A tangent formula derived from Patterson-function arguments .4. The solution of difference structures directly from superstructure reflections

The possibility of solving by direct methods the difference structure delta of a superstructure using only the intensities of the superstructure reflections h is shown. The refinement of the phases Phi of the strong superstructure E values, which are normalized separately from the E values of the main or substructure reflections, is carried out maximizing the function Z(delta P') Sigma(h)(E(h) - [E(h)])C-h(Phi), where C-h(Phi) is the amplitude of the structure factor of delta(3) expressed in terms of Phi. The maximization is performed with a new tangent formula that only differs from that given previously [Rius (1993). Acta Cryst. A49, 406-409] by an extra summation, i.e. the phase information is now derived from quartets and negative quintets instead of triplets and negative quartets. A preliminary test calculation demonstrates the capability of this tangent formula to solve the difference structure of the mineral wermlandite using only the measured superstructure intensities. Although more tests covering a variety of situations are still required to allow for a generalization, this result seems to confirm the viability of determining the internal structure of reconstructed surfaces by interpreting the corresponding three-dimensional difference Patterson function by direct methods. Access to this function is now possible, since, with the advent of intense synchrotron sources, not only in-plane intensity data but also the corresponding diffraction rods can be measured.