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

  • A. F. Mishnev
Part of the NATO ASI Series book series (NSSB, volume 274)


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

$$ \left| \begin{gathered} F\left( 0 \right) + \frac{{2{ \in _0}}} {\Gamma }\quad F\left( { - H} \right)\quad F\left( { - K} \right) \hfill \\ F\left( H \right)\quad \in F\left( 0 \right) + \frac{{2{ \in _0}}} {\Gamma }\quad F\left( {H - K} \right) \hfill \\ F\left( H \right)\quad F\left( {K - H} \right)\quad \frac{{2{ \in _0}}} {\Gamma } \hfill \\ \end{gathered} \right| = 0 $$
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
$$ \tau _0^3 = {\tau _0}\left( {\sum\limits_{i = 1}^3 {{{\left| {{F_i}} \right|}^2}} } \right) + 2\left| {F\left( H \right)F\left( K \right)F\left( {K - H} \right)} \right|\cos {\Phi _3} = 0 $$
$$ {\tau _0} = F\left( 0 \right) + \frac{{2{ \in _{_0}}}} {\Gamma }\quad and\quad {\Phi _3} = \Phi \left( H \right) + \Phi \left( { - K} \right) + \Phi \left( {K - H} \right) $$


  1. Ewald, P.P., and Heno, Y., 1968, X-ray diffraction in the case of three strong rays. I. Crystal composed of non-absorbing point atoms, Acta Cryst., A24:5.CrossRefGoogle Scholar
  2. Gerber, R.B., and Karplus, M., 1972, On the determination of the phases of electromagnetic scattering amplitudes from experimental data, J. Chem. Phys., 56:1921.CrossRefGoogle Scholar
  3. Gerber, R.B., and Karplus, M., 1972, Derivation of phase-determining relations from the unitary theorem of electromagnetic scattering (unpublished).Google Scholar
  4. Post, B., 1979, A solution of the X-ray ‘phase problem’, Acta Cryst., A35:17.CrossRefGoogle Scholar
  5. Schenk, H., 1981, The negative-quartet relation from electron-density considerations, Acta Cryst., A37:573.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • A. F. Mishnev
    • 1
  1. 1.Institute of Organic SynthesisLatvian SSR Academy of SciencesRigaUSSR

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