Crystal truncation rods from miscut surfaces with alternating terminations

A long-standing experimental challenge has been to identify the orientation of the {\alpha} and \{beta} terraces on basal plane surfaces of crystals with hexagonal close-packed and related structures. To demonstrate how surface X-ray scattering can be sensitive to such {\alpha} vs. \{beta} terminations, we develop a general theory for the intensity distributions along crystal truncation rods (CTRs) for miscut surfaces with a combination of two terminations. We consider fractional-unit-cell-height steps, and variation of the coverages of the terraces above each step. Example calculations are presented for the GaN (0001) surface with various reconstructions. These show which CTR positions are most sensitive to the fractional coverage of the two terminations. We compare the CTR profiles for exactly oriented surfaces to those for vicinal surfaces having a small miscut angle, and investigate the circumstances under which the CTR profile for an exactly oriented surface is equal to the sum of the intensities of the corresponding family of CTRs for a miscut surface.