Crystallography and Computational Quantum Mechanics Part IV:

Rotational Symmetry

In previous talks we showed that lattices have translational symmetry: a three-dimensional lattice is defined by the primitive vectors, a₁, a₂, and a₃ which determine the periodicity of the lattice. If we have an atom at any point R in space, then an identical atom is at the point

${\bf R}' = {\bf R} + n_{1} {\bf a}_{1} + n_{2} {\bf a}_{2} + n_{3} {\bf a}_{3}$   ,   (1) where n₁, n₂, and n₃ span the whole set of positive integers. This is true for both conventional and primitive lattices.

A lattice, along with its accompanying "basis can have many other symmetries. We briefly touched on rotational symmetry in our discussion of crystal systems, and we have talked about the screw axis, a combination of rotational and translational symmetry. Both operations, as well as mirror planes and glide planes, result in a crystal structure which is identical to the original structure.

In this tutorial we will discuss allowed rotations in a crystal structure (or lattice). It turns out that only rotations of certain angles, specified by the Crystallographic Restriction Theorem, are allowed. In and two and three dimensions these angles are 30°, 60°, 90°, 120° and 180°. We have discussed rotational symmetry in conjunction with two-dimensional systems, but here we will generalize to three dimensions. Fortunately our drawing skills will not be taxed very much, as rotation in three dimensions takes place in a plane perpendicular to an axis. As a result, most of the drawings shown below will be two dimensional, you'll have to imagine the axis pointing out of the page or screen.

What is Rotational Symmetry?

Simply put, an object has rotational symmetry if a rotation does not change its appearance. A simple example is shown in Fig. 1. If your browser works correctly, the square will eventually rotate by 90° and stop before starting again. If you look away from the picture for 15-20 seconds and then look back, you can not tell if the rotation happened or not. The square is invariant to a rotation of 90°. Since four such rotations bring it back to its original position, this is a 4-fold rotation, and the axis is called a 4-fold axis.

This works with real atoms as well. Fig. 2 shows an a view of a cesium chloride crystal looking down a Cartesian axis. If we rotate the picture by 90° around the central cesium atom (sorry, no animation this time) we get an image that looks exactly the same. What's more, if we re-centered the picture on one of the other atoms and did the rotation we'd still get exactly the same picture. Cesium chloride has a bunch of 4-fold rotation axes.

Of course not every rotation is invariant – if you rotate either Fig. 1 or Fig. 2 by, say, 57.3°, you won't see an identical picture. In two- and three- dimensional periodic crystals there are only a few rotations that can yield a crystal structure with rotational symmetry: 180°, 120°, 90° and 60°, or 2-fold, 3-fold, 4-fold and 6-fold rotations. You might think there could be a 72° (5-fold) rotation axis, but that's forbidden by the Crystallographic Restriction Theorem, which is a subject for another time.

Not all crystal structures have rotational symmetry, and not all rotations are allowed for everywhere – for example, only hexagonal crystals can have a 6-fold rotation axis, and only tetragonal and cubic crystals can have a 4-fold rotational axis.

What's more, not all crystals in a given crystal system have all the rotations allowed by that system. As an example see PrRu₄P₁₂. It is cubic, like cesium chloride, but it does not have a 4-fold rotation axis.

In the following we'll look at the allowed rotations and note the crystal classes where they can appear.

2-Fold (180°) Rotations

A 2-fold rotation is a rotation of 180° around one axis – called that because making two rotations takes you back to the original structure. These rotations can occur any crystal system except triclinic. Fig. 3 shows a projection of either an orthorhombic crystal looking down a Cartesian axis or a tetragonal crystal looking down the a- or b- axis. Rotating by 180° results in a structure indistinguishable from the original.

What may not be so obvious is that there is another 2-fold axis in this picture. Fig. 4 shows another view of the same structure, but doubled unit cell in the a-c plane. Here we can see that there is a 2-fold axis at the point of the central lithium atom (but not the other Li atom, nor the tin atoms). This multiplicity occurs in many crystals.

