Special Issue on Materials Science on Hypermaterials
The Local Structure of the Fibonacci Chain and the Penrose Tiling from X-Ray Fluorescence Holography
2021 Volume 62 Issue 3 Pages 342-349
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2021 Volume 62 Issue 3 Pages 342-349
Structural investigations based on X-ray fluorescence holography can add a new perspective to the research of aperiodic systems. This technique can reconstruct the average 3-dimensional (3-D) local structure around specific elements directly from the experimental data. However, these structures can be difficult to disentangle for the complex arrangements found in quasicrystals. Therefore, we illustrate this point of view by considering appropriate reference models with a 1-D model of a Fibonacci chain and a 2-D Penrose tiling. It is demonstrated that the holographic reconstructions correspond to a projection of the average structure. The results from X-ray fluorescence holography can then be interpreted as statistical information on inter-atomic connections in the system.
Since the discovery of quasicrystals, the question about the actual atomic arrangements (“Where are the atoms?”) is a main driving force of the research on their structure.1–4) Quasicrystals consist of long-range ordered, but aperiodic, atomic arrangements. These complex structures are conventionally investigated by X-ray and/or electron diffraction experiments in combination with high-dimensional modeling. This procedure is necessary since quasicrystals are periodic only in a high-dimensional hyperspace, of which the experiment can provide only lower-dimensional section.
A new perspective on the structure of quasicrystals can be gained from X-ray fluorescence holography (XFH) experiments. In contrast to diffraction techniques, XFH is able to visualize the atomic scale structure in physical space directly, i.e. without prior assumptions about the structure and without modeling.5,6) XFH circumvents the phase problem of traditional crystallography by measuring the interference between a reference wave (incident X-ray beam) and an object wave (scattered X-rays), using fluorescent X-rays as a probe. These aspects render XFH unique as a tool to investigate the structure of quasicrystals.
The conventional XFH method has so far mainly been used for periodic systems, and could provide valuable information on the local structure e.g. concerning the characterization of lattice distortions around the Ga impurity atoms in an InSb single crystal,7) the determination of the spontaneous formation of a suboxidic coordination around Co in ferromagnetic rutile,8) or the determination of different dopant sites in topological insulators like Mn:Bi2Te39) and In:Bi2Se3.10) Furthermore, detailed information on the atomic shifts in a Si1−xGex single crystal11) and about the in-plane disordering of Zn–Y clusters in a novel Zn,Y doped Mg alloy were obtained.12) For aperiodic systems, however, a new complication arises, which is related to the missing translational invariance. XFH measures the average structure over all atoms (of the same element) – in the case of a periodic crystal, this can justifiably be assumed to be the same for each target atom. In contrast, in a quasicrystal structure, every site is different. Thus, the average structure is much more complex compared to usual (periodic) crystals, even though a distinct local order is still maintained. In this paper, we describe the local order observed in the average structure for several typical quasicrystalline reference systems, and show the corresponding results of (computer simulated) holograms and their reconstructions in real space. For this, a series of two systems with rising dimensionality is considered, starting from a 1-D model of a Fibonacci chain to a 2-D Penrose tiling.
