Short- and Long-Range Order Parameters in Disordered Solid Solutions

For a crystalline solid made up of a partially ordered array of structural units of more than one type it is shown that the state of order may be described and defined in terms of the self-convolution, or Patterson, function of the structure. The peak heights of the Patterson function are proportional to the short-range order parameters, ${\alpha }_i$. The periodic component is described by the long-range order parameters, defined as the limiting values of the ${\alpha }_i$ at large distances from the origin. The limitations of previous definitions of long-range order parameters are discussed.

The interpretation of the Patterson function as a vector distribution function forms a basis for the calculation of configurational free energy and hence the derivation of equations giving the order parameters as functions of temperature. Some numerical results are obtained for short- and long-range order parameters for the alloys ${\mathrm{Cu}}_3$Au and $\beta$-CuZn.

For ${\mathrm{Cu}}_3$Au, and for most other binary alloys, it is shown that two long-range order parameters, $s_1$ and $s_2$, are required to specify the state of long-range order, rather than the single parameter previously used. The fact that these two parameters are not simply related is interpreted as evidence for nonrandom fluctuations in the composition of the alloy, probably associated with the presence of out-of-phase domain boundaries which may be rich in one or other of the component types of atoms. It is suggested that a regular superlattice of out-of-phase domains may be present under equilibrium conditions for most alloys.