Abstract
The microscopic theory of elasticity was reconsidered in detail in an ordered alloy system following the line of reasoning pursued by the present author. This has made it possible to express more clearly the physical meaning of solute-lattice energy and to describe solute-lattice coupling parameters (CP) Ψi(nγ−n′γ′) in terms of lattice CP (atomic force constants) Φij(nγ−n′γ′) and the stress-free strain ηj1 either in Fourier space or in real space:
Ψi(γ;γ′⁄q)=(i⁄2)(a⁄2π)ηj1[∂Φij(γ;γ′⁄q)⁄∂q1], in Fourier space,
Ψi(nγ−n′γ′)=−1⁄2Φij(nγ−n′γ′)ηj1x1(nγ−n′γ′), in real space,
where Ψi(γ; γ′⁄q) and Φij(γ; γ′⁄q) are the Fourier representation of solute-lattice CP and lattice CP, respectively; the latter CP is identical to the element of the dynamical matrix of the system. In the Fourier space representation, the notation 2πq1⁄a represents the allowed reciprocal point along the l direction in the first Brillouin zone. In the real space representation, x1(nγ−n′γ′) indicates the inter-atomic distance along the l direction between the lattice points x1(nγ) and x1(n′γ′), where (nγ) indicates the γth lattice site in the nth unit cell in the crystal.
As the result, all CP, i.e., solute, solute-lattice and lattice CP, which are usually treated as parameters in the microscopic theory of elasticity, are correctly described in terms of the force constants or the dynamical matrix and stress-free strains in the system. The estimation of elastic free energy associated with continuous ordering or continuous clustering is thus attributed to obtain experimentally the force constants or the dynamical matrix in the system.
Quantitative calculations of the elastic free energy in β-brass are made on the basis of the theoretical argument developed using the force constants. The contour maps of the elastic free energy in k-space are very much like to those estimated previously, using the experimentally obtained solute-lattice CP, in the sense that the significant anisotropy and the wave vector dependence of the elastic free energy appear in the contour maps in the first Brillouin zone. A deep valley of negative values with the minimum point at 0.03, 0, 0 is exhibited in the contour map of the elastic free energy along the [100]* direction in the (001)* plane. This result assures the spinodal decomposition with the wave vector of λ=9 nm exhibited experimentally in β-brass. In contrast, the prohibition of secondary ordering from the B2 to D03 phase in β-brass was consistently explained by the large positive value of elastic free energy at 1/2 1/2 1/2 point in k-space.