As described above, holography in general exploits the interference of two waves, which are called reference wave and object wave. In the case of the so-called inverse mode of XFH (further details on the distinction between different X-ray holography modes are given elsewhere6,13)), these are an incident X-ray beam and the scattered X-rays. The process can be described as follows. The incident wave is a simple plane wave, determined by the wavevector k and the position vector r:
| \begin{equation} \psi_{\textit{inc}}(\boldsymbol{k},\boldsymbol{r}) = e^{i\boldsymbol{k}\cdot \boldsymbol{r}} \end{equation} | (1) |
When this wave is scattered by atoms located at a position a, the scattered wave is given by
| \begin{equation} \psi (\boldsymbol{k},\boldsymbol{r},\boldsymbol{a}) = e^{i\boldsymbol{k}\cdot \boldsymbol{a}}f_{\boldsymbol{k},\boldsymbol{a}}\frac{e^{ik| \boldsymbol{r} - \boldsymbol{a} |}}{| \boldsymbol{r} - \boldsymbol{a} |} \end{equation} | (2) |
| \begin{align} I(\boldsymbol{k}) &= \left| \psi_{\textit{inc}} + \sum_{h}\psi \right|_{r = 0}^{2}{} \approx 1 + \sum_{h}2\cdot \mathit{Re}\left[f_{\boldsymbol{k},\boldsymbol{a}_{h}}\frac{e^{i\boldsymbol{k}\cdot \boldsymbol{a}_{h} + ka_{h}}}{a_{h}} \right] \\ &= 1 + \chi (\boldsymbol{k}) \end{align} | (3) |
| \begin{equation} G(\boldsymbol{r}) = r \int \chi (\boldsymbol{k})\mathit{Re}[e^{i(kr - \boldsymbol{k}\cdot \boldsymbol{r})}]d\boldsymbol{k}. \end{equation} | (4) |
Equation (3) implies that the hologram of a complex structure is just the sum of the fundamental holograms of each scatterer. This property makes it possible to calculate the hologram for arbitrarily complex structures in a simple approach.
In the actual XFH experiment, characteristic X-rays that are emitted from a specific element (when the incident X-rays have an energy above the respective absorption edge) are detected, which makes the technique intrinsically element-selective regarding the emitter atoms. In practice, the experiment is conducted at a synchrotron facility in order to achieve a high flux rate and to be able to tune the energy of the incident X-rays appropriately.
2.2 Computational detailsThe simulation of the holograms was done using the freely available 3DAirImage software.13) For each system, a set of 39 holograms with incident energies between 8.5 to 18 keV were calculated, in steps of 0.25 keV. The energy range is comparable to typical XFH experiments, while the number of holograms is larger than the number that can usually be collected during an experiment (typically around 8 holograms).
The latter condition was chosen to be (as much as possible) free from Fourier transform errors resulting from a limited energy range. The problem can be seen when regarding eq. (4), which nominally demands an infinite energy range for an exact reconstruction, which is necessarily substituted with a finite sum of holograms in the experiment.
2.3 The projection of the average structureAs a reference for the XFH reconstructions, we describe the average structure around atoms of a specific element. This is obtained by projecting each atom of the element K into the origin and mapping its environment. For a set of i atoms located at positions ri, with the electron number Z(ri) and the atomic distribution ρa(ri), the projection is obtained as follows:
| \begin{equation} P_{\text{K}}(\boldsymbol{r}) = \sum_{\boldsymbol{r}_{\text{K},i}}Z(\boldsymbol{r})\rho_{a}(\boldsymbol{r} + \boldsymbol{r}_{\text{K},i}) \end{equation} | (5) |
| \begin{equation} P_{\text{Patterson}}(\boldsymbol{r}) = \sum_{\boldsymbol{r}_{i}}\rho_{e}(\boldsymbol{r}_{i})\rho_{e}(\boldsymbol{r} + \boldsymbol{r}_{i}), \end{equation} | (6) |
| \begin{equation} P_{\text{Patterson}}(\boldsymbol{r}) = \text{Z} \sum_{r_{i}}Z(\boldsymbol{r})\rho_{a}(\boldsymbol{r} + \boldsymbol{r}_{i}), \end{equation} | (7) |
Similar to the Patterson function, the projection PK(r) also indicates inter-atomic connections. The intensity of the signals in PK(r) depends on the number of atoms with the same inter-atomic vector. In the following, this will be referred to as atomic density of the average structure. As will be shown below, the projection functions can be directly related to the reconstructions from X-ray fluorescence holograms.
Suitable reference systems for quasicrystals are so-called approximant (AP) phases, which are periodic structures close to the aperiodic quasicrystalline ones. They can be described as projections of the quasicrystalline hyper-structure along rational directions in hyperspace, with a rational slope approximating the (irrational) golden ratio $\tau = \frac{1 + \sqrt{5} }{2} \approx 1.618$. The approximants are denoted by the slope in hyperspace as 1/0, 1/1, 2/1, 3/2, 5/3 etc., i.e. with a set of adjacent Fibonacci numbers Fn of the sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34… (each number is the sum of the two preceding numbers), with the limit of $\lim _{n \to \infty }( \frac{F_{n + 1}}{F_{n}} ) = \tau $. The better the rational slope approximates τ, the closer is the structural relationship between the approximant and the quasicrystal.
