## CHAPTER 2. Symmetry and Groups, and Crystal Structures

#### 2.1. The Seven Crystal Systems

The unit cell. The unit cell of a mineral is the smallest divisible unit of a mineral that possesses the symmetry and properties of the mineral. It is a small group of atoms, from four to as many as 1000, that have a fixed geometry relative to one another. The atoms are arranged in a "box" with parallel sides called the unit cell which is repeated by simple translations to make up the crystal. The atoms may be at the corners, on the edges, on the faces, or wholly enclosed in the box, and each cell in the crystal is identical. This is what was meant by an "ordered internal arrangement" in our definition of a mineral. It is the reason why crystals have such nice faces, cleavages, and regular properties.

The box of the unit cell is, in general, a parallel-piped with no constraints on the lengths of the axes or the angles between the axes. The box is defined by three axes or cell edges, termed a, b, and c and three inter-axial angles alpha, beta, and gamma, such that alpha is the angle between b and c, beta between a and c, and gamma between a and b. The presence of internal symmetry in the unit cell may place constraints on the geometry of the unit cell. The different kinds of symmetry possible place different constraints on the unit cell geome tries giving rise to characteristic cell geometries for each of the seven Crystal Systems. These are outlined in Table 2.1.

Table 2.1 Unit Cell Axial Constraints, and Allowed Symmetry Operations of the Seven Crystal Systems.
```System		Constraints			Operations

Triclinic	None				1, -1

Monoclinic	alpha = gamma = 90		1, -1, 2, -2(m)

Orthorhombic	alpha = beta = gamma = 90	1, -1, 2, -2(m)

Trigonal	alpha = beta = 90 gamma = 120 	1, -1, 2, -2(m), 3, -3
a = b

Hexagonal	alpha = beta = 90 gamma = 120 	1, -1, 2, -2(m), 3, -3, 6, -6
a = b

Tetragonal	alpha = beta = gamma = 90	1. -1, 2, -2(m), 4, -4
a = b

Cubic		alpha = beta = gamma = 90	1. -1, 2, -2(m), 3, -3, 4, -4
a = b = c

```

#### 2.2. Symmetry Operations

A symmetry operation is a transposition of an object These may be of three distinct types: rotations, inversions (including roto-inversions i.e. improper rotations), or translations, or combinations thereof. We will discuss symmetry groups made up of rotation and inversion operations only which are called the point groups, each of which is one of the 32 crystal classes. We will also discuss the groups made up from all three types of opera tion which give rise to the 230 space groups.

#### 2.2.2. Rotations

Permissible rotations - Proper
```1-fold	360 º 	I	Identity

2-fold	180 º 	2

3-fold	120 º 	3

4-fold	 90 º 	4

6-fold	 60 º 	6	```

Permissible rotations - Improper (result in enantiomorphs).
```1-fold	360 º  + i	i

2-fold	180 º  + i	-2 = m

3-fold	120 º  + i	-3

4-fold	 90 º  + i	-4

6-fold	 60 º  + i	-6	```

#### 2.2.3. Translations

Permissible translations are unit cell translations or fractions thereof that are consistent with the rotational symmetry (e.g. 1/2, 1/3, 1/4, and 1/6), plus combinations.

#### 2.3. Stereographic Projections.

There are three common methods to graphically display orientations of vectors in three dimen sions. In order to describe faces of crystals we would like to plot the vectors normal (perpendicular) to crystal faces.

2.3.1. Spherical. The simplest to visualize is the spherical projection. In this method, the vector is merely projected vertically onto the equatorial plane of the sphere. However, this method has the serious disadvantage of compressing the low-angle vectors onto the outside of the plot.

2.3.2. Stereographic. This disadvantage can be avoided if, instead of projecting vertically, one projects radially to the pole of the opposite hemisphere. This is the standard stereographic or equal-area plot that we will use to plot poles (perpendiculars) to faces of crystals. The plot is also sometimes called the "Wulff net". If the angle made by a vector with the vertical is r, then the distance from the center of the plot is R tan r/2, where R is the radius. 2.3.3. Gnomonic. There is also a third type of plot called the gnomonic projection in which the vector is extended till it intersects a plane tangent to the sphere at the north pole. This has the disadvantage of not being able to display horizontal vectors. It is, however, what arises naturally from x-ray and electron diffraction experiments where each node corresponds to a lattice plane in real space and results in a reciprocal lattice.

