Alloy order parameter

In eq. (1) the ETT order parameter z = s(p-Pc) measures, in a convenient direction, the chemical potential from that corresponding to the ETT. From the values given in Table I for the above s and q, we readily see that the occurrence of the ETTs discussed in this paper always implies an increase of the alloy free energy. Thus, CuPt random alloys, that just below and above the equiatomic concentration present both the relevant ETT s, are less stable than CuPd or AgPd and, thus more likely to be destabilised. Moreover, the proximity to both the critical concentrations implies large contributions to the BSE from the X and L points. Now, the concentration wave susceptibility, Xcc(q). as observed by Gyorffy and Stocks, is proportional to... 

The order-disorder transition of a binary alloy (e.g. CuZn) provides another instructive example. The body-centred lattice of this material may be described as two interpenetrating lattices, A and B. In the disordered high-temperature phase each of the sub-lattices is equally populated by Zn and Cu atoms, in that each lattice point is equally likely to be occupied by either a Zn or a Cu atom. At zero temperature each of the sub-lattices is entirely occupied by either Zn or Cu atoms. In terms of fractional occupation numbers for A sites, an appropriate order parameter may be defined as... 

Naturally, the fixed composition phase transformations treated in this section can be accompanied by local fluctuations in the composition field. Because of the similarity of Fig. 17.3 to a binary eutectic phase diagram, it is apparent that composition plays a similar role to other order parameters, such as molar volume. Before treating the composition order parameter explicitly for a binary alloy, a preliminary distinction between types of order parameters can be obtained. Order parameters such as composition and molar volume are derived from extensive variables any kinetic equations that apply for them must account for any conservation principles that apply to the extensive variable. Order parameters such as the atomic displacement 77 in a piezoelectric transition, or spin in a magnetic transition, are not subject to any conservation principles. Fundamental differences between conserved and nonconserved order parameters are treated in Sections 17.2 and 18.3. 

For a binary A-B alloy, another independent parameter, Xb (or = 1 — Xg) must be added to the fixed-stoichiometry order parameters in the preceding section. The phenomenological form of the Landau expansion, Eq. 17.2, can be extended to include Xb and has been used to catalog the conditions for many transitions in two-component systems [3]. 

Because composition is a locally-conserved order parameter, it cannot change in one location without affecting its neighborhood—fluxes are required to change a composition field. For example, in a binary alloy, the concentration field cb is re-... 

The Cahn-Hilliard equation applies to conserved order-parameter kinetics. For the binary A-B alloy treated in Section 18.1, the quantity in Eq. 18.22 is the change in homogeneous and gradient energy due to a change of the local concentration cB and is related to flux by... 

A more complicated but solvable problem is a definition of the order parameter for antiferromagnetics, binary alloys, superconductors etc. The dimensionless units T/Tc and 77/770 (Fig. 1.4) allow us to present the behaviour of the order parameter rj = r) T) in a form universal for many quite different systems. Moreover, in some cases even quantitative similarities hold which concerns in particular the value of the exponent (3. (The value of 77 = 0 characterizes always disordered phase.)... 

The range of coherence follows naturally from the BCS theory, and we see now why it becomes short in alloys. The electron mean free path is much shorter in an alloy than in a pure metal, and electron scattering tends to break up the correlated pairs, so dial for very short mean free paths one would expect die coherence length to become comparable to the mean free path. Then the ratio k i/f (called the Ginzburg-Landau order parameter) becomes greater than unity, and the observed magnetic properties of alloy superconductors can be derived. The two kinds of superconductors, namely those with k < 1/-/(2T and those with k > l/,/(2j (the inequalities follow from the detailed theory) are called respectively type I and type II superconductors. 

The ordering of atoms on the sites of AB3 alloys of DOi9 structure increases the solubility of interstitial impurity, if C atoms are distributed over the octahedral interstitial sites and ae parameter is negative (P < a). In the rest cases the atomic order prevents the dissolution of interstitial impurity. At the boundary values of order parameter ae = -1/3 and +1 the concentration of interstitial atoms in T positions reduces to zero. 

The order parameter is essentially a kinematic measure, describing the state of order within a system without any intrinsic reference to what factors drove the system to the state of interest. For example, in thinking about the transition between the ordered and disordered states of an alloy, it is useful to define an order parameter that measures the occupation probabilities on different sublattices. Above the order-disorder temperature, the sublattice occupations are random, while below the critical temperature, there is an enhanced probability of finding a particular species on a particular sublattice. The conventional example of this thinking is that provided by brass which is a mixture of Cu and Zn atoms in equal concentrations on a bcc lattice. The structure can be interpreted as two interpenetrating simple cubic lattices where it is understood that at high temperatures we are as likely to find a Cu atom on one sublattice as the other. A useful choice for the order parameter, which we denote by r], is... 

Fig. 12.4. Schematic illustration of conserved and nonconserved order parameters for characterizing the internal state of a binary alloy (adapted from Chen and Wang (1996)) (a) disordered phase (rj = 0) with uniform composition cq, (b) two-phase mixture consisting of disordered phases (rj = 0) with composition Cq, and c, (c) ordered single phase (ri = 1) of single composition ci with an antiphase domain boundary.

---