Some geometrical relations in dislocated crystals

When a single crystal deforms by glide which is unevenly distributed over the glide surfaces the lattice becomes curved. The constant feature of distortion by glide on a single set of planes is that the orthogonal trajectories of the deformed glide planes (the c-axes in hexagonal metals) are straight lines. This leads to the conclusion that in polygonisation experiments on single hexagonal metal crystals the polygon walls are planes, while the glide planes are deformed into cylinders whose sections are the involutes of a single curve. The analysis explains West's observation that the c-axes in bent crystals of corundum are straight lines. For double glide on two orthogonal sets of planes there is a complete analogy between the geometrical properties of the distorted glide planes and those of the “slip-lines” in the mathematical theory of plasticity. More general cases are discussed and formulae are derived connecting the density of dislocations with the lattice curvatures. For a three-dimensional network of dislocations the “state of dislocation” of a region is shown to be specified by a second-rank tensor, which has properties like those of a stress tensor except that it is not symmetrical.