Showing papers in "Journal of Applied Crystallography in 2006"

Mercury: visualization and analysis of crystal structures

Abstract: Since its original release, the popular crystal structure visualization program Mercury has undergone continuous further development. Comparisons between crystal structures are facilitated by the ability to display multiple structures simultaneously and to overlay them. Improvements have been made to many aspects of the visual display, including the addition of depth cueing, and highly customizable lighting and background effects. Textual and numeric data associated with structures can be shown in tables or spreadsheets, the latter opening up new ways of interacting with the visual display. Atomic displacement ellipsoids, calculated powder diffraction patterns and predicted morphologies can now be shown. Some limited molecular-editing capabilities have been added. The object-oriented nature of the C++ libraries underlying Mercury makes it easy to re-use the code in other applications, and this has facilitated three-dimensional visualization in several other programs produced by the Cambridge Crystallographic Data Centre.

Reduction and analysis of SANS and USANS data using IGOR Pro

Abstract: A software package is presented for performing reduction and analysis of small-angle neutron scattering (SANS) and ultra-small-angle neutron scattering (USANS) data. A graphical interface has been developed to visualize and quickly reduce raw SANS and USANS data into one- or two-dimensional formats for interpretation. The resulting reduced data can then be analyzed using model-independent methods or non-linear fitting to one of a large and growing catalog of included structural models. The different instrumental smearing effects for slit-smeared USANS and pinhole-smeared SANS data are handled automatically during analysis. In addition, any number of SANS and USANS data sets can be analyzed simultaneously. The reduction operations and analysis models are written in a modular format for extensibility, allowing users to contribute code and models for distribution to all users. The software package is based on Igor Pro, providing freely distributable and modifiable code that runs on Macintosh and Windows operating systems.

Co‐refinement of multiple‐contrast neutron/X‐ray reflectivity data using MOTOFIT

Abstract: The contrast-variation technique is employed in multiple-contrast neutron/X-ray reflectometry experiments to highlight scattering from different structural components that are present at a surface or interface. The advantage of this technique is that the structural model used to describe the interfacial scattering length density profile must apply to all the contrasts measured. A new reflectivity analysis package, MOTOFIT, which runs in the IGOR Pro environment (http://www.wavemetrics.com), has been created to aid the simultaneous fitting (with the same structural model) of these multiple-contrast data, using an intuitive graphical user interface, which most co-refinement packages do not possess. MOTOFIT uses a slab-model approach with the Abeles matrix method, and extensions for surface roughness to perform non-linear least-squares regression on the experimental reflectivity curves. Other features, such as the ability to create complicated interparameter constraints or analyse reflectivity from multilayers, simulated annealing, etc., make MOTOFIT a powerful reflectometry analysis package

TLSMD web server for the generation of multi-group TLS models

Abstract: The TLSMD web server extracts information about dynamic properties of a protein based on information derived from a single-crystal structure. It does so by analyzing the spatial distribution of individual atomic thermal parameters present in an input structural model. The server partitions the protein structure into multiple, contiguous chain segments, each segment corresponding to one group in a multi-group description of the protein's overall dynamic motion. For each polypeptide chain of the input protein, the analysis generates the optimal partition into two segments, three segments, … up to 20 segments. Each such partition is optimal in the sense that it is the best approximation of the overall spatial distribution of input thermal parameters in terms of N chain segments, each acting as a rigid group undergoing TLS (translation/libration/screw) motion. This multi-group TLS model may be used as a starting point for further crystallographic refinement, or as the basis for analyzing inter-domain and other large-scale motions implied by the crystal structure.

ISODISPLACE: a web-based tool for exploring structural distortions

Abstract: ISODISPLACE is a new internet-server tool for exploring structural phase transitions. Given parent-phase structural information, it generates atomic displacement patterns induced by irreducible representations of the parent space-group symmetry and allows a user to visualize and manipulate the amplitude of each distortion mode interactively. ISODISPLACE is freely accessible at http://stokes.byu.edu/isodisplace.html via common internet browsers.

ATSAS 2.1, a program package for small‐angle scattering data analysis

Abstract: The program package ATSAS 2.1 for small-angle X-ray and neutron scattering data analysis is presented. The programs included in the package cover the major processing and interpretation steps from primary data reduction to three-dimensional modelling. This system is primarily oriented towards the analysis of biological macromolecules, but could also be used for non-biological isotropic and partially ordered objects (nanoparticle systems, colloidal solutions, polymers in solution and bulk). Recent developments in the programs included in ATSAS 2.1 are highlighted. The main programs run on multiple hardware platforms, including Windows PC, Linux RedHat and Suse, DEC Alpha, SGI IRIX and Mac OSX.

