3.4: Group Properties

Every space group \(\mathcal{G}\) is the semi-direct product of two sets, \(\mathcal{R\ \bigotimes\ L}\):

• \(\mathcal{L =}\left\{ \left( 1 \middle| \boldsymbol{T}_{mnp} \right) \right\}\), the set of \(N\) Bravais lattice vectors that is an invariant subgroup of \(\mathcal{G}\). The order of \(\mathcal{L}\) is determined by setting periodic boundary conditions for the crystalline lattice. Since these conditions generally involve many unit cells, \(N\) is a very large value.
• \(\mathcal{R =}\left\{ \left( R_{i} \middle| \boldsymbol{\tau}_{i} \right):i = 1,\ldots,h \right\}\), a set of \(h\) rotation-displacement operations called essential symmetry operations that is not an invariant subgroup. \(\mathcal{R}\) contains the identity operation \(\left( 1 \middle| \boldsymbol{0} \right) = \left( R_{1} \middle| \boldsymbol{\tau}_{1} \right)\), and any nonzero displacement \(\boldsymbol{\tau}_{i}\) are not lattice vectors. Therefore, \(\mathcal{R}\) can include proper or improper rotations, screw rotations, and glide reflections. The related set \(\mathcal{R}_{0} = \left\{ R_{i}:i = 1,\ldots,h \right\}\), consisting of the \(h\) rotations, is one of the 32 crystallographic point groups.

As a result, the order of the space group \(\mathcal{G}\) is \(hN\), and every member has the form

\[\left( R_{i} \middle| R_{i}\boldsymbol{T}_{mnp} + \boldsymbol{\tau}_{i} \right) = \left( R_{i} \middle| \boldsymbol{\tau}_{i} \right)\left( 1 \middle| \boldsymbol{T}_{mnp} \right), \nonumber \]

which emphasizes the compatibility between the Bravais lattice and rotational symmetry because \(R_{i}\boldsymbol{T}_{mnp}\) must be a lattice vector. By expanding the product of sets,

\[\begin{align*} \mathcal{G} &= \mathcal{R\ \bigotimes\ L} \\[4pt] &= \left\{ \left( 1 \middle| \boldsymbol{0} \right),\left( R_{2} \middle| \boldsymbol{\tau}_{2} \right),\ldots,\left( R_{h} \middle| \boldsymbol{\tau}_{h} \right) \right\}\ \bigotimes_{}^{}{\ \left\{ \left( 1 \middle| \boldsymbol{T}_{mnp} \right) \right\}} \\[4pt] &= \left( 1 \middle| \boldsymbol{0} \right) \times \left\{ \left( 1 \middle| \boldsymbol{T}_{mnp} \right) \right\}\ \bigoplus_{}^{}{\ \left( R_{2} \middle| \boldsymbol{\tau}_{2} \right) \times \left\{ \left( 1 \middle| \boldsymbol{T}_{mnp} \right) \right\}}\ \bigoplus\cdots\bigoplus_{}^{}{\ \left( R_{h} \middle| \boldsymbol{\tau}_{h} \right) \times \left\{ \left( 1 \middle| \boldsymbol{T}_{mnp} \right) \right\}} \\[4pt] &= \left\{ \left( 1 \middle| \boldsymbol{T}_{mnp} \right) \right\}\ \bigoplus_{}^{}{\ \left\{ \left( R_{2} \middle| R_{2}\boldsymbol{T}_{mnp} + \boldsymbol{\tau}_{2} \right) \right\}}\ \bigoplus\cdots\bigoplus_{}^{}{\ \left\{ \left( R_{h} \middle| R_{h}\boldsymbol{T}_{mnp} + \boldsymbol{\tau}_{h} \right) \right\}} \end{align*} \]

the space group \(\mathcal{G}\) is also the sum of \(h\) cosets of \(\mathcal{G}\) with respect to the invariant subgroup \(\mathcal{L}\). These \(h\) cosets form the factor group \(\mathcal{"G/L"}\) of the space group \(\mathcal{G}\) with respect to the Bravais lattice \(\mathcal{L}\):

\[\mathcal{G/L =}\left\{ \left\{ \left( 1 \middle| \boldsymbol{T}_{mnp} \right) \right\},\left\{ \left( R_{2} \middle| R_{2}\boldsymbol{T}_{mnp} + \boldsymbol{\tau}_{2} \right) \right\},\ldots,\left\{ \left( R_{h} \middle| R_{h}\boldsymbol{T}_{mnp} + \boldsymbol{\tau}_{h} \right) \right\} \right\}. \nonumber \]

