A hidden Markov model for describing turbostratic disorder applied to carbon blacks and graphene
rresearch papers
Acta Crystallographica Section A
Foundations ofCrystallography
ISSN 0108-7673© 2018 International Union of CrystallographyPrinted in Singapore – all rights reserved
A hidden Markov model for describing turbostraticdisorder applied to carbon blacks and graphene
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 present a mathematical framework to represent turbostratic disorder in mate-rials like carbon blacks, smectites, and twisted n -layer graphene. In particular, theset of all possible disordered layers, including rotated, shifted, and curved layersform a stochastic sequence governed by a hidden Markov model. The probabilitydistribution over the set of layer types is treated as an element of a Hilbert space,and using tools of Fourier analysis and functional analysis, we develop expres-sions for the scattering cross sections of a broad class of disordered materials.
1. Introduction
Warren (1941) was ahead of his time when he observed that cer-tain heat treated carbon blacks appear to comprise a sequenceof equally spaced graphite layers with some random rotationand parallel translation between them. In a paper published thenext year, Biscoe & Warren (1942) decided to name this typeof disorder among layers turbostratic disorder . The authorsused the word to describe “graphite layers stacked togetherroughly parallel and equidistant, but with each layer having acompletely random orientation about the layer normal” whileDritz & Tchoubar (1990) state “When all the layers are ran-domly misoriented, the stack is (ideally) turbostratic and thereare no Bragg reflections other than those belonging to the lseries.” In the ensuing years, the meaning of the word tur-bostratic has evolved, with a widely cited paper by Li et al. (2007) allowing turbostratic disorder to include layers that areshifted, rotated and curved over some non-uniform probabilitydistribution. This broad definition of the word turbostratic is thedefinition we will use in this paper, with the goal of bringingtogether a broad range of disorders under the same mathemati-cal framework.The notion of turbostratic disorder was developed by Shi(1993) to model carbon blacks as a sequence of turbostrati-cally disordered carbon layers that each depend on their pre-ceding layer. Moreover, Shi et al. (1993) have written a pro-gram CARBONXS that computes the scattering cross sectionof their theoretical carbon blacks. The performance of CAR-BONXS has been compared by Zhou et al. (2014) to GSAS, atraditional Rietveld refinement program that does not take tur-bostratic effects into account, suggesting that an appreciation ofturbostratic disorder is necessary to obtain a good fit to X-raydiffraction data.Since Shi’s models are a good fit for carbon blacks, the mate-rial appears to truly comprise a Markov chain of turbostraticallydisordered carbon layers. However, since Shi’s model allowscarbon layers to be shifted both parallel and perpendicular tothe basal plane by any magnitude, there is an uncountable infin-ity of positions a carbon layer may find itself in. Carbon blackstherefore comprise a Markov chain of layers with infinite state space, and are therefore not fully understood by the analysis ofchains with finite state space explored by Riechers et al. (2015)Varn & Crutchfield (2016) and Hart et al. (2018). The propertiesof carbon blacks are well worth exploring, as the material hasmany applications including the moderation of neutrons (Zhou et al. , 2014), lithium-ion batteries (Shi, 1993), and the manu-facture of rubber (Ung´ar et al. , 2002)Aside from carbon blacks, it is now well known (Huang et al. , 2017) (Razado-Colambo et al. , 2016) that layers of graphene can be stacked atop one another with rotation betweenthem adopting an angle θ ∈ [ − π , π ] taking one of an uncount-able infinity of values. The rotation can adopt any angle, butthe 6 fold rotational symmetry of graphene allows us to work inthe restricted range θ ∈ [ − π , π ] . It is intriguing that a particu-lar countably infinite subset of [ − π , π ] has been the object ofgreat interest and fruitful research among the nanoscience com-munity; specifically the countable set of angles θ i for which apair of layers differing by these angles form a moir´e pattern.Under these angles a twisted bilayer forms a superlattice, andthe resultant crystal takes on a so called commensurate struc-ture. Lopes dos Santos et al. (2007) derived an expression forthe moir´e angles θ i as exactly the set of angles satisfyingcos ( θ i ) = i + i + i + i + i = , , , ... . An illustration of the superlattice producedwhen a pair of layers differ by an angle θ is shown in Fig-ure 1. The electronic properties of moir´e graphene are rich andexotic, and have become subject of a huge international researcheffort; see for example Huang et al. (2017), Razado-Colambo et al. (2016) Brown et al. (2012), Havener et al. (2012), orCao et al. (2018), where the most recent authors identified amagic angle where a twisted bilayer becomes a superconductor.Though the moir´e angles are importantly distinct from the otherrotation angles, the latter are still of interest, in fact Bistritzer &MacDonald (2011) have remarked that for all other angles θ , atwisted bilayer has no unit cell, but has instead a quasi-periodicstructure with its own set of properties. Acta Cryst. (2018). A , 000000 Hart, Hansen and Kuhs · possibly a subtitle a r X i v : . [ phy s i c s . c h e m - ph ] F e b esearch papersresearch papers Figure 1
An example of a moir´e superlattice with angle θ ≈ . ◦ . Most of the relevant nanoscience literature is focused on thesimplest interesting model - a single twisted bilayer - but bystacking several layers atop eachother one can form twisted n -layer graphene (Havener et al. , 2012). Assuming any one of the n layers’ angle of rotation depends only on a previous layer’sangle of rotation, twisted n -layer graphene can be described byMarkov chain with either a countable or an uncountable num-ber of layer types; depending on whether we insist the rotationangles are moir´e, or allow them to take any value in [ − π , π ] . Inany case, the rotation angles would follow a probability distri-bution, which has been sought experimentally by Brown et al. (2012) who attempted to infer it from scattering data. Theirempirical distribution is compared to torque each atom is sub-ject to, as well as the potential energy per atom.Smectites (clays) are another class of turbostratically dis-ordered materials. They have been scrutinised under Rietveldrefinement by Ufer et al. (2008) and Ufer et al. (2009) but theturbostratic effects have not been treated rigorously, and mayhave a more natural description in the framework presentedhere.
2. The scattering cross section
For a crystal composed of otherwise identical layers that differonly by rotation, translation, or change in curvature, the struc-ture factor of the layers are related too. In particular, a layer’sstructure factor is related by a Fourier transform to the layer’satomic positions, which are related by some rotation, translationor curvature map to the atomic positions of some other layer.To formalise this idea, suppose an arbitrarily chosen referencelayer is composed of a periodic array of unit cells. Then for agiven unit cell, we express positions in reciprocal space (cid:126) Q withreciprocal primitive lattice vectors (cid:126) a ∗ ,(cid:126) b ∗ , (cid:126) c ∗ and real numbers h , k , and l such that (cid:126) Q = π ( h (cid:126) a ∗ + k (cid:126) b ∗ + l (cid:126) c ∗ ) . (2)Hence, a unit cell comprising N atoms, each with an atomicform factor f j and position (cid:126) r j with j = ... N has structurefactor F unit given by F unit = N (cid:88) j = f j e − i (cid:126) Q · (cid:126) r j . (3)The structure factor of an entire layer of unit cells is F = (cid:88) ( m , m ) ∈D N (cid:88) j = f j e − i (cid:126) Q · ( (cid:126) r j + m (cid:126) a + m (cid:126) b ) (4) = F unit (cid:88) ( m , m ) ∈D e − i ( m (cid:126) Q · (cid:126) a + m (cid:126) Q · (cid:126) b ) (5) = F unit (cid:88) ( m , m ) ∈D e − π i ( m h + m k ) (6)where (cid:126) a and (cid:126) b are the primitive lattice vectors that span the basalplane and D is some subset of Z defining the shape of the layer.If for example the layers are elliptical then D ⊂ Z representssome set of lattice nodes enclosed by an ellipse, which we mightexpect for layers of carbon black crystallites given that Ung´ar et al. (2002) found the crystallites themselves to be ellipsoidal.If we consider a simpler case of each layer being rectangularwith equal dimensions D = [ , ..., N a − ] × [ , ..., N b − ] (7)then we obtain the structure factor of a single layer F = F unit (cid:88) ( m , m ) ∈D e − π i ( m h + m k ) (8) = F unit N a − (cid:88) m = e − π im h N b − (cid:88) m = e − π im k (9) = F unit sin ( N a π h ) sin ( π h ) sin ( N b π k ) sin ( π k ) e − i ( N a − ) π h e − i ( N b − ) π k (10)where the last line follows from the definition of the Dirichletkernel. The contribution of this layer to the scattering pattern S is | F | allowing us to recover a perhaps familiar expression S = | F unit | sin ( N a π h ) sin ( π h ) sin ( N b π k ) sin ( π k ) . (11)The term η ( (cid:126) Q ) = sin ( N a π h ) sin ( π h ) sin ( N b π k ) sin ( π k ) (12)is called the shape function, can be modified to represent thedifferent shapes crystallites can take. This is discussed by Shi et al. (1993), Warren (1969) and Ergun (1976).Next, we consider the structure factors of two layers that dif-fer by some rotation. Suppose the rotation is defined by theorthonormal matrix X , then we multiply X to the position (cid:126) r j Hart, Hansen and Kuhs · possibly a subtitle Acta Cryst. (2018). A , 000000 Figure 1
An example of a moir´e superlattice with angle θ ≈ . ◦ . Most of the relevant nanoscience literature is focused on thesimplest interesting model - a single twisted bilayer - but bystacking several layers atop eachother one can form twisted n -layer graphene (Havener et al. , 2012). Assuming any one of the n layers’ angle of rotation depends only on a previous layer’sangle of rotation, twisted n -layer graphene can be described byMarkov chain with either a countable or an uncountable num-ber of layer types; depending on whether we insist the rotationangles are moir´e, or allow them to take any value in [ − π , π ] . Inany case, the rotation angles would follow a probability distri-bution, which has been sought experimentally by Brown et al. (2012) who attempted to infer it from scattering data. Theirempirical distribution is compared to torque each atom is sub-ject to, as well as the potential energy per atom.Smectites (clays) are another class of turbostratically dis-ordered materials. They have been scrutinised under Rietveldrefinement by Ufer et al. (2008) and Ufer et al. (2009) but theturbostratic effects have not been treated rigorously, and mayhave a more natural description in the framework presentedhere.
