Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 594-597   | 1 | 2 |

Section 1D quasiperiodic structures

W. Steurera* and T. Haibacha

aLaboratory of Crystallography, Department of Materials, ETH Hönggerberg, HCI G 511, Wolfgang-Pauli-Strasse 10, CH-8093 Zurich, Switzerland
Correspondence e-mail: 1D quasiperiodic structures

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The Fibonacci sequence, the best investigated example of a 1D quasiperiodic structure, can be obtained from the substitution rule σ: [\hbox{S} \rightarrow \hbox{L}], [\hbox{L} \rightarrow \hbox{LS}], replacing the letter S by L and the letter L by the word LS (see e.g. Luck et al., 1993[link]). Applying the substitution matrix [S = \pmatrix{0 &1\cr 1 &1\cr}]associated with σ, this rule can be written in the form [\pmatrix{\hbox{S}\cr \hbox{L}\cr} \rightarrow \pmatrix{0 &1\cr 1 &1\cr} \pmatrix{\hbox{S}\cr \hbox{L}\cr} = \pmatrix{\hbox{L}\cr \hbox{L} + \hbox{S}\cr}.]S gives the sum of letters, [\hbox{L} + \hbox{S} = \hbox{S} + \hbox{L}], and not their order. Consequently, the same substitution matrix can also be applied, for instance, to the substitution [\sigma']: [\hbox{S} \rightarrow \hbox{L}], [\hbox{L} \rightarrow \hbox{SL}]. The repeated action of S on the alphabet [A = \{\hbox{S, L}\}] yields the words [A_{n} = \sigma^{n}(\hbox{S})] and [B_{n} = \sigma^{n}(\hbox{L}) = A_{n + 1}] as illustrated in Table[link]. The frequencies of letters contained in the words [A_{n}] and [B_{n}] can be calculated by applying the nth power of the transposed substitution matrix on the unit vector. From [\openup3pt\pmatrix{\nu_{n + 1}^{A}\cr \nu_{n + 1}^{B}\cr} = S^{T} \pmatrix{\nu_{n}^{A}\cr \nu_{n}^{B}\cr}]it follows that [\openup3pt\pmatrix{\nu_{n}^{A}\cr \nu_{n}^{B}\cr} = (S^{T})^{n} \pmatrix{1\cr 1\cr}.]In the case of the Fibonacci sequence, [v_{n}^{B}] gives the total number of letters S and L, and [v_{n}^{A}] the number of letters L.

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Expansion of the Fibonacci sequence [B_{n} = \sigma^{n}(\hbox{L})] by repeated action of the substitution rule σ: [\hbox{S} \rightarrow \hbox{L}], [\hbox{L} \rightarrow \hbox{LS}]

[\nu_{\rm L}], [\nu_{\rm S}] are the frequencies of the letters L and S in word [B_{n}].

[B_{n}][\nu_{\rm L}][\nu_{\rm S}]n
L 1 0 0
LS 1 1 1
LSL 2 1 2
LSLLS 3 2 3
  [\vdots] [\vdots] [\vdots]
  [F_{n + 1}] [F_{n}] n

An infinite Fibonacci sequence, i.e. a word [B_{n}] with [n \rightarrow \infty], remains invariant under inflation (deflation). Inflation (deflation) means that the number of letters L, S increases (decreases) under the action of the (inverted) substitution matrix S. Inflation and deflation represent self-similarity (scaling) symmetry operations on the infinite Fibonacci sequence. A more detailed discussion of the scaling properties of the Fibonacci chain in direct and reciprocal space will be given later.