However, for our purposes of displaying the orientations of crystal faces we will use the stereo graphic projection exclusively.

#### 2.4. Allowable Rotations.

Illustrated below are stereographic projections of general crystal forms that have the allowable rotation operations that are consistent with translation symmetry.

Proper Rotations: Improper Rotations: Each of these ten allowable rotations generates, by itself, a unique point group. In addition, there are 22 possible combinations of rotation operations, giving a total of 32 possible 3-dimensional point groups. Each point group corresponds to different crystal class. Each crystal class places constraints on the axial geometry such that each of these 32 classes may be associated into one of the 7 crystal systems, each having different constraints on the axial lengths and inter-axial angles.

#### 2.5. GROUPS

A set of elements (operations) is a group if the following properties hold:

1. Closure: combining any two elements of the group gives a third element of the group.

2. Association: For any three elements of the group (ab)c = a(bc). Note: not necessarily commutative (ab = ba). If it is true for all members of the group, the group is called Abelian.

3. Identity: There is an element of the group, I, such that aI = Ia = a for each element of the group.

4. Inverses: For each element, a, there is another element, b, such that ab = I = ba.

#### 2.6. Crystal Morphology

At this point it is useful to develop some of the formalism of crystal morphology (shapes). The morphology of a perfect crystal (i.e., our wooden blocks), in general, reflects the maximum symme try that a crystal can have. That is, there may be portions of the crystal structure that violate some of the apparent symmetry, but if high-symmetry forms (crystal faces) are present, the crystal is likely to have high symmetry. (e. g., if the crystal is a cube, it is most probably isometric.)

A crystal form is a crystal face plus its symmetric equivalents. For example, a cube is a crystal form made up of six symmetrically equivalent faces.

A special form is a crystal form that is repeated by the symmetry operations onto itself so that there are fewer faces than the order of the point group. The projections of special forms or special faces will lie on symmetry operations in our stereographic projections.

A general form is one that is not repeated onto itself by the symmetry operations so that it has the same number of faces as the order of the group.

Forms are either general or special. In our stereographic projections, we will plot only the general form because this defines the point group. In addition to being special or general, forms may also be open or closed.

A closed form is one that encloses a volume; (e.g., a cube, tetrahedron, octahedron, etc). A closed form may then be the only form present on a perfect crystal.

An open form is one that does not enclose a volume; (e.g., prism, pinacoid, etc.). A crystal that has an open form must have more than one form present.

#### 2.7. Miller Indices 2.7.1. Planes A crystal face (or plane) cuts the crystallographic axes at , 2, and 1. These intersections are called intercepts. Because symbols are cumbersome, these intercepts are in verted and all fractions are cleared, as shown below. , 1/2, 1/1 = 0, 1/2, 1 = (0 1 2)

These operations give us the Miller indices of any plane. These planes may be a cleavage plane, a crystal face, or any diffracting X-ray plane. Thus, a cube face is (0 0 1), the octahedron (1 1 1), and a dodecahedron (1 1 0). There may also be negative (0 0 -1) Miller indices. Miller indices are always in relation to the crystallographic axes, not any orthogonal system of convenience. The general form for Miller indices is (h k l).

For hexagonal axes the general form is (h k i l). However, you will note in the example below that h + k + i = 0. This is always the case so the i index is superfluous. Hence, we can merely use (2 1 0). -1/2, 1, 1, 0 = (-2 1 1 0) = (-2 1 0) In general, crystal faces, diffracting X-ray planes,and cleavages will be denoted with simple parentheses, e.g. (2 1 0). However, a crystal form (a face plus its symmetric equivalents will be denoted with curly brackets, e.g. {2 1 0}. Hence the cube, {1 0 0} is made up of faces (1 0 0), (0 1 0), (0 0 1), (-1 0 0), ( 0 -1 0), and (0 0 -1).

2.7.2. Directions