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Strong and Weak Structure Factor Magnitudes F_H

If, in a crystal structure, atoms lie in the neighbourhood of a set of planes H, as indicated in Fig. 1a, then reflection by planes H is strong and hence the intensity I_H is large. Of course, the converse is also true: if one observes a large intensity I_H, then the atoms lie near planes as indicated in Fig. 1a. This statement follows also from the structure-factor expression:

$$F_H = \vert F_H\vert\exp (i{\phi}_H) = \sum^N_{j=1} f_j \exp 2{\pi}i(hx_j + ky_j + lz_j).$$

A large F_H will be found if (hx_j + ky_j + lz_j) mod 1 is approximately constant for all j; or, in other words, if all atoms lie near one of the planes H. The phase $\phi_H$ depends on the value of the constant and changes with the origin.

Figure 1: A reflected beam H is strong when the atoms lie in the neighbourhood of the set of planes H (a) and low when the atoms are spread out with respect to the planes H (b).

Conversely, a structure-factor magnitude |F_H| is small, if the atoms are randomly distributed with respect to the planes H, as shown in Fig. 1b.

The electron density can be thought of as a superposition of density waves parallel to lattice planes, the amplitudes of which are the |F_H|-values, the relative phases being given by the $\phi_H$-values. We will see later that these density waves afford a physical picture of the phase relationships used in Direct Methods.

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