Fundamentals of the Physics of Solids / 10-The Structure of Noncrystalline Solids

The indices h and k run over all (positive and negative) integers, so, due to the irrationality of τ , the allowed values of K make up a quasicontinuous spectrum. However – as already mentioned in relation to Fig. 10.9 – the intensity is large only at peaks for which the ratio of the indices is close to τ – that is, for which the two integers are subsequent Fibonacci numbers.

To provide an intuitively clear interpretation of this result, we shall return to the representation of the Fibonacci chain by the projection of the points in a strip of a square lattice. The reciprocal lattice of the square lattice is made up by the points K = 2aπ (h, k). In the coordinate system rotated by an angle φ these points are given as

Then, apart from the phase factors, the expression for the structure amplitude is written as

This shows that the scattered intensity receives contributions from each point K of the reciprocal lattice. The scattering “vector” K, which determines the position of the di raction peak associated with the reciprocal-lattice point K is the projection Khk of K on a line of irrational slope, while the amplitude of the di raction peak is determined by its projection along the perpendicular direction. This amplitude is small when the projection is large, i.e., when K is not close to the specified line. To understand this result recall that the Fibonacci chain was constructed using the points in a narrow strip (of width l = a cos φ) in the direct lattice, therefore di raction by the Fibonacci chain can be interpreted as scattering by a specially oriented narrow strip of a square lattice. In contrast to the usual, rather sharp di raction peaks that appear for macroscopic samples, the peaks now have a finite width in the direction perpendicular to the strip. The di raction pattern is shown in Fig. 10.12. Di raction peaks far from the line K = 0 hardly contribute. This leads to a relatively sparse distribution of sharp peaks.

10.2.5 Penrose Tiling of the Plane

The two-dimensional generalization of the quasiperiodic Fibonacci chain is the quasiperiodic covering (tiling) of the plane with two objects. As demonstrated by R. Penrose in 1974, this is possible with two rhombi of equal sides. The angles of the thinner rhombus are 36◦ and 144◦, while those of the thicker are 72◦ and 108◦. These figures are called Penrose tiles. To rule out periodic tilings, sides are marked with single or double arrows, and the common sides

324 10 The Structure of Noncrystalline Solids

Fig. 10.12. Di raction pattern of a finite strip of a square lattice and its section along the line K = 0

of neighboring tiles are required to have the same number of arrows in the same direction. Figure 10.13 shows the elementary rhombi and some simple allowed matches.

Fig. 10.13. The 36◦ and 72◦ rhombi used in Penrose tiling and some simple matches

The rule for matching tiles is often expressed in an alternative way: each tile has a marked vertex – the one where the two sides with double arrows join –, and tiles can be matched only if the joining vertices are either all marked or all unmarked. Figure 10.14 shows a possible tiling of the plane that

respects matching rules. For clarity, thin tiles are shaded gray. The presence of local fivefold symmetry in many places is obvious at first sight. In spite of the lack of strict long-range periodicity, the tile edges (or bonds between atoms at the vertices) are all along five specific directions, thus the pattern shows long-range directional order.

Fig. 10.14. Penrose tiling of the plane using 36◦ and 72◦ rhombi

The position vectors of the vertices of the rhombi can all be written as

are the vectors drawn from the center of a circle of radius a to the vertices of an inscribed regular pentagon. Two neighboring vectors ei plus their resultant and the origin make up a thick rhombus, while second-neighbors give a thin rhombus. The five vectors are obviously not linearly independent, e.g.,

If the ni in (10.2.46) could take any integer values, points rn would fill the plane densely because of the irrationality of τ . To obtain a Penrose tiling, only certain integers are allowed, ensuring finite distances between lattice points.

To establish the selection rule for the integers note that if the primitive vectors of a five-dimensional hypercubic lattice are projected on a particular

10.2.6 Three-Dimensional Quasicrystals

In three dimensions quasiperiodic filling of space with simple objects is much more di cult to illustrate, therefore we shall adopt the opposite approach. First we shall investigate what symmetries may be exhibited by the structures obtained via the projection of a part of a higher-dimensional lattice onto three dimensions, and only then shall we turn to the spatial arrangement of atoms in three-dimensional quasicrystals.

