A Markov theoretic description of stacking disordered aperiodic crystals including ice and opaline silica
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Acta Crystallographica Section A
Foundations ofCrystallography
ISSN 0108-7673© 2017 International Union of CrystallographyPrinted in Singapore – all rights reserved
A Markov theoretic description of stacking disorderedaperiodic crystals including ice and opaline silica
A. G. Hart, a * T. C. Hansen b and W. F. Kuhs c a University of Bath, Bath, UK, ab Institut Laue-Langevin, Grenoble, France, and c GZG Abt. Kristallogra-phie, Universit¨at G¨ottingen, Germany. Correspondence e-mail: [email protected]
We review the Markov theoretic description of 1D aperiodic crystals, describ-ing the stacking-faulted crystal polytype as a special case of an aperiodic crystal.Under this description we generalise the centrosymmetric unit cell underlying atopologically centrosymmetric crystal to a reversible Markov chain underlying areversible aperiodic crystal. We show that for the close-packed structure, almostall stackings are irreversible when the interaction reichweite s >
4. Moreover,we present an analytic expression of the scattering cross section of a large classof stacking disordered aperiodic crystals, lacking translational symmetry of theirlayers, including ice and opaline silica (opal CT). We then relate the observedstackings and their underlying reichweite to the physics of various nucleation andgrowth processes of disordered ice. We proceed by discussing how the derivedexpressions of scattering cross sections could significantly improve implementa-tions of Rietveld’s refinement scheme and compare this Q -space approach to thepdf-analysis of stacking disordered materials.
1. Introduction
1D aperiodic crystals are similar to ordinary crystals by virtueof being translationally symmetric in two independent direc-tions yet differ by being aperiodic in the third. Consequently,they cannot be described by just an underlying unit cell and lat-tice, suggesting the need to go beyond the usual language andformalism of crystallography to describe them.The contemporary description of these crystals begins with apaper by Hendricks & Teller (1942), proposing that the aperi-odic direction is composed of a sequence of layers where theprobability of a layer being a certain type depends on somefinite number of near neighbour layers. Jagodzinski (1949) laterassumed this dependence was caused by an interatomic interac-tion with constant range he called reichweite , which he denotedwith the integer s if and only if a layer’s type depends on s pre-ceding layers. This sequence of layers was described by Marti et al. (1981) as a sequence of random variables where a givenvariable depends on only finitely many preceding variables;which the authors recognised as a Markov chain of order s .They successfully analysed this description of aperiodic crystalsby partitioning the layer chain into blocks comprising s adja-cent layers and noting each block depends only on the imme-diately preceding block, hence reducing the chain to first order.The approach taken by Marti et al. (1981) has been used practi-cally by Cherepanova & Tsybulya (2004) to model graphite likematerials as Markov chains, allowing them to study their X-Raydiffraction pattern. Hostettler et al. (2002) expressed the aver-age structure factor of the aperiodic orange crystal HgI usingthe same principle, hence could study the crystal’s diffractionpattern.The probability of a particular layer being a certain typedepending on some nearest neighbour interaction is strongly analogous to the axial next nearest neighbour Ising (ANNNI)model, as noticed by Christy (1989) and Shaw & Heine (1990).Christy (1989) went further and used the ANNNI model to com-pare the suite of polytypes generated for sapphirine and wollas-tonite each with reichweite equal to 4. An overview of aperiodiccrystal theory is provided by Estevez-Rams et al. (2007) and theelementary treatment of Markov theory to describe 1D aperiod-icity is presented by Welberry (2004).In addition to simple crystal structures, aperiodic polytypeshave been studied by Varn & Crutchfield (2004) who simulatedtransforms between ordered and disordered polytypes, specif-ically the solid state transformation of annealed ZnS crystalsfrom a 2H to a 3C. Furthermore, Varn et al. (2013) and Varn &Crutchfield (2004) have made significant progress relating thestatistics of stacking faults distributed throughout a general ape-riodic crystal to the crystal’s scattering pattern using the tools ofinformation theory. The authors cast the problem of finding thesimplest possible process that could give rise to an aperiodiccrystal exhibiting the observed scattering pattern as definingthe crystal’s ε -machine, which may be represented as a directedgraph or a hidden Markov model (HMM). The HMM describesa Markov process with states hidden from the observer, but eachemitting an observable (in this context a layer type) followingsome probability distribution depending on the state. The resul-tant sequence of observables (layer types) may not be a Markovchain of any order, and in this case would describe an aperiodiccrystal with infinite reichweite . However, all Markov chains offinite order can be described by some HMM, so the formula-tion of an aperiodic crystal as a HMM generalises the Markovtheoretic description of aperiodic crystals.This work on creating an information theoretic descriptionof aperiodic crystals has culminated in the new field of chaotic Acta Cryst. (2017). A , 000000 Hart, Hansen and Kuhs · Aperiodic crystals as Markov chains a r X i v : . [ phy s i c s . c h e m - ph ] F e b esearch papers crystallography summarised elegantly by Varn & Crutchfield(2015), who invite the reader to view the work on aperiodiccrystals as the foundations of a new generalised crystallogra-phy, encompassing the study of disordered materials as well theas translationally symmetric and ordered structures studied byso-called classical crystallography .Our present paper builds upon chaotic crystallography byfirst reviewing the description of 1D aperiodic crystals thatencompasses both periodic and aperiodic polytypes. Under thisdescription we generalise the notion of a topologically cen-trosymmetric crystal having an underlying centrosymmetricunit cell to a reversible aperiodic crystal having an underlyingreversible Markov chain. By extending the notion of acentricityto aperiodic crystals, it may be possible to explain the physicalproperties of aperiodic crystals in formal analogy to the prop-erties of acentric periodic crystals. Examples include the piezo-electric effect exhibited by acentric crystals which may have ananalogous effect exhibited by aperiodic crystals with an under-lying irreversible Markov chain.This paper also adds to the chaotic crystallographer’s mathe-matical toolbox. We use the matrix formulation of Markov the-ory to build on the work of Jagodzinski (1954) to constructnew analytic expressions for the differential scattering crosssection of 1D aperiodic crystals with finite reichweite s . Sev-eral similar derivations do of course exist, including that ofKakinoki & Komura (1954) who reached a different expres-sion, that is mathematically interesting but more cumbersomethan ours. There is also the expression for the average struc-ture factor product of a single layer derived by Allegra (1961),which is superseded by our expression for the entire scatter-ing cross section. More recently Riechers et al. (2015) pro-duced a deep theoretical paper relating the scattering pattern ofa close-packed structure to its underlying ε -machine, in doingso, produced the most general possible expression for the scat-tering cross section of a close-packed structure. However, theirexpression was derived by exploiting a feature of the close-packed structure that does not apply to all aperiodic crystals,namely that the layer types are translations of one another in realspace. This assumption is very powerful, as the authors demon-strate by deriving closed form expressions for the diffractionpattern of crystals with independent and identically distributedlayers, crystals exhibiting random growth and stacking faults,and crystals exhibiting Shockley-Frank stacking faults. How-ever, the assumption that a crystal’s constituent layers differonly by a translation in real space does not hold for the ape-riodic ice described by Hansen et al. (2008 a ), nor does it holdfor aperiodic opal, composed of a disordered sequence of cristo-balite and tridymite layers. Motivated by aperiodic ice and opal,we have derived a general expression for the cross section foran aperiodic crystal with finite s that does not assume differentlayer types have translational symmetry. Since we are interestedin finite s , we need only consider Markov theory without wor-rying about the theory of HMMs. However, for completeness,we do include a derivation in Appendix A of the cross sectionof the general aperiodic crystal described by a HMM withoutthe assumption of translationally symmetric layer types. The computational cost of evaluating our cross section iscompared to the cross section developed by Hendricks & Teller(1942) and Warren (1959) then implemented by Treacy et al. (1991) in the software package DIFFaX.We argue that our expression is computationally efficient, andcould be used to significantly improve several modern uses ofRietveld (1969)’s scheme to refine theoretical cross sectionsof real aperiodic crystals like ice or opal. Examples includeKuhs et al. (2012), Berliner & Werner (1986) and Hansen et al. (2008 a ) who needlessly employed a Monte Carlo simulation toestimate the scattering cross section of ice which could havebeen estimated more accurately at a much greater speed usingour expression for the cross section.
