  # The -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: (8)
to the structure factor equation. For instance the partitioning of the unitary structure-factor equation in into: (9)
such that aj = n1/2j and leads to (10)
From the definition of the unitary structure factor it follows that (11)
and the second factor can be reduced as follows (12)
These results used in (10) give (13)

In case then 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 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 . 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 0.5 + 0.3 which is certainly true, and S2H = - to 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 .In conclusion the mathematical treatment leads to the same result as the graphic explanation from the preceding paragraph: the relationship.   Next: Large |EH|, |EK| and |E-H-K|: The Up: An Introduction to Direct Methods. The Previous: The |E|'s of H and 2H: