The structure factor for infinite periodic systems

Consider an infinite array of diffracting points, with regular spacings in three directions of length a, a2, and as along x, y, and z respectively (Fig. 5.12). After choosing one point as the origin, the position of any other point in the array can be expressed as 

The symbol Skj means that the product is unit when k = j and zero otherwise. It is easy to show that any distance d in real space has its counterpart in reciprocal space  

With the artifice of reciprocal space, finding the conditions for in-phase scattering by a periodic array of points of scattering factor / becomes very easy. In analogy with equation 5.19  

The quantity (hn + km + Ip) must be an integer for constructive interference. Since n, m, and p are integers, it follows that also h, k, and I must be integers. Diffracted intensities are non-zero only for integer values of h, k, and Z, that is, when the r vector lands on a node of the reciprocal lattice. 

In real crystals the translationally independent unit is not a single scattering point, but an ensemble of molecules in the unit cell, each of which is described by the molecular structure factor. When the molecular structure factor, equation 5.19, is combined with the condition 5.26 for constructive interference of a translation-ally periodical lattice, the final expression for the crystallographic structure factor results  

---