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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Heisenberg ferromagnet ： ウィキペディア英語版 Spin wave Spin waves are propagating disturbances in the ordering of magnetic materials. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known as magnons, which are boson modes of the spin lattice that correspond roughly to the phonon excitations of the nuclear lattice. As temperature is increased, the thermal excitation of spin waves reduces a ferromagnet's spontaneous magnetization. The energies of spin waves are typically only in keeping with typical Curie points at room temperature and below. The discussion of spin waves in antiferromagnets is beyond the scope of this article. ==Theory== The simplest way of understanding spin waves is to consider the Hamiltonian $\backslash mathcal$ for the Heisenberg ferromagnet: :$\backslash mathcal\; =\; -\backslash frac\; J\; \backslash sum\_\; \backslash mathbf\_i\; \backslash cdot\; \backslash mathbf\_j\; -\; g\; \backslash mu\_B\; \backslash sum\_i\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\_i$ where is the exchange energy, the operators represent the spins at Bravais lattice points, is the , is the Bohr magneton and is the internal field which includes the external field plus any "molecular" field. Note that in the classical continuum case and in dimensions Heisenberg ferromagnet equation has the form :$\backslash mathbf\_t=\backslash mathbf\; \backslash times\; \backslash mathbf\_.$ In and dimensions this equation admits several integrable and non-integrable extensions like the Landau-Lifshitz equation, the Ishimori equation and so on. For a ferromagnet and the ground state of the Hamiltonian $|0\backslash rangle$ is that in which all spins are aligned parallel with the field . That $|0\backslash rangle$ is an eigenstate of $\backslash mathcal$ can be verified by rewriting it in terms of the spin-raising and spin-lowering operators given by: :$S^\backslash pm\; =\; S^x\; \backslash pm\; i\; S^y$ resulting in :$\backslash mathcal\; =\; -\backslash frac\; J\; \backslash sum\_\; S^z\_i\; S^z\_j\; -\; g\; \backslash mu\_B\; H\; \backslash sum\_i\; S^z\_i\; -\; \backslash frac\; J\; \backslash sum\_\; S^-\_i\; S^+\_j$ where has been taken as the direction of the magnetic field. The spin-lowering operator annihilates the state with minimum projection of spin along the -axis, while the spin-raising operator annihilates the ground state with maximum spin projection along the -axis. Since :$S^z\_i|0\backslash rangle\; =\; s|0\backslash rangle$ for the maximally aligned state, we find :$\backslash mathcal\; |0\backslash rangle\; =\; \backslash left\; (-Js^2\; -g\; \backslash mu\_B\; H\; s\; \backslash right\; )N|0\backslash rangle$ where N is the total number of Bravais lattice sites. The proposition that the ground state is an eigenstate of the Hamiltonian is confirmed. One might guess that the first excited state of the Hamiltonian has one randomly selected spin at position rotated so that :$S^z\_i\; |1\backslash rangle\; =\; (s-1)|1\backslash rangle,$ but in fact this arrangement of spins is not an eigenstate. The reason is that such a state is transformed by the spin raising and lowering operators. The operator $S^+\_i$ will increase the -projection of the spin at position back to its low-energy orientation, but the operator $S^\_j$ will lower the -projection of the spin at position . The combined effect of the two operators is therefore to propagate the rotated spin to a new position, which is a hint that the correct eigenstate is a spin wave, namely a superposition of states with one reduced spin. The exchange energy penalty associated with changing the orientation of one spin is reduced by spreading the disturbance over a long wavelength. The degree of misorientation of any two near-neighbor spins is thereby minimized. From this explanation one can see why the Ising model magnet with discrete symmetry has no spin waves: the notion of spreading a disturbance in the spin lattice over a long wavelength makes no sense when spins have only two possible orientations. The existence of low-energy excitations is related to the fact that in the absence of an external field, the spin system has an infinite number of degenerate ground states with infinitesimally different spin orientations. The existence of these ground states can be seen from the fact that the state $|0\backslash rangle$ does not have the full rotational symmetry of the Hamiltonian $\backslash mathcal$, a phenomenon which is called spontaneous symmetry breaking. In this model the magnetization :$M\; =\; \backslash frac$ where is the volume. The propagation of spin waves is described by the Landau-Lifshitz equation of motion: :$\backslash frac\; =\; -\backslash gamma\backslash mathbf\; M\; \backslash times\backslash mathbf\; H\; -\; \backslash frac$ where is the gyromagnetic ratio and is the damping constant. The cross-products in this forbidding-looking equation show that the propagation of spin waves is governed by the torques generated by internal and external fields. (An equivalent form is the Landau-Lifshitz-Gilbert equation, which replaces the final term by a more "simply looking" equivalent one.) The first term on the r.h.s. describes the precession of the magnetization under the influence of the applied field, while the above-mentioned final term describes how the magnetization vector "spirals in" towards the field direction as time progresses. In metals the damping forces described by the constant are in many cases dominated by the eddy currents. One important difference between phonons and magnons lies in their dispersion relations. The dispersion relation for phonons is to first order linear in wavevector , namely , where is frequency, and is the velocity of sound. Magnons have a parabolic dispersion relation: where the parameter represents a "spin stiffness." The form is the third term of a Taylor expansion of a cosine term in the energy expression originating from the dot-product. The underlying reason for the difference in dispersion relation is that ferromagnets violate time-reversal symmetry. Two adjacent spins in a solid with lattice constant that participate in a mode with wavevector have an angle between them equal to . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Spin wave」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.