This factor group has order \(h\) and is isomorphic (i.e., follows the same multiplication table) with the crystallographic point group \(\mathcal{R}_{0}\). The identity member is the subgroup \(\mathcal{L =}\left\{ \left( 1 \middle| \boldsymbol{T}_{mnp} \right) \right\}\).

Factor Groups

To form a factor group of any group \(\mathcal{G}\), \(\mathcal{G}\) must have an invariant subgroup, which is a subset of members that form complete classes of \(\mathcal{G}\). To illustrate important features of factor groups applied to space groups, consider the point group \(\mathcal{C}_{4v} = \left\{ 1,4_{z},2_{z},4_{z}^{3},m_{x},m_{y},m_{x + y},m_{x - y} \right\}\), which consists of 8 members and 5 classes:

• \(\mathcal{K}_{1} = \left\{ 1 \right\}\)
• \(\mathcal{K}_{2} = \left\{ 4_{z},4_{z}^{3} \right\}\)
• \(\mathcal{K}_{3} = \left\{ 2_{z} \right\}\)
• \(\mathcal{K}_{4} = \left\{ m_{x},m_{y} \right\}\) and
• \(\mathcal{K}_{5} = \left\{ m_{x + y},m_{x - y} \right\}\).

\(\mathcal{C}_{4} = \left\{ 1,4_{z},2_{z},4_{z}^{3} \right\} = \mathcal{K}_{1}\bigoplus_{}^{}\mathcal{K}_{2}\bigoplus_{}^{}\mathcal{K}_{3}\) is an invariant subgroup (order 4) of \(\mathcal{C}_{4v}\) because it consists of three complete classes. By considering how the members of \(\mathcal{C}_{4v}\) multiply with each other, then \(\mathcal{C}_{4}\) can be factored out of \(\mathcal{C}_{4v}\) as follows:

\[\begin{align*} \mathcal{C}_{4v} &= \left\{ 1,4_{z},2_{z},4_{z}^{3} \right\}\ \bigoplus_{}^{}{\ \left\{ m_{x},m_{x + y},m_{y},m_{x - y} \right\}} = \left\{ 1,4_{z},2_{z},4_{z}^{3} \right\}\ \bigoplus_{}^{}{\ \left\{ m_{x} \times \left\{ 1,4_{z},2_{z},4_{z}^{3} \right\} \right\}}\\ &= \left\{ 1,m_{x} \right\}\ \bigotimes_{}^{}{\ \left\{ 1,4_{z},2_{z},4_{z}^{3} \right\}} = \mathcal{C}_{s}\bigotimes_{}^{}\mathcal{C}_{4}. \end{align*} \]

So, according to the first line, \(\mathcal{C}_{4v}\) is the sum of 2 (= 8/4) cosets with respect to the subgroup \(\mathcal{C}_{4}\). We selected \(m_{x}\) as the coset representative in the factoring, but any one of \(m_{x},m_{y},m_{x + y},m_{x - y}\) would serve that purpose. Therefore, the second line shows that \(\mathcal{C}_{4v}\) is also the product of two subgroups \(\mathcal{C}_{s}\) and \(\mathcal{C}_{4}\). However, \(\mathcal{C}_{s}\) is not an invariant subgroup of \(\mathcal{C}_{4v}\) because \(m_{x}\) is not the only member of class \(\mathcal{K}_{4}\). Accordingly, the factor group of \(\mathcal{C}_{4v}\) with respect to \(\mathcal{C}_{4}\) is

\[\mathcal{C}_{4v} / \mathcal{C}_{4} = \left\{ \left\{ 1,4_{z},2_{z},4_{z}^{3} \right\},\left\{ m_{x},m_{y},m_{x + y},m_{x - y} \right\} \right\}, \nonumber \]

which contains 2 members. The member \(\left\{ 1,4_{z},2_{z},4_{z}^{3} \right\} = \mathcal{C}_{4}\) is the identity, and this 2-member factor group is isomorphic with \(\mathcal{C}_{s}\).