2. The scattering cross section
For a crystal composed of otherwise identical layers that differonly by rotation, translation, or change in curvature, the struc-ture factor of the layers are related too. In particular, a layer’sstructure factor is related by a Fourier transform to the layer’satomic positions, which are related by some rotation, translationor curvature map to the atomic positions of some other layer.To formalise this idea, suppose an arbitrarily chosen referencelayer is composed of a periodic array of unit cells. Then for agiven unit cell, we express positions in reciprocal space (cid:126) Q withreciprocal primitive lattice vectors (cid:126) a ∗ ,(cid:126) b ∗ , (cid:126) c ∗ and real numbers h , k , and l such that (cid:126) Q = π ( h (cid:126) a ∗ + k (cid:126) b ∗ + l (cid:126) c ∗ ) . (2)Hence, a unit cell comprising N atoms, each with an atomicform factor f j and position (cid:126) r j with j = ... N has structurefactor F unit given by F unit = N (cid:88) j = f j e − i (cid:126) Q · (cid:126) r j . (3)The structure factor of an entire layer of unit cells is F = (cid:88) ( m , m ) ∈D N (cid:88) j = f j e − i (cid:126) Q · ( (cid:126) r j + m (cid:126) a + m (cid:126) b ) (4) = F unit (cid:88) ( m , m ) ∈D e − i ( m (cid:126) Q · (cid:126) a + m (cid:126) Q · (cid:126) b ) (5) = F unit (cid:88) ( m , m ) ∈D e − π i ( m h + m k ) (6)where (cid:126) a and (cid:126) b are the primitive lattice vectors that span the basalplane and D is some subset of Z defining the shape of the layer.If for example the layers are elliptical then D ⊂ Z representssome set of lattice nodes enclosed by an ellipse, which we mightexpect for layers of carbon black crystallites given that Ung´ar et al. (2002) found the crystallites themselves to be ellipsoidal.If we consider a simpler case of each layer being rectangularwith equal dimensions D = [ , ..., N a − ] × [ , ..., N b − ] (7)then we obtain the structure factor of a single layer F = F unit (cid:88) ( m , m ) ∈D e − π i ( m h + m k ) (8) = F unit N a − (cid:88) m = e − π im h N b − (cid:88) m = e − π im k (9) = F unit sin ( N a π h ) sin ( π h ) sin ( N b π k ) sin ( π k ) e − i ( N a − ) π h e − i ( N b − ) π k (10)where the last line follows from the definition of the Dirichletkernel. The contribution of this layer to the scattering pattern S is | F | allowing us to recover a perhaps familiar expression S = | F unit | sin ( N a π h ) sin ( π h ) sin ( N b π k ) sin ( π k ) . (11)The term η ( (cid:126) Q ) = sin ( N a π h ) sin ( π h ) sin ( N b π k ) sin ( π k ) (12)is called the shape function, can be modified to represent thedifferent shapes crystallites can take. This is discussed by Shi et al. (1993), Warren (1969) and Ergun (1976).Next, we consider the structure factors of two layers that dif-fer by some rotation. Suppose the rotation is defined by theorthonormal matrix X , then we multiply X to the position (cid:126) r j Hart, Hansen and Kuhs · possibly a subtitle Acta Cryst. (2018). A , 000000 esearch papers of each atom in the unit cell, as well as the lattice vectors them-selves, and find the structure factor F X of the rotated layer is F X = (cid:88) ( m , m ) ∈D N (cid:88) j = f j e − i (cid:126) Q · ( X (cid:126) r j + m X (cid:126) a + m X (cid:126) b ) (13) = F X unit (cid:88) ( m , m ) ∈D e − i ( m (cid:126) Q · ( X (cid:126) a )+ m (cid:126) Q · ( X (cid:126) b )) (14)where F X unit is the unit cell of a rotated layer with expression F X unit = N (cid:88) j = f j e − i (cid:126) Q · ( X (cid:126) r j ) . (15)It follows that the structure factor of a rotated layer that is alsorectangular is F X = F unit sin ( N a π h X ) sin ( π h X ) sin ( N b π k X ) sin ( π k X ) e − i ( N a − ) π h X e − i ( N b − ) π k X (16)where h X = (cid:126) Q · ( X (cid:126) a ) (17) k X = (cid:126) Q · ( X (cid:126) b ) l X = (cid:126) Q · ( X (cid:126) c ) . The next example is a layer that differs only by a translation (cid:126) v from a layer with structure factor F . The translated layer hasstructure factor F (cid:126) v related to F via the simple relation F (cid:126) v = Fe − i (cid:126) Q · (cid:126) v . (18)Usefully, when layers differ by some linear transformation(rotation or translation) the structure factor of each layer canbe expressed as a periodic arrangement of unit cells with trans-lational symmetry, where each layer type’s unit cell is related bysome transformation to the unit cell of another layer type. How-ever, this feature does not apply to layers that differ by somenonlinear transformation like change in curvature, which wasdiscussed by Li et al. (2007) when describing disordered layersof graphite. In fact for a reference layer with structure factor F , a second layer differing from the reference by a nonlineartransformation φ has structure factor F φ = (cid:88) ( m , m ) ∈D N (cid:88) j = f j e − i (cid:126) Q · (cid:0) φ ( (cid:126) r j + m (cid:126) a + m (cid:126) b ) (cid:1) . (19)The nonlinearity of φ means we cannot factorise out the struc-ture factor of a unit cell; which is consistent with physicalintuition. One would not expect a curved layer to comprise aperiodic array of identical unit cells because some cells wouldbe curved more than others. Consequently, instead of thinkingabout aperiodic crystals as comprised of unit cells, it is safer tothink of them as comprised of layers, that cannot (in general) bebroken down into constituent cells. With the preamble about structure factors out the way, we arein a position to approach the differential scattering cross section(or scattering pattern) of an aperiodic crystal. First of all, sup-pose a crystal is composed of a sequence of layers, labelled inorder from n = , ..., N c . Each layer has a type (or structure fac-tor) indexed by the set A . If the number of layer types is finite,then A is some finite subset of the positive integers N . If A iscountably infinite then we let A = N and if uncountably infi-nite we allow A ⊂ R n to be open and connected. The structurefactor of each layer is labelled F n , so the structure factor of theentire crystal ψ is ψ = N c (cid:88) n = F n e π inl (20)and it follows that the cross section has expression d σ d Ω = | ψ | = N c (cid:88) n = N c (cid:88) m = F n F ∗ m e π i ( n − m ) l . (21)Now, F n F ∗ m is the average structure factor product Y m − n dis-cussed by Berliner & Werner (1986), obtained by taking expec-tation over the distribution of structure factor pairs separated by m − n layers. When A is countable, we can write this down as Y m − n = F n F ∗ m = (cid:88) x ∈A (cid:88) y ∈A F ( x ) G m − n ( x , y ) F ∗ ( y ) (22)where G m ( x , y ) is the pair correlation function between layers x , y ∈ A where y is m layers ahead of x , and F ( x ) and F ( y ) are the structure factors of x and y respectively. If A ⊂ R n thensimilarly Y m − n = (cid:90) A (cid:90) A F ( x ) G m − n ( x , y ) F ∗ ( y ) dxdy . (23)We can then recover the expression for the differential scatter-ing cross section presented by Berliner & Werner (1986) andderived by Wilson (1942) d σ d Ω = N c (cid:88) n = N c (cid:88) m = Y m − n e π i ( n − m ) l (24) = N c (cid:88) m = − N c ( N c − | m | ) Y m e π im l . (25)For completeness, we note that the dimension of the crystal N a , N b , N c may not be the same for every crystal in a sample, butcould in general follow some distribution. Ung´ar et al. (2002)for example, report that carbon blacks have log-normal size dis-tribution. In this case the observed scattering pattern would beobtained by summing the cross section over each dimensiontimes the probability of a crystallite adopting that dimension. Suppose a powder sample of a crystal is placed into a flat traywith normal vector ˆ n . The powder average I ( Q ) of the cross sec-tion is given by I ( Q ) = (cid:90) ∂ B Q d σ d Ω ( (cid:126) Q ) ω ( (cid:126) Q ) dS ( (cid:126) Q ) (26) Acta Cryst. (2018). A , 000000 Hart, Hansen and Kuhs · possibly a subtitle esearch papers where we are integrating over the sphere of radius Q = | (cid:126) Q | andhave introduced the spherically symmetric preferred orientationfunction ω ( (cid:126) Q ) to represent the probability density that crystal-lite’s normal vector is rotated by angles ( θ, ϕ ) from ˆ n where ( θ, ϕ ) are the spherical polar angles of the vector (cid:126) Q . An illus-tration of this is shown in Figure 2. research papers where we are integrating over the sphere of radius Q = | (cid:126) Q | andhave introduced the spherically symmetric preferred orientationfunction ω ( (cid:126) Q ) to represent the probability density that crystal-lite’s normal vector is rotated by angles ( θ, ϕ ) from ˆ n where ( θ, ϕ ) are the spherical polar angles of the vector (cid:126) Q . An illus-tration of this is shown in Figure 2. h (cid:48) k (cid:48) Sample Tray l (cid:48) (cid:126) Q ϕ θ Figure 2
The probability density that a crystallite centred at the origin is oriented suchthat it’s normal vector is in the direction of the vector (cid:126) Q with polar angle ( θ , ϕ ) is ω ( (cid:126) Q ) . A point in the reciprocal lattice coordinates (cid:126) Q = ( h , k , l ) is representedin Cartesian coordinates by ( h (cid:48) , k (cid:48) , l (cid:48) ) . The vector normal to the sample tray ˆ n is parallel to l (cid:48) . The preferred orientation function ω is introduced becausecrystallites in a container often align with the geometry of thecontainer, resulting in some orientations being more likely thanothers. In the special case that all orientations are equally likely, I ( Q ) = π Q (cid:90) ∂ B Q d σ d Ω ( (cid:126) Q ) dS ( (cid:126) Q ) . (27)Numerically computing either of these integrals is not easybecause the the cross section d σ d Ω ( (cid:126) Q ) is, roughly speaking, closeto zero everywhere except for points surrounding (cid:126) Q where h , k , l are all integers. At these points the cross section is highlypeaked. As the size of a crystallite grows the peaks becometaller and thinner, converging to delta functions in the limit ofinfinite crystallite size. Na¨ıve quadrature does not perform wellon integrands with many thin peaks, so should be avoided forcomputing the powder average of big crystallites. If the crys-tals are indeed big, a common method of computing the pow-der average is to numerically integrate each peak separatelyand sum the contributions. One can also employ the tangent-cylinder approximation derived by Brindley & Mring (1951)and discussed by Shi et al. (1993) to speed up the integration ofeach peak.An alternative to numerical integration is to derive an expres-sion for the powder average using a Harmonic expansion, which does not require numerical integration over the sphere! We shallpresent a version of this for the simplest case that ω ( (cid:126) Q ) = π Q , which may be adequate for a highly disordered mate-rial. Let L m i j ( x , y ) denote the distance between the i th atom in alayer of type x and the j th atom of a layer type y for layers x , y separated vertically by m layers. When we talk about distance,we assume a unit length is 2 π | (cid:126) c ∗ | . It is shown in Appendix Athat I ( Q ) = N c (cid:88) m = − N c ( N c − | m | ) (cid:90) A (cid:90) A G m ( x , y ) (28) × n x (cid:88) i = n y (cid:88) j = f i f j sinc (cid:0) QL m i j ( x , y ) (cid:1) dxdy . The term n x (cid:88) i = n y (cid:88) j = f i f j sinc (cid:0) QL m i j ( x , y ) (cid:1) (29)is highly related to Debye’s equation, who’s 100th birthday wasrecently celebrated by Scardi et al. (2016).
