Interlacing of growth steps on crystal surfaces as a consequence of crystallographic symmetry

During crystal growth, concentric steps of unit-layer thickness [=d(hkl)u, with the surfaceʹs hkl Miller indices corrected according to the selection rules for non-primitive lattices] are often found to split into lower steps in a regular fashion [Frank (1951). Phil. Mag. 42, 1014-1021]. These `interlacedʹ step patterns are introduced by a stacking of two or more growth layers, with different lateral anisotropy in step velocity within each unit layer. In this paper, a general relation between the symmetry of the crystal surface and the configuration of the concentric step patterns thereon is derived and is used to give theoretical shapes of spirals, growth hillocks and etch pits. It is shown that many of the interlaced patterns and their details are imposed by the presence of screw axes and/or glide planes perpendicular to the crystal surface. Finally, the results are compared with the patterns of unit-layer height and lower steps observed by optical and atomic force microscopy on crystals such as SiC, GaN, potash alum, garnet and NiSO4·6H2O.