Quantization of the Frank-Bilby equation for misfit dislocation arrays in interfaces

A quantization procedure for the Frank-Bilby equation to calculate the periodic structure of interphase interfaces and grain boundaries with one, two and three sets of discrete misfit dislocation arrays is presented mainly in terms of the basic O-lattice equation. The line directions and spacings of misfit dislocation arrays are derived to account for the full relaxation of the long-range strains in the interface without the ad hoc assumption proposed by Bollmann who assumed that misfit dislocation arrays are the lines of intersection between the O-cell walls and the interface. It has been found that the discrete dislocation arrangement derived from the above Bollmann's assumption cannot always ensure that the interface is free of long-range strains.