2. Forging the Markov chain
We begin by representing a 1D aperiodic crystal as a sequenceof layers, each of a finite set of distinct layer types. The proba-bility that a layer is a particular type is assumed to depend on thetype of each of finitely many consecutive preceding layers. Thenumber of interacting preceding layers is denoted s and called reichweite as coined by Jagodzinski (1949).Following the method employed by Marti et al. (1981), thecrystal is partitioned into blocks each comprising s consecutivelayers - then the set of distinct blocks is indexed with the set B . Blocks go by different names in different papers, including structural motifs in the work of Michels-Clark et al. (2013), butwe will continue to refer to them as blocks . We notice that ablock’s type depends only on the block immediately precedingit, revealing that a complete sequence of blocks is generated bya first order Markov process. For those interested in forming ananalogy with the ANNNI model, consider a nearest neighbourIsing chain with m discrete spin states, rather than just 2 states;up and down. The spin state at each site is analogous to theblock type of a particular set of s consecutive layers, hence m is equal to the number of distinct block types | B | . The m stateANNNI model is usually analysed using the transfer matrix method (Baxter, 2007), which is essentially what we proceedwith here. Making use of the probabilist’s lexicon, we definethe transition matrix (another name for the transfer matrix) Ξ with elements ξ i j representing the probability that a block istype j ∈ B given its predecessor is type i ∈ B . Next, elementaryresults of Markov theory reveal that the probability of a blockbeing type j given that the ν th preceding block is type i is the i j th element of the matrix Ξ ν ; the transition matrix raised to thepower ν .We now seek to compute the probability π i that a block sam-pled from an aperiodic crystal is type i . Here, to sample a crystal is to randomly, and with equi-probability, select a position froman infinite crystal and observe which type of block is at thatposition. By assumption, the probability of sampling a j blockis related to the probability of the previous block being type i ,which gives rise to the self-consistency condition π j = (cid:88) i ∈ B π i ξ i j (1)which may be expressed in matrix notation π Ξ = π (2) Hart, Hansen and Kuhs · Aperiodic crystals as Markov chains
Acta Cryst. (2017). A , 000000 esearch papers where π i are elements of the so-called stationary distributionvector π , a left eigenvector of the transition matrix Ξ with cor-responding eigenvalue 1. The row vector π contains the ele-ments of a probability distribution, so satisfies the normalisationcondition (cid:88) i ∈ B π i = . (3) The pair-correlation function G i j ( ν ) with i j ∈ B is defined asthe probability that a layer block sampled from a crystal is type i and the block ν blocks ahead is type j . We observe that G i j ( ν ) are the components of the matrix G ( ν ) = Diag ( π ) Ξ ν (4)where Diag ( π ) is the diagonal matrix with diagonal entries thecomponents of the vector π . We shallanalyse G ( ν ) as a sequence in ν under the technical conditionsthat the transition matrix Ξ generates a chain that is positiverecurrent , irreducible and aperiodic ; this allows us to build adescription of aperiodic crystals which can be later extendedto polytypes. We justify these three conditions by assuming anaperiodic crystal has three respective physical properties. Thefirst, is that after observing that a block is type i , the mean num-ber of blocks after which another is type i is finite. The second,is that after observing that a block is type i there will certainlybe a block of type j somewhere ahead of the block i . Thirdly,all block types have period k =
1, where the period k is defined k = gcd { ν > ( Ξ ν ) ii > } (5)where gcd stands for greatest common divisor. It suffices toassume any block type has non-zero probability of followingany other to satisfy all three condition. For a chain that is posi-tive recurrent, irreducible and aperiodic, each row of the matrix G ( ν ) converges as ν → ∞ with exponential order to the sameunique steady state row vector π ; specifically the convergence is O ( | λ | ν ) where λ is the second largest eigenvalue of the tran-sition matrix Ξ . Proof is included in many books on Markovtheory like Norris (1998). This result is useful because it allowsa crystallographer to estimate how distant a pair of blocks mustbe before it becomes reasonable to approximate their separa-tion as infinite, and therefore assume their types are uncorre-lated. There is however some subtlety to this, which is coveredin more detail by Riechers et al. (2015). Marti et al. (1981) observed that 1D aperiodic crystals can be representedby a finite order Markov chain, a special type of an ε -machineconsidered by Varn & Crutchfield (2015) in their discussionof chaotic crystallography. The entropic density h discussed byVarn & Crutchfield can be expressed in terms of the matrix for- malism so far developed h = − (cid:88) i j π i ξ i j log ( ξ i j ) (6)for ease of computation in practical scenarios. The entropy ratecan be informally thought of as measuring an aperiodic crys-tal’s disorder per unit length in the aperiodic direction, max-imising at h = log ( | B | ) when any block could follow its pre-decessor with equi-probability, and minimising at h = h can be com-puted in the same way for ensembles that are usually thoughtof as periodic crystals with stack defects, and provides a mea-sure of how much order there is to the distribution of stackdefects. For example, a crystal comprising stack defects thattend to be nearly equally spaced will have a lower entropy ratethan one with the same frequency of stack defects distributedwith equi-probability throughout the crystal. One might expectfrom thermodynamic considerations that the entropic densitywill increase with increasing temperature. Varn et al. (2013) and Riechers et al. (2015) have remarked that a polytypecomprising a periodic array of unit cells is in fact a special caseof an aperiodic crystal. In the formalism used here, a Markovchain with transition matrix Ξ underlies a polytype if and only ifits state space can be reduced to a closed communicating class C with the additional property that ξ i j ∈ { , } ∀ i j ∈ C . Less for-mally, any Markov chain that eventually underlies a sequence ofblocks, where the probability of some block following the nextis either unity or zero is a polytype.Now, recall that 1D aperiodic crystals are described by pos-itive recurrent , irreducible and aperiodic chains and note thatthis cannot be said of the chains that underlie polytypes. How-ever, we can reduce a polytype’s underlying chain to the closedcommunicating class C which is both positive recurrent and irreducible forming sufficient conditions for the stationary statevector π to exist (Norris, 1998). Conveniently, a Markov chainunderlying a polytype differs from that underlying an aperiodiccrystal by reducing to a chain that is periodic . Neutron and X-ray diffraction experiments can measure a 1Daperiodic crystal’s differential scattering cross section, whichwe seek to express as concisely as possible. We begin by notingthat a 1D aperiodic crystal possesses long range order acrossthe basel plane spanned by two primitive lattice vectors (cid:126) a and (cid:126) b and is aperiodic along the direction plane’s normal (cid:126) c . We let N a and N b denote the number of unit cells across the respectivelattice vectors spanning the basal plane, and note that the crys-tal’s cross sectional area is therefore N a × N b . Our notation hasbeen standard so far - but we now break convention and let N c denote the number of blocks (rather than unit cells) stacked inthe (cid:126) c direction, which together comprise the entire crystal (Weshould note here that our crystals are now cuboids). Returning Acta Cryst. (2017). A , 000000 Hart, Hansen and Kuhs · Aperiodic crystals as Markov chains esearch papers to convention, we denote positions in reciprocal space by (cid:126) Q = π ( h (cid:126) a ∗ + k (cid:126) b ∗ + l (cid:126) c ∗ ) (7)where (cid:126) a ∗ , (cid:126) b ∗ and (cid:126) c ∗ are primitive lattice vectors and h , k , l arereal numbers. For a macroscopic crystal where N a and N b arelarge we have an expression for the differential cross sectiondeveloped by Berliner & Werner (1986) d σ ( (cid:126) Q ) d Ω = sin ( N a π h ) sin ( π h ) sin ( N b π k ) sin ( π k ) × (8) N c (cid:88) m = − N c ( N c − | m | ) Y m ( (cid:126) Q ) e π im l . Here, Y m ( (cid:126) Q ) is the average structure factor product which hasexpression Y m ( (cid:126) Q ) = (cid:88) i ∈ B (cid:88) j ∈ B G i j ( m ) F i ( (cid:126) Q ) F ∗ j ( (cid:126) Q ) (9)where G i j ( m ) is exactly the pair-correlation discussed in sec-tion 2.2; the probability of finding a layer block of type j sepa-rated by a distance m c from a block of type i . Finally, F i is thestructure factor of a type i block, F ∗ i is its complex conjugate,and h , k are nodal lines. Berliner and Werner’s expression(8) was derived using the works of Wilson (1942), Hendricks& Teller (1942) and Jagodzinski (1949). Further, (8) has beenused practically to study the cross section of computer gener-ated statistically faulted crystals by Berliner & Werner (1986)themselves in an effort to analyse stacking-faults in the 9R lat-tice and compare the results of simulation to those measured forLi at 20K. More recently, in a pair of papers by Hansen et al. (2008 a ) & (2008 b ), equation (8) was used to compute the dif-ferential scattering cross section of stack-faulted cubic ice.Since (8) is widely used in simulations, it would be useful touse the matrix formalism of Markov theory to derive an equiv-alent analytic expression that can be efficiently evaluated by acomputer. Starting from equations (8) and (9) then building onwork by Jagodzinski (1954) it can be shown d σ d Ω = sin ( N a π h ) sin ( π h ) sin ( N b π k ) sin ( π k ) Re (cid:26) Tr (cid:0) ˆ P ( ˆ S + N c I ) ˆ F (cid:1)(cid:27) (10)where the terms are defined as follows. First, let Q be the invert-ible square matrix transforming Ξ into its Jordan Normal form