Asymptotic analysis for the mean first passage time in finite or spatially periodic 2D domains with a cluster of small traps

A hybrid asymptotic-numerical method is developed to approximate the mean first passage time (MFPT) and the splitting probability for a Brownian particle in a bounded two-dimensional (2D) domain that contains absorbing disks, referred to as "traps”, of asymptotically small radii. In contrast to previous studies that required traps to be spatially well separated, we show how to readily incorporate the effect of a cluster of closely spaced traps by adapting a recently formulated least-squares approach in order to numerically solve certain local problems for the Laplacian near the cluster. We also provide new asymptotic formulae for the MFPT in 2D spatially periodic domains where a trap cluster is centred at the lattice points of an oblique Bravais lattice. Over all such lattices with fixed area for the primitive cell, and for each specific trap set, the average MFPT is smallest for a hexagonal lattice of traps. doi:10.1017/S1446181121000018

Asymptotics of the principal eigenvalue of the Laplacian in 2D periodic domains with small traps

We derive and numerically implement various asymptotic approximations for the lowest or principal eigenvalue of the Laplacian with a periodic arrangement of localised traps of small \[\mathcal{O}(\varepsilon )\] spatial extent that are centred at the lattice points of an arbitrary Bravais lattice in \[{\mathbb{R}^2}\] . The expansion of this principal eigenvalue proceeds in powers of \[\nu \equiv - 1/\log (\varepsilon {d_c})\] , where d c is the logarithmic capacitance of the trap set. An explicit three-term approximation for this principal eigenvalue is derived using strong localised perturbation theory, with the coefficients in this series evaluated numerically by using an explicit formula for the source-neutral periodic Green’s function and its regular part. Moreover, a transcendental equation for an improved approximation to the principal eigenvalue, which effectively sums all the logarithmic terms in powers of v, is derived in terms of the regular part of the periodic Helmholtz Green’s function. By using an Ewald summation technique to first obtain a rapidly converging infinite series representation for this regular part, a simple Newton iteration scheme on the transcendental equation is implemented to numerically evaluate the improved ‘log-summed’ approximation to the principal eigenvalue. From a numerical computation of the PDE eigenvalue problem defined on the fundamental Wigner–Seitz (WS) cell for the lattice, it is shown that the three-term asymptotic approximation for the principal eigenvalue agrees well with the numerical result only for a rather small trap radius. In contrast, the log-summed asymptotic result provides a very close approximation to the principal eigenvalue even when the trap radius is only moderately small. For a circular trap, the first few transcendental correction terms that further improves the log-summed approximation for the principal eigenvalue are derived. Finally, it is shown numerically that, amongst all Bravais lattices with a fixed area of the primitive cell, the principal eigenvalue is maximised for a regular hexagonal arrangement of traps.

Crystallographic analysis of the lattice metric (CALM) from single electron backscatter diffraction or transmission Kikuchi diffraction patterns

A new software is presented for the determination of crystal lattice parameters from the positions and widths of Kikuchi bands in a diffraction pattern. Starting with a single wide-angle Kikuchi pattern of arbitrary resolution and unknown phase, the traces of all visibly diffracting lattice planes are manually derived from four initial Kikuchi band traces via an intuitive graphical user interface. A single Kikuchi bandwidth is then used as reference to scale all reciprocal lattice point distances. Kikuchi band detection, via a filtered Funk transformation, and simultaneous display of the band intensity profile helps users to select band positions and widths. Bandwidths are calculated using the first derivative of the band profiles as excess-deficiency effects have minimal influence. From the reciprocal lattice, the metrics of possible Bravais lattice types are derived for all crystal systems. The measured lattice parameters achieve a precision of <1%, even for good quality Kikuchi diffraction patterns of 400 × 300 pixels. This band-edge detection approach has been validated on several hundred experimental diffraction patterns from phases of different symmetries and random orientations. It produces a systematic lattice parameter offset of up to ±4%, which appears to scale with the mean atomic number or the backscatter coefficient.

Distribution rules of systematic absences and generalized de Wolff figures of merit applied to electron backscatter diffraction ab initio indexing

A new method for electron backscatter diffraction ab initio indexing is reported that adopts several methods originally invented for powder indexing. Distribution rules of systematic absences and error-stable Bravais lattice determination are used to eliminate the negative influence of non-visible bands and erroneous information from visible bands. In addition, generalized versions of the de Wolff figures of merit are proposed as a new sorting criterion for the obtained unit-cell parameters, which can be used in both orientation determination and ab initio indexing from Kikuchi patterns. Computational results show that the new figures of merit work well, similar to the original de Wolff Mn . The ambiguity of the indexing solutions is also pointed out, which happens in particular for low-symmetry cells and may generate multiple distinct solutions even if very accurate positions of band centre lines and the projection centre are given. It is supposed that this is the reason why indexing was successful in an orthorhombic case but not in a triclinic cell.