3. Finitely many hidden states
It remains now to define the pair correlation function G m ( x , y ) which captures the probability of sampling from a crystal alayer of type x , then finding a layer of type y m layers aheadof x . To this end, we maintain the assumption of Varn et al. (2013), Riechers et al. (2015), and Varn & Crutchfield (2015)that the sequence of layers follows a Hidden Markov Model. Inparticular when the set of hidden states S and layer types A arefinite, the Hidden Markov Model (HMM) is an ordered quin-tuple Γ = ( A , S , µ , T , V ) where the terms are exactly thosedefined by Hart et al. (2018) in their Appendix A.In particular the probability of a layer adopting a hidden state j ∈ S can be represented as the element of a vector v . Giventhe hidden state of the HMM is i ∈ S , then the probability of atransition to j ∈ S is the i j th element of a transition matrix T .This matrix represents an operator which maps a distribution ofhidden states v of some layer to the distribution of hidden states w of the next layer, which is to say T v = w . (30)For a layer with hidden states following a distribution v , thelayer found m layers ahead has hidden state following the dis-tribution u which is related to v by T m v = u . (31)The stationary distribution π of T represents the probability dis-tribution over the set of hidden states obtained by sampling alayer from the crystal. A sufficient condition for π to exist andbe unique is that the Markov Chain induced by T is positiverecurrent, which means from any state s the probability of even-tual return to s state is unity. Further since, T π = π (32) Hart, Hansen and Kuhs · possibly a subtitle Acta Cryst. (2018). A , 000000 Figure 2
The probability density that a crystallite centred at the origin is oriented suchthat it’s normal vector is in the direction of the vector (cid:126) Q with polar angle ( θ , ϕ ) is ω ( (cid:126) Q ) . A point in the reciprocal lattice coordinates (cid:126) Q = ( h , k , l ) is representedin Cartesian coordinates by ( h (cid:48) , k (cid:48) , l (cid:48) ) . The vector normal to the sample tray ˆ n is parallel to l (cid:48) . The preferred orientation function ω is introduced becausecrystallites in a container often align with the geometry of thecontainer, resulting in some orientations being more likely thanothers. In the special case that all orientations are equally likely, I ( Q ) = π Q (cid:90) ∂ B Q d σ d Ω ( (cid:126) Q ) dS ( (cid:126) Q ) . (27)Numerically computing either of these integrals is not easybecause the the cross section d σ d Ω ( (cid:126) Q ) is, roughly speaking, closeto zero everywhere except for points surrounding (cid:126) Q where h , k , l are all integers. At these points the cross section is highlypeaked. As the size of a crystallite grows the peaks becometaller and thinner, converging to delta functions in the limit ofinfinite crystallite size. Na¨ıve quadrature does not perform wellon integrands with many thin peaks, so should be avoided forcomputing the powder average of big crystallites. If the crys-tals are indeed big, a common method of computing the pow-der average is to numerically integrate each peak separatelyand sum the contributions. One can also employ the tangent-cylinder approximation derived by Brindley & M´ering (1951)and discussed by Shi et al. (1993) to speed up the integration ofeach peak. An alternative to numerical integration is to derive an expres-sion for the powder average using a Harmonic expansion, whichdoes not require numerical integration over the sphere! We shallpresent a version of this for the simplest case that ω ( (cid:126) Q ) = π Q , which may be adequate for a highly disordered mate-rial. Let L m i j ( x , y ) denote the distance between the i th atom in alayer of type x and the j th atom of a layer type y for layers x , y separated vertically by m layers. When we talk about distance,we assume a unit length is 2 π | (cid:126) c ∗ | . It is shown in Appendix Athat I ( Q ) = N c (cid:88) m = − N c ( N c − | m | ) (cid:90) A (cid:90) A G m ( x , y ) (28) × n x (cid:88) i = n y (cid:88) j = f i f j sinc (cid:0) QL m i j ( x , y ) (cid:1) dxdy . The term n x (cid:88) i = n y (cid:88) j = f i f j sinc (cid:0) QL m i j ( x , y ) (cid:1) (29)is highly related to Debye’s equation, who’s 100th birthday wasrecently celebrated by Scardi et al. (2016).
3. Finitely many hidden states
It remains now to define the pair correlation function G m ( x , y ) which captures the probability of sampling from a crystal alayer of type x , then finding a layer of type y m layers aheadof x . To this end, we maintain the assumption of Varn et al. (2013), Riechers et al. (2015), and Varn & Crutchfield (2015)that the sequence of layers follows a Hidden Markov Model. Inparticular when the set of hidden states S and layer types A arefinite, the Hidden Markov Model (HMM) is an ordered quin-tuple Γ = ( A , S , µ , T , V ) where the terms are exactly thosedefined by Hart et al. (2018) in their Appendix A.In particular the probability of a layer adopting a hidden state j ∈ S can be represented as the element of a vector v . Giventhe hidden state of the HMM is i ∈ S , then the probability of atransition to j ∈ S is the i j th element of a transition matrix T .This matrix represents an operator which maps a distribution ofhidden states v of some layer to the distribution of hidden states w of the next layer, which is to say T v = w . (30)For a layer with hidden states following a distribution v , thelayer found m layers ahead has hidden state following the dis-tribution u which is related to v by T m v = u . (31)The stationary distribution π of T represents the probability dis-tribution over the set of hidden states obtained by sampling alayer from the crystal. A sufficient condition for π to exist andbe unique is that the Markov Chain induced by T is positiverecurrent, which means from any state s the probability of even-tual return to s state is unity. Further since, T π = π (32) Hart, Hansen and Kuhs · possibly a subtitle Acta Cryst. (2018). A , 000000 esearch papers we have that π is an eigenvector of T with eigenvalue 1.Every hidden state emits a symbol from the alphabet accord-ing to some distribution that depends on the hidden state. Evenif the number of hidden states is finite, the alphabet of symbols A could be finite, countably infinite or uncountably infinite. Thetheory presented by Riechers et al. (2015) and Hart et al. (2018)assumes A is finite, and therefore that the probability distribu-tion over symbols from the hidden state s ∈ S is a vector v s ∈ V .Further, the probability of emitting a symbol x ∈ A is one ofthe entries of the vector v s , denoted v s ( x ) . This present paperextends the existing theory by stating that if A is countably infi-nite then v s is an infinite sequence with x th term v s ( x ) and if A ⊂ R n is uncountably infinite, then v s is a probability densityfunction v s ( x ) . To distinguish these cases, the ordered quintu-ple Γ defining the HMM either contains vectors, sequences, orprobability density functions for A finite, countably infinite anduncountably infinite respectively. Whatever the cardinality of A , the pair correlation function G m ( x , y ) is given by G m ( x , y ) = (cid:88) r ∈ S (cid:88) s ∈ S v r ( x ) π r T mrs v s ( y ) (33)where T mrs is the rs th element of the matrix T m . With the expres-sion for the pair correlation (33) and cross section (25) together,we obtain a direct expression for the cross section of a crystalwith finitely many hidden states, and any of finitely, countablyinfinitely or uncountably infinitely layer types. Section 3.1 runsthrough an application of this expression. Suppose we have a finite state space and uncountable alpha-bet. Then for each state r ∈ S there is a probability densityfunction v r ( x ) over the alphabet of symbols A ⊂ R n . It is shownin Appendix B that the cross section for such a crystal can beexpressed d σ d Ω = Re (cid:26) Tr (cid:0) Diag ( π ) H ( S + N c I ) (cid:1)(cid:27) (34)where H is a Hermitian matrix with dimension equal to that of T with rs th element h rs = (cid:90) A v r ( x ) F ( x ) dx (cid:90) A v s ( y ) F ∗ ( y ) dy . (35)Moreover S = N c (cid:88) m = ( N c − m )( T e π il ) m (36)while Diag ( π ) is the diagonal matrix with elements the station-ary vector π , I is the identity matrix, Tr is the trace operator and Re { z } denotes the real part z ∈ C . It may be useful to note that S = N c (cid:88) m = ( N