A new method of calculating interplanar spacing: the position-factor method

When the formulae for calculating the interplanar spacing d′_hkl for the seven primary lattices among the 14 Bravais lattices are used for crystal structures containing an additional atom, d′_hkl must be corrected by multiplying by a correction coefficient. Based on a simple principle of analytical geometry, a new position-factor method has been developed to calculate these correction coefficients. The position factor P [where P = hx + ky + lz = s + m/q, the scalar product of the position (x, y, z) of an additional atom in a unit cell and any plane (hkl) passing through the additional atom] indicates the following characteristics of the (hkl) plane: whether there is an additional plane (m ≠ 0) or not (m = 0), what the correction coefficient (m/q) is for the interplanar spacing if m ≠ 0, and what the distance [(s + m/q)d′_hkl] is from the plane to the origin. In order to provide two examples of calculations using P, six possible interplanar spacings were obtained for a hexagonal close-packed crystal, and four possible interplanar spacings and the structure factor were obtained for a diamond crystal. In addition, the 12 undetermined plane indices in X-ray diffraction card 1-1249 (PDF-2, ICDD, Newtown Square, PA, USA) for diamond were determined using the new calculation.