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Debye Function
In mathematics, the family of Debye functions is defined by :D_n(x) = \frac \int_0^x \frac\,dt. The functions are named in honor of Peter Debye, who came across this function (with ''n'' = 3) in 1912 when he analytically computed the heat capacity of what is now called the Debye model. Mathematical properties Relation to other functions The Debye functions are closely related to the polylogarithm. Series expansion They have the series expansion :D_n(x) = 1 - \frac x + n \sum_^\infty \frac x^, \quad , x, 0, Derivative The derivative obeys the relation :xD^_n(x) = n(B(x)-D_n(x)), where B(x)=x/(e^x-1) is the Bernoulli function. Applications in solid-state physics The Debye model The Debye model has a density of vibrational states :g_(\omega)=\frac for 0\le\omega\le\omega_ with the ''Debye frequency'' ''ω''D. Internal energy and heat capacity Inserting ''g'' into the internal energy :U=\int_0^\infty d\omega\,g(\omega)\,\hbar\omega\,n(\omega) with th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
X-ray Diffraction
X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles and intensities of these diffracted beams, a crystallographer can produce a three-dimensional picture of the density of electrons within the crystal. From this electron density, the mean positions of the atoms in the crystal can be determined, as well as their chemical bonds, their crystallographic disorder, and various other information. Since many materials can form crystals—such as salts, metals, minerals, semiconductors, as well as various inorganic, organic, and biological molecules—X-ray crystallography has been fundamental in the development of many scientific fields. In its first decades of use, this method determined the size of atoms, the lengths and types of chemical bonds, and the atomic-scale differences among vari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GNU Scientific Library
The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C; wrappers are available for other programming languages. The GSL is part of the GNU Project and is distributed under the GNU General Public License. Project history The GSL project was initiated in 1996 by physicists Mark Galassi and James Theiler of Los Alamos National Laboratory.GSL homepage They aimed at writing a modern replacement for widely used but somewhat outdated Fortran libraries such as . They carried out the overall design and wrote early modules; with that ready they recruited other scient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero-point Energy
Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly Quantum fluctuation, fluctuate in their lowest energy state as described by the Heisenberg uncertainty principle. Therefore, even at absolute zero, atoms and molecules retain some vibrational motion. Apart from atoms and molecules, the empty space of Vacuum state, the vacuum also has these properties. According to quantum field theory, the universe can be thought of not as isolated particles but continuous fluctuating Field (physics), fields: matter fields, whose Quantum, quanta are fermions (i.e., leptons and quarks), and Force field (physics), force fields, whose quanta are bosons (e.g., photons and gluons). All these fields have zero-point energy. These fluctuating zero-point fields lead to a kind of reintroduction of an Luminiferous aether, aether in physics since some systems can detect the existence of this energy. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Physics
Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the previous theories, or new theories based on the older paradigm, will often be referred to as belonging to the area of "classical physics". As such, the definition of a classical theory depends on context. Classical physical concepts are often used when modern theories are unnecessarily complex for a particular situation. Most often ''classical physics'' refers to pre-1900 physics, while ''modern physics'' refers to post-1900 physics which incorporates elements of quantum mechanics and relativity. Overview Classical theory has at least two distinct meanings in physics. In the context of quantum mechanics, classical theory refers to theories of physics that do not use the quantisation paradigm, which includes classical mechanics a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Squared Displacement
In statistical mechanics, the mean squared displacement (MSD, also mean square displacement, average squared displacement, or mean square fluctuation) is a measure of the deviation of the position of a particle with respect to a reference position over time. It is the most common measure of the spatial extent of random motion, and can be thought of as measuring the portion of the system "explored" by the random walker. In the realm of biophysics and environmental engineering, the Mean Squared Displacement is measured over time to determine if a particle is spreading slowly due to diffusion, or if an advective force is also contributing. Another relevant concept, the variance-related diameter (VRD, which is twice the square root of MSD), is also used in studying the transportation and mixing phenomena in the realm of environmental engineering. It prominently appears in the Debye–Waller factor (describing vibrations within the solid state) and in the Langevin equation (describing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neutron Diffraction
Neutron diffraction or elastic neutron scattering is the application of neutron scattering to the determination of the atomic and/or magnetic structure of a material. A sample to be examined is placed in a beam of thermal or cold neutrons to obtain a diffraction pattern that provides information of the structure of the material. The technique is similar to X-ray diffraction but due to their different scattering properties, neutrons and X-rays provide complementary information: X-Rays are suited for superficial analysis, strong x-rays from synchrotron radiation are suited for shallow depths or thin specimens, while neutrons having high penetration depth are suited for bulk samples.Measurement of residual stress in materials using neutrons |
Bose–Einstein Distribution
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Bose–Einstein may refer to: * Bose–Einstein condensate ** Bose–Einstein condensation (network theory) * Bose–Einstein correlations * Bose–Einstein statistics In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Debye
Peter Joseph William Debye (; ; March 24, 1884 – November 2, 1966) was a Dutch-American physicist and physical chemistry, physical chemist, and List of Nobel laureates in Chemistry, Nobel laureate in Chemistry. Biography Early life Born Petrus Josephus Wilhelmus Debije in Maastricht, Netherlands, Debye enrolled in the Aachen University of Technology in 1901. In 1905, he completed his first degree in electrical engineering. He published his first paper, a mathematically elegant solution of a problem involving eddy currents, in 1907. At Aachen, he studied under the theoretical physicist Arnold Sommerfeld, who later claimed that his most important discovery was Peter Debye. In 1906, Sommerfeld received an appointment at Ludwig Maximilian University of Munich, Munich, Kingdom of Bavaria, Bavaria, and took Debye with him as his assistant. Debye got his Doctor of Philosophy, Ph.D. with a dissertation on radiation pressure in 1908. In 1910, he derived the Planck's law, Planck radiat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Of States
In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states in the system of volume V whose energies lie in the range from E to E+\delta E. It is mathematically represented as a distribution by a probability density function, and it is generally an average over the space and time domains of the various states occupied by the system. The density of states is directly related to the dispersion relations of the properties of the system. High DOS at a specific energy level means that many states are available for occupation. Generally, the density of states of matter is continuous. In isolated systems however, such as atoms or molecules in the gas phase, the density distribution is discrete, like a spectral density. Local variations, most often due to distortions of the original system, are often re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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