Frustration

Ferromagnetic interactions are satisfied by a parallel alignment of the atomic moments. Antiferromagnetic interactions may not be so easily appeased. In structures with odd-membered rings it is impossible to satisfy all the antiferromagnetic interactions simultaneously. A consequence is that T_N << |θ_p|.
Examples of crystal lattices where nearest-neighbour interactions are naturally frustrated include the triangular, kagom´e, fcc and tetrahedral lattices. The 16d sites of the spinel structure form a tetrahedral lattice; each tetrahedron has four triangular faces.

As an illustration of frustration, consider the lowest-energy configurations for three-, four- and five-membered rings in Fig. 1. The exchange is supposed to be of the form −2 S_i · S_j with either Ising spins, which have only one component (S = (0,±1)) or vector spins (S = (S_x, S_y ), with S²_x+ S²_y= 1).

Exchange in the odd-membered rings is frustrated, which means that the total exchange energy divided by the number of bonds is less than . Besides a low ordering temperature, the hallmarks of frustration are increased degeneracy of the ground state and a tendency to formnoncollinear spin structures, epitomized by the three- and five-membered rings in Fig. 1(b).

Figure 1: Lowest-energy configurations for (a) Ising and (b) classical 2D vector spins forming three-, fouror five-membered rings. Energy per bond in units of J and the degree of degeneracy for Ising spins are shown.