Quartets and Other Phase Relationships

Abstract

In earlier chapters it was shown that the phase sum

$$ {\varphi _3} = \varphi H + \varphi K + \varphi - H - K$$

((1))

is independent of the choice of origin. In direct methods quantities which are independent of the choice of origin are called structure invariants. When it is possible to estimate the phase sum of an invariant, this sum is generally called a phase relationship. Although in principle a phase relationship is always a structure invariant, it is sensible to make the distinction between all invariants and those, for which it is possible to estimate phase sums. Of course, for direct methods the latter are the only important invariants, because reliable phase relationships are the basis of all existing phasing methods, whereas for any structure there are millions more of useless structure invariants.