X-Ray Diffraction

X-rays diffraction is fundamental to understanding the structure and crystallography of biological, geological, or technological materials. X-rays scatter predominantly by the electrons in solids, and have an elastic (coherent, Thompson) and an inelastic (incoherent, Compton) component. The atomic scattering factor is largest (= Z) for forward scattering, and decreases with increasing scattering angle and decreasing wavelength. The amplitude of the diffracted wave is the structure factor, F hkl, and its square gives the intensity. In practice, intensities are modified by temperature (Debye-Waller), absorption, Lorentz-polarization, and the multiplicity of the lattice planes involved in diffraction. Diffraction patterns reflect the symmetry (point group) of the crystal; however, they are centrosymmetric (Friedel law) even if the crystal is not. Systematic absences of reflections in diffraction result from glide planes and screw axes. In polycrystalline materials, the diffracted beam is affected by the lattice strain or grain size (Scherrer equation). Diffraction conditions (Bragg Law) for a given lattice spacing can be satisfied by varying θ or λ — for study of single crystals θ is fixed and λ is varied (Laue), or λ is fixed and θ varied to study powders (Debye-Scherrer), polycrystalline materials (diffractometry), and thin films (reflectivity). X-ray diffraction is widely applied.

Diffraction of Electrons and Neutrons

Electron scattering, significantly stronger than that for X-rays, is sensitive to surfaces and small volumes of materials. Low-energy electron diffraction (LEED) provides information on surface reconstruction and the arrangement of adsorbed atoms. Reflection high energy electron diffraction (RHEED) probes surface crystallography, and monitors, in situ, mechanisms of thin film growth. Transmission electron diffraction reveals a planar cross-section of the reciprocal lattice, where intensities are products of the structure and lattice amplitude factors determined by the overall shape of the specimen. The amplitude of any diffracted beam at the exit surface oscillates with thickness (fringes) and the excitation error (bend contours). Selected area diffraction produce spot or ring patterns, where low-index zone-axis orientations reflect the symmetry of the crystal, and double-diffraction shows positive intensities even for reflections forbidden by extinction rules. Kikuchi lines appear as pairs of dark and bright lines, and help in tilting the specimen. A focused probe produces convergent beam electron diffraction (CBED), useful for symmetry analysis at nanoscale resolution. Neutrons interact with the nucleus and the magnetic moment of the atom via the spin of the neutron; the latter finds particular use in studies of magnetic order. The atomic scattering factor for neutrons shows negligible angular dependence.

Generating the atomic pair distribution function without instrument or emission profile contributions

A method for generating the atomic pair distribution function (PDF) from powder diffraction data by the removal of instrument contributions, such as Kα2 from laboratory instruments or peak asymmetry from neutron time-of-flight data, has been implemented in the computer programs TOPAS and TOPAS-Academic. The resulting PDF is sharper, making it easier to identify structural parameters. The method fits peaks to the reciprocal-space diffraction pattern data whilst maximizing the intensity of a background function. The fit to the raw data is made `perfect' by including a peak at each data point of the diffraction pattern. Peak shapes are not changed during refinement and the process is a slight modification of the deconvolution procedure of Coelho [J. Appl. Cryst. (2018), 51, 112–123]. Fitting to the raw data and subsequently using the calculated pattern as an estimation of the underlying signal reduces the effects of division by small numbers during atomic scattering factor and polarization corrections. If the peak shape is sufficiently accurate then the fitting process should also be able to determine the background if the background intensity is maximized; the resulting calculated pattern minus background should then comprise coherent scattering from the sample. Importantly, the background is not allowed complete freedom; instead, it comprises a scan of an empty capillary sample holder with a maximum of two additional parameters to vary its shape. Since this coherent scattering is a calculated pattern, it can be easily recalculated without instrumental aberrations such as capillary sample aberration or Kα2 from laboratory emission profiles. Additionally, data reduction anomalies such as incorrect integration of data from two-dimensional detectors, resulting in peak position errors, can be easily corrected. Multiplicative corrections such as polarization and atomic scattering factors are also performed. Once corrected, the pattern can be scaled to produce the total scattering structure factor F(Q) and from there the sine transform is applied to obtain the pair distribution function G(r).

