Crystallography and Computational Quantum Mechanics Part II

Crystal Systems and Conventional Cells

In Part I of this tutorial we showed how any periodic crystal can be defined by a set of primitive vectors, which describe the periodicity of the lattice, and a basis, which describes the positions of the atoms within the unit cell defined by the primitive vectors. A crystal has translational symmetry: if R is a point in a lattice defined by the primitive vectors a₁, a₂, and a₃, then the point

$\mathbf{R}' = \mathbf{R} + n_{1} \, \mathbf{a}_{1} + n_{2} \, \mathbf{a}_{2} + n_{3} \, \mathbf{a}_{3} $   (1) is indistinguishable from R, no matter which basis vectors decorate the lattice.

Crystals have other symmetries as well. Fig. 1 shows the cubic perovskite structure looking down one of the cubic axes. If someone rotated this picture by 90° around the center line you could not tell the difference. There is a similar symmetry along the axes through the green calcium atoms, though you have to expand the picture in your mind to see it. The perovskite structure obviously has rotational symmetry around these axes — and it actually has many more symmetries.

Some crystals will have no rotational symmetry except a complete (360°, or 1-fold) rotation. Others will have symmetry that allows 180°, 120°, 90° and/or 60° rotations that do not change the crystal. These are known as 2-, 3-, 4-, and 6-fold rotations.^* The point group of a lattice which describes its rotational symmetry is known as its holohedry. All lattices with the same holohedry belong to the same crystal system. We will define the seven three-dimensional crystal systems below.

Two things should be mentioned before we start:

1. The holohedry and crystal class designations are properties of the lattice alone, not the crystal (lattice plus basis). Some crystals, such as the perovskite structure shown in Fig. 1, have the full rotational symmetry of the lattice, including the 4-fold rotation axes discussed above. Others, such as β-manganese, shown in Fig. 2, have no 4-fold rotation axis, but the lattice vectors are cubic, so β-manganese belongs to the cubic crystal system.



2. Some crystals in a given class may have additional translational symmetry. For example, in the cubic crystal system we have simple cubic crystals (what you'd think), body-centered cubic crystals (an additional lattice point at the center of the cube), and face-centered crystals (extra lattice points at the center of each cubic face). These crystals all have different primitive lattice vectors and unit cells, but each could be described by a simple cubic lattice and so belongs the the cubic crystal system. This will be discussed in detail in the next section, when we cover Conventional and Primitive Lattices, in particular the fourteen Bravais Lattices.

The Three-Dimensional Crystal Systems

There are seven crystal systems in three dimensions. Each has an allowed set of rotational symmetries and can be defined by a standard conventional cell. We delineate the classes.

System I: The Triclinic Crystal System

A crystal in the triclinic system has no rotational symmetry except the 360° rotation. As such it can be specified just about any way, but the Encyclopedia uses a standard set of primitive vectors:

$\begin{array}{ccc} {\bf A}_{1} & = & a \, \hat{x} \\ {\bf A}_{2} & = & b \, \cos\gamma \, \hat{x} + b \, \sin\gamma \, \hat{y} \\ {\bf A}_{3} & = & c_{x} \, \hat{x} + c_{y} \, \hat{y} + c_{z} \, \hat{z} \end{array}$   ,   (2) where $\begin{array}{ccc} c_{x} & = & c \, \cos\beta \\ c_{y} & = & c \, (\cos\alpha - \cos\beta \cos\gamma) / \sin\gamma \\ c_{z} & = & \sqrt{c^2 - c_{x}^2 - c_{y}^2} \end{array}$   ,   (3) with a unit cell volume $V = a \, b \, c_{z} \, \sin\gamma$   .   (4) The lattice constants a, b, c, α, β, and γ are defined in the previous article. In general these can have any values that do not lead to a zero-volume crystal, but specific values are associated with higher-symmetry crystal systems, as we will see below.

Here we use capital letters A₁, A₂, and A₃ to designate the conventional lattice describing the crystal system. This will make it easier to distinguish between the conventional and primitive cells of a given crystal system. This will be important in later sections.

A unit cell associated with triclinic system is shown in Fig. 3, using albite ($S6_{8}$) as an example. The primitive vectors are shown as red, green and blue lines. The unit cell defined directly by these unit vectors is shown as a box, and the Wigner-Seitz cell, the set of all points closer to the origin than to any other lattice point. You can view the Wigner-Seitz cell in any orientation by going to the albite page and clicking the Wigner-Seitz button.^†

System II: The Monoclinic Crystal System

The monoclinic crystal system has a 2-fold rotation (180°) axis, but no higher rotation. This is usually drawn by taking one of the primitive vectors perpendicular to the other two, with the 2-fold rotation along that primitive vector. The standard representation used in the Encyclopedia is called unique axis b, with primitive vectors

$\begin{array}{ccc} {\bf A}_1 & = & a \, \hat{x} \\ {\bf A}_2 & = & b \, \hat{y} \\ {\bf A}_3 & = & c \, \cos\beta \, \hat{x} + c \, \sin\beta \, \hat{z} \end{array}$   .   (5) This implies that α = γ = 90°. Some papers, particularly older ones, use “unique axis c,” where α = β = 90° and γ is freely chosen: $\begin{array}{ccc} {\bf A}_1 & = & a \, \hat{x} \\ {\bf A}_2 & = & b \, \sin\beta \hat{y} + b \, \cos\beta \hat{z} \\ {\bf A}_3 & = & c \hat{z} \end{array}$   .   (5') We have not seen anyone use “unique axis a.”