ˆ Ξ like so Ξ = Q − ˆ Ξ Q (11)allowing us to define ˆ P = Q − Diag ( π ) Q (12)for Diag ( π ) the diagonal matrix with entries the components ofthe stationary distribution vector π . Next, we call F the struc-ture matrix with elements F i j = F i F ∗ j where F i and F ∗ i arethe structure factor and conjugate structure factor of an i blockrespectively. We then let ˆ F = Q − FQ (13) and note that Tr denotes the trace. Finally, if Ξ is diagonalisablethen ˆ S is a diagonal matrix with entries s n = (cid:40) N c ( N c − ) if λ n e π il = λ n e π il ( λ Ncn e π ilNc + N c ( − λ n e π il ) − )( − λ n e π il ) otherwise, (14)where λ n are the eigenvalues shared by the transition matrix Ξ and its diagonal representation ˆ Ξ . On the other hand, if Ξ is not diagonalisable (defective), then ˆ S is an upper triangularmatrix with a more complicated expression. See Appendix B fora derivation of equation (10), where the unlikely and somewhatpathological case of a defective transition matrix Ξ is also cov-ered. By unlikely , we mean that almost all transition matricesare diagonalisable, in the measure theoretic sense that almostall real numbers between 0 and 1 are not equal to . The rea-son we bother with the defective case at all is to ensure thatan optimisation algorithm seeking to find the set of transitionprobabilites that best fits experimental data would not ‘break’ ifthe algorithm were to try to evaluate the cross section using anear defective transition matrix; where we informally define anear defective matrix as one for which the process of diagonali-sation is numerically unstable. With equation (10) now defined,we use Berliner & Werner (1986)’s observation that the crosssection of a macroscopic crystal can be well approximated bytaking the limit as N a and N b tend to infinity, revealing d σ d Ω = N a N b δ ( h − h ) δ ( k − k ) Re (cid:26) Tr (cid:0) ˆ P ( ˆ S + N c I ) ˆ F (cid:1)(cid:27) (15)We note for experimentalists here that taking a powder aver-age of equation (10) can be problematic near 2 θ = θ = π (measured during backscattering experiments) where broad-ened Bragg spots cut the Ewald shell almost tangentially andare therefore in significant need of a Lorentz correction. On theother hand, the Lorentz correction hardly varies at angles wellaway form 2 θ = θ = π allowing one to easily bringexpression (10) to bear on powder averaged samples for manyvalues of θ .This expression allows the differential scattering cross sec-tion to be computed much more efficiently than a popularmethod employed by Berliner & Werner (1986) to investigateLithium at 20K, as well as by Hansen et al. (2008 a ) and byKuhs et al. (2012) to study so-called cubic ice. The popularmethod estimates the pair-correlation functions G i j ( m ) , whichform part of the Berliner and Werner’s expression for the aver-age structure factor product (9) and scattering cross section (8).This section compares this method’s performance to evaluatingexpression (10) directly. The popular method begins by sam-pling block of layers from the crystal. Then a randomly selectedlayer is fixed atop the seed with probability dependant on pre-ceding s layers, which are initially the set of layers composingthe seed. The algorithm continues to grow the crystal by itera-tively affixing a new layer atop the last with layer determined bythe previous s layers; this process continues until the crystallitesize is of the order of the coherently scattering domain.The estimate for the pair-correlation G i j ( m ) is taken by sim-ply counting up the number of times a layer of of type j is found Hart, Hansen and Kuhs · Aperiodic crystals as Markov chains
Acta Cryst. (2017). A00
Acta Cryst. (2017). A00 , 000000 esearch papers m layers ahead of one of type i , then dividing by the total num-ber of trials. Since the estimate is reached by sampling a phasespace (the set of possible layer sequences) then taking an aver-age to estimate a thermodynamic variable (probability of pairoccurrence), this method is a Monte Carlo (MC) simulation. The traditional MC algorithm estimates the pair-correlationfunctions with an uncertainty converging toward 0 as the num-ber of crystals grown, hence samples taken, tends to infinity.The convergence is asymptotically proportional to the recipro-cal square root of the number of samples taken. In big O nota-tion, the convergence is O ( N − ) where N is the number ofsamples (Rubinstein & Kroese, 2008). Consequently high accu-racy comes at a heavy computational cost, specifically everyadditional correct decimal point appended to the estimate forthe pair-correlation function requires that 100 times more sam-ples be taken, requiring that the simulation be run for 100 timeslonger. Having obtained an estimate for the pair-correlationfunction, the MC algorithm can compute the scattering crosssection at each value of l in O ( | B | ) time, where | B | is the num-ber of distinct layer blocks. On the other hand, the time takento evaluate equation (10) for many values of l does not dependon the size of the crystal N , instead we compute (10) in O ( | B | ) for each value of l .Further, since we express (10) exactly, it can be evaluatedby a computer as precisely as compounded rounding errorson floating point arithmetic allow. This is deceptively usefulbecause the Levenberg-Marquardt (1963), or similar, algorithmemployed by Rietveld’s refinement scheme (Rietveld, 1969)requires that the sum of square residuals be differentiated withrespect to each free parameter and evaluated at every data pointfor each iteration. The square residual sum is expressed in termsof the cross section, hence Rietveld’s method demands that thecross section be differentiated with respect each free parame-ter including the transition probabilities. Now, it is shown inAppendix C that analytic derivatives with respect to parame-ters of the structure factor do exist for (10), but neither theMonte Carlo algorithm nor (10) easily yield analytic derivativeswith respect to the transition probabilities. Consequently, thesederivatives must be evaluated numerically using a finite differ-ence. The finite difference approximation evaluates a functionat two (or more) points, separated by a small, but finite, lengththen uses the values to estimate the derivative at a nearby point.Crucially, the approximation is better as the separation betweenpoints gets smaller, tending to the exact derivative as the sepa-ration tends to 0. Consequently, computing the cross section toa high precision permits a smaller finite difference, producing abetter approximated matrix of derivatives, resulting in a muchbetter behaved descent algorithm. Treacy et al. (1991) have written a user friendly and flexi-ble program in FORTRAN computing the X-ray/neutron scat-tering pattern of stack faulted crystal structures. The program:Diffracted intensities from faulted Xtals (DIFFaX), computes a crystal’s differential scattering cross section using an equa-tion that is fundamentally the same as that developed by Hen-dricks & Teller (1942), Warren (1959), and us in this paper.However, their derivation and heuristic is somewhat different.Roughly speaking, Treacy et al. (1991) consider that the scat-tering pattern for a crystal is equal to the pattern of the samecrystal shifted upward by one layer superposed with the patternproduced by a single layer placed under the shifted crystal. Byrelating the scattering pattern to itself recursively, a set of simul-taneous equations can set up and solved to yield the scatteringpattern of a stack faulted or aperiodic crystal. Solving this setof simultaneous equations has computational complexity | B | ,as does evaluating (10). However, the time taken to evaluate theexponential function is the most time intensive process if thenumber of block types | B | is small, which is true for low inter-action range and low number of layer types. Though the origi-nal DIFFaX does not support a least squares refinement of freeparameters, the work in progress DIFFaX+ being developed byLeoni (2017) does support Rietveld refinement.
3. Close-packed structures
The topologically close-packed aperiodic crystal is ubiqui-tous in nature and has some interesting mathematical features.First of all, the atomic coordination of a close-packed struc-ture is determined by assuming a crystal layers’ constituentatoms are equally sized hard spheres that are physically stackedbetween other layers. The layers (so-called modular layers ) areconstrained by the structure’s geometry to 1 of 3 distinct equi-librium positions labelled A , B and C such that adjacent layerscannot have the same relative position. A cannot follow A , B cannot follow B , nor can C follow C . We denote a sequence oflayers by printing the positions A , B and C and the order theseappear in the crystal, exemplified in figure 1. ... ABCABCBACBA ...
Figure 1
A subset of a layer sequence.
It can be seen from figure 2 that if a layer is flanked by lay-ers of the same type, the resultant lattice is hexagonal. If it isflanked by layers of different types, the resultant lattice is cubic.We label hexagonally stacked layers H and cubic ones K inaccordance with Wyckoff-Jagodzinski notation, then notice alllayer sequences have unique representation as a stack sequence.Under this representation, the layer sequence in figure 1 has aunique stack sequence illustrated and explained in figure 3. Acta Cryst. (2017). A , 000000 Hart, Hansen and Kuhs · Aperiodic crystals as Markov chains esearch papers Figure 2
Stacking sequences of close-packed layers of atoms. A-first layer (with out-lines of atoms shown); B-second layer; C-third layer. Reproduced from Ham-mond (2009) with permission of the International Union of Crystallography.More images and a short overview of the close-packed structure is provided byKrishna & Pandey (1981).
We recall that for a 1D aperiodic crystal, a given layers’ typedepends on finitely many consecutive preceding layers s . Fora close-packed 1D aperiodic crystal, we observe that the stacktype of a layer depends on the stack type of s − reichweites equal to 4 can be characterised by the transition probabilities α , β , γ and δ that a K stack follows an HH , HK , KH and KK pair of stacks respectively. ... KKKKHKKKK ...
