Crystallography and Computational Quantum Mechanics Part V:

The Crystallographic Restriction Theorem and Quasicrystals

In the article on rotations, we found that in two or three dimensions there can be periodic crystals that have 2-, 3-, 4-, and 6-fold axes with rotational symmetry – that is, if we rotate the crystal by 180° (2-fold axis), 120° (3-fold), 90° (4-fold), or 60° (6-fold), the resulting crystal is identical to the original – you can't tell if a rotation occurred or not.

Missing from this list is a 5-fold (72°) rotation. It turns out that you cannot have a periodic crystal with 5-fold rotational symmetry in two or three dimensions. This is known as the Crystallographic Restriction Theorem. In this article we'll show why a 5-fold axis is forbidden.

Even though 5-fold rotational symmetry is forbidden, X-ray diffraction experiments show systems that look like crystals – they have well defined diffraction spots – but show 5-fold symmetry. We'll talk about these quasicrystals at the end of the article.

First, however, we want to understand why periodic crystals can't have a 5-fold rotation axis.

What Makes an Allowed Rotation?

We already showed that periodic systems can have certain types of rotational symmetry, mainly 2-, 3-, 4-, and 6-fold axes. All of these systems have some similar properties. Let's look at that now.

       

Fig. 1 shows a 2-dimensional cross section of the Wigner-Seitz cells from three different crystal systems:

• Orthorhombic (left), which has 2-fold rotational axis at the center of each cell,
• Hexagonal (center), which has 3- and 6- fold rotational axis at the center of each cell, and
• Cubic (right), which has a 4-fold rotational axis at the center of each cell.

What we get out of this is that each crystal system has a particular shape for its Wigner-Seitz cells: orthorhombic systems are parallelepipeds, trigonal/hexagonal systems are hexagonal prisms, and cubic systems are cubes. (Tetragonal systems are a combination of orthorhombic and cubic, a tetragonal prism.) Furthermore, the Wigner-Seitz cells tile the lattice – there is no empty space.

       

We can define a set of primitive vectors for each lattice, as shown in Fig. 2. (The third vector points out of the screen in each case.) If we start a primitive vector at the center of a Wigner-Seitz cell, it points to the center of another Wigner-Seitz cell.

       

Finally, as we learned at the beginning of this series, any integer linear combination of primitive vectors, e.g.,

n₁ a₁ + n₂ a₂ + n₃ a₃   (1) also points to the center of another Wigner-Seitz cell. We show examples in Fig. 3.

What About 5-fold Rotations?

Suppose we could have a system with a 5-fold rotation axis. Then the two dimensional projection of the Wigner-Seitz cell would have to be a regular pentagon, as shown in Fig. 4. The question is, can we tile a bunch of pentagons?

The answer is no, we cannot. Fig. 5 shows an attempt to to this, and it fails. There is blank space. The simple explanation is that when we have an intersection of three or four unit cells their interior angles must add up to 360°. The interior angles of a regular pentagon are 108°, and 360/108 = 3.333… is not an integer.

But — hear me out here — suppose we could find a pattern of pentagons that tiles the lattice. What would the primitive vectors of the lattice look like?

We can answer this fairly simply. For the proposed regular pentagon Wigner-Seitz cell shown in Fig. 4 the primitive vectors in the plane^† must emanate from the center of the cell, pass through the center of each line at the zone boundary, and have a length equal to twice the distance from the center to the zone boundary. This gives five possible primitive vectors, as shown in Fig. 6. We arbitrarily chose two of them to be the candidate primitive vectors a₁ and a₂.

According to (1), all the vectors shown in Fig. 6 must be integer combinations of the others. In particular, the combination a₁ + a₂ should be another one of the vectors in the figure. We see how that works out in Fig. 7

This fails, though not as badly as you might have expected. a₁ + a₂ is in the direction of another lattice vector, but it comes up about 40% short. This is just another nail in the coffin – there cannot be a 5-fold rotation axis in a three dimensional periodic lattice.

In fact, the full Crystallographic Restriction Theorem says that you can't have any other rotation axes in two or three dimensions. So no 7-fold, 8-fold, 257-fold axes, either. In higher dimensions, however, you can have some of these rotation axes. In particular you can have a 5-fold rotation axis in four dimensions, which has interesting consequences.

Quasicrystals^‡

This all seems pretty cut-and-dried: the universe hates pentagonal systems. Nature is not that boring, however. Often x-ray diffraction patterns look like Fig. 8, which shows an electron diffraction study of a sample of an Zn-Mg-Ho alloy. There is a definite 5-fold (10?) rotational symmetry around the center spot, apparently violating the Crystallographic Restriction Theorem. Similar results are found in materials which have been rapidly heated. Interestingly, the first artificial material of this type seems to in the 1945 Trinity atomic bomb test, although this was not realized until 2021. (Bindi, 2021). Since these structures are almost crystals, they came to be known as quasicrystals.

