Umklapp collisions and thermal conductivity

“Umklapp collisions and thermal conductivity” deals with heat conduction in a dielectric solid. Collisions of phonons are divided into Umklapp and normal according as a reciprocal lattice vector is or is not involved in the phonon momentum balance. A local temperature is defined by appeal to local thermodynamic equilibrium. An equilibrium phonon distribution can be off-centred, yet non-decaying, if the only collisions are “normal”, conserving the total phonon momentum. Then heat flow does not decay, even if a representative collision reverses the phonon group velocity. Conversely, in an Umklapp collision it is the non-conservation of phonon momentum that causes heat flow to decay.

On the use of the scattering amplitude in coherent X-ray Bragg diffraction imaging

Lens-less imaging of crystals with coherent X-ray diffraction offers some unique possibilities for strain-field characterization. It relies on numerically retrieving the phase of the scattering amplitude from a crystal illuminated with coherent X-rays. In practice, the algorithms encode this amplitude as a discrete Fourier transform of an effective or Bragg electron density. This short article suggests a detailed route from the classical expression of the (continuous) scattering amplitude to this discrete function. The case of a heterogeneous incident field is specifically detailed. Six assumptions are listed and quantitatively discussed when no such analysis was found in the literature. Details are provided for two of them: the fact that the structure factor varies in the vicinity of the probed reciprocal lattice vector, and the polarization factor, which is heterogeneous along the measured diffraction patterns. With progress in X-ray sources, data acquisition and analysis, it is believed that some approximations will prove inappropriate in the near future.

Generalized absolute values, ideals and homomorphisms in mixed lattice groups

A mixed lattice group is a generalization of a lattice ordered group. The theory of mixed lattice semigroups dates back to the 1970s, but the corresponding theory for groups and vector spaces has been relatively unexplored. In this paper we investigate the basic structure of mixed lattice groups, and study how some of the fundamental concepts in Riesz spaces and lattice ordered groups, such as the absolute value and other related ideas, can be extended to mixed lattice groups and mixed lattice vector spaces. We also investigate ideals and study the properties of mixed lattice group homomorphisms and quotient groups. Most of the results in this paper have their analogues in the theory of Riesz spaces.

A Peek into Theory of Automation with Applications to Various Branches of Science

In this paper, we shall be concerned with the existence and uniqueness of solution to three- point boundary value problems associated with a system of first order matrix difference system. Shortest and Closest Lattice vector methods are used as a tool to obtain the best least square solution of the three-point boundary value problems when the characteristic matrix D is rectangular. In this paper, we shall be concerned with the existence and uniqueness of solution to three- point boundary value problems associated with a system of first order matrix difference system. Shortest and Closest Lattice vector methods are used as a tool to obtain the best least square solution of the three-point boundary value problems when the characteristic matrix D is rectangular.

Self-dual DeepBKZ for finding short lattice vectors

In recent years, the block Korkine-Zolotarev (BKZ) and its variants such as BKZ 2.0 have been used as de facto algorithms to estimate the security of a lattice-based cryptosystem. In 2017, DeepBKZ was proposed as a mathematical improvement of BKZ, which calls LLL with deep insertions (DeepLLL) as a subroutine alternative to LLL. DeepBKZ can find a short lattice vector by smaller blocksizes than BKZ. In this paper, we develop a self-dual variant of DeepBKZ, as in the work of Micciancio and Walter for self-dual BKZ. Like DeepBKZ, our self-dual DeepBKZ calls both DeepLLL and its dual variant as main subroutines in order to accelerate to find a very short lattice vector. We also report experimental results of DeepBKZ and our self-dual DeepBKZ for random bases on the Darmstadt SVP challenge.

Pattern-matching indexing of Laue and monochromatic serial crystallography data for applications in materials science

Serial crystallography data can be challenging to index, as each frame is processed individually, rather than being processed as a whole like in conventional X-ray single-crystal crystallography. An algorithm has been developed to index still diffraction patterns arising from small-unit-cell samples. The algorithm is based on the matching of reciprocal-lattice vector pairs, as developed for Laue microdiffraction data indexing, combined with three-dimensional pattern matching using a nearest-neighbors approach. As a result, large-bandpass data (e.g. 5–24 keV energy range) and monochromatic data can be processed, the main requirement being prior knowledge of the unit cell. Angles calculated in the vicinity of a few theoretical and experimental reciprocal-lattice vectors are compared, and only vectors with the highest number of common angles are selected as candidates to obtain the orientation matrix. Global matching on the entire pattern is then checked. Four indexing options are available, two for the ranking of the theoretical reciprocal-lattice vectors and two for reducing the number of possible candidates. The algorithm has been used to index several data sets collected under different experimental conditions on a series of model samples. Knowing the crystallographic structure of the sample and using this information to rank the theoretical reflections based on the structure factors helps the indexing of large-bandpass data for the largest-unit-cell samples. For small-bandpass data, shortening the candidate list to determine the orientation matrix should be based on matching pairs of reciprocal-lattice vectors instead of triplet matching.