Before beginning our detailed discussion of point groups and space groups, let me explain their notation. While there are several conventions for notating point groups and space groups, here I will explain the Hermann–Mauguin1 symbol (hereafter referred to as HM symbol), which is the most widely used notation in crystallography.
HM symbols are expressed by combining the symbols for symmetry elements explained on the page “1.2. Symmetry and Symmetry Operations/Elements“. First, I will explain the notation for point groups, and then proceed to explain the notation for space groups.
Point Groups
As explained on the previous page, a point group is a set of symmetry operations that have a fixed point. While point groups exist infinitely, here I will limit the explanation to crystallographic point groups (those composed only of operations compatible with translational symmetry; for details see the separate page).
Relationship between Point Groups and Unit Cells
Before diving into the main topic, let me organize the discussion somewhat deductively. All crystals are classified into 230 types of space groups, which are further classified into 32 types of crystallographic point groups. These crystallographic point groups are classified into 7 crystal systems, which in turn are classified into 6 crystal families2. In other words, if we move upward from crystallographic point groups, we reach a crystal with some unit cell, and if we move downward, we reach the lowest-level classification: crystal families. A crystal family can be understood as a classification based on the shape of the unit cell, formed by combining the trigonal and hexagonal crystal systems from the 7 crystal systems. Thus, while point groups and unit cells are originally unrelated concepts, for crystallographic point groups specifically, we can relate them to the unit cell.
Each crystal family possesses a unique symmetry element (a rotation axis or rotoinversion axis) that characterizes the shape of the unit cell3, and the direction of this element is called the principal axis. When there are two or more unique symmetry elements, their directions are called secondary axis and tertiary axis, respectively. Rules are established for associating the directions of the principal and secondary/tertiary axes with the directions of the unit cell edges \(a, b, c\) as follows.
| Crystal Family | Primary Axis | Secondary Axis | Tertiary Axis |
|---|---|---|---|
| Triclinic | – | – | – |
| Monoclinic | \(\textbf{b}\) | – | – |
| Orthorhombic | \(\textbf{a}\) | \(\textbf{b}\) | \(\textbf{c}\) |
| Tetragonal | \(\textbf{c}\) | \(\textbf{a}\) = \(\textbf{b}\) | \([110]\) = \([1\bar{1}0]\) |
| Hexagonal4 | \(\textbf{c}\) | \(\textbf{a}\) = \(\textbf{b}\) = \([\bar{1}\bar{1}0]\) | \([1\bar{1}0]\) = \([120]\) = \([\bar{2}\bar{1}0]\) |
| Cubic | \(\textbf{a}\) = \(\textbf{b}\) = \(\textbf{c}\) | \([111]\) = \([\bar{1}11]\) = \([1\bar{1}1]\) = \([11\bar{1}]\) | \([110]\) = \([1\bar{1}0]\) = \([011]\) = \([01\bar{1}]\) = \([101]\) = \([\bar{1}01]\) |
The equals sign = indicates crystallographically equivalent directions. For example, in the cubic family, when written as \(a\) = \(b\) = \(c\), this means that \(a\), \(b\), and \(c\) possess identical properties as a result of the threefold rotation about the \([111]\) direction, which is the unique symmetry element of the cubic family. Therefore, the principal axes of the cubic family are the three axes \(a\), \(b\), and \(c\). The same applies to the secondary and tertiary axes of the tetragonal, hexagonal, and cubic families.
Notation for Point Groups
Now for the main topic. In HM notation, point groups are expressed by arranging (at most) three elements as follows: $$\Large X\ Y\ Z$$
The three elements \(X, Y, Z\) correspond respectively to the principal (primary), secondary, and tertiary axes defined for each crystal family. Since symmetry elements exist along these directions, they are also called symmetry directions. In the positions \(X, Y, Z\), we write the symbols for symmetry elements: \(1, 2, 3, 4, 6, \bar{1}, m, \bar{3}, \bar{4}, \bar{6}\). When rotation \(2, 3, 4, 6\) and reflection \(m\)5 exist simultaneously in the same symmetry direction, they are connected with a forward slash “/”, as in \(2/m\)6. For all crystal families except triclinic, the symmetry element in the principal axis direction is written in \(X\). Triclinic has no principal axis, but instead the identity element \(1\) or inversion center \(\bar{1}\) is written in \(X\). That is, \(X\) is never blank. On the other hand, when there is no symmetry element on a secondary axis, \(Y, Z\) are omitted. That is all the rules. All crystallographic point groups and the crystal families (systems) to which they belong are summarized on a separate page; here I will merely explain a couple of examples.