c − m )( T e π il ) m (37) = T e π il (cid:0) ( T e π il ) N c − N c ( T e π il − I ) − I (cid:1)(cid:0) T e π il − I (cid:1) − (38) when (cid:0) T e π il − I (cid:1) − exists. This model includes a class of crys-tals that, in the absence of turbostratic disorder, comprise a finitenumber of layer types, where the probability of some layer typefollowing another depends on the previous layer type. Each hid-den state represents a layer type without turbostratic disorder,while the distribution over the alphabet of symbols representsthe distribution over possible disorders a particular layer typecould adopt. A simple, if perhaps unrealistic, example is a crys-tal composed of 2 layer types labelled A and B , which adoptsome turbostratic disorder like a rotation, translation, or nonlin-ear deformation over some distributions v ( x ) and v ( x ) respec-tively. Suppose the probability given a layer is type A that thenext is also type A is α and the probability that if a layer is type B that the next will be type A is β . Then the transition matrixbetween hidden states A and B takes the form T = (cid:20) α − αβ − β (cid:21) . (39)For this toy model, we can expand the expression for the crosssection (34) and arrive at d σ d Ω = ( − α )( β − α + ) (40) × Re (cid:26) s (cid:2) β ( h − h ) + ( − α )( h − h ) (cid:3) + s (cid:2) ( − α )( h + h ) + β ( h + h ) (cid:3)(cid:27) + h + h − α where s = (cid:40) N c ( N c − ) if e π il = e π il ( e π ilNc + N c ( − e π il ) − )( − e π il ) otherwise, (41)and s = (cid:40) N c ( N c − ) if ( α − β ) e π il = 1 ( α − β ) e π il (( α − β ) Nc e π ilNc + N c ( − ( α − β ) e π il ) − )( − ( α − β ) e π il ) otherwise.(42)We can see that multiplying out the matrices and taking thetrace generates an expression that is long and hard to read evenfor the simplest case of a crystal with 2 hidden states! Conse-quently, we consider a cross section defined once we have deter-mined the transition matrix T , the stationary distribution π andthe matrix H . We will now explore a more sophisticated model,with a concrete application to carbon blacks. Shi (1993) wrote a thesisabout the crystal structure of disordered carbons to better under-stand their role as an electrode in lithium-ion batteries. Part ofthis document includes two sophisticated models of turbostraticcarbon blacks, which can be fitted to scattering data using theprogram CARBONX written by Shi et al. (1993). CARBONXwas recently picked up by Zhou et al. (2014) who compared the
Acta Cryst. (2018). A , 000000 Hart, Hansen and Kuhs · possibly a subtitle esearch papers performance of Shi’s model to the standard Rietveld refinementprogram GSAS for describing the cross section of disorderedcarbons obtained from a range of sources. Zhou et al. (2014)found that Shi’s account of turbostratic disorder improved thefit, suggesting the turbostratic disorder is much like Shi (1993)describes.The remainder of this section will express both Shi’s 1 layermodel and 2 layer model as hidden Markov models, where eachhidden states emits a disordered layer over some distributiondependent on the state. For both of these models, we will obtainthe transition matrix T , the matrix H and stationary vector π ,hence arrive at an expression for the cross section. We’ll startwith the 1 layer model, noting that these carbon blacks have 4hidden states, we will label 1, 2, 3, 4. States 1 , , A , B , C while the hidden state 4 represents a layerthat has slipped across the basal plane in a random directionwith random magnitude with uniform probability. According toShi’s 1 layer model model, for some probability P of slippageacross the basal plane, the transition matrix looks like T = − P − P P − P − P P − P − P P − P − P − P P (43)which has stationary vector π = . (44)In addition to the possibility of a layer slipping across the basalplane, Shi’s model stipulates that all layers may be shifted inthe direction orthogonal to the basal plane. The probability ofno shift occurring is denoted g , but if some shift does occur, theshift adopts a magnitude following a normal distribution cen-tred at zero with variance σ . The probability density of a layer n = , , z in the direction orthogonal tothe basal plane therefore has expression w ( z ) = g δ ( z ) + ( − g ) √ πσ exp (cid:18) − z σ (cid:19) . (45)The alphabet A for Shi’s 1 layer model is uncountable and com-prises ordered pairs x = ( z , n ) where n ∈ { , , , } denoteswhether the 0 disorder layer is type A , B , C or the 4th type thatslipped across the basal plane, while z is the displacement ofthat layer orthogonal to the basal plane and follows distribution(45).The structure factor of a layer x ≡ ( n , z ) can therefore bewritten F ( x ) = F ( z , n ) = F n ( z ) (46)and we have that a layer A (which has hidden state 1) with 0displacement orthogonal to the basal plane has unit cells with astructure factor F unit1 ( ) = f cos (cid:18) π ( h + k ) (cid:19) (47) where f is the form factor of a carbon atom. Consequently, ifwe make the simplifying assumption that all layers are rectan-gular with the same dimensions, then the structure factor of thelayer A with 0 orthogonal displacement is (by equation (10)) F ( ) = f cos (cid:18) π ( h + k ) (cid:19) sin ( N a π h ) sin ( π h ) sin ( N b π k ) sin ( π k ) (48) × e − i ( N a − ) π h e − i ( N b − ) π k . The layers A , B , C with displacement z have structure factors F ( z ) = F ( ) e π izl (49) F ( z ) = F ( ) e π i ( zl +( h + k ) /3 ) F ( z ) = F ( ) e π i ( zl − ( h + k ) /3 ) respectively. Now the probability of a hidden state n emitting asymbol x ∈ A is given by the probability density function v n ( x ) so (cid:90) A v n ( x ) F ( x ) dx = (cid:90) R w ( z ) F n ( z ) dz (50) = e π i φ n F ( ) (cid:90) R w ( z ) e π izl dz (51) = e π i φ n F ( ) F [ w ]( l ) (52)where F [ w ] is the Fourier transform of w F [ w ]( l ) = g + ( − g ) exp (cid:18) − σ l (cid:19) (53)and φ n = n = ( h + k ) /3 if n = − ( h + k ) /3 if n = v ( x ) and F ( x ) such that (cid:90) A v ( x ) F ( x ) dx = . (55)This gives us an expression for HH = | F ( ) F [ w ]( l ) | (56) × e − π i ( h + k ) e π i ( h + k ) e π i ( h + k ) e − π i ( h + k ) e − π i ( h + k ) e π i ( h + k ) . With T , π and H we have all we need to evaluate equation (34)and obtain the cross section for Shi’s 1 layer model.Shi’s 2 layer model is similar, and in the formalism of thispaper has 7 hidden states each comprising pairs of conventionallayers AB , AC , BA , BC , CA , CB as well as a layer XX translated Hart, Hansen and Kuhs · possibly a subtitle Acta Cryst. (2018). A , 000000 esearch papers somewhere across the basal plane. Like the 1 layer model, lay-ers are displaced in the direction orthogonal to the basal planeaccording to distribution (45), but this time with g =
0. Weenumerate these layer types from 1 to 7 and obtain the transi-tion matrix according to Shi’s description T = P t P P P t ¯ P P P t P P ¯ P P t P P P t P P P t P − P − P − P − P − P − P P (57)where ¯ P = − P t − P and P t , P and ¯ P are probabilities summingto 1. The stationary vector is π = . (58)Now to obtain H , first let ϕ = + e π i ( cl +( h + k ) /3 ) (59) ϕ = + e π i ( cl − ( h + k ) /3 ) ϕ = e π i ( h + k ) /3 + e π icl ϕ = e π i ( h + k ) /3 + e π i ( cl − ( h + k ) /3 ) ϕ = e − π i ( h + k ) /3 + e π icl ϕ = e − π i ( h + k ) /3 + e π i ( cl +( h + k ) /3 ) ϕ = , which we have introduced for notational convenience. Thestructure factor for layer types ( n , z ) are F n ( z ) = F ( ) e π izl ϕ n (60)for n = H has elements h nm = | F ( ) F [ w ]( l ) | ϕ n ϕ m , (61)where we have for n > m > h nm = (cid:90) A v ( x ) F ( x ) dx = . (62)Both of Shi’s models make specific assumptions that simplifythe mathematics and allow the models to be expressed con-cisely, but are not necessarily physically principled. For exam-ple the 2 layer model accounts for normally distributed tur-bostratic spacing between pairs of layers, but not for disorderwithin a pair of layers. Moreover, certain transitions e.g. AB to AC are assumed impossible, even though they are physicallyplausible. By framing Shi’s model in the HMM framework, wecan straight forwardly modify the model to encompass any dis-order we like, while retaining a neat expression for the crosssection. Recommending specific improvements to Shi’s modelis beyond the scope of this paper, which instead presents theseexamples to demonstrate that the HMM framework is flexi-ble and general enough to describe a wide range of turbostaticmaterials.
