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Nearly Free Electron Model
In solid-state physics, the nearly free electron model (or NFE model and quasi-free electron model) is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model is closely related to the more conceptual empty lattice approximation. The model enables understanding and calculation of the electronic band structures, especially of metals. This model is an immediate improvement of the free electron model, in which the metal was considered as a non-interacting electron gas and the ions were neglected completely. Mathematical formulation The nearly free electron model is a modification of the free-electron gas model which includes a ''weak'' periodic perturbation meant to model the interaction between the conduction electrons and the ions in a crystalline solid. This model, like the free-electron model, does not take into account electron–electron interactions; that is, the independent el ...
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Solid-state Physics
Solid-state physics is the study of rigid matter, or solids, through methods such as solid-state chemistry, quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from their atomic-scale properties. Thus, solid-state physics forms a theoretical basis of materials science. Along with solid-state chemistry, it also has direct applications in the technology of transistors and semiconductors. Background Solid materials are formed from densely packed atoms, which interact intensely. These interactions produce the mechanical (e.g. hardness and Elasticity (physics), elasticity), Heat conduction, thermal, Electrical conduction, electrical, Magnetism, magnetic and Crystal optics, optical properties of solids. Depending on the material involved and the conditions in which it was formed, the atoms may be arranged in a regular, geometric patt ...
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Bloch State
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves modulated by periodic functions. The theorem is named after the Swiss physicist Felix Bloch, who discovered the theorem in 1929. Mathematically, they are written where \mathbf is position, \psi is the wave function, u is a periodic function with the same periodicity as the crystal, the wave vector \mathbf is the crystal momentum vector, e is Euler's number, and i is the imaginary unit. Functions of this form are known as Bloch functions or Bloch states, and serve as a suitable basis for the wave functions or states of electrons in crystalline solids. The description of electrons in terms of Bloch functions, termed Bloch electrons (or less often ''Bloch Waves''), underlies the concept of electronic band structures. These eigenstates are written with subscripts as \psi_, where n is a discrete index, called the band index, ...
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Pauli Exclusion Principle
In quantum mechanics, the Pauli exclusion principle (German: Pauli-Ausschlussprinzip) states that two or more identical particles with half-integer spins (i.e. fermions) cannot simultaneously occupy the same quantum state within a system that obeys the laws of quantum mechanics. This principle was formulated by Austrian physicist Wolfgang Pauli in 1925 for electrons, and later extended to all fermions with his spin–statistics theorem of 1940. In the case of electrons in atoms, the exclusion principle can be stated as follows: in a poly-electron atom it is impossible for any two electrons to have the same two values of ''all'' four of their quantum numbers, which are: ''n'', the principal quantum number; ', the azimuthal quantum number; ''m'', the magnetic quantum number; and ''ms'', the spin quantum number. For example, if two electrons reside in the same orbital, then their values of ''n'', ', and ''m'' are equal. In that case, the two values of ''m''s (spin) pair must be ...
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Wave Vector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), and its direction is perpendicular to the wavefront. In isotropic media, this is also the direction of wave propagation. A closely related vector is the angular wave vector (or angular wavevector), with a typical unit being radian per metre. The wave vector and angular wave vector are related by a fixed constant of proportionality, 2 radians per cycle. It is common in several fields of physics to refer to the angular wave vector simply as the ''wave vector'', in contrast to, for example, crystallography. It is also common to use the symbol for whichever is in use. In the context of special relativity, a '' wave four-vector'' can be defined, combining the (angular) wave vector and (angular) frequency. Definition The terms ...
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Band Gap
In solid-state physics and solid-state chemistry, a band gap, also called a bandgap or energy gap, is an energy range in a solid where no electronic states exist. In graphs of the electronic band structure of solids, the band gap refers to the energy difference (often expressed in electronvolts) between the top of the valence band and the bottom of the conduction band in insulators and semiconductors. It is the energy required to promote an electron from the valence band to the conduction band. The resulting conduction-band electron (and the electron hole in the valence band) are free to move within the crystal lattice and serve as charge carriers to conduct electric current. It is closely related to the HOMO/LUMO gap in chemistry. If the valence band is completely full and the conduction band is completely empty, then electrons cannot move within the solid because there are no available states. If the electrons are not free to move within the crystal lattice, then there ...
