Brillouin Zones of Integer Lattices and Their Perturbations

TL;DR

This work analyzes Brillouin zones Zone_k for locally finite sets A ⊂ ℝ^d with a focus on the integer lattice and its perturbations. It develops a geometric-combinatorial framework via bisector arrangements and Voronoi tessellations, establishes stability of Zone_k under bounded perturbations, and proves a linear upper bound on the number of chambers in Zone_k for planar lattices while giving higher-dimensional bounds via inversion to k-sets. It further shows that the maximum chamber diameter tends to zero for the unperturbed lattice but can be kept bounded away from zero under certain perturbations, highlighting a delicate balance between order and perturbation. The results have relevance for understanding medium- and long-range order in crystalline and aperiodic structures and provide computational data and methods to explore these Brillouin structures in practice.

Abstract

For a locally finite set, $A \subseteq \mathbb{R}^d$, the $k$-th Brillouin zone of $a \in A$ is the region of points $x \in \mathbb{R}^d$ for which $\|x-a\|$ is the $k$-th smallest among the Euclidean distances between $x$ and the points in $A$. If $A$ is a lattice, the $k$-th Brillouin zones of the points in $A$ are translates of each other, which tile space. Depending on the value of $k$, they express medium- or long-range order in the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone under perturbations, a linear upper bound on the number of chambers in a zone for lattices in $\mathbb{R}^2$, and the convergence of the maximum volume of a chamber to zero for the integer lattice.

Figures (9)

• Figure 1: Left: the arrangement of bisectors defined by the point in the center and all other points in the integer lattice. Starting with the second, every fourth Brillouin zone is colored dark blue alternating with light blue. Middle: the $6$-th Brillouin zone sandwiched between two circles centered at the point in the center. Right: the order-$k$ Brillouin tessellation of the integer lattice obtained by overlaying the order-$5$ with the order-$6$ Voronoi tessellations or, equivalently, by drawing the $6$-th Brillouin zones of all points in the integer lattice.
• Figure 1: The solid curves show the min and max distances, $r_k$ and $R_k$, of the $k$-th Brillouin zones of $0 \in {\mathbb Z}{\hbox{${\mathbb Z}$}}^2$ from $0$, together with their lower and upper bounds. For comparison, the dotted curves show the same information for a strong perturbation of ${\mathbb Z}{\hbox{${\mathbb Z}$}}^2$, so the lowest and highest curves are the bounds from \ref{['eqn:perturbed_lower_and_upper_bound']}. As detailed in Appendix \ref{['app:A']}, the dotted curves graphing the min and max distances of the $k$-th Brillouin zone in the perturbed integer lattice are provably correct up to $k = 34$ and may possibly be contaminated by missing bisectors starting from $k = 35$ onward.
• Figure 1: The open blue ball, $B_0$, is used to define the finite set $A$. The open yellow ball, $B(0,1)$, does not contain any integer points apart from $0$. Assuming $\tau \leq \min ( d_1 , d_2)$, the points of $A$ and their perturbations avoid the magenta half-space, $H$, which is necessary to apply Lemma \ref{['lem:stability_of_crossing']}. In fact, in order to remove the dependence of Lemma \ref{['lem:stability_of_crossing']} on $u$ and $A$, we need to choose $\tau$ even smaller to avoid the gray half-space, $H_\ell$. Note that the distances $d_1, d_2$ remain constant as we rotate $u$.
• Figure 1: The image of the integer lattice under inversion. The dotted lines and circles are the images of nine vertical and nine horizontal integer lines. The blue unit circle is preserved by the inversion. Since all integer points other than $0$ lie on or outside the unit circle, their images under the inversion all lie on or inside the unit circle.
• Figure 1: By assumption, $S(y,R)$ is no smaller than $S(x,r)$. The half-circle of points $a \in S(y,R)$ with ${\langle a-y , y-x \rangle} \geq 0$ is highlighted. The cone with angle $\theta_0 = \arccos \sqrt{2}/6$ is guaranteed to contain at least one of the four poles of $S(y,R)$ together with the nearby integer point.

Theorems & Definitions (34)

• Definition 1: Brillouin Zones
• Lemma 2: Thickened Sphere
• Proof 1
• Proposition 3: Plane Arrangements
• Proposition 4: Spherical Order-$k$ Voronoi Tessellation
• Corollary 5: Spherical Order-$k$ Brillouin Tessellation
• Proof 2
• Theorem 1: Width for Integer Lattices
• Proof 3
• Theorem 2: Width for Perturbed Integer Lattices