The use of high-resolution imaging methods in scanning transmission electron microscopy (STEM) is limited in many cases by the sensitivity of the sample to the beam and the onset of electron beam damage (for example, in the study of organic systems, in tomography and during in situ experiments). To demonstrate that alternative strategies for image acquisition can help alleviate this beam damage issue, here we apply compressive sensing via Bayesian dictionary learning to high-resolution STEM images. These computational algorithms have been applied to a set of images with a reduced number of sampled pixels in the image. For a reduction in the number of pixels down to 5% of the original image, the algorithms can recover the original image from the reduced data set. We show that this approach is valid for both atomic-resolution images and nanometer-resolution studies, such as those that might be used in tomography datasets, by applying the method to images of strontium titanate and zeolites. As STEM images are acquired pixel by pixel while the beam is scanned over the surface of the sample, these postacquisition manipulations of the images can, in principle, be directly implemented as a low-dose acquisition method with no change in the electron optics or the alignment of the microscope itself.

Metal-organic frameworks (MOFs), with their well-ordered pore networks and tunable surface chemistries, offer a versatile platform for preparing well-defined nanostructures wherein functionality such as catalysis can be incorporated. Notably, atomic layer deposition (ALD) in MOFs has recently emerged as a versatile approach to functionalize MOF surfaces with a wide variety of catalytic metal-oxo species. Understanding the structure of newly deposited species and how they are tethered within the MOF is critical to understanding how these components couple to govern the active material properties. By combining local and long-range structure probes, including X-ray absorption spectroscopy, pair distribution function analysis, and difference envelope density analysis, with electron microscopy imaging and computational modeling, we resolve the precise atomic structure of metal-oxo species deposited in the MOF NU-1000 through ALD. These analyses demonstrate that deposition of NiOH clusters occurs selectively within the smallest pores of NU-1000, between the zirconia nodes, serving to connect these nodes along the c-direction to yield heterobimetallic metal-oxo nanowires. This bridging motif perturbs the NU-1000 framework structure, drawing the zirconia nodes closer together, and also underlies the sintering resistance of these clusters during the hydrogenation of light olefins.

While aberration correction for scanning transmission electron microscopes (STEMs) dramatically increased the spatial resolution obtainable in the images of materials that are stable under the electron beam, the practical resolution of many STEM images is now limited by the sample stability rather than the microscope. To extract physical information from the images of beam sensitive materials, it is becoming clear that there is a critical dose/dose-rate below which the images can be interpreted as representative of the pristine material, while above it the observation is dominated by beam effects. Here, we describe an experimental approach for sparse sampling in the STEM and in-painting image reconstruction in order to reduce the electron dose/dose-rate to the sample during imaging. By characterizing the induction limited rise-time and hysteresis in the scan coils, we show that a sparse line-hopping approach to scan randomization can be implemented that optimizes both the speed of the scan and the amount of the sample that needs to be illuminated by the beam. The dose and acquisition time for the sparse sampling is shown to be effectively decreased by at least a factor of 5× relative to conventional acquisition, permitting imaging of beam sensitive materials to be obtained without changing the microscope operating parameters. The use of sparse line-hopping scan to acquire STEM images is demonstrated with atomic resolution aberration corrected the Z-contrast images of CaCO3, a material that is traditionally difficult to image by TEM/STEM because of dosage issues.

We describe rippling: a tactic for the heuristic control of the key part of proofs by mathematical induction. This tactic significantly reduces the search for a proof of a wide variety of inductive theorems. We first present a basic version of rippling, followed by various extensions which are necessary to capture larger classes of inductive proofs. Finally, we present a generalised form of rippling which embodies these extensions as special cases. We prove that generalised rippling always terminates, and we discuss the implementation of the tactic and its relation with other inductive proof search heuristics.