The Propagation of Order in Crystal Lattices

Zernike's problem of the propagation of order in a binary crystal alloy is discussed by means of the matrix formalism recently developed for treating cooperative phenomena. It is proved generally that the existence of long range order over any temperature range implies a degeneracy of the maximum characteristic value of the fundamental matrix over the same temperature range. For a two-dimensional crystal, the special form of the matrix theory developed by Kramers and Wannier has been used to obtain explicitly the probabilities for finding an $A$ or $B$ atom at any lattice site if it is known that there is an $A$ (or $B$) atom at a certain site (correlation probabilities, or intermediate range order). The results are obtained in the form of power series valid at low temperatures by a perturbation treatment of the highest characteristic values and characteristic vectors. It is seen that the maximum characteristic value is doubly degenerate from $T=0\mathrm{}$ to some finite temperature, consistent with the existence of long range order at low temperature. The absence of long range order at high temperatures is proved by showing that the maximum characteristic value is non-degenerate at sufficiently high temperatures. By comparing the solutions of Zernike's approximate equations for the correlation probabilities in two dimensions with our exact solution, and his expressions for the long range order and energy (short range order) in three dimensions with those given by van der Waerden, we find that Zernike's approximation is better for a two- than a three-dimensional crystal. The problem of propagation of order is generalized by an investigation of the correlation probabilities for more complicated configurations of fixed atoms. For example, the ordering influence of an adjacent pair of disordered atoms ("dipole") is found as a function of position in the lattice. It is found here, as might be expected, that the ordering influence falls off rapidly with the distance of the site from the dipole. Other cases are treated as well.