3.1 The 1-D case: The Fibonacci chainThe Fibonacci chain (FC) is the most basic representation of an aperiodic system. It includes two different distances between points, commonly denoted as S (short) and L (long) distances. Starting with two points at a distance S, the FC can be constructed by certain substitution rules, for which in every step
| \begin{equation} \begin{split} \text{S} & \to \text{L}\\ \text{L} & \to \text{LS} \end{split} \end{equation} | (8) |
We have simulated FC approximants from the 1/0 to the 34/21 AP, for a set of Co atoms with the distances S = 2 Å and L = 2τ Å ≈ 3.236 Å. The construction scheme is displayed for the first 5 APs in Fig. 1(a). The projection of the average structure after eq. (5) is illustrated in Fig. 1(b). Note that the different signal intensities correspond to the different densities of atoms for the individual connections. For example, in the 1/1 AP, each atom has exactly one S and one L connection, but two LS connections. The projection for the 34/21 AP is additionally displayed at the bottom of the Figure. Labels for the types of connections that produce the individual signals are denoted for reference. Note that the unit cell length of this AP is about 152 Å, so the displayed section corresponds to only about one fifth of the unit cell. However, the intensity ratios of the signals are comparable to the 5/3 AP.
(a) The Fibonacci chain approximants from the 1/0 to the 5/3 AP. Bonds are shown between two atoms with the distance S. Dashed lines indicate the unit cells of the approximant. (b) Projections of the average structure after eq. (5). The white dashed lines indicate the corners of the unit cells. Additionally, a projection of the 34/21 AP is displayed (the scale of 30 Å corresponds to about one fifth of the unit cell length).
Then, for each AP, a set of 39 holograms was calculated as described above. Two exemplary holograms, for the 1/1 AP and the 34/21 AP, are shown in Fig. 2. They are drawn in an orthographic projection, where the radial and angular directions indicate θ and φ, respectively. The holograms follow a simple line pattern, which is typical for 1-D structures.6,13) The differences in the patterns are difficult to perceive by eye. They are more readily observed in the XFH reconstructions of the real space structure, as shown in Fig. 3. Note that the intensity of atomic images in XFH reconstructions decreases linearly with the distance from the emitter atom (which is located at the origin). The intensity rate in Fig. 3 was therefore scaled with the distance from the origin.
Exemplary holograms (at 12 keV) of FC approximants, for the 1/1 AP (a) and the 34/21 AP (b).
XFH reconstructions for subsequently larger APs of the FC, from the 1/0 to the 34/21 AP.
Clearly, the atomic images for the 1/0 AP in Fig. 3 are an example for a simple regular periodic arrangement of atoms, in which each atomic image is located at a constant distance (here: L) from each other and for which each atomic image has a similar intensity. The reconstruction of the 1/1 AP already captures several of the important building blocks of the quasi-periodic arrangement, in particular S and L units. In contrast, the higher AP’s include atomic images for more different units (e.g. “LL”), and the intensity ratios of the individual signals gradually approach the ratios observed for the 34/21 AP. The intensity ratios of the atomic images in the XFH reconstructions and the signals in the projections of the average structure displayed in Fig. 1(b) match well, indicating a correspondence of both approaches. Thus, the XFH reconstructions have the same information content as this kind of atomic density map, i.e. statistical information on the location and density of interatomic connections in the average structure.
