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Before introducing a model for the binary alloy, let us describe some of the salient features of an actual binary alloy, ,a-brass, which is a body-centered cubic lattice made up of Zn and Cu atoms. A unit cell of this lattice in its completel: ordered state, which exists only at absolute zero, is shown in Fig. 14.3, where an open circle indicates a Zn atom and a solid circle a Cu atom. As the temperature is increased some Zn atoms will exchange positions with Cu atoms, but the probability of finding a Zn atom in the "right" place is greater than ~. Above the critical temperature of 742 K, however, the Zn and Cu atoms are thoroughl: mixed, and the probability of finding a Zn atom in the "right" place becomes exactly !. This transition can be discovered experimentally through the Bragg

reflection of X rays from the crystal. In the ordered state X-ray reflection will reveal that there are two sets of atomic planes with spacing d, whereas in the state of disorder there is only one set of atomic planes with spacing d/2. It is observed experimentally that the specific heat Cp approaches infinity as the temperature approaches the critical temperature from both sides. A model for a binary alloy follows. Let there be two kinds of atoms, called 1 and 2, of which there are N I and N z, respectively. Let their positions be confined to the lattice sites of a given lattice, with y nearest neighbors to each lattice site. At each lattice site there shall be one and only one atom. Thus the total number of sites is N = N I + N z. There are three types of nearest-neighbor pairs: (11), (22), and (12). The pair (12) is not distinguished from the pair (21). Let a configuration of the system be such that the number of pairs of each type present is, respectively, NIl> N zz , N 12 . Neglecting the kinetic energy of the atoms and all but nearest-neighbor interactions, we take the energy of the system to be EA(Nn , N zz , N 12 ) = INn + zNn + 12N12 (14.20)

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where the subscript A stands for "alloy." Obviously E A is in general degenerate. Moreover, the numbers N n , N zz , N 12 are not independent of one another. By analogy with (14.7) we have the relations yNI

FIGURE 6.24 To compare the current chart to another security, select Comparison Chart from the Chart Settings menu.

(14.21)

(14.22)

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Hence the energy depends only on one variable, N ll : EA(Nn )

+ ty zN]

(14.23)

where the term in brackets is a constant. A correspondence between the binary alloy and the lattice gas may be established by identifying N I with N a , which in turn is identified with N + of the Ising model. Comparison of (14.23) and the exponent in (14.19) immediately leads to the correspondence summarized in the accompanying table:

N1 N - N 1 = Nz 1 + z - 2 12

(Helmholtz free energy)

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