Those aren't the only two rotation axes in this picture. Fig. 5 shows the even more 2-fold rotation axes in this structure. The vaguely football-shaped marker is the symbol for a 2-fold rotation axis. Many crystals have a large number of rotation axes. We won't point them all out, but you should be aware they are present. Standard crystallography resources have diagrams showing all possible rotation axes (and much more) in every space group.

3-Fold (120°) Rotations

A three-fold rotation axis requires three 120° rotations to return to the original orientation, but you wouldn't notice it because an object with a 3-fold axis looks exactly the same after one rotation as it does with 2, 3, 4, or however many. The trigonal and cubic crystal systems have 3-fold rotation axes.^†

Let's start with the 3-fold axis. Fig. 6 shows several Wigner-Seitz cells for a hexagonal lattice with a basis that gives it trigonal symmetry,^‡ shown looking down the rotational axis.

There are two 3-fold axis in a hexagonal (or rhombohedral) lattice: at the center of the conventional Wigner-Seitz cell and at the three-cornered boundaries between the cells. Unlike the 2-fold axis there aren't multiple centers. We should use a triangle to denote the rotation axes, but since we already used the triangle to mark the “atom” positions we won't bother here. Just remember that space group diagrams use triangles to mark 3-fold axes.

A cubic lattice has a three-fold axis along the body diagonal ([111] axis) of the conventional unit cell, even if the actual lattice is body-centered or face-centered. The left half of Fig. 7 shows a view of the cubic “A15” structure, Cr₃Si, looking down that axis. The 3-fold rotation symmetry is obvious.

           

4-Fold (90°) Rotations

Four-fold rotation axes can only occur in the tetragonal and cubic crystal systems. Fig. 8 shows an example system. It could be a tetragonal crystal, looking down the z- ([001]) axis of the conventional tetragonal cell, or a cubic crystal, looking down the x-, y-, or z-axis ([100], [010], or [001], respectively) of the conventional cubic cell. The figure shows two 4-fold axes, indicated by black diamonds: one axis is at the center of a unit cell, and one on the corner between four cells.

This particular cubic system also has a number of 2-fold axes, as shown in Fig. 9.

The cubic crystal system is defined by 3-fold rotation and a 4-fold rotation. We might expect that this requires Cr₃Si to have a 4-fold axis, but it does not. As seen on the right side of Fig. 7, which looks down the [100] axis, Cr₃Si has a 2-fold rotation axis, but not a 4-fold one. This is actually the minimal requirement for a cubic crystal: all cubic crystals have a 2-fold axis along the [100] directions, and a 3-fold axis along the [111] directions. The cubic system has the maximal rotational symmetry of a cubic lattice, which includes 4-fold rotations along the [100] directions.

5-Fold (72°) Rotations

Logically we should have a 5-fold, or pentagonal, rotational symmetry. In that case the Wigner-Seitz cell for the conventional pentagonal lattice would look like Fig. 10:

Unfortunately, the Crystallographic Restriction Theorem (CRT) prohibits this rotational lattice in a periodic crystal. In fact, in two and three dimensions only 2-, 3-, 4- and 6-fold rotations are allowed. The CRT article will discuss seeming exceptions to this rule. For periodic systems, however,

6-Fold (60°) Rotations

Six-fold, or 60° rotations, are permitted symmetries in two and three dimensions. These rotations can only occur in the hexagonal crystal system, with a hexagonal lattice. In fact, the existence of a 6-fold rotation axis defines the hexagonal crystal system, which is otherwise trigonal with a 3-fold axis. Of course a hexagonal crystal does have a 120° rotational symmetry, but that's a combination of two 60° rotations.

In fact there are two 6-fold axes in a hexagonal crystal: one at the center of the Wigner-Seitz cell, and one at the three-cell junctions between the cells, as shown in Fig. 12.

HOWEVER: There are Defects

This all comes with a caveat. Figures 1-8 and 10 show well ordered, well behaved crystals. This would seem to be the default. Every crystal in the Cambridge Structural Database (CSD) and the Inorganic Crystal Structure Database (ICSD) has a well-defined unit cell. There may be some uncertainty about which kind of atom occupies a particular site, but in general everything forms a nice, periodic crystal.

