The transition metal atoms on the pentagon axes are located about 0. In each column the shorter axial separation is opposite the longer separation in the three neighboring prisms. However, examination of his pattern shows that a basic repeating unit can be outlined by connecting the star centers to form the elongated hexagons marked in Fig. These hexagon units have exactly the same shape and packing arrangement as those formed by connecting the pentagon centers in Fig.
The pattern formed by connecting the pentagrams in a can be continued indefinitely, as in Fig. Flipping some of the pentagons at random will produce an aperiodic tiling. Curiously, four of these cells contain the same isomer. Each decagonal cavity can be fitted with three pentagons and each pair of fused decagons with six.
The 3 pentagons in each decagon can be placed in 10 different orientations conserving the number of possible edge-to-edge contacts. There are 52 ways in which the 6 pentagons can be fitted into the fused decagon pair conserving edge-to-edge contacts. There are 12 more ways that these pentagons can be fitted without overlap that sacrifice an edge-to-edge contact, but these are excluded by our local packing rule for optimizing contacts. There are, however, only combinations with non-identical autocorrelation functions, counting up-down and enantiomorphic pairs only once. For example, the five pentagons bordering each pentagram can be flipped in 15 different combinations.
We will compare diffraction patterns and Patterson autocorrelation functions of different pentilings after more systematic analysis of their designs and packing properties. An obvious strategy is that used by Penrose 3 to generate aperiodic tilings with pentagons: starting with a pentagonal array of pentagons, each pentagon was subdivided into six smaller pentagons and the gaps in the array between the larger pentagons were filled with the smaller ones; this process was iterated.
Geometers call this process substitution tiling 2.
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In our application of this substitution tiling strategy to enumerate periodic pentilings, we call the process pentangulation by analogy with the triangulation process used to enumerate possible icosahedral surface lattice designs in the quasiequivalence theory of icosahedral virus construction As illustrated in Fig. The series 2, 13, 89, , , … are the fourth order Fibonacci numbers starting with 2.
The packing density in any periodic pentiling can be calculated by dividing the area of the pentagons in the unit cell by the total pentagon and gap areas. Periodic pentilings with more than two pentagons per unit cell. Pentagons in a repeating unit are shaded and unit cells are marked by solid lines. Different pentiling lattices can be twinned with themselves or each other in various combinations.
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Furthermore, these pentilings can all be twinned with themselves, or twinned with each other in various combinations, as illustrated in Fig. It is evident from Figs. Other parameters listed in Table 1 that are important for characterizing the Fibonacci pentile packings are the mean coordination C n , which defines the average number of edge-to-edge contacts per pentagon and V n , which defines the number of pentagon vertices contained within the unit cell.
Thus, less ordered packing arrangements of the pentagonal columns of transition metal atoms can lead to slightly denser M atom packing while maintaining the same nearest neighbor separations. Such a density increase would be accommodated by a reduction in aluminum content. Furthermore, the occurrence of only a small number of local motifs in these large period pentilings suggests that ordered lattices with moderate size periods may provide adequate models for less ordered states. The larger the pentile number, the larger the number of isomers and the greater the possibilities for introducing discrete disorder by flipping pentagons without altering the mean coordination or packing density see Fig.
Such local displacive rearrangements can also transform one ordered pentiling into another with comparable packing density. The superposition in Fig. These pentagon flips should correspond to energetically equivalent local packing arrangements. A pentagon flip in atomic terms corresponds to interchanging a pentagon center and vertex in one orientation for a vertex and center in the other.
The activation energy for such coordinated movements in columns of atoms might be very high, but could be facilitated by lattice defects designated as phasons in quasicrystallography 6.
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Even if such flips are rare in the locally well-ordered condensed state, during crystallization columns of atoms would have to choose between pentagon axial and vertex sites. As we will show, these lattices have virtually indistinguishable diffraction patterns. Thus, the Fourier transforms have mm symmetry and the single quadrant displayed for each model in Fig.
The Patterson functions Fig. All unit cells have the same golden ratio shape because the Fibonacci pentile numbers all have even order indices. Diffraction patterns of regular pentilings. Model electron density maps similar to Fig. The pentagon edge length E in this construction is 4. Fast Fourier transforms of these maps were calculated to 0. The quadrants, which include all independent Fourier coefficients for each pattern, are aligned with the 0, k axis horizontal. The patterns from the higher order pentilings have evident fold symmetry.
The structure factors, which are discrete points in reciprocal space, are depicted as Gaussian disks with radii proportional to the amplitudes as indicated in the F scale at the top. Patterson maps of regular pentilings. The circle has radius In both reciprocal and Patterson space, the patterns for P 4 , P 6 , and P 8 show dominant fold symmetry, characteristic of the averaged pentagonal packing of the pentagons in these moderate size unit cells.