DASH: a program for crystal structure determination from powder diffraction data

Abstract: DASH is a user-friendly graphical-user-interface-driven computer program for solving crystal structures from X-ray powder diffraction data, optimized for molecular structures. Algorithms for multiple peak fitting, unit-cell indexing and space-group determination are included as part of the program. Molecular models can be read in a number of formats and automatically converted to Z-matrices in which flexible torsion angles are automatically identified. Simulated annealing is used to search for the global minimum in the space that describes the agreement between observed and calculated structure factors. The simulated annealing process is very fast, which in part is due to the use of correlated integrated intensities rather than the full powder pattern. Automatic minimization of the structures obtained by simulated annealing and automatic overlay of solutions assist in assessing the reproducibility of the best solution, and therefore in determining the likelihood that the global minimum has been obtained.

ENGIN-X: a third-generation neutron strain scanner

Abstract: ENGIN-X, a new time-of-flight (TOF) neutron diffractometer optimized to measure elastic strains at precise locations in bulky specimens recently commissioned at the ISIS Facility in the Rutherford Laboratory, UK, is described. Fast counting times, together with a flexible and accurate definition of the instrumental gauge volume are the main requirements of neutron strain scanning and have been addressed on ENGIN-X through the design of a novel TOF diffractometer with a tuneable resolution and interchangeable radial collimators. Further, the routine operation of the instrument has been optimized by creating a virtual instrument, i.e. a three-dimensional computer representation of the diffractometer and samples, which assists in the planning and execution of experiments. On comparing ENGIN-X with its predecessor ENGIN, a 25× gain in performance is found, which has allowed the determination of stresses up to 60 mm deep in steel specimens. For comparison with constant-wavelength diffractometers, special attention has been paid to the absolute number of counts recorded during the experiments. A simple expression is presented for the estimation of counting times in TOF neutron strain scanning experiments.

SHADE web server for estimation of hydrogen anisotropic displacement parameters

Abstract: The SHADE web server estimates anisotropic displacement parameters for hydrogen atoms by combining a rigid-body analysis of the non-hydrogen-atom anisotropic displacement parameters (ADPs) with a contribution from internal atomic motion. The contributions from internal mean square displacements are based on a previously compiled database derived from analysis of neutron diffraction experiments. The estimated hydrogen-atom ADPs can be used as fixed parameters in advanced applications of high-resolution X-ray diffraction, such as electron density studies using multipole modelling. The resulting electron density models have been shown to be in excellent agreement with reference models based on atomic motion derived from neutron diffraction experiments.

MolIso– a program for colour‐mapped iso‐surfaces

Abstract: The program MolIso allows the visualization of colour-mapped iso-surfaces from XD grid or GAUSSIAN CUBE files. It is written in C++ and uses the OpenGL and GLUT libraries, and is therefore portable. Linux and Windows versions are currently available. Optically correct looking, transparent visualizations are possible. The molecule inside the iso-surface can be depicted as ball-and-stick or atomic displacement parameter ellipsoid–line models. Screenshots can easily be taken in Portable Pix Map format. Short video clips, with various codecs for presentations, are easy to create.

Enhancement of crystalline perfection by organic dopants in ZTS, ADP and KHP crystals as investigated by high‐resolution XRD and SEM