Note that this is a special case of the triclinic unit cell, as will be the case for every lattice below.

The volume cell is

$V = a \, b \, c\, \sin\beta$   (6) for unique axis b and $V = a \, b \, c\, \sin\gamma$   (6') for unique axis c.

An example of a monoclinic system is shown in Fig. 4, using the Cr$_{2}$Te$_{4}$O$_{11}$ lattice as an example. The “b” axis is the 2-fold rotation axis.

System III: The Orthorhombic Crystal System

While the monoclinic crystal system has one, and only one, 2-fold rotation axis, the orthorhombic system has three. The obvious way to orient the system is to place the rotation axes along the Cartesian axes, so the primitive vectors are

$\begin{array}{ccc} {\bf A}_1 & = & a \, \hat{x} \\ {\bf A}_2 & = & b \, \hat{y} \\ {\bf A}_3 & = & c \, \hat{z} \end{array}$   ,   (7) with unit cell volume $V = a \, b \, c$   .   (8) Obviously α = β = γ = 90°, so this is a special case of the monoclinic lattice.

Fig. 5 shows an example an orthorhombic lattice, using Re$_{2}$O$_{7}$ as a reference. The unit cell is just a rectangular prism, and the conventional unit cell is identical in size and shape to the Wigner-Seitz cell, just offset. All lattices in the orthorhombic system will have this behavior.

Systems V and VI: The Trigonal and Hexagonal Crystal Systems (The Hexagonal Crystal Family)

The numbering looks out of order, but we have our reasons. The trigonal crystal system has a 3-fold (120°) rotation axis and the hexagonal system has a 6-fold (60°) rotation axis, and we place them here because we are going through the systems in the order of increasing rotational symmetry. We'll talk the two systems together because they can be represented by the same lattice vectors. The lattice is universally called the hexagonal lattice, with primitive vectors

$\begin{array}{ccc} {\bf A}_1 & = & \frac12 \, a \, \hat{x} - \frac{\sqrt{3}}{2} \, a \hat{y} \\ {\bf A}_2 & = & \frac12 \, a \, \hat{x} + \frac{\sqrt{3}}{2} \, a \hat{y} \\ {\bf A}_3 & = & c \, \hat{z} \end{array}$   ,   (9) with unit cell volume $V = \frac{\sqrt{3}}{2} a^2 \, c$   .   (10) This lattice can be generated from the monoclinic lattice (unique axis c^‡) by setting a = b, α = β = 90°, and γ = 120°. Fig. 6 shows a representative unit cell, in this case hexagonal magnesium ($A3$). In both cases the Wigner-Seitz cell is a hexagonal prism, and the two systems together are sometimes referred to as the hexagonal crystal family.

In addition the the 3- or 6-fold rotation, this lattice has multiple 2-fold axes, emanating from the origin through the centers of the faces of each side of the prism and through through the edges between the faces.

We can see the difference in Fig. 7, which shows fictitious trigonal and hexagonal Wigner-Seitz cells looking down the rotation axis. The difference between the 3-fold and 6-fold symmetries is obvious.

                               

Is it necessary to make this distinction? The easy answer is yes, because we define crystal systems by their holohedry, and the two systems have different allowed rotations. In practice this is a little confusing until we get to the Bravais lattices, which we'll cover in the next tutorial. There we will see that the trigonal system actually has two lattices one of which is definitely not hexagonal, while the hexagonal system has only one.

System IV: The Tetragonal Crystal System

The tetragonal system as a 4-fold (90°) rotation axis along one (and only one) direction. You can generate this from the orthorhombic lattice (7) by setting b = a. This makes the primitive vectors

$\begin{array}{ccc} {\bf A}_1 & = & a \, \hat{x} \\ {\bf A}_2 & = & a \, \hat{y} \\ {\bf A}_3 & = & c \, \hat{z} \end{array}$   ,   (11) with unit cell volume $V = a^{2} \, c$   .   (12) Fig. 8 shows a representative unit cell, using CuAu $(L1_{0})$ as an example. The Wigner-Seitz cell is a tetragonal prism. Looking at the picture you can see that there are also 2-fold rotation axes along the $\hat{x}$ and $\hat{y}$ directions.

System VII: The Cubic Crystal System

If we set c = a in (11) we have three mutually orthogonal primitive vectors with the same length. The unit cell for this system is cubic, so this defines the cubic crystal system, with primitive vectors

$\begin{array}{ccc} {\bf A}_1 & = & a \, \hat{x} \\ {\bf A}_2 & = & a \, \hat{y} \\ {\bf A}_3 & = & a \, \hat{z} \end{array}$   ,   (13) with unit cell volume $V = a^{3}$   .   (14) As shown in Fig. 9, the Wigner-Seitz cell is a cube.