Figure 3
All layers are flanked by layers of different type, hence they are cubic, exceptfor the layer C at the centre of the subsequence ... BCB ... which is hexagonal.
The Markov process generating the stack sequence is inti-mately related to that producing the layer sequence. Afterall, the labels A , B and C are deployed arbitrarily to rep-resent a sequence that can be equally well represented inWyckoff-Jagodzinski HK notation. In order to formally relatethese equivalent representations, we first declare that two layersequences l and l belong to the same equivalence class l ∼ l if and only if there exists a permutation on the set of layer types φ such that l = φ ◦ l . We note that all layer sequences belong-ing to the same class have the same stack sequence representa-tion, consequently, the sequences belonging to the same equiv-alence class are described by the same set of transition proba-bilities; hence they are represented by the same Markov chain.Figure 4 displays a stack sequence on its right hand side withmembers of the corresponding layer sequence equivalence classon its left. We let the crystal’s stack block Markov chain havetransition matrix X and stationary state vector p . ... ABCABCABCABC ......
ACBACBACBACB ......
BACBACBACBAC ......
BCABCABCABCA ......
CABCABCABCAB ......
CBACBACBACBA ...
Figure 4
These six layer sequences belong to the same equivalence class because thereis a permutation on the set of layer types that will map any one of the sequencesto any other. For example, the permutation φ : { A , B , C } → { A , B , C } defined φ ( A ) = A , φ ( B ) = C , φ ( C ) = B is employed to map the first sequence inthe list to the second. This equivalence class comprises all layer sequences withperfectly cubic stack sequence KKKKKKKKKK . Spherically close-packed structures are not the only struc-tures that can be described in
ABC or HK notation. For example,1D aperiodic ice crystals are open-packed, but placing hypo-thetical spheres with appropriate radii at the midpoint of eachof an ice crystal’s hydrogen bonds results in an ensemble ofspheres that is close-packed. Hence, we say that cubic ice istopologically close-packed and describe it using the same ABC or HK notation as real close-packed structures. Similarly, sil-icon carbide layers are composed of tetrahedra arranged withspherical close-pack topology, as can be seen in figure 5. Infor-mally, any aperiodic 1D crystal that can be described in ABC or HK notation has the same spherical close-pack topology. Hart, Hansen and Kuhs · Aperiodic crystals as Markov chains
Acta Cryst. (2017). A , 000000 esearch papers Figure 5
Here the labels 1 , , A , B , C . Notice the arrangement of tetrahedra isthe same as the arrangement of spheres dipslayed in figure 2. Reproduced froman educational paper by Ortiz et al. (2013) on the prolific polytypism of siliconcarbide, with permission of the International Union of Crystallography. The simplest close-packed structure is a crystal composed ofthree layer types A , B , an C that are identical up to some trans-lation in real space, so their structure factors are equal up tosome rotation in reciprocal space, as observed by Yi & Canright(1996). This structure has been studied extensively by manyauthors including Berliner & Werner (1986), Yi & Canright(1996), and Riechers et al. (2014), but is insufficient to describethe aperiodic ice studied by Hansen et al. (2008 a ) and Hansen et al. (2015). In fact, Hansen et al. explain that the content ofan ice layer depends on that of its neighbours; so are forced toconsider the existence of 6 distinct layers AB , AC , BA , BC , CA ,and CB each with their own structure factor. These layers areobtained by considering a pair of layers A and B for example,then shifting the borders of the layer A in the aperiodic direc-tion by half a unit cell, obtaining a layer containing half the A layer and half the B layer; then labelling this layer AB . This isillustrated in figure 6. ... AB (cid:122)(cid:125)(cid:124)(cid:123) B C A B C A B (cid:124) (cid:123)(cid:122) (cid:125) BC ... Figure 6
New layer types from old
Hansen et al. (2008 a ) provide complete molecular detailsincluding the structure factor of each of the 6 layer types,including the important observation that the layers are not sim-ply identical up to translation. The authors further assume thata layer depends on some finite number s of previous layers,allowing ice to be described as an aperiodic crystal built from asequence of layer blocks each with length s . Under the condi-tions that A cannot follow A , B cannot follow B nor can C follow C , ice comprises | B | = × s distinct block types. The crosssection of ice is therefore given by equation (10), which is interms of an appropriate transition matrix Ξ and structure matrix F . The remainder of this section describes how one might con-struct these matrices, though before we begin, we note that forlow s the resulting matrices are small, and it may be possible tocleverly construct the transition matrix using methods similarto those deployed by Riechers et al. (2015) to produce analyticexpressions for the eigenvalues and cross section.First of all, we need to define what is meant by a block oftype i when discussing ice. To do this, we index the set of blocktypes with a subset of the integers { , , ... , × s } (cid:51) i usingthe following scheme. First, find the quotient q and remainder r upon dividing i − i − = q + r . The remainderis either 0 , , , , , or 5; so for each of these numbers respec-tively, set the first layer in the block as AB , AC , BA , BC , CA , or CB . Now express the quotient in binary, producing a sequenceof s − X , Y , Z ∈ { A , B , C } andobserve that if the n th layer of the block is the layer XY , theneither it is the last layer of the block, or the next layer is Y Z for Z taking one of 2 values (since Z (cid:54) = Y ). If the n th digit of thebinary representation of q is 0 then choose Z depending on Y asfollows. If Y = A , set Z = B , if Y = B , set Z = C , and if Y = C ,set Z = A . On the other hand if the n th digit of the binary rep-resentation of q is 1, then if Y = A , set Z = C , if Y = B , set Z = A , and if Y = C , set Z = B . This procedure determines thefirst of a block’s layers, then constructs the n th layer from the n − AB , AC , BA , BC , CA , or CB so set the remain-der r to the respective index 0 , , , , , or 5 depending on thefirst layer. Next, define a sequence of s − n + AB , BC , or CA set the n th binary digit to 0, otherwise, set the digit to 1. The sequenceof digits is a binary representation of a number q . With q and r determined, set the index i = q + r + reichweite , it is now clearwhat we mean when referring to a block of type i . For example,if s = BA AB BA AC , which canbe seen by noting 21 − = × +
2, so r =
2, hence the
Acta Cryst. (2017). A , 000000 Hart, Hansen and Kuhs · Aperiodic crystals as Markov chains esearch papers first of the block’s layers is BA , and the quotient 3 has binaryrepresentation 011, fixing the next three layers as AB , BA , and AC .Next, we need the structure factor of the i th block F i whichwe can find as follows. Let f in be the structure factor of the n thlayer in the block of type i ; which has expression found in thework of Hansen et al. (2008 a ). Now a block’s structure factor isjust a superposition of the structure factor of each layer shiftedto the right position, so F i = s − (cid:88) n = f in exp (cid:18) π im lns (cid:19) . (16)Now we can fully define the structure matrix F by recalling its i j th elements are just F i F ∗ j .Next on the agenda is to obtain the i j th element of the transi-tion matrix. These elements can be discovered by noting that ablock i defines a stack sequence ( ) in HK notation with length s −
1. The concatenation of blocks i and j defines a second stacksequence ( ) with length 2 s − s − i . Suppose an arbitrary stack sequenceof length 2 s − s − ( ) , then the probability that the remaining stacksform sequence ( ) is the probability that block i is followedby block j ; which specifies the i j th element of the transitionmatrix.For example, when s = Ξ has element ξ
21 35 representing the probability that a block of type 35 willfollow one of type 21. Using the index scheme, notice this is theprobability that the block
CA AC CA AC will follow the block
BA AB BA AC . This is exactly the probability of
ACAC follow-ing
BABAC which equals the probability of
HHHH following
HHK . Recalling that the probability of K following HH , HK , KH , and KK is given by α , β , γ , and δ respectively, we havethat ξ
21 35 = ( − β )( − γ )( − α ) .With the structure matrix F and transition matrix Ξ nowdefined, computing the cross section of aperiodic ice is just amatter of evaluating equation (10). It is also possible to com-pute the entropic density h with only the transition matrix. There is ample X-ray diffraction data (Graetsch et al. ,1994),(Guthrie et al. , 1995) to suggest that the opal CT formof silica is composed of disordered (cubic) cristobalite and(hexagonal) tridymite layers, in topological identity to disor-dered ice. Like ice, these opaline forms of silica are topologi-cally close-packed stackings of
ABC (cristobalite) and AB layers(tridymite), in which the packing centers are located at the linearmidpoint of the Si-O-Si covalent bond chains, i.e. close to the(likely disordered) oxygen position. This view is supported byHRTEM studies in which the stacking disorder could be directlyvisualised (Elzea & Rice, 1996). It was also recognised that thecrystallite sizes of the stacking disordered opals is nanoscopic(Guthrie et al. , 1995) leading to diffraction broadening. It wasfurther suggested that additionally disordered, more amorphousregions could be an intrinsic part of opal CT. The various micro-structural aspects of opal CT were recently discussed by Wilson (2014), who highlighted some discrepancies between diffrac-tion and spectroscopic findings. As suspected by Guthrie et al. (1995), a straightforward assignment of cubic and hexagonalpeak intensities in the complex first diffraction (triplet) peakto the relative proportions of tridymite and cristobalite layersseems unjustified; a statement supported by the work of Elzea& Rice (1996).Arasuna et al. (2013) have suggested that a largely amor-phous water-containing opal (so-called opal A) that undergoesheating and annealing will transform (under loss of water) con-tinuously into a progressively more crystalline opal CT form,finally becoming a material dominated by cristobalite stackings;though the authors adopt a simplified view of stacking disorderwhich is unlikely to be quantitatively correct. A more involvedtreatment of stacking disorder and micro-crystallinity as pre-sented here and in the past for ice (Hansen et al. , 2008 a ) (Kuhs et al. , 2012) has not yet been applied to opal but appears highlydesirable as it may well resolve some of the open issues on thenature of disorder in opal CT as discussed by Wilson (2014). Inany case, the annealing of amorphous silica via stacking disor-dered opal CT into a largely crystalline form close to the melt-ing point of silica shows a close resemblance to the annealing ofamorphous ice with the main difference that opals drive towardsa cubic form close to melting while ice prefers a hexagonalarrangement.It is also noteworthy that Guthrie et al. (1995) were inter-ested in the X-ray diffraction pattern of opal containing watermolecules. Our formalism could capture this by letting an opalunit cell containing an H O molecule at a particular position andorientation within the cell be a new layer type with appropriatestructure factor.
