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Reciprocal Lattice
In physics, the reciprocal lattice emerges from the Fourier transform of another lattice. 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, known as reciprocal space or k space, which is the dual of ''physical space'' considered as a vector space, and the reciprocal lattice is the sublattice of that space that is dual to the ''direct lattice''. In quantum physics, reciprocal space is closely related to momentum space according to the proportionality \mathbf = \hbar \mathbf, where \mathbf is the momentum vector and \hbar is the reduced Planck constant. The reciprocal lattice of a reciprocal lattice is equivalent to the original direct lattice, because the defining equations are symmetrical with respect to the vectors in real and reciprocal space. Mathematically, direct and reciprocal lattice vectors represent co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reciprocal Monoclinic Lattice
Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another polynomial by reversing its coefficients * Reciprocal rule, a technique in calculus for calculating derivatives of reciprocal functions * Reciprocal spiral, a plane curve * Reciprocal averaging, a statistical technique for aggregating categorical data In science and technology * Reciprocal aircraft heading, 180 degrees (the opposite direction) from a stated heading * Reciprocal lattice, a basis for the dual space of covectors, in crystallography * Reciprocal length, a measurement used in science * Reciprocating engine or piston engine * Reciprocating oscillation in physical wave theory Life sciences and medicine * Hybrid (biology), in genetics, the result of a reciprocal pair of crossings, forming ''reciprocal hybrids'' * Reciprocal altr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, 2π 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 ''k'' for whichever is in use. In the context of special relativity, ''wave vector'' can refer to a four-vector, in which the (angular) wave vector and (angular) frequency are combined. Def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angular Wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temporal frequency, which is defined as the number of wave cycles per unit time (''ordinary frequency'') or radians per unit time (''angular frequency''). In multidimensional systems, the wavenumber is the magnitude of the ''wave vector''. The space of wave vectors is called ''reciprocal space''. Wave numbers and wave vectors play an essential role in optics and the physics of wave scattering, such as X-ray diffraction, neutron diffraction, electron diffraction, and elementary particle physics. For quantum mechanical waves, the wavenumber multiplied by the reduced Planck's constant is the ''canonical momentum''. Wavenumber can be used to specify quantities other than spatial frequency. For example, in optical spectroscopy, it is often used a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase (waves)
In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it varies by one full turn as the variable t goes through each period (and F(t) goes through each complete cycle). It may be measured in any angular unit such as degrees or radians, thus increasing by 360° or 2\pi as the variable t completes a full period. This convention is especially appropriate for a sinusoidal function, since its value at any argument t then can be expressed as \phi(t), the sine of the phase, multiplied by some factor (the amplitude of the sinusoid). (The cosine may be used instead of sine, depending on where one considers each period to start.) Usually, whole turns are ignored when expressing the phase; so that \phi(t) is also a periodic function, with the same period as F, that repeatedly scans the same range of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sinusoidal Plane Wave
In physics, a sinusoidal (or monochromatic) plane wave is a special case of plane wave: a field whose value varies as a sinusoidal function of time and of the distance from some fixed plane. For any position \vec x in space and any time t, the value of such a field can be written as :F(\vec x, t)=A \cos\left(2\pi \nu (\vec x \cdot \vec n - c t) + \varphi\right)\, where \vec n is a unit-length vector, the direction of propagation of the wave, and "\cdot" denotes the dot product of two vectors. The parameter A, which may be a scalar or a vector, is called the amplitude of the wave; the coefficient \nu, a positive scalar, its spatial frequency; and the adimensional scalar \varphi, an angle in radians, is its initial phase or phase shift. The scalar quantity d = \vec x \cdot \vec n gives the (signed) displacement of the point \vec x from the plane that is perpendicular to \vec n and goes through the origin of the coordinate system. This quantity is constant over each plane perpendic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frequency Domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. A frequency-domain representation can also include information on the phase shift that must be applied to each sinusoid in order to be able to recombine the frequency components to recover the original time signal. A given function or signal can be converted between the time and frequency domains with a pair of mathematical operators called transforms. An example is the Fourier transform, which converts a time function into a complex valued sum or integral of sine waves of different frequencies, with amplitudes and phases, each of which represents a frequency component. The "spectrum" of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wigner–Seitz Cell
The Wigner–Seitz cell, named after Eugene Wigner and Frederick Seitz, is a primitive cell which has been constructed by applying Voronoi decomposition to a crystal lattice. It is used in the study of crystalline materials in crystallography. The unique property of a crystal is that its atoms are arranged in a regular three-dimensional array called a lattice. All the properties attributed to crystalline materials stem from this highly ordered structure. Such a structure exhibits discrete translational symmetry. In order to model and study such a periodic system, one needs a mathematical "handle" to describe the symmetry and hence draw conclusions about the material properties consequent to this symmetry. The Wigner–Seitz cell is a means to achieve this. A Wigner–Seitz cell is an example of a primitive cell, which is a unit cell containing exactly one lattice point. For any given lattice, there are an infinite number of possible primitive cells. However there is only one W ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brillouin Zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. 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 origin of the reciprocal lattice than they are to any other reciprocal lattice points (see the derivation of the Wigner–Seitz cell). Another definition is as the set of points in ''k''-space that can be reached from the origin without crossing any Bragg plane. Equivalently, this is the Vor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laue Equations
In crystallography and solid state physics, the Laue equations relate incoming waves to outgoing waves in the process of elastic scattering, where the photon energy or light temporal frequency does not change by scattering, by a crystal lattice. They are named after physicist Max von Laue (1879–1960). The Laue equations can be written as \mathbf= \mathbf_ - \mathbf_ = \mathbf as the condition of elastic wave scattering by a crystal lattice, where \mathbf_, \mathbf k_, and \mathbf are an incoming (to the crystal) wavevector, an outgoing (from the crystal by scattering) wavevector, and a reciprocal lattice vector for the crystal respectively. Due to elastic scattering , \mathbf_, ^2=, \mathbf_, ^2, three vectors. \mathbf, \mathbf_, and -\mathbf_ , form a rhombus if the equation is satisfied. If the scattering satisfies this equation, all the crystal lattice points scatter the incoming wave toward the scattering direction (the direction along \mathbf k_). If the equation is not sati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffraction
Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a secondary source of the propagating wave. Italian scientist Francesco Maria Grimaldi coined the word ''diffraction'' and was the first to record accurate observations of the phenomenon in 1660. In classical physics, the diffraction phenomenon is described by the Huygens–Fresnel principle that treats each point in a propagating wavefront as a collection of individual spherical wavelets. The characteristic bending pattern is most pronounced when a wave from a coherent source (such as a laser) encounters a slit/aperture that is comparable in size to its wavelength, as shown in the inserted image. This is due to the addition, or interference, of different points on the wavefront (or, equivalently, each wavelet) that travel by paths of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
X-ray
An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 picometers to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz ( to ) and energies in the range 145 eV to 124 keV. X-ray wavelengths are shorter than those of UV rays and typically longer than those of gamma rays. In many languages, X-radiation is referred to as Röntgen radiation, after the German scientist Wilhelm Conrad Röntgen, who discovered it on November 8, 1895. He named it ''X-radiation'' to signify an unknown type of radiation.Novelline, Robert (1997). ''Squire's Fundamentals of Radiology''. Harvard University Press. 5th edition. . Spellings of ''X-ray(s)'' in English include the variants ''x-ray(s)'', ''xray(s)'', and ''X ray(s)''. The most familiar use of X-rays is checking for fractures (broken bones), but X-rays are also used in other ways. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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