Direct determination of triplet phases and enantiomorphs of non-centrosymmetric structures. II. Experimental results

The minimum strategy for distinguishing enantiomorphs by dynamical diffraction is determined. It is found that, in the absence of anomalous dispersion, it is possible to determine the absolute hand of an enantiomorphic crystal by three-beam dynamical X-ray or electron diffraction in a general orientation only if a fourth noncoplanar reciprocal-lattice point can be identified. Three-beam dynamical diffraction alone is unable to distinguish enantiomorphic forms. Identification is possible using four or more dynamical beams, in general, unless all relevant structure factors lie on a plane in reciprocal space passing through the origin. Supporting computations are given.It has recently been suggested that, because of the ambiguity in the sign of indices, N-beam X-ray multiple scattering conditions do not allow enantiomorphs to be distinguished in the absence of anomalous dispersion (Shen & Colella, 1986). By contrast, Hummer, Weckert & Bondza (1989) and Chien, Tang & Chang (1989) have recently claimed to determine the hand of crystals using experimental three-beam X-ray measurements. The purpose of this note is to show that three dynamical beams alone are not sufficient to distinguish enantiomorphs in the absence of dispersion (or 'absorption' for electron diffraction), whereas four or more beams may be used for this purpose. We are concerned only with the predictions of exact many-beam theory without dispersion; structure factors therefore satisfy F 9 = F_*g.For a crystal A with charge density p(r), we define crystal B with p'(r) = p(-r) as the enantiomorph of A. Crystal B is then defined to be in the 'same' orientation as crystal A; its structure factors are the conjugates of those of A and lie on the same lattice (see Fig. 1). As a consequence of Friedel's law, kinematic X-ray and electron diffraction patterns are therefore insensitive to the hand of crystals. The center of symmetry in these patterns results in an ambiguity in sign for all indices, reflecting the loss of information on hand. This information is recovered under certain multiple scattering conditions, since multiple scattering renders the diffracted intensities sensitive to the phase of the structure factors. As shown in Figs. 1 (a) and (c), crystal B may be brought into a mirror relationship with (but not into coincidence with) crystal A by twofold rotation.Since It has been known for many years that X-ray diffraction patterns affected by anomalous dispersion may be used to determine hand (DeVries, 1958). Then, the absorptive part of the optical potential results in intensity differences between conjugate reflections. In the electron diffraction literature, examples of the determination of hand using multiple scattering are given by Goodman & Secomb (1977) for dextrorotatory quartz and Tanaka, Takayoshi, Ishida & Endoh (1985) for MnSi. Depending on crystal thickness and beam energy, transmission electron diffraction patterns may show single scattering, multiple scattering without absorption or multiple scattering with absorption. In gene...

mentioning

confidence: 99%

On the minimum number of beams needed to distinguish enantiomorphs in X-ray and electron diffraction

Spence¹,

Zuo²,

O’Keeffe³

et al. 1994

“…The information on the sine of the relative phase, which is directly related to the noncentrosymmetry of a crystal, is still missing in such an experiment. Although such phase information may exist at the center of a multiple-reflection peak (Hummer & Billy, 1986;Chang & Tang, 1988;Hummer, Weckert & Bondza, 1989), it is usually affected by the kinematic effect if one uses the dynamical theory or the extinction effect if one uses the kinematic theory. Recently it has been demonstrated by Shen & Finkelstein (1990) that another way to obtain the noncentrosymmetric phase information is to utilize circularly polarized X-rays as the incident beam in a multiple-beam diffraction experiment.…”

Section: Gaas 442: Solving Acentric Phase Problem~mentioning

confidence: 99%

“…It has been known that the interference effect in a multiple-beam diffraction process can be used to extract lost crystallographic phase information from diffraction intensities (Colella, 1974; Post, 1977;Chapman, Yoder & Colella, 1981;Chang, 1982;Juretschke, 1982;Shen, 1986;Shen & Colella, 1986;Hummer & Billy, 1986;Shen & Colella, 1987, 1988Chang & Tang, 1988;Hummer, Weckert & Bondza, 1989;Shen & Finkelstein, 1990). As realized by several authors, when linearly polarized or unpolarized X-rays are used as the incident beam, the multibeam interference effect on the wings of a multiple reflection peak contains information on only the cosine of the relative phase or the real part of the phase factor.…”

Section: Gaas 442: Solving Acentric Phase Problem~mentioning

confidence: 99%

Effects of a general X-ray polarization in multiple-beam Bragg diffraction

Shen¹

1993

The formalism of the N-beam dynamical theory of X-ray diffraction is extended to include all possible incident and diffracted polarizations. With this new formalism it is shown that the intensity of a simultaneously excited Bragg reflection can be described through a polarization density matrix that involves the Stokes-Poincar6 parameters. In particular, the multibeam diffracted intensity is sensitive to the circularly polarized component in the incident beam and the structure-factor phases of the diffracting crystal. Experimental results on the GaAs 442 and Ge 333 reflections confirm the theoretical calculations. © 1993 International Union of Crystallography Printed in Great Britain -all rights reservedThis kind of measurement can provide useful acentric phase information and can also be used for circular X-ray polarimetry. Another feature of N-beam diffraction is its ability to turn a linear polarization into an elliptical polarization, which means it can be used as an X-ray phase plate. IntroductionX-ray polarization plays an important role in every scattering and diffraction experiment. In crystallography, one needs to use the polarization-factor correction in order to obtain structure factors from diffracted intensities (Warren, 1969). In X-ray physics and Acta Crystallographica Section A ISSN 0108-7673 ©1993 606 X-RAY POLARIZATION IN MULTIPLE-BEAM BRAGG DIFFRACTION material science, polarization analysis can often reveal interesting features of the magnetic properties and atomic-site symmetries of the diffracting material (Templeton & Templeton, 1985;Blume & Gibbs, 1988;Belyakov & Dmitrienko, 1989;Kirfel, Petcov & Eichhorn, 1991;Finkelstein, Shen & Shastri, 1992). The increasingly available energy and polarization tunabilities of many intense synchrotron sources promise to provide experimenters with an extra dimension in the analysis of X-ray polarization in future diffraction experiments. It is, therefore, worthwhile to re-examine and extend some of the well established fields in X-ray diffraction. One such field is that of multiple-beam Bragg diffraction.Because of the co-existing effects of multibeam interference and polarization mixing (Shen, 1991), some new and interesting features in multibeam diffraction can emerge if one includes general elliptically polarized X-rays in the incident beam. These new features include phase determination on noncentrosymmetric crystals (Shen & Finkelstein, 1990), complete characterization of an elliptical polarization of an X-ray beam and utilization of multibeam diffraction as an X-ray circular phase plate.Multiple-beam diffraction occurs in a crystal when two or more atomic planes satisfy the Bragg conditions simultaneously. The process usually gives rise to a secondary peak (Renninger peak) in the intensity of a Bragg reflection, H, when the diffraction crystal is rotated around the scattering vector H (Renninger, 1937;Cole, Chambers & Dunn, 1962). The reflection H is usually termed the main reflection and the rotation is described by an azimuthal angle, ~o...

“…7. The experiments have been published by Hammer, Weckert & Bondza (1989). The following parameters were used in the calculation: symmetrical Laue case, thickness of the crystal plate d --0.07 mm, wavelength ~.…”

Section: Fa(hkl) =Fa(-h-k K -1) Exp 2zrt~ • T Where H = (Hkl)mentioning

confidence: 99%

Enantiomorphism and three-beam X-ray diffraction: determination of the absolute structure

Hümmer¹,

Weckert²

1995

Self Cite

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.