2044:
1214:
2118:. The energy of the demagnetizing field favors magnetic configurations that minimize magnetic charges. In particular, on the edges of the sample, the magnetization tends to run parallel to the surface. In most cases it is not possible to minimize this energy term at the same time as the others. The static equilibrium then is a compromise that minimizes the total magnetic energy, although it may not minimize individually any particular term.
4206:
3792:
1809:
2607:
1670:
3959:
4021:
4400:
The interaction of micromagnetics with mechanics is also of interest in the context of industrial applications that deal with magneto-acoustic resonance such as in hypersound speakers, high frequency magnetostrictive transducers etc. FEM simulations taking into account the effect of magnetostriction
3289:
4419:
Another application that has emerged in the last decade is the application of micromagnetics towards neuronal stimulation. In this discipline, numerical methods such as finite-element analysis are used to analyze the electric/magnetic fields generated by the stimulation apparatus; then the results
3617:
3803:
5048:
Cortés-Ortuño, David; Beg, Marijan; Nehruji, Vanessa; Breth, Leoni; Pepper, Ryan; Kluyver, Thomas; Downing, Gary; Hesjedal, Thorsten; Hatton, Peter; Lancaster, Tom; Hertel, Riccardo; Hovorka, Ondrej; Fangohr, Hans (2018-11-12). "Proposal for a micromagnetic standard problem for materials with
2039:{\displaystyle U(\mathbf {r} )={\frac {1}{4\pi }}\left(-\int _{V}{\frac {\nabla '\cdot \mathbf {M} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} V+\int _{\partial V}{\frac {\mathbf {n} \cdot \mathbf {M} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} S\right).}
3453:
4390:
3305:
The purpose of dynamic micromagnetics is to predict the time evolution of the magnetic configuration. This is especially important if the sample is subject to some non-steady conditions such as the application of a field pulse or an AC field. This is done by solving the
2624:
The magnetoelastic energy describes the energy storage due to elastic lattice distortions. It may be neglected if magnetoelastic coupled effects are neglected. There exists a preferred local distortion of the crystalline solid associated with the magnetization director
3580:
571:
2471:
1554:
1316:
4420:
are validated or explored further using in-vivo or in-vitro neuronal stimulation. Several distinct set of neurons have been studied using this methodology including retinal neurons, cochlear neurons, vestibular neurons, and cortical neurons of embryonic rats.
4408:
and antivortex states; or even 3d-Bloch points, where, for example, the magnetization leads radially into all directions from the origin, or into topologically equivalent configurations. Thus in space, and also in time, nano- (and even pico-)scales are used.
3842:
322:
4201:{\displaystyle {\frac {\partial \mathbf {m} }{\partial t}}=-{\frac {|\gamma |}{1+\alpha ^{2}}}\mathbf {m} \times \mathbf {H} _{\mathrm {eff} }-{\frac {\alpha |\gamma |}{1+\alpha ^{2}}}\mathbf {m} \times (\mathbf {m} \times \mathbf {H} _{\text{eff}}),}
706:
The exchange energy tends to favor configurations where the magnetization varies slowly across the sample. This energy is minimized when the magnetization is perfectly uniform. The exchange term is isotropic, so any direction is equally acceptable.
2430:
1137:
3090:
2754:
2833:
2203:
3023:
2301:
3787:{\displaystyle \mathbf {H} _{\mathrm {eff} }={\frac {2A}{\mu _{0}M_{s}}}\nabla ^{2}\mathbf {m} -{\frac {1}{\mu _{0}M_{s}}}{\frac {\partial F_{\text{anis}}}{\partial \mathbf {m} }}+\mathbf {H} _{\text{a}}+\mathbf {H} _{\text{d}}}
807:
2132:
This interaction arises when a crystal lacks inversion symmetry, encouraging the magnetization to be perpendicular to its neighbours. It directly competes with the exchange energy. It is modelled with the energy contribution
2968:
3345:
1494:
4295:
3806:
The terms of the Landau-Lifshitz-Gilbert equation: precession (red) and damping (blue). The trajectory of the magnetization (dotted spiral) is drawn under the simplifying assumption that the effective field
3338:
fields however are the magnetostatic field and the applied field. It can be described informally as the derivative of the magnetic energy density with respect to the orientation of the magnetization, as in:
998:
1396:
3484:
2602:{\displaystyle E_{\text{DMI}}=\int _{V}D\mathbf {m} \cdot \left({\frac {\partial \mathbf {m} }{\partial x}}\times {\hat {x}}-{\frac {\partial \mathbf {m} }{\partial y}}\times {\hat {y}}\right).}
1665:{\displaystyle U_{\text{out}}=U_{\text{in}},\quad {\frac {\partial U_{\text{in}}}{\partial \mathbf {n} }}-{\frac {\partial U_{\text{out}}}{\partial \mathbf {n} }}=\mathbf {M} \cdot \mathbf {n} }
431:
1440:
211:
2112:
26:
dealing with the prediction of magnetic behaviors at sub-micrometer length scales. The length scales considered are large enough for the atomic structure of the material to be ignored (the
3954:{\displaystyle {\frac {\partial \mathbf {m} }{\partial t}}=-|\gamma |\mathbf {m} \times \mathbf {H} _{\mathrm {eff} }+\alpha \mathbf {m} \times {\frac {\partial \mathbf {m} }{\partial t}}}
1233:
1543:
2078:
2654:
1348:
1169:
1217:
Example of micromagnetic configuration. Compared to a uniform state, the flux closure structure lowers the energy of the demagnetizing field, at the expense of some exchange energy.
226:
2138:
370:
2781:
4287:
2338:
408:
1771:
901:
864:
837:
133:
3085:
2645:
2227:
1797:
1695:
1061:
923:
701:
97:
3611:
From the expression of the different contributions to the magnetic energy, the effective field can be found to be (excluding the DMI and magnetoelastic contributions):
4404:
Apart from conventional magnetic domains and domain-walls, the theory also treats the statics and dynamics of topological line and point configurations, e.g. magnetic
3043:
1196:
4249:
4229:
4006:
3982:
2786:
2463:
679:
652:
625:
163:
3063:
1728:
2330:
1036:
594:
2973:
2235:
4412:
The corresponding topological quantum numbers are thought to be used as information carriers, to apply the most recent, and already studied, propositions in
3284:{\displaystyle E_{\text{m-e}}=\int _{V}{\frac {\lambda }{2}}{\mbox{tr}}^{2}+\mu \,{\mbox{tr}}-3\mu E{\big \{}{\mbox{tr}}-{\frac {1}{3}}{\mbox{tr}}{\big \}}.}
5186:
Miehe, Christian; Ethiraj, Gautam (2011-10-15). "A geometrically consistent incremental variational formulation for phase field models in micromagnetics".
2838:
743:
4973:
Hoffmann, Markus; Zimmermann, Bernd; MĂĽller, Gideon P.; SchĂĽrhoff, Daniel; Kiselev, Nikolai S.; Melcher, Christof; BlĂĽgel, Stefan (2017-08-21).
5460:
5111:
4951:
4829:
4491:
2783:
is the isotropic magnetostrictive constant. The elastic energy density is assumed to be a function of the elastic, stress-producing strains
3448:{\displaystyle \mathbf {H} _{\mathrm {eff} }=-{\frac {1}{\mu _{0}M_{s}}}{\frac {\mathrm {d} ^{2}E}{\mathrm {d} \mathbf {m} \mathrm {d} V}}}
4385:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}|\mathbf {m} |^{2}=2\mathbf {m} \cdot {\frac {\partial \mathbf {m} }{\partial t}}=0.}
69:. In the 1970's computational methods were developed for the analysis of recording media due to the introduction of personal computers.
1448:
5239:
5557:
5576:
931:
65:. According to D. Wei, Brown introduced the name "micromagnetics" in 1958. The field prior to 1960 was summarised in Brown's book
3819:
3307:
2651:
model, one can assume this strain to be isochoric and fully isotropic in the lateral direction, yielding the deviatoric ansatz
4590:
Bar'yakhtar, V. G.; Ivanov, B. A. (2015-09-01). "The Landau-Lifshitz equation: 80 years of history, advances, and prospects".
3025:
is the fourth-order elasticity tensor. Here the elastic response is assumed to be isotropic (based on the two Lamé constants
3575:{\displaystyle \mathrm {d} E=-\mu _{0}M_{s}\int _{V}(\mathrm {d} \mathbf {m} )\cdot \mathbf {H} _{\text{eff}}\,\mathrm {d} V}
1360:
566:{\displaystyle E_{\text{exch}}=A\int _{V}\left((\nabla m_{x})^{2}+(\nabla m_{y})^{2}+(\nabla m_{z})^{2}\right)\mathrm {d} V}
3311:
1055:
The Zeeman energy is the interaction energy between the magnetization and any externally applied field. It is written as:
724:
4659:
KruzĂk, Martin; Prohl, Andreas (2006). "Recent
Developments in the Modeling, Analysis, and Numerics of Ferromagnetism".
3832:
term arising from the coupling of the magnetic system to the environment. The equation can be written in the so-called
1404:
1041:
The anisotropy energy favors magnetic configurations where the magnetization is everywhere aligned along an easy axis.
4401:
into micromagnetics are of importance. Such simulations use models described above within a finite element framework.
