3.8: Group-Subgroup Relationships

A useful section for each space group in the International Tables lists certain subgroups and supergroups, which can be helpful for studying solid-solid phase transitions. A set \(\mathcal{S}\) is a subgroup of \(\mathcal{G}\) if all members of \(\mathcal{S}\) are contained in \(\mathcal{G}\). \(\mathcal{S}\) is a proper subgroup if \(\mathcal{G}\) contains members that are not in \(\mathcal{S}\). \(\mathcal{S}\) is a maximal subgroup of \(\mathcal{G}\) if there are no other subgroups of \(\mathcal{G}\) for which \(\mathcal{S}\) is also a proper subgroup. This often occurs by removing inversion, a reflection, or a C₂ rotation from \(\mathcal{G}\) so that most maximal subgroups \(\mathcal{S}\) have one-half the number of operations of the original group \(\mathcal{G}\). Among point groups, consider \(\mathcal{D}_{4h}\) with order 16 consisting of 8 proper and 8 improper rotations. Each of the 4 groups in the second row has order 8 and is a maximal subgroup of \(\mathcal{D}_{4h}\). Are there any other maximal subgroups of \(\mathcal{D}_{4h}\)?

@ >p(- 6) * >p(- 6) * >p(- 6) * >p(- 6) * @

()

\(\mathcal{D}_{\mathbf{4}\boldsymbol{h}}\):

&

Figure 3.38

&
() ()

\(\mathcal{D}_{\mathbf{4}\boldsymbol{h}}\):

&

Figure 3.38

&
() \(\mathcal{C}_{\mathbf{4}\mathbf{h}}\) & \(\mathcal{D}_{\mathbf{4}}\) & \(\mathcal{C}_{\mathbf{4}\mathbf{v}}\) & \(\mathcal{D}_{\mathbf{2}\mathbf{d}}\)

Figure 3.39 &

Figure 3.40 &

Figure 3.41 &

Figure 3.42
\(\small{\begin{Bmatrix} \boldsymbol{E} & \boldsymbol{C}_{\mathbf{4}\boldsymbol{z}} \\ \boldsymbol{C}_{\mathbf{4}\boldsymbol{z}}^{\mathbf{2}} & \boldsymbol{C}_{\mathbf{4}\boldsymbol{z}}^{\mathbf{3}} \\ \boldsymbol{i} & \boldsymbol{\sigma}_{\boldsymbol{z}} \\ \boldsymbol{S}_{\mathbf{4}\boldsymbol{z}} & \boldsymbol{S}_{\mathbf{4}\boldsymbol{z}}^{\mathbf{3}} \\ \end{Bmatrix}}\) & \(\small{\begin{Bmatrix} \boldsymbol{E} & \boldsymbol{C}_{\mathbf{4}\boldsymbol{z}} \\ \boldsymbol{C}_{\mathbf{4}\boldsymbol{z}}^{\mathbf{2}} & \boldsymbol{C}_{\mathbf{4}\boldsymbol{z}}^{\mathbf{3}} \\ \boldsymbol{C}_{\mathbf{2}\boldsymbol{x}} & \boldsymbol{C}_{\mathbf{2}\boldsymbol{y}} \\ \boldsymbol{C}_{\mathbf{2}\left( \boldsymbol{x + y} \right)} & \boldsymbol{C}_{\mathbf{2}\left( \boldsymbol{x - y} \right)} \\ \end{Bmatrix}}\) & \(\small{\begin{Bmatrix} \boldsymbol{E} & \boldsymbol{C}_{\mathbf{4}\boldsymbol{z}} \\ \boldsymbol{C}_{\mathbf{4}\boldsymbol{z}}^{\mathbf{2}} & \boldsymbol{C}_{\mathbf{4}\boldsymbol{z}}^{\mathbf{3}} \\ \boldsymbol{\sigma}_{\boldsymbol{x}} & \boldsymbol{\sigma}_{\boldsymbol{y}} \\ \boldsymbol{\sigma}_{\left( \boldsymbol{x + y} \right)} & \boldsymbol{\sigma}_{\left( \boldsymbol{x - y} \right)} \\ \end{Bmatrix}}\) & \(\small{\begin{Bmatrix} \boldsymbol{E} & \boldsymbol{C}_{\mathbf{4}\boldsymbol{z}}^{\mathbf{2}} \\ \boldsymbol{C}_{\mathbf{2}\boldsymbol{x}} & \boldsymbol{C}_{\mathbf{2}\boldsymbol{y}} \\ \boldsymbol{S}_{\mathbf{4}\boldsymbol{z}} & \boldsymbol{S}_{\mathbf{4}\boldsymbol{z}}^{\mathbf{3}} \\ \boldsymbol{\sigma}_{\left( \boldsymbol{x + y} \right)} & \boldsymbol{\sigma}_{\left( \boldsymbol{x - y} \right)} \\ \end{Bmatrix}}\)
order = 8 & order = 8 & order = 8 & order = 8
()