We are interested in whether a 1D aperiodic crystal isreversible, which we define informally as whether it looks thesame (in some statistical sense) upside down. Before proceed-ing, we should declare that we have appropriated the word reversible from Markov theory and do not mean it in the ther-modynamic sense of a process maintaining a constant entropy.We also restrict ourselves to the special case of a topologi-cally close-packed 1D aperiodic crystal in order to convenientlyrelate this theory to the experimental findings of Hansen et al. (2008 a ) & (2008 b ); but note that the idea of a reversible crystalis quite general. Further, we restrict ourselves to crystals withlayer types that are individually inversion symmetric. Next, weobserve that since a set of layer chain equivalence classes isrelated to a unique stack chain, we say that a layer chain equiv-alence class contains reversible layer chains if and only if itsrelated stack chain is reversible.With the preamble out of the way, we define a reversible crys-tal as one for which the probability of sampling a type i blockof stacks and discovering the next block is type j matches theodds of sampling from the crystal a block with stack sequencethe reverse of j , and noting its successor is the block type withstack sequence reversing that of block type i . In other words,sampling a two block long sequence of stacks running from the Hart, Hansen and Kuhs · Aperiodic crystals as Markov chains
Acta Cryst. (2017). A , 000000 esearch papers beginning of an i block’s sequence to the end of a j block’s is aslikely as the sequence running from the end of the j block’s tothe beginning to the i block’s. Formally, an aperiodic crystal isreversible if and only if its underlying Markov process satisfiesthe reversibility condition p i X i j = p σ ( j ) X σ ( j ) σ ( i ) ∀ i j (17)where σ : B → B is the involution mapping a block type indexto the index of the block type with the reverse stack sequence.Note that we are not using Einstein’s sum notation; reversibilityentails that equation (17) holds pointwise over all i and j .We are interested in whether the s = i j satisfy-ing σ ( i ) = j . Next, using that this crystal’s involution map isdefined σ ( ) = , σ ( ) = , σ ( ) = , σ ( ) = p X = ( − γ )( − α ) p = p X = p σ ( ) X σ ( ) σ ( ) (18) p X = ( − γ )( − β ) p = p X = p σ ( ) X σ ( ) σ ( ) p X = ( − γ ) β p = p X = p σ ( ) X σ ( ) σ ( ) p X = ( − β ) γ p = p X = p σ ( ) X σ ( ) σ ( ) p X = βδ p = p X = p σ ( ) X σ ( ) σ ( ) p X = γβ p = p X = p σ ( ) X σ ( ) σ ( ) revealing the remarkable fact that all s = α = γ and β = δ the probability that a stack is type K depends onlyon a single preceding stack, which is to say s =
3, and the crys-tal is still reversible. Now if we let α = β = γ = δ we areleft with a sequence of independent and identically distributedrandom variables where s = < s ≤ s >
4. Specif-ically, for crystals with reichweite greater than 4 some setsof transition probabilities ( α, β, γ etc) satisfy condition (17)while others do not. We can see that there exist reversiblecrystals with s > s > s = K following HHH be 1 , HHK be 1 , HKK be 0 , KKH be 1 , KHK be 0and the probability be for all other blocks. Note that theblocks HHH , KKH , KHH , HKK , HKH , HHK , KHK forma closed communicating class where the probability of one fol-lowing the other is either 0 or 1, so represents a polytype withstack sequence representation shown in figure 7. The polytypedoes not satisfy the reversibility condition and is therefore irre-versible. ...
HHHKKHK (cid:124) (cid:123)(cid:122) (cid:125) polytypic lattice
HHHKKHK (cid:124) (cid:123)(cid:122) (cid:125) polytypic lattice ...
Figure 7
This crystal is irreversible because the block
HHH is certainly followed theblock
KKH , but in reverse
HHH is certainly followed by
KHK . This result can be extended to any s > H stacks in a row before being followed bythe sequence KKHK . In summary, we have established that fora the topologically close-packed crystal with symmetric layertypes, if s ≤ s > s > s ≤ s > s > s > et al. (2009). The Markov theoretic considerations in the previous chap-ter may shed some light on the process by which some class ofcrystals form. In particular, if an aperiodic crystal is irreversible,then its formation process must have an intrinsic directionality.Contrapositively, any formation process that does not have anyparticular or special directionality should produce a reversibleaperiodic crystal.This observation can be applied to ice, sometimes called I ch (Hansen et al. , 2015) or ice I sd (Malkin et al. , 2012), a mate-rial which can be well described without requiring a reichweites > et al. , 2008 a ), for which there is no recent evi-dence for the existence of polytypes (Hansen et al. , 2008 a );an earlier specific search for polytypes was similarly fruit-less (Kuhs et al. , 1987). Having no experimental evidence forgrowth processes with s > Acta Cryst. (2017). A , 000000 Hart, Hansen and Kuhs · Aperiodic crystals as Markov chains esearch papers trast to other materials with longer-ranged or even infinite reich-weite discussed by Varn & Crutchfield (2016). Can we learnsomething else for the growth physics from a Markov theoreticdescription of 1D periodic crystals? First of all, the stacking dis-order in ice I ch is very strongly influenced by the parent phase,both in its reichweite and in the frequency of the distinguish-able stackings observed; indeed, the stacking disorder providesvery clear and reproducible fingerprints to trace back the parentphase after its transformation into ice I ch (Kuhs et al. , 2012).This information is wiped out only upon prolonged annealing(Hansen et al. , 2008 b ), (Kuhs et al. , 2012); this is understood tobe a consequence of annihilation of various partial dislocations(Hondoh, 2015), a process which also depends on the lateralextent of the stacks. This process proceeds in a discontinuousmanner and eventually yields good hexagonal ice on approach-ing 240K within a laboratory timescales of seconds to minutes(Kuhs et al. , 2004) in full agreement with Hondoh (2015). Inter-estingly, a satisfactory description of ice I ch obtained from highpressure ices (recovered to ambient pressure at low tempera-ture) or obtained from water vapour requires s =
4, makinguse of 4 parameters α, β, γ and δ (Kuhs et al. , 2012), but theformation from super-cooled water is adequately described by s =
2, making use of single stacking fault parameter (Malkin et al. , 2012) (Amaya et al. , 2017). An explanation for this dif-ference is certainly worth pursuing.Kuhs et al. (2012) introduced the term cubicity to describethe proportion of cubic sequences in a crystal, which has beenfound to be almost 80% when freezing very small (15 nm)droplets at 225K (Amaya et al. , 2017), while larger (900 nm)drops freeze to a 50%:50% mixture of cubic and hexagonalsequences at 232K (Malkin et al. , 2012). It turns out that highestcubicities are obtained when no time is allowed for any anneal-ing of stacking faults, like in the very fast (timescale of µ s)freezing achieved by Amaya et al. (2017). This poses a ques-tion on the nature of the initially formed nucleus: is it cubic,hexagonal or stacking-disordered? While the bulk crystal in itsstable form is certainly hexagonal, the reasons for this prefer-ence are somewhat less clear: On the basis of quantum mechan-ical calculations Engel et al. (2015) have suggested that theanharmonic vibrational energies favour the hexagonal form as aconsequence of differences in the fourth nearest-neighbour pro-tons, related to the occurrence of the topologically different boatand chair-forms of the 6-membered water rings in cubic andhexagonal ice. Still, as nucleation (and growth) for super-cooledwater is kinetically controlled, freezing may well start also witha cubic or a stacking disordered nucleus (Lupi et al. , 2017).The subsequent growth appears experimentally to follow a fast s = et al. , 1980), which likelyhave formed from a cubic (or stacking-faulty) nucleus growingalong directions separated by the octahedral angle of 70 . ◦ (i.e. the angle between two or more cubic [111] directions).Maintaining a larger macroscopic snow-crystal in a stacking disordered state is energetically expensive due to the develop-ment of large-angle grain boundaries between several stackingdirections (Kobayashi & Kuroda, 1987); so individual branchesdevelop by further growth from the gas phase after the initialfreezing. Moreover, at high enough temperatures the stackingdisorder will quickly anneal as discussed above. Thus, the onlytraces left of the earlier stacking disorder are these multiplytwinned, branched hexagonal crystals.But why is vapour-grown ice I ch so complex that a satisfac-tory description demands that s =