Quasicrystals with icosahedral symmetry make up the most characteristic class of three dimensional quasicrystals. Their di raction patterns display two-, three-, as well as fivefold rotation axes. The icosahedral point groups

¯¯

I (235) and Ih (m35) are known to be incompatible with three-dimensional spatial periodicity. However in six-dimensional space the six fivefold axes of the icosahedral point group appear naturally. As a generalization of the foregoing, icosahedral quasicrystals may therefore be regarded as the three-dimensional projections of a part of a six-dimensional cubic lattice.

For a precise mathematical formulation, consider the projection of a sixdimensional hypercubic lattice on a particular three-dimensional subspace that is perpendicular to the direction [111111] and spanned by the vectors

The projections of the primitive vectors a(100000), a(010000), . . . of the hy-

and e6 = a(0, 0, 1). It is readily seen that these projected vectors have the same magnitude, too. For an even clearer manifestation of the symmetries the vectors are rotated around the y-axis through φ = arctan(1/τ ). When this new set of basis vectors,

and their opposites are all drawn from a common origin, their tips are seen to point into the vertices of the regular icosahedron inscribed in a sphere of radius a (see Fig. 5.12).

328 10 The Structure of Noncrystalline Solids

The primitive vectors of the reciprocal lattice of a six-dimensional hypercubic lattice generate another 6D hypercubic lattice. Projection on threedimensional space and then rotation leads to the vectors

Di raction peaks therefore appear at those values of K that can be written as

i=1

where the hi are integers. A simple rearrangement of the terms leads to the form

The distribution of the di raction peaks of an icosahedral quasicrystal is such that along each of the three directions the observed di raction pattern is similar to that of a Fibonacci chain. Consequently, di raction peaks are specified by six indices. In principle, arbitrarily close-lying Bragg peaks should also be observed, however only a few of the peaks are su ciently intense. This is clearly seen on the powder di raction pattern shown in Fig. 10.16. In comparison, the di raction pattern obtained after annealing is shown in the bottom part of the figure. Di raction peaks in the annealed sample correspond to scattering by orthorhombic Al6Mn, while the Bragg peaks of the quenched sample may only be interpreted in terms of a quasicrystalline structure, and should be indexed by six parameters.

It is much more di cult to visualize the atomic arrangement in a threedimensional quasicrystal than in the oneand two-dimensional cases. The 3D generalization of planar Penrose tiles is a pair of rhombohedra, spanned by the vectors given in (10.2.52). From the arrangement of the vectors it is easy to show that one type is defined by three vectors pointing to three adjacent vertices of the icosahedron, e.g., −e1, −e2, and e6, while the other by three vectors pointing to three nonadjacent vertices, e.g., e1, e3, and e6. These rhombohedra are shown in Fig. 10.17. The volume ratio of the two rhombohedra is precisely τ , so they are often referred to as “thin” and “thick”. When matching conditions are suitably chosen, only quasiperiodic filling of the space is possible with them.

Figure 10.18 shows the di raction patterns calculated from the Fourier spectra of quasicrystals obtained by filling the space with such rhombohedra. The three patterns correspond to di erent orientations.

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minum atoms all other marked sites – nevertheless occupation should always be understood in an average sense.

Just as real crystals are never perfectly periodic because of the large number of defects they contain, orientational order always extends only over finite regions in quasicrystals as well. Defects and disordered regions inevitably appear in them.

Further Reading

1.N. E. Cusack, The Physics of Structurally Disordered Matter: An Introduction, Graduate Student Series in Physics, Adam Hilger, Bristol (1987).

2.Introduction to Quasicrystals, Edited by M. V. Jarić, Academic Press, Inc., Boston (1988).

3.C. Janot, Quasicrystals, A Primer, Second Edition, Clarendon Press, Oxford (1997).

4.G. Venkataraman, D. Sahoo, and V. Balakrishnan, Beyond the Crystalline State: An Emerging Perspective, Springer Series in Solid-State Sciences, Vol. 84., Springer-Verlag Berlin (1989).

5.Y. Waseda, The Structure of Non-Crystalline Materials: Liquids and Amorphous Solids, McGraw-Hill, New York (1980).

6.R. Zallen, The Physics of Amorphous Solids, John Wiley & Sons, New York (1998).

7.J. M. Ziman, Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems, Cambridge University Press, Cambridge (1979).