A new method of calculating planar density: the position-duplication-number method

The traditional method of calculating planar density requires an auxiliary sectional view showing the arrangement of atoms on the plane, this being very difficult for those planes with higher indices. In the present paper, a new position-duplication-number method is developed to calculate the planar density of all Bravais lattices and all crystal structure types using the formula ρ_(hkl)^(m/q) = N_(hkl)^(m/q)d′_(hkl)/V_cell. The new method is illustrated for calculating planar densities of the (110) and (579) planes of the β-cristobalite SiO₂ structure, with additional examples in hexagonal close-packed, face-centred cubic and body-centred cubic systems. These examples show the advantages of the new method: it does not require any auxiliary sectional view, and it is suitable for all crystal structures and all plane indices.