The statistical information can be illustrated more clearly by considering the integrated intensity of each site. This is shown in Fig. 4 (for several selected APs), in which the signal of the S sites was normalized to 1. In the 1/0 AP (pink bars), only L sites (LL, LLL etc.) appear, with an equal atomic density. In the 1/1 AP (green bars), the atomic densities at the S and L sites are equal, and the LS site has twice the number of atoms, as discussed before for the projection in Fig. 1(b). In the ideal FC, the ratio of L sites over S sites is exactly τ, which is approximated well in the atomic density of the 34/21 AP (blue bars) with actually 34/21 ≈ 1.619. The relationship to the other building blocks can also be seen in the figure. Major components are LS blocks and LLS blocks (which can be realized as any permutation, i.e. SLL, LSL or LLS in the actual structure). The XFH reconstruction of this AP (red bars) follow the same distribution; the differences are related to difficulties to clearly determine the spurious background in the reconstructions.
Integrated intensities for the first 8 images in the XFH reconstructions of some APs and atomic density map of the FC. The bar of the S unit is normalized to 1.
Due to the projection into an average structure, “new” pseudo-units appear (which are not expected in the actual local environment). For the FC, these are for instance the appearance of both S and L signals near the origin. This leads to atomic images with an apparent distance of (L-S) = 1.236 Å. In the actual local environment, no atoms at this distance will be observed, nor will any atom have neighbors both at S and L distances in the same direction. These features appear only in the projection of the average structure, and are therefore classified as pseudo-units. The nature of the pseudo-units is easily comprehended in the case of the FC, therefore it is instructive to point it out here. Similar pseudo-units also appear in the higher-dimensional examples in the following sections, but become progressively more complex.
3.1.2 Limitations for XFHAn apparent obstacle for the XFH reconstructions are the tailing features, which are formed in between the intense signals of the actual atomic positions (at y ≠ 0), as seen in Fig. 3. These are artifacts caused by the still imperfect Fourier transform of a limited number of holograms with a finite energy step. In typical XFH reconstructions (of periodic systems), these tails are lost in the background noise, which can reach levels of 10–20% of the intensity of the atomic images. The intensity threshold is then adjusted to cut off signals under this level. For aperiodic systems, particularly for ‘real’ examples with more than 1 dimension, the tails can be more problematic, because there are intrinsically weak signals related to positions with low atomic density in the average structure, which can be difficult to distinguish from this background. The problem can be reduced by including more holograms or by more sophisticated reconstruction algorithms, e.g. based on sparse modeling.17)
3.2 2-D case: The Penrose tilingThe Penrose tiling (PT) is a common example of a 2-D quasicrystal. It has a 10-fold (or decagonal) symmetry. Popular representatives that include PTs as structural motifs are Al–Co–Ni or Al–Co–Cu systems, but many others are known.18) The tiling can be constructed by a set of two different rhombi, which are called thick (acute angle of γ = 72°) and thin rhombi (γ = 36°). An example for a PT (with atoms decorated on each vertex) is displayed in Fig. 5(a), and the geometric relationships of the two kinds of rhombi are illustrated in Fig. 5(b). There are 5 different distances to be considered: the edge length a, and the 2 diagonals in each rhombus; the long diagonal in the thick rhombus and the short diagonal in the thin rhombus have lengths of τa and a/τ, respectively, and the other diagonals are denoted with e and f, and can also be expressed in terms of τ, with $e = 2a\sqrt{ {1 - ( \frac{1}{2}\tau )^{2}}\mathstrut} $ and $f = 2a\sqrt{ {1 - ( \frac{1}{2\tau } )^{2}}\mathstrut} $. For the displayed case with a decoration of atoms on the vertices of the rhombi, the shortest interatomic distance is the a/τ. For our model system, we chose an edge length of a = 4 Å, giving a shortest interatomic distance of 4/τ Å ≈ 2.472 Å, which is in the order of magnitude of typical atomic distances in intermetallic alloys.
An example of a section of a Penrose tiling with atoms decorated on the vertex of each rhombus (a) and the relationship between the different distances in the two kinds of rhombi (b). Also shown is the unit cell of a 1/1, 1/1 AP (c).