The Fibonacci numbers [F_{n} = F_{n - 1} + F_{n - 2}] form a series with [\lim_{n \rightarrow \infty} (F_{n + 1}/F_{n}) = \tau] {the golden mean [\tau=[1 + (5)^{1/2}]/2] [= 2 \cos(\pi /5) = 1.618 \ldots\]}. The ratio of the frequencies of L and S in the Fibonacci sequence converges to τ if the sequence goes to infinity. The continued fraction expansion of the golden mean τ, [\tau = 1 + {1 \over 1 + {\displaystyle{1 \over 1 + {\displaystyle{1 \over 1 + \ldots}}}}},]contains only the number 1. This means that τ is the `most irrational' number, i.e. the irrational number with the worst truncated continued fraction approximation to it. This might be one of the reasons for the stability of quasiperiodic systems, where τ plays a role. The strong irrationality may impede the lock-in into commensurate systems (rational approximants).

By associating intervals (e.g. atomic distances) with length ratio τ to 1 to the letters L and S, a quasiperiodic structure [s({\bf r})] (Fibonacci chain) can be obtained. The invariance of the ratio of lengths [\hbox{L/S} = (\hbox{L} + \hbox{S})/\hbox{L} = \tau] is responsible for the invariance of the Fibonacci chain under scaling by a factor [\tau^{n}, n \in {\bb Z}]. Owing to a minimum atomic distance S in real crystal structures, the full set of inverse symmetry operators [\tau^{-n}] does not exist. Consequently, the set of scaling operators [s = \{\tau^{0} = 1, \tau^{1}, \ldots \}] forms only a semi-group, i.e. an associative groupoid. Groupoids are the most general algebraic sets satisfying only one of the group axioms: the associative law. The scaling properties of the Fibonacci sequence can be derived from the eigenvalues of the scaling matrix S. For this purpose the equation [\det |S - \lambda I| = 0]with eigenvalue λ and unit matrix I has to be solved. The evaluation of the determinant yields the characteristic polynomial [\lambda^{2} - \lambda - 1 = 0,]yielding in turn the eigenvalues [\lambda_{1} = [1 + (5)^{1/2}]/2 = \tau], [\lambda_{2} = [1 - (5)^{1/2}]/2 = - 1/\tau] and the eigenvectors[\hbox{\bf w}_{1} = \pmatrix{1\cr \tau\cr},\quad \hbox{\bf w}_{2} = \pmatrix{1\cr - 1/\tau\cr}.]Rewriting the eigenvalue equation gives for the first (i.e. the largest) eigenvalue [\pmatrix{0 &1\cr 1 &1\cr} \pmatrix{1\cr \tau\cr} = \pmatrix{\tau\cr 1 + \tau\cr} = \pmatrix{\tau\cr \tau^{2}\cr} = \tau \pmatrix{1\cr \tau\cr}.]Identifying the eigenvector[\pmatrix{1\cr \tau\cr}]with[\pmatrix{\hbox{S}\cr \hbox{L}\cr}]shows that an infinite Fibonacci sequence [s({\bf r})] remains invariant under scaling by a factor τ. This scaling operation maps each new lattice vector [\tau {\bf r}] upon a vector r of the original lattice: [s(\tau {\bf r}) = s({\bf r}).]Considering periodic lattices, these eigenvalues are integer numbers. For quasiperiodic `lattices' (quasilattices) they always correspond to algebraic numbers (Pisot numbers). A Pisot number is the solution of a polynomial equation with integer coefficients. It is larger than one, whereas the modulus of its conjugate is smaller than unity: [\lambda_{1} \,\gt \,1] and [|\lambda_{2}| \,\lt \,1] (Luck et al., 1993[link]). The total lengths [l_{n}^{A}] and [l_{n}^{B}] of the words [A_{n}, B_{n}] can be determined from the equations [l_{n}^{A} = \lambda_{1}^{n} l^{A}] and [l_{n}^{B} = \lambda_{1}^{n} l^{B}] with the eigenvalue [\lambda_{1}]. The left Perron–Frobenius eigenvector [{\bf w}_{1}] of S, i.e. the left eigenvector associated with [\lambda_{1}], determines the ratio S:L to 1:[\tau]. The right Perron–Frobenius eigenvector [{\bf w}_{1}] of S associated with [\lambda_{1}] gives the relative frequencies, 1 and τ, for the letters S and L (for a definition of the Perron–Frobenius theorem see Luck et al., 1993[link], and references therein).

The general case of an alphabet [A = \{\hbox{L}_{1} \ldots \hbox{L}_{k}\}] with k letters (intervals) [\hbox{L}_{i}], of which at least two are on incommensurate length scales and which transform with the substitution matrix S, [{\rm L}'_{i} \rightarrow {\textstyle\sum\limits_{j = 1}^{k}} S_{ij} {\rm L}_{j},]can be treated analogously. S is a [k \times k] matrix with non-negative integer coefficients. Its eigenvalues are solutions of a polynomial equation of rank k with integer coefficients (algebraic or Pisot numbers). The dimension n of the embedding space is generically equal to the number of letters (intervals) k involved by the substitution rule. From substitution rules, infinitely many different 1D quasiperiodic sequences can be generated. However, their atomic surfaces in the nD description are generically of fractal shape (see Section[link]).

The quasiperiodic 1D density distribution [\rho ({\bf r})] of the Fibonacci chain can be represented by the Fourier series [\rho ({\bf r}) = (1/V) \,{\textstyle\sum\limits_{{\bf H}^{\parallel}}}\, F ({\bf H}^{\parallel}) \exp (-2\pi i {\bf H}^{\parallel} \cdot {\bf r}),]with [{\bf H}^{\parallel} \in {\bb R}] (the set of real numbers). The Fourier coefficients [F({\bf H}^{\parallel})] form a Fourier module [M^{*} = \{{\bf H}^{\parallel} = {\textstyle\sum_{i = 1}^{2}} h_{i} {\bf a}_{i}^{*} | h_{i} \in {\bb Z}\}] equivalent to a [{\bb Z}] module of rank 2. Thus a periodic function in 2D space can be defined by [\rho ({\bf r}^{\parallel}, {\bf r}^{\perp}) = (1/V) \,{\textstyle\sum\limits_{{\bf H}}}\, F({\bf H}) \exp [-2\pi i ({\bf H}^{\parallel} \cdot {\bf r}^{\parallel} + {\bf H}^{\perp} \cdot {\bf r}^{\perp})],]where [{\bf r} = ({\bf r}^{\parallel}, {\bf r}^{\perp}) \in \Sigma] and [{\bf H} = ({\bf H}^{\parallel}, {\bf H}^{\perp}) \in \Sigma^{*}] are, by construction, direct- and reciprocal-lattice vectors (Figs.[link] and[link]): [\displaylines{{\bf r} = n_{1} {\bf d}_{1} + n_{2} {\bf d}_{2}, \hbox{ with } {\bf d}_{1} = {1 \over a^{*} (2 + \tau)} \pmatrix{1\cr -\tau\cr}_{V},\cr {\bf d}_{2} = {1 \over a^{*} (2 + \tau)} \pmatrix{\tau\cr 1\cr}_{V}{\semi} \cr {\bf H} = h_{1} {\bf d}_{1}^{*} + h_{2} {\bf d}_{2}^{*}, \hbox{ with } {\bf d}_{1}^{*} = a^{*} \pmatrix{1\cr - \tau\cr}_{V}, {\bf d}_{2}^{*} = a^{*} \pmatrix{\tau\cr 1\cr}_{V}. \cr}]The 1D Fibonacci chain results from the cut of the parallel (physical) space with the 2D lattice Σ decorated with line elements for the atomic surfaces (acceptance domains). In this description, the atomic surfaces correspond simply to the projection of one 2D unit cell upon the perpendicular-space coordinate. This satisfies the condition that each unit cell contributes exactly to one point of the Fibonacci chain (primitive unit cell). The physical space [{\bf V}^{\parallel}] is related to the eigenspace of the substitution matrix S associated with its eigenvalue [\lambda_{1} = \tau]. The perpendicular space [{\bf V}^{\perp}] corresponds to the eigenspace of the substitution matrix S associated with its eigenvalue [\lambda_{2} = - 1/\tau]. Thus, the physical space scales to powers of τ and the perpendicular space to powers of [- 1/\tau].


