  The properties of crystals consisting of several components can be widely
tuned. Often solid solutions are produced, where substitutional or
interstitional disorder determines the crystal thermodynamic and mechanical
properties. The chemical and structural disorder impedes the study of the
elasticity of such solid solutions, since standard procedures like potential
expansions cannot be applied. We present a generalization of a
density-functional based approach recently developed for one-component crystals
to multi-component crystals. It yields expressions for the elastic constants
valid in solid solutions with arbitrary amounts of point defects and up to the
melting temperature. Further, both acoustic and optical phonon eigenfrequencies
can be computed in linear response from the equilibrium particle densities and
established classical density functionals. As a proof of principle, dispersion
relations are computed for two different binary crystals: A random fcc crystal
as an example for a substitutional, and a disordered sodium chloride structure
as an example of an interstitial solid solution. In cases where one of the
components couples only weakly to the others, the dispersion relations develop
characteristic signatures. The acoustic branches become flat in much of the
first Brillouin zone, and a crossover between acoustic and optic branches takes
place at a wavelength which can far exceed the lattice spacing.*