We are not proposing that any particular crystal model represents the actual atomic arrangement of a quasicrystal. What we have demonstrated is that a wide range of crystal models with common local pentagonal coordination relations have nearly indistinguishable decagonal Fourier transforms and autocorrelation functions.
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Periodic pentiling models with moderate size unit cells provide a rational foundation for crystallographically refining the actual local atomic arrangements that build up the quasiperiodic decagonal crystal structures. For a crystallographer, a crystal is like an orderly forest that is useful for determining the average structure of the trees. The repeating unit may be a clump of trees related by noncrystallographic symmetry or constrained to grow in non-equivalent configurations. These complexities can aid the crystallographer in seeing the trees more clearly.
Quasicrystallographers have, however, had difficulty seeing the trees for the forest. The aperiodic space-filling and periodic higher dimensional representations of quasicrystalline forests are mathematically elegant, but these abstractions have tended to obscure sight of the trees. It is evident that these atomic trees are locally ordered in clusters, which are arranged quasiperiodically.
How can the structure of these clumps be most clearly visualized? One way to approach the question of how the atoms are arranged in a quasicrystal is to look at the Patterson function that epitomizes all the information about the interatomic vectors contained in the diffraction pattern. The central portion of this Patterson, which is repeated at the quasiequivalent origins, is indistinguishable from that of any pentiling with comparable or larger cell dimensions or from a five-fold average of any of these Pattersons which would represent the autocorrelation function of the pentiling with infinite size unit cell that is the ultimate quasiperiodic lattice.
This data can be analyzed crystallographically.
Unit cell axes are marked and the a axis is vertical. The centers of symmetry located at the left and right margins are also marked. The quadrant in Fig.
The thick lines, which are parallel to the crystal axes and are spaced at intervals of consecutive Fibonacci numbers, pass through rows of lattice points including the most intense spots. Quasicrystallographers have concluded that the diffraction maxima in decagonal patterns, such as in Fig. There are, however, no incommensurate periodicities in the world of experimental distance measurements. For example, the diffraction pattern from a helix with an irrational screw axis would be indexed by an experimentalist as the Fourier-Bessel transform of a helix with u units in t turns.
The exact values of u and t depend on the accuracy with which the ratio of the layer-line spacings corresponding to the helix pitch and unit axial translation can be measured. The more precise this measurement, the larger the calculated repeat distance. Indexing a large helix repeat distance does not imply an implausible range of interaction, but does indicate the skill of the fiber diffractionist in making precision measurements.
The same considerations can be applied to indexing experimentally recorded quasicrystal diffraction patterns.
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A set of selection rules for indexing the diffraction maxima in the decagonal patterns could be formulated as for helical diffraction, but all that is needed is the choice of pentile model and the rule for indexing the lowest resolution set of strong reflections. Odd and even order pentile numbers correspond to fat and thin golden ratio rhombic unit cells as already noted, and either will represent the decagonal quasicrystal symmetry. For the even order P 2m Fibonacci pentile diffraction patterns illustrated in Fig.
These three reflections have very nearly the same Bragg spacing. For higher resolution decagonally related reflections or for higher order pentiling lattices, these ratios approach much closer to 1 e. An experimental diffraction pattern can be represented as the 5-fold average of a pentile model pattern that best fits the measured width of the Bragg reflections.
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A curious feature of quasicrystal diffraction patterns, associated with what is called phason disorder, is a sharpening of the diffraction peaks with increasing angle Such resolution-dependent peak sharpening for decagonal quasicrystals can be accounted for by pentagonal averaging of diffraction from pentile lattice domains with relatively small pentile number. For a multiply twinned mosaic of such domains see Fig.
This peak broadening would progressively diminish for higher resolution reflections. The two-dimensional pentile lattices appear to provide reasonable trial models for the projected arrangement of the transition metal atoms in the plane perpendicular to the 5-fold axes of decagonal quasicrystals such as Al 65 Cu 20 Co 15 7 and Al 70 Ni 15 Co 15 8. The Bragg spacings for the first equatorial decagonal set of strong reflections—which have been given five-dimensional indices —are 3. The indexing of a pentile lattice diffraction pattern Fig.
Thus, the mean pentagon edge lengths corresponding to the Bragg spacings of 3. These values are in good accord with the density maps projected from the calculated five-dimensional Fourier 7 , 8 and the mean edge length of 4. There are many fascinating aspects to the decagonal quasicrystal diffraction patterns obtained by Steurer and colleagues 7 , 8. The satellite reflections around the strong Bragg maxima are suggestive of ghost spectra 21 due to microdomain structure or directional quasiperiodic density fluctuations on a long distance scale.
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