Examples of Point Groups
\(2/m\)
This point group belongs to the monoclinic family. Therefore, this notation means that a twofold rotation \(2\) and reflection \(m\) exist simultaneously in the direction of the \(b\) axis, which is the principal axis.
\(3\ m\)
This point group belongs to the hexagonal family. Therefore, this notation means that a threefold rotation \(3\) exists in the direction of the \(c\) axis, which is the principal axis, and reflection \(m\) exists in all directions of the secondary axis: \(a\), \(b\), and \([\bar{1}\bar{1}0]\). The tertiary axis has only the identity element \(1\). While writing \(3\ m\ 1\) is not incorrect, it is conventionally written as \(3\ m\).
Now, how should one write a case where the secondary axis is \(1\) and the tertiary axis is \(m\)? One might momentarily think \(3\ 1\ m\) would be appropriate, but \(3\ 1\ m\) is completely equivalent (isomorphic) to \(3\ m\) when rotated by 1/6 turn around the \(c\) axis, so there is no need to distinguish them. However, as will be explained later, for space groups this distinction becomes necessary.
\(\bar{6}\ m\ 2\)
This point group belongs to the hexagonal family. Therefore, this notation means that sixfold rotoinversion \(\bar{6}\) exists in the direction of the \(c\) axis, which is the principal axis; reflection \(m\) exists in all directions of the secondary axis: \(a\), \(b\), and \([\bar{1}\bar{1}0]\); and twofold rotation \(2\) exists in all directions of the tertiary axis: \([1\bar{1}0]\), \([120]\), and \([\bar{2}\bar{1}0]\).
Space Groups
Notation for Space Groups
Space groups are expressed by arranging (at most) four elements as follows: $$\Large W\ X\ Y\ Z$$
\(W\) denotes the lattice type (\(P, A, B, C, I, F, R\)). It is never blank. The \(X, Y, Z\) follow basically the same rules as point group notation, but there are several points requiring attention.
First, the symmetry directions to which \(X, Y, Z\) correspond follow the rules of point groups. However, there is one exception: when the monoclinic family is written in full symbol notation (details below), the rule deviates from point group conventions, and \(X, Y, Z\) are made to correspond to the \(a, b, c\) axis directions instead. The symmetry elements written in \(X, Y, Z\) can include rotations, rotoversions (rotoinversions), inversion centers, reflections, screws, and glide reflections. When a rotation or screw exists in the same symmetry direction as a reflection or glide reflection, they are connected by a forward slash “/” as in the point group case.
Additionally, in space groups it can occur that multiple symmetry elements exist in the same symmetry direction but do not intersect7. Rules are established to determine which symmetry element should be preferentially notated in such cases, as follows:
- When rotation, rotoinversion, or screw exists in the same symmetry direction, prioritize the element with the highest order.
- When rotation, rotoinversion, or screw of the same order exists in the same symmetry direction, prioritize rotation.
- When reflection and glide reflection exist in the same symmetry direction, prioritize reflection.
Furthermore, while in point groups the identity element \(1\) appears only in the point group \(1\), in space groups \(1\) can appear even in space groups other than \(P1\) in order to clarify its relationship with axes.
Several examples are given below. All 230 space groups are compiled on a separate page.
Examples
\(C\ 1\ 2/m\ 1\)
This space group belongs to the monoclinic family. The first character \(C\) indicates a base-centered lattice, meaning that translation vectors exist at \(\textbf{a}\), \(\textbf{b}\), \(\textbf{c}\), and \(1/2(\textbf{a}+\textbf{b})\). The \(1\ 2/m\ 1\) following \(C\) is a full symbol notation, which is the exceptional case mentioned above. That is, \(1\), \(2/m\), and \(1\) correspond respectively to the \(\textbf{a}\), \(\textbf{b}\), and \(\textbf{c}\) directions. In essence, this space group can be interpreted as having \(2/m\) along the \(\textbf{b}\) axis in a \(C\) base-centered lattice.
Incidentally, this space group possesses not only \(2\) and \(m\) along the \(\textbf{b}\) axis direction, but also a screw \(2_1\) and glide reflection \(a\). Despite this, one does not write \(C\ 1\ 2_1/m\ 1\) or \(C\ 1\ 2/a\ 1\). This is because the prioritization rules mentioned above apply.
\(P\ 3\ 1\ m\)
This space group belongs to the hexagonal family. The first character \(P\) indicates a primitive lattice, with translation vectors at \(\textbf{a}\), \(\textbf{b}\), and \(\textbf{c}\). The principal axis \(c\) has \(3\), the secondary axis has \(1\), and the tertiary axis directions \([1\bar{1}0]\), \([120]\), and \([\bar{2}\bar{1}0]\) have \(m\).