The construction of approximant structures for the Penrose tiling is more complicated than that for the Fibonacci chain. In the following, we will use the formalism of Ref. 19) to construct a 1/1, 1/1 approximant (the two sets of numbers denote the rational slope in hyperspace along the two orthogonal directions in real space). An example is illustrated in Fig. 5(c); the rhombi edge length is also a = 4 Å, and the dimensions of the unit cell are x1/1 = 37.889 Å and y1/1 = 12.311 Å. Furthermore, a large model of a Penrose tiling with 601 vertices was generated,20) which provides a reasonably well representation of an actual complete Penrose tiling, at least concerning the average local structure, so it will be referred to as PT. An exemplary hologram for the PT is shown in Fig. 6. The overall symmetry of the hologram is 10-fold (decagonal), which can be well observed from the intense line-like features (X-ray standing wavelines).
Exemplary hologram of a Penrose tiling (at 18 keV incident energy).
The XFH reconstructions of the 1/1, 1/1 AP and the PT are displayed in Fig. 7. They are compared with the projections of the average structure. The average tilings in the two structures are illustrated by the dashed lines; they consist of pseudo-tiles with an edge length of a/τ2, i.e. about 1.528 Å. The motifs are periodic for the AP with the unit cell dimensions, as indicated by the thick dashed lines in Fig. 7(a), (c). Also, it should again be emphasized that no two atoms at the distance of 1.528 Å can be observed in the actual local environment, nor will there appear a structure corresponding to the new pseudo-tiles; these features are solely a representation of the average structure of the Penrose tiling. They correspond to the pseudo-units of the FC in the 1-D case as described above.
XFH reconstructions for the 1/1, 1/1 AP (a) and the PT (b), as well as the respective projections of the average structure (c), (d). For the AP, the size of the unit cell (half of the unit cell length in x direction) is shown by the thick dashed rectangles. Additionally, the average tilings are illustrated by the dashed lines (note that the tilings are different for the AP and the PT). The intensity in the XFH reconstructions is scaled with the distance from the origin.
The reconstructions from the holographic data can also again be interpreted as statistical information of the atomic distributions. Most features that appear in the PT (d) can also be seen in the approximant (c), with some differences in the relative atomic densities and the arrangements of the pseudo-tiles. The latter is periodic for the AP, which leads to differences in the tiling of the AP and the PT, particularly at distances from the origin larger than $\frac{1}{2}y_{1/1}$.
The different distances of the rhombi of the Penrose tiling as illustrated in Fig. 5(b) are naturally reflected in the average structure; they are denoted explicitly in Fig. 7(d). For example, the first and second atomic images (viewed from the origin) are related to the distances of “a/τ” and “a” (corresponding to about 2.472 Å and 4.0 Å, respectively). The atomic images in Fig. 7 are a convenient and graphical way to express the statistics of the individual specific connections. Another perspective is given by integrating the signals (similar to Fig. 4), thereby providing numerical values for these statistics. This is displayed in Fig. 8, for the XFH reconstruction of the PT and for the corresponding density map.
Integrated intensities for the images in the XFH reconstruction and for the projection of the average structure of the PT. The bar of the unit length a is normalized to 1.
To rationalize these values, it is helpful to consider the relationships between the tiles in an ideal Penrose tiling. The ratio of the number of thick and thin tiles is $\frac{N_{\textit{thick}}}{N_{\textit{thin}}} = \tau $. The total number of tiles can therefore be expressed as
| \begin{equation} N_{\textit{total}} = N_{\textit{thick}} + N_{\textit{thin}} = (1 + \tau)N_{\textit{thin}} = \tau N_{\textit{thick}} \end{equation} | (9) |
| \begin{equation} \begin{split} \frac{N_{a/\tau}}{N_{a}} & = \frac{N_{\textit{thin}}}{2N_{\textit{total}}} = \frac{1}{2(1 + \tau)} = \frac{1}{2\tau^{2}} \approx 0.191\\ \frac{N_{e}}{N_{a}} & = \frac{N_{\textit{thick}}}{2N_{\textit{total}}} = \frac{1}{2\tau} \approx 0.309 \end{split} \end{equation} | (10) |
For the other types of connections, pairs (or clusters) of tiles have to be taken into account. In the PT, only eight different ways are possible in which tiles can meet at a vertex, which simplifies the evaluation.21,22) Limiting the discussion to connections of the types “τa” and “f” as examples, we can describe the important configurations (illustrated in Fig. 9): The “τa” connection can be realized as a single “τa” unit or via a pair of tiles (1 thick + 1 thin tile) as “a + a/τ” (red, dashed arrows). Similarly, the “f” connections (blue, dotted arrows) can also be realized by any pair of two thick rhombi, or by a cluster of one thick and one or two thin rhombi as illustrated in the figure. The multiplicity of these connections explains the comparably large atomic densities in the average structure as seen in Figs. 7 and 8.