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2D embedding of the Fibonacci chain. The short and long distances S and L, generated by the intersection of the atomic surfaces with the physical space [{\bf V}^{\parallel}], are indicated. The atomic surfaces are represented by bars parallel to [{\bf V}^{\perp}]. Their lengths correspond to the projection of one unit cell (shaded) upon [{\bf V}^{\perp}].


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Schematic representation of the reciprocal space of the embedded Fibonacci chain depicted in Fig.[link]. The physical-space reciprocal basis [{\bf a}_{1}^{*}] and [{\bf a}_{2}^{*}] is marked. The diameters of the filled circles are roughly proportional to the reflection intensities. One 2D reciprocal-lattice unit cell is shadowed. The actual 1D diffraction pattern of the 1D Fibonacci chain results from a projection of the 2D reciprocal space onto the parallel space. The correspondence between 2D reciprocal-lattice positions and their projected images is indicated by dashed lines.

By block-diagonalization, the reducible substitution (scaling) matrix S can be decomposed into two non-equivalent irreducible representations. These can be assigned to the two 1D orthogonal subspaces [{\bf V}^{\parallel}] and [{\bf V}^{\perp}] forming the 2D embedding space [{\bf V} = {\bf V}^{\parallel} \oplus {\bf V}^{\perp}]. Thus, using [WSW^{-1} = S_{V} = S_{V}^{\parallel} \oplus S_{V}^{\perp}], where [W = \pmatrix{1 &\tau\cr -\tau &1\cr} = \pmatrix{{\bf d}_{1}^{*} &{\bf d}_{2}^{*}},]one obtains [\eqalign{\pmatrix{1 &\tau\cr -\tau &1\cr} \pmatrix{0 &1\cr 1 &1\cr}_{D} \pmatrix{1 &\tau\cr -\tau &1\cr}^{-1} &= \pmatrix{\tau{\,\,\vrule height 6pt depth 6pt} &{\hskip -13pt}0\cr\noalign{\vskip -5pt}\noalign{\vskip 5pt}\noalign{\hrule}\cr\noalign{\vskip -5pt}0{\,\,\vrule height 10pt depth 3pt} &{\hskip -8pt}-1/\tau\cr}_{V}\cr &= \pmatrix{S^{\parallel}{\hskip 3.6pt}{\vrule height 6pt depth 6pt} &{\hskip -5.8pt}0\cr\noalign{\hrule} 0{\hskip 7.9pt\vrule height 10pt depth 4pt} &S^{\perp}\cr}_{V},}]the scaling operations [S^{\parallel}] and [S^{\perp}] in parallel and in perpendicular space as indicated by the partition lines.

The metric tensors for the reciprocal and the direct 2D square lattices read [G^{*} = |a^{*}|^{2} (2 + \tau) \pmatrix{1 &0\cr 0 &1\cr} \hbox{ and } G = {1 \over |a^{*}|^{2} (2 + \tau)} \pmatrix{1 &0\cr 0 &1\cr}.]The short distance S of the Fibonacci sequence is related to [a^{*}] by [\eqalign{S &= 1/[a^{*} (2 + \tau)]\cr &= \min \left\{\big|\pi^{\parallel} ({\bf d}_{i} - {\bf d}_{j})\big| \bigg| \big|\pi^{\perp} ({\bf d}_{i} - {\bf d}_{j})\big| \,\lt\, \Omega_{\rm AS} \wedge i, j \in {\bb Z}\right\},\cr}]with the projectors [\pi^{\parallel}] and [\pi^{\perp}] onto [{\bf V}^{\parallel}] and [{\bf V}^{\perp}]. The point density [\rho_{\rm p}] of the Fibonacci chain, i.e. the number of vertices per unit length, can be calculated using the formula [\rho_{\rm p} = {\Omega_{\rm AS} \over \Omega_{\rm UC}} = {(1 + \tau)/[a^{*} (2 + \tau)] \over 1/[|a^{*}|^{2} (2 + \tau)]} = a^{*} \tau^{2},]where [\Omega_{\rm AS}] and [\Omega_{\rm UC}] are the areas of the atomic surface and of the 2D unit cell, respectively.