The cubic crystal system has three 4-fold rotation axes, along the primitive vectors (13). It also has four 3-fold rotation axes along the (111) diagonals.

Relationships Between the Crystal Systems

We conclude this section by noting that we can generate all the lattices from the triclinic lattice (2) by taking special values of the lattice constants. Table 1 shows these relationships for the standard lattices, ignoring the problems of “unique axis b” and “unique axis c.” In this table we present the lattices in their traditional order as found in the International Tables of Crystallography (Hahn, 2002) rather the 1-fold, 2-fold, 3-fold, 4-fold order we use above.

Table 1: Values of the standard lattice constants needed to produce a lattice in the given crystal system. For the monoclinic system we only show the standard unique axis b representation. The final column shows the rotation axes that define each system, including the multiplicity of axes for higher axes. All crystal systems higher than orthorhombic have multiple 2-fold axes, so we do not enumerate them.

Number	System	a	b	c	α	β	γ	Rotations
I	Triclinic	a	b	c	α	β	γ	1-fold
II	Monoclinic	a	b	c	90°	β	90°	2-fold
III	Orthorhombic	a	b	c	90°	90°	90°	2-fold (3)
IV	Tetragonal	a	a	c	90°	90°	90°	2-fold; 4-fold
V	Trigonal	a	a	c	90°	90°	60° or 120°	2-fold; 3-fold
VI	Hexagonal	a	a	c	90°	90°	60° or 120°	2-fold; 6-fold
VII	Cubic	a	a	a	90°	90°	90°	2-fold; 3-fold (4); 4-fold (3)

Next

In next section we'll show how to construct primitive lattices for real crystals from these conventional lattices. It turns out that every conventional lattice is also a primitive lattice, but not every primitive lattice is a conventional lattice. In fact, although there are six independent conventional lattices there are fourteen primitive lattices, so we have some explaining to do.

Further Reading

For an earlier and much more formal version of much of this discussion, see The Library of Crystallographic Prototypes: Parts 1 (Mehl, 2017) and 2 (Hicks, 2019). We also have a tutorial on crystal systems in two dimensions.

Ashcroft and Mermin discuss the various lattices and their Wigner-Seitz cells in Chapter 4.

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.

gnuplot

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

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:

Bravais Lattice:

In three dimensions, one of the fourteen allowed lattices. Each Bravais lattice belongs to one of the crystal systems. See the next section, Conventional and Primitive Lattices, for more information.

Conventional Cell:

The unit cell describing all of crystals in a given crystal class. A lattice in this system may have additional translational symmetry, which leads to a different primitive lattice. We will discuss this in the next section, Conventional and Primitive Lattices.

Crystal:

A periodically repeated collection of objects in n-dimensions.

Crystallographic Information File (CIF):

The Crystallographic Information File (CIF) is a standard format for presenting the structure of a crystal, incliding information on the stoichiometry, lattice, basis, thermal displacement of the atoms, and other experimental information. All the structures found in the Encyclopedia of Crystallographic Prototypes are generated using CIF files.

Crystal System:

The collection of all lattices with the same holohedry.

Holohedry:

The point group of a lattice which describes its rotational symmetry, without translations, mirrors, glides, or inversion. In two dimensions the only possibilities are 1-, 2-, 3-, 4-, and 6-fold rotations (rotations by 360°, 180°, 120°, 90°, and 60°, respectively) about the origin.

Lattice:

A periodically repeated collection of points in n-dimensions.

n-fold Rotation Axis:

A rotation of the crystal by 360°/n which replicates the original crystal. The only allowed values of n are 1, 2, 3, 4, and 6.

Primitive Cell:

The lattice vectors describing a given crystal system. The primitive lattice may be the conventional lattice for the crystal system, or it may contain additional translational symmetry. This will be covered in the next section, Conventional and Primitive Lattices.

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.

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

^* These are the only crystal-preserving rotations which maintain symmetry in a system with translational invariant in 2- or 3- dimensions. This is known as the Crystallographic Restriction Theorem. That tutorial also explains the reason for the awkward statement about “system(s) with translational symmetry.”

^† It is not obvious how to generate the Wigner-Seitz cell in Jmol. We do this with the command

load '' {444 555 1}};unitcell 0.02;axes 0.02; polyhedron id 'WS' wigner;color $WS translucent;display within(0, $ws) entered in the console or in a macro file.

^‡ Why is this “unique axis γ” rather than “unique axis β” as in the monoclinic system? Convention. Also known as “Because we say so.” Another convention is taking γ = 120°, as γ = 60° will generate the identical lattice. We will leave this proof as an exercise for the reader. (This hint may help.)

References

• 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.
• T. Hahn, ed., International Tables of Crystallography. Volume A: Spacegroup symmetry (Kluwer Academic publishers, International Union of Crystallography, Chester, England, 2002).
  For a free version of most of this information see the Bilbao Crystallographic Server
• 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)
• 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)