Robust matching rules for real quasicrystals
aa r X i v : . [ c ond - m a t . o t h e r] S e p Robust matching rules for real quasicrystals
Pavel Kalugin and Andr´e Katz Laboratoire de Physique des Solides, CNRS, Universit´e Paris-Sud, Universit´e Paris-Saclay,F-91405 Orsay, France. Directeur de recherche honoraire, CNRS, FranceE-mail: [email protected]
Abstract.
We consider the problem of extraction and validation of matching rules, directlyfrom the phased diffraction data of a quasicrystal, and propose an algorithmic procedure toproduce the rules of the shortest possible range. We have developed a geometric framework toexpress such rules together with a homological mechanism enforcing the long-range quasiperiodicorder. This mechanism tolerates the presence of defects in a robust way.
1. Introduction
It is commonly acknowledged that the long-range order in quasicrystals depends on ahypothetical order propagation mechanism usually referred to as matching rules . However,so far, this understanding has been applied to the structure determination on a case by casebasis only, for instance by trying to interpret the observed structure as a decoration of a tilingalready known to have matching rules. In [1], we suggested that the exploration of matchingrules should instead be the primary goal in solving quasicrystalline structures. This article isa short presentation of this program; for further details and bibliography see [1] and referencestherein.
2. Homology-based matching rules
By the very nature of our program, the model should be locally derivable from the atomicstructure only. This requirement leads naturally to modeling matching rules in terms of simplicialtilings with vertices located at the atomic positions and labeled by their local environment. Still,even a locally deterministic triangulation may yield ambiguous results in degenerate cases (e.g.there exist two ways to cut a square in two triangles). This justifies the use of homologicalmethods, since they allow for construction of matching rules inherently insensitive to suchartificial ambiguities.Let us recall some definitions and notations of [1]. We encode the matching rules of asimplicial tiling in a geometrical object obtained by gluing together all prototiles by theirmatching faces. This yields a finite cellular complex B equipped with the metric data inheritedfrom the d -dimensional physical space E . In [1] we call B a flat-branched semisimplicial complex(or FBS-complex ). We assume that the tiling of E can be lifted to a corrugated d -surface in alarger space E ⊕ F of dimension n > d , which can be seen as a graph of the “phason coordinatefunction” ϕ : E → F . This graph is a subset of a periodic pattern in E ⊕ F . Factoring the entireconstruction over the translations of the corresponding lattice L ⊂ E ⊕ F results in wrapping a) (b) (c) Figure 1.
Three views of a square patch of a simplicial tiling for d = 2: (a) represents theoriginal patch, (b) depicts the lifted tiling with only one dimension of the phason coordinate ϕ shown and (c) represents the projection of the patch on the plane ( x, ϕ ) (only the shadow ofthe lifted patch and the projection of its boundary are shown). The oriented area of the liftedpatch projected onto the plane ( x, ϕ ) equals the integral of the mixed form ω = dϕ ∧ dx overthe lifted patch. If β : B → T n is slope locking, this integral evaluates to a boundary term andits absolute value is bounded by Kr for some K >
0. Therefore, the difference of the value of ϕ averaged over the edges of the square parallel to the x axis is bounded by K .the lifted tiling over T n = ( E ⊕ F ) / L . This wrapping can be pulled back to the FBS-complex B via the lifting map β : B → T n .To illustrate the key idea of the homology-based matching rules, let us assume for themoment that the phason gradient is constant (that is ϕ is affine). Since the phason gradientdepends on d ( n − d ) real parameters, one needs the same number of conditions to fix it. Inparticular, the condition of zero phason gradient is equivalent to the annihilation of a certain d ( n − d )-dimensional space T of d -forms on E ⊕ F by the graph of ϕ . This space (we shall refer toits elements as mixed forms ) is spanned by products of constant 1-forms in F and ( d − E .For the actual tiling models, individual lifted tiles do not generally annihilate T , and thematching rules imply only an asymptotic annihilation of T by large patches of the lifted tiling.We shall thus require such annihilation for the image of every d -cycle on B under the lifting map β (for brevity we do not make distinction between constant forms on the space E ⊕ F and thoseon its factor T n ). If β has such property (called slope locking in [1]), then for any mixed form ω ∈ T , its pullback on B is a coboundary and the integral of ω over the graph of ϕ is reducedto a boundary term. As illustrated by Figure 1, in this case the difference between the valuesof ϕ averaged over the opposite faces of an arbitrarily oriented cube of edge length r in E isbounded by some constant K independent on r . Therefore the difference between the values of ϕ averaged over any two cubes sharing a common face is also bounded by K . By partitioninga cube of edge length r into 2 n cubes of edge length r/ Kd for the difference between values of ϕ averaged over the original cube and that averaged over any of the cubes of the partition. Thisprocedure can be iterated down to the scale of individual tiles. Since the number of iterationsgrows as log ( r ) as r → ∞ , one has k ϕ ( a ) − ϕ ( b ) k < Kd log ( r ) + const, yielding k ϕ ( x ) k = O (log( k x k )) (1)and thus fixing the slope of the lifted tiling.Let us show now that the homology-based matching rules are robust with respect to thepresence of defects, which means that a small concentration of defects results in a small overallhason gradient. Defects can be conveniently introduced by assuming that the FBS-complex B of the perfect structure (the one for which the lifting map β is slope locking) is contained in alarger FBS-complex ˇ B ⊃ B , such that the extension of β to ˇ B may not be slope locking. Thedefects thus correspond to the simplices of the complement ˇ B \ B (the corresponding tiles areshown shaded on Figure 3). Let ε stand for the concentration of the defects. Then, following thereasoning above, we obtain that the difference of ϕ averaged over the opposite faces of a cubeof edge length r is bounded by K + εK r for some positive real K and K , and the maximalvalue of the phason gradient is limited by a term proportional to the density of defects: k ϕ ( x ) k = O (max(log( k x k ) , ε k x k ) . Figure 2. A d -dimensional cube ofedge length r and two sequences of nestedcubes obtained by repeated partitioning andconverging to the points a and b (fiveiterations are shown). Figure 3.
