On the extension of Frank's formula to crystals with different lattice parameters

The concepts involved in Frank's [1] formula and the treatment of surface dislocations by Bilby et al. [4] are combined to develop a vector equation for the closure failure associated with a boundary between crystals with different lattice parameters. The relation between the extended Frank's formula and Bollmann's work [13 to 15], coincidence boundaries, and moire fringes is discussed. A tilt-misfit and a twist-misfit phase boundary are treated. The range of applicability of the equations is limited to low-angle phase boundaries including low-angle grain boundaries.