Breaking the rules: aperiodic structures, incommensurate materials, and quasicrystals

3.6.1 Incommensurate or modulated structures

We will find in Chapter 6 that the experimental technique of diffraction gives information about the crystal structure. For normal ordered crystals, the periodicity and symmetry of the crystal lattice are reflected in the diffraction data, and as a result it is possible to use the signatures in the diffraction data to identify the size, shape, and symmetry of the unit cell. For many years it was recognized that some materials give diffraction data that suggest that the periodicity of their lattices are modulated by one or more superimposed waves with a repeat distance that does not match (isincommensurate with) the repeat distance of the crystal lattice. Often these incommensurate phases are associated with a normal phase transition (Chapter 12), with a temperature range between

the temperatures over which the two normal phases are stable. For example, quartz has an incommensurate phase that is stable for just 1.5 K sandwiched between the high-temperature hexagonal β-phase and the low-temperature trigonal α-phase. In some cases the range of stability of the incommensurate phase is much larger. In Chapter 12 we will show how an incommensurate phase can arise as a near-inevitability of the constraints on the normal phase transition due to symmetry, but there are several other mechanisms for the stability of incommensurate phases, some of which are associated with site-ordering processes, and some of which are associated with the distribution of electron energies.

Although the three-dimensionality of the crystal is broken by the incom-mensurate modulation, it is possible to describe the overall symmetry of an incommensurate structure in terms of higher-dimensional space groups. Not surprisingly, there are large number of space groups in each of the higher dimensions.

3.6.2 Quasicrystals

In 1985 the worlds of condensed matter physics, materials science, chemistry, and crystallography were stunned by the publication of the results of diffraction experiments on alloys that appeared to show the existence of 5-fold and 10-fold rotational symmetry. An example is shown in Fig. 3.16. From the basic

Fig. 3.16 Electron diffraction pattern of an Al–Mn alloy, showing 10-fold rotation symmetry. (Photograph courtesy of Kevin Knowles, University of Cambridge.)

considerations discussed in this chapter, the existence of 5-fold and 10-fold rotation symmetry is inconsistent with long-range lattice symmetry, yet the form of the diffraction data looked exactly like the combination of the rotation symmetry and the long-range crystallographic order. There have now been many experiments that have confirmed these initial results, and the challenge now is to understand the origin of these intriguing results. There are a number of theoretical ideas that have been proposed. The most compelling idea is that the alloy can form units that can be described by cells with angles that are multiples of 36^◦. Two such objects can be tiled together to fill space completely without the generation of a regular lattice – these are known asPenrose tiles. The challenge then is how to decorate these objects with atoms. One mathematical approach is to follow the practice in the study of incommensurate crystals and develop a formalism based on higher-dimensional space groups, with the projection onto the three-dimensions of the experimental data giving the impression of an aperiodic structure.

Summary of chapter

• The crystal lattice is an infinite periodic of points. The lattice can be fully described by three basis vectors.

• The unit cell is the region of space bounded by eight lattice points that forms a parallelepiped. If the unit cell has the same volume as the space associated with a single lattice point, this is said to be aprimitive unit

cell. It is often convenient to define a unit cell that has a volume that is a multiple of the volume of the primitive unit cell, when the shape of the unit cell most clearly reflects the symmetry of the lattice.

• The lattice parameters are the three distinct lengths of the edges of the unit cell, and the three distinct angles subtended by the edges of the unit cell.

• The constraints on the possible symmetry of a lattice lead to the definitions of seven types of lattice. These can be combined with the possibilities of forming non-primitive unit cells to give the definitions of 14Bravais lattices.

• The positions of atoms within the unit cell are conveniently described usingfractional coordinates, in which each position is represented as a combination of vectors along each of the three lattice vectors of length that is a fraction of the lattice repeat distance.

• A point is conveniently represented by the Dirac delta function. This enables a point to be treated using the same mathematics as used for continuous functions.

• The crystal structure can be described mathematically as a set of atom positions, thebasis or motif, which are convoluted with the crystal lattice.

• Point groups are defined as combinations of rotation axes, mirror planes, rotoinversion axes, and a centre of symmetry. There are 32 independent crystallographic point groups. Point groups can be used to describe a wide range of crystal properties and behaviour.

