On the origin of crystallinity: a lower bound for the regularity radius of Delone sets

Abstract

The mathematical conditions for the origin of long‐range order or crystallinity in ideal crystals are one of the very fundamental problems of modern crystallography. It is widely believed that the (global) regularity of crystals is a consequence of `local order', in particular the repetition of local fragments, but the exact mathematical theory of this phenomenon is poorly known. In particular, most mathematical models for quasicrystals, for example Penrose tiling, have repetitive local fragments, but are not (globally) regular. The universal abstract models of any atomic arrangements are Delone sets, which are uniformly distributed discrete point sets in Euclidean d space. An ideal crystal is a regular or multi‐regular system, that is, a Delone set, which is the orbit of a single point or finitely many points under a crystallographic group of isometries. The local theory of regular or multi‐regular systems aims at finding sufficient local conditions for a Delone set X to be a regular or multi‐regular system. One of the main goals is to estimate the regularity radius for Delone sets X in terms of the radius R of the largest `empty ball' for X. The celebrated `local criterion for regular systems' provides an upper bound for for any d. Better upper bounds are known for d ≤ 3. The present article establishes the lower bound for all d, which is linear in d. The best previously known lower bound had been for d ≥ 2. The proof of the new lower bound is accomplished through explicit constructions of Delone sets with mutually equivalent (2dR − ϵ)‐clusters, which are not regular systems. The two‐ and three‐dimensional constructions are illustrated by examples. In addition to its fundamental importance, the obtained result is also relevant for the understanding of geometrical conditions of the formation of ordered and disordered arrangements in polytypic materials.

Boric Acid (H3(BO3): Recognize The Molecular Interactions in Solutions

“…Unit sel asam borat (164) Gambar 11. Ikatan hidrogen asam borat (164) H 3 (BO 3 ) terdiri atas tiga atom penyusun yaitu H, B dan O jari-jari ketiga atom ini memiliki perbedaan jari-jari. Jari-jari atom merupkan jarak dari inti ke elektron terluar dari suatu atom.…”

On the regularity radius of Delone sets in $\mathbb{R}^3$

“…For dimensions 1 and 2 the exact values are known: ρ1 = 2R and ρ2 = 4R. For d = 1 the verification is straightforward, and for d = 2 the proof follows from the Local Criterion as well as the general lower bound ρd ≥ 4R, d ≥ 2, described in [4,5,1] (or its generalization ρd ≥ 2dR, d ≥ 2, proved in [1]).…”

“…The property whether or not a Delone set in Euclidean space is a regular system or a multi-regular system is invariant under scaling. Thus, beginning with Section 3, we will focus entirely on Delone sets with parameters 1 2 and R (that is, with r = 1 2 ); these are the sets with distance between points at least 1.…”

Embedding parallelohedra into primitive cubic networks and structural automata description

“…The problem of crystallization is one of the fundamental and difficult problems of crystallography. The geometric approach for modeling crystalline structures is a local theory of regular systems that was developed by B. N. Delone, his associates and followers (Delone et al, 1976;Dolbilin, 1976;Baburin et al, 2018). Though significant progress has been made in developing the local theory, rigorously proven results have not yet been used by practicing crystallographers.…”