The periodic unit of doubly-diffracted reflections from periodic grain boundaries in cubic crystals and its relationship with coincident site lattice

A geometrical formulation of how the double diffraction scattering from constituting crystals of a periodic grain boundary in cubic crystal yields a doubly-diffracted reflection unit within the composite diffraction pattern of the boundary is provided. The doubly-diffracted reflection unit formed represents a primitive displacement shift complete unit of zero-layer reciprocal lattices (DSCZLRL) of the coincident site unit of zero-layer reciprocal lattices (CSZLRL). Its reciprocal transformation yields a real space unit of coincident site lattices that includes a coincident site of zero-layer lattice (CSZLL) of the periodic boundary as its basic unit. The formulation is applied in a case study on the determination of the coincident site lattice (CSL) from the double diffraction electron pattern of a Sigma 3 [101]/(-111) grain boundary of aluminium.