In reality this is not the case. In fact, we are compelled to issue a warning:

Real “crystals” have many imperfections. Fig. 14 looks much like Fig. 3, but

• There is a substitutional impurity in the top right cell: a green atom appears, seemingly randomly.
• The third and fourth rows of cells do not line up with the other rows. This is a stacking fault.
• In the fourth row a blue atom is missing in the furthest right cell. This is a vacancy.
• There are other possible imperfections. One is a dislocation: a partial row of atoms jammed between two of the rows in this picture. We didn't draw that because it's really difficult.

All of these defects destroy the periodicity of the crystal. So why do we see “ordered” crystals? We see them because these defects are local: averaged over many, many cells, the entire crystal looks periodic.

The End

That's the end of our brief review of possible rotational symmetries in crystals. The important conclusions are:

• In a two or three dimensional crystal, only 2-, 3-, 4-, and 6-fold rotations can preserve the symmetry.
• Even the, some rotations are forbidden for a given crystal class: e.g., a monoclinic crystal can't have a 3-, 4, or 6-fold rotation.
• Just because a rotation is allowed does not mean it is mandatory. A cubic crystal may have a 4-fold rotation axis, but it need not have one.
• In reality, a crystal is a mess, with defects all over the place. It's only when we look at a large number of cells that we see an average symmetry. So

  

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.

Cambridge Crystallographic Data Centre (CCDC)

The Crystallographic Data Centre (CCDC) hosts both the organic Cambridge Structural Database and the Inorganic Crystal Structure Database, with a search engine which allows free, albeit somewhat restricted, access to structures in both the CSD and the CCDC.

Cambridge Structural Database (CSD)

Cambridge Structural Database (CSD) contains three-dimensional structural data for organic and metal organic systems. As of 1 January 2025 it contained 1,359,039 structures. There is a paywall, which can be worked around using the CCDC search engine described above.

gnuplot

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

Inorganic Crystal Structure Database (ICSD)

The Inorganic Crystal Structure Database (ICSD) contains structural data for inorganic crystals, though the occasional organic crystal slips in. In early 2025 the ICSD had information for 318901 structures, though many are duplicates. Like the CSD this is paywalled, but you can get any structure from the CCDC search engine if you are patient.

Jmol

Jmol is an open-source Java viewer which can be used to visualize crystal structures as well as molecules. Many of the figures shown here were drawn with Jmol.

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°.

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.

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

^† Technically the hexagonal crystal system has a 3-fold axis as well, but it's a combination of two 60° (6-fold axis) rotations, so we won't count it.

^‡ As we've established in Part II and Part IV, a hexagonal lattice can have either trigonal (3-fold axis) or hexagonal (6-fold axis) symmetry, depending on the basis. Here the basis (two sets of triangles) is chosen to give trigonal symmetry.

References

1. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Publishing, Orlando, 1976), chap. 4, pp. 73–75. A downloadable copy is available through the Internet Archive.
2. T. Hahn, ed., International Tables of Crystallography. Volume A: Spacegroup symmetry (Kluwer Academic publishers, International Union of Crystallography, Chester, England, 2002).
   For free versions of most of this information see the Bilbao Crystallographic Server and the Hypertext Book of Crystallographic Space Group Diagrams and Tables.
3. D. Hicks, M. J. Mehl, E. Gossett, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 2, Comput. Mater. Sci. 161, S1–S1011 (2019), doi:10.1016/j.commatsci.2018.10.043. (arXiv link)
4. M. J. Mehl, D. Hicks, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 1, Comput. Mater. Sci. 136, S1–S828 (2017), doi:10.1016/j.commatsci.2017.01.017. (arXiv link)
5. D. Hicks, M. J. Mehl, E. Gossett, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 2, Comput. Mater. Sci. 161, S1–S1011 (2019), doi:10.1016/j.commatsci.2018.10.043. (arXiv link)