|
Orbital Magnetization
In quantum mechanics, orbital magnetization, Morb, refers to the magnetization induced by orbital motion of charged particles, usually electrons in solids. The term "orbital" distinguishes it from the contribution of spin degrees of freedom, Mspin, to the total magnetization. A nonzero orbital magnetization requires broken time-reversal symmetry, which can occur spontaneously in ferromagnetic and ferrimagnetic materials, or can be induced in a non- magnetic material by an applied magnetic field. Definitions The orbital magnetic moment of a finite system, such as a molecule, is given classically by : \mathbf_ = \frac\int \mathbf\times\mathbf(\mathbf) \ d^3\mathbf where J(r) is the current density at point r. (Here SI units are used; in Gaussian units, the prefactor would be 1/2''c'' instead, where ''c'' is the speed of light.) In a quantum-mechanical context, this can also be written as : \mathbf_ = \frac \langle\Psi \vert\mathbf \vert\Psi\rangle where −''e'' and ''me'' are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary (macroscopic and Microscopic scale, (optical) microscopic) scale, but is not sufficient for describing them at very small submicroscopic (atomic and subatomic) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales. Quantum systems have Bound state, bound states that are Quantization (physics), quantized to Discrete mathematics, discrete values of energy, momentum, angular momentum, and ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
G-factor (physics)
A ''g''-factor (also called ''g'' value) is a dimensionless quantity that characterizes the magnetic moment and angular momentum of an atom, a particle or the nucleus. It is the ratio of the magnetic moment (or, equivalently, the gyromagnetic ratio) of a particle to that expected of a classical particle of the same charge and angular momentum. In nuclear physics, the nuclear magneton replaces the classically expected magnetic moment (or gyromagnetic ratio) in the definition. The two definitions coincide for the proton. Definition Dirac particle The spin magnetic moment of a charged, spin-1/2 particle that does not possess any internal structure (a Dirac particle) is given by \boldsymbol \mu = g \mathbf S , where ''μ'' is the spin magnetic moment of the particle, ''g'' is the ''g''-factor of the particle, ''e'' is the elementary charge, ''m'' is the mass of the particle, and S is the spin angular momentum of the particle (with magnitude ''ħ''/2 for Dirac particles). B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barnett Effect
The Barnett effect is the magnetization of an uncharged body when spun on its axis. It was discovered by American physicist Samuel Jackson Barnett, Samuel Barnett in 1915. An uncharged object rotating with angular velocity tends to spontaneously magnetize, with a magnetization given by : M = \chi \omega / \gamma, where is the gyromagnetic ratio for the material, is the magnetic susceptibility. The magnetization occurs parallel to the axis of spin. Barnett was motivated by a prediction by Owen Richardson in 1908, later named the Einstein–de Haas effect, that magnetizing a ferromagnet can induce a mechanical rotation. He instead looked for the opposite effect, that is, that spinning a ferromagnet could change its magnetization. He established the effect with a long series of experiments between 1908 and 1915. See also * London moment References Further reading * Magnetism {{CMP-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyromagnetic Ratio
In physics, the gyromagnetic ratio (also sometimes known as the magnetogyric ratio in other disciplines) of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol , gamma. Its SI unit is the reciprocal second per tesla (s−1⋅T−1) or, equivalently, the coulomb per kilogram (C⋅kg−1). The -factor of a particle is a related dimensionless value of the system, derived as the ratio of its gyromagnetic ratio to that which would be classically expected from a rigid body of which the mass and charge are distributed identically, and for which total mass and charge are the same as that of the system. For a classical rotating body Consider a nonconductive charged body rotating about an axis of symmetry. According to the laws of classical physics, it has both a magnetic dipole moment due to the movement of charge and an angular momentum due to the movement of mass arising from its rotation. It can be shown that as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berry Connection And Curvature
In physics, Berry connection and Berry curvature are related concepts which can be viewed, respectively, as a local gauge potential and gauge field associated with the Berry phase or geometric phase. The concept was first introduced by S. Pancharatnam as geometric phase and later elaborately explained and popularized by Michael Berry in a paper published in 1984 emphasizing how geometric phases provide a powerful unifying concept in several branches of classical and quantum physics. Berry phase and cyclic adiabatic evolution In quantum mechanics, the Berry phase arises in a cyclic adiabatic evolution. The quantum adiabatic theorem applies to a system whose Hamiltonian H(\mathbf R) depends on a (vector) parameter \mathbf R that varies with time t. If the n'th eigenvalue \varepsilon_n(\mathbf R) remains non-degenerate everywhere along the path and the variation with time ''t'' is sufficiently slow, then a system initially in the normalized eigenstate , n(\mathbf R(0))\rangle w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavevector