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Bragg Plane
In physics, a Bragg plane is a Plane (geometry), plane in reciprocal space which bisects a reciprocal lattice vector, \scriptstyle \mathbf, at right angles. The Bragg plane is defined as part of the Von Laue condition for Interference (wave propagation), diffraction peaks in X-ray crystallography, x-ray diffraction crystallography. Considering the adjacent diagram, the arriving x-ray plane wave is defined by: :e^ = \cos + i\sin Where \scriptstyle \mathbf is the incident wave vector given by: :\mathbf = \frac\hat where \scriptstyle \lambda is the wavelength of the incident photon. While the Bragg's law, Bragg formulation assumes a unique choice of direct lattice planes and specular reflection of the incident X-rays, the Von Laue formula only assumes monochromatic light and that each scattering center acts as a source of secondary wavelets as described by the Huygens principle. Each scattered wave contributes to a new plane wave given by: :\mathbf = \frac\hat^\prime The con ...
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Brillouin Zone
In mathematics and solid state physics, the first Brillouin zone (named after Léon Brillouin) is a uniquely defined primitive cell in reciprocal space Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray diffraction, X-ray and Electron diffraction, electron diffraction as well as the Electronic band structure, e .... In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice is broken up into Brillouin zones. The boundaries of this cell are given by planes related to points on the reciprocal lattice. The importance of the Brillouin zone stems from the description of waves in a periodic medium given by Bloch's theorem, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone. The first Brillouin zone is the locus of points in reciprocal space that are closer to the or ...
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Degenerate Energy Level
In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the ''degree of degeneracy'' (or simply the ''degeneracy'') of the level. It is represented mathematically by the Hamiltonian (quantum mechanics), Hamiltonian for the system having more than one linear independence, linearly independent eigenstate with the same energy eigenvalue. When this is the case, energy alone is not enough to characterize what state the system is in, and other quantum numbers are needed to characterize the exact state when distinction is desired. In classical mechanics, this can be understood in terms of different possible trajectories corresponding to the same energy. Degeneracy plays a fu ...
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Reciprocal Lattice
Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray and electron diffraction as well as the energies of electrons in a solid. It emerges from the Fourier transform of the lattice associated with the arrangement of the atoms. The ''direct lattice'' or ''real lattice'' is a periodic function in physical space, such as a crystal system (usually a Bravais lattice). The reciprocal lattice exists in the mathematical space of spatial frequencies or wavenumbers ''k'', known as reciprocal space or ''k'' space; it is the dual of physical space considered as a vector space. In other words, the reciprocal lattice is the sublattice which is dual to the direct lattice. The reciprocal lattice is the set of all vectors \mathbf_m, that are wavevectors k of plane waves in the Fourier series of a spatial function whose periodicity is the same as that of a direct lattice \mathbf_n. Each plane wave in th ...
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Screening Effect
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 mediums, such as ionized gases (classical plasmas), electrolytes, and electronic conductors (semiconductors, metals). In a fluid, with a given permittivity , composed of electrically charged constituent particles, each pair of particles (with charges and ) interact through the Coulomb force as \mathbf = \frac\hat, where the vector is the relative position between the charges. This interaction complicates the theoretical treatment of the fluid. For example, a naive quantum mechanical calculation of the ground-state energy density yields infinity, which is unreasonable. The difficulty lies in the fact that even though the Coulomb force diminishes with distance as , the average number of particles at each distance is proportional to , assuming the fluid is fairly isotropic. As a result, a charge fluctuation at any on ...
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Fourier Series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always Convergent series, converge. Well-behaved functions, for example Smoothness, smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric func ...
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Gibbs Paradox
In statistical mechanics, a semi-classical derivation of entropy that does not take into account the Identical particles, indistinguishability of particles yields an expression for entropy which is not extensive variable, extensive (is not proportional to the amount of substance in question). This leads to a physical paradox, paradox known as the Gibbs paradox, after Josiah Willard Gibbs, who proposed this thought experiment in 1874‒1875. The paradox allows for the entropy of closed systems to decrease, violating the second law of thermodynamics. A related paradox is the "Gibbs paradox#Mixing paradox, mixing paradox". If one takes the perspective that the definition of entropy must be changed so as to ignore particle permutation, in the thermodynamic limit, the paradox is averted. Illustration of the problem Gibbs considered the following difficulty that arises if the ideal gas entropy is not extensive. Reprinted in and in Two containers of an ideal gas sit side-by-side. ...
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