Cation distribution in Cu2ZnSnSe4, Cu2FeSnS4 and Cu2ZnSiSe4 by multiple-edge anomalous diffraction

Multiple-Edge Anomalous Diffraction (MEAD) has been applied to various quaternary sulfosalts belonging to the adamantine compound family in order to validate the distribution of copper, zinc and iron cations in the structure. Semiconductors from this group of materials are promising candidates for photovoltaic applications. Their properties strongly depend on point defects, in particular related to cation order–disorder. However, Cu+, Zn2+ and Fe2+ have very similar scattering factors and are all but indistinguishable in usual X-ray diffraction experiments. Anomalous diffraction utilizes the dependency of the atomic scattering factors f′ and f′′ of the energy of the radiation, especially close to the element-specific absorption edges. In the MEAD technique, individual Bragg peaks are tracked over an absorption edge. The intensity changes depending on the structure factor can be highly characteristic for Miller indices selected for a specific structural problem, but require very exact measurements. Beamline KMC-2 at synchrotron BESSY II, Berlin, has been recently upgraded for this technique. Anomalous X-ray powder diffraction and XAFS compliment the data. Application of this technique confirmed established cation distribution in Cu2ZnSnSe4 (CZTSe) and Cu2FeSnS4 (CFTS). In contrast to the literature, cation distribution in Cu2ZnSiSe4 (CZSiSe) is shown to adopt a highly ordered wurtz-kesterite structure type.

Refinement of organic crystal structures with multipolar electron scattering factors

A revolution in resolution is occurring now in electron microscopy arising from the development of methods for imaging single particles at cryogenic temperatures and obtaining electron diffraction data from nanocrystals of small organic molecules or macromolecules. Near-atomic or even atomic resolution of molecular structures can be achieved. The basis of these methods is the scattering of an electron beam due to the electrostatic potential of the sample. To analyse these high-quality experimental data, it is necessary to use appropriate atomic scattering factors. The independent atom model (IAM) is commonly used although various more advanced models, already known from X-ray diffraction, can also be applied to enhance the analysis. In this study a comparison is presented of IAM and TAAM (transferable aspherical atom model), the latter with the parameters of the Hansen–Coppens multipole model transferred from the University at Buffalo Databank (UBDB). By this method, TAAM takes into account the fact that atoms in molecules are partially charged and are not spherical. Structure refinements were performed on a carbamazepine crystal using electron structure-factor amplitudes determined experimentally [Jones et al. (2018). ACS Cent. Sci. 4, 1587–1592] or modelled with theoretical quantum-mechanical methods. The results show the possibilities and limitations of the TAAM method when applied to electron diffraction. Among others, the method clearly improves model fitting statistics, when compared with IAM, and allows for reliable refinement of atomic thermal parameters. The improvements are more pronounced with poorer-resolution diffraction data.

Synthesis, Characterization, and Crystal Structure of N-(3-nitrophenyl)cinnamamide

N-(3-nitrophenyl)cinnamamide 1 with formula C15H12N2O3 was synthesized, and its crystal structure was determined by single-crystal X-ray diffraction analysis. Compound 1 crystallizes in the monoclinic space group P21/n with unit cell dimensions: a = 6.7810 (5) Å, b = 23.0913 (15) Å, c = 8.2079 (5) Å, V = 1282.76 (15) Å3, Z = 4, determined at 150 K with MoKα radiation. The experimental structure refined against atomic scattering factors is compared with the structure obtained using a Hirshfeld Atom Refinement (HAR) approach and Density Functional Theory (DFT) geometry optimizations.