Abstract: To reveal the influence of complexing agents on crystalline perfection, tristhiourea zinc(II) sulfate (ZTS), ammonium dihydrogen phosphate (ADP) and potassium hydrogen phthalate (KHP) crystals grown by slow-evaporation solution growth technique using low concentrations (5 x 10 -3 M) of dopants like ethylenediamminetetraacetic acid (EDTA) and 1,10-phenanthroline (phen) were characterized by high-resolution X-ray diffractometry (XRD) and scanning electron microscopy (SEM). High-resolution diffraction curves (DCs) recorded for ZTS and ADP crystals doped with EDTA show that the specimen contains an epilayer, as observed by the additional peak in the DC, whereas undoped specimens do not have such additional peaks. On etching the surface layer, the additional peak due to the epilayer disappears and a very sharp DC is obtained, with full width at half-maximum (FWHM) of less than 10 arcsec, as expected from the plane wave dynamical theory of X-ray diffraction for an ideally perfect crystal. SEM micrographs also confirm the existence of an epilayer in doped specimens. The ZTS specimen has a layer with a rough surface morphology, having randomly oriented needles, whereas the ADP specimen contains a layer with dendric structure. In contrast to ADP and ZTS crystals, the DC of phen-doped KHP shows no additional peak, but it is quite broad (FWHM = 28 arcsec) with a high value of integrated intensity, p (area under the DC). The broadness of the DC and the high value of p indicate the formation of a mosaic layer on the surface of the crystal. However, similar to ADP and ZTS, the DC recorded after etching the surface layer of the KHP specimen shows a very sharp peak with an FWHM of 8 arcsec. An SEM photograph of phen-doped KHP shows deep cracks on the surface, confirming the mosaicity. After removing the surface layer, the SEM pictures reveal a smooth surface. A similar trend is observed with other complexing agents, like oxalic acid, bipy and picolinic acid. However, only typical examples are described in the present article where the effects were observed prominently. The investigations on ZTS, ADP and KHP crystals, employing high-resolution XRD and SEM studies, revealed that some organic dopants added to the solution during the growth lead to the formation of a surface layer, due to complexation of these dopants with the trace metal ion impurities present in the solution, which prevents the entry of impurities, including the solvent, into the crystal, thereby assisting crystal growth with high crystalline perfection. The influence of organic dopants on the second harmonic generation efficiency is also investigated.

A new experimental station for simultaneous X-ray microbeam scanning for small- and wide-angle scattering and fluorescence at BESSY II

Abstract: A new instrument for simultaneous microbeam small- and wide-angle X-ray scattering and X-ray fluorescence (SAXS/WAXS/XRF) is presented. The instrument is installed at the microfocus beamline at BESSY II and provides a beam of 10 µm size with a flux of about 109 photons s−1. A SAXS resolution up to 500 A d-spacing and a range of scattering vectors of almost three orders of magnitude are reached by using a large-area high-resolution CCD-based detector for simultaneous SAXS/WAXS. The instrument is particularly suited for scanning SAXS/WAXS/XRF experiments on hierarchically structured biological tissues. The necessary infrastructure, such as a cryo-stream facility and an on-site preparation laboratory for biological specimens, are available.

SXD – the single-crystal diffractometer at the ISIS spallation neutron source

Abstract: SXD, the single-crystal diffractometer at the ISIS spallation neutron source, uses an array of two-dimensional position-sensitive detectors and the neutron time-of-flight technique to measure diffraction data throughout very large volumes of reciprocal space for each fixed orientation of a single-crystal sample. This paper describes SXD in detail, following major improvements to the instrument. Particular emphasis is placed on the range of science possible, using recent results as examples, and the opportunities for future experiments.

High-pressure X-ray investigation of zincite ZnO single crystals using diamond anvils with an improved shape

Abstract: A high-pressure cell for single-crystal investigations requires large opening angles. If it does not contain any beryllium but only two diamonds mounted on steel backing plates, normally only lower pressures can be attained without risk of disruption of the diamond anvils [Ahsbahs (2004). Z. Kristallogr. 219, 305–308]. In order to enable pressures up to at least 10 GPa, specially shaped anvils were designed. In a first study with such anvils, the high-pressure behaviour of zincite ZnO was investigated up to the transition pressure of about 9.5 GPa. A slight decrease of the axial ratio c/a with increasing pressure was observed. The fit of the Birch–Murnaghan equation of state gave a bulk modulus K = 146.5 (8) GPa with a fixed pressure derivative K′ = 4. During the phase transformation, the single crystals were destroyed. The pronounced preferred orientation of the obtained material, however, enabled the determination of the orientation relations between the low-pressure wurtzite-type and the high-pressure NaCl-type phase. The orientation relations are similar to those in CdS and CdSe.

Grazing‐incidence small‐angle X‐ray scattering from thin polymer films with lamellar structures – the scattering cross section in the distorted‐wave Born approximation

Abstract: Grazing-incidence small-angle X-ray or neutron scattering of thin polymer films reveals information about the ordering and preferential orientations of the phase-separated microdomains within the films. The grazing-incidence geometry enhances the surface sensitivity; however, the scattering has to be treated within the framework of the distorted-wave Born approximation. In this work, the case of thin films with lamellar mesostructure is studied, where the orientation of the lamellae is either perpendicular or parallel to the film interfaces. For perpendicular lamellae, Bragg rods are found, which are extended along the film normal, whereas for parallel lamellae, peaks along the film normal appear. The positions of the maxima present in the latter case are explained by accounting for refraction at the film surface and reflection at the film–substrate interface. The results are relevant for thin films of lamellar diblock copolymers.