Subgroups of space groups arise by losing translations or rotations, leading to two major types of maximal subgroups:

Type I: Translation equivalent

Type I: Translation equivalent (“Translationengleiche”) subgroups retain all translations, but they lose some rotational symmetry. Changing the point group may or may not retain the crystal system. One example is CaCl₂, which adopts a high-temperature form with space group \(P4_{2}/mnm\) and a low-temperature form with space group \(Pnnm\). In the low-temperature form, the 4₂-screw and reflections \(..m\) are lost, but the glide planes \(.n.\) and reflections \(/m\) perpendicular to the original 4₂-axis are retained. As a result, the point group of the space group drops from \(\mathcal{D}_{4h}\) (order = 16) to \(\mathcal{D}_{2h}\) (order = 8).

Figure 3.43

Type II: Class equivalent

Type II: Class equivalent (“klassengleiche”) subgroups preserve the point group of the space group, but they lose certain lattice translations. There are three subdivisions of this type:

1. If a space group involves a centered lattice, then this subgroup arises by removing all lattice centering translations. One example occurs during the order-disorder transition of β-CuZn. The high-temperature form is completely disordered Cu and Zn in a BCC-packing, space group \(Im\overline{3}m\), whereas the low-temperature form exhibits ordering of Cu and Zn atoms, observed by neutron diffraction, space group \(Pm\overline{3}m\).

Figure 3.44

2. This subgroup differs from the original space group and the number of atoms in the unit cell of the subgroup is an integral multiple of those in the unit cell of the original space group. For example, YbGa₂ at ambient pressure is hexagonal \(P6_{3}/mmc\). On increasing pressure, it transforms to a structure in which the c-axis has approximately halved, space group \(P6/mmm\). Both space groups have the same point group \(\mathcal{D}_{6h} = 6/mmm\), but the high-pressure form has twice as many lattice translations as the ambient-pressure form.

Figure 3.45

Figure 3.46

1. The space group and its subgroup are the same group, but some translations of the original group have been lost. The space group \(P4_{2}/mnm\) has only Type IIc klassengleiche subgroups, an example of which is the trirutile structure of WV₂O₆ in which only threefold multiples of \(\mathbf{c}\) of the rutile structure are retained. Ordering of W and V atoms create the subgroup.

Solid-solid phase transitions often involve both translation equivalent and class equivalent components simultaneously. These kinds of phase transitions may follow different possible pathways and belong to first-order phase transitions. On the other hand, continuous symmetry-breaking phase transitions, often called second-order, may be analyzed using Landau theory and belong to a single irreducible representation of the space group. Such transitions occur according to a single pathway, by either losing translations (class equivalent) or rotations (translation equivalent).

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1. J.-L. Mi, et al., Chem. Mater. 2017, 29, 6378-6388.↩
2. G.B. Carpenter, Acta Crystallogr. 1952, 5, 132-135.↩
3. N.W. Alcock, Acta Crystallogr. Sect. A 1969, A25, S101-S101.↩
4. P.D. Battle, et al. J. Solid State Chem. 2002, 167, 188-195.↩