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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:

$$\big\vert\sum^N_{j=1} a_jb_j\big\vert^2 \leq \sum^N_{j=1}\vert a_j\vert^2 \sum^N_{j=1}\vert b_j\vert^2$$ (8)

to the structure factor equation. For instance the partitioning of the unitary structure-factor equation in $P\overline{1}$ into:

$$U_H = \sum_{j=1} n_j \cos 2 {\pi} H{\cdot}r = \sum_{j=1} a_jb_j$$ (9)

such that a_j = n1/2_j and $b_j = n^{1/2}_j \cos 2 {\pi}H{\cdot}r$leads to

$$U^2_H \leq \big(\sum^N_{j=1} n_j\big)\big(\sum^N_{j=1} n_j \cos^2 2{\pi}H{\cdot}r\big).$$ (10)

From the definition of the unitary structure factor it follows that

$$\sum^N_{j=1} n_j = 1$$ (11)

and the second factor can be reduced as follows

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

These results used in (10) give

$$U^2_H \leq \frac{1}{2}(1 + U_{2H}).$$ (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 |U_2H| 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 |U_2H| is large, expression (13) requires the sign of 2H to be positive even if U_H is somewhat smaller than $\frac{1}{2}$. Moreover, when |U_H| and |U_2H| are reasonably large, but at the same time (13) is fulfilled for both signs of 2H, it is still more likely that S_2H = + than that S_2H = -. For example, for |U_H| = 0.4 and |U_2H| = 0.3, S_2H = + leads in (13) to 0.16 $\leq$ 0.5 + 0.3 which is certainly true, and S_2H = - to $0.16 \leq 0.5 - 0.3$ which is also true. Then probability arguments indicate that still S_2H = + is the more likely sign. The probability is a function of the magnitudes |U_H| and |U_2H| and in this example the probability of S_2H = + 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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