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Next: Large |EH|, |EK| and |E-H-K|: The Up: An Introduction to Direct Methods. The Previous: The |E|'s of H and 2H:

The $\sum_1$-Relation from a Harker-Kasper Inequality

In 1948 Harker and Kasper published their paper on inequality relationships, which actually opened the field of direct methods. They applied the Cauchy inequality:

\begin{displaymath}
\big\vert\sum^N_{j=1} a_jb_j\big\vert^2 \leq \sum^N_{j=1}\vert a_j\vert^2 \sum^N_{j=1}\vert b_j\vert^2\end{displaymath} (8)
to the structure factor equation. For instance the partitioning of the unitary structure-factor equation in $P\overline{1}$ into:

\begin{displaymath}
U_H = \sum_{j=1} n_j \cos 2 {\pi} H{\cdot}r = \sum_{j=1} a_jb_j\end{displaymath} (9)
such that aj = n1/2j and $b_j = n^{1/2}_j \cos 2 {\pi}H{\cdot}r$leads to

\begin{displaymath}
U^2_H \leq \big(\sum^N_{j=1} n_j\big)\big(\sum^N_{j=1} n_j \cos^2
2{\pi}H{\cdot}r\big).\end{displaymath} (10)
From the definition of the unitary structure factor it follows that

\begin{displaymath}
\sum^N_{j=1} n_j = 1\end{displaymath} (11)
and the second factor can be reduced as follows

\begin{displaymath}
\begin{array}
{@{}r@{}l}
\displaystyle
\sum_{j=1} n_j \cos^2...
 ...\frac{1}{2}(1+U_{2H}). \hbox{\rule{0cm}{12pt}\hfill}\end{array}\end{displaymath} (12)
These results used in (10) give

\begin{displaymath}
U^2_H \leq \frac{1}{2}(1 + U_{2H}).\end{displaymath} (13)

In case $U^2_H \gt \frac{1}{2}$ then $U_{2H}\geq 0$ or in other words the sign of reflection 2H is positive whatsoever its |U2H| value is. Note that the sign of H may have both values. In practice $U^2_H \gt \frac{1}{2}$ does not often occur. However, when |U2H| is large, expression (13) requires the sign of 2H to be positive even if UH is somewhat smaller than $\frac{1}{2}$. Moreover, when |UH| and |U2H| are reasonably large, but at the same time (13) is fulfilled for both signs of 2H, it is still more likely that S2H = + than that S2H = -. For example, for |UH| = 0.4 and |U2H| = 0.3, S2H = + leads in (13) to 0.16 $\leq$ 0.5 + 0.3 which is certainly true, and S2H = - to $0.16 \leq 0.5 - 0.3$ which is also true. Then probability arguments indicate that still S2H = + is the more likely sign. The probability is a function of the magnitudes |UH| and |U2H| and in this example the probability of S2H = + being correct is $\gt 90\%$.In conclusion the mathematical treatment leads to the same result as the graphic explanation from the preceding paragraph: the $\sum_1$ relationship.


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Next: Large |EH|, |EK| and |E-H-K|: The Up: An Introduction to Direct Methods. The Previous: The |E|'s of H and 2H:

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