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, 2 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 for whichever is in use. In the context of special relativity, a '' wave four-vector'' can be defined, combining the (angular) wave vector and (angular) frequency. Definition The terms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wannier Function
The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier in 1937. Wannier functions are the localized molecular orbitals of crystalline systems. The Wannier functions for different lattice sites in a crystal are orthogonal, allowing a convenient basis for the expansion of electron states in certain regimes. Wannier functions have found widespread use, for example, in the analysis of binding forces acting on electrons. Definition Although, like localized molecular orbitals, Wannier functions can be chosen in many different ways, the original, simplest, and most common definition in solid-state physics is as follows. Choose a single band in a perfect crystal, and denote its Bloch states by :\psi_(\mathbf) = e^u_\mathbf(\mathbf) where ''u''k(r) has the same periodicity as the crystal. Then the Wannier functions are defined by :\phi_(\mathbf) = \frac \sum_ e^ \psi_(\mathbf), where * R is any lattice ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Muffin-tin Approximation
The muffin-tin approximation is a shape approximation of the potential well in a crystal lattice. It is most commonly employed in quantum mechanical simulations of the electronic band structure in solids. The approximation was proposed by John C. Slater. Augmented plane wave method (APW) is a method which uses muffin-tin approximation. It is a method to approximate the energy states of an electron in a crystal lattice. The basic approximation lies in the potential in which the potential is assumed to be spherically symmetric in the muffin-tin region and constant in the interstitial region. Wave functions (the augmented plane waves) are constructed by matching solutions of the Schrödinger equation within each sphere with plane-wave solutions in the interstitial region, and linear combinations of these wave functions are then determined by the variational method. Many modern electronic structure methods employ the approximation. Among them APW method, the linear muffin-tin orbit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brillouin Zone
In mathematics and solid state physics, the first Brillouin zone (named after Léon Brillouin) is a uniquely defined primitive cell in reciprocal space Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray diffraction, X-ray and Electron diffraction, electron diffraction as well as the Electronic band structure, e .... 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 or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Momentum Operator
In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimension, the definition is: \hat = - i \hbar \frac where is the reduced Planck constant, the imaginary unit, is the spatial coordinate, and a partial derivative (denoted by \partial/\partial x) is used instead of a total derivative () since the wave function is also a function of time. The "hat" indicates an operator. The "application" of the operator on a differentiable wave function is as follows: \hat\psi = - i \hbar \frac In a basis of Hilbert space consisting of momentum eigenstates expressed in the momentum representation, the action of the operator is simply multiplication by , i.e. it is a multiplication operator, just as the position operator is a multiplication operator in the position representation. Note that the definition ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystal Momentum
In solid-state physics, crystal momentum or quasimomentum is a Momentum#Momentum in quantum mechanics, momentum-like Vector (geometric), vector associated with electrons in a Crystal structure, crystal lattice. It is defined by the associated Reciprocal lattice, wave vectors \mathbf of this lattice, according to \mathbf_ \equiv \hbar \mathbf (where \hbar is the reduced Planck constant). Frequently, crystal momentum is Momentum#Conservation, conserved like mechanical momentum, making it useful to physicists and materials scientists as an analytical tool. Lattice symmetry origins A common method of modeling crystal structure and behavior is to view electrons as Quantum mechanics, quantum mechanical particles traveling through a fixed infinite periodic potential V(x) such that V(\mathbf + \mathbf) = V(\mathbf), where \mathbf is an arbitrary Bravais lattice, lattice vector. Such a model is sensible because crystal Ion, ions that form the lattice structure are typically on the order ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bloch's Theorem
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves modulated by periodic functions. The theorem is named after the Swiss physicist Felix Bloch, who discovered the theorem in 1929. Mathematically, they are written where \mathbf is position, \psi is the wave function, u is a periodic function with the same periodicity as the crystal, the wave vector \mathbf is the crystal momentum vector, e is Euler's number, and i is the imaginary unit. Functions of this form are known as Bloch functions or Bloch states, and serve as a suitable basis for the wave functions or states of electrons in crystalline solids. The description of electrons in terms of Bloch functions, termed Bloch electrons (or less often ''Bloch Waves''), underlies the concept of electronic band structures. These eigenstates are written with subscripts as \psi_, where n is a discrete index, called the band index, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
