Space Groups: Symmorphic vs. Non-Symmorphic

Defining the Space Group

A space group $G$ is the group of all isometries that map a crystal structure onto itself. An element of a space group, $g\in G$, can be represented by a pair $(A|\vec{u})$, where $A$ is a $3\times 3$ orthogonal matrix representing a rotation or reflection, and $\vec{u}$ is a translation vector. The action of this operator on a point $\vec{r}$ in space is defined as:

This notation, known as Seitz notation, makes the composition of two such operations clear. Given your background in linear algebra and group theory, you can verify that the group multiplication law is:

This shows that space groups are not, in general, simple direct products of their translational and rotational parts due to the $B\vec{u}$ term.

The set of all pure lattice translations, $(I|\vec{R})$, where $\vec{R}$ is a Bravais lattice vector, forms a subgroup $T$ of the space group $G$. As we saw in the previous module, this subgroup is abelian. It is also a normal subgroup, meaning that for any $g\in G$ and $t\in T$, the conjugate element $gt{g}^{-1}$ is also in $T$. The set of all rotational parts $\{A\}$ for a given space group forms the point group $P$, which is isomorphic to the quotient group $G/T$.

Symmorphic Space Groups: The Semidirect Product

The simplest way to combine translations and point operations is to form a symmorphic space group. There are 73 such groups. A space group is defined as symmorphic if there exists a point in the crystal (which we can choose as the origin) such that the site symmetry group at that point is isomorphic to the crystal's point group $P$.

This has a more direct group-theoretic consequence: a space group $G$ is symmorphic if it contains a subgroup isomorphic to its point group $P$. In our Seitz notation, this means that for every operation $A$ in the point group $P$, the element $(A|\vec{0})$ (a pure rotation or reflection about the origin) is itself an element of the space group $G$.

In this case, any element of the space group can be written as a product of a pure lattice translation and a pure point operation about the common origin:

This structure is known as a semidirect product, denoted $G=T⋊P$. The symmorphic space groups represent the straightforward combination of one of the 14 Bravais lattices with one of the 32 compatible crystallographic point groups.

Non-Symmorphic Space Groups: Screw Axes and Glide Planes

The majority of space groups (157 out of 230) are non-symmorphic. In these groups, it is impossible to find an origin such that all point group operations can be performed without an accompanying fractional translation.

For a non-symmorphic group, there is at least one point operation $F\in P$ such that $(F|\vec{0})$ is not a symmetry of the crystal. Instead, the corresponding symmetry operation in the space group must take the form $(F|\vec{u})$, where $\vec{u}$ is a translation vector that is not a full Bravais lattice vector. These composite operations are the source of new symmetry elements that do not exist in point groups:

• Screw Axis: A rotation followed by a fractional translation parallel to the axis of rotation. For example, a $2_1$ screw axis represents a 180° rotation followed by a translation of half a lattice vector along the axis.
• Glide Plane: A reflection across a plane followed by a fractional translation parallel to that plane.

It is crucial to understand that the mere presence of screw axes or glide planes does not automatically make a space group non-symmorphic. A symmorphic group can generate such operations through the combination of its fundamental symmetries (pure point operations and pure lattice translations). A space group is truly non-symmorphic only if these fractional translations are intrinsic and cannot be eliminated by any choice of origin.

Deconstructing a Non-Symmorphic Space Group: The P2${}_1$/c Example

To make these ideas concrete, let's examine the structure of a common non-symmorphic space group, P2${}_1$/c (monoclinic, space group No. 14). This is one of the most frequently observed space groups in molecular crystals.

The Hermann-Mauguin notation "P2${}_1$/c" tells us:

• P: The Bravais lattice is Primitive.
• 2${}_1$/c: This describes the symmetry elements. By convention in the monoclinic system, this refers to the unique $b$-axis.
  • 2${}_1$: There is a two-fold screw axis parallel to the $b$-axis.
  • c: There is a c-glide plane perpendicular to the $b$-axis (i.e., the ac-plane). The translation component is along the $c$-axis.
  • /: The slash indicates that the glide plane is perpendicular to the screw axis.

The point group is found by "removing" the translational components of the symmetry elements: $2_1\to 2$ and $c\to m$. Thus, the point group for P2${}_1$/c is 2/m.

In this space group, neither a pure two-fold rotation about the $b$-axis nor a pure reflection in the $ac$-plane are symmetry operations on their own. They must be combined with fractional translations to map the crystal onto itself. This is the essence of a non-symmorphic space group. Furthermore, combinations of these non-symmorphic elements generate other symmetries, like inversion centers, which are displaced from the origin.

Conclusion

In this lesson, we have moved beyond simple translational symmetry to embrace the full symmetry of crystalline solids as described by space groups. You now have the tools to define and distinguish between the two fundamental classes of space groups.

Key Takeaways:

• Space Group: The group of all isometries (translations, rotations, reflections, and their combinations) that leave a crystal lattice invariant. Elements can be written as $(A|\vec{u})$.
• Symmorphic Space Group (73 total): A space group that contains its point group as a subgroup. It can be described as a semidirect product of the translation group and the point group, $G=T⋊P$. All point operations can be performed about a common origin.
• Non-Symmorphic Space Group (157 total): A space group in which at least one point operation must be combined with a fractional (non-lattice) translation to be a symmetry. This is true regardless of the choice of origin. These groups contain intrinsic screw axes and/or glide planes.

With this complete description of crystal symmetry, we are now positioned to explore its consequences. In the next lesson, we will investigate how the full space group symmetry, particularly at high-symmetry points in the Brillouin zone, governs the classification and degeneracies of electronic band structures.