4? The observed preferentialstacking sequences (Kuhs et al. , 2012) indicate a persistenceof hexagonal or cubic consecutive stackings rather than a fre-quent switching between them and an overall preponderance ofhexagonal stackings. Such a persistence can easily be explainedby the growth of stackings around screw dislocations (Th¨urmer& Nie, 2013), a growth mechanism which avoids a costly layer-by-layer nucleation along the stack. Recovered high-pressurephases of ice (like ice IX or ice V) when transformed into iceI ch do not show strong indications for persistence nor for alter-nating stackings. Rather, the stackings developed could wellreflect orientational relationships with their parent phase. Sucha topological inheritance has been demonstrated in the ice I h → ice II transition (Bennett et al. , 1997) and is manifest inthe observed textural relationships. It is well conceivable thatstructural inheritance could also express itself in certain stack-ing topologies to minimize the bond-breaking as well as strivingfor the shortest pathways for (multistage) diffusionless recon-structive phase transitions (Christy, 1993).It is also worth mentioning that the topology of the ice IXphase is acentric, with water molecules arranged along a 4-foldscrew-axis. In particular, there are two enantiomorphic forms ofice IX (and the same is true for ice III) with left and right handedforms occurring in nature with equal probability. Consequently,a naturally occurring sample of ice IX is expected have equalproportions of right and left handed crystallites. Now, we expecta sequence of right handed layers to be the reverse of a sequenceof left handed layers, and since we expect a sample to containboth forms in equal proportions, an irreversible ice IX crystalwould not be simply distinguished from a reversible counter-part by examining their X-ray or neutron powder diffractionpatterns.Further work is undoubtedly needed to elucidate the myriadof transitions between the many forms of ice in search for anexplanation for the observed stacking probabilities.
4. The pair distribution function of a 1D aperiodiccrystal
The scattering-length density function β ( (cid:126) r ) describes the distri-bution of scatterers of the ensemble when centred at the origin(Sivia, 2011). The autocorrelation function of β ( (cid:126) r ) , which isgiven by its convolution with its complex conjugate, producesa pair-correlation function that we will denote g ( (cid:126) r ) . There areseveral expressions for pair-correlation functions that differ intheir normalisations, and they can be written either in vectorialor in orientationally-averaged form. See Fischer et al. (2005)and Keen (2001). Hart, Hansen and Kuhs · Aperiodic crystals as Markov chains
Acta Cryst. (2017). A00
Acta Cryst. (2017). A00 , 000000 esearch papers The pair-correlation function g ( (cid:126) r ) of an aperiodic ensembleof scatterers (atoms) gives the probability density of finding anatom a vector distance (cid:126) r from an ensemble-averaged atom atthe origin. It can be obtained by Fourier transforming the totalscattered intensity I ( (cid:126) Q ) and is usually separated into a (trivial)self-correlation part and a structure-dependent so-called distinctpart g ( (cid:126) r ) sel f + g ( (cid:126) r ) distinct ∝ (cid:90) Q − space I ( (cid:126) Q ) e − π i (cid:126) Q · (cid:126) r d (cid:126) Q . (19) I ( (cid:126) Q ) is obtained experimentally as the total differential scat-tering cross section into solid angle d Ω as e.g. measured ina diffraction experiment; this measured total intensity is com-posed of the trivial self-scattering part and the structurally moreinteresting distinct part: I ( (cid:126) Q ) = d σ ( (cid:126) Q ) d Ω = I ( (cid:126) Q ) sel f + I ( (cid:126) Q ) distinct . (20)For powders with a random orientation of particles, the diffrac-tion data obtained are usually 1D averages of I ( (cid:126) Q ) like thoseobtained for amorphous materials or liquids; consequently onlythe isotropic function g ( r ) can be accessed experimentally. Yet,for known atomic arrangements of the powder crystallites theisotropic average of their 3D pair-correlation functions can beobtained by integration over the 3D shell at constant r . Theisotropic g ( r ) , which by choice contains only the distinct part,can then be renormalised into a Radial Distribution Function orRDF ( r ) from which coordination numbers can be obtained byintegration. It is also possible to normalise g ( r ) into the densityfunction D ( r ) whose slope at small r is proportional to the sam-ple’s atomic number density. It is this function D ( r ) , when gen-eralised to polyatomic systems, that is frequently called the PairDistribution Function, or PDF ( r ) . Note that the PDF ( r ) con-verges to zero at large r , since it represents fluctuations aroundthe average atomic density.PDF-analysis has developed into an important tool foranalysing the often defective atomic arrangements of nanomate-rials (Neder, 2014), (Egami & Billinge, 2012). Nanocrystallinematerials often exhibit stacking-faults as a consequence of theirmanufacturing procedures (ball milling, mechanical alloying)or crystal growth (Zehetbauer & Zhu, 2009). Such materialsshow very broad reflections as a consequence of crystal sizebroadening and stacking-faults as well as microstrains, and thusare not routinely accessible by Rietveld analyses if these con-tributions are not disentangled (Gayle & Biancaniello, 1995). APDF-analysis of stacking-faults in nanomaterials seems a viablealternative and was performed e.g. for CdSe by Yang et al. (2013) as well as by Masadeh et al. (2007) and Gawai et al. (2016) for ZnS nanocubes and nanowires. The stacking-faultmodel used in these works is rather simple and limited to mix-ture models of the pure cubic and hexagonal constituents via asingle stacking-fault probability parameter. That said, a moresophisticated treatment of disordered ZnS was presented byVarn et al. (2002) who offer a broad statistical description of theclose-packed topology, which includes ZnS. The authors findthe minimum effective memory length for stacking sequences in close-packed structures and discuss how to infer the ε -machinefrom scattering data.The pair distribution function g ( r ) is sensitive to the next-nearest neighbour arrangements, consequently also to the reich-weite of the layer interactions, it is in principle possible toextract detailed information on the nature of stacking faultsfrom an experimental g ( r ) using a Markov chain approach.Obtaining g ( r ) via a Fourier transformation of the (incom-pletely and imperfectly) measured I meas ( Q ) results in noiseand artefacts, so it might well be worth calculating g ( r ) fromthe direct-space structure model scattered intensity I model ( Q ) ,accounting for instrument resolution etc, then comparing with I meas ( Q ) , as has been done for nanocrystalline, stacking-faultyice by Hansen et al. (2008 a ), (2008 b ), and Kuhs et al. (2012).Indeed, a Q -space based approach like Rietveld refinement isthe only way to proceed in cases where high Q data are notavailable for making a meaningful Fourier transformation toobtain g ( r ) from I ( Q ) .We should stress that neither Q -space nor PDF-analysis isgenerally better than the other, but that they are chosen care-fully depending on different experimental situations: For exam-ple, a low density of defects in an otherwise crystalline systemis better analysed using Q -space refinement since S ( Q ) displayslong-range correlations of defects as diffuse scattering near thebase of Bragg peaks. On the other hand, for a high density ofdefects, especially when one begins to see broad and/or asym-metric intrinsic profiles of Bragg peaks, which can also happenfor quasi-2D or quasi-1D systems, then PDF-analysis is in prin-ciple better than Rietveld refinement. In these cases, the meth-ods of efficiently calculating PDFs of aperiodic crystals devel-oped by Varn et al. (2013) and Riechers et al. (2015) may comein handy. Moreover, the resolution of the neutron diffractome-ter plays a big role in deciding between PDF and Q -space anal-ysis, at least for reactor-based diffractometers where there is atradeoff between high Q (needed for good PDF-analysis andresolution in R-space), and good Q -space resolution (neededfor seeing diffuse scattering at the base of Bragg peaks). It isonly with some spallation-source diffractometers that one canachieve very good Q -space resolution in addition to a high Q max (at the expense of counting rate at high Q ), and in that case onecould conceivably attempt both PDF and Q -space analysis.
