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Classical Heisenberg Model
The Classical Heisenberg model, developed by Werner Heisenberg, is the n = 3 case of the n-vector model, one of the models used in statistical physics to model ferromagnetism, and other phenomena. Definition It can be formulated as follows: take a d-dimensional lattice, and a set of spins of the unit length :\vec_i \in \mathbb^3, , \vec_i, =1\quad (1), each one placed on a lattice node. The model is defined through the following Hamiltonian: : \mathcal = -\sum_ \mathcal_ \vec_i \cdot \vec_j\quad (2) with : \mathcal_ = \begin J & \mboxi, j\mbox \\ 0 & \mbox\end a coupling between spins. Properties * The general mathematical formalism used to describe and solve the Heisenberg model and certain generalizations is developed in the article on the Potts model. * In the continuum limit the Heisenberg model (2) gives the following equation of motion :: \vec_=\vec\wedge \vec_. :This equation is called the continuous classical Heisenberg ferromagnet equation or shortly Heisenberg model ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Werner Heisenberg
Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series of papers with Max Born and Pascual Jordan, during the same year, his matrix formulation of quantum mechanics was substantially elaborated. He is known for the uncertainty principle, which he published in 1927. Heisenberg was awarded the 1932 Nobel Prize in Physics "for the creation of quantum mechanics". Heisenberg also made contributions to the theories of the hydrodynamics of turbulent flows, the atomic nucleus, ferromagnetism, cosmic rays, and subatomic particles. He was a principal scientist in the German nuclear weapons program during World War II. He was also instrumental in planning the first West German nuclear reactor at Karlsruhe, together with a research reactor in Munich, in 1957. Following World War II, he was appointed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical XY Model
The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's ''n''-vector model for . Definition Given a -dimensional lattice , per each lattice site there is a two-dimensional, unit-length vector The ''spin configuration'', is an assignment of the angle for each . Given a ''translation-invariant'' interaction and a point dependent external field \mathbf_=(h_j,0), the ''configuration energy'' is : H(\mathbf) = - \sum_ J_\; \mathbf_i\cdot\mathbf_j -\sum_j \mathbf_j\cdot \mathbf_j =- \sum_ J_\; \cos(\theta_i-\theta_j) -\sum_j h_j\cos\theta_j The case in which except for nearest neighbor is called ''nearest neighbor'' case. The ''configuration probability'' is given by the Boltzmann distribution with inverse temperature : :P(\mathbf)=\frac \qquad Z=\int_ \prod_ d\theta_j\;e^. where is the normalization, or partition fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Models
Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally biased portrayal of something Spin, spinning or spinnin may also refer to: Physics and mathematics * Spin, the rotation of an object around a central axis * Spin (physics) or particle spin, a fundamental property of elementary particles * Spin group, a particular double cover of the special orthogonal group SO(''n'') * Spin tensor, a tensor quantity for describing spinning motion in special relativity and general relativity * Spin (aerodynamics), autorotation of an aerodynamically stalled aeroplane * SPIN bibliographic database, an indexing and abstracting service focusing on physics research Textile arts * Spinning (polymers), a process for creating polymer fibres * Spinning (textiles), the creation of yarn or thread by twistin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetic Ordering
Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles give rise to a magnetic field, which acts on other currents and magnetic moments. Magnetism is one aspect of the combined phenomena of electromagnetism. The most familiar effects occur in ferromagnetic materials, which are strongly attracted by magnetic fields and can be magnetized to become permanent magnets, producing magnetic fields themselves. Demagnetizing a magnet is also possible. Only a few substances are ferromagnetic; the most common ones are iron, cobalt, and nickel and their alloys. The rare-earth metals neodymium and samarium are less common examples. The prefix ' refers to iron because permanent magnetism was first observed in lodestone, a form of natural iron ore called magnetite, Fe3O4. All substances exhibit some type of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ferromagnetism
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 ferrimagnetis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetism
Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles give rise to a magnetic field, which acts on other currents and magnetic moments. Magnetism is one aspect of the combined phenomena of electromagnetism. The most familiar effects occur in ferromagnetic materials, which are strongly attracted by magnetic fields and can be magnetized to become permanent magnets, producing magnetic fields themselves. Demagnetizing a magnet is also possible. Only a few substances are ferromagnetic; the most common ones are iron, cobalt, and nickel and their alloys. The rare-earth metals neodymium and samarium are less common examples. The prefix ' refers to iron because permanent magnetism was first observed in lodestone, a form of natural iron ore called magnetite, Fe3O4. All substances exhibit some type of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical XY Model