Line profile analysis: pattern modelling versus profile fitting

Abstract: Powder diffraction data collected on a nanocrystalline ceria sample within a round robin conducted by the IUCr Commission on Powder Diffraction were analysed by two alternative approaches: (i) whole-powder-pattern modelling based upon a fundamental microstructural parameters approach, and (ii) a traditional whole-powder-pattern fitting followed by Williamson–Hall and Warren–Averbach analysis. While the former gives results in close agreement with those of transmission electron microscopy, the latter tends to overestimate the domain size effect, providing size values about 20% smaller. The origin of the discrepancy can be traced back to a substantial inadequacy of profile fitting with Voigt profiles, which leads to systematic errors in the following line profile analysis by traditional methods. However, independently of the model, those systematic errors seem to have little effect on the volume-weighted mean size.

Automated diffraction image analysis and spot searching for high-throughput crystal screening

Abstract: A new software package, DISTL (Diffraction Image Screening Tool and Library), for the rapid analysis of X-ray diffraction patterns collected from macromolecular crystals is presented. Within seconds, the program characterizes the strength and quality of the Bragg spots, determines the limiting resolution of the image, and identifies deleterious features such as ice-rings and intense salt reflections. The procedure also generates a reliable set of intense peaks for auto-indexing. The ability to classify a large number of crystals quickly will be especially useful at synchrotron and home-laboratory X-ray sources where automated crystal screening and data collection systems have been implemented.

Effects of cryoprotectant concentration and cooling rate on vitrification of aqueous solutions

Abstract: Vitrification of aqueous cryoprotectant mixtures is essential in cryopreservation of proteins and other biological samples. Systematic measurements of critical cryoprotective agent (CPA) concentrations required for vitrification during plunge-cooling from T = 295 K to T = 77 K in liquid nitrogen are reported. Measurements on fourteen common CPAs, including alcohols (glycerol, methanol, 2-propanol), sugars (sucrose, xylitol, dextrose, trehalose), polyethylene glycols (ethylene glycol, PEG 200, PEG 2000, PEG 20000), glycols [dimethyl sulfoxide (DMSO), 2-methyl-2,4-pentanediol (MPD)], and salt (NaCl), were performed for volumes ranging over four orders of magnitude from ∼1 nl to 20 µl, and covering the range of interest in protein crystallography. X-ray diffraction measurements on aqueous glycerol mixtures confirm that the polycrystalline-to-vitreous transition occurs within a span of less than 2% w/v in CPA concentration, and that the form of polycrystalline ice (hexagonal or cubic) depends on CPA concentration and cooling rate. For most of the studied cryoprotectants, the critical concentration decreases strongly with volume in the range from ∼5 µl to ∼0.1 µl, typically by a factor of two. By combining measurements of the critical concentration versus volume with cooling time versus volume, the function of greatest intrinsic physical interest is obtained: the critical CPA concentration versus cooling rate during flash-cooling. These results provide a basis for more rational design of cryoprotective protocols, and should yield insight into the physics of glass formation in aqueous mixtures.

Development of an ultra-small-angle X-ray scattering instrument for probing the microstructure and the dynamics of soft matter

Abstract: A Bonse–Hart camera suitable for studying dilute and weakly scattering systems has been developed. The required level of reduction of the parasitic background scattering was realised by replacing channel-cut crystals by a pair of polished and deeply etched parallel crystals. The performance of this ultra-small-angle X-ray scattering instrument is illustrated by means of two examples: the microstructure of a short-range attractive colloidal system; and the growth dynamics of soot particles in a diffusion flame. In addition, the feasibility of performing dynamic X-ray scattering experiments is demonstrated.

XRD2DScan: new software for polycrystalline materials characterization using two‐dimensional X‐ray diffraction

Abstract: XRD2DScan is a Windows application for displaying and analyzing two-dimensional X-ray diffraction patterns collected with an area detector. This software allows users to take full advantage of diffractometers that are equipped with an area detector but that cannot readily process the information contained in diffraction patterns from polycrystalline materials. XRD2DScan has many capabilities for generating different types of scans (2θ scan, ψ scan, d spacing versus ψ angle), which allows users to extract the maximum amount of information from two-dimensional patterns. Analyses of multiple data files can be fully automated using batch processing. The use of the software is illustrated through several examples.