2648:
1311:{\displaystyle E_{\text{demag}}=-{\frac {\mu _{0}}{2}}\int _{V}\mathbf {M} \cdot \mathbf {H} _{\text{d}}\mathrm {d} V}
839:, the anisotropy energy density, is a function of the orientation of the magnetization. Minimum-energy directions for
171:
2087:
1227:
The demagnetizing field is the magnetic field created by the magnetic sample upon itself. The associated energy is:
5700:
5695:
41:, by minimizing the magnetic energy, and with dynamic behavior, by solving the time-dependent dynamical equation.
1505:
4231:
is the
Gilbert Damping constant, characterizing how quickly the damping term takes away energy from the system (
2052:
136:
58:
317:{\displaystyle E=E_{\text{exch}}+E_{\text{anis}}+E_{\text{Z}}+E_{\text{demag}}+E_{\text{DMI}}+E_{\text{m-e}},}
1324:
1145:
5705:
4633:
Landau, L; Lifshitz, E (1935). "On the theory of magnetic permeability dispersion in ferromagnetic solids".
31:
165:. The problem then consists in finding the spatial orientation of the magnetization, which is given by the
4547:
Brown, William Fuller (1978-03-01). "Domains, micromagnetics, and beyond: Reminiscences and assessments".
4434:
4413:
3467:
2127:
873:
734:
330:
38:
2759:
2425:{\displaystyle E_{\text{DMI}}=\int _{V}D(\mathbf {m} \cdot \nabla m_{z}-m_{z}\nabla \cdot \mathbf {m} ),}
4853:
Brown, William Fuller (1958-03-01). "Rigorous
Approach to the Theory of Ferromagnetic Microstructure".
5217:
Komineas, Stavros; Papanicolaou, Nikos (2007). "Dynamics of vortex-antivortex pairs in ferromagnets".
4254:
375:
5595:
5586:
Tiersten, H. F. (1964). "Coupled
Magnetomechanical Equations for Magnetically Saturated Insulators".
5528:
5479:
5429:
5302:
5254:
5191:
5150:
5068:
4996:
4862:
4746:
4703:
4599:
4556:
4516:
1733:
1132:{\displaystyle E_{\text{Z}}=-\mu _{0}\int _{V}\mathbf {M} \cdot \mathbf {H} _{\text{a}}\mathrm {d} V}
879:
842:
815:
422:
62:
1548:
outside of the body. These are supplemented with the boundary conditions on the surface of the body
106:
4737:
Brown, William Fuller (1941-07-15). "The Effect of
Dislocations on Magnetization Near Saturation".
3802:
1351:
1222:
1199:
716:
77:
The purpose of static micromagnetics is to solve for the spatial distribution of the magnetization
27:
3068:
2749:{\displaystyle \mathbf {\varepsilon } _{0}(\mathbf {m} )={\frac {3}{2}}\lambda _{\text{s}}\,\left}
2628:
2210:
1776:
1678:
906:
684:
80:
5503:
5419:
5218:
5140:
5058:
4986:
3985:
4907:. Interscience tracts on physics and astronomy. Vol. 18. Interscience Publishers. p. 7
3608:. This is usually not a problem, as this component has no effect on the magnetization dynamics.
2828:{\displaystyle \mathbf {\varepsilon } _{e}:=\mathbf {\varepsilon } -\mathbf {\varepsilon } _{0}}
5648:
5572:
5553:
5495:
5470:
Gilbert, Thomas L. (2004). "A Phenomenological Theory of
Damping in Ferromagnetic Materials".
5456:
5383:. 2019 9th International IEEE/EMBS Conference on Neural Engineering (NER). pp. 1758–761.
5361:
5238:
Thiaville, AndrĂ©; GarcĂa, JosĂ©; Dittrich, Rok; Miltat, Jacques; Schrefl, Thomas (March 2003).
5168:
5107:
5084:
5030:
5012:
4947:
4878:
4835:
4825:
4797:
4762:
4719:
4676:
4615:
4572:
4487:
3825:
3028:
2613:
1174:
730:
720:
100:
4941:
4234:
4214:
3991:
3967:
2198:{\displaystyle E_{\text{DMI}}=\int _{V}\mathbf {D} :(\nabla \mathbf {m} \times \mathbf {m} )}
5638:
5628:
5603:
5536:
5487:
5437:
5384:
5351:
5341:
5310:
5270:
5262:
5199:
5158:
5076:
5020:
5004:
4870:
4789:
4754:
4711:
4668:
4607:
4564:
4524:
3294:
2438:
1697:
is the unit normal to the surface. Furthermore, the potential satisfies the condition that
657:
630:
603:
141:
4975:"Antiskyrmions stabilized at interfaces by anisotropic Dzyaloshinskii-Moriya interactions"
3048:
1700:
1442:. This field depends on the magnetic configuration itself, and it can be found by solving
1213:
421:
The exchange energy is a phenomenological continuum description of the quantum-mechanical
54:
2309:
5599:
5532:
5483:
5433:
5306:
5258:
5195:
5154:
5072:
5000:
4866:
4750:
4707:
4603:
4560:
4520:
3018:{\displaystyle \mathbb {C} :=\lambda \mathbf {1} \otimes \mathbf {1} +2\mu \mathbb {I} }
2296:{\displaystyle E_{\text{DMI}}=\int _{V}D\mathbf {m} \cdot (\nabla \times \mathbf {m} ),}
5643:
5616:
5356:
5329:
5025:
4974:
1800:
1205:
The Zeeman energy favors alignment of the magnetization parallel to the applied field.
1021:
579:
4528:
5689:
5381:
Magnetic
Stimulation of Dissociated Cortical Neurons on a Planar Mulitelectrode Array
4694:
Brown, William Fuller (1940-10-15). "Theory of the
Approach to Magnetic Saturation".
3829:
1050:
5507:
802:{\displaystyle E_{\text{anis}}=\int _{V}F_{\text{anis}}(\mathbf {m} )\mathrm {d} V}
61:
in several works in 1940-1941 using energy expressions taken from a 1938 paper by
5667:
5442:
5407:
5163:
5128:
4902:
4486:(second ed.). Oxford ; New York: Oxford University Press. p. 135.
4251:= 0, no damping, permanent precession). These equations preserve the constraint
2229:
is the spiralization tensor, that depends upon the crystal class. For bulk DMI,
5633:
5517:"Recent Developments in the Modeling, Analysis, and Numerics of Ferromagnetism"
5346:
5266:
5080:
5008:
5540:
5203:
4672:
50:
5499:
5491:
5388:
5172:
5088:
5016:
4882:
4839:
4801:
4766:
4723:
4680:
4619:
4576:
5408:"Micromagnetics and spintronics: models and numerical methods (open access)"
4429:
5652:
5365:
5034:
4793:
4758:
4715:
99:
at equilibrium. In most cases, as the temperature is much lower than the
5617:"Modeling intracochlear magnetic stimulation: a Finite-Element Analysis"
5330:"Modeling intracochlear magnetic stimulation: a Finite-Element Analysis"
4011:
It can be shown that this is mathematically equivalent to the following
5275:
23:
5607:
5314:
4874:
4611:
4824:. Heidelberg ; New York: Springer Science & Business Media.
4568:
4405:
5223:
4507:
Aharoni, Amikam (2001). "Micromagnetics: past, present and future".
3828:
of the magnetization around the effective field, with an additional
3824:
This is the equation of motion of the magnetization. It describes a
3314:
describing the evolution of the magnetization in terms of the local
2963:{\displaystyle E_{\text{m-e}}={\frac {1}{2}}\int _{V}:\mathbb {C} :}
5424:
5145:
5063:
4991:
220:
The static equilibria are found by minimizing the magnetic energy,
5621:
IEEE Transactions on Neural
Systems and Rehabilitation Engineering
5334:
IEEE Transactions on Neural
Systems and Rehabilitation Engineering
5240:"Micromagnetic study of Bloch-point-mediated vortex core reversal"
3801:
1489:{\displaystyle \nabla ^{2}U_{\text{in}}=\nabla \cdot \mathbf {M} }
1212:
4780:
Elmore, W. C. (1938-05-01). "The Magnetic Structure of Cobalt".
3597:. Then the above definition leaves unspecified the component of
703:; and the integral is performed over the volume of the sample.
5129:"Micromagnetics and spintronics: models and numerical methods"
993:{\displaystyle F_{\text{anis}}(\mathbf {m} )=-K_{1}m_{z}^{2},}
135:
of the magnetization is assumed to be everywhere equal to the
57:
on antidomain walls. Micromagnetics was then expanded upon by
5673:
5293:
Döring, W. (1968). "Point Singularities in Micromagnetism".
5516:
30:), yet small enough to resolve magnetic structures such as
5679:
5106:. Berlin New York: Springer Science & Business Media.
