Comparison of calculations for interplanar distances in a crystal lattice

The derivations for the general formulae of lattice interplanar distances are reviewed along with the methods using elementary geometry, intermediate Cartesian axes, and reciprocal lattice vectors. To highlight the characteristics of these three methods and the connections between them, examples for the simple cases such as orthorhombic, hexagonal, and rhombohedral systems are included. Calculations from reciprocal space are established from those from direct space with heavily involved mathematics for which details are seldom included in crystallography monographs. The only geometric method found in the literature for the interplanar distances in a crystal lattice is derived for a few specific simple cases with cos²α?+?cos²β?+?cos²γ?=?1, where α, β, and γ are the angles between the normal to the plane and the axes of a orthogonal system. However, the geometric method introduced in this work is a newly developed method and this method is complementary to other methods including the advanced contemporary reciprocal method. The connections between Cartesian and crystal coordinate systems, for angular relationships and the volume of unit cell are revealed. The interplanar spacing in non-primitive lattice and crystals are also discussed.