4. Uncountably many hidden states
Having examined a HMM with a finite number of hidden states,we will now move on the stranger world of uncountably manyhidden states. If S is uncountably infinite then a probability dis-tribution over S is given by some probability density function v . We suppose S ⊂ R n is open, connected and bounded. Sincethe integral of v over S must equal unity, v is necessarily squareintegrable and therefore in the Hilbert space of square integrablefunctions L . Given the states are distributed according to v , thedistribution over hidden states at the next layer w ∈ L is (cid:90) S k ( r , s ) v ( r ) dx = w ( s ) (63)where k ( r , s ) represents the probability density of s ∈ S follow-ing r ∈ S and is called the transition kernel. This gives rise toan integral operator T : L → L defined ( T v )( s ) = (cid:90) S k ( r , s ) v ( r ) dx . (64)The probability of sampling from the crystal a layer with hiddentype r is given by the probability density function π ( r ) whichexists, is unique and satisfies T π = π (65)if the transition kernel k ( r , s ) is positive recurrent. If it exists,the stationary distribution π is an eigenvector of the operator T with eigenvalue one. Given a distribution over hidden states v ,the distribution over hidden states of a layer w after m transi-tions satisfies T m v = w . (66)The pair correlation function for a crystal with uncountablymany hidden states is therefore G m ( x , y ) = (cid:90) S (cid:90) S v ( r , x ) π ( r )( T m δ r )( s ) v ( s , y ) drds (67)where δ r ( s ) is the shifted delta function δ ( r − s ) where we inter-pret ( T m δ r )( s ) = ( T m − T δ r )( s ) (68) = ( T m − k r )( s ) (69)where k r is the probability density function k r ( s ) ≡ k ( r , s ) . Acta Cryst. (2018). A , 000000 Hart, Hansen and Kuhs · possibly a subtitle esearch papers Suppose the probability of a layer being a certain typedepends only on the type of the previous layer, then we havea Markov chain of layer types. This is a special case of a HMMwhere every hidden state emits a symbol with probability 1 andno two states emit the same symbol. Formally, this is obtainedby letting S = A and letting V be the identity map. For aMarkov chain of layers adopting one of uncountably many layertypes, the pair correlation function reduces to G m ( x , y ) = π ( x )( T m δ x )( y ) . (70)With this, we show in Appendix C that the cross section of acrystal described by a Markov chain, with an uncountable infi-nite of layer types is d σ d Ω = Re (cid:40) (cid:90) A (cid:90) A F ( x ) F ∗ ( y ) π ( x ) Z δ x ( y ) dxdy (cid:41) (71) + N c (cid:90) A | F ( x ) | π ( x ) dx where Re { z } represents the real part of the complex number z ∈ C , while δ x ( y ) is the shifted delta function δ ( x − y ) and Z : L → L is a linear operator defined Z ≡ N c (cid:88) m = ( N c − | m | )( e π il T ) m (72)where we interpret Z δ x ( y ) as the evaluation at y of the function Z δ x .Expression (71) for the cross section is quite unwieldy,demanding the evaluation of both a double integral and repeatedapplication of the operator T . Using numerical integration forthis task may not be a good idea. A possible approach is toapproximate the infinite state space as large but finite, hence dis-cretising the structure factors and state distributions - collapsingthe problem to the case of a large but finite state space.Alternatively, one can follow the lead of Berliner & Werner(1986), Hansen et al. (2008 a ), Hansen et al. (2008 b ) and com-pute the cross section using a Monte Carlo simulation. Whenthe state space S is finite, Hart et al. (2018) argue that the MonteCarlo approach is much slower than computing the cross sectionexplicitly using matrix operations. However, for an uncountablestate space the problem of computing the cross section explic-itly boils down to recursively computing integrals, which is gen-erally much harder. In the upcoming sections, we consider spe-cial cases of (71) that admit to further analysis and yield expres-sions that are faster to compute and perhaps more informative. The first of these cases requires that T is a compact, self-adjoint operator on the Hilbert space of square integrable func-tions L . These conditions hold for a crystal where the prob-ability density of a state y following state x is equal to theprobability density of state x following y ; which is to say k ( x , y ) = k ( y , x ) . This condition is sufficient (but not necessary) to imply that the Markov chain of layers describing the crystal is reversible , representing a special type of crystal that appears thesame (in some statistical sense) when turned upside-down. Thesignificance of these reversible crystals is treated extensively byEllison et al. (2009) and discussed in the context of ice and opalby Hart et al. (2018). For a crystal comprising layers that differ(for example) only by some rotation or translation, the proba-bility density of a layer rotated to angle or position y followinga layer rotated to angle or position x must equal the probabilitydensity of a layer at angle or position x following one at angle orposition y . We note here that this idea could in principle applyto a much broader class of disorders.With the assumption that T is a compact self-adjoint oper-ator on some Hilbert space, the Spectral Theorem provides anexpression for the cross section d σ d Ω = Re (cid:40) (cid:88) n ∈ Λ s n (cid:90) A F ( x ) π ( x ) u n ( x ) dx (cid:90) A F ∗ ( y ) u n ( y ) dy (cid:41) (73) + N c (cid:90) A π ( x ) | F ( x ) | dx where 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, (74)and u n and λ n e π il the eigenvectors and eigenvalues of e π il T indexed by the set Λ which repeats eigenvalues accordingto their algebraic multiplicity. The details are fleshed out inAppendix C.2. The second case applies to a so-called convolution kernel k on an uncountable state space, which requires for some n ∈ N that A is the open hypercube of dimension n denoted A =( , ) n and that k ( x , y ) ≡ (cid:88) m ∈ Z n P ( m + y − x ) (75)for some probabilty distribution P in the Hilbert space of squareintegrable functions L . If we return to the example of a crys-tal composed of layers that differ only by a rotation, we caninterpret condition (75) as insisting that the angle of rotationbetween any pair of layers follows the same probability distribu-tion P . The summation over m represents the fact that a rotationto angle θ is equal to a rotation to angle θ + m π for all m ∈ Z so we let the space A = ( , ) and interpret for x ∈ A that 2 π x is a layer’s angle of rotation. Now n represents the dimensionof the state space A , and equals 1 here. If for example, layerswere identical up to some translation in any of three directions,then the state space A would be three dimensional and n wouldadopt the value 3 and A = ( , ) .We note that P ( y − x ) = P ( x − y ) does not hold in general,so T is not necessarily self-adjoint even if it has a convolu-tion kernel. We also remark that the stationary distribution π of the operator T with convolution kernel is uniform. With this Hart, Hansen and Kuhs · possibly a subtitle Acta Cryst. (2018). A , 000000 esearch papers established, we present in Appendix C a derivation for the crosssection of a crystal with kernel Pd σ d Ω = Re (cid:40) (cid:90) A F ( x )( F ∗ (cid:126) s )( x ) + N c | F ( x ) | dx (cid:41) (76)with a (cid:126) b representing the convolution of a with b and thefunction s ∈ L satisfying F [ s ] = (cid:40) N c ( N c − ) if F [ P ] e π il = F [ P ] e π il ( F [ P ] Nc e π ilNc + N c ( −F [ P ] e π il ) − )( −F [ P ] e π il ) otherwise, (77)where F [ φ ] represents the Fourier transform of φ ∈ L . Givena choice of P , the function s ∈ L does not have an analyticform in general, but can be approximated numerically using atmost 2 Fast Fourier Transforms (FFTs). The first FFT is used tocompute F [ P ] , if the transform cannot be obtained analytically,from which we can find F [ s ] via equation (77). The second FFTis used to find the inverse transform of F [ s ] , yielding s . Thatsaid, computing s explicitly may not even be necessary if weobserve that F ∗ (cid:126) s = F − (cid:2) F [ F ∗ ] F [ s ] (cid:3) (78)by the convolution theorem. Application to twisted n -layer graphene With the theoryoutlined, we now have a lens through which to examine a toymodel of twisted n -layer graphene. Suppose first of all thatthe layers of graphene can adopt any of the uncountably manyangles of rotation θ ∈ [ − π , − π ] = A relative to some arbitrary2D coordinate system. We assume that the probability of a rota-tion to angle y given a previous layer is at angle x is given by asymmetric function P ∈ L such that P ( y − x ) ≡ P ( x − y ) . Con-sequently, the cross section of this model satisfies both equa-tions (73) and (76). In order to express the cross section moreconcretely, we first note that the structure factor of the grapheneunit cell has expression F unit ( ) = f e π i ( h + k ) + f e π i ( h + k ) (79)so the structure factor of the unit cell at some arbitrary rotation θ is therefore F unit ( θ ) = f e π i ( h ( cos ( θ ) − sin ( θ ))+ k ( cos ( θ )+ sin ( θ ))) (80) + f e π i ( h ( cos ( θ ) − sin ( θ ))+ k ( cos ( θ )+ sin ( θ ))) . Then by the derivation of equation (16), the structure factor ofa graphene layer is F ( θ ) = F unit ( θ ) sin ( N a π h θ ) sin ( π h θ ) sin ( N b π k θ ) sin ( π k θ ) (81) × e − i ( N a − ) π h θ e − i ( N b − ) π k θ where h θ = (cid:126) Q · ( X (cid:126) a ) (82) k θ = (cid:126) Q · ( X (cid:126) b ) where X is the rotation matrix X = (cid:20) cos ( θ ) − sin ( θ ) sin ( θ ) cos ( θ ) (cid:21) . (83)Next, we observe that since T has a convolution kernel P ( y − x ) ,the operator T has a uniform stationary distribution π , whichintegrates to unity over its domain [ − π , π ] , so we deduce π ( θ ) ≡ π . (84)Now, the angle of rotation between some pair of layers follows adistribution P , which would ideally be chosen with some phys-ical motivation, and be consistent with empirical data, like thatpresented by Brown et al. (2012). With a choice of P , we havean expression for the cross section of n -layer twisted graphene d σ d Ω = π Re (cid:40) π (cid:90) − π F ( θ )( F ∗ (cid:126) s )( θ ) + N c | F ( θ ) | d θ (cid:41) (85)which can be numerically integrated in good time to high preci-sion. We considered in section 4.3 a crystal with transition opera-tor T imbued with a convolution kernel where layers can exhibita broad range of turbostratic disorder. In this section we zoominto a special case where all layers are identical up to sometranslation, and show in Appendix C.3 that the cross section ofthese crystals has expression d σ d Ω = | F ( ) | (cid:18) Re (cid:8) F [ s ]( (cid:126) Q ) (cid:9) + N c (cid:19) . (86)One can intuit the relevance of this model by considering a crys-tal where a layer may slip across the basal plane by some mag-nitude following some distribution. For example the centre ofone layer may slip by some magnitude away from the centreof the next layer. As we move up through the crystal, the cen-tre of each layer performs a random walk, and the centre of the n th layer will gradually drift away from the centre of the 1stlayer as n grows. Alternatively, one might consider a sequenceof layers with expected vertical separation c (where vertical isorthogonal to the basal plane) but due to the effects of disorder,a layer is separated vertically from its predecessor by some ran-dom value following a normal distribution centred at c . Dritz &Tchoubar (1990) and Guinier (1964) describe this type of disor-der as disorder of the second type . This is subtly different fromShi’s model, where the layers adopt positions following inde-pendent and identical normal distribution centred at each of thelayers’ expected position, an example of disorder of the firsttype . Figure 3 illustrates this difference. Acta Cryst. (2018). A , 000000 Hart, Hansen and Kuhs · possibly a subtitle esearch papers
0c 1c 2c 3c 4c 5c 6c1
Figure 3
Here, the layers nearest the axis perpendicular to the basel plane have posi-tion normally distributed about their expected positions 0 , c , c , c ... so exhibitdisorder of the first type. The layers furthest from the axis have normally dis-tributed pairwise separation hence undergo disorder of the second type and forma less coherent scattering pattern. The distributions have the same variance 0 . c . This distinction is important because the different disorderswould arise from different physics, and the different disordersgive rise to different scattering patterns. In particular, Shi’smodel of disorder suggests that if a layer is separated by itsneighbour by some distance approximately c , then the next layeris separated by approximately 2 c , and the next approximately3 c , and this continues for arbitrary nc , without reduction in theaccuracy of the approximation. However for the model incor-perating disorder of the second type, this approximation wouldgradually get worse with increasing n . This suggests that theform of the scattering pattern, which depends strongly on theperiodicity of layers, would differ, and this is reflected in thedifferent expressions for the cross section.To provide a specific example of disorder of the second type,suppose we have a sequence of graphite layers that are identical,except for some vertical displacement z that follows a distribu-tion P ( z ) = g δ ( z − c ) + ( − g ) √ πσ exp (cid:18) ( z − c ) σ (cid:19) (87)inspired by Shi’s 1 layer model. Then by noting | F ( ) | = f cos (cid:18) π ( h + k ) (cid:19) sin ( N a π h ) sin ( π h ) sin ( N b π k ) sin ( π k ) (88)we have all we need to compute the cross section explicitly. Thismodel is overly simple of course, but admits to much extension,and could therefore capture a large range of possible disorders.
