Riemannian-Geometrical Interpretation of Quantum Motion of a Particle in a Dislocated Crystal

Recently, Kawamura developed a theory of scattering of an electron in a crystal due to the presence of dislocations. ' ) One of the essential features of dislocated crystals is the lack of global translational symmetry although the local features of such crystals are not different from those of perfect crystals. The consequence of this feature is that locally the quantum propagation of an electron is in the form of a plane wave; namely if the phase of the wave is once determined at a lattice point, then it is also determined uniquely at other lattice points in the neighborhood. However, in a global sense, the phase of the wave is not uniquely determined in dislocated crystals. It depends on the path along which the wave propagates. Thus the mismatching of phases of waves, which have propagated along different paths, takes place at a lattice point where the waves meet each other. Detailed description of such an interesting feature is given in Kawamura's paper in the case of screw dislocations. ' ) In this paper, we discuss this problem in a general framework by using the continuum description of deformed crystals. Suppose a particle moves in a deformed crystal, whose deformation is described in terms of a strain field ;3o,(x). Namely, when two points at x and x + dx in a perfect crystal are displaced by amounts u and u + tiu respectively with du i = PJ' (x) dxj, we call the resulting crystal a deformed crystal with a strain field ;30i( x )2) Suppose a particle moves over a small distance dx in a deformed crystal. Then the particle is considered to move over a distance dy, whose components are given by