Magnetism and local molecular field.
Despite its somewhat naive simplicity, the method of the local molecular field has had undeniable success in satisfactorily explaining a large number of previously known facts and in opening the way for the discovery of new facts. Let us note, however, that all the structures that have been discussed above are collinear structures: on the average (in time) all the atomic magnets pointing in one or the opposite direction are parallel to a single direction. However, the local molecular field method can also be extended to noncollinear structures such as that of helimagnetism, which Yoshimori and Villain discovered independently in an absolutely unexpected manner; one can thus interpret phenomena in a remarkably simple and concrete manner. Nevertheless, the method can hardly be recommended for more complex structures such as the umbrella structure, which requires the decomposition of the principal crystal lattice into a large number of sublattices. Indeed, under these conditions an atom belonging to a given sublattice has only a very small number of neighbors (one or two) in each of the other sublattices, and the molecular field method, which consists in replacing the instantaneous action of an atom by that of an average atom, will be more likely to yield a correct result, the larger the number of atoms to which it is applied. Its correctness probably also increases as the atomic spin becomes larger. Independently of this problem, the method applied to a large number of sublattices completely loses its chief advantage, simplicity. The method also involves more insidious traps. If a judicious choice of parameters is made, the method can lead one to calculate curves and thermal variations of the spontaneous magnetization or paramagnetic susceptibility that coincide remarkably well with the experimental results, for example, to within a few thousandths. Under these conditions, one could expect that the elementary interaction energies deduced from these parameters would correspond to the actual values with the same accuracy. This is not so; errors of 10 to 20 percent and even greater are frequently made in this manner. A certain amount of caution thus becomes imperative. On the other hand, recourse to the local molecular field seems indispensable since more rigorous methods lead to insurmountable complications. Consider for example that the rigorous solution is not yet known for the simplest case, that of a simple cubic lattice with identical atoms of spin 1/2, and interactions reduced to those present between nearest-neighbor atoms. How then should one treat the case of garnets with 160 atoms in the unit cell, spins up to 5/2, and at least six different coupling constants? One must therefore be lenient toward the imperfections of the molecular field methods, considering the simplicity with which the successes recalled in the first few lines of these conclusions were obtained.
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