5. A countable infinity of hidden states
Having delved into both uncountable and finite state spaces,this section presents a short treatment of countably infinite statespaces. Suppose S is countably infinite, then the probability ofa HMM adopting each state is enumerated as a sequence. Sincethis sequence sums to 1 it is necessarily square summable hencean element of the Hilbert space of square summable sequences (cid:96) . For a probability distribution v ∈ (cid:96) over hidden states, theprobability distribution over states for the next state w is givenby (cid:88) i ∈ N k i j v j = w i (89)where k i j is the transition kernel denoting the probability of thestate j following the state i . Much like HMMs with finite anduncountable hidden states, the transition kernel gives rise to thetransition operator T : (cid:96) → (cid:96) with stationary distributionan eigensequence with associated eigenvalue 1. The alphabetof symbols can be finite, countably infinite or uncountably infi-nite. In the special case that every state emits a unique symbolwith probability 1, the HMM is just a Markov chain. This formsa simple model of n -layer moir´e graphene, where the set of alllayer pairs forming a moir´e pattern is countably infinite. We calleach of these layer pairs a superlattice and suppose each super-lattice is labelled by some i ∈ N and given that a superlattice istype i the probability that the next superlattice is type j dependsonly on i and j . Then the sequence of superlattices forms aMarkov chain with uncountable state space S . The cross sec-tion of a Markovian crystal with countably infinite state spaceis shown in Appendix C to satisfy d σ d Ω = (cid:32) Re (cid:40) (cid:88) i ∈ N (cid:88) j ∈ N F i F ∗ j π i Z δ i j (cid:41) + N c (cid:88) i ∈ N F ∗ i F i π i (cid:33) where F i is structure factor of the state indexed by i , π i , is the i th element of the stationary distribution π , δ i j is the Kroneckerdelta, and Z : (cid:96) → (cid:96) is defined Z ≡ N c (cid:88) m = ( N c − | m | )( e π il T ) m . (90)We note here that if π is uniform, then for any sequence φ i ∈ (cid:96) we interpret (cid:88) i ∈ N π i φ i ≡ lim n →∞ n n (cid:88) i = φ i . (91)Moreover, if k i j = k ji then by the Spectral Theorem d σ d Ω = Re (cid:40) (cid:88) n ∈ Λ s n (cid:88) i ∈ N F i π i u in (cid:88) j ∈ N F ∗ j u jn (cid:41) + N c (cid:88) i ∈ N π i F ∗ i F i (92)with 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, (93)where u n and λ n are the eigensequences and eigenvalues of T indexed by Λ repeating eigenvalues according to their algebraicmultiplicity. A derivation is presented in Appendix C.1.
6. Outlook
We began to extend the study of chaotic crystallography to crys-tals with infinitely many layer types - describing turbostratically Hart, Hansen and Kuhs · possibly a subtitle Acta Cryst. (2018). A , 000000 esearch papers disordered materials like carbon blacks, smectites, and n -layergraphene. In particular, we derived an explicit cross section forcarbon blacks related to the two models proposed by Shi (1993).There are many other disordered materials that could be exam-ined under the framework presented here, smectites for exam-ple, are often studied with a qualitative look at diffraction peaks(Ufer et al. , 2008) (Ufer et al. , 2009), suggesting a mathematicalframework could be well received. It may be that the HMM isa good setting to formulate a well principled model of disorder,then test its validity by comparing theoretical and experimentalcross sections.This framework is open to much theoretical development. Forexample, for a crystal with uncountably many hidden states,the operator T is a Fredholm integral operator connecting themathematical theory to the well developed mathematical field ofFredholm theory. Exploring this connection may answer somepractical questions, like how the eigenvalues of T are related tothe convergence of the state distribution to steady state, which isdiscussed for the case of finitely many hidden states by Riechers et al. (2015). These authors pose the major problem of chaoticcrystallography as reconstructing a crystal’s ε -machine fromscattering data, where the ε -machine is (roughly speaking) themost information theoretically simple process that could giverise to the observed scattering pattern. Developing a theory ofhow to construct the ε -machine of a turbostratic material couldbe of great use to any community studying a turbostraticallydisordered material.Moreover, our treatment of countably infinite spaces is brief,and could plausibly be developed into something more readilyapplicable to crystals with a countably infinite number of layertypes like moir´e graphene. Bringing moir´e graphene into thepurview of chaotic crystallography could shed some new lighton the mysterious material, and be a fruitful area of research. Appendix APowder average
To make sense of our expression, consider a Cartesian coor-dinate system, where the unit length is equal to 2 π | (cid:126) c ∗ | . A point (cid:126) Q = ( h , k , l ) in the reciprocal lattice coordinates will be denoted (cid:126) Q (cid:48) = ( h (cid:48) , k (cid:48) , l (cid:48) ) in the Cartesian coordinates. Suppose a layerof type x is composed of n x atoms and the i th atom is foundat position (cid:0) a i ( x ) , b i ( x ) , c i ( x ) (cid:1) in these Cartesian coordinates.Then the square distance between the i th atom in a layer of type x and the j th atom in a layer of type y for layers x , y separatedby m is (cid:0) L m i j ( x , y ) (cid:1) = ( a i ( x ) − a j ( y )) + ( b i ( x ) − b j ( y )) (94) +( c i ( x ) − c j ( y ) + m ) . In the Cartesian system the structure factor of a layer x is F ( x ) = n x (cid:88) i = f i e i ( a i ( x ) h (cid:48) + b i ( x ) k (cid:48) + c i ( x ) l (cid:48) (95) and the structure factor product F ( x ) F ∗ ( y ) = n x (cid:88) i = n y (cid:88) j = f i f j e i (cid:0) ( a i ( x ) − a j ( y )) h (cid:48) +( b i ( x ) − b j ( y )) k (cid:48) +( c i ( x ) − c j ( y )) l (cid:48) (cid:1) (96)Now, to derive our expression for the cross section, we willuse the Harmonic expansion presented by Ba˘zant & Oh (1986)14 π Q (cid:90) ∂ B Q d σ d Ω dS = ∞ (cid:88) n = Q n ( n + ) ! ∆ n (cid:18) d σ d Ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126) Q (cid:48) = (97)where ∆ n = (cid:18) ∂ ∂ h (cid:48) + ∂ ∂ k (cid:48) + ∂ ∂ l (cid:48) (cid:19) n . (98)We will begin by writing down the cross section d σ d Ω = N c (cid:88) m = − N c ( N c − m ) (cid:90) A (cid:90) A G m ( x , y ) F ( x ) F ∗ ( y ) e π im l dxdy (99)and then seek a nice expression for ∆ n (cid:0) d σ d Ω (cid:1)(cid:12)(cid:12) (cid:126) Q (cid:48) = . So we con-sider ∆ n (cid:18) d σ d Ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126) Q (cid:48) = = N c (cid:88) m = − N c ( N c − m ) (cid:90) A (cid:90) A G m ( x , y ) (100) × ∆ n (cid:18) F ( x ) F ∗ ( y ) e π im l (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126) Q (cid:48) = dxdy and bring our attention to ∆ n (cid:18) F ( x ) F ∗ ( y ) e π im l (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126) Q (cid:48) = (101) = n x (cid:88) i = n y (cid:88) j = ∆ n (cid:18) f i f j exp (cid:0) i (( a i ( x ) − a j ( y )) h (cid:48) +( b i ( x ) − b j ( y )) k (cid:48) +( c i ( x ) − c j ( y ) + m ) l (cid:48) ) (cid:1)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126) Q (cid:48) = n x (cid:88) i = n y (cid:88) j = f i f j ( − ) n (cid:0) ( a i ( x ) − a j ( y )) (102) +( b i ( x ) − b j ( y )) +( c i ( x ) − c j ( y ) + m ) (cid:1) n = n x (cid:88) i = n y (cid:88) j = f i f j ( − ) n (cid:0) L m i j ( x , y ) (cid:1) n . Acta Cryst. (2018). A , 000000 Hart, Hansen and Kuhs · possibly a subtitlepossibly a subtitle