5. Conclusion and outlook
We have mildly generalised the cross section derived by Riech-ers et al. (2015) to reach an expression for the scattering crosssection of crystals including ice and opal. These crystals haveclose-packed topology, so we studied the close-packed topol-ogy in more detail, finding that those topologically close-packedcrystals with reichweite reichweite greater than 4 are almost surely irre-versible.Our expression for the cross section provides the experimen-tal crystallographer with a description that fully accounts forthe stacking disorder of ice and opal (and possibly more) whenapplying a Rietveld-like analysis. This could be used to estimatetransition probabilities, and understand the distribution of stack-ing faults much more accurately than trying to estimate them
Acta Cryst. (2017). A , 000000 Hart, Hansen and Kuhs · Aperiodic crystals as Markov chains esearch papers via MC simulation. Moreover, one could seek to determine the reichweite of opal in the same way that Hansen et al. (2008 a )and Kuhs et al. (2012) measured the reichweite of ice, whichwas fundamentally similar to the method outlined by Varn et al. (2002) to determine the ε -machine of a close-packed crystal.Roughly speaking both approaches involve attempting to fit amodel with s = good enough by some metric, the methods increment s until thefit becomes good enough; though the ε -machine reconstructionof Varn et al. (2002) is somewhat more sophisticated. Furtherto this, one could examine how the transition probabilities andentropic density of opal evolve under change in temperature, orany other variable. Obtaining the reichweite of opal could pro-vide information about its reversibility, providing clues aboutits formation process.A fruitful direction of future theoretical work may be toextend the theory so far explored to crystals composed ofinfinitely many kinds of layer, which could be applied to acrystal composed of layers that are identical to their imme-diate predecessor up to some rotation, translation, change incurvature, or shift orthogonal to the basel plane, which takesone of infinitely many values. Such crystals possess so-calledturbostratic disorder, and include a range of materials includ-ing smectites (Ufer et al. , 2008), (Ufer et al. , 2009), carbonblacks (Shi, 1993) (Zhou et al. , 2014) and possibly n -layergraphene; a novel material that has captured the attention ofthe nanoscience community (Razado-Colambo et al. , 2016)(Huang et al. , 2017). Such an extension of existing theory maybe achievable by replacing the transition matrix (an operator ona finite dimensional vector space) with a transition operator onan infinite dimensional Banach space. This functional analytictreatment could be extended to hidden Markov models withinfinitely many alphabetical symbols as well as an infinite statespace.We thank the Institut Laue Langevin for funding A. G.Hart’s internship. Further, we thank Joellen Preece for adviceon Markov theory and probability, as well as Henry Fischerfor guidance on PDF analysis. Further thanks are owed to theanonymous reviewers for their knowledgeable and detailed sug-gestions which helped to considerably improve the manuscript.We extend our gratitude to Chris Cook, Michael Green,Matthew Hill, Daniel Hoare, and Lucy Roche for offering cor-rections and criticism. Appendix ACross section of a 1D aperiodic crystal describedby a HMM
HMMs are an ordered quintuple Γ = ( A , S , µ , T , V ) where S is the state space of some hidden Markov process giving riseto a sequence of states S n satisfying the Markov property. T isthe transition matrix between states of the hidden Markov pro-cess, with elements T rs representing the probability that a hid- den state r ∈ S will transition to another hidden state s ∈ S . µ is some initial probability distribution over the state space S . A is the alphabet of symbols, which are not hidden, and representthe set of distinct layer types. At every state s ∈ S some symbolfrom the alphabet A will be emitted with probability followinga distribution dependant only on s . Specifically, the probabilitythat the state s ∈ S will emit a symbol x ∈ A is the element V sx of the matrix V . The sequence of hidden states S n gives rise toa sequence of symbols X n ; which is of course the sequence oflayer types that compose a crystal.Our definition of a HMM is presented differently to that ofRiechers et al. (2015), but is equivalent. In fact, we can definea set of matrices T with elements T [ x ] for each x ∈ A withcomponents T [ x ] rs = T rs V sx representing the probability of botha transition to state s ∈ S from state r ∈ S and an emissionof symbol x ∈ A from state s . Then we recover the quadruple ( A , S , µ , T ) used by Riechers et al. (2015) to define the HMM.Now, to derive the cross section we first consider the aver-age structure factor product Y m expressed by Berliner & Werner(1986) in terms of structure factors and pair correlation func-tions of the layer types comprising a crystal. To this end we let F x represent the structure factor of the layer type x ∈ A , and F be a matrix with entries F xy = F x F ∗ y . Further we let π rep-resent the stationary distribution of the hidden Markov process,which exists if the hidden Markov process is positive recurrentand irreducible. Starting from the average structure factor prod-uct Y m provided by Berliner & Werner (1986), making use ofBayes’ theorem and the Markov property of the sequence S m ,we have for m > Y m = (cid:88) x ∈A (cid:88) y ∈A P ( X = x , X m = y ) F xy (21) = (cid:88) x ∈A (cid:88) y ∈A P ( X = x ) P ( X m = y | X = x ) F xy (22) = (cid:88) x ∈A (cid:88) y ∈A (cid:88) r ∈ S (cid:88) s ∈ S P ( S = r ) P ( X = x | S = r ) (23) × P ( S m = s | S = r ) P ( X m = y | S m = s ) F xy = (cid:88) x ∈A (cid:88) y ∈A (cid:88) r ∈ S (cid:88) s ∈ S π s v rx T msr v sy F xy (24) = (cid:88) x ∈A (cid:88) y ∈A (cid:88) r ∈ S (cid:88) s ∈ S π s T msr v rx F xy v sy (25) = Tr (cid:0) Diag ( π ) T m V FV T (cid:1) . (26)Now, Y − m = (cid:88) x ∈A (cid:88) y ∈A P ( X = x , X m = y ) F yx (27) = (cid:88) x ∈A (cid:88) y ∈A P ( X = x , X m = y ) F ∗ xy (28)because F is Hermitian, so Y − m = Tr (cid:0) Diag ( π ) T m V F ∗ V T (cid:1) , (29) Hart, Hansen and Kuhs · Aperiodic crystals as Markov chains
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Acta Cryst. (2017). A00 , 000000 esearch papers and last of all Y = (cid:88) x ∈A (cid:88) y ∈A P ( X = x , X = y ) F xy (30) = (cid:88) x ∈A (cid:88) y ∈A π s δ sr v rx F xy v sy (31) = Tr (cid:0) Diag ( π ) V FV T (cid:1) . (32)Next, N (cid:88) m = ( N − | m | ) Y m e π iml (33) = Tr (cid:18) Diag ( π ) N (cid:88) m = ( N − m ) (cid:0) T e π il (cid:1) m V FV T (cid:19) = Tr (cid:18) Diag ( π ) SV FV T (cid:19) (34)where S = N (cid:88) m = ( N − m ) (cid:0) T e π il (cid:1) m (35) = T e π il ( T N e π ilN + N ( I − T e π il ) − I )( I − T e π il ) − if ( I − T e π il ) − exists. Similarly − (cid:88) m = − N ( N − | m | ) Y m e π iml (36) = N (cid:88) m = ( N − m ) Y − m e − π iml = Tr (cid:18) Diag ( π ) S ∗ V F ∗ V T (cid:19) (37)so N (cid:88) m = − N ( N − | m | ) Y m e π iml (38) = Tr (cid:18) Diag ( π ) SV FV T (cid:19) + Tr (cid:18) Diag ( π ) S ∗ V F ∗ V T (cid:19) (39) + Tr (cid:18) Diag ( π ) V FV T (cid:19) = Re (cid:26) Tr (cid:0) Diag ( π ) SV FV T (cid:1)(cid:27) + Tr (cid:0) Diag ( π ) V FV T (cid:1) . (40)The general expression for the cross section given by Berliner& Werner (1986) d σ d Ω = sin ( N a π h ) sin ( π h ) sin ( N b π k ) sin ( π k ) N c (cid:88) m = − N c ( N c − | m | ) Y m e π im l (41) completes the derivation d σ d Ω = sin ( N a π h ) sin ( π h ) sin ( N b π k ) sin ( π k ) (42) × (cid:18) Re (cid:26) Tr (cid:0) Diag ( π ) SV FV T (cid:1)(cid:27) + N c Tr (cid:0) Diag ( π ) V FV T (cid:1)(cid:19) = sin ( N a π h ) sin ( π h ) sin ( N b π k ) sin ( π k ) Re (cid:26) Tr (cid:0) Diag ( π )( S + N c I ) V FV T (cid:1)(cid:27) . (43) Appendix BExpressing the differential scattering crosssection of 1D aperiodic crystals
