Theoretical and experimental advances in determination of three-phase invariants by multiple-beam X-ray diffraction are reviewed. The fundamental physics and mathematical analyses are explained. Plane-wave dynamical theory for the interpretation of multiple-beam interference is summarized. New results of its applications to the solution of the enantiomorphism problem of light-atom structures and to the solution of the structure of macromolecules by means of measured phases in conjunction with statistical methods are reported. Practical aspects of applying the three-beam diffraction technique to proteins are emphasized.After obtaining his first degree in material sciences, Edgar Weckert received his doctorate in 1988 from the Faculty of Mathematics and Physics at the University of Erlangen, Germany. His thesis on three-beam X-ray diffraction using synchrotron radiation was the first application to the solution of the enantiomorphism problem of non-centrosymmetric low-Z structures. Since then, while a postdoctoral assistant at the University of Karlsruhe, he has been developing this technique both theoretically and experimentally and applying it to aperiodic structures and macromolecular crystals. His work on phasing macromolecular structures by means of physically estimated reflection phases in conjunction with maximum-entropy methods very recently won him the first Max yon Laue prize of the Deutsche Gesellschaft far Kristallographie awarded to young scientists. Kurt Hammer received his doctorate in 1971 and his habilitation in 1978 from the

Determination of triplet phases of macromolecular structures by means of ψ -scan experiments are reported. The ways in which difficulties can be overcome when investigating macromolecular structures (numerous overlap of multiple beam diffraction profiles, overall weak scattering power and radiation damage) are discussed. It is shown theoretically and experimentally that measurable interference effects exist if the crystal dimensions are smaller than the Pendellösung length. The effect of the unavoidable overlapping of weak three-beam cases is negligible if a three-beam case with strong reflections is selected for phase determination. Data recorded at the HASYLAB synchrotron radiation source DORIS III are presented. In case of lysozyme it was possible to measure a set of 80 triplet invariants with a mean phase error of about 17°. First experiments to study the feasibility of phase determination of large proteins, e. g. of catalase (unit cell volume 1200 nm 3 ), are presented.

Recent progress in experimental triplet phase determination by the method of three-beam diffraction for 0108-7673/90/050393-10503.00 non-centrosymmetric light-atom structures is reported. The measurements were carried out with a special 0-circle diffractometer installed at the DORIS II storage ring in Hamburg. Experimental results O 1990 International Union of Crystallography 394 DIRECT MEASUREMENTS OF TRIPLET PHASE RELATIONSHIPS confirm the theoretical considerations. In general, the ~-scan profiles consist of a phase-independent Umweganregung or AuJhellung part superimposed on a phase-dependent part due to the three-beam interference, which contains the phase information.Experimentally, Umweganregung and Aufhellung effects can be evaluated by comparison of the two centrosymmetrically related three-beam cases. It is shown that with moderate phase-independent effects the triplet phase can be determined with an accuracy of about 45 ° . IntroductionIn a recent paper (Hiimmer, Weckert & Bondza, 1989) we reported the direct measurements of triplet phases by means of three-beam interference experiments for two non-centrosymmetric test structures with relatively small unit cells using Cu radiation from a rotating-anode generator. It was shown that the threebeam interference effect leads to typical 0-scan profiles for triplet phases near 0, 180, +90 or -90 °.On the other hand, the theoretical considerations in the preceding paper (Weckert & Hiimmer, 1990) show that it should be possible to achieve a higher precision in experimental triplet phase determination in spite of phase-independent Umweganregung and Aufhellung effects superimposed on the interference effect. The three-beam interference between the primary diffracted wave and the Umweg wave gives rise to the typical 'ideal' 0-scan profiles which carry the phase information. It should be possible to evaluate the troublesome Umweganregung and Aufhellung effects by comparing the 0-scan profiles of two centrosymmetrically related three-beam cases 0/h/g and O/-h/-g which involve equal triples of structurefactor magnitudes but triplet phases of opposite sign if anomalous-dispersion effects can be neglected. However, because of the experimental systematic uncertainties the Umweganregung and Aufhellung should be avoided. Thus, the basic requirement for this procedure is that the primary, secondary and the coupling reflections have nearly equal intensities.Our method of quantitative determination of triplet phases is in some important aspects different from that proposed by Shen & Colella (1988) and Tang & Chang (1988).In the experiments of Shen & Collella the intensity change of very weak primary reflections caused by strong multiple reflections is used. The phase information is obtained from the asymmetry in the wings of the multiple-reflection peaks which are dominated by strong Umweganregung. The asymmetry decreases with decreasing cosine of the triplet phase. So the authors stated that only cos ~b can be determined.In the paper of Tang & Chang (1988) the ...

No abstract

It is shown that three-beam X-ray diffraction provides a means of resolving the enantiomorphism problem. It is based on the fact that three-beam interference leads to significantly different ~-scan profiles for triplet phases close to +90 or -90 ° , which are selectors between enantiomorphs. Since this method works without the need of anomalous scattering, it is particularly suitable for resolving the absolute structure of light-atom compounds. The application of this method is discussed in detail. Its capability of distinguishing between enantiomorphs has been rejected in a recent paper by Colella [Acta Cryst. (1994), A50, 55-57]. Detailed comments on the invalid arguments of Colella's analysis are presented.