To make sense of our expression, consider a Cartesian coor-dinate system, where the unit length is equal to 2 π | (cid:126) c ∗ | . A point (cid:126) Q = ( h , k , l ) in the reciprocal lattice coordinates will be denoted (cid:126) Q (cid:48) = ( h (cid:48) , k (cid:48) , l (cid:48) ) in the Cartesian coordinates. Suppose a layerof type x is composed of n x atoms and the i th atom is foundat position (cid:0) a i ( x ) , b i ( x ) , c i ( x ) (cid:1) in these Cartesian coordinates.Then the square distance between the i th atom in a layer of type x and the j th atom in a layer of type y for layers x , y separatedby m is (cid:0) L m i j ( x , y ) (cid:1) = ( a i ( x ) − a j ( y )) + ( b i ( x ) − b j ( y )) (94) +( c i ( x ) − c j ( y ) + m ) . In the Cartesian system the structure factor of a layer x is F ( x ) = n x (cid:88) i = f i e i ( a i ( x ) h (cid:48) + b i ( x ) k (cid:48) + c i ( x ) l (cid:48) (95) and the structure factor product F ( x ) F ∗ ( y ) = n x (cid:88) i = n y (cid:88) j = f i f j e i (cid:0) ( a i ( x ) − a j ( y )) h (cid:48) +( b i ( x ) − b j ( y )) k (cid:48) +( c i ( x ) − c j ( y )) l (cid:48) (cid:1) (96)Now, to derive our expression for the cross section, we willuse the Harmonic expansion presented by Ba˘zant & Oh (1986)14 π Q (cid:90) ∂ B Q d σ d Ω dS = ∞ (cid:88) n = Q n ( n + ) ! ∆ n (cid:18) d σ d Ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126) Q (cid:48) = (97)where ∆ n = (cid:18) ∂ ∂ h (cid:48) + ∂ ∂ k (cid:48) + ∂ ∂ l (cid:48) (cid:19) n . (98)We will begin by writing down the cross section d σ d Ω = N c (cid:88) m = − N c ( N c − m ) (cid:90) A (cid:90) A G m ( x , y ) F ( x ) F ∗ ( y ) e π im l dxdy (99)and then seek a nice expression for ∆ n (cid:0) d σ d Ω (cid:1)(cid:12)(cid:12) (cid:126) Q (cid:48) = . So we con-sider ∆ n (cid:18) d σ d Ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126) Q (cid:48) = = N c (cid:88) m = − N c ( N c − m ) (cid:90) A (cid:90) A G m ( x , y ) (100) × ∆ n (cid:18) F ( x ) F ∗ ( y ) e π im l (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126) Q (cid:48) = dxdy and bring our attention to ∆ n (cid:18) F ( x ) F ∗ ( y ) e π im l (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126) Q (cid:48) = (101) = n x (cid:88) i = n y (cid:88) j = ∆ n (cid:18) f i f j exp (cid:0) i (( a i ( x ) − a j ( y )) h (cid:48) +( b i ( x ) − b j ( y )) k (cid:48) +( c i ( x ) − c j ( y ) + m ) l (cid:48) ) (cid:1)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126) Q (cid:48) = n x (cid:88) i = n y (cid:88) j = f i f j ( − ) n (cid:0) ( a i ( x ) − a j ( y )) (102) +( b i ( x ) − b j ( y )) +( c i ( x ) − c j ( y ) + m ) (cid:1) n = n x (cid:88) i = n y (cid:88) j = f i f j ( − ) n (cid:0) L m i j ( x , y ) (cid:1) n . Acta Cryst. (2018). A , 000000 Hart, Hansen and Kuhs · possibly a subtitlepossibly a subtitle esearch papers Next, we notice ∞ (cid:88) n = Q n ( n + ) ! ∆ n (cid:18) F ( x ) F ∗ ( y ) e π im l (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126) Q (cid:48) = (103) = ∞ (cid:88) n = Q n ( n + ) ! n x (cid:88) i = n y (cid:88) j = f i f j ( − ) n (cid:0) L m i j ( x , y ) (cid:1) n = n x (cid:88) i = n y (cid:88) j = f i f j ∞ (cid:88) n = ( QL m i j ( x , y )) n ( n + ) ! ( − ) n = n x (cid:88) i = n y (cid:88) j = f i f j sinc ( QL m i j ( x , y )) (104)where we have used the series expansion of sincsinc ( x ) = ∞ (cid:88) n = x n ( n + ) ! ( − ) n . (105)Putting all this together14 π Q (cid:90) ∂ B Q d σ d Ω dS = ∞ (cid:88) n = Q n ( n + ) ! ∆ n (cid:18) d σ d Ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126) Q (cid:48) = (106) = N c (cid:88) m = − N c ( N − m ) (cid:90) A (cid:90) A G m ( x , y ) (107) × ∞ (cid:88) n = Q n ( n + ) ! ∆ n (cid:18) F ( x ) F ∗ ( y ) e π im l (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126) Q (cid:48) = dxdy = N c (cid:88) m = − N c ( N c − m ) (cid:90) A (cid:90) A G m ( x , y ) (108) × n x (cid:88) i = n y (cid:88) j = f i f j sinc ( QL m i j ( x , y )) dxdy . Appendix BFinite state space, uncountable alphabet
Suppose we have a finite state space S and alphabet A ⊂ R n an open connected set. Then by combining the expression forthe pair correlation function (33) with the average structure fac-tor product (23) we arrive at Y m = (cid:90) A (cid:90) A (cid:88) r ∈ S (cid:88) s ∈ S π s T msr v r ( x ) F ( x ) F ∗ ( y ) v s ( y ) dxdy (109) = (cid:88) r ∈ S (cid:88) s ∈ S π s T msr (cid:90) A v r ( x ) F ( x ) dx (cid:90) A v s ( y ) F ∗ ( y ) dy (110) = (cid:88) r ∈ S (cid:88) s ∈ S π s T msr h rs (111) = Tr (cid:0) Diag ( π ) T m H (cid:1) (112) where H is a Hermitian matrix with dimension equal to thenumber of states in the space S , and has elements h rs = (cid:90) A v r ( x ) F ( x ) dx (cid:90) A v s ( y ) F ∗ ( y ) dy . (113)Next, we note that Y m = Y ∗− m so d σ d Ω = N c (cid:88) m = − N c ( N c − | m | ) Y m e π im l (114) = N c (cid:88) m = ( N c − m ) Y m e π im l (115) + N c Y + N c (cid:88) m = ( N c − m ) Y ∗ m e − π im l = N c (cid:88) m = ( N c − m ) Tr (cid:0) Diag ( π ) T m H (cid:1) e π im l (116) + N c Tr (cid:0) Diag ( π ) H (cid:1) + N c (cid:88) m = ( N c − m ) Tr (cid:0) Diag ( π ) T m H ∗ (cid:1) e − π im l = Tr (cid:32) Diag ( π ) (cid:18) N c (cid:88) m = ( N c − m )( T e π il ) m (cid:19) H (cid:33) (117) + N c Tr (cid:0) Diag ( π ) H (cid:1) + Tr (cid:32) Diag ( π ) (cid:18) N c (cid:88) m = ( N c − m )( T e − π il ) m (cid:19) H ∗ (cid:33) = Tr (cid:0) Diag ( π ) SH (cid:1) (118) + N c Tr (cid:0) Diag ( π ) H (cid:1) + Tr (cid:0) Diag ( π ) S ∗ H ∗ (cid:1) = Re (cid:26) Tr (cid:0) Diag ( π ) H ( S + N c I ) (cid:1)(cid:27) (119)and we arrive at the expression for cross section. Appendix CInfinite state space
This section describes a crystal comprising infinitely manylayer types, where the probability distribution over the set oflayers follows a Markov chain. In some cases, the argumentsfor an uncountable space represented by R n (open and con-nected) and a countable space represented by N are essen-tially the same, and in these cases arguments may be madeover a general Hilbert space H that apply to both the squaresummable sequences (cid:96) and the square integrable functions L ,representing distributions over the countable state space N oruncountable state space R n (open and connected) respectively. Hart, Hansen and Kuhs · possibly a subtitle Acta Cryst. (2018). A , 000000 esearch papers For brevity, we allow the symbols of integration (cid:90) · dx (120)to represent either integration over the open connected set A ⊂ R n or summation over N . C.1. The most general kernel
The cross section of a crystal with a countable infinity oruncountable infinity of hidden layers is N c (cid:88) m = − N c ( N c − | m | ) (cid:90) (cid:90) F ( x ) F ∗ ( y ) G m ( x , y ) e π im l dxdy , which we can split into three terms N c (cid:88) m = − N c ( N c − | m | ) (cid:90) (cid:90) F ( x ) F ∗ ( y ) G m ( x , y ) e π im l dxdy (121) = N c (cid:88) m = ( N c − | m | ) (cid:90) (cid:90) F ( x ) F ∗ ( y ) G m ( x , y ) e π im l dxdy + N c (cid:90) (cid:90) F ( x ) F ∗ ( y ) G ( x , y ) dxdy + N c (cid:88) m = ( N c − | m | ) (cid:90) (cid:90) F ( x ) F ∗ ( y ) G − m ( x , y ) e − π im l dxdy . Now the second term of the RHS of equation (121) requires anexpression for G ( x , y ) , which represents the probability (den-sity) of sampling a layer that is type x , and given it is type x thatit is itself type y . In an uncountable space we let δ a ( y ) ≡ δ ( a − y ) for a ∈ A represent the shifted Dirac delta function evaluatedat y ∈ A and notice G ( x , y ) = π ( x ) δ x ( y ) . In a countable space, G ( i , j ) = π i δ i j for i j ∈ N by the same arguments. Again, toavoid writing essentially the same thing twice, we continue byallowing δ x ( y ) to represent δ xy for