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Ising Model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The model allows the identification of phase transitions as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition. The Ising model was invented by the physicist , who gave it as a prob ...
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Rudolf Peierls
Sir Rudolf Ernst Peierls, (; ; 5 June 1907 – 19 September 1995) was a German-born British physicist who played a major role in Tube Alloys, Britain's nuclear weapon programme, as well as the subsequent Manhattan Project, the combined Allied nuclear bomb programme. His obituary in ''Physics Today'' described him as "a major player in the drama of the eruption of nuclear physics into world affairs". Peierls studied physics at the University of Berlin, at the University of Munich under Arnold Sommerfeld, the University of Leipzig under Werner Heisenberg, and ETH Zurich under Wolfgang Pauli. After receiving his DPhil from Leipzig in 1929, he became an assistant to Pauli in Zurich. In 1932, he was awarded a Rockefeller Fellowship, which he used to study in Rome under Enrico Fermi, and then at the Cavendish Laboratory at the University of Cambridge under Ralph H. Fowler. Because of his Jewish background, he elected to not return home after Adolf Hitler's rise to power in 19 ...
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Ernst Ising
Ernst Ising (; May 10, 1900 in Cologne, Germany – May 11, 1998 in Peoria, Illinois, USA) was a German physicist, who is best remembered for the development of the Ising model. He was a professor of physics at Bradley University until his retirement in 1976. Life Ernst Ising was born in Cologne in 1900. Ernst Ising's parents were the merchant Gustav Ising and his wife Thekla Löwe. After school, he studied physics and mathematics at the University of Göttingen and University of Hamburg. In 1922, he began researching ferromagnetism under the guidance of Wilhelm Lenz. He earned a Ph.D in physics from the University of Hamburg in 1924 when he published his doctoral thesis (an excerpt or a summary of his doctoral thesis was published as an article in a scientific journal in 1925 and this has led many to believe that he published his full thesis in 1925, see,). His doctoral thesis studied a problem suggested by his teacher, Wilhelm Lenz. He investigated the special case of a ...
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Magnetic Moment
In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnets), permanent magnets, elementary particles (such as electrons), various molecules, and many astronomical objects (such as many planets, some moons, stars, etc). More precisely, the term ''magnetic moment'' normally refers to a system's magnetic dipole moment, the component of the magnetic moment that can be represented by an equivalent magnetic dipole: a magnetic north and south pole separated by a very small distance. The magnetic dipole component is sufficient for small enough magnets or for large enough distances. Higher-order terms (such as the magnetic quadrupole moment) may be needed in addition to the dipole moment for extended objects. The magnetic dipole moment of an object is readily defined in terms of the torque that the ...
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Correlation Function
A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables representing the same quantity measured at two different points, then this is often referred to as an autocorrelation function, which is made up of autocorrelations. Correlation functions of different random variables are sometimes called cross-correlation functions to emphasize that different variables are being considered and because they are made up of cross-correlations. Correlation functions are a useful indicator of dependencies as a function of distance in time or space, and they can be used to assess the distance required between sample points for the values to be effectively uncorrelated. In addition, they can form the basis of rules for interpolating values at points for which there are no observations. Correlation functions us ...
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Disordered Phase
In physics, the terms order and disorder designate the presence or absence of some symmetry or correlation in a many-particle system. In condensed matter physics, systems typically are ordered at low temperatures; upon heating, they undergo one or several phase transitions into less ordered states. Examples for such an order-disorder transition are: * the melting of ice: solid-liquid transition, loss of crystalline order; * the demagnetization of iron by heating above the Curie temperature: ferromagnetic-paramagnetic transition, loss of magnetic order. The degree of freedom that is ordered or disordered can be translational (crystalline ordering), rotational (ferroelectric ordering), or a spin state ( magnetic ordering). The order can consist either in a full crystalline space group symmetry, or in a correlation. Depending on how the correlations decay with distance, one speaks of long range order or short range order. If a disordered state is not in thermodynamic equilibriu ...
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Ordered Phase
In physics, the terms order and disorder designate the presence or absence of some symmetry or correlation in a many-particle system. In condensed matter physics, systems typically are ordered at low temperatures; upon heating, they undergo one or several phase transitions into less ordered states. Examples for such an order-disorder transition are: * the melting of ice: solid-liquid transition, loss of crystalline order; * the demagnetization of iron by heating above the Curie temperature: ferromagnetic-paramagnetic transition, loss of magnetic order. The degree of freedom that is ordered or disordered can be translational (crystalline ordering), rotational (ferroelectric ordering), or a spin state ( magnetic ordering). The order can consist either in a full crystalline space group symmetry, or in a correlation. Depending on how the correlations decay with distance, one speaks of long range order or short range order. If a disordered state is not in thermodynamic equilibriu ...
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Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denot ...
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Complement Graph
In the mathematical field of graph theory, the complement or inverse of a graph is a graph on the same vertices such that two distinct vertices of are adjacent if and only if they are not adjacent in . That is, to generate the complement of a graph, one fills in all the missing edges required to form a complete graph, and removes all the edges that were previously there.. The complement is not the set complement of the graph; only the edges are complemented. Definition Let be a simple graph and let consist of all 2-element subsets of . Then is the complement of , where is the relative complement of in . For directed graphs, the complement can be defined in the same way, as a directed graph on the same vertex set, using the set of all 2-element ordered pairs of in place of the set in the formula above. In terms of the adjacency matrix ''A'' of the graph, if ''Q'' is the adjacency matrix of the complete graph of the same number of vertices (i.e. all entries are unity ...
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Vertex (graph Theory)
In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another. From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects. The two vertices forming an edge are said to be the endpoints of this edge, and the edge is said to be incident to the vertices. A vertex ''w'' is said to be ...
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Antiferromagnetic
In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usually related to the spins of electrons, align in a regular pattern with neighboring spins (on different sublattices) pointing in opposite directions. This is, like ferromagnetism and ferrimagnetism, a manifestation of ordered magnetism. The phenomenon of antiferromagnetism was first introduced by Lev Landau in 1933. Generally, antiferromagnetic order may exist at sufficiently low temperatures, but vanishes at and above the Néel temperature – named after Louis Néel, who had first identified this type of magnetic ordering. Above the Néel temperature, the material is typically paramagnetic. Measurement When no external field is applied, the antiferromagnetic structure corresponds to a vanishing total magnetization. In an external magnetic field, a kind of ferrimagnetic behavior may be displayed in the antiferromagnetic phase, with the absolute value of one of the sublattice magne ...
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Ferromagnetic
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials are the familiar metals noticeably attracted to a magnet, a consequence of their large magnetic permeability. Magnetic permeability describes the induced magnetization of a material due to the presence of an ''external'' magnetic field, and it is this temporarily induced magnetization inside a steel plate, for instance, which accounts for its attraction to the permanent magnet. Whether or not that steel plate acquires a permanent magnetization itself, depends not only on the strength of the applied field, but on the so-called coercivity of that material, which varies greatly among ferromagnetic materials. In physics, several different types of material magnetism are distinguished. Ferromagnetism (along with the similar effect ferrimagneti ...
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