• Distortions of a structure can be described in terms of symmetry opera-tions that are preserved or destroyed. The different combinaopera-tions of preserved and broken symmetry operations are called irreducible representations.

• Rotational and reflection symmetry operations can be combined with translations along a fraction of the lattice repeat to give the translational symmetry elements calledscrew axes and glide planes respectively.

• The point and translational symmetry operations can be combined with the Bravais lattices to give 230space groups.

• Although the formalism developed to describe crystal structures is comprehensive, it is found that there are some materials that appear to break the rules. Two examples arequasicrystals, which appear to have 5-fold and 10-fold rotational symmetry, andincommensurate crystals in which the periodicity in one or more directions is modulated by a wave whose wavelength has no relation to the repeat lengths of the lattice.

Further reading

An introduction to the formalism of the crystalline state is given by both Kittel (1996) and Ashcroft and Mermin (1976). More detailed treatments can be found in Alcock (1990), Giacovazzoet al. (1992), Hammond (2001), Ladd (1999) and Windle (1977). Janot (1997) and Senechal (1995) are primary source references for the physics of quasicrystals.

Exercises

(3.1) Show that the presence of a plane of reflection at x= 0 in a crystal of orthorhombic symmetry will be accompanied by a mirror plane at x= 1/2. Similarly show that if there is a plane of reflection at x = 1/4 there will also be a plane of reflection at x = 3/4. (Hint: You will need to recognise the role of translational symmetry.)

(3.2) Show that if there is a centre of symmetry at 0, 0, 0 in an orthorhombic crystal there will also be centres of symmetry at seven other positions in the unit cell. (Hint:

You will need to recognise the role of planes of reflection.) (3.3) Show that if there is a rotation axis parallel to one of the axes in an orthorhombic crystal passing through the origin there will be three other parallel rotation axes in the unit cell. Show that this is also true if the rotation axis passes not through the origin but through 1/4, 0, 0.

(3.4) Show that the point group mmm contains three orthogonal 2-fold axes.

(3.5) Show that the point group 4mm contains four mirror planes.

(3.6) Show that I -centred and F -centred tetragonal lattices are different descriptions of the same lattice.

(3.7) The figure below shows a plan of the crystal structure of the andalusite polymorph of Al₂SiO₅.

x

Identify all the symmetry operations, deduce the space group, and list the general positions in terms of fractional coordinates. Compare the structure with that of the sillimanite polymorph given in Problem 2.8.

(3.8) List the set of fractional atomic coordinates related by the following symmetry operations, assuming each passes through the origin and is oriented parallel to z:

(a) 3-fold rotation axis; (b) 6-fold rotation axis; (c) 3 axis;

(d) 6 axis. (Note: in each case use the axes of the lattice, for which the angle between x and y is 120^◦, rather than orthogonal axes.)

(3.9) List the set of fractional atomic coordinates related by the following screw symmetry operations, assuming each passes through the origin and is oriented parallel to z: (a) 3₁; (b) 6₁; (c) 6₂axes.

(3.10) The figure below shows the hexagonal crystal structure of Na₂CO₃.

Identify the point symmetry of the sites occupied by the different atoms. (Hint: identify all point symmetry operations of the crystal systematically, and deduce the point symmetry of the atoms as arising from the symmetry elements that intersect each atom.) We will meet Na₂CO₃ again in Chapter 12, where you will be able to appreciate this author’s interest in this material!

(3.11) Show that two orthogonal 2-fold rotation axes parallel to [100] and [010] that intersect at the origin of the unit cell will generate a third 2-fold rotation axis parallel to[001]

which also passes through the origin. Compare this with the case where the 2-fold rotation parallel to[010] passes through the point 0, 0,¹₄. List the general coordinates for both cases.

(3.12) List the general coordinates of the following orthorhom-bic space groups, ensuring that if a centre of symmetry is to present it lies on the origin of the unit cell: (a) P mma,

(b) P ccb, (c) P 2₁mn, (d) P bmb. In each case, state the symmetry elements not explicitly stated in the space group label.

(3.13) The two space groups P mna and P nma sound similar, but are listed as separate space groups. What are the

differences between these two space groups? (Note: it is not adequate to simply state that the mirror plane affects a different direction, since how we specify the directions in the orthorhombic group is arbitrary. The important point is the actual symmetry.)