Another invariant subgroup of \(\mathcal{C}_{4v}\) is \(\mathcal{C}_{2v} = \left\{ 1,2_{z},m_{x},m_{y} \right\} = \mathcal{K}_{1}\bigoplus_{}^{}\mathcal{K}_{3}\bigoplus_{}^{}\mathcal{K}_{4}\). In this case

\[\begin{align*} \mathcal{C}_{4v} &= \left\{ 1,2_{z},m_{x},m_{y} \right\}\ \bigoplus_{}^{}{\ \left\{ 4_{z},4_{z}^{3},m_{x + y},m_{x - y} \right\}} = \left\{ 1,2_{z},m_{x},m_{y} \right\}\ \bigoplus_{}^{}{\ \left\{ 4_{z} \times \left\{ 1,2_{z},m_{x},m_{y} \right\} \right\}} \\[4pt] &= \left\{ 1,4_{z} \right\}\ \bigotimes_{}^{}{\ \left\{ 1,2_{z},m_{x},m_{y} \right\}} = \left\{ 1,4_{z} \right\}\bigotimes_{}^{}\mathcal{C}_{2v}.\end{align*} \]

As above, \(\mathcal{C}_{4v}\) is the sum of 2 (= 8/4) cosets. However, by choosing \(4_{z}\) as a coset representative, \(\mathcal{C}_{4v}\) is the product of the set \(\left\{ 1,4_{z} \right\}\), which is not a group, and the invariant subgroup \(\mathcal{C}_{2v}\). The factor group of \(\mathcal{C}_{4v}\) with respect to \(\mathcal{C}_{2v}\) is

\[\mathcal{C}_{4v} / \mathcal{C}_{2v} = \left\{ \left\{ 1,2_{z},m_{x},m_{y} \right\},\left\{ 4_{z},4_{z}^{3},m_{x + y},m_{x - y} \right\} \right\}, \nonumber \]

and contains 2 members. The member \(\left\{ 1,2_{z},m_{x},m_{y} \right\} = \mathcal{C}_{2v}\) is the identity. This 2-member factor group is also isomorphic with the group \(\mathcal{C}_{s}\).

This example summarizes important features of factor groups that are relevant for space groups:

Example \(\PageIndex{1}\)

For a space group \(\mathcal{G}\) (order \(hN\)) with the lattice group \(\mathcal{L}\) (order \(N\)) as an invariant subgroup, then

1. The factor group \(\mathcal{G/L}\) is the set of the \(h (= hN/N)\) different cosets of \(\mathcal{G}\) with respect to \(\mathcal{L}\);
2. The identity member of the factor group \(\mathcal{G/L}\) is the set \(\mathcal{L}\);
3. The subset \(\mathcal{R}\) of \(\mathcal{G}\), such that \(\mathcal{G = R\ \bigotimes\ L}\), consists of h coset representatives of \(\mathcal{G}\) with respect to \(\mathcal{L}\);
4. The subset \(\mathcal{R}\) of \(\mathcal{G}\) may or may not be a group, but it always contains the identity \(\left( 1 \middle| \mathbf{0} \right)\).

As mentioned above, the factor group \(\mathcal{G/L}\) of space group \(\mathcal{G}\) is isomorphic with one of the crystallographic point groups \(\mathcal{R}_{0}\) because they have the same abstract group multiplication table. Therefore, \(\mathcal{R}_{0}\) is called the point group of the space group \(\mathcal{G}\) and it describes the spatial characteristics of all vector or tensor properties of a crystal with space group \(\mathcal{G}\). Examples of such properties include crystalline shapes, electrical resistivities, and magnetic susceptibilities.

