In solid-state physics, the free electron model is a simple model for the behaviour of valence electrons in a crystal structure of a metallic solid. It was developed principally by Arnold Sommerfeld who combined the classical Drude model with quantum mechanical Fermi-Dirac statistics. Given its simplicity, it is surprisingly successful in explaining many experimental phenomena, especially Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. ... In chemistry, valence electrons are the electrons contained in the outermost, or valence, electron shell of an atom. ... Enargite crystals In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ... For alternative meanings see metal (disambiguation). ... For other uses, see Solid (disambiguation). ... Arnold Johannes Wilhelm Sommerfeld (December 5, 1868 in Königsberg, East Prussia – April 26, 1951 in Munich, Germany) was a German physicist who introduced the fine-structure constant in 1919. ... Classical physics is physics based on principles developed before the rise of quantum theory, usually including the special theory of relativity and general theory of relativity. ... The Drude model of electrical conduction was developed in the 1900s by Paul Drude to explain the transport properties of electrons in materials (especially metals). ... Fig. ... Fermi-Dirac distribution as a function of ε/μ plotted for 4 different temperatures. ...

• the Wiedemann-Franz law which relates electrical conductivity and thermal conductivity;
• the temperature dependence of the heat capacity;
• the shape of the electronic density of states;
• the range of binding energy values;
• electrical conductivities.

In physics, the Wiedemann-Franz law states that the ratio of the thermal conductivity (K) to the electrical conductivity (σ) of a metal is proportional to the temperature (T). ... Electrical conductivity or specific conductivity is a measure of a materials ability to conduct an electric current. ... In physics, thermal conductivity, k, is the intensive property of a material that indicates its ability to conduct heat. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... 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. ...

Ideas and assumptions

As in the Drude model, valence electrons are assumed to be completely detached from their ions ("electron gas"). As in an ideal gas, electron-electron interactions are completely neglected (they are weak because of the shielding effect). In chemistry, valence electrons are the electrons contained in the outermost, or valence, electron shell of an atom. ... “Multivalent” redirects here. ... An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. ... The electronic Hamiltonian for a multi-electron molecule in atomic units is: where is the vector position of electron with vector components in Bohr radii, is the charge of fixed nucleus a in units of the elementary charge, is the vector position of nucleus with vector components in Bohr radii. ... The shielding effect or screening effect is an effect which occurs on a subatomic level between electrons occupying energy levels and is caused by repulsive forces of other electrons between it and the nucleus. ...

The crystal lattice is not explicitely taken into account. A quantum-mechanical justification is given by Bloch's Theorem: an unbound electron moves in a constant periodic potential as a free electron in vacuum, except for the electron mass m becoming an effective mass m* which may deviate considerably from m (one can even use negative effective mass to describe conduction by electron holes. Effective masses can be derived from band structure computations. While the static lattice does not hinder the motion of the electrons, they can well be scattered by impurities and by phonons; these two interactions determine electrical and thermal conductivity (superconductivity requires more refined theory than the free electron model). A Bloch wave or Bloch state is the wavefunction of a particle (usually, an electron) placed in a periodic potential. ... Properties The electron is a fundamental subatomic particle which carries a negative electric charge. ... In solid state physics, a particles effective mass is the mass it seems to carry in the semiclassical model of transport in a crystal. ... For the following two reasons the electron hole was introduced into calculations: If an electron is excited into higher state it leaves a hole in its old state. ... In solid state physics, the electronic band structure, or simply band structure, refers to the dispersion relation (the relation between energy versus momentum) of electrons in a crystal. ... Normals modes of vibration progression through a crystal. ... A magnet levitating above a high-temperature superconductor, cooled with liquid nitrogen. ...

According to the Pauli exclusion principle, each phase space element (Δk)³(Δx)³ can be occupied only by two electrons (one per spin quantum number). This restriction of available electron states is taken into account by Fermi-Dirac statistics (see also Fermi gas). Main predictions of the free electron model are derived by the Sommerfeld expansion of the Fermi-Dirac occupancy for energies around the Fermi level. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ... For other senses of this term, see phase space (disambiguation). ... The terms spin and SPIN have several meanings, including those primarily discussed as spinning: For spin in sub-atomic physics, see spin (physics) For the stalled aircraft maneuver or any of several forms of loss of control in aircraft, see spin (flight) For the periodical, see Spin Magazine For the... A quantum number describes the energies of electrons in atoms. ... Fermi-Dirac distribution as a function of ε/μ plotted for 4 different temperatures. ... A Fermi gas is a collection of non-interacting fermions. ... In quantum mechanics, particles with a half-integer spin, usually spin 1/2 (for example electrons) follow the Pauli exclusion principle, which states that no two particles may occupy the same quantum state. ...

Technicalities

Effective mass

A band structure computation actually yields a dispersion relation E(k) between electron wave vector k and energy E. An effective mass is obtained by approximating the true dispersion relation in the limit of small k by the free-electron form In solid state physics, the electronic band structure, or simply band structure, refers to the dispersion relation (the relation between energy versus momentum) of electrons in a crystal. ... The relation between the energy of a system and its corresponding momentum is known as its dispersion relation. ... A wave vector is a vector that represents two properties of a wave: the magnitude of the vector represents wavenumber (inversely related to wavelength), and the vector points in the direction of wave propagation. ...

(with the free-electron mass m replaced by m*). A lattice electron with a fictitious mass can be seen as a quasiparticle (though there is a one-to-one correspondence to the real particle which is not the case for other quasiparticles such as phonons). In physics, a quasiparticle refers to a particle-like entity arising in certain systems of interacting particles. ... Normals modes of vibration progression through a crystal. ...

Relation with other electron models

The assumption of electrons that move freely through a periodic potential should be contrasted with the tight-binding model, which uses the opposite simplification of treating the electrons as tightly bound to the atomic cores. (Coulomb interactions between electrons are still neglected.) The predictions of these two complementary models are reassuringly similar. Taking into account the specifities of the potential in a real, three-dimensional crystal lattice leads to more complicated dispersion relations and to band theory. In the tight binding model, electrons are treated as highly localised, and expanded as single electron wavefunctions in terms of atomic orbitals. ... In solid state physics, band theory is the theory of the behavior of the electrons in solids. ...

See also

• Nearly-free electron model
• Free electron laser

In solid-state physics, the nearly free electron model is a model of electron behavior in solids that enables understanding the electronic band structure of crystalline materials. ... This article needs to be cleaned up to conform to a higher standard of quality. ...

External articles and references

• Ashcroft, Mermin: Solid State Physics.