The multiplicity of the τa (red, dashed line) and f (blue, dotted line) connections in clusters of tiles.
The two systems described above represent fundamental reference models for the structural analysis of quasicrystals by subsequent XFH experiments at synchrotron facilities. They can be used as a starting point to interpret the results of measurements e.g. on decagonal quasicrystals. The structure of a real sample will naturally be more complex than the simplified model systems described above. For instance, decorations of the Penrose tiles should then be taken into account, which may complicate the interpretation of the average structure. A simple model for this case is illustrated in Fig. 10, where we assume a decoration of atoms at the center of each thick rhombus of the 1/1, 1/1 AP of a Penrose tiling. As a result, two cases have to be distinguished for the projections of the average structure: either the decorated atoms are of a different element than the atoms at the vertices of the tiles (hetero-decoration, Fig. 10(b)) or of the same element (homo-decoration, Fig. 10(c)). The former case corresponds to e.g. a decoration of lighter atoms like Al into a tiling of metal elements (or vice versa). This case is able to highlight the advantage of element-selectivity in the XFH approach, because the projection (and correspondingly, a reconstruction from XFH data) is centered on a specific element, see the sum in eq. (5). Consequently, only a few new signals appear at the centers and edges of the pseudo-tiles. The latter case with a homo-decoration (c) is more complex, because a whole new sub-tiling is included in the projection of the average structure, as indicated by the red dashed lines. In the reconstruction of the structure of real samples from XFH data, the comparison with model systems as described here will be able to support the interpretation of the obtained images.
An example of a decorated 1/1, 1/1 AP of a Penrose tiling. Atoms are decorated at the center of the thick rhombi, as shown by the red atoms in (a). The projections of the average structure are displayed for a hetero-decoration (the red atoms are of a different element than the blue atoms) in (b) and a homo-decoration (the red atoms are of the same element as the blue atoms) in (c). The red dashed lines illustrate the new sub-tiling established by the additional emitter atoms.
Furthermore, the insight gained from XFH may be particularly interesting for icosahedral quasicrystals (which can be regarded as 3D quasicrystals), which are mainly realized as ternary alloys suffering from chemical disorder. Additionally, even for a simpler binary system, like Yb–Cd, positional disorder can play a significant role in determining the structural features.23) XFH can address these issues: it provides 3-dimensional information, which are intrinsically element-selective due to the use of characteristic X-rays, and the XFH signal intensity is also closely linked to the positional disorder of the atoms.7) The average structure of these systems can be described with 3D maps analogous to the 2-D maps shown in the examples above. They will exhibit similar structural pseudo-features, realized as pseudo-clusters. The concepts outlined here will be useful for the understanding of the complex structural features observed in this type of quasicrystal.
A theoretical approach to X-ray holography for quasicrystals is presented. Two fundamental model systems (Fibonacci chain in 1-D and Penrose tiling in 2-D) are considered. A projection scheme for the average structure is described, using maps of the average atomic density. It is demonstrated that this projection corresponds to the results from XFH reconstructions. The XFH results can therefore be viewed as a statistics of specific atomic correlations, which are categorized as ‘pseudo-building blocks’ due to their appearance in the average structure.
The fundamental structural features described here also represent a basis for the structural analysis of quasicrystals by subsequent XFH experiments at synchrotron facilities.
This work was supported by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research on Innovative Areas ‘Hypermaterials’ (Grant Numbers 19H05819 and 20H05273) and the JSPS Grant-in-Aid 20K15027. JRS is grateful to Benedict Paulus (Philipps University of Marburg, Germany) for fruitful discussions on the projection of the average structure.