Also, characteristics of the set \(\mathcal{R}\) are important when considering irreducible representations of a space group. If \(\mathcal{R}\) is a group, then \(\boldsymbol{\tau}_{i} = \boldsymbol{0}\) for every member \(\left( R_{i} \middle| \boldsymbol{\tau}_{i} \right)\) of \(\mathcal{R}\), i.e., \(\mathcal{R =}\left\{ \left( 1 \middle| \boldsymbol{0} \right),\left( R_{2} \middle| \boldsymbol{0} \right),\ldots,\left( R_{h} \middle| \boldsymbol{0} \right) \right\}\). Therefore, \(\mathcal{R}\) is one of the 32 crystallographic point groups, and the corresponding space group is called symmorphic. On the other hand, if \(\mathcal{R}\) is not a group, then the displacement component \(\boldsymbol{\tau}_{i} \neq \boldsymbol{0}\) for some \(\left( R_{i} \middle| \boldsymbol{\tau}_{i} \right)\). Such space groups are called nonsymmorphic. Of the 230 3-d space groups, 73 are symmorphic and 157 are nonsymmorphic. Among the 17 2-d space groups, also called plane groups, 13 are symmorphic and 4 are nonsymmorphic.

Space Group Notation

The International symbolism is preferred over the Schönflies notation because it conveys information about how symmetry operations are oriented with respect to the unit cell axes. Every space group symbol is based on the product \(\mathcal{R\ \bigotimes\ L}\) because it consists of two parts: (1) a lattice designator; and (2) a point group or point group-like symbol that indicates the essential symmetry operations. For 3-d space groups, the lattice designators include primitive (P), base-centered (A, B, or C), body-centered (I), face-centered (F), or rhombohedral (R), and the point group or point group-like symbol is derived from one of the 32 crystallographic point groups. For symmorphic groups, such as \(C2/m\), \(I4/mmm\), and \(Fm\overline{3}m\), the symbols show only proper rotations, improper rotations, or reflections. For nonsymmorphic groups, such as \(P2_{1}/c\), \(P6_{3}/mmc\), and \(Fd\overline{3}m\), the symbol explicitly includes screw rotations and/or glide planes. From every space group symbol, three important characteristics are evident:

1. Point Group of the Space Group: For a symmorphic space group, it is the symbol for \(\mathcal{R}\); for a nonsymmorphic space group, it is the symbol arising by converting all screw rotations \(n_{j}\) to rotations \(n\) and all glide reflection \((a,b,c,n,or\ d)\) to reflections \(m\) in the symbol for \(\mathcal{R}\).
2. Crystal Class: Determined by the point group of the space group.
3. Lattice Type: Given by the lattice designator.

Plane Groups (Space Groups in 2-d)

There are five possible rotational symmetries: 1-, 2-, 3-, 4-, and 6-fold with their axes perpendicular to the plane, but there are no screw rotations and no improper rotations \(\overline{n}\), because inversion is the 2-fold rotation. There can also be vertical reflection planes (lines) \(m\) and a single type of glide reflection, designated by \(g\) (there are no diagonal glides in 2-d). Translational periodicity is specified by either primitive p or centered rectangular c lattices. Combining the 11 possible 2-d point symmetries with these lattices generates five crystal classes and the following 17 plane groups:

Crystal Class	Space Group	Comments
Oblique	p1 p2	Asymmetric 2-fold rotation is also the inversion in 2-d (holohedral)
Rectangular	pm pg p2mm p2mg p2gg cm c2mm	No inversion center but reflection symmetry Nonsymmorphic; no inversion center, but glide reflections Full lattice symmetry (holohedral)     Nonsymmorphic; reflections do not intersect 2-fold axis; equivalent to p2gm Nonsymmorphic; glide reflections do not intersect 2-fold axis Equivalent to “cg” because c-centering generates glide reflections Equivalent to “c2mg” and “c2gg”
Trigonal	p3 p3m1 p31m	3-fold rotation; no inversion center Reflections perpendicular to a-, b-, a+b-axes Reflections perpendicular to 2a+b-, a+2b-, a–b-directions
Tetragonal	p4 p4mm p4gm	4-fold rotation Full lattice symmetry (holohedral) Nonsymmorphic; glide reflections perpendicular to a-, b-axes
Hexagonal	p6 p6mm	6-fold rotation Full lattice symmetry (holohedral)

Plane groups p2mg and p2gm are equivalent because the assignment of a- and b-axes in the rectangular system is arbitrary from the perspective of abstract mathematical groups.

 

Example \(\PageIndex{1}\)

Determine the 2-d space group symbol for graphene, which consists of regular hexagons of three-connected carbon atoms fused along every edge to create a honeycomb network (shown to the right).

Figure 3.10: Structure of graphene.