countable spaces. With this,we have that the second term satisfies N c (cid:90) (cid:90) F ( x ) F ∗ ( y ) G ( x , y ) dxdy (122) = N c (cid:90) π ( x ) F ∗ ( x ) F ( x ) dx . Next, we focus on the third term, noting the probability (den-sity) of sampling a state x then finding a state y after movingforward m blocks is the same as sampling a state y then findinga state x after moving backward m blocks. Thus F ( x ) F ∗ ( y ) G m ( x , y ) = F ∗ ( x ) F ( y ) G − m ( x , y ) (123) so we have that the LHS of (121) equals N c (cid:88) m = ( N c − | m | ) (cid:90) (cid:90) F ( x ) F ∗ ( y ) G m ( x , y ) e π im l dxdy (124) + N c (cid:90) π ( x ) F ∗ ( x ) F ( x ) dx + N c (cid:88) m = ( N c − | m | ) (cid:90) (cid:90) F ∗ ( x ) F ( y ) G m ( x , y ) e − π im l dxdy hence we can see that the third term is just the complex conju-gate of the first. With this information, note that the sum of thefirst and third term is just twice the real part of the first, so weproceed by only considering the first term and noting G m ( x , y ) ≡ π ( x ) T m δ x ( y ) for m > N c (cid:88) m = ( N c − | m | ) (cid:90) (cid:90) F ( x ) F ∗ ( y ) G m ( x , y ) e π im l dxdy (126) = (cid:90) (cid:90) F ( x ) F ∗ ( y ) π ( x ) N c (cid:88) m = ( N c − | m | )( e π il T ) m δ x ( y ) dxdy = (cid:90) (cid:90) F ( x ) F ∗ ( y ) π ( x ) Z δ x ( y ) dxdy (127)where z ∈ C and Z is an operator with expression Z ≡ N c (cid:88) m = ( N c − | m | )( e π il T ) m . (128)Putting all this together, we have d σ d Ω = Re (cid:40) (cid:90) (cid:90) F ( x ) F ∗ ( y ) π ( x ) Z δ x ( y ) dxdy (cid:41) (129) + N c (cid:90) | F ( x ) | π ( x ) dx where Re { z } represents the real part of the complex number z ∈ C . This completes the derivation of the general cross sec-tion for both countable and uncountable state spaces. C.2. The case of a symmetric kernel
With the general expression established for both countableand uncountable spaces, we consider the special case where k ( x , y ) = k ( y , x ) so T is self-adjoint. Since T is compact, weobserve by the Spectral Theorem for compact self-adjoint oper-ators that ( e π il T ) m δ x ( y ) = (cid:88) n ∈ Λ (cid:104) δ x , u n (cid:105) ( λ n e π il ) m u n ( y ) (130) = (cid:88) n ∈ Λ u n ( x ) u n ( y )( λ n e π il ) m (131)where u n and λ n e π il are the eigenvectors and eigenvalues of e π il T indexed by the set Λ which repeats eigenvalues accord-ing to their algebraic multiplicity. Here we have also used (cid:104) φ, ψ (cid:105) Acta Cryst. (2018). A , 000000 Hart, Hansen and Kuhs · possibly a subtitlepossibly a subtitle
With the general expression established for both countableand uncountable spaces, we consider the special case where k ( x , y ) = k ( y , x ) so T is self-adjoint. Since T is compact, weobserve by the Spectral Theorem for compact self-adjoint oper-ators that ( e π il T ) m δ x ( y ) = (cid:88) n ∈ Λ (cid:104) δ x , u n (cid:105) ( λ n e π il ) m u n ( y ) (130) = (cid:88) n ∈ Λ u n ( x ) u n ( y )( λ n e π il ) m (131)where u n and λ n e π il are the eigenvectors and eigenvalues of e π il T indexed by the set Λ which repeats eigenvalues accord-ing to their algebraic multiplicity. Here we have also used (cid:104) φ, ψ (cid:105) Acta Cryst. (2018). A , 000000 Hart, Hansen and Kuhs · possibly a subtitlepossibly a subtitle esearch papers to denote inner product on the Hilbert space H of φ, ψ ∈ H .With this, we can proceed from equation (126) and deduce (cid:90) (cid:90) F ( x ) F ∗ ( y ) π ( x ) × (132) N c (cid:88) m = ( N c − | m | )( e π il T ) m δ x ( y ) dxdy = (cid:90) (cid:90) F ( x ) F ∗ ( y ) π ( x ) × N c (cid:88) m = ( N c − | m | ) (cid:88) n ∈ Λ u n ( x ) u n ( y )( λ n e π il ) m dxdy = (cid:88) n ∈ Λ (cid:90) F ( x ) π ( x ) u n ( x ) (cid:90) F ∗ ( y ) u n ( y ) × (133) N c (cid:88) m = ( N c − | m | )( λ n e π il ) m dxdy = (cid:88) n ∈ Λ s n (cid:90) F ( x ) π ( x ) u n ( x ) dx (cid:90) F ∗ ( y ) u n ( y ) dy (134)where s n = (cid:40) N c ( N c − ) if l ∈ Z and λ n = λ n e π il ( λ Ncn e π ilNc + N c ( − λ n e π il ) − )( − λ n e π il ) otherwise, (135)and the expression for the cross section follows immediately.We can evaluate (134) approximately by summing over only thefirst few values of n , requiring only a few eigenvalues and eigen-vectors. Unfortunately, there is no method of deriving closedform solutions for u n for a general kernel k ; but analytic solu-tions do exist in some special cases. In the case of an uncount-able space represented by A it may be fruitful to note that solu-tions to the eigenvalue equation λ n u n ( x ) = (cid:90) A k ( x , y ) u n ( y ) dy (136)also satisfy Lu n = λ − n u n (137)for L a differential operator with kernel k . Finding the eigen-values and eigenvectors of L is then a question of solving thedifferential equation (137). C.3. The case of a convolution kernel
Let A = ( , ) n , φ ∈ L and k ( x , y ) = (cid:88) m ∈ Z n P ( m + y − x ) (138)where P is a square integrable probability distribution over R n .Then T φ = (cid:90) A k ( x , y ) φ ( x ) dx (139) = (cid:88) m ∈ Z n (cid:90) A P ( m + y − x ) φ ( x ) dx (140) = (cid:90) R n P ( y − x ) φ ( x ) dx ≡ P (cid:126) φ (141) where (cid:126) denotes the convolution. Now define the sequence offunctions T n + φ = P (cid:126) T n φ (142)for which we denote the n th term by P (cid:126) n φ , and identify thissequence with T n φ . Next, we denote the Fourier transform by F : L → L and use the convolution theorem to deduce F (cid:2) P (cid:126) n φ (cid:3) = ( F [ P ]) n F [ φ ] . (143)We now observe that Z δ x = N c (cid:88) m = ( N c − | m | )( e π il T ) m δ x (144) = N c (cid:88) m = ( N c − | m | ) e π im l P (cid:126) m δ x (145) = F − (cid:20) N c (cid:88) m = ( N c − | m | ) e π im l F (cid:2) P (cid:126) m δ x (cid:3)(cid:21) (146) = F − (cid:20) N c (cid:88) m = ( N c − | m | ) e π im l F [ P ] m F [ δ x ] (cid:21) (147) = F − (cid:20) F [ s ] F [ δ x ] (cid:21) (148) = s (cid:126) δ x (149) = s x (150)where s ∈ L is defined s ≡ F − (cid:20) N c (cid:88) m = ( N c − | m | ) e π im l F [ P ] m (cid:21) (151)so satisfies F [ s ] = (cid:40) N c ( N c − ) if F [ P ] e π il = F [ P ] e π il ( F [ P ] Nc e π ilNc + N c ( −F [ P ] e π il ) − )( −F [ P ] e π il ) otherwise,(152)and s x ( y ) ≡ s ( y − x ) allowing us to arrive at an expression forthe cross section d σ d Ω = Re (cid:40) (cid:90) A (cid:90) R n F ( x ) F ∗ ( y ) s ( y − x ) dxdy (cid:41) (153) + N c (cid:90) A | F ( x ) | dx = Re (cid:40) (cid:90) A F ∗ ( x )( F (cid:126) s )( x ) dy + N c | F ( x ) | dx (cid:41) . (154) Hart, Hansen and Kuhs · possibly a subtitle Acta Cryst. (2018). A , 000000 esearch papers If we make the further assumption that layers are identical up totranslation, then starting from equation (153) d σ d Ω = Re (cid:26) (cid:90) A (cid:90) R n F ( x ) F ∗ ( y ) s ( y − x ) dxdy (cid:27) (155) + N c (cid:90) A | F ( x ) | dx = Re (cid:26) | F ( ) | (cid:90) A (cid:90) R n e π i (cid:126) x · (cid:126) Q e − π i (cid:126) y · (cid:126) Q s ( y − x ) dxdy (cid:27) (156) + N c | F ( ) | (cid:90) A dx = Re (cid:26) | F ( ) | (cid:90) A e π i (cid:126) x · (cid:126) Q (cid:90) R n e − π i (cid:126) y · (cid:126) Q s ( y − x ) dydx (cid:27) (157) + N c | F ( ) | = Re (cid:26) | F ( ) | (cid:90) A e π i (cid:126) x · (cid:126) Q F [ s ]( (cid:126) Q ) e − π i (cid:126) x · (cid:126) Q dx (cid:27) (158) + N c | F ( ) | = Re (cid:26) | F ( ) | F [ s ]( (cid:126) Q ) (cid:90) A dx (cid:27) (159) + N c | F ( ) | = | F ( ) | (cid:18) Re (cid:8) F [ s ]( (cid:126) Q ) (cid:9) + N c (cid:19) (160)and we arrive at the expression for the cross section of a crystalwith convolution kernel composed of layers that are identicalup to translation. References
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