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

Abstract

When the formulae for calculating the interplanar spacingd′hklfor the seven primary lattices among the 14 Bravais lattices are used for crystal structures containing an additional atom,d′hklmust 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 factorP[whereP=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 ifm≠ 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 usingP, 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.