dimensional electron gas as well.^{ [4] }

**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.

**Fermi–Dirac statistics** is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac distribution of particles over energy states. It is named after Enrico Fermi and Paul Dirac, each of whom derived the distribution independently in 1926. Fermi–Dirac statistics is a part of the field of statistical mechanics and uses the principles of quantum mechanics.

In physics, **mean free path** is the average distance over which a moving particle travels before substantially changing its direction or energy, typically as a result of one or more successive collisions with other particles.

A **Fermi gas** is an idealized model, an ensemble of many 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** is defined by small shifts in otherwise degenerate energy levels and the resulting splittings in those energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the nucleus and electron clouds.

In solid-state physics, the **electron mobility** characterises how quickly an electron can move through a metal or semiconductor when pulled by an electric field. There is an analogous quantity for holes, called **hole mobility**. The term **carrier mobility** refers in general to both electron and hole mobility.

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. The study of the Fermi surfaces of materials is called **fermiology**.

In solid-state physics, the **free electron model** is a quantum mechanical 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, the **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 plasmas and electrolytes, the **Debye length****, is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists. With each Debye length the charges are increasingly electrically screened and the electric potential decreases in magnitude by 1/e. A ****Debye sphere** is a volume whose radius is the Debye length. 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.

In physics, the **gyromagnetic ratio** of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol γ, gamma. Its SI unit is the radian per second per tesla (rad⋅s^{−1}⋅T^{−1}) or, equivalently, the coulomb per kilogram (C⋅kg^{−1}).

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 works on Brownian motion. The more general form of the equation in the classical case 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.

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 Thomas–Fermi 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.

In condensed matter physics, **Lindhard theory** 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. It is named after Danish physicist Jens Lindhard, who first developed the theory in 1954.

**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 **electronic specific heat**, sometimes called the **electron heat capacity**, is the specific heat of an electron gas. 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.

- 1 2 3 4 5 N. W. Ashcroft and N. D. Mermin,
*Solid State Physics*(Thomson Learning, Toronto, 1976) - ↑ Stanton, Liam G.; Murillo, Michael S. (2016-04-08). "Ionic transport in high-energy-density matter".
*Physical Review E*. American Physical Society (APS).**93**(4): 043203. Bibcode:2016PhRvE..93d3203S. doi: 10.1103/physreve.93.043203 . ISSN 2470-0045. PMID 27176414. - ↑ Yu Liu and Jianzhong Wu, J. Chem. Phys.
**141**064115 (2014) - ↑ François Perrot and M. W. C. Dharma-wardana, Phys. Rev. Lett.
**87**, 206404 (2001)

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