|
Madelung Constant
The Madelung constant is used in determining the electrostatic potential of a single ion in a crystal by approximating the ions by point charges. It is named after Erwin Madelung, a German physicist. Because the anions and cations in an ionic solid attract each other by virtue of their opposing charges, separating the ions requires a certain amount of energy. This energy must be given to the system in order to break the anion–cation bonds. The energy required to break these bonds for one mole of an ionic solid under standard conditions is the lattice energy. Formal expression The Madelung constant allows for the calculation of the electric potential of the ion at position due to all other ions of the lattice :V_i = \frac \sum_ \frac\,\! where r_ = , r_i-r_j, is the distance between the th and the th ion. In addition, * number of charges of the th ion * the elementary charge, 1.6022 C * ; is the permittivity of free space. If the distances are normalized to the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sodium Chloride
Sodium chloride , commonly known as Salt#Edible salt, edible salt, is an ionic compound with the chemical formula NaCl, representing a 1:1 ratio of sodium and chloride ions. It is transparent or translucent, brittle, hygroscopic, and occurs as the mineral halite. In its edible form, it is commonly used as a condiment and curing (food preservation), food preservative. Large quantities of sodium chloride are used in many industrial processes, and it is a major source of sodium and chlorine compounds used as feedstocks for further Chemical synthesis, chemical syntheses. Another major application of sodium chloride is deicing of roadways in sub-freezing weather. Uses In addition to the many familiar domestic uses of salt, more dominant applications of the approximately 250 million tonnes per year production (2008 data) include chemicals and de-icing.Westphal, Gisbert ''et al.'' (2002) "Sodium Chloride" in Ullmann's Encyclopedia of Industrial Chemistry, Wiley-VCH, Weinheim . Chem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rocksalt
Halite ( ), commonly known as rock salt, is a type of salt, the mineral (natural) form of sodium chloride ( Na Cl). Halite forms isometric crystals. The mineral is typically colorless or white, but may also be light blue, dark blue, purple, pink, red, orange, yellow or gray depending on inclusion of other materials, impurities, and structural or isotopic abnormalities in the crystals. It commonly occurs with other evaporite deposit minerals such as several of the sulfates, halides, and borates. The name ''halite'' is derived from the Ancient Greek word for "salt", ἅλς (''háls''). Occurrence Halite dominantly occurs within sedimentary rocks where it has formed from the evaporation of seawater or salty lake water. Vast beds of sedimentary evaporite minerals, including halite, can result from the drying up of enclosed lakes and restricted seas. Such salt beds may be hundreds of meters thick and underlie broad areas. Halite occurs at the surface today in playas in region ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Crystal System
In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of these crystals: *Primitive cubic (abbreviated ''cP'' and alternatively called simple cubic) *Body-centered cubic (abbreviated ''cI'' or bcc) *Face-centered cubic (abbreviated ''cF'' or fcc) Note: the term fcc is often used in synonym for the ''cubic close-packed'' or ccp structure occurring in metals. However, fcc stands for a face-centered cubic Bravais lattice, which is not necessarily close-packed when a motif is set onto the lattice points. E.g. the diamond and the zincblende lattices are fcc but not close-packed. Each is subdivided into other variants listed below. Although the ''unit cells'' in these crystals are conventionally taken to be cubes, the primitive unit cells often are not. Bravais lattices The three Bravais latices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ewald Summation
Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions (e.g. electrostatic interactions) in periodic systems. It was first developed as the method for calculating the electrostatic energies of ionic crystals, and is now commonly used for calculating long-range interactions in computational chemistry. Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an equivalent summation in Fourier space. In this method, the long-range interaction is divided into two parts: a short-range contribution, and a long-range contribution which does not have a singularity. The short-range contribution is calculated in real space, whereas the long-range contribution is calculated using a Fourier transform. The advantage of this method is the rapid convergence of the energy compared with that of a direct summation. This means that the method has high accuracy and reasonable spee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Transform
In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the ''inverse transform''. General form An integral transform is any transform ''T'' of the following form: :(Tf)(u) = \int_^ f(t)\, K(t, u)\, dt The input of this transform is a function ''f'', and the output is another function ''Tf''. An integral transform is a particular kind of mathematical operator. There are numerous useful integral transforms. Each is specified by a choice of the function K of two variables, that is called the kernel or nucleus of the transform. Some kernels have an associated ''inverse kernel'' K^( u,t ) which (roughly speaking) yields an inverse transform: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where the infinite series representation which initially defined the function becomes divergent. The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of singularities. The case of several complex variables is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of sheaf cohomology. Initial discussion Suppose ''f'' is an analytic function defined on a non-empty open subset ''U'' of the complex plane If ''V'' is a larger open subset of containing ''U'', and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jonathan Borwein
