Bounding the regularity radius for regular crystals

A central question in crystallography is how (or if) a globally crystallographic pattern – a crystal – can be determined by local rules. Therefore it is natural to ask `how far is local?'. In the previous issue of Acta Cryst. A, Baburin et al. (2018) give a partial answer for a particular class of (mathematical, idealized) crystals: they consider only `regular systems'. A regular system is essentially a crystal with only one type of atom, where each atom is surrounded in the same way by its neighbours. They show that in arbitrary dimension d `local' means at least 2dR. Here R denotes the radius of the largest empty ball in the crystal (compare the grey ball in Fig. 1). For instance, for the primitive lattice in dimension two we obtain d = 2 and R = 2^1/2/2, and hence 2dR = 2 (2^1/2). The result says for this example that we need to know at least all neighbourhoods of each atom with radius 2 (2^1/2) in order to ensure that it is indeed a crystal.

Figure 1 The orange points form a non-crystalline point set. The green balls of radius 2 (2^1/2) can detect the non-crystallinity of the point set. Smaller balls like the blue one cannot: they all look alike.

Commonly used mathematical models for crystals (or quasicrystals, or more general structures) are Delone sets. A Delone set is an (infinite, discrete) point set X such that (i) there is such that each open ball of radius r contains at most one point of X, and (ii) there is such that each closed ball of radius R contains at least one point of X. The points of X represent the (ideal) atomic positions of some structure. One milestone in mathematical crystallography is the Local Theorem (Delone et al., 1976): it provides a necessary and sufficient local condition for a Delone set X being a crystal. In a nutshell this result states that the Delone set X is a crystal if and only if the number of different local patterns of X of radius ρ stays bounded if ρ tends to infinity. For a more precise version of the statement see Delone et al. (1976).

In order to count the number of different local patterns properly it is useful to define the cluster-counting function. For let denote the (centred) ρ-cluster , where denotes the ball of radius ρ centred in x. The cluster-counting function is the number of different (centred) ρ-clusters in X. Note that it is important to consider centred clusters: for instance, in the integer lattice there are several different ρ-clusters of the form if x is arbitrary, but there is only one kind of ρ-cluster for any particular value ρ if x is required to lie in .

From now on we consider this particular case where there is only one kind of ρ-cluster. If ρ is very small (e.g. ) this does not imply anything on X: all ρ-clusters in X consist of only one point. If ρ is large, and there is only one kind of ρ-cluster in X (up to congruence), then by the Local Theorem X is necessarily crystallographic. Hence it is natural to ask for good (upper and lower) bounds on the value such that, if there is only one ρ-cluster in some Delone set X in d-dimensional Euclidean space, then X is necessarily crystallographic. Thus depends on the dimension d, but is universal for all Delone sets X in d-dimensional Euclidean space.

A commonly used mathematical model of a crystal is the orbit of one point, or of several (inequivalent) points under a crystallographic group in d-di­men­sio­nal Euclidean space. In the first case the corresponding (infinite) Delone set is called a regular system, in the latter case the Delone set is called a multiregular system. One particular instance of the question of the origin of crystallinity is to find good bounds on the value described above for regular systems. Let us call the smallest such the regularity radius (of all regular systems X in a given dimension d).

Since we may scale any crystallographic Delone set X arbitrarily, the bounds on ought to be expressed in terms of R (the radius of the largest empty ball in X). In dimensions d = 1 and d = 2 the exact values of the corresponding are known: and (see e.g. Barburin et al., 2018; Dolbilin, 2018). Similar arguments as in Dolbilin (2018) yield that is at least 4R for any . Engel conjectured that in dimension three we have (Engel, 1986).

A good lower bound on for Delone sets in arbitrary dimension is obtained in Barburin et al. (2018): it is shown that is at least 2dR (Theorem 5.8). In particular, grows at least linearly in the dimension d. Hence there is no general bound on independent of d. The result is obtained by a sophisticated construction of Delone sets X (`Engel sets') with only one kind of centred ρ-cluster of radius such that X still is not a regular system. The construction works in any dimension .

This recent result shows that there are still profound questions and answers found in mathematical crystallography today. A next step might be to treat the corresponding question for multiregular systems. Here one cannot expect a regularity radius such that beyond that radius there exists only one congruence class of ρ-clusters. Rather one would require that beyond there are at most m types of ρ-clusters, where m is the number of different orbits with respect to the underlying crystallographic group.