The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's ''n''-vector model for . Definition Given a -dimensional lattice , per each lattice site there is a two-dimensional, unit-length vector The ''spin configuration'', is an assignment of the angle for each . Given a ''translation-invariant'' interaction and a point dependent external field \mathbf_=(h_j,0), the ''configuration energy'' is : H(\mathbf) = - \sum_ J_\; \mathbf_i\cdot\mathbf_j -\sum_j \mathbf_j\cdot \mathbf_j =- \sum_ J_\; \cos(\theta_i-\theta_j) -\sum_j h_j\cos\theta_j The case in which except for nearest neighbor is called ''nearest neighbor'' case. The ''configuration probability'' is given by the Boltzmann distribution with inverse temperature : :P(\mathbf)=\frac \qquad Z=\int_ \prod_ d\theta_j\;e^. where is the normalization, or partition fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heisenberg Model (quantum)
The quantum Heisenberg model, developed by Werner Heisenberg, is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spins of the magnetic systems are treated quantum mechanically. It is related to the prototypical Ising model, where at each site of a lattice, a spin \sigma_i \in \ represents a microscopic magnetic dipole to which the magnetic moment is either up or down. Except the coupling between magnetic dipole moments, there is also a multipolar version of Heisenberg model called the multipolar exchange interaction. Overview For quantum mechanical reasons (see exchange interaction or ), the dominant coupling between two dipoles may cause nearest-neighbors to have lowest energy when they are ''aligned''. Under this assumption (so that magnetic interactions only occur between adjacent dipoles) and on a 1-dimensional periodic lattice, the Hamiltonian can be written in the form :\hat H = -J \sum_^ \sig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dipole Phase
In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways: *An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system is a pair of charges of equal magnitude but opposite sign separated by some typically small distance. (A permanent electric dipole is called an electret.) *A magnetic dipole is the closed circulation of an electric current system. A simple example is a single loop of wire with constant current through it. A bar magnet is an example of a magnet with a permanent magnetic dipole moment. Dipoles, whether electric or magnetic, can be characterized by their dipole moment, a vector quantity. For the simple electric dipole, the electric dipole moment points from the negative charge towards the positive charge, and has a magnitude equal to the strength of each charge times the separation between the charges. (To be precise: for the definition of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ishimori Equation
The Ishimori equation is a partial differential equation proposed by the Japanese mathematician . Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable . Equation The Ishimori equation has the form : \frac = \mathbf\wedge \left(\frac + \frac\right)+ \frac\frac + \frac\frac,\qquad (1a) : \frac-\alpha^2 \frac=-2\alpha^2 \mathbf\cdot\left(\frac\wedge \frac\right).\qquad (1b) Lax representation The Lax representation :L_t=AL-LA\qquad (2) of the equation is given by :L=\Sigma \partial_x+\alpha I\partial_y,\qquad (3a) :A= -2i\Sigma\partial_x^2+(-i\Sigma_x-i\alpha\Sigma_y\Sigma+u_yI-\alpha^3u_x\Sigma)\partial_x.\qquad (3b) Here :\Sigma=\sum_^3S_j\sigma_j,\qquad (4) the \sigma_i are the Pauli matrices and I is the identity matrix. Reductions The Ishimori equation admits an important reduction: in 1+1 dimensions it reduces to the continuous classical Heisenberg ferromagnet equation (CCHFE). The CCHFE is integrable. Equiva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-vector Model
In statistical mechanics, the ''n''-vector model or O(''n'') model is a simple system of interacting spins on a crystalline lattice. It was developed by H. Eugene Stanley as a generalization of the Ising model, XY model and Heisenberg model. In the ''n''-vector model, ''n''-component unit-length classical spins \mathbf_i are placed on the vertices of a ''d''-dimensional lattice. The Hamiltonian of the ''n''-vector model is given by: :H = -J_\mathbf_i \cdot \mathbf_j where the sum runs over all pairs of neighboring spins \langle i, j \rangle and \cdot denotes the standard Euclidean inner product. Special cases of the ''n''-vector model are: :n=0: The self-avoiding walk :n=1: The Ising model :n=2: The XY model :n=3: The Heisenberg model :n=4: Toy model for the Higgs sector of the Standard Model The general mathematical formalism used to describe and solve the ''n''-vector model and certain generalizations are developed in the article on the Potts model. Continuum limi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
