핵심 개념
Fundamental properties and stability of Brillouin zones in integer lattices.
초록
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.
통계
Depending on the value of k, they express medium- or long-range order in the set.