For an infinite Fibonacci sequence generated from the intervals S and L an average distance d can be calculated: [d = \lim\limits_{n \rightarrow \infty} {F_{n}\hbox{S} + F_{n + 1}\hbox{L} \over F_{n} + F_{n + 1}} = \lim\limits_{n \rightarrow \infty} {F_{n} (\hbox{S} + \tau \hbox{L}) \over F_{n} (1 + \tau)} = {\hbox{S}(1 + \tau^{2}) \over (1 + \tau)} = (3 - \tau)\hbox{S}.]Therefrom, the point density can also be calculated: [\rho_{\rm p} = 1/d = 1/[(3 - \tau)\hbox{S}] = [a^{*} (2 + \tau)]/(3 - \tau) = a^{*} \tau^{2}.]

An approximant structure of the Fibonacci sequence with a unit cell containing m intervals L and n intervals S can be generated by shearing the 2D lattice Σ by the shear matrix [S_{m}], [S_{m} = {1 \over 2 + \tau} \pmatrix{\tau^{2} + x\tau + 1 &-x\cr x\tau^{2} &\tau^{2} -x\tau + 1\cr}_{D},]where [x = (n\tau - m)/(m\tau + n)]: [\eqalign{{\bf d}'_{i} &= {\textstyle\sum\limits_{j = 1}^{2}} S_{mij} {\bf d}_{j}\semi\cr {\bf d}'_{1} &= {1 \over 2 + \tau} \left[\vphantom{1 \over 2 + \tau}(\tau^{2} + x\tau + 1) {\bf d}_{1} - x{\bf d}_{2}\right]\cr &= {1 \over (2 + \tau)a^{*}} \pmatrix{1\cr -\tau -x\cr}_{V}\cr &= {1 \over (2 + \tau)a^{*}} {\let\normalbaselines\relax\openup8pt\pmatrix{1\cr - {\displaystyle{2n\tau + m\tau \over m\tau + n}}\cr}_{V}},\cr {\bf d}'_{2} &= {1 \over 2 + \tau} \left[\vphantom{1 \over 2 + \tau}x\tau^{2} {\bf d}_{1} + (\tau^{2} - x\tau + 1){\bf d}_{2}\right]\cr &= {1 \over (2 + \tau)a^{*}} \pmatrix{\tau\cr -x\tau + 1\cr}_{V}\cr &= {1 \over (2 + \tau)a^{*}} {\let\normalbaselines\relax\openup8pt\pmatrix{\tau\cr {\displaystyle{2m\tau - n\tau \over m\tau + n}}\cr}}_{V}.}]This shear matrix does not change the magnitudes of the intervals L and S. In reciprocal space the inverted and transposed shear matrix is applied on the reciprocal basis, [(S_{m}^{-1})^{T} = {1 \over 2 + \tau} \pmatrix{\tau^{2} -x\tau + 1 &-x\tau^{2}\cr x &\tau^{2} + x\tau + 1\cr}_{D},]where [x = (n\tau - m)/(m\tau + n)]: [\eqalign{{\bf d}_{i}^{*'} &= {\textstyle\sum\limits_{j = 1}^{2}} (S_{m}^{-1})_{ij}^{T} {\bf d}_{j}^{*}\semi\cr {\bf d}_{1}^{*'} &= {1 \over 2 + \tau} \left[\vphantom{1 \over 2 + \tau}(\tau^{2} - x\tau + 1){\bf d}_{1}^{*} - x\tau^{2} {\bf d}_{2}^{*}\right]\cr &= a^{*} \pmatrix{1 - x\tau\cr -\tau\cr}_{V}\cr &= a^{*} {\let\normalbaselines\relax\openup8pt\pmatrix{{\displaystyle{2m\tau - n\tau \over m\tau + n}}\cr -\tau\cr}}_{V},\cr {\bf d}_{2}^{*'} &= {1 \over 2 + \tau} \left[\vphantom{1 \over 2 + \tau}x{\bf d}_{1}^{*} + (\tau^{2} + x\tau + 1){\bf d}_{2}^{*}\right]\cr &= a^{*} \pmatrix{\tau + x\cr 1\cr}_{V}\cr &= a^{*} {\let\normalbaselines\relax\openup8pt\pmatrix{{\displaystyle{2n\tau + m\tau \over m\tau + n}}\cr 1\cr}}_{V}.}]

