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In condensed matter physics, the term geometrical frustration (or in short: frustration〔The psychological side of this problem is treated in a different article, frustration〕) refers to a phenomenon, where atoms tend to stick to non-trivial positions or where, on a regular crystal lattice, conflicting inter-atomic forces (each one favoring rather simple, but different structures) lead to quite complex structures. As a consequence of the frustration in the geometry or in the forces, a plenitude of distinct ground states may result at zero temperature, and usual thermal ordering may be suppressed at higher temperatures. Much studied examples are amorphous materials, glasses, or dilute magnets. The term ''frustration'', in the context of magnetic systems, has been introduced by Gerard Toulouse (1977).〔 G. Toulouse, ''Commun. Phys.''2, 115 (1977) 115〕 Indeed, frustrated magnetic systems had been studied even before. Early work includes a study of the Ising model on a triangular lattice with nearest-neighbor spins coupled antiferromagnetically, by G. H. Wannier, published in 1950. Related features occur in magnets with ''competing interactions'', where both ferro- as well as antiferromagnetic couplings between pairs of spins or magnetic moments are present, with the type of interaction depending on the separation distance of the spins. In that case commensurability, such as helical spin arrangements may result, as had been discussed originally, especially, by A. Yoshimori, T. A. Kaplan, R. J. Elliott, and others, starting in 1959, to describe experimental findings on rare-earth metals. A renewed interest in such spin systems with frustrated or competing interactions arose about two decades later, beginning in the 70s of the 20th century, in the context of spin glasses and spatially modulated magnetic superstructures. In spin glasses, frustration is augmented by stochastic disorder in the interactions, as may occur, experimentally, in non-stoichiometric magnetic alloys. Carefully analyzed spin models with frustration include the Sherrington-Kirkpatrick model, describing spin glasses, and the ANNNI model, describing commensurability magnetic superstructures. ==Magnetic ordering== Geometrical frustration is an important feature in magnetism, where it stems from the relative arrangement of spins. A simple 2D example is shown in Figure 1. Three magnetic ions reside on the corners of a triangle with antiferromagnetic interactions between them; the energy is minimized when each spin is aligned opposite to neighbors. Once the first two spins align anti-parallel, the third one is ''frustrated'' because its two possible orientations, up and down, give the same energy. The third spin cannot simultaneously minimize its interactions with both of the other two. Since this effect occurs for each spin, the ground state is sixfold degenerate. Only the two states where all spins are up or down have more energy. Similarly in three dimensions, four spins arranged in a tetrahedron (Figure 2) may experience geometric frustration. If there is an antiferromagnetic interaction between spins, then it is not possible to arrange the spins so that all interactions between spins are antiparallel. There are six nearest-neighbor interactions, four of which are antiparallel and thus favourable, but two of which (between 1 and 2, and between 3 and 4) are unfavourable. It is impossible to have all interactions favourable, and the system is frustrated. Geometrical frustration is also possible if the spins are arranged in a non-collinear way. If we consider a tetrahedron with a spin on each vertex pointing along the ''easy axis'' (that is, directly towards or away from the centre of the tetrahedron), then it is possible to arrange the four spins so that there is no net spin (Figure 3). This is exactly equivalent to having an antiferromagnetic interaction between each pair of spins, so in this case there is no geometrical frustration. With these axes, geometric frustration arises if there is a ferromagnetic interaction between neighbours, where energy is minimized by parallel spins. The best possible arrangement is shown in Figure 4, with two spins pointing towards the centre and two pointing away. The net magnetic moment points upwards, maximising ferromagnetic interactions in this direction, but left and right vectors cancel out (i.e. are antiferromagnetically aligned), as do forwards and backwards. There are three different equivalent arrangements with two spins out and two in, so the ground state is three-fold degenerate. 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「Geometrical frustration」の詳細全文を読む スポンサード リンク
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