Solution

The rotational and translational symmetries of a crystal are closely related. For 2-d crystals, it is often easiest to identify the crystal system first, then the lattice, and then the set of essential symmetry operations.

Crystal System and Lattice

Figure 3.11: Structure of graphene with one unit cell and 6-fold axis noted.

Because 6-fold rotational symmetry occurs at the centers of each hexagon, the system is hexagonal. As a result, the unit cell is the rhombus with a = b, γ = 120°, highlighted in yellow.

Essential Symmetry Operations

Figure 3.12: Structure of graphene with one unit cell, 6-fold axis, and 6 vertical mirror planes noted.

The 6-fold axis intersects each lattice point, so this operation is \(\left( 6 \middle| \mathbf{0} \right)\). As a result, five additional operations are: \(\left( 6 \middle| \mathbf{0} \right)^{2} = \left( 3 \middle| \mathbf{0} \right)\), \(\left( 6 \middle| \mathbf{0} \right)^{3} = \left( 2 \middle| \mathbf{0} \right)\), \(\left( 6 \middle| \mathbf{0} \right)^{4} = \left( 3^{2} \middle| \mathbf{0} \right)\), \(\left( 6 \middle| \mathbf{0} \right)^{5} = \left( 6^{5} \middle| \mathbf{0} \right)\), and \(\left( 6 \middle| \mathbf{0} \right)^{6} = \left( 1 \middle| \mathbf{0} \right)\).

There is also reflection symmetry and these vertical planes (lines) intersect lattice points. There are 3 reflections perpendicular to a-, b-, and a+b-directions (blue; parallel to C–C bonds): \(\left( m_{\boldsymbol{a}} \middle| \mathbf{0} \right)\), \(\left( m_{\boldsymbol{b}} \middle| \mathbf{0} \right)\), \(\left( m_{\boldsymbol{a} + \boldsymbol{b}} \middle| \mathbf{0} \right)\); and 3 reflections perpendicular to 2a+b-, a+2b-, and a–b-directions (green; bisecting C–C bonds): \(\left( m_{2\boldsymbol{a} + \boldsymbol{b}} \middle| \mathbf{0} \right)\), \(\left( m_{\boldsymbol{a} + 2\boldsymbol{b}} \middle| \mathbf{0} \right)\), \(\left( m_{\boldsymbol{a} - \boldsymbol{b}} \middle| \mathbf{0} \right)\).

Therefore, the set of essential symmetry operations contains 12 operations

\[\mathcal{R =}\begin{Bmatrix} \left( 1 \middle| \mathbf{0} \right),\left( 6 \middle| \mathbf{0} \right),\left( 3 \middle| \mathbf{0} \right),\left( 2 \middle| \mathbf{0} \right),\left( 3^{2} \middle| \mathbf{0} \right),\left( 6^{5} \middle| \mathbf{0} \right), \\ \left( m_{\boldsymbol{a}} \middle| \mathbf{0} \right),\left( m_{\boldsymbol{b}} \middle| \mathbf{0} \right),\left( m_{\boldsymbol{a} + \boldsymbol{b}} \middle| \mathbf{0} \right),\left( m_{2\boldsymbol{a} + \boldsymbol{b}} \middle| \mathbf{0} \right),\left( m_{\boldsymbol{a} + 2\boldsymbol{b}} \middle| \mathbf{0} \right),\left( m_{\boldsymbol{a} - \boldsymbol{b}} \middle| \mathbf{0} \right) \\ \end{Bmatrix} \nonumber \]

and is the point group \(6mm = \mathcal{C}_{6v}\). The Bravais lattice set is:

\[\mathcal{L =}\left\{ \left( 1 \middle| n_{1}\boldsymbol{a} + n_{2}\boldsymbol{b} \right):\ n_{1},n_{2} = \text{integers};a = b,\ \gamma = 120{^\circ} \right\},\]

which is a primitive (p) lattice. The space group is \(\mathcal{G = R}\bigotimes_{}^{}\mathcal{L} = \boldsymbol{p}\boldsymbol{6mm}\) and is symmorphic. To verify that space group \(p6mm\) is the product of the point group \(6mm\mathcal{= R}\) and the lattice \(\mathcal{L}\), determine the rotation-translation operations for positions (1)-(5) in graphene. These sites are not at the origin point, so we must apply the approach \(\left( R \middle| \boldsymbol{\tau} \right) = \left( 1 \middle| \boldsymbol{t} \right)\left( R \middle| \boldsymbol{\tau}_{R} \right)\left( 1 \middle| \boldsymbol{-t} \right)\):

Figure 3.13: Structure of graphene with 5 rotation axes noted.