How does this occur? When we look back at the Crystallographic Restriction Theorem we find that it only holds in periodic systems. The proof outlined in the last section depends on this. In particular, Fig. 7 shows that a system with 5-fold symmetry cannot be periodic.

What happens if the system is not periodic? Can we still have rotational symmetry? The answer is yes. Perhaps the most famous (modern example) was discovered by Sir Roger Penrose, who found that the two dimensional plane can be tiled by carefully connecting two different shapes. The Penrose tiling is non-periodic, but definitely has 5-fold rotation axes all over the plane, as shown in Fig. 9.

It turns out that we can think of these structures as two or three dimensional projections of higher dimensional periodic systems. Since 5-fold rotation axes can exist in higher dimensions there is no violation of the Crystallographic Restriction Theorem.

Unfortunately no one has written a code which will perform quantum mechanical calculations in four or more dimensions and project the results back into three dimensions. For that reason computational materials calculations can't directly access quasicrystalline systems, although clever researchers can approximate them over small volumes using periodic three-dimensional cells.

One more thing

We have not proved the Crystallographic Restriction Theorem, we have only given examples of why 5-fold rotation axes are forbidden in two and three dimensions. A formal proof of the 2- and 3-d case can be found in (Yue, 2019).

Resources

AFLOW

AFLOW (Automatic FLOW) is an open-source package which can be used to generate and run first-principles electronic structure calculations for a variety of codes. It can also be used to analyze and compare crystal structures, including the production of Crystallographic Information Files (CIFs). This code is the primary resource used to generate the structures in the Encyclopedia of Crystallographic Prototypes. Unfortunately it is not programmed for quasicrystals.

gnuplot

gnuplot is a freely-distributable code for plotting graphs. We use it extensively in these tutorials and in other sections of the Encyclopedia.

Glossary

Here is a brief definition of some of the terms used in this article:

Basis:

The collection of items (atoms, pixels, paint drops) that decorate a lattice to produce a crystal or a wallpaper. Every object in a crystal structure is part of the basis.

Basis Vectors:

The vectors pointing from the origin of the lattice to the individual members of the basis.

Cartesian (Basis) Coordinates:

The positions of the basis vectors relative to the origin given on a standard Cartesian grid.

Crystal:

A periodically repeated collection of objects in n-dimensions.

Crystallographic Restriction Theorem:

In two or three dimensions symmetry is only preserved for rotations (and screw rotations) with angles of 30°, 60°, 90°, 120° and 180°. For a longer discussion of this theorem, see the Wikipedia page and especially references there-in.

Lattice:

A periodically repeated collection of points in n-dimensions.

Lattice Coordinates:

The positions of the basis vectors expressed relative to the chosen primitive vectors of the system.

Primitive Vectors:

A set of vectors that defines the allowed shifts in the origin of the lattice that do not violate translational symmetry.

Quasicrystal:

A non-periodic structure that nevertheless contains axes with 5-fold (or 7+-fold, but usually 5) rotational symmetry in two or three dimensions.

Rotational Symmetry:

A rotation of the crystal about an axis which produces a structure indistinguishable from the original.

Screw Axis:

A combination of translational and rotational symmetry: a translation along an axis of some amount is combined with a rotation around that axis leads to a structure which is identical the first structure.

Translational Symmetry:

A shift of the origin of a crystal that produces a structure indistinguishable from the original.

Unit Cell:

The (non-unique) smallest area (smallest volume in three dimensions) of space that reproduces all of the information about the crystal structure, and which can be periodically tiled to create the entire structure.

Wigner-Seitz Cell

A uniquely defined unit cell consisting of all spatial points closer to a given lattice point than to any other lattice point.

Footnotes

^† Remember that the third vector is perpendicular to the screen.

^‡ Not to be confused with supramolecular quasi-crystals. The adjective and the hyphen are important.

References

1. L. Bindi, William Kolb, G. Nelson Eby, Paul D. Asimow, Terry C. Wallace, and Paul J. Steinhardt, Accidental synthesis of a previously unknown quasicrystal in the first atomic bomb test, Proceedings of the National Academy of Sciences 118, e2101350118 (2021), doi:10.1073/pnas.210135011.
2. Z. Yue, A rigorous proof on the crystallographic restriction theorem, International Journal of Physics and Mathematics 1, 1 (2019). A copy of the paper is available here.