Earlier, I explained that for point groups \(3\ m\) and \(3\ 1\ m\) are not distinguished. One might think the same applies here. However, in space groups we must distinguish between \(P\ 3\ m\ 1\) and \(P\ 3\ 1\ m\). This issue will be discussed in detail on another page, but the space group notation includes the identity element \(1\) to clarify the symmetry directions, as in this example.
\(I4/m\ 2/m\ 2/m\)
This space group belongs to the tetragonal family. The first character \(I\) indicates a body-centered lattice, with translation vectors at \(\textbf{a}\), \(\textbf{b}\), and \(\textbf{c}\). The principal axis \(c\) has \(4/m\), the secondary axes \(\textbf{a}\) and \(\textbf{b}\) have \(2/m\), and the tertiary axis directions \([110]\) and \([1\bar{1}0]\) also have \(2/m\). This space group contains screw and glide reflection symmetry elements as well, but due to the prioritization rules mentioned above, only rotations and reflections appear in the symbol.
Full Symbol / Short Symbol
The point group and space group notations explained so far are called full symbols. However, note that despite being called “full”, they do not literally enumerate all symmetry elements; the identity element \(1\) and (following the prioritization rules) screw and glide reflection elements may be omitted.
In contrast to the full symbol, a short symbol (short symbol) is a notation expressing point groups and space groups more concisely. For example, the point group \(2/m\ 2/m\ 2/m\) (full symbol) becomes \(m\ m\ m\) (short symbol). The rules for converting from full symbol to short symbol are as follows:
- When rotation or screw and reflection or glide reflection exist in the same direction, the rotation or screw is omitted and only the reflection or glide reflection is notated. For example, \(2/m \rightarrow m\) or \(4_1/a \rightarrow a\).
- However, if this omission makes it impossible to completely express the properties of the point group or space group, omission is not applied.
Regarding the first point, why prioritize “reflection or glide reflection”? When placing a symmetry element along a direction in space, rotation or screw axes require two parameters, while reflection or glide reflection planes require only one. In other words, reflection or glide reflection planes can express the nature of symmetry more clearly.
Regarding the second point, further explanation is needed. The HM symbol is designed, whether in full or short symbol form, so that the entire group can be generated by treating the symmetry elements (and their contained symmetry operations) in the symbol as generators8. Generators are, as their name suggests, the elements used to generate an entire group (see details elsewhere). For example, the point group \(2/m\ 2/m\ 2/m\) is a point group with a twofold rotation \(2\) and reflection \(m\) in each of three mutually perpendicular directions. Even if we remove the three \(2\) rotations in these three directions, the combined action of the three remaining reflection planes \(m\) inevitably produces \(2\). That is, the point group \(2/m\ 2/m\ 2/m\) is completely generated by the three reflections \(m\) in mutually perpendicular directions, so it is expressed in short symbol notation as \(m\ m\ m\). Conversely, if we were to retain only reflection \(m\) for the point group \(4/m\), we would have merely the point group \(m\), which has entirely different properties from the original. Therefore, the fourfold rotation \(4\) cannot be omitted, and the short symbol remains \(4/m\).
The full and short symbols for all crystallographic point groups and space groups are compiled on separate pages (crystallographic point groups and space groups), so please refer to them.
Footnotes
- The Hermann-Mauguin symbol was proposed by Hermann (1928, 1931) and Mauguin (1931), and was introduced in Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935), a precursor to the ITA. ↩︎
- This classification discussion will be explained in detail on the “3.7. Classification of Space Groups” page. Note that both “crystallographic point groups” and “crystal families” contain the word “crystal” in Japanese, but this is merely a matter of Japanese translation. In English, the former is “crystallographic point groups” and the latter is “crystal families”. ↩︎
- To be precise, a crystal family is a concept that resolves the discrepancy between crystal systems and lattice systems. This too will be explained on the “3.7. Classification of Space Groups” page. ↩︎
- For clarification, the hexagonal family is a classification that combines the trigonal and hexagonal crystal systems. ↩︎
- For reflections (and glide reflections), the direction of the normal to the mirror plane is the direction of the symmetry element. ↩︎
- Even if a rotoinversion and reflection exist in the same direction, they can ultimately be expressed as a combination of rotation and reflection, so consideration of them is unnecessary: \(\bar{1}/m=2/m,\ \bar{3}/m=6/m,\ \bar{4}/m =4/m,\ \bar{6}/m =3/m\) ↩︎
- In point groups, all symmetry elements pass through the fixed point. ↩︎
- This does not mean that the HM symbol is necessarily composed of only the minimal set of generators. ↩︎