(1) \(\left( 6 \middle| \boldsymbol{\tau}_{\mathbf{1}} \right)=\left( 1 \middle| \boldsymbol{a} \right)\left( 6 \middle| \mathbf{0} \right)\left( 1 \middle| \boldsymbol{-a} \right) = \left( 1 \middle| \boldsymbol{a} \right)\left( 6 \middle| \boldsymbol{-a} - \boldsymbol{b} \right)\)
\(\left( 6 \middle| \boldsymbol{\tau}_{\mathbf{1}} \right)=\left( 6 \middle| \boldsymbol{-b} \right) = \left( 1 \middle| \boldsymbol{-b} \right)\left( 6 \middle| \mathbf{0} \right)\)

(2) \(\left( 6 \middle| \boldsymbol{\tau}_{\mathbf{2}} \right)=\left( 1 \middle| \boldsymbol{a} + \boldsymbol{b} \right)\left( 6 \middle| \mathbf{0} \right)\left( 1 \middle| - \boldsymbol{a} - \boldsymbol{b} \right)\)
\(\left( 6 \middle| \boldsymbol{\tau}_{\mathbf{2}} \right)=\left( 1 \middle| \boldsymbol{a} + \boldsymbol{b} \right)\left( 6 \middle| \boldsymbol{-b} \right)=\left( 6 \middle| \boldsymbol{a} \right) = \left( 1 \middle| \boldsymbol{a} \right)\left( 6 \middle| \mathbf{0} \right)\)

(3) \(\left( 6 \middle| \boldsymbol{\tau}_{\mathbf{3}} \right)=\left( 1 \middle| \boldsymbol{b} \right)\left( 6 \middle| \mathbf{0} \right)\left( 1 \middle| \boldsymbol{-b} \right) = \left( 1 \middle| \boldsymbol{b} \right)\left( 6 \middle| \boldsymbol{a} \right)\)
\(\left( 6 \middle| \boldsymbol{\tau}_{\mathbf{3}} \right)=\left( 6 \middle| \boldsymbol{a} + \boldsymbol{b} \right) = \left( 1 \middle| \boldsymbol{a} + \boldsymbol{b} \right)\left( 6 \middle| \mathbf{0} \right)\)

(4) \(\left( 3 \middle| \boldsymbol{\tau}_{\mathbf{4}} \right)=\left( 1 \middle| ⅔\boldsymbol{a} + ⅓\boldsymbol{b} \right)\left( 3 \middle| \mathbf{0} \right)\left( 1 \middle| - ⅔\boldsymbol{a} - ⅓\boldsymbol{b} \right)\)
\(\left( 3 \middle| \boldsymbol{\tau}_{\mathbf{4}} \right)=\left( 3 \middle| \boldsymbol{a} \right) = \left( 1 \middle| \boldsymbol{a} \right)\left( 3 \middle| \mathbf{0} \right)\)

(5) \(\left( 3 \middle| \boldsymbol{\tau}_{\mathbf{5}} \right)=\left( 1 \middle| ⅓\boldsymbol{a} + ⅔\boldsymbol{b} \right)\left( 3 \middle| \mathbf{0} \right)\left( 1 \middle| - ⅓\boldsymbol{a} - ⅔\boldsymbol{b} \right)\)
\(\left( 3 \middle| \boldsymbol{\tau}_{\mathbf{5}} \right)=\left( 3 \middle| \boldsymbol{a} + \boldsymbol{b} \right) = \left( 1 \middle| \boldsymbol{a} + \boldsymbol{b} \right)\left( 3 \middle| \mathbf{0} \right)\)

Each of these operations is the product of a Bravais lattice vector \(\left( 1 \middle| n_{1}\boldsymbol{a} + n_{2}\boldsymbol{b} \right)\) and an essential symmetry operation \(\left( R \middle| \mathbf{0} \right)\).