We begin with the expression for the differential scatteringcross developed by Berliner & Werner (1986) d σ d Ω = sin ( N a π h ) sin ( π h ) sin ( N b π k ) sin ( π k ) × (44) N c (cid:88) m = − N c ( N c − | m | ) Y m e π im l and proceed by splitting the sum N c (cid:88) m = − N c ( N c − | m | ) Y m e π im l (45) = Y N c + N c (cid:88) m = ( N c − m ) Y m e π im l + N c (cid:88) m = ( N c − m ) Y − m e − π im l . Now, according to Berliner & Werner (1986), the average struc-ture factor product has expression Y m ( (cid:126) Q ) = (cid:88) i ∈ B (cid:88) j ∈ B F i ( (cid:126) Q ) F ∗ j ( (cid:126) Q ) G i j ( m ) (46)where F i ( (cid:126) Q ) and F ∗ i ( (cid:126) Q ) are the structure factor and conju-gate structure factor of an i block respectively. For brevity, let F i j = F i ( (cid:126) Q ) F ∗ j ( (cid:126) Q ) be the i j th components of what we will callthe structure matrix F . Next, G i j ( m ) denotes the probabilitythat a j block is m blocks ahead of an i block so we have that G i j ( m ) = π i ξ m i j for positive m . Postponing the case where m is negative, we proceed by noting Y m ( (cid:126) Q ) = (cid:88) i ∈ B (cid:88) j ∈ B F i j π i ξ m i j , m > = (cid:88) i ∈ B (cid:88) j ∈ B F Tji π i ξ m i j (48) Acta Cryst. (2017). A , 000000 Hart, Hansen and Kuhs · Aperiodic crystals as Markov chains esearch papers and use that F is Hermitian, which is to say F T = F ∗ , to deduce Y m ( (cid:126) Q ) = (cid:88) i ∈ B (cid:88) j ∈ B F ∗ ji π i ξ m i j (49)which we identify as Y m ( (cid:126) Q ) = Tr (cid:18) F ∗ Diag ( π ) Ξ m (cid:19) (50)where Tr is the trace. Now, the probability of sampling an i block from a crystal then finding a j block m blocks behindthe first block, is equal to the probability of sampling a j blockfrom the crystal then finding a type i block m blocks ahead ofthe former block, consequently Y − m ( (cid:126) Q ) = Tr (cid:18) F Diag ( π ) Ξ m (cid:19) , m > . (51)Now, continuing from equation (45) N c (cid:88) m = − N c ( N c − | m | ) Y m e π im l (52) = N c Tr (cid:18) F Diag ( π ) (cid:19) + N c (cid:88) m = ( N c − m ) Tr (cid:18) F ∗ Diag ( π ) Ξ m (cid:19) e π im l + N c (cid:88) m = ( N c − m ) Tr (cid:18) F Diag ( π ) Ξ m (cid:19) e − π im l . Since the two summands on the RHS of the previous equationare complex conjugate, we seek an expression for only one ofthem, from which we can deduce the other easily. We do this byinvoking the linearity of the trace to deduce N c (cid:88) m = ( N c − m ) Tr (cid:18) F ∗ Diag ( π ) Ξ m (cid:19) e π im l (53) = Tr (cid:18) F ∗ Diag ( π ) N c (cid:88) m = ( N c − m ) Ξ m e π im l (cid:19) then we consider 2 cases, the first is for Ξ a diagonalisablematrix, where N c (cid:88) m = ( N c − m ) Ξ m e π im l = N c (cid:88) m = ( N c − m ) Q λ m
0. . .0 λ m n Q − e π im l (54) = Q (55) N c (cid:88) m = ( N c − m ) λ m e π im l
0. . .0 ( N c − m ) λ m n e π im l Q − for some invertible matrix Q . Now the sum s n = N c (cid:88) m = ( N c − m ) λ m n e π im l (56)has analytic expression s n = (cid:40) N c ( N c − ) if λ n e π il = λ n e π il ( λ Ncn e π ilNc + N c ( − λ n e π il ) − )( − λ n e π il ) otherwise, (57)so we can let ˆ S be the diagonal matrix with components s n .Next, we consider the case of a defective Ξ . Though we can-not diagonalise Ξ , we can express it terms of its Jordan Canon-ical form J = Q − Ξ Q (58)where J is a block diagonal matrix comprised of Jordan blocks J p ; which are themselves upper triangular matrices each asso-ciated with an eigenvalue λ p of Ξ . The columns of the matrix Q are the eigenvectors and generalised eigenvectors of Ξ , so Q is necessarily invertible. We proceed by first noting N c (cid:88) m = ( N c − m ) Ξ m e π im l = N c (cid:88) m = ( N c − m ) Q J m
0. . .0 J m p Q − e π im l (59) = Q (60) N c (cid:88) m = ( N c − m ) J m e π im l
0. . .0 ( N c − m ) J m p e π im l Q − . Now the weighted sum of Jordan blocks S p = N c (cid:88) m = ( N c − m ) J m p e π im l (61)has expression S p = J p e π il ( J N c p e π ilN c + N c ( I − J p e π il ) − I ) (62) × ( I − J p e π il ) − (63)under the condition that the resolvent (cid:0) I − J p e π il (cid:1) is invertible,which is true unless λ p e π il =
1. In the case of an invertibleresolvent, it is noteworthy for ease of computation that (cid:18) I − J p e π il (cid:19) − i j = (cid:40) ( − λ p e π il ) − + i − j if i ≤ j (cid:18) J p e π il (cid:19) N c i j = (cid:40) ( λ p e π il ) N c λ i − j (cid:0) N c j − i (cid:1) if i ≤ j Hart, Hansen and Kuhs · Aperiodic crystals as Markov chains
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Acta Cryst. (2017). A00 , 000000 esearch papers where (cid:0) ab (cid:1) are binomial coefficients. On the other hand, if λ p e π il = (cid:0) I − J p e π il (cid:1) is singular andwe note (cid:18) N c (cid:88) m = ( N c − m ) J m p (cid:19) i j = N c (cid:88) m = ( N c − m ) (cid:18) m j − i (cid:19) . (66)We can now define the matrix ˆ S as the block diagonal matrixcomposed of the blocks S p Having considered both cases of Ξ diagonalisable and not,we proceed by recalling equation (53) and note N c (cid:88) m = ( N c − m ) Tr (cid:18) F ∗ Diag ( π ) Ξ m (cid:19) e π im l (67) = Tr (cid:18) F ∗ Diag ( π ) Q − ˆ SQ (cid:19) hence we can express equation (52) N c (cid:88) m = − N c ( N c − | m c | ) Y m e π im l (68) = Tr (cid:18) F Diag ( π ) (cid:19) + Tr (cid:18) F ∗ Diag ( π ) Q − ˆ SQ (cid:19) + Tr (cid:18) F Diag ( π ) Q − ˆ S ∗ Q (cid:19) then using equation (44), linearity of the trace, and the traceoperator’s cyclic permutation propertyTr ( ABC ) = Tr ( CBA ) , (69)we deduce d σ d Ω = sin ( N a π h ) sin ( π h ) sin ( N b π k ) sin ( π k ) × (70)Tr (cid:18) Diag ( π )( N c F + Q − ˆ SQF ∗ + Q − ˆ S ∗ QF ) (cid:19) . Finally we deploy the change of basisDiag ( π ) = Q − ˆ PQ (71) F = Q − ˆ FQ (72)and once again use the linearity and cyclic permutation propertyof the trace to deduce d σ d Ω = sin ( N a π h ) sin ( π h ) sin ( N b π k ) sin ( π k ) × (73)Tr (cid:18) ˆ P ( N c ˆ F + ˆ S ˆ F ∗ + ˆ S ∗ ˆ F ) (cid:19) = sin ( N a π h ) sin ( π h ) sin ( N b π k ) sin ( π k ) Re (cid:26) Tr (cid:0) ˆ P ( ˆ S + N c I ) ˆ F ) (cid:1)(cid:27) . (74) Appendix CAnalytic derivatives of the scattering crosssection
We may be interested in using scattering data to measure cer-tain molecular quantities encoded in the structure factor. Forexample, the separation between a particular pair of atoms ina block of type i . We can represent these unknown molecularquantities as free parameters that we attempt to estimate by find-ing values for them that best fit experimental scattering data.This sort of refinement often requires the evaluation of a Jacobimatrix of derivatives. Consequently, it may be useful to havean expression for the analytic derivatives of the scattering crosssection with respect to the structure factor’s free parameters.Letting τ be such a free parameter, we have that ∂∂ τ (cid:18) d σ d Ω (cid:19) = N a N b δ ( h − h ) δ ( k − k ) × (75)Tr (cid:18) Diag ( π )( ∂ F + Q − SQ ∂ F ∗ + Q − S ∗ Q ∂ F ) (cid:19) where ∂ F is a matrix with elements ∂ F i j = ∂ F i ∂ τ F ∗ j + F i ∂ F ∗ j ∂ τ . (76)Here, the derivative ∂ F i ∂ τ may be expressed analytically if possi-ble or approximated using a finite difference if necessary. References
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