Average unit cell of a square Fibonacci tiling

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Abstract

Square Fibonacci tiling is an example of a quasicrystalline structure with classical crystallographic 4-fold symmetry. Using a probabilistic approach, an average unit cell has been constructed in physical space. This allows one to calculate the diffraction patterns in different directions. The projected structure into different directions can be also regarded as decorated Fibonacci chain. The diffraction pattern of the square Fibonacci chain has been fully analyzed in the average unit cell approach.

Introduction

Lifshitz [1] recently defines the square Fibonacci tiling consisting of an infinite set of lines, along x- and y-directions, respectively, with tetragonal point group symmetry 4 mm. Its generalization to 3D is straightforward. The inter-line spacings follow the well-known Fibonacci sequence of short (S) and long (L) distances. The lengths of these distances are 1 and the golden mean value τ≈1.618, respectively. Consequently, the square Fibonacci tiling consists of three different tiles: a small square of dimensions 1 × 1, denoted by S2; a large square of dimensions τ×τ, denoted by L2; and a rectangle of dimensions 1×τ, denoted by R. There are only three allowed vertex configurations and minimal covering cluster of nine tiles, which has been discussed in [1]. Following the rules for the 1D sequence: SL, LSL, it is easy to find the substitution rules for the square tiling: S2L2, L2L2+S2+2R and RL2+S2. There are τ2 times as many L2 tiles as there are S2 tiles, and there are 2τ times as many R tiles as there are S2 tiles. The square Fibonacci tiling inherits its τ-scaling or inflation symmetry directly from the 1D Fibonacci sequences. Fig. 1 shows the square Fibonacci tiling with the ‘atoms’ placed at each vertex of the tiling. Its 2D diffraction pattern is shown in Fig. 2, where different cutting-lines: [1,1], [1,τ] and [1,2] have been also marked.

The average unit cell has been constructed using the statistical approach and the reference lattice concept [2]. The reference lattice is a set of parallel lines periodically arranged in 2D and perpendicular to the corresponding scattering vector. The position of any atom with respect to the reference lattice is given by the variable u with the probability P(u) defined for the whole structure. The structure factor for the given scattering vector k0 and its higher harmonics (k=nk0) then reads (for the atomic form-factor equals to 1 and λ=2π/k):F(k)=∫0λP(u)exp(i(nk0u))du.

There are two properties of the structure factor, which are very important for the present concept of the average unit cell. The first one is the scalar product between the wave vector k and positional vector r, which means that for a chosen scattering vector the whole structure is then projected onto the direction pointed out by that particular wave vector. The second property is that these scalar products determinate the phase shifts of the individual components and the information concerning the atomic positions can be reduced to a single unit cell, i.e. modulo the corresponding wavelength. In the meaning of the present approach the diffraction intensity measures the quality of the adjustment of the reference lattice to the real structure. The diffraction peaks indicate a kind of a similarity resonance between the reference lattice (linked to the scattering vector) and the real structure. One should remember that the concept of the average unit cell based on the reference lattice is not an approximation and such a unit cell can be constructed for arbitrary chosen scattering vector. However, more appropriate are those wave vectors corresponding to significantly non-zero intensities of diffraction patterns. For them the similarity resonance of the reference lattice and the real structure are really observed as well-developed diffraction peaks.

For the modulated structures, including quasicrystals (like the Fibonacci chain), satellite reflections are also observed in the diffraction pattern. For incommensurate modulation this requires another wave vector q0 – the modulation vector. To reconstruct the diffraction pattern of such structure one has to use two average unit cells for two different wave vectors [2], [3], [4]. Then the structure factor for k=nk0+mq0 where n is an index of the main reflection and m – of the satellite, readsF(k)=∫0λP(u,v)exp(i(nk0u+mq0v))dudv.

Usually there is a simple relation between the corresponding probability distributions (like a linear relation for the Fibonacci chain [3] or sinusoidal relation for the harmonic modulations [4]), making the calculations easily accomplished. Recently, it has been proved [5] that the average unit cell approach for quasicrystals and other modulated structures is equivalent to the higher dimensional analysis [6], [7], [8].

Section snippets

Average unit cell for the Fibonacci chain

If the lengths of the two building elements of the Fibonacci chain are equal to 1 and τ, the average distance is then equal to a=1+1/τ2≈1.382, and the corresponding scattering vector k0=2π/a≈4.547. The probability distribution, and also the average unit cell, for k0 is a flat function bounded within (−u0+u0), where u0=1/(2τ)≈0.309, as it is shown in Fig. 3. The same function describes all higher harmonics of k0(k=nk0) and its Fourier transform isF(k)=Nsin(ku0)ku0,which leads to the following

Average unit cell for the square Fibonacci chain

Diffraction pattern of the square Fibonacci tiling can be considered in two different ways. The first way is to notice that for each xn the atomic positions along the y-direction are the same, which leads to the following expression for the intensities:I(kx,ky)=Ix(kx)·Iy(ky)=sin(wx)wx2·sin(wy)wy2,wherewi=(ki−miq1i)ui,fori=x,y.The second way is to construct the average unit cell by the projection of the whole structure onto the given direction. For this particular structure the average unit cell

Conclusions

In the present paper the average unit cell has been constructed for the square Fibonacci tiling using the statistical approach. For arbitrary chosen 2D scattering vector the corresponding average unit cell is given by the product of two 1D distributions: one calculated for the x-component of the scattering vector and the other for y-component. Such 2D average unit cell can then be projected onto the direction pointed out by the chosen scattering vector. The obtained probability distribution has

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