CMPZ – an algorithm for the efficient comparison of periodic structures

Abstract: The systematic comparison of the atomic structure of solid compounds has become an important task in crystallography, chemistry, physics and materials science, in particular in the context of structure prediction and structure determination of crystalline solids. In this work, an efficient and robust algorithm for the comparison of periodic structures is presented, which is based on the mapping of the point patterns of the two structures into each other. This algorithm has been implemented as the module CMPZ in the structure visualization and analysis program KPLOT.

On the absolute calibration of bench-top small-angle X-ray scattering instruments: a comparison of different standard methods

Abstract: Absolute calibration relates the measured (arbitrary) intensity to the differential scattering cross section of the sample, which contains all of the quantitative information specific to the material The importance of absolute calibration in small-angle scattering experiments has long been recognized This work details the absolute calibration procedure of a small-angle X-ray scattering instrument from Bruker AXS The absolute calibration presented here was achieved by using a number of different types of primary and secondary standards The samples were: a glassy carbon specimen, which had been independently calibrated from neutron radiation; a range of pure liquids, which can be used as primary standards as their differential scattering cross section is directly related to their isothermal compressibility; and a suspension of monodisperse silica particles for which the differential scattering cross section is obtained from Porod's law Good agreement was obtained between the different standard samples, provided that care was taken to obtain significant signal averaging and all sources of background scattering were accounted for The specimen best suited for routine calibration was the glassy carbon sample, due to its relatively intense scattering and stability over time; however, initial calibration from a primary source is necessary Pure liquids can be used as primary calibration standards, but the measurements take significantly longer and are, therefore, less suited for frequent use

Structural redetermination, thermal expansion and refractive indices of KLu(WO4)2

Abstract: The crystal structure of monoclinic KLu(WO4)2 (KLuW) crystals was determined at room temperature by using single-crystal X-ray diffraction data. The unit-cell parameters were a = 10.576 (7), b = 10.214 (7), c = 7.487 (2) A, β = 130.68 (4)°, with Z = 4, in space group C2/c. The unit-cell parameters of KLu1−xYbx(WO4)2 were determined in relation to Yb concentration. Vickers micro-indentations were used to study the microhardness of KLuW. The linear thermal expansion tensor was determined and the principal axis with maximum thermal expansion (\alpha_{33}' = 16.72 × 10−6 K−1), X_3', was located 13.51° from the c axis. The room-temperature optical tensor was studied in the near-infrared (NIR) and visible range. The principal optical axis with maximum refractive index (ng = 2.113), Ng, was located 18.5° from the c axis at 632.8 nm. Undoped and ytterbium-doped KLuW crystals were grown by the TSSG (top-seeded-solution growth) slow-cooling method. The crystals show {110}, {\bar{1}11}, {010} and {310} faces that basically constitue the habit of the KLuW crystals.

Microradian X-ray diffraction in colloidal photonic crystals

Abstract: Ultra-high-resolution small-angle X-ray scattering in various colloidal photonic crystals is reported It is demonstrated that an angular resolution of about two microradians is readily achievable at a third-generation synchrotron source using compound refractive optics The scheme allows fast acquisition of two-dimensional X-ray diffraction data and can be realised at sample–detector separations of only a few metres As a result, diffraction measurements in colloidal crystals with interplanar spacings larger than a micrometre, as well as determination of the range of various order parameters from the width of the Bragg peaks, are made possible

A method for the non-destructive analysis of gradients of mechanical stresses by X-ray diffraction measurements at fixed penetration/information depths

Abstract: A rigorous measurement strategy for (X-ray) diffraction stress measurements at fixed penetration/information depths has been developed. Thereby errors caused by lack of penetration-depth control in traditional (X-ray) diffraction (sin2ψ) measurements have been annulled. The range of accessible penetration/information depths and experimental aspects have been discussed. As a practical example, the depth gradient of the state of residual stress in a sputter-deposited nickel layer of 2 µm thickness has been investigated by diffraction stress measurements with uncontrolled penetration/information depth and two controlled penetration/information depths corresponding to about one quarter and one tenth of the layer thickness, respectively. The decrease of the planar tensile stress in the direction towards the surface could be well established quantitatively.

