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The influence of frozen-in topological defects in a crystal on the long-wavelength quantum states of a particle is considered. In the continuum limit of a conveniently defined tight-binding model one is led to a covariant Schrödinger equation on a Riemann-Cartan manifold. When the tight-binding transfer energies are assumed to depend on the local lattice deformations caused by the defects, additional noncovariant terms are generated in the Hamiltonian. These terms generate bound states of the particle to edge dislocations and enhance the scattering of particles on screw dislocations. [S0031-9007(98)05432-5]

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Starting from a phase-field description of the isothermal solidification of a dilute binary alloy, we establish a model where capillary waves of the solidification front interact with the diffusive concentration field of the solute. The model does not rely on the sharp-interface assumption and includes nonequilibrium effects, relevant in the rapid-growth regime. In many applications it can be evaluated analytically, culminating in the appearance of an instability that, interfering with the Mullins-Sekerka instability, is similar to that found by Cahn in grain-boundary motion.