ASYMPTOTIC ANALYSIS FOR THE MEAN FIRST PASSAGE TIME IN FINITE OR SPATIALLY PERIODIC 2D DOMAINS WITH A CLUSTER OF SMALL TRAPS

A hybrid asymptotic-numerical method is developed to approximate the mean first passage time (MFPT) and the splitting probability for a Brownian particle in a bounded two-dimensional (2D) domain that contains absorbing disks, referred to as “traps”, of asymptotically small radii. In contrast to previous studies that required traps to be spatially well separated, we show how to readily incorporate the effect of a cluster of closely spaced traps by adapting a recently formulated least-squares approach in order to numerically solve certain local problems for the Laplacian near the cluster. We also provide new asymptotic formulae for the MFPT in 2D spatially periodic domains where a trap cluster is centred at the lattice points of an oblique Bravais lattice. Over all such lattices with fixed area for the primitive cell, and for each specific trap set, the average MFPT is smallest for a hexagonal lattice of traps.

Computational scanning tunneling microscope image database

We introduce the systematic database of scanning tunneling microscope (STM) images obtained using density functional theory (DFT) for two-dimensional (2D) materials, calculated using the Tersoff-Hamann method. It currently contains data for 716 exfoliable 2D materials. Examples of the five possible Bravais lattice types for 2D materials and their Fourier-transforms are discussed. All the computational STM images generated in this work are made available on the JARVIS-STM website (https://jarvis.nist.gov/jarvisstm). We find excellent qualitative agreement between the computational and experimental STM images for selected materials. As a first example application of this database, we train a convolution neural network model to identify the Bravais lattice from the STM images. We believe the model can aid high-throughput experimental data analysis. These computational STM images can directly aid the identification of phases, analyzing defects and lattice-distortions in experimental STM images, as well as be incorporated in the autonomous experiment workflows.

Mathematical Tools that Connect Different Indexing Analyses

As mathematical tools that can be commonly used for indexing analyses from different types of experimental patterns, we have recently developed (i) rules on forbidden hkl’s that can be used even when the space group and setting are unknown, (ii) an algorithm for error-stable Bravais lattice determination, (iii) generalization of the de Wolff figure of merit for powder diffraction (1D data) to data in higher-dimensions such as Kikuchi patterns (2D data) by electron backscatter diffraction (EBSD). In particular, (ii) could be used in a variety of situations, not just for indexing. It is explained how (i)–(iii) are used in the mathematical framework of our indexing algorithms.

Nonadiabatic Atomic-Like State Stabilizing Antiferromagnetism and Mott Insulation in MnO

This paper reports evidence that the antiferromagnetic and insulating ground state of MnO is caused by a nonadiabatic atomic-like motion, as is evidently the case in NiO. In addition, it is shown that experimental findings on the displacements of the Mn and O atoms in the antiferromagnetic phase of MnO corroborate the presented suggestion that the rhombohedral-like distortion in antiferromagnetic MnO, as well as in antiferromagnetic NiO is an inner distortion of the monoclinic base-centered Bravais lattice of the antiferromagnetic phases.

Computational Design Generation and Evaluation of Beam-Based Tetragonal Bravais Lattice Structures for Tissue Engineering

The study and application of computational design is gaining importance in biomedical engineering as medical devices are becoming more complex, especially with the emergence of 3D printed scaffold structures. Scaffolds are medical devices that act as temporary mechanical support and facilitate biological interactions to regenerate damaged tissues. Past computational design studies have investigated the influence of geometric design in lattice structured scaffolds to investigate mechanical and biological behavior. However, these studies often focus on symmetric cubic structures leaving an opportunity for investigating a larger portion of the design space to find favorable scaffold configurations beyond these constraints. Here, tissue growth behavior is investigated for tetragonal Bravais lattice structured scaffolds by implementing a computational approach that combines a voxel-based design generation method, curvature-based tissue growth modeling, and a design mapping technique for selecting scaffold designs. Results show that tetragonal unit cells achieve higher specific tissue growth than cubic unit cells when investigated for a constant beam width, thus demonstrating the merits in investigating a larger portion of the design space. It is seen that cubic structures achieve around 50% specific growth, while tetragonal structures achieve more than 60% specific growth for the design space investigated. These findings demonstrate the need for continued adaption and use of computational design methodologies for biomedical applications, where the discovery of favorable solutions may significantly improve medical outcomes.