4943:
Magnetic Domains: The Analysis of Magnetic Microstructures
4896:
4894:
4892:
1401:
and hence can be written as the gradient of a potential
1391:{\displaystyle \nabla \times \mathbf {H} _{\text{d}}=0}
3254:
3204:
3160:
3129:
5188:
Computer Methods in Applied Mechanics and Engineering
4935:
4933:
4931:
4929:
4927:
4925:
4923:
4921:
4298:
4257:
4237:
4217:
4024:
3994:
3970:
3845:
3620:
3487:
3348:
3093:
3071:
3051:
3031:
2976:
2841:
2789:
2762:
2657:
2631:
2474:
2441:
2341:
2312:
2238:
2213:
2141:
2090:
2055:
1812:
1779:
1736:
1703:
1681:
1557:
1508:
1451:
1407:
1363:
1327:
1236:
1177:
1148:
1064:
1024:
934:
909:
882:
845:
818:
746:
687:
660:
633:
606:
582:
434:
378:
333:
229:
174:
144:
109:
83:
5680:
MuMax -- GPU-accelerated Micromagnetic Modeling Tool
2835:. A quadratic form for the magnetoelastic energy is
413:
The contributions to this energy are the following:
4477:
4475:
4473:
4471:
4469:
729:Magnetic anisotropy arises due to a combination of
4542:
4540:
4538:
4467:
4465:
4463:
4461:
4459:
4457:
4455:
4453:
4451:
4449:
4384:
4281:
4243:
4223:
4200:
4000:
3976:
3953:
3786:
3574:
3447:
3283:
3079:
3057:
3037:
3017:
2962:
2827:
2775:
2748:
2639:
2601:
2457:
2424:
2324:
2295:
2221:
2197:
2106:
2072:
2038:
1791:
1765:
1722:
1689:
1664:
1537:
1488:
1434:
1390:
1342:
1310:
1190:
1163:
1131:
1030:
992:
917:
895:
858:
831:
801:
695:
673:
646:
619:
588:
565:
402:
364:
316:
205:
157:
127:
91:
1435:{\displaystyle \mathbf {H} _{\text{d}}=-\nabla U}
3474:of the magnetization and the associated change d
4815:
4813:
4811:
3087:, we obtain the invariant-based representation
206:{\displaystyle \mathbf {m} =\mathbf {M} /M_{s}}
49:Micromagnetics originated from a 1935 paper by
5102:Miyazaki, Terunobu; Jin, Hanmin (2012-08-22).
4654:
4652:
4650:
4648:
3065:). Taking into account the constant length of
2107:{\displaystyle \mathbf {M} \cdot \mathbf {n} }
5668:ÎĽMAG -- Micromagnetic Modeling Activity Group
5550:Continuum mechanics of electromagnetic solids
3273:
3198:
8:
4484:Introduction to the Theory of Ferromagnetism
2612:This term is important for the formation of
1538:{\displaystyle \nabla ^{2}U_{\text{out}}=0}
2073:{\displaystyle -\nabla \cdot \mathbf {M} }
5642:
5632:
5571:(1. Aufl. ed.). Stuttgart: Teubner.
5441:
5423:
5355:
5345:
5288:
5286:
5274:
5222:
5162:
5144:
5062:
5024:
4990:
4360:
4354:
4346:
4334:
4329:
4323:
4318:
4307:
4301:
4299:
4297:
4268:
4263:
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4236:
4216:
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4135:
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2520:
2514:
2501:
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2411:
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2386:
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2256:
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2237:
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2212:
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2179:
2165:
2159:
2146:
2140:
2099:
2091:
2089:
2065:
2054:
2020:
2012:
2003:
1994:
1989:
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1958:
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1886:
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1811:
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1507:
1481:
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1121:
1115:
1110:
1101:
1095:
1085:
1069:
1063:
1023:
981:
976:
966:
948:
939:
933:
910:
908:
887:
881:
850:
844:
823:
817:
791:
783:
774:
764:
751:
745:
688:
686:
665:
659:
638:
632:
611:
605:
581:
555:
544:
534:
515:
505:
486:
476:
455:
439:
433:
389:
384:
379:
377:
356:
344:
339:
334:
332:
305:
292:
279:
266:
253:
240:
228:
197:
188:
183:
175:
173:
149:
143:
120:
115:
110:
108:
84:
82:
2122:Dzyaloshinskii–Moriya Interaction Energy
1799:. The solution of these equations (c.f.
103:of the material considered, the modulus
4445:
3478:of the magnetic energy are related by:
1343:{\displaystyle \mathbf {H} _{\text{d}}}
1164:{\displaystyle \mathbf {H} _{\text{a}}}
4822:Micromagnetics and Recording Materials
2435:and for materials with symmetry class
5515:Kruzik Martin, Prohl Andreas (2006).
2332:plane interfacial DMI takes the form
7:
5674:OOMMF -- Micromagnetic Modeling Tool
5049:Dzyaloshinskii–Moriya interaction".
365:{\displaystyle |\mathbf {M} |=M_{s}}
37:Micromagnetics can deal with static
5451:Brown, William Fuller Jr. (1978) .
2776:{\displaystyle \lambda _{\text{s}}}
4367:
4357:
4308:
4302:
4112:
4109:
4106:
4038:
4028:
3942:
3932:
3909:
3906:
3903:
3859:
3849:
3743:
3728:
3679:
3635:
3632:
3629:
3565:
3536:
3489:
3435:
3425:
3409:
3363:
3360:
3357:
2567:
2557:
2527:
2517:
2405:
2379:
2276:
2176:
2059:
2021:
1947:
1932:
1867:
1786:
1752:
1635:
1620:
1603:
1588:
1510:
1475:
1453:
1426:
1364:
1301:
1122:
792:
737:. It can be generally written as:
556:
527:
498:
469:
14:
1209:Energy of the demagnetizing field
4940:Hubert, A.; Schäfer, R. (1998).
4361:
4347:
4324:
4282:{\displaystyle |\mathbf {m} |=1}
4264:
4182:
4173:
4162:
4100:
4091:
4032:
3936:
3922:
3897:
3888:
3853:
3820:Landau-Lifshitz-Gilbert equation
3798:Landau-Lifshitz-Gilbert equation
3774:
3759:
3747:
3689:
3623:
3553:
3541:
3430:
3351:
3308:Landau-Lifshitz-Gilbert equation
3293:This energy term contributes to
3230:
3222:
3073:
2997:
2989:
2950:
2902:
2737:
2719:
2711:
2674:
2633:
2561:
2521:
2502:
2412:
2372:
2283:
2266:
2215:
2188:
2180:
2166:
2100:
2092:
2066:
2004:
1995:
1976:
1967:
1959:
1915:
1906:
1887:
1878:
1820:
1683:
1658:
1650:
1639:
1607:
1482:
1410:
1372:
1330:
1290:
1281:
1151:
1111:
1102:
1014:. In this approximation, called
949:
925:. The simplest such function is
911:
784:
689:
403:{\displaystyle |\mathbf {m} |=1}
385:
340:
184:
176:
116:
85:
5588:Journal of Mathematical Physics
5133:The European Physical Journal B
3334:by the magnetization. The only
1766:{\displaystyle |r^{2}\nabla U|}
1584:
896:{\displaystyle F_{\text{anis}}}
859:{\displaystyle F_{\text{anis}}}
832:{\displaystyle F_{\text{anis}}}
5472:IEEE Transactions on Magnetics
5190:. 245–246. Elsevier: 331–347.
4330:
4319:
4269:
4259:
4192:
4169:
4136:
4128:
4065:
4057:
4008:the Gilbert damping constant.
3883:
3875:
3545:
3532:
3268:
3260:
3237:
3234:
3218:
3210:
3181:
3166:
3149:
3141:
2957:
2954:
2946:
2923:
2909:
2906:
2898:
2875:
2678:
2670:
2585:
2545:
2416:
2368:
2287:
2273:
2192:
2173:
2013:
1990:
1984:
1971:
1924:
1901:
1895:
1882:
1824:
1816:
1783:
1759:
1738:
1716:
1705:
953:
945:
788:
780:
541:
524:
512:
495:
483:
466:
390:
380:
345:
335:
167:magnetization direction vector
128:{\displaystyle |\mathbf {M} |}
121:
111:
16:Magnetism of sub-micron scales
1:
5104:The Physics of Ferromagnetism
4529:10.1016/S0921-4526(01)00954-1
3312:partial differential equation
2756:where the material parameter
725:Magnetocrystalline anisotropy
5569:Computational micromagnetism
5552:. Amsterdam: North-Holland.
3080:{\displaystyle \mathbf {m} }
2640:{\displaystyle \mathbf {m} }
2222:{\displaystyle \mathbf {D} }
1792:{\displaystyle r\to \infty }
1690:{\displaystyle \mathbf {n} }
918:{\displaystyle \mathbf {m} }
696:{\displaystyle \mathbf {m} }
92:{\displaystyle \mathbf {M} }
5412:European Physical Journal B
4509:Physica B: Condensed Matter
3593:is always perpendicular to
2465:the energy contribution is
2306:and for a thin film in the
5722:
5634:10.1109/TNSRE.2016.2624275
5548:Maugin, GĂ©rard A. (1988).
5443:10.1140/epjb/e2019-90599-6
5347:10.1109/TNSRE.2016.2624275
5295:Journal of Applied Physics
5267:10.1103/PhysRevB.67.094410
5164:10.1140/epjb/e2019-90599-6
5127:Abert, Claas (June 2019).
5009:10.1038/s41467-017-00313-0
4855:Journal of Applied Physics
4549:Journal of Applied Physics
3817:
3466:is the energy density. In
2125:
1220:
1048:
714:
327:subject to the constraint
5541:10.1137/S0036144504446187
5204:10.1016/j.cma.2012.03.021
4673:10.1137/S0036144504446187
4482:Aharoni, Amikam (2007) .
