the order parameter
How ordered is an ordering alloy — or how far along is any structural transition? We want a single dial that reads zero in the high-symmetry, disordered phase and grows to one (or some maximum) as the low-symmetry, ordered phase develops. That dial is the order parameter. It is the quantity that measures how much of the new phase's distinguishing feature has appeared — the chemical order, the atomic displacement, the strain, the polarisation — and it is the central variable of the whole modern theory of transitions.
For an ordering alloy the long-range order parameter is defined so that S = 1 when every atom is on its correct sublattice and S = 0 when occupation is completely random. One common form: S = (r minus x)/(1 minus x), where r is the fraction of a given sublattice actually occupied by its right species and x is that species' overall concentration. As the alloy cools below the transition, S climbs from 0 toward 1. For a displacive transition the order parameter is instead the size of the atomic shift (or the spontaneous polarisation, or strain); the idea is the same.
The order parameter is powerful because its behaviour near the transition tells you the KIND of transition. In a second-order (continuous) transition it rises smoothly from zero, typically like S proportional to (Tc minus T)^(1/2) just below the critical temperature Tc; in a first-order transition it jumps discontinuously from zero to a finite value. Landau theory is built entirely by expanding the free energy in powers of this one variable, which is why identifying the right order parameter is the first step in understanding any structural transition.
Define the long-range order parameter S for beta-brass: S = 0 when completely random, S = 1 when copper and zinc each sit on their correct sites. Measuring the superlattice-reflection intensity (proportional to S^2) at different temperatures traces S versus T: it rises smoothly from 0 at 454 degrees C toward 1 as the alloy cools — the hallmark of a second-order transition.
The order parameter: a single number that is 0 in the disordered phase and grows in the ordered one; how it varies with T reveals whether the transition is second- or first-order.
The order parameter is 0 in the high-symmetry (disordered) phase and grows in the low-symmetry (ordered) one; HOW it varies with temperature reveals the type of transition — continuous from zero (often like (Tc minus T)^(1/2)) for second-order, a jump to a finite value for first-order.