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23-853: [REDACTED] Look up llg in Wiktionary, the free dictionary. LLG or variation , may refer to: The Landau–Lifshitz–Gilbert equation , used in micromagnetics Local government Local-level governments of Papua New Guinea The Logical Language Group Lole language (ISO 639 code: llg ) Lycée Louis-le-Grand , a well known public high school in Paris Chillagoe Airport , IATA airport code "LLG" See also [ edit ] [REDACTED] Search for "llg" on Misplaced Pages. All pages with titles beginning with LLG All pages with titles containing LLG 2LG LG (disambiguation) Topics referred to by

46-414: A single magnetic domain and to magnetic domains in larger ferromagnets. The demagnetizing field of an arbitrarily shaped object requires a numerical solution of Poisson's equation even for the simple case of uniform magnetization. For the special case of ellipsoids (including infinite cylinders) the demagnetization field is linearly related to the magnetization by a geometry dependent constant called

69-410: A surface pole ) is analogous to a bound surface electric charge. Although the magnetic charges do not exist, it can be useful to think of them in this way. In particular, the arrangement of magnetization that reduces the magnetic energy can often be understood in terms of the pole-avoidance principle , which states that the magnetization tries to reduce the poles as much as possible. One way to remove

92-418: A body with no electric currents . These are Ampère's law and Gauss's law The magnetic field and flux density are related by where μ 0 {\displaystyle \mu _{0}} is the permeability of vacuum and M is the magnetisation . The general solution of the first equation can be expressed as the gradient of a scalar potential U ( r ) : Inside

115-463: A sphere in SI units. Note that in cgs units γ assumes values between 0 and 4 π . This equation can be generalized to include ellipsoids having principal axes in x, y, and z directions such that each component has a relationship of the form: Other important examples are an infinite plate (an ellipsoid with two of its axes going to infinity) which has γ = 1 (SI units) in a direction normal to

138-433: Is a dimensionless constant called the damping factor. The effective field H eff is a combination of the external magnetic field, the demagnetizing field , and various internal magnetic interactions involving quantum mechanical effects, which is typically defined as the functional derivative of the magnetic free energy with respect to the local magnetization M . To solve this equation, additional conditions for

161-502: Is different from Wikidata All article disambiguation pages All disambiguation pages Landau%E2%80%93Lifshitz%E2%80%93Gilbert equation The various forms of the LLG equation are commonly used in micromagnetics to model the effects of a magnetic field and other magnetic interactions on ferromagnetic materials . It provides a practical way to model the time-domain behavior of magnetic elements. Recent developments generalizes

184-563: Is the derivative with respect to distance from the surface. The outer potential U out must also be regular at infinity : both | r U | and | r U | must be bounded as r goes to infinity. This ensures that the magnetic energy is finite. Sufficiently far away, the magnetic field looks like the field of a magnetic dipole with the same moment as the finite body. Any two potentials that satisfy equations ( 5 ), ( 6 ) and ( 7 ), along with regularity at infinity, have identical gradients. The demagnetizing field H d

207-411: Is the dimensionless damping parameter, τ ⊥ {\displaystyle \tau _{\perp }} and τ ∥ {\displaystyle \tau _{\parallel }} are driving torques, and x is the unit vector along the polarization of the current. Demagnetizing field The demagnetizing field , also called the stray field (outside

230-405: Is the gradient of this potential (equation 4 ). The energy of the demagnetizing field is completely determined by an integral over the volume V of the magnet: Suppose there are two magnets with magnetizations M 1 and M 2 . The energy of the first magnet in the demagnetizing field H d of the second is The reciprocity theorem states that Formally, the solution of

253-421: The demagnetizing factor . Since the magnetization of a sample at a given location depends on the total magnetic field at that point, the demagnetization factor must be used in order to accurately determine how a magnetic material responds to a magnetic field. (See magnetic hysteresis .) In general the demagnetizing field is a function of position H ( r ) . It is derived from the magnetostatic equations for

