03 | Elements of symmetry

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The simplest and most direct way to describe the symmetry of an object is to list all of the ways we can transform the object as to leave it indistinguishable from its original reference state. Point symmetries are those symmetry operations that do not contain any translational symmetry. In Hermann–Mauguin notation[1] every point symmetry operation can be describe as a form of rotation or rotary inversion (rotation + inversion).

The identity operation

The simplest symmetry elements is the identity operation, symbolized as \(1\) in Hermann–Mauguin notation. It can be thought of as a 1-fold or 0° rotation and it is the transformation that does nothing to an object. All objects, compounds, and materials have at least one symmetry operation and it is the identity operation, \(\textbf{1}\).

Proper n-fold rotations

The \(\textbf{n}\)-fold rotations are the next simplest symmetry operations to visualize. These rotational symmetry operations are defined such that \(\textbf{n}\) is equivalent to the number of times the operation much be applied to complete a circle, that is sweep \(\textbf{n} = \frac{2\pi}{n}\; rad \). These n-fold symmetry operations act about a symmetry element called a rotation axis.

For example, shown below is a 90° (4-fold) rotation of a square for which the start and end positions are indistinguishable. In Hermann–Mauguin notation this symmetry operation is represented by the the symbol \(\textbf{4}\).

If two 4-fold rotations are applied sequentially the resulting operation is equivalent to applying a 2-fold rotation once. Of course, in Hermann–Mauguin notation a 2-fold rotation is represented with the symbol \(\textbf{2}\). Mathematically, the relationship between the 4-fold rotation and the 2-fold rotation can be represented as \(\textbf{4}^2 \equiv \textbf{2}\).

Improper rotations

The improper rotations behave as though a combination of a proper rotation about a rotation axis plus inversion through a point, a rotary inversion. While it is easy to visualize rotary inversion as a composition of multiple symmetry elements, these improper rotations are as fundamental a set of symmetry operations as are all the proper rotations. Their symbols are analogous to those of the proper rotation: \(\bar{1} \equiv i\), \(\bar{2} \equiv m\), \(\bar{3}\), \(\bar{4}\), and \(\bar{6}\). Two of these rotary inversions stand out.

Inversion

The \(\bar{1}\) operation is indistinguishable from an inversion center and is sometimes given the alternative symbol \(i\) for this reason. The symmetry operation of inversion negates the coordinates of an object. In illustrations an inversion center is shown as either a dot or open circle.

Reflection

Likewise, it can be easily shown that a 2-fold proper rotation followed by inversion—the \(\bar{2}\) operation—is equivalent to a reflection through the plane that is perpendicular to the axis of rotation and for this reason is sometime given the alternative symbol \(m\). The symmetry operation of reflection negates the coordinate along the axes of improper rotation. This is the same as the coordinate whose axes is perpendicular to the corresponding “plane of reflection”. While in Hermann–Mauguin notation, reflection is considered a rotary inversion, in illustrations, the reflection plane is diagramed as a solid line if it is perpendicular to the plane of the illustration as shown below for a square.

Higher order rotary inversions

In contrast the \(\bar{3}\), \(\bar{4}\), and \(\bar{6}\) are unique and not reducible to actions about simpler elements of symmetry.[2]

Illustrating symmetry elements

To illustrate symmetry element in an illustration unique glyphs are used for each element. These are summarized below.

H-M Symbol	Element Glyph	Description
\(1\)		identity operation
\(2\)	fi	2-fold (180°) rotation
\(3\)	3	3-fold (120°) rotation
\(4\)	4	4-fold (90°) rotation
\(6\)	6	6-fold (60°) rotation
\(\bar{1} \equiv i\)	o	inversion
\(\bar{2} \equiv m\)	—	reflection
\(\bar{3}\)	`	3-fold rotary inversion (120° rotation + inversion)
\(\bar{4}\)	T	4-fold rotary inversion (90° rotation + inversion)
\(\bar{6}\)	$	6-fold rotary inversion (60° rotation + inversion)

Reading and Additional Resources

1. Ladd, M. F. C.; Palmer, R. A. Structure determination by X-ray crystallography Chapt. 1.4: External Symmetry of Crystals. New York: Plenum Press, 1977, 23–43.

2. Visualizations of molecular symmetry by Dean Johnson at Otterbein University using Schönflies notation

Notes

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[1]

Throughout we will use Hermann–Mauguin notation. This is the preferred symmetry notation of crystallography, solid state physics, and materials chemistry. Here elements of symmetry are all elegantly described as either proper rotations or improper rotations(rotary inversions). The other common symmetry notation, Schönflies, is popular in molecular chemistry and spectroscopy. While some helpful resources are provided to show the relationships between these two notation systems only H-M notation will be use in this course.

[2]

However, rotary inversions can be reframed as rotary reflections as is done in Schönflies notation. Nonetheless the rotation axis plus point of inversion about which a rotary inversion acts is arguably simpler than the the rotation axis plus reflection plane required to describe a rotary reflection.