A relation between structure factor, triple products and a single Patterson vector, and its application to sign determination

The Fourier coefficients D h of a hypersection of the double Patterson function at a fixed value of U can be written as D h = (1/V) Σ h′ F h F h′ F_{\overline {\bf h + h^\prime}} cos 2π h′. U/cos 2π h. (U/2). It is shown that also D h = F h G h, where G h is the Fourier coefficient of a reduced structure derived from the original structure by positioning an atom with form factor Σ h′ f i (h′)fj ({\overline {\bf h + h^\prime}}) halfway in between those atoms i and j, which are a vector U apart. If U is a single vector between two atoms with form factor f(h), then the reduced structure contains only one atom, at the origin, and G h = (1/V) Σ h′ f(h′)f({\overline {\bf h + h^\prime}}). We obtain the relation Σ h′ F h F h′ F_{\overline {\bf h + h^\prime}} cos 2 π h′. U/[cos 2π h. (U/2) Σ h′ f(h′)f{\bf h + h^\prime;}] = F h . Using this formula with probabilities for the signs of the triple products, the single vectors U are found by a scanning process, as we demonstrate on a heavy atom structure. The signs of the structure factors have thus been calculated directly from the triple products.