Some Considerations Concerning the Physical Interpretation of Sayre’s Equation and Phase Triplets in Direct Methods

Abstract

Despite of the permanent progress in the theory of direct methods a problem of physical interpretation of phase relations is still not clear enough. Schenk (1981) has given a graphic explanation of phase triplets and quartets on the basis of electron density considerations. Such a geometrical approach is very convenient and viseable, however it does not take into account a real process of X-ray scattering by crystal. In direct methods a crystal is considered as an ideal infinite periodic structure. Therefore, direct methods are also valid for a large perfect crystal and one can compare phase relations of direct methods with inferences of the dynamic theory of X-ray diffraction. For this purpose we use Ewald’s dynamical theory in the case of three strong coplanar beams (Ewald & Heno, 1968; Post, 1979). The condition of compatibility of the dynamical equations has the form

$$ \left| \begin{gathered} F\left( 0 \right) + \frac{{2{ \in _0}}} {\Gamma }\quad F\left( { - H} \right)\quad F\left( { - K} \right) \hfill \\ F\left( H \right)\quad \in F\left( 0 \right) + \frac{{2{ \in _0}}} {\Gamma }\quad F\left( {H - K} \right) \hfill \\ F\left( H \right)\quad F\left( {K - H} \right)\quad \frac{{2{ \in _0}}} {\Gamma } \hfill \\ \end{gathered} \right| = 0 $$

((1))

where ∈₀ is a resonance error, Г ≃ e²λ²/mc²πV, V is the volume of the unit cell. Determinant (1) (except the diagonal terms) is identical to the Karle — Hauptman’s one. If we had taken more waves into consideration we would have got a determinant of a higher order. Unfortunately, ∈₀ cannot be measured or calculated for an unknown structure. Nevertheless, we can use the determinant (1) for some illustrations. Expansion of the latter yields the dispersion equation

$$ \tau _0^3 = {\tau _0}\left( {\sum\limits_{i = 1}^3 {{{\left| {{F_i}} \right|}^2}} } \right) + 2\left| {F\left( H \right)F\left( K \right)F\left( {K - H} \right)} \right|\cos {\Phi _3} = 0 $$

((2))

where

$$ {\tau _0} = F\left( 0 \right) + \frac{{2{ \in _{_0}}}} {\Gamma }\quad and\quad {\Phi _3} = \Phi \left( H \right) + \Phi \left( { - K} \right) + \Phi \left( {K - H} \right) $$

.