Paracrystalline: Amorphous silicon has traditionally been represented by a continuous random network model in which there is no long-range ordering for the atoms, and some have less than fourfold coordination, which form dangling bonds—a type of defect. Treacy and Borisenko (p. 950 ; see the Perspective by Gibson ) used fluctuation electron microscopy to explain that models including regions of crystalline order are needed to fit the observed local variations in structure. Thus, on the 1- to 2-nanometer-length scale, this material should be thought of as having a paracrystalline structure containing localized crystalline regions.

The geometry of the most regular polycatenanes and weavings, as an extended family of discrete knots and catenanes, is described in terms of sticks and corners in their optimal embeddings.

Using electron correlograph analysis we show that coherent nanodiffraction patterns from sputtered amorphous silicon indicate that there is more local crystallinity in unannealed amorphous silicon than was previously suspected. By comparing with simulations for various models we show that within a typical unannealed amorphous silicon film a substantial volume fraction (>50%) is topologically crystalline with correlation lengths up to 2~nm. Electron correlograph analysis is a variant of the fluctuation electron microscopy technique and its sensitivity to local crystalline ordering is derived from its sensitivity to four-body correlations.

We explore the flexibility windows of the 194 presently-known zeolite frameworks. The flexibility window represents a range of densities within which an ideal zeolite framework is stress-free. Here, we consider the ideal zeolite to be an assembly of rigid corner-sharing perfect tetrahedra. The corner linkages between tetrahedra are hard-sphere oxygen atoms, which are presumed to act as freely-rotating, force-free, spherical joints. All other inter-tetrahedral forces, such as coulomb interactions, are ignored. Thus, the flexibility window represents the null-space of the kinematic matrix that governs the allowable internal motions of the ideal zeolite framework. We show that almost all of the known aluminosilicate or aluminophosphate zeolites exhibit a flexibility window. Consequently, the presence of flexibility in a hypothetical framework topology promises to be a valuable indicator of synthetic feasibility. We describe computational methods for exploring the flexibility window, and discuss some of the exceptions to this flexibility rule.

A simple parameterization is presented for the elastic electron scattering cross sections from single atoms into the annular dark-field (ADF) detector of a scanning transmission electron microscope (STEM). The dependence on atomic number,Z, and inner reciprocal radius of the annular detector,q₀, of the cross section σ(Z,q₀) is expressed by the empirical relation whereA(q₀) is the cross section for hydrogen (Z= 1), and the detector is assumed to have a large outer reciprocal radius. Using electron elastic scattering factors determined from relativistic Hartree-Fock simulations of the atomic electron charge density, values of the exponentn(Z,q₀) are tabulated as a function ofZandq₀, for STEM probe sizes of 1.0 and 2.0 Å.Comparison with recently published experimental data for single-atom scattering [Krivanek et al. (2010).Nature464, 571–574] suggests that experimentally measured exponent values are systematically lower than the values predicted for elastic scattering from low-Zatoms. It is proposed that this discrepancy arises from the inelastic scattering contribution to the ADF signal. A simple expression is proposed that corrects the exponentn(Z,q₀) for inelastic scattering into the annular detector.