1171:is the applied field and
5492:10.1109/TMAG.2004.836740
5389:10.1109/NER.2019.8717125
5081:10.1088/1367-2630/aaea1c
4901:Brown Jr., W.F. (1963).
4441:Footnotes and references
3038:{\displaystyle \lambda }
1499:inside of the body and
1191:{\displaystyle \mu _{0}}
137:saturation magnetization
59:William Fuller Brown Jr.
5567:Prohl, Andreas (2001).
4820:Wei, Dan (2012-04-28).
4592:Low Temperature Physics
4244:{\displaystyle \alpha }
4224:{\displaystyle \alpha }
4001:{\displaystyle \alpha }
3977:{\displaystyle \gamma }
3836:(or implicit form) as:
1018:, the easy axis is the
903:is an even function of
28:continuum approximation
5051:New Journal of Physics
4794:10.1103/PhysRev.53.757
4759:10.1103/PhysRev.60.139
4716:10.1103/PhysRev.58.736
4435:Magnetic nanoparticles
4414:information technology
4386:
4283:
4245:
4225:
4202:
4002:
3978:
3955:
3815:
3788:
3576:
3449:
3301:Dynamic micromagnetics
3285:
3081:
3059:
3039:
3019:
2964:
2829:
2777:
2750:
2641:
2603:
2459:
2458:{\displaystyle D_{2d}}
2426:
2326:
2297:
2223:
2199:
2128:Antisymmetric exchange
2116:surface charge density
2108:
2074:
2040:
1793:
1767:
1724:
1691:
1666:
1539:
1490:
1436:
1392:
1354:. The field satisfies
1344:
1312:
1218:
1192:
1165:
1133:
1032:
994:
919:
897:
874:Time-reversal symmetry
860:
833:
803:
735:spin-orbit interaction
697:
681:are the components of
675:
648:
621:
590:
567:
404:
366:
318:
207:
159:
129:
93:
5406:Abert, Claas (2019).
4979:Nature Communications
4387:
4284:
4246:
4226:
4203:
4003:
3979:
3956:
3805:
3789:
3577:
3450:
3286:
3082:
3060:
3040:
3020:
2965:
2830:
2778:
2751:
2642:
2620:Magnetoelastic Energy
2604:
2460:
2427:
2327:
2298:
2224:
2200:
2109:
2082:volume charge density
2075:
2041:
1794:
1768:
1725:
1692:
1667:
1540:
1491:
1437:
1393:
1345:
1313:
1216:
1193:
1166:
1134:
1033:
995:
920:
898:
861:
834:
804:
698:
676:
674:{\displaystyle m_{z}}
649:
647:{\displaystyle m_{y}}
622:
620:{\displaystyle m_{x}}
591:
568:
405:
367:
319:
215:reduced magnetization
208:
160:
158:{\displaystyle M_{s}}
130:
94:
73:Static micromagnetics
4296:
4255:
4235:
4215:
4022:
4015:(or explicit) form:
3992:
3968:
3843:
3618:
3604:that is parallel to
3485:
3346:
3091:
3069:
3058:{\displaystyle \mu }
3049:
3029:
2974:
2839:
2787:
2760:
2655:
2629:
2472:
2439:
2339:
2310:
2236:
2211:
2139:
2088:
2080:is often called the
2053:
1810:
1777:
1734:
1723:{\displaystyle |rU|}
1701:
1679:
1555:
1506:
1449:
1405:
1361:
1325:
1234:
1175:
1146:
1062:
1022:
932:
907:
880:
843:
816:
744:
685:
658:
631:
604:
580:
432:
425:. It is written as:
423:exchange interaction
376:
331:
227:
172:
142:
107:
81:
63:William Cronk Elmore
5615:Mukesh, S. (2017).
5600:1964JMP.....5.1298T
5533:2006SIAMR..48..439K
5484:2004ITM....40.3443G
5455:. New York: Wiley.
5434:2019EPJB...92..120A
5379:Mukesh, S. (2019).
5328:Mukesh, S. (2017).
5307:1968JAP....39.1006D
5259:2003PhRvB..67i4410T
5196:2012CMAME.245..331M
5155:2019EPJB...92..120A
5073:2018NJPh...20k3015C
5001:2017NatCo...8..308H
4867:1958JAP....29..470B
4751:1941PhRv...60..139B
4708:1940PhRv...58..736B
4604:2015LTP....41..663B
4561:1978JAP....49.1937B
4521:2001PhyB..306....1A
3589:is a unit vector, d
3330:is the local field
2325:{\displaystyle x-y}
1352:demagnetizing field
1223:Demagnetizing field
1200:vacuum permeability
1016:uniaxial anisotropy
1012:anisotropy constant
986:
717:Magnetic anisotropy
4382:
4279:
4241:
4221:
4198:
3998:
3986:gyromagnetic ratio
3974:
3951:
3816:
3784:
3572:
3445:
3281:
3258:
3208:
3164:
3133:
3077:
3055:
3035:
3015:
2960:
2825:
2773:
2746:
2637:
2614:magnetic skyrmions
2599:
2455:
2422:
2322:
2293:
2219:
2195:
2104:
2070:
2036:
1789:
1773:remain bounded as
1763:
1720:
1687:
1662:
1535:
1486:
1432:
1388:
1340:
1308:
1219:
1188:
1161:
1129:
1028:
990:
972:
915:
893:
856:
829:
799:
693:
671:
644:
617:
586:
563:
400:
362:
314:
203:
155:
125:
89:
5701:Magnetic ordering
5696:Dynamical systems
5608:10.1063/1.1704239
5462:978-0-88275-665-3
5315:10.1063/1.1656144
5247:Physical Review B
5113:978-3-642-25583-0
4953:978-3-540-64108-7
4875:10.1063/1.1723183
4831:978-3-642-28577-6
4612:10.1063/1.4931649
4493:978-0-19-850809-0
4374:
4316:
4189:
4159:
4088:
4045:
3949:
3866:
3826:Larmor precession
3781:
3766:
3752:
3738:
3723:
3676:
3560:
3470:terms, a change d
3443:
3402:
3257:
3251:
3207:
3163:
3132:
3125:
3101:
2863:
2849:
2770:
2734:
2701:
2692:
2588:
2574:
2548:
2534:
2482:
2349:
2246:
2149:
2018:
1929:
1843:
1644:
1630:
1612:
1598:
1578:
1565:
1526:
1469:
1417:
1379:
1337:
1297:
1268:
1244:
1158:
1118:
1072:
1031:{\displaystyle z}
942:
890:
853:
826:
777:
754:
731:crystal structure
721:Anisotropy energy
711:Anisotropy energy
598:exchange constant
589:{\displaystyle A}
442:
308:
295:
282:
269:
256:
243:
101:Curie temperature
5713:
5656:
5646:
5636:
5627:(8): 1353–1362.
5611:
5594:(9): 1298–1318.
5582:
5563:
5544:
5511:
5478:(6): 3443–3449.
5466:
5447:
5445:
5427:
5393:
5392:
5376:
5370:
5369:
5359:
5349:
5340:(8): 1353–1362.
5325:
5319:
5318:
5301:(2): 1006–1007.
5290:
5281:
5280:
5278:
5244:
5235:
5229:
5228:
5226:
5214:
5208:
5207:
5183:
5177:
5176:
5166:
5148:
5124:
5118:
5117:
5099:
5093:
5092:
5066:
5045:
5039:
5038:
5028:
4994:
4970:
4964:
4963:
4961:
4960:
4937:
4916:
4915:
4913:
4912:
4898:
4887:
4886:
4850:
4844:
4843:
4817:
4806:
4805:
4777:
4771:
4770:
4734:
4728:
4727:
4691:
4685:
4684:
4656:
4643:
4642:
4630:
4624:
4623:
4587:
4581:
4580:
4569:10.1063/1.324811
4555:(3): 1937–1942.
