[IUCr Home Page] [Commission Home Page]

next up previous
Next: The |E|'s of H and 2H: Up: An Introduction to Direct Methods. The Previous: Strong and Weak Structure Factor Magnitudes FH

Normalized Structure Factors EH

Note: in this text FH 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

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

\begin{displaymath}
\vert E_H\vert^2 = \frac{\vert F_H\vert^2}{\sum^N_{j=1}f^2_j}\end{displaymath} (2)
It is obvious from a comparison of (1) and (2) that $\langle$E2H$\rangle$ = 1 for all values of $\theta$.

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

\begin{displaymath}
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).\end{displaymath} (3)
If the form factor fj has the same shape for all atoms (fj = Zjf), expression (3) can be written as

\begin{displaymath}
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).\end{displaymath} (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{displaymath}
\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}\end{displaymath} (5)
The maximum possible value of |EH| is N/N1/2 = N1/2.

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

\begin{displaymath}
\vert U_H\vert=\frac{\vert F^2_H\vert}{\sum^N_{j=1}f_j}.\end{displaymath} (6)
The denominator represents the maximum possible value of FH and thus UH varies between 0 and 1. In the equal atom case the relation between UH and |EH| is given by

|EH|2 = N|UH|2.

(7)

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


next up previous
Next: The |E|'s of H and 2H: Up: An Introduction to Direct Methods. The Previous: Strong and Weak Structure Factor Magnitudes FH

Copyright © 1984, 1998 International Union of Crystallography

IUCr Webmaster