A small-angle neutron scattering study of fine-scale NbC precipitation kinetics in the α-Fe–Nb–C system

Abstract: The fine-scale precipitation of NbC in ferrite has been quantitatively characterized in the temperature range 873-1073 K for two alloy compositions, containing respectively 800 p.p.m. Nb and 400 p.p.m. Nb (by weight). Transmission electron microscopy (TEM) has revealed that the precipitates are located on dislocations, and have a plate-like morphology with an average aspect ratio between 2 and 3. Small-angle neutron scattering (SANS) has been systematically used to determine the precipitation kinetics. The validity of the quantitative SANS measurements of size and volume fraction has been assessed by TEM image analysis and chemical dissolution experiments. The precipitation kinetics is observed to depend strongly on temperature but to be similar for the two alloy compositions. From the measurements, it is inferred that precipitate nucleation is extremely rapid, in relation to the nature of the nucleation sites. A time-temperature transformation diagram is built from the kinetic data, showing a maximum reaction rate between 973 and 1073 K.

Application of Bayesian analysis to indirect Fourier transformation in small-angle scattering

Abstract: Using Bayesian analysis for indirect Fourier transformation (IFT) of data from small-angle scattering (SAS) leads to probability distributions for parameters describing the experimental data. This quantification may provide extra information about the scattering system. The shape of the probability distribution for the maximum diameter of the scatterer may contain information about e.g. the heterogeneity of the scattering sample. The information content in the experimental data can be quantified as an `effective number of parameters' which can be determined from the data. The applicability of the Bayesian approach to IFT in SAS is demonstrated using simulated as well as experimental data.

Hyperquenching for protein cryocrystallography

Abstract: When samples having volumes characteristic of protein crystals are plunge cooled in liquid nitrogen or propane, most cooling occurs in the cold gas layer above the liquid. By removing this cold gas layer, cooling rates for small samples and modest plunge velocities are increased to 1.5 × 104 K s−1, with increases of a factor of 100 over current best practice possible with 10 µm samples. Glycerol concentrations required to eliminate water crystallization in protein-free aqueous mixtures drop from ∼28% w/v to as low as 6% w/v. These results will allow many crystals to go from crystallization tray to liquid cryogen to X-ray beam without cryoprotectants. By reducing or eliminating the need for cryoprotectants in growth solutions, they may also simplify the search for crystallization conditions and for optimal screens. The results presented here resolve many puzzles, such as why plunge cooling in liquid nitrogen or propane has, until now, not yielded significantly better diffraction quality than gas-stream cooling.

Method of separated form factors for polydisperse vesicles

Abstract: Use of the Schulz or Gamma distribution in the description of particle sizes facilitates calculation of analytic polydisperse form factors using Laplace transforms, {\cal L}[f(u)]. Here, the Laplace transform approach is combined with the separated form factor (SFF) approximation [Kiselev et al. (2002). Appl. Phys. A, 74, S1654–S1656] to obtain expressions for form factors, P(q), for polydisperse spherical vesicles with various forms of membrane scattering length density (SLD) profile. The SFF approximation is tested against exact form factors that have been numerically integrated over the size distribution, and is shown to represent the vesicle form factor accurately for typical vesicle sizes and membrane thicknesses. Finally, various model SLD profiles are used with the SFF approximation to fit experimental small-angle neutron scattering (SANS) curves from extruded unilamellar vesicles.

Observations of the Gibeon meteorite and the inverse Greninger-Troiano orientation relationship

Abstract: All the possible parallelism conditions associated with low-index crystallographic planes and directions in face- and body-centred cubic crystals are enumerated in terms of their minimum angle–axis misorientations. These include the Kurdjumov–Sachs (K-S), Nishiyama–Wassermann (N-W) and Pitsch orientation relationships, which are expressed in terms of close-packed (and near-close-packed) planes, close-packed directions, or both in the two crystal systems. By classifying these into either coplanar or codirectional relationships, it is possible to specify intermediate relations with either high-index directions or planes. In such a way, the Greninger–Troiano (G-T) relationship is approximated as a coplanar relation lying between the K-S and N-W relations. Based on geometric symmetry, a further correspondence relationship, which falls into the codirectional category and is located between the K-S and Pitsch relations, is represented both in pole-figure form and in Rodrigues–Frank space. It is referred to here as the inverse G-T, or G-T′. Experimental evidence from the plessite regions of the Gibeon meteorite is presented to support the validity of this relationship. It is shown that, in plessite, all the well known models, as well as their intermediates, are applicable to the γ-to-α transformation. By contrast, in the Widmanstatten portions, the misorientations are largely limited to the K-S, G-T and N-W relations.