Point Symmetry in 2- and 3-D

Where are we going over the next 2 weeks?

We will begin our study of crystallography by examining the properties of symmetry operations that do not involve translation. In two dimensions we will examine rotations and reflections and combinations of these elements. We will see that there are 10 possible outcomes - 10 possible symmetry contents for planar motifs or patterns which can become part of ordered 2-D patterns - 10 planar point groups. Moving into three dimensions allows the introduction of inversion, rotoinversion, perpendicular mirrors and multiple axes of rotation combined at some special angles. There are 32 unique combinations in three dimensions and these are the 32 3-D point groups also called the 32 crystal classes. Every crystal has one of these 32 point groups as its underlying structure - if the crystal is able to form unhindered, its outward morphology (shape) should be identifiable as belonging to one of these 32 classes.

Because crystals have long-range 3-D order we need to identify the possible ways this internal order can be developed. As above we will start with 2-D translationally periodic patterns and then move to 3-D. There are five 2-D ways to repeat lattice points in space and these can be combined with the 10 planar point groups to generate 17 2-D space groups. There are 14 ways to translationally repeat points in 3-D and, when combined with the 32 3-D point groups, 230 3-D space groups result.

Specifically then:

[1] (2nd class - Friday (today)) Combinations of a single rotation axis perpendicular to the paper with or without parallel mirrors (perpendicular to the paper) give us the 10 2-D point groups.

[2] (3rd class - Monday) Add center of inversion, rotoinversion axes, perpendicular mirrors, and additional axes of rotation to the first 10 point groups gives the 32 3-D point groups or Crystal Classes.

[3] (4th class - Wednesday) Add crystallographic axes to the point groups, discuss their axial ratios and angular relationships, Miller Indices, and the ideas of forms.

[4] (5th class - Friday) Derive the 5 2-D plane lattices and explore the result of combining these lattices with the 10 2-D point groups: the 17 2-D plane or space groups.

[5] (6th class - Monday) Derive the 14 3-D (Bravais) lattices from the 5 2-D lattices and touch on the result of combining these with the 32 3-D point groups: the 230 3-D space groups. Here we will also add the glide plane and the screw axis to our symmetry "building blocks". Solving a crystal structure.

Quick Bonding Review

Before getting going here, lets quickly review the different bonding types we need to be concerned with in minerals (should have covered this on day one):

ionic: attraction between cations and anions due to their electric charges

covalent: sharing of unfilled (usually) d-orbitals

metallic: sea of electrons washing back and forth through the valence band of most metals

van der Waals: long range (and thus weak) bonds between plus and minus charges

hydrogen: equivalent in magnitude to van der Waals, specifically involving hydrogen

Symmetry operations/elements (without translation):

• Rotation: Rotation of a motif (unit of pattern) about an imaginary axis produces one or more motifs which are said to be "symmetrically equivalent" to the initial motif. Rotational symmetry is commonly expressed as a whole number from 1 to infinity where the number refers to the number of motifs in the finished pattern. We can draw or make single objects with rotational symmetries anywhere between the extremes of 1 and infinity. However if add the constraint that the pattern of motifs must also be translationally periodic (have an ordered pattern and completely fill 2- or 3-D space, then the only rotational symmetries permissible are 1, 2, 3, 4, and 6. These are expressed as 1-fold (360 degrees), 2-fold (180 degrees), 3-fold (120 degrees), 4-fold (90 degrees) and 6-fold (60 degrees) axes. Note that rotation does not change the motif's "handedness" (a pattern made by rotating a left hand about an axis will be composed of all left hands). The motifs are then said to be congruent as distinct from the pairs of motifs created by reflection. These are the first 5 planar point groups.

• Reflection: A reflection across a mirror (in 3-D) or a mirror line (in 2-D) generates a new motif that has the opposite handedness and the two are said to be "enantiomorphic". This is the 6th planar point group.

• Rotation axis combined with parallel mirror planes (or perpendicular mirror lines): The other four planar point groups are generated by adding a perpendicular mirror line to each of the 5 rotational groups and letting the combined symmetry generate a pattern. (The combination of a mirror parallel to a 1-fold axis is the same as the mirror by itself and thus we don't count it again.) Note in the cases of 2mm, 4mm and 6mm, the addition of a single mirror (the first 'm' in the symbol), when coupled with the rotation axis, generates another mirror or set of mirrors half way between the mirrors in the first set. When we define coordinate axes to define these symmetries in a week or so we will find that the first 'm' refers to the symmetry along the axes while the 2nd 'm' refers to the symmetry in the interaxial directions. In the case of 3m, additional interaxial mirrors are not generated.

• Inversion: Inversion through the center of an object produces an inverted object that is "facing the other way". Inversion thus produces an enantiomorphic pair of motifs.

• Rotoinversion: The combination of rotation and inversion produces several new point groups in three dimensions.

  1-fold plus inversion: this combination is the same as simply inverting through the center of the object and thus the 'bar 1' operation (see p.23) is also referred to as 'i' which stands for 'center of symmetry'.

  The 'bar 2', 'bar 3', 'bar 4' and 'bar 6' operations are depicted in Fig. 2.12 in your text. Remember that in any of these cases, the rotation is immediately followed by inversion before you stop to generate a new face on the crystal (even if there happens to be a face present prior to inversion). The 'bar 2' operation is equivalent to a mirror plane and thus doesn't add a new point group and this notation is not used. Also the 'bar 6' operation is equivalent to a 3-fold axis perpendicular to a mirror (see below) and thus some authors use one notation while others will use the other. The 'bar 2' produces 2 motifs, the 'bar 4' produces 4 motifs and 'bar 6' produces 6 motifs. The 'bar 3' operation however also produces 6 motifs and is equivalent to a 3-fold and a center of symmetry.

• Combinations of Rotations: Some combinations of symmetry axes are permissible - the angular relationships have to chosen carefully so as not to generate an infinite set of axes. The combination of a 4-fold axis with a perpendicular 2-fold generates a second 2-fold at 90 degrees to the first and then, by inspection, a second set of two 2-folds is observed half way between the first set. All these 2-fold axes lie in the equatorial plane perpendicular to the 4-fold. This construction is shown in Fig. 2.14 in the book and is called 422 - do you see the parallelism with the planar group 4mm? In addition to 422 we can derive 622, 222, 32, 432, and 23. See the book for details (p. 26-27).

• Rotations plus perpendicular Mirrors: Mirrors can be added perpendicular to the 2, 3, 4, and 6-fold axes and thus generate the 2/m, 3/m, 4/m and 6/m point groups. Similar mirrors can be added to several other of the point groups mentioned in the paragraph above and again generate new point groups. For instance 422 plus mirrors gives 4/m2/m2/m. Compare this to 4mm (which we deduced from analysis of 2-D plane patterns) - the distribution of symmetrically related points in these 3 closely related point groups is different (Fig. 2.18).

• 32 point groups: At this point we have constructed most of the 32 point groups. These are subdivided in Table 2.1 on the basis of increasing rotational symmetry. These 32 point groups are also known as the 32 crystal classes which have been grouped in Table 2.2 into 6 crystal systems based on the major or special rotational rotation axis. Finally, Fig. 2.21 shows all 32 of these crystal classes and their appropriate symmetries.

After the overview, during the rest of the 2nd lecture we derived the 2mm, 3m, 4mm and 6mm point groups. In the last 5 minutes we began to develop the concept of rotoinversion.

Remember the meaning of the 2 "m's" in each of the notations.