4544:
4533:
4532:
4504:
4498:
4497:
4479:
4391:
4389:
4388:
4383:
4375:
4373:
4365:
4364:
4355:
4350:
4339:
4338:
4333:
4327:
4322:
4317:
4315:
4311:
4305:
4300:
4288:
4286:
4285:
4280:
4272:
4267:
4262:
4250:
4248:
4247:
4242:
4230:
4228:
4227:
4222:
4207:
4205:
4204:
4199:
4191:
4190:
4187:
4185:
4176:
4165:
4160:
4158:
4157:
4156:
4140:
4139:
4131:
4122:
4117:
4116:
4115:
4103:
4094:
4089:
4087:
4086:
4085:
4069:
4068:
4060:
4054:
4046:
4044:
4036:
4035:
4026:
4007:
4005:
4004:
3999:
3984:is the electron
3983:
3981:
3980:
3975:
3960:
3958:
3957:
3952:
3950:
3948:
3940:
3939:
3930:
3925:
3914:
3913:
3912:
3900:
3891:
3886:
3878:
3867:
3865:
3857:
3856:
3847:
3793:
3791:
3790:
3785:
3783:
3782:
3779:
3777:
3768:
3767:
3764:
3762:
3753:
3751:
3750:
3741:
3740:
3739:
3736:
3726:
3724:
3722:
3721:
3720:
3711:
3710:
3697:
3692:
3687:
3686:
3677:
3675:
3674:
3673:
3664:
3663:
3653:
3645:
3640:
3639:
3638:
3626:
3581:
3579:
3578:
3573:
3568:
3562:
3561:
3558:
3556:
3544:
3539:
3531:
3530:
3521:
3520:
3511:
3510:
3492:
3454:
3452:
3451:
3446:
3444:
3442:
3438:
3433:
3428:
3422:
3418:
3417:
3412:
3405:
3403:
3401:
3400:
3399:
3390:
3389:
3376:
3368:
3367:
3366:
3354:
3295:magnetostriction
3290:
3288:
3287:
3282:
3277:
3276:
3267:
3259:
3255:
3252:
3244:
3233:
3225:
3217:
3209:
3205:
3202:
3201:
3180:
3179:
3174:
3165:
3161:
3148:
3140:
3139:
3134:
3130:
3126:
3118:
3116:
3115:
3103:
3102:
3099:
3086:
3084:
3083:
3078:
3076:
3064:
3062:
3061:
3056:
3044:
3042:
3041:
3036:
3024:
3022:
3021:
3016:
3014:
3000:
2992:
2981:
2969:
2967:
2966:
2961:
2953:
2945:
2944:
2939:
2930:
2919:
2905:
2897:
2896:
2891:
2882:
2874:
2873:
2864:
2856:
2851:
2850:
2847:
2834:
2832:
2831:
2826:
2824:
2823:
2818:
2809:
2801:
2800:
2795:
2782:
2780:
2779:
2774:
2772:
2771:
2768:
2755:
2753:
2752:
2747:
2745:
2741:
2740:
2735:
2727:
2722:
2714:
2703:
2702:
2699:
2693:
2685:
2677:
2669:
2668:
2663:
2647:. For a simple
2646:
2644:
2643:
2638:
2636:
2608:
2606:
2605:
2600:
2595:
2591:
2590:
2589:
2581:
2575:
2573:
2565:
2564:
2555:
2550:
2549:
2541:
2535:
2533:
2525:
2524:
2515:
2505:
2497:
2496:
2484:
2483:
2480:
2464:
2462:
2461:
2456:
2454:
2453:
2431:
2429:
2428:
2423:
2415:
2404:
2403:
2391:
2390:
2375:
2364:
2363:
2351:
2350:
2347:
2331:
2329:
2328:
2323:
2302:
2300:
2299:
2294:
2286:
2269:
2261:
2260:
2248:
2247:
2244:
2228:
2226:
2225:
2220:
2218:
2204:
2202:
2201:
2196:
2191:
2183:
2169:
2164:
2163:
2151:
2150:
2147:
2113:
2111:
2110:
2105:
2103:
2095:
2079:
2077:
2076:
2071:
2069:
2045:
2043:
2042:
2037:
2032:
2028:
2024:
2019:
2017:
2016:
2011:
2007:
1998:
1993:
1987:
1983:
1979:
1970:
1962:
1956:
1954:
1953:
1935:
1930:
1928:
1927:
1922:
1918:
1909:
1904:
1898:
1894:
1890:
1881:
1873:
1864:
1862:
1861:
1844:
1842:
1831:
1823:
1798:
1796:
1795:
1790:
1772:
1770:
1769:
1764:
1762:
1751:
1750:
1741:
1729:
1727:
1726:
1721:
1719:
1708:
1696:
1694:
1693:
1688:
1686:
1671:
1669:
1668:
1663:
1661:
1653:
1645:
1643:
1642:
1633:
1632:
1631:
1628:
1618:
1613:
1611:
1610:
1601:
1600:
1599:
1596:
1586:
1580:
1579:
1576:
1567:
1566:
1563:
1544:
1542:
1541:
1536:
1528:
1527:
1524:
1518:
1517:
1495:
1493:
1492:
1487:
1485:
1471:
1470:
1467:
1461:
1460:
1441:
1439:
1438:
1433:
1419:
1418:
1415:
1413:
1397:
1395:
1394:
1389:
1381:
1380:
1377:
1375:
1349:
1347:
1346:
1341:
1339:
1338:
1335:
1333:
1317:
1315:
1314:
1309:
1304:
1299:
1298:
1295:
1293:
1284:
1279:
1278:
1269:
1264:
1263:
1254:
1246:
1245:
1242:
1197:
1195:
1194:
1189:
1187:
1186:
1170:
1168:
1167:
1162:
1160:
1159:
1156:
1154:
1138:
1136:
1135:
1130:
1125:
1120:
1119:
1116:
1114:
1105:
1100:
1099:
1090:
1089:
1074:
1073:
1070:
1037:
1035:
1034:
1029:
999:
997:
996:
991:
985:
980:
971:
970:
952:
944:
943:
940:
924:
922:
921:
916:
914:
902:
900:
899:
894:
892:
891:
888:
865:
863:
862:
857:
855:
854:
851:
838:
836:
835:
830:
828:
827:
824:
808:
806:
805:
800:
795:
787:
779:
778:
775:
769:
768:
756:
755:
752:
702:
700:
699:
694:
692:
680:
678:
677:
672:
670:
669:
653:
651:
650:
645:
643:
642:
626:
624:
623:
618:
616:
615:
595:
593:
592:
587:
572:
570:
569:
564:
559:
554:
550:
549:
548:
539:
538:
520:
519:
510:
509:
491:
490:
481:
480:
460:
459:
444:
443:
440:
409:
407:
406:
401:
393:
388:
383:
371:
369:
368:
363:
361:
360:
348:
343:
338:
323:
321:
320:
315:
310:
309:
306:
297:
296:
293:
284:
283:
280:
271:
270:
267:
258:
257:
254:
245:
244:
241:
212:
210:
209:
204:
202:
201:
192:
187:
179:
164:
162:
161:
156:
154:
153:
134:
132:
131:
126:
124:
119:
114:
98:
96:
95:
90:
88:
5721:
5720:
5716:
5715:
5714:
5712:
5711:
5710:
5686:
5685:
5664:
5659:
5614:
5585:
5579:
5566:
5560:
5547:
5514:
5469:
5463:
5450:
5405:
5401:
5399:Further reading
5396:
5378:
5377:
5373:
5327:
5326:
5322:
5292:
5291:
5284:
5242:
5237:
5236:
5232:
5216:
5215:
5211:
5185:
5184:
5180:
5126:
5125:
5121:
5114:
5101:
5100:
5096:
5047:
5046:
5042:
4972:
4971:
4967:
4958:
4956:
4954:
4939:
4938:
4919:
4910:
4908:
4900:
4899:
4890:
4852:
4851:
4847:
4832:
4819:
4818:
4809:
4782:Physical Review
4779:
4778:
4774:
4739:Physical Review
4736:
4735:
4731:
4696:Physical Review
4693:
4692:
4688:
4658:
4657:
4646:
4632:
4631:
4627:
4589:
4588:
4584:
4546:
4545:
4536:
4506:
4505:
4501:
4494:
4481:
4480:
4447:
4443:
4426:
4398:
4366:
4356:
4328:
4306:
4294:
4293:
4253:
4252:
4233:
4232:
4213:
4212:
4180:
4148:
4141:
4123:
4098:
4077:
4070:
4055:
4037:
4027:
4020:
4019:
4013:Landau-Lifshitz
3990:
3989:
3966:
3965:
3941:
3931:
3895:
3858:
3848:
3841:
3840:
3822:
3813:
3800:
3772:
3757:
3742:
3731:
3727:
3712:
3702:
3701:
3678:
3665:
3655:
3654:
3646:
3621:
3616:
3615:
3603:
3551:
3522:
3512:
3502:
3483:
3482:
3423:
3407:
3406:
3391:
3381:
3380:
3349:
3344:
3343:
3328:effective field
3324:
3322:Effective field
3316:effective field
3303:
3169:
3127:
3107:
3094:
3089:
3088:
3067:
3066:
3047:
3046:
3027:
3026:
2972:
2971:
2934:
2886:
2865:
2842:
2837:
2836:
2813:
2790:
2785:
2784:
2763:
2758:
2757:
2709:
2705:
2694:
2658:
2653:
2652:
2627:
2626:
2622:
2566:
2556:
2526:
2516:
2513:
2509:
2488:
2475:
2470:
2469:
2442:
2437:
2436:
2395:
2382:
2355:
2342:
2337:
2336:
2308:
2307:
2252:
2239:
2234:
2233:
2209:
2208:
2155:
2142:
2137:
2136:
2130:
2124:
2086:
2085:
2051:
2050:
2002:
1988:
1974:
1957:
1942:
1913:
1899:
1885:
1866:
1865:
1853:
1849:
1845:
1835:
1808:
1807:
1775:
1774:
1742:
1732:
1731:
1699:
1698:
1677:
1676:
1634:
1623:
1619:
1602:
1591:
1587:
1571:
1558:
1553:
1552:
1519:
1509:
1504:
1503:
1462:
1452:
1447:
1446:
1408:
1403:
1402:
1370:
1359:
1358:
1328:
1323:
1322:
1288:
1270:
1255:
1237:
1232:
1231:
1225:
1211:
1178:
1173:
1172:
1149:
1144:
1143:
1109:
1091:
1081:
1065:
1060:
1059:
1053:
1047:
1020:
1019:
1008:
962:
935:
930:
929:
905:
904:
883:
878:
877:
846:
841:
840:
819:
814:
813:
770:
760:
747:
742:
741:
727:
715:Main articles:
713:
683:
682:
661:
656:
655:
634:
629:
628:
607:
602:
601:
578:
577:
540:
530:
511:
501:
482:
472:
465:
461:
451:
435:
430:
429:
419:
417:Exchange energy
374:
373:
352:
329:
328:
301:
288:
275:
262:
249:
236:
225:
224:
193:
170:
169:
145:
140:
139:
105:
104:
79:
78:
75:
55:Evgeny Lifshitz
47:
17:
12:
11:
5:
5719:
5717:
5709:
5708:
5706:Magnetostatics
5703:
5698:
5688:
5687:
5684:
5683:
5677:
5671:
5663:
5662:External links
5660:
5658:
5657:
5612:
5583:
5577:
5564:
5559:978-0444703996
5558:
5545:
5527:(3): 439–483.