Jonathan Michael Borwein (20 May 1951 – 2 August 2016) was a Scottish mathematician who held an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia. He was a close associate of David H. Bailey, and they have been prominent public advocates of experimental mathematics. Borwein's interests spanned pure mathematics (analysis), applied mathematics (optimization), computational mathematics (numerical and computational analysis), and high performance computing. He authored ten books, including several on experimental mathematics, a monograph on convex functions, and over 400 refereed articles. He was a co-founder in 1995 of software company MathResources, consulting and producing interactive software primarily for school and university mathematics. He was not associated with MathResources at the time of his death. Borwein was also an expert on the number pi and especially its computation. Early life and education Borwein was born in St. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Borwein
David Borwein (March 24, 1924 – September 3, 2021) was a Lithuanian-born Canadian mathematician, known for his research in the summability theory of series and integrals. He also did work in measure theory and probability theory, number theory, and approximate subgradients and coderivatives. He latterly collaborated with his son, Jonathan Borwein, and with B.A. Mares Jr. on the properties of single-variable and many-variable sinc integrals. Biography Borwein was born in March 1924 in Lithuania to an Ashkenazi Jewish family. He formerly resided and worked in St. Andrews, Scotland, before moving to London, Ontario where he eventually became Head of Mathematics at the University of Western Ontario. He was also the president of the Canadian Mathematical Society (CMS). The David Borwein Distinguished Career Award given out by the CMS is named after him. He was an active researcher in summability theory, classical analysis, inequalities, matrix transformations, and was profess ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Nachrichten
''Mathematische Nachrichten'' (abbreviated ''Math. Nachr.''; English: ''Mathematical News'') is a mathematical journal published in 12 issues per year by Wiley-VCH GmbH. It should not be confused with the ''Internationale Mathematische Nachrichten'', an unrelated publication of the Austrian Mathematical Society. It was established in 1948 by East German mathematician Erhard Schmidt, who became its first editor-in-chief. At that time it was associated with the German Academy of Sciences at Berlin, and published by Akademie Verlag. After the fall of the Berlin Wall, Akademie Verlag was sold to VCH Verlagsgruppe Weinheim, which in turn was sold to John Wiley & Sons. According to the 2020 edition of Journal Citation Reports, the journal had an impact factor of 1.228, ranking it 111th among 333 journals in the category "Mathematics". As of 2021, Ben Andrews, Robert Denk, Klaus Hulek and Frédéric Klopp are the editors-in-chief of the journal. References External links * * P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergent Series
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_1, a_2, a_3, \ldots) defines a series that is denoted :S=a_1 + a_2 + a_3 + \cdots=\sum_^\infty a_k. The th partial sum is the sum of the first terms of the sequence; that is, :S_n = a_1 +a_2 + \cdots + a_n = \sum_^n a_k. A series is convergent (or converges) if and only if the sequence (S_1, S_2, S_3, \dots) of its partial sums tends to a limit; that means that, when adding one a_k after the other ''in the order given by the indices'', one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if and only if there exists a number \ell such that for every arbitrarily small positive number \varepsilon, there is a (sufficiently large) integer N such that for all n \ge N, :\left , S_n - \ell \right , 1 produce a convergent series: *: ++++++\cdots = . * Alternating the signs of reciprocals of powers o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Introduction To Solid State Physics
''Introduction to Solid State Physics'', known colloquially as ''Kittel'', is a classic condensed matter physics textbook written by American physicist Charles Kittel in 1953. The book has been highly influential and has seen widespread adoption; Marvin L. Cohen remarked in 2019 that Kittel's content choices in the original edition played a large role in defining the field of solid-state physics. It was also the first proper textbook covering this new field of physics. The book is published by John Wiley and Sons and, as of 2018, it is in its ninth edition and has been reprinted many times as well as translated into over a dozen languages, including Chinese, French, German, Hungarian, Indonesian, Italian, Japanese, Korean, Malay, Romanian, Russian, Spanish, and Turkish. In some later editions, the eighteenth chapter, titled ''Nanostructures'', was written by Paul McEuen. Along with its competitor '' Ashcroft and Mermin'', the book is considered a standard textbook in condensed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
