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. Solid-state physics, the largest branch of Condensed matter physics, is the study of rigid Matter, or Solids The bulk of solid-state physics theory and In chemistry valence electrons are the Electrons contained in the outermost or valence, Electron shell of an Atom. In Mineralogy and Crystallography, a crystal structure is a unique arrangement of Atoms in a Crystal. The M acro E xpansion T emplate A ttribute L anguage complements TAL, providing macros which allow the reuse of code across A solid' object is in the States of matter characterized by resistance to Deformation and changes of Volume. It was developed principally by Arnold Sommerfeld who combined the classical Drude model with quantum mechanical Fermi-Dirac statistics. Arnold Johannes Wilhelm Sommerfeld (5 December 1868 &ndash 26 April 1951 was a German theoretical Physicist who pioneered developments in atomic The Drude model of Electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of Electrons in materials (especially metals Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Statistical mechanics, Fermi-Dirac statistics is a particular case of Particle statistics developed by Enrico Fermi and Paul Dirac that Given its simplicity, it is surprisingly successful in explaining many experimental phenomena, especially
- 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 ( &sigma) Electrical conductivity or specific conductivity is a measure of a material's ability to conduct an Electric current. In Physics, thermal conductivity, k is the property of a material that indicates its ability to conduct Heat. Specific heat capacity, also known simply as specific heat, is the measure of the heat energy required to increase the Temperature of a unit quantity In statistical and Condensed matter physics, the density of states ( DOS) of a system describes the number of states at each energy level that are available
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Ideas and assumptions
As in the Drude model, valence electrons are assumed to be completely detached from their ions ("electron gas"). In chemistry valence electrons are the Electrons contained in the outermost or valence, Electron shell of an Atom. An ion is an Atom or Molecule which has lost or gained one or more Valence electrons giving it a positive or negative electrical charge As in an ideal gas, electron-electron interactions are completely neglected (they are weak because of the shielding effect). These four properties that constitute an ideal gas can be easily remembered by the acronym RIPE which stands for - R andom Motion (molecules are in constant random motion In Atomic molecular and optical physics as well as in Quantum chemistry, molecular Hamiltonian is the name given to the Hamiltonian representing the The shielding effect describes the decrease in attraction between an Electron and the nucleus in any atom with more than one electron shell
The crystal lattice is not explicitly 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). A Bloch wave or Bloch state, named after Felix Bloch, is the Wavefunction of a particle (usually an Electron) placed in a periodic potential The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J In Solid state physics, a particle's effective mass is the Mass it seems to carry in the semiclassical model of transport in a Crystal. An electron hole is the conceptual and mathematical Opposite of an Electron, useful in the study of Physics and Chemistry. Effective masses can be derived from band structure computations. In Solid-state physics, the electronic band structure (or simply band structure) of a Solid describes ranges of Energy that an Electron 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). In Physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the Atomic lattice of a Solid Superconductivity is a phenomenon occurring in certain Materials generally at very low Temperatures characterized by exactly zero electrical resistance
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). The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin Quantum numbers describe values of conserved numbers in the dynamics of the Quantum system. This restriction of available electron states is taken into account by Fermi-Dirac statistics (see also Fermi gas). In Statistical mechanics, Fermi-Dirac statistics is a particular case of Particle statistics developed by Enrico Fermi and Paul Dirac that A Fermi gas, or Free electron gas, is a collection of non-interacting Fermions. 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 Fermi energy is a concept in Quantum mechanics usually referring to the energy of the highest occupied Quantum state in a system of Fermions at
Technicalities
Effective mass
A band structure computation actually yields a dispersion relation E(k) between electron wave vector k and energy E. In Solid-state physics, the electronic band structure (or simply band structure) of a Solid describes ranges of Energy that an Electron Dispersion relations describe the ways that wave propagation varies with the Wavelength or Frequency of a wave. A wave vector is a vector representation of a Wave. The wave vector has magnitude indicating Wavenumber (reciprocal of Wavelength) and the An effective mass is obtained by approximating the true dispersion relation in the limit of small k by the free-electron form
(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 In Physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the Atomic lattice of a Solid
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. In the tight binding model for a solid-state lattice of atoms it is assumedthat the full Hamiltonian H of the system may be approximated by theHamiltonian of an isolated (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 Solid-state physics, the electronic band structure (or simply band structure) of a Solid describes ranges of Energy that an Electron
See also
External articles and references
- Ashcroft, Mermin: Solid State Physics. 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 A free-electron laser, or FEL is a Laser that shares the same optical properties as conventional lasers such as emitting a Beam consisting of coherent
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