5512:
5467:
5461:
5453:Micromagnetics
5448:
5402:
5400:
5397:
5395:
5394:
5371:
5320:
5282:
5230:
5209:
5178:
5119:
5112:
5094:
5057:(11): 113015.
5040:
4965:
4952:
4917:
4904:Micromagnetics
4888:
4861:(3): 470–471.
4845:
4830:
4807:
4788:(9): 757–764.
4772:
4745:(2): 139–147.
4729:
4702:(8): 736–743.
4686:
4667:(3): 439–483.
4644:
4625:
4598:(9): 663–669.
4582:
4534:
4499:
4492:
4444:
4442:
4439:
4438:
4437:
4432:
4425:
4422:
4397:
4394:
4393:
4392:
4381:
4378:
4372:
4369:
4363:
4359:
4353:
4349:
4345:
4342:
4337:
4332:
4326:
4321:
4314:
4310:
4304:
4278:
4275:
4271:
4266:
4261:
4240:
4220:
4209:
4208:
4197:
4194:
4184:
4179:
4175:
4171:
4168:
4164:
4155:
4151:
4147:
4144:
4138:
4134:
4130:
4126:
4120:
4114:
4111:
4108:
4102:
4097:
4093:
4084:
4080:
4076:
4073:
4067:
4063:
4059:
4052:
4049:
4043:
4040:
4034:
4030:
3997:
3973:
3962:
3961:
3947:
3944:
3938:
3934:
3928:
3924:
3920:
3917:
3911:
3908:
3905:
3899:
3894:
3890:
3885:
3881:
3877:
3873:
3870:
3864:
3861:
3855:
3851:
3818:Main article:
3811:
3799:
3796:
3795:
3794:
3776:
3771:
3761:
3756:
3749:
3745:
3734:
3730:
3719:
3715:
3709:
3705:
3700:
3695:
3691:
3685:
3681:
3672:
3668:
3662:
3658:
3652:
3649:
3643:
3637:
3634:
3631:
3625:
3601:
3583:
3582:
3571:
3567:
3555:
3550:
3547:
3543:
3538:
3534:
3529:
3525:
3519:
3515:
3509:
3505:
3501:
3498:
3495:
3491:
3456:
3455:
3441:
3437:
3432:
3427:
3421:
3416:
3411:
3398:
3394:
3388:
3384:
3379:
3374:
3371:
3365:
3362:
3359:
3353:
3323:
3320:
3318:acting on it.
3302:
3299:
3280:
3275:
3270:
3266:
3262:
3250:
3247:
3242:
3239:
3236:
3232:
3228:
3224:
3220:
3216:
3212:
3200:
3195:
3192:
3189:
3186:
3183:
3178:
3173:
3168:
3157:
3154:
3151:
3147:
3143:
3138:
3124:
3121:
3114:
3110:
3106:
3097:
3075:
3054:
3034:
3013:
3009:
3006:
3003:
2999:
2995:
2991:
2987:
2984:
2980:
2959:
2956:
2952:
2948:
2943:
2938:
2933:
2929:
2925:
2922:
2918:
2914:
2911:
2908:
2904:
2900:
2895:
2890:
2885:
2881:
2877:
2872:
2868:
2862:
2859:
2854:
2845:
2822:
2817:
2812:
2808:
2804:
2799:
2794:
2766:
2744:
2739:
2733:
2730:
2725:
2721:
2717:
2713:
2708:
2697:
2691:
2688:
2683:
2680:
2676:
2672:
2667:
2662:
2635:
2621:
2618:
2610:
2609:
2598:
2594:
2587:
2584:
2578:
2572:
2569:
2563:
2559:
2553:
2547:
2544:
2538:
2532:
2529:
2523:
2519:
2512:
2508:
2504:
2500:
2495:
2491:
2487:
2478:
2452:
2449:
2445:
2433:
2432:
2421:
2418:
2414:
2410:
2407:
2402:
2398:
2394:
2389:
2385:
2381:
2378:
2374:
2370:
2367:
2362:
2358:
2354:
2345:
2321:
2318:
2315:
2304:
2303:
2292:
2289:
2285:
2281:
2278:
2275:
2272:
2268:
2264:
2259:
2255:
2251:
2242:
2217:
2194:
2190:
2186:
2182:
2178:
2175:
2172:
2168:
2162:
2158:
2154:
2145:
2126:Main article:
2123:
2120:
2114:is called the
2102:
2098:
2094:
2068:
2064:
2061:
2058:
2047:
2046:
2035:
2031:
2027:
2023:
2015:
2010:
2006:
2001:
1997:
1992:
1986:
1982:
1978:
1973:
1969:
1965:
1961:
1952:
1949:
1945:
1941:
1938:
1934:
1926:
1921:
1917:
1912:
1908:
1903:
1897:
1893:
1889:
1884:
1880:
1876:
1872:
1869:
1860:
1856:
1852:
1848:
1841:
1838:
1834:
1829:
1826:
1822:
1818:
1815:
1801:magnetostatics
1788:
1785:
1782:
1761:
1757:
1754:
1749:
1745:
1740:
1718:
1714:
1711:
1707:
1685:
1673:
1672:
1660:
1656:
1652:
1648:
1641:
1637:
1626:
1622:
1616:
1609:
1605:
1594:
1590:
1583:
1574:
1570:
1561:
1546:
1545:
1534:
1531:
1522:
1516:
1512:
1497:
1496:
1484:
1480:
1477:
1474:
1465:
1459:
1455:
1431:
1428:
1425:
1422:
1412:
1399:
1398:
1387:
1384:
1374:
1369:
1366:
1332:
1319:
1318:
1307:
1303:
1292:
1287:
1283:
1277:
1273:
1267:
1262:
1258:
1252:
1249:
1240:
1221:Main article:
1210:
1207:
1185:
1181:
1153:
1140:
1139:
1128:
1124:
1113:
1108:
1104:
1098:
1094:
1088:
1084:
1080:
1077:
1068:
1049:Main article:
1046:
1043:
1027:
1010:is called the
1006:
1001:
1000:
989:
984:
979:
975:
969:
965:
961:
958:
955:
951:
947:
938:
913:
886:
849:
822:
810:
809:
798:
794:
790:
786:
782:
773:
767:
763:
759:
750:
712:
709:
691:
668:
664:
641:
637:
614:
610:
585:
574:
573:
562:
558:
553:
547:
543:
537:
533:
529:
526:
523:
518:
514:
508:
504:
500:
497:
494:
489:
485:
479:
475:
471:
468:
464:
458:
454:
450:
447:
438:
418:
415:
399:
396:
392:
387:
382:
359:
355:
351:
347:
342:
337:
325:
324:
313:
304:
300:
291:
287:
278:
274:
265:
261:
252:
248:
239:
235:
232:
213:, also called
200:
196:
191:
186:
182:
178:
152:
148:
123:
118:
113:
87:
74:
71:
67:Micromagnetics
46:
43:
22:is a field of
20:Micromagnetics
15:
13:
10:
9:
6:
4:
3:
2:
5718:
5707:
5704:
5702:
5699:
5697:
5694:
5693:
5691:
5681:
5678:
5675:
5672:
5669:
5666:
5665:
5661:
5654:
5650:
5645:
5640:
5635:
5630:
5626:
5622:
5618:
5613:
5609:
5605:
5601:
5597:
5593:
5589:
5584:
5580:
5578:9783519003588
5574:
5570:
5565:
5561:
5555:
5551:
5546:
5542:
5538:
5534:
5530:
5526:
5522:
5518:
5513:
5509:
5505:
5501:
5497:
5493:
5489:
5485:
5481:
5477:
5473:
5468:
5464:
5458:
5454:
5449:
5444:
5439:
5435:
5431:
5426:
5421:
5417:
5413:
5409:
5404:
5403:
5398:
5390:
5386:
5382:
5375:
5372:
5367:
5363:
5358:
5353:
5348:
5343:
5339:
5335:
5331:
5324:
5321:
5316:
5312:
5308:
5304:
5300:
5296:
5289:
5287:
5283:
5277:
5272:
5268:
5264:
5260:
5256:
5253:(9): 094410.
