Convergent-Beam X-ray Crystallography

TL;DR

Convergent-beam X-ray diffraction (CBXD) integrates high-convergence optics with diffraction topography to map Bragg reflections across a crystal volume, enabling spatially resolved structure-factor measurements. The authors develop a geometric and data-analysis framework that combines snapshot and rotation CBXD, including tomographic reconstruction of crystal morphology and the computation of structure factors as a function of position within the crystal. They validate the approach on silicon and vitamin B$_{12}$ crystals, achieving structure-factor accuracy within a few percent after applying corrections for polarization, Lorentz factor, and absorption, and they demonstrate 3D reconstruction of crystal shape via tomography. The method offers a versatile route to study diffusion, binding, growth, and dynamic responses in molecular, polymeric, and MOF crystals, potentially enabling region-specific crystallography and dose-aware analyses in time-resolved experiments.

Abstract

Molecular and polymeric crystals show a wide range of functional properties that arise from the interplay between the atomic-scale structure of their constituent molecules and the organization of these molecules within the crystal lattice at macroscopic length scales. X-ray diffraction can provide structural information at these disparate length scales, but usually only through experiments that address one or the other of molecular (or unit-cell) structure versus crystal structure. Consequently, the accuracy of determined molecular or polymer structures may be limited by unaccounted crystal inhomogeneities of the crystal lattice and the characterization of crystalline materials might not reveal the underlying causes of crystal morphology. Here we introduce X-ray convergent-beam diffraction to obtain spatially-resolved structural information from crystals by projection topographic imaging. Using highly focusing X-ray multilayer Laue lenses, we show that Bragg reflections can be mapped into tomographic images of the crystal, for the characterization of strain and defects at high resolution. We demonstrate how the crystal morphology obtained this way can be accounted for when determining structure factors as a function of position in the crystal. The approach may assist in studies such as diffusion and binding in MOFS, protein-drug binding, crystal growth, and the mechanical responses of photo-reactive or thermally driven dynamic crystals.

Figures (9)

• Figure 1: The geometry of CBXD. (a) A crystal is placed in the beam focused by an X-ray lens (here, a pair of MLLs) and the diffraction pattern is recorded on a plane pixel-array detector. (b) Of all rays provided by the lens, only those that satisfy the diffracting condition for a RLP $\mathbf{G}_{{hk\ell}}$ will reflect. A ray originating from an angle $\phi_y$ to the optical axis maps to a position $\Delta f\tan \phi_y$ on a crystal defocused by $\Delta f$. (c) A volume in reciprocal space (blue shading) is formed by Ewald spheres rotated over a range of $\mathbf{k}_\mathrm{in}$ wave-vectors as supplied by the lens. Any $\mathbf{G}_{{hk\ell}}$ within this volume will produce a reflection, $\mathbf{k}_\mathrm{out}$. (d) The diffraction condition for $\mathbf{G}_{{hk\ell}}$ is satisfied for all wave-vectors that lie on a Kossel circle $\mathbf{k}_K(\chi)$ (shown in orange) in a plane perpendicular to and bisecting $\mathbf{G}_{{hk\ell}}$. A Bragg streak and a deficiency line are generated if $\mathbf{k}_K(\chi)$ passes through the lens aperture. (e) The deficiency line (in the projected lens pupil) and Bragg streak are approximately parallel on the detector and separated by an angle $2\theta_B$. (f) Crystal diffraction for a given incident ray path is found by integrating over all paths $L_\mathrm{in} + L_\mathrm{out}$ for all possible scattering locations $s_0$ along the incident path, assuming single scattering.
• Figure 2: Vitamin B$_{12}$ snapshot convergent-beam diffraction. (a) A calculated diffraction pattern for a photon energy of 17.5k, with Bragg streaks coloured according to the position of the $\mathbf{k}_\mathrm{in}$ wave-vectors in the lens aperture as mapped in (b). The corresponding deficiency lines are mapped across the lens pupil in (c). (d) Experimental CBXD pattern of a vitamin B$_{12}$ crystal recorded with a lens of $\mathrm{NA} = 0.028$ at a defocus of $\Delta f =3.5\mm$, and in the same orientation as calculated in (a). (e) The mapping of the measured Bragg streaks to their corresponding incident $\mathbf{k}_\mathrm{in}$ wave-vectors in the lens pupil. (f) Optical image of the crystal from the beamline in-line microscope for the crystal orientation used for (d).
• Figure 3: Estimates of a topograph of a vitamin B$_{12}$ crystal determined from a single snapshot CBXD pattern. (a) Mapping of measured Bragg streaks to the positions of their deficiency lines, before normalisation. (b)--(d) Iterates of the topograph, as consistency between overlapping deficiency lines, and reflection intensities of same reflections across patterns is enforced. (e) Scaled intensity (blue) and relative error (red) of the 641 reflection, showing convergence after 8 iterations. (f) Histograms of the relative error of all observed reflections for various numbers of iterations.
• Figure 4: Rotation CBXD topographs of a silicon wedged single crystal with a hole, cut as a perpendicular lamella from a (100) Si wafer using a focused ion beam (FIB). (a) Scanning electron micrographs of the crystal. (b) The deficiency line and Bragg streak remain separated by $2\theta$ and move together (here horizontally) as the crystal is rotated, to map out the projected view of a defocused crystal. (c) Rotation diffraction pattern formed by summing exposures over a range of $\pm15°$ about the vertical axis with a step size of 0.02°. (d) A distorted topogram from the $\bar{1}1\bar{3}$ reflection. (e) Line profiles of the measured and simulated topogram intensity along the positions indicated in (d). Experimental (f) and simulated (g) rotation topographs obtained with a coarse rotation step of 0.2°.
• Figure 5: Coarse-step rotation topographs recorded with 0.2° step size. (a) In-situ micrograph of a needle-like vitamin B$_{12}$ crystal near the orientation used to collect a series of diffraction patterns (b), in which the positions of four topographs are indicated, shown in detail in (c) to (f). The detector pixel coordinates of the topographs are displayed.