Escape from the second dimension: A topological distinction between edge and screw dislocations

Volterra's definition of dislocations in crystals geometrically distinguishes edge and screw defects according to whether the Burgers vector is perpendicular or parallel to the defect. A homotopy-theoretic analysis of dislocations as topological defects fails to differentiate edge and screw. Here we bridge the gap between the geometric and topological descriptions by demonstrating that there is a topological difference between screw and edge defects. Our construction distinguishes edge and screw based on the disclination-line pairs at the core of smectic dislocations. By exploiting the connection between topology and geometry in the form of Gaussian curvature, this analysis results in an invariant for dislocations in the saddle-splay vector. This construction can be generalized to crystals with triply periodic order.

Figure 1

Dislocations and the decomposition of their cores into disclination pairs, reproduced from Ref. [14]. (Left) Screw dislocation core composed of two helically winding $+\frac{1}{2}$ disclination lines. (Right) Edge dislocation comprised of a $\pm \frac{1}{2}$ disclination dipole. Both dislocations have Burgers scalar $+10$, and the insets show the infinite-compression phase structure of ${\mathrm{\Phi }}_{\text{screw}}$ and ${\mathrm{\Phi }}_{\text{screw}}$ before replacing their cores. Note the radial nature of the smectic layers entering the screw dislocation core, ensuring the geometry of a $+1$ disclination, compared to the disclination-charge-neutral edge dislocation.

Figure 2

The Gauss map on the helicoid for contours at varying distances from the core. (Top) Near the core the geometric texture of the director winds by $2\pi$ and maps the equator on the unit sphere. (Bottom) Far from the core the director and normals align along $\widehat{z}$ in accordance with boundary conditions. Connecting the two contours requires sweeping out the entire upper hemisphere under the Gauss map.