5252:
5248:
5241:
5234:
5231:
5225:
5220:
5213:
5210:
5205:
5201:
5197:
5193:
5189:
5182:
5179:
5174:
5170:
5165:
5160:
5156:
5152:
5147:
5142:
5138:
5134:
5130:
5123:
5120:
5115:
5109:
5105:
5098:
5095:
5090:
5086:
5082:
5078:
5074:
5070:
5065:
5060:
5056:
5052:
5044:
5041:
5036:
5032:
5027:
5022:
5018:
5014:
5010:
5006:
5002:
4998:
4993:
4988:
4984:
4980:
4976:
4969:
4966:
4955:
4949:
4945:
4944:
4936:
4934:
4932:
4930:
4928:
4926:
4924:
4922:
4918:
4906:
4905:
4897:
4895:
4893:
4889:
4884:
4880:
4876:
4872:
4868:
4864:
4860:
4856:
4849:
4846:
4841:
4837:
4833:
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4823:
4816:
4814:
4812:
4808:
4803:
4799:
4795:
4791:
4787:
4783:
4776:
4773:
4768:
4764:
4760:
4756:
4752:
4748:
4744:
4740:
4733:
4730:
4725:
4721:
4717:
4713:
4709:
4705:
4701:
4697:
4690:
4687:
4682:
4678:
4674:
4670:
4666:
4662:
4655:
4653:
4651:
4649:
4645:
4640:
4636:
4629:
4626:
4621:
4617:
4613:
4609:
4605:
4601:
4597:
4593:
4586:
4583:
4578:
4574:
4570:
4566:
4562:
4558:
4554:
4550:
4543:
4541:
4539:
4535:
4530:
4526:
4522:
4518:
4514:
4510:
4503:
4500:
4495:
4489:
4485:
4478:
4476:
4474:
4472:
4470:
4468:
4466:
4464:
4462:
4460:
4458:
4456:
4454:
4452:
4450:
4446:
4440:
4436:
4433:
4431:
4428:
4427:
4423:
4421:
4417:
4415:
4410:
4407:
4402:
4395:
4379:
4376:
4370:
4351:
4343:
4340:
4335:
4312:
4292:
4291:
4290:
4276:
4273:
4238:
4218:
4195:
4177:
4166:
4153:
4149:
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4142:
4132:
4124:
4118:
4095:
4082:
4078:
4074:
4071:
4061:
4050:
4047:
4041:
4018:
4017:
4016:
4014:
4009:
3995:
3987:
3971:
3945:
3926:
3918:
3915:
3892:
3879:
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3868:
3862:
3839:
3838:
3837:
3835:
3831:
3827:
3821:
3810:
3804:
3797:
3769:
3754:
3732:
3717:
3713:
3707:
3703:
3698:
3693:
3683:
3670:
3666:
3660:
3656:
3650:
3647:
3641:
3614:
3613:
3612:
3609:
3607:
3600:
3596:
3592:
3588:
3569:
3548:
3527:
3523:
3517:
3513:
3507:
3503:
3499:
3496:
3493:
3481:
3480:
3479:
3477:
3473:
3469:
3465:
3461:
3439:
3419:
3414:
3396:
3392:
3386:
3382:
3377:
3372:
3369:
3342:
3341:
3340:
3337:
3333:
3329:
3321:
3319:
3317:
3313:
3310:, which is a
3309:
3300:
3298:
3296:
3291:
3278:
3264:
3248:
3245:
3240:
3226:
3214:
3193:
3190:
3187:
3184:
3176:
3171:
3155:
3152:
3145:
3136:
3122:
3119:
3112:
3108:
3104:
3095:
3052:
3032:
3007:
3004:
3001:
2993:
2985:
2982:
2941:
2936:
2931:
2927:
2920:
2912:
2893:
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2883:
2879:
2870:
2866:
2860:
2857:
2852:
2843:
2820:
2815:
2810:
2806:
2802:
2797:
2792:
2764:
2742:
2731:
2728:
2723:
2715:
2706:
2695:
2689:
2686:
2681:
2665:
2660:
2650:
2619:
2617:
2615:
2596:
2592:
2582:
2576:
2570:
2551:
2542:
2536:
2530:
2510:
2506:
2498:
2493:
2489:
2485:
2476:
2468:
2467:
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2447:
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2249:
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2231:
2230:
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2160:
2156:
2152:
2143:
2134:
2129:
2121:
2119:
2117:
2096:
2083:
2062:
2056:
2049:The quantity
2033:
2029:
2025:
2008:
1999:
1980:
1963:
1950:
1943:
1939:
1936:
1919:
1910:
1891:
1874:
1870:
1858:
1854:
1850:
1846:
1839:
1836:
1832:
1827:
1813:
1806:
1805:
1804:
1802:
1780:
1755:
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1654:
1646:
1624:
1614:
1592:
1581:
1572:
1568:
1559:
1551:
1550:
1549:
1532:
1529:
1520:
1514:
1502:
1501:
1500:
1478:
1472:
1463:
1457:
1445:
1444:
1443:
1429:
1423:
1420:
1385:
1382:
1367:
1357:
1356:
1355:
1353:
1305:
1285:
1275:
1271:
1265:
1260:
1256:
1250:
1247:
1238:
1230:
1229:
1228:
1224:
1215:
1208:
1206:
1203:
1201:
1183:
1179:
1126:
1106:
1096:
1092:
1086:
1082:
1078:
1075:
1066:
1058:
1057:
1056:
1052:
1051:Zeeman energy
1045:Zeeman energy
1044:
1042:
1039:
1025:
1017:
1013:
1009:
987:
982:
977:
973:
967:
963:
959:
956:
936:
928:
927:
926:
884:
876:ensures that
875:
871:
869:
847:
820:
796:
771:
765:
761:
757:
748:
740:
739:
738:
736:
732:
726:
722:
718:
710:
708:
704:
666:
662:
639:
635:
612:
608:
599:
583:
560:
551:
545:
535:
531:
521:
516:
506:
502:
492:
487:
477:
473:
462:
456:
452:
448:
445:
436:
428:
427:
426:
424:
416:
414:
411:
397:
394:
357:
353:
349:
311:
302:
298:
289:
285:
276:
272:
263:
259:
250:
246:
237:
233:
230:
223:
222:
221:
218:
216:
198:
194:
189:
180:
168:
150:
146:
138:
102:
72:
70:
68:
64:
60:
56:
52:
44:
42:
40:
35:
34:or vortices.
33:
29:
25:
21:
5624:
5620:
5591:
5587:
5568:
5549:
5524:
5520:
5475:
5471:
5452:
5415:
5411:
5380:
5374:
5337:
5333:
5323:
5298:
5294:
5250:
5246:
5233:
5212:
5187:
5181:
5136:
5132:
5122:
5103:
5097:
5054:
5050:
5043:
4982:
4978:
4968:
4957:. Retrieved
4946:. Springer.
4942:
4909:. Retrieved
4903:
4858:
4854:
4848:
4821:
4785:
4781:
4775:
4742:
4738:
4732:
4699:
4695:
4689:
4664:
4660:
4638:
4634:
4628:
4595:
4591:
4585:
4552:
4548:
4515:(1–4): 1–9.
4512:
4508:
4502:
4483:
4418:
4411:
4403:
4399:
4396:Applications
4210:
4012:
4010:
3963:
3834:Gilbert form
3833:
3823:
3814:is constant.
3808:
3610:
3605:
3598:
3594:
3590:
3586:
3584:
3475:
3471:
3463:
3459:
3457:
3335:
3331:
3327:
3325:
3315:
3304:
3292:
2649:small-strain
2623:
2611:
2434:
2305:
2206:
2135:
2131:
2115:
2081:
2048:
1674:
1547:
1498:
1400:
1320:
1226:
1204:
1141:
1054:
1040:
1015:
1011:
1004:
1002:
872:
867:
811:
728:
705:
597:
575:
420:
412:
326:
219:
214:
166:
76:
66:
48:
36:
32:domain walls
19:
18:
5521:SIAM Review
5276:10261/25225
5224:0712.3684v1
4661:SIAM Review
3468:variational
866:are called
5690:Categories
5425:1810.12365
5418:(6): 120.
5146:1810.12365
5139:(6): 120.
5064:1803.11174
4992:1702.07573
4985:(1): 308.
4959:2024-06-23
4911:2024-06-14
4641:: 153–166.