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276-553: The LL equation, the precessional term γ' depends on the damping term. This better represents the behavior of real ferromagnets when the damping is large. In 1996 John Slonczewski expanded the model to account for the spin-transfer torque , i.e. the torque induced upon the magnetization by spin -polarized current flowing through the ferromagnet. This is commonly written in terms of the unit moment defined by m = M / M S : where α {\displaystyle \alpha }

299-417: The LLG equation to include the influence of spin-polarized currents in the form of spin-transfer torque . In a ferromagnet , the magnitude of the magnetization M at each spacetime point is approximated by the saturation magnetization M s (although it can be smaller when averaged over a chunk of volume). The Landau-Lifshitz equation, a precursor of the LLG equation, phenomenologically describes

322-573: The demagnetizing field must be included to accommodate the geometry of the material. In 1955 Gilbert replaced the damping term in the Landau–Lifshitz (LL) equation by one that depends on the time derivative of the magnetization: This is the Landau–Lifshitz–Gilbert (LLG) equation, where η is the damping parameter, which is characteristic of the material. It can be transformed into the Landau–Lifshitz equation: where In this form of

345-414: The equations for the potential is where r ′ is the variable to be integrated over the volume of the body in the first integral and the surface in the second, and ∇ ′ is the gradient with respect to this variable. Qualitatively, the negative of the divergence of the magnetization − ∇ · M (called a volume pole ) is analogous to a bulk bound electric charge in the body while n · M (called

368-489: The interfaces ( domain walls ) between domains. However, these poles vanish if the magnetic moments on each side of the domain wall meet the wall at the same angle (so that the components n · M are the same but opposite in sign). Domains configured this way are called closure domains . An arbitrarily shaped magnetic object has a total magnetic field that varies with location inside the object and can be quite difficult to calculate. This makes it very difficult to determine

391-468: The magnet), is the magnetic field (H-field) generated by the magnetization in a magnet . The total magnetic field in a region containing magnets is the sum of the demagnetizing fields of the magnets and the magnetic field due to any free currents or displacement currents . The term demagnetizing field reflects its tendency to act on the magnetization so as to reduce the total magnetic moment . It gives rise to shape anisotropy in ferromagnets with

414-448: The magnetic body, the potential U in is determined by substituting ( 3 ) and ( 4 ) in ( 2 ): Outside the body, where the magnetization is zero, At the surface of the magnet, there are two continuity requirements: This leads to the following boundary conditions at the surface of the magnet: Here n is the surface normal and ∂ / ∂ n {\textstyle \partial /\partial n}

437-400: The magnetic poles inside a ferromagnet is to make the magnetization uniform. This occurs in single-domain ferromagnets. This still leaves the surface poles, so division into domains reduces the poles further . However, very small ferromagnets are kept uniformly magnetized by the exchange interaction . The concentration of poles depends on the direction of magnetization (see the figure). If

460-408: The magnetic properties of a material such as, for instance, how the magnetization of a material varies with the magnetic field. For a uniformly magnetized sphere in a uniform magnetic field H 0 the internal magnetic field H is uniform: where M 0 is the magnetization of the sphere and γ is called the demagnetizing factor, which assumes values between 0 and 1, and equals 1/3 for

483-446: The magnetization is along the longest axis, the poles are spread across a smaller surface, so the energy is lower. This is a form of magnetic anisotropy called shape anisotropy . If the ferromagnet is large enough, its magnetization can divide into domains . It is then possible to have the magnetization parallel to the surface. Within each domain the magnetization is uniform, so there are no volume poles, but there are surface poles at

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506-457: The rotation of the magnetization in response to the effective field H eff which accounts for not only a real magnetic field but also internal magnetic interactions such as exchange and anisotropy. An earlier, but equivalent, equation (the Landau–Lifshitz equation) was introduced by Landau & Lifshitz (1935) : where γ is the electron gyromagnetic ratio and λ is a phenomenological damping parameter, often replaced by where α

529-403: The same term [REDACTED] This disambiguation page lists articles associated with the title LLG . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=LLG&oldid=1078747751 " Category : Disambiguation pages Hidden categories: Short description

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