Core Concepts

Fundamental properties and stability of Brillouin zones in integer lattices.

Abstract

The article delves into Brillouin zones, focusing on the integer lattice and its perturbations. It explores geometric and combinatorial aspects, stability under perturbations, and bounds on chamber numbers. The study includes Voronoi tessellations, plane arrangements, and Gauss circle problem analysis.
Introduction to Brillouin zones for locally finite sets in Rd.
Geometric background on bisector arrangements and Voronoi tessellations.
Types of sets like Delone sets, periodic sets, and lattices discussed.
Definition of Brillouin zones based on distances to points in a set.
Stability analysis under perturbations for integer lattices.
Bounds on distance, width, number of chambers in Brillouin zones.
Experimental illustrations of area changes in perturbed zones.
Theoretical proofs for stability and distortion of Brillouin zones.

Stats

Depending on the value of k, they express medium- or long-range order in the set.

Quotes

Key Insights Distilled From

by Herbert Edel... at **arxiv.org** 03-22-2024

Deeper Inquiries

Perturbations can affect the area exchange in Brillouin zones by causing some oscillation around the expected value of 1.0 for the area of each zone. This is due to the fact that perturbations introduce noise and minor displacements in the lattice structure, leading to deviations from the ideal volume of 1.0 for each Brillouin zone. The area exchange becomes a zero-sum game, with some fluctuation around 1.0 as perturbations impact the geometry of the zones.

The stability results regarding Brillouin zones have significant implications for crystallography studies. In crystallography, understanding and analyzing atomic arrangements within crystals are crucial for determining their properties and behavior. The stability of Brillouin zones under perturbations ensures that even small variations or disturbances in lattice structures do not drastically alter key geometric properties such as distances and widths within these zones. This stability allows researchers to confidently study crystal structures modeled as lattices, providing insights into their quantum properties and structural characteristics.

To rigorously prove the convergence of distortion to 4/π in relation to Brillouin zone boundaries, one would need to establish a formal mathematical proof based on geometric principles and analytical techniques. The proof would involve demonstrating how the length distortion between boundaries approaches this universal constant as defined by previous analyses on curve distortions. By utilizing advanced mathematical methods such as differential geometry, calculus, and geometrical transformations, one could systematically show how this convergence occurs across different boundary shapes within Brillouin zones through a rigorous deductive argument supported by mathematical reasoning and calculations.

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