51:Lev Landau
39:equilibria
5500:0018-9464
5173:1434-6028
5089:1367-2630
5017:2041-1723
4883:0021-8979
4840:779865001
4802:0031-899X
4767:0031-899X
4724:0031-899X
4681:0036-1445
4635:Sov. Phys
4620:1063-777X
4577:0021-8979
4430:Magnetism
4368:∂
4358:∂
4352:⋅
4239:α
4219:α
4178:×
4167:×
4150:α
4133:γ
4125:α
4119:−
4096:×
4079:α
4062:γ
4051:−
4039:∂
4029:∂
3996:α
3972:γ
3943:∂
3933:∂
3927:×
3919:α
3893:×
3880:γ
3872:−
3860:∂
3850:∂
3744:∂
3729:∂
3704:μ
3694:−
3680:∇
3657:μ
3549:⋅
3524:∫
3504:μ
3500:−
3383:μ
3373:−
3265:ε
3241:−
3227:⊗
3215:ε
3191:μ
3185:−
3172:ε
3156:μ
3146:ε
3120:λ
3109:∫
3053:μ
3033:λ
3008:μ
2994:⊗
2986:λ
2937:ε
2932:−
2928:ε
2889:ε
2884:−
2880:ε
2867:∫
2816:ε
2811:−
2807:ε
2793:ε
2765:λ
2724:−
2716:⊗
2696:λ
2661:ε
2586:^
2577:×
2568:∂
2558:∂
2552:−
2546:^
2537:×
2528:∂
2518:∂
2507:⋅
2490:∫
2409:⋅
2406:∇
2393:−
2380:∇
2377:⋅
2357:∫
2317:−
2280:×
2277:∇
2271:⋅
2254:∫
2185:×
2177:∇
2157:∫
2097:⋅
2063:⋅
2060:∇
2057:−
2000:−
1964:⋅
1948:∂
1944:∫
1911:−
1875:⋅
1868:∇
1855:∫
1851:−
1840:π
1787:∞
1784:→
1753:∇
1655:⋅
1636:∂
1621:∂
1615:−
1604:∂
1589:∂
1511:∇
1479:⋅
1476:∇
1454:∇
1427:∇
1424:−
1368:×
1365:∇
1286:⋅
1272:∫
1257:μ
1251:−
1180:μ
1107:⋅
1093:∫
1083:μ
1079:−
960:−
868:easy axes
762:∫
528:∇
499:∇
470:∇
453:∫
5653:27831887
5508:35628797
5366:27831887
5035:28827700
4424:See also
2009:′
1981:′
1920:′
1892:′
1871:′
5644:5750049
5596:Bibcode
5529:Bibcode
5480:Bibcode
5430:Bibcode
5357:5750049
5303:Bibcode
5255:Bibcode
5192:Bibcode
5151:Bibcode
5069:Bibcode
5026:5566362
4997:Bibcode
4863:Bibcode
4747:Bibcode
4704:Bibcode
4600:Bibcode
4557:Bibcode
4517:Bibcode
3830:damping
3458:where d
1350:is the
1198:is the
596:is the
45:History
24:physics
5651:
5641:
5575:
5556:
5506:
5498:
5459:
5364:
5354:
5171:
5110:
5087:
5033:
5023:
5015:
4950:
4881:
4838:
4828:
4800:
4765:
4722:
4679:
4618:
4575:
4490:
4406:vortex
4211:where
3964:where
3585:Since
2970:where
2207:where
2084:, and
1803:) is:
1675:where
1321:where
1142:where
1038:axis.
1003:where
812:where
723:, and
576:where
5504:S2CID
5420:arXiv
5243:(PDF)
5219:arXiv
5141:arXiv
5059:arXiv
4987:arXiv
4289:, as
1243:demag
281:demag
5649:PMID
5573:ISBN
5554:ISBN
5496:ISSN
5457:ISBN
5362:PMID
5169:ISSN
5108:ISBN
5085:ISSN
5031:PMID
5013:ISSN
4948:ISBN
4879:ISSN
4836:OCLC
4826:ISBN
4798:ISSN
4763:ISSN
4720:ISSN
4677:ISSN
4616:ISSN
4573:ISSN
4488:ISBN
3988:and
3737:anis
3336:real
3332:felt
3326:The
3045:and
1730:and
941:anis
889:anis
852:anis
825:anis
776:anis
753:anis
733:and
654:and
441:exch
255:anis
242:exch
53:and
5639:PMC
5629:doi
5604:doi
5537:doi
5488:doi
5438:doi
5385:doi
5352:PMC
5342:doi
5311:doi
5271:hdl
5263:doi
5200:doi
5159:doi
5077:doi
5021:PMC
5005:doi
4871:doi
4790:doi
4755:doi
4712:doi
4669:doi
4608:doi
4565:doi
4525:doi
4513:306
4188:eff
3812:eff
3602:eff
3559:eff
3100:m-e
2848:m-e
2481:DMI
2348:DMI
2245:DMI
2148:DMI
1629:out
1564:out
1525:out
372:or
307:m-e
294:DMI
5692::
5647:.
5637:.
5625:25
5623:.
5619:.
5602:.
5590:.
5535:.
5525:48
5523:.
5519:.
5502:.
5494:.
5486:.
5476:40
5474:.
5436:.
5428:.
5416:92
5414:.
5410:.
5360:.
5350:.
5338:25
5336:.
5332:.
5309:.
5299:39
5297:.
5285:^
5269:.
5261:.
5251:67
5249:.
5245:.
5198:.
5167:.
5157:.
5149:.
5137:92
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5131:.
5083:.
5075:.
5067:.
5055:20
5053:.
5029:.
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5003:.
4995:.
4981:.
4977:.
4920:^
4891:^
4877:.
4869:.
4859:29
4857:.
4834:.
4810:^
4796:.
4786:53
4784:.
4761:.
4753:.
4743:60
4741:.
4718:.
4710:.
4700:58
4698:.
4675:.
4665:48
4663:.
4647:^
4637:.
4614:.
4606:.
4596:41
4594:.
4571:.
4563:.
4553:49
4551:.
4537:^
4523:.
4511:.
4448:^
4416:.
4380:0.
3462:/d
3297:.
3256:tr
3206:tr
3162:tr
3131:tr
2983::=
2803::=
2616:.
1597:in
1577:in
1468:in
1202:.
870:.
719:,
627:,
600:;
410:.
217:.
5682:.
5676:.
5670:.
5655:.
5631::
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5543:.
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5368:.
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5313::
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5091:.
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4989::
4983:8
4962:.
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4873::
4865::
4842:.
4804:.
4792::
4769:.
4757::
4749::
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4683:.
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4579:.
4567::
4559::
4531:.
4527::
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4496:.
4377:=
4371:t
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4348:m
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4341:=
4336:2
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4325:m
4320:|
4313:t
4309:d
4303:d
4277:1
4274:=
4270:|
4265:m
4260:|
4196:,
4193:)
4183:H
4174:m
4170:(
4163:m
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4146:+
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4083:2
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4058:|
4048:=
4042:t
4033:m
3946:t
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3916:+
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3869:=
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3809:H
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3733:F
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3708:0
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3684:2
3671:s
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3651:A
3648:2
3642:=
3636:f
3633:f
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3606:m
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3591:m
3587:m
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3542:m
3537:d
3533:(
3528:V
3518:s
3514:M
3508:0
3497:=
3494:E
3490:d
3476:E
3472:m
3464:V
3460:E
3440:V
3436:d
3431:m
3426:d
3420:E
3415:2
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3370:=
3364:f
3361:f
3358:e
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3279:.
3274:}
3269:]
3261:[
3249:3
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3238:]
3235:)
3231:m
3223:m
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3211:[
3199:{
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3188:3
3182:]
3177:2
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3153:+
3150:]
3142:[
3137:2
3123:2
3113:V
3105:=
3096:E
3074:m
3012:I
3005:2
3002:+
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2990:1
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2958:]
2955:)
2951:m
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2913::
2910:]
2907:)
2903:m
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2712:m
2707:[
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2675:m
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2494:V
2486:=
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2413:m
2401:z
2397:m
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2373:m
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2284:m
2274:(
2267:m
2263:D
2258:V
2250:=
2241:E
2216:D
2193:)
2189:m
2181:m
2174:(
2171::
2167:D
2161:V
2153:=
2144:E
2101:n
2093:M
2067:M
2034:.
2030:)
2026:S
2022:d
2014:|
2005:r
1996:r
1991:|
1985:)
1977:r
1972:(
1968:M
1960:n
1951:V
1940:+
1937:V
1933:d
1925:|
1916:r
1907:r
1902:|
1896:)
1888:r
1883:(
1879:M
1859:V
1847:(
1837:4
1833:1
1828:=
1825:)
1821:r
1817:(
1814:U
1781:r
1760:|
1756:U
1748:2
1744:r
1739:|
1717:|
1713:U
1710:r
1706:|
1684:n
1659:n
1651:M
1647:=
1640:n
1625:U
1608:n
1593:U
1582:,
1573:U
1569:=
1560:U
1533:0
1530:=
1521:U
1515:2
1483:M
1473:=
1464:U
1458:2
1430:U
1421:=
1416:d
1411:H
1386:0
1383:=
1378:d
1373:H
1336:d
1331:H
1306:V
1302:d
1296:d
1291:H
1282:M
1276:V
1266:2
1261:0
1248:=
1239:E
1184:0
1157:a
1152:H
1127:V
1123:d
1117:a
1112:H
1103:M
1097:V
1087:0
1076:=
1071:Z
1067:E
1026:z
1007:1
1005:K
988:,
983:2
978:z
974:m
968:1
964:K
957:=
954:)
950:m
946:(
937:F
912:m
885:F
848:F
821:F
797:V
793:d
789:)
785:m
781:(
772:F
766:V
758:=
749:E
690:m
667:z
663:m
640:y
636:m
613:x
609:m
584:A
561:V
557:d
552:)
546:2
542:)
536:z
532:m
525:(
522:+
517:2
513:)
507:y
503:m
496:(
493:+
488:2
484:)
478:x
474:m
467:(
463:(
457:V
449:A
446:=
437:E
398:1
395:=
391:|
386:m
381:|
358:s
354:M
350:=
346:|
341:M
336:|
312:,
303:E
299:+
290:E
286:+
277:E
273:+
268:Z
264:E
260:+
251:E
247:+
238:E
234:=
231:E
199:s
195:M
190:/
185:M
181:=
177:m
151:s
147:M
122:|
117:M
112:|
86:M
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