The point [x_{n}(t)] of the nth interval L or S of an infinite Fibonacci sequence is given by [x_{n}(t) = \{x_{0} + n(3 - \tau) - (\tau - 1) [\hbox{frac} (n\tau + t) - (1/2)]\}\hbox{S},]where t is the phase of the modulation function [y(t) =] [{(\tau - 1)[\hbox{frac} (n\tau + t) - (1/2)]}] (Janssen, 1986[link]). Thus, the Fibonacci sequence can also be dealt with as an incommensurately modulated structure. This is a consequence of the fact that for 1D structures only the crystallographic point symmetries 1 and [\bar{1}] allow the existence of a periodic average structure.

The embedding of the Fibonacci chain as an incommensurately modulated structure can be performed as follows:

  • (1) select a subset [ \Lambda^{*} \subset M^{*}] of strong reflections for main reflections [{\bf H} = h{\bf a}^{*}, h \in {\bb Z}];

  • (2) define a satellite vector [{\bf q} = \alpha {\bf a}^{*}] pointing from each main reflection to the next satellite reflection.

One possible way of indexing based on the same [{\bf a}^{*}] as defined above is illustrated in Fig.[link]. The scattering vector is given by [{\bf H}^{\parallel} = h(\tau + 1){\bf a}^{*} + mq], where [{\bf q} = \tau {\bf a}^{*}], or, in the 2D representation, [{\bf H} = h_{1}{\bf d}_{1}^{*} + h_{2}{\bf d}_{2}^{*}], where[{\bf d}_{1}^{*} = a^{*} \pmatrix{1 + \tau\cr 0\cr}_{V}]and[{\bf d}_{2}^{*} = a^{*} \pmatrix{\tau\cr 1\cr}_{V},]with the direct basis [{\bf d}_{1} = {1 \over a^{*} (1 + \tau)} \pmatrix{1\cr - \tau\cr}_{V}, \quad{\bf d}_{2} = {1 \over a^{*}} \pmatrix{0\cr 1\cr}_{V}.]The modulation function is saw-tooth-like (Fig.[link].


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Reciprocal space of the embedded Fibonacci chain as a modulated structure. Several main and satellite reflections are indexed. The square reciprocal lattice of the quasicrystal description illustrated in Fig.[link] is indicated by grey lines. The reflections located on [{\bf V}^{\parallel}] can be considered to be projected either from the 2D square lattice of the embedding as for a QS or from the 2D oblique lattice of the embedding as for an IMS.


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2D direct-space embedding of the Fibonacci chain as a modulated structure. The average period is [(3 - \tau)\hbox{S}]. The square lattice in the quasicrystal description shown in Fig.[link] is indicated by grey lines. The rod-like atomic surfaces are now inclined relative to [{\bf V}^{\parallel}] and arranged so as to give a saw-tooth modulation wave.


Janssen, T. (1986). Crystallography of quasicrystals. Acta Cryst. A42, 261–271.
Luck, J. M., Godréche, C., Janner, A. & Janssen, T. (1993). The nature of the atomic surfaces of quasiperiodic self-similar structures. J. Phys. A Math. Gen. 26, 1951–1999.

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