Normalized Structure Factors E_H

Note: in this text F_H designates the structure factor corrected for thermal motion and brought to an absolute scale; generally this is done using a Wilson plot. Since the scattering factor of any atom decreases for larger reflection angle $\theta$, and the expected intensity $\langle \vert F\vert^2\rangle_{\theta}$ of a reflection is given by

$$\langle\vert F\vert^2\rangle_{\theta} = \sum^N_{j=1} f^2_j(\theta)$$ (1)

reflections measured at different $\theta$-values can not be compared directly. Expression (1) can be used to calculate the so called normalized structure factor

$$\vert E_H\vert^2 = \frac{\vert F_H\vert^2}{\sum^N_{j=1}f^2_j}$$ (2)

It is obvious from a comparison of (1) and (2) that $\langle$E²_H$\rangle$ = 1 for all values of $\theta$.

The structure-factor expression in terms of the normalized structure factor is then:

$$E_H = \frac{1}{(\sum^N_{j=1} f^2_j)^{1/2}} \sum^N_{j=1} f_j \exp 2 {\pi}i(hx_j + ky_j + lz_j).$$ (3)

If the form factor f_j has the same shape for all atoms (f_j = Z_jf), expression (3) can be written as

$$E_H = \frac{1}{(\sum^N_{j=1} f^2_j)^{1/2}} \sum^N_{j=1} f_j \exp 2 {\pi}i(hx_j + ky_j + lz_j).$$ (4)

This is clearly the structure factor formula of a point atom structure, because no $\theta$-dependent factors are present any more.

In order to find the maximum value of |E|, let us consider an equal atom structure for which the structure factor (4) further reduces to

$$\begin{array} {@{}r@{}l} E_H&\displaystyle= \frac{1}{(NZ^2)^... ...}i(hx_j + ky_j + lz_j).\hbox{\rule{0cm}{12pt}\hfill}\end{array}$$ (5)

The maximum possible value of |E_H| is N/N^1/2 = N^1/2.

The unitary structure factor U was used extensively in the early literature on Direct Methods:

$$\vert U_H\vert=\frac{\vert F^2_H\vert}{\sum^N_{j=1}f_j}.$$ (6)

The denominator represents the maximum possible value of F_H and thus U_H varies between 0 and 1. In the equal atom case the relation between U_H and |E_H| is given by

|E_H|² = N|U_H|².

which can easily be verified by the reader from (6) and (2).

Copyright © 1984, 1998 International Union of Crystallography