A square patch of a simplicialtiling with defects. The thick line representsthe oriented boundary of its defect-free part.Its length scales with r as O (max( εr d , r d − )),where ε is the density of defects.
3. Working with real quasicrystals
The exploration of matching rules starts with the phased diffraction data, which is seen notas an approximation to the actual structure but as a source of information about the localenvironments. We start by assigning to each atomic surface an anchoring point in T n . Then weset up the vertices of the future tiling in the physical space, placing them roughly at the at thepeaks in the phased density (some peaks should be skipped to avoid too short distances betweenvertices, see Figure 4). The vertices are also adjusted to the projections of the anchoring pointsof the corresponding atomic surfaces. Applying Delaunay triangulation to the vertex set yieldsa simplicial tiling T of finite local complexity (note that in practice we work with a finite patchof T ). The vertices of T are characterized by an atomic surface label and a translation ofthe lattice L . We call two simplices having all their vertices related by the same translation L -equivalent. Factoring T over the L -equivalence yields an FBS-complex B together with the a bc Figure 4.
The contour plot of theFourier synthesis of the phased densityfor the icosahedral quasicrystal Cd . Yb(obtained from the diffraction data of[2]), restricted to a square 30˚A × L -equivalent. Thedumbbell-shaped peak near the label(c) arises from cutting of two differentatomic surfaces. The two maxima ofthis peak are too close to each otherfor both to be retained in T . Insuch situations, the retained maximumis chosen at random, and this choicemay be different for L -equivalent peaks.Thus, the FBS-complex B will include2-simplices corresponding to both of thetriangles (c).lifting map β : B → T n (the latter takes each vertex of B to the anchoring point of thecorresponding atomic surface).In the (unlikely) case of β being slope locking, our goal is achieved. Otherwise we haveto proceed further with reduction and refinement of B . The reduction consists in gradualelimination of simplexes of B , starting with those having vertices located at the peaks ofdensity of uncertain shape or those occurring rarely in T , until the restriction of β on theremaining sub-complex B ⊂ B becomes slope-locking. It may occur that the slope locking isnot achieved before too many simplices are eliminated and B does not allow for tiling of theentire space (this is the case, for instance, when B does not admit a real cycle which imagein T n corresponds to the winding of E over T n ). In this case, B has to be refined and thenthe reduction start over again. The refinement consists in enriching the vertex labels by thelabels of neighboring vertices in T . This procedure can be repeated, each time encoding into B the information about larger local configurations. Naturally, the first time the slope lockingconditions are achieved would correspond to the matching rules of the shortest possible range.Suppose now that the proposed algorithm yielded an FBS-complex B with a slope-lockinglifting map β : B → T n . To produce an actual structure model, the matching rules encodedby B require a validation. An ultimate way to validate the matching rules would consist inconstructing a tiling obeying them and comparing the predicted diffraction intensities with theexperimental data. This is an intricate problem, since there is no systematic way to constructa tiling satisfying a given set of matching constraints. It might be therefore interesting beforetrying this to perform a partial validation, by checking other predictions of the model, namely thevalue of the total atomic density and the distribution of the density between different atomicsurfaces (see [1]). Finally, even if the existence of an infinite tiling free of matching defectsremains unproven, the robust nature of the proposed matching rules makes tilings with a smallconcentration of defects a valid structure model as well. References [1] Kalugin P and Katz A 2019
Acta Crystallographica Section A Nature materials6