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.

In determining a point group, one must have diagnostic faces such as the general form. For example, if you have a cube, it can occur in several point groups as a special form. Thus there is no way to uniquely determine the point group. If (when) you encounter this situation in the lab, assign the highest point group symmetry (i.e. 4/m -3 2/m for a cube).

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.

The order of the group is the number of elements of the group. We will first consider groups made up of all allowable combinations of rotation and inversion operations to make up the point groups in two dimensions and in three dimensions. There are ten possible 2-dimensional point groups and 32 possible 3-dimensional point groups. Each of these 32 3-D point groups corresponds to one of the crystal classes. We will then combine these with the possible translation operations to form the 17 2-dimensional space groups and 230 3-D space groups.

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

Directions in a crystal are merely the vector components with respect to the crystallographic axes that have been reduced to the smallest whole numbers. These are given in square brackets [1 3 0], [0 1 0], etc. In general, the [1 1 1] is not normal to the (1 1 1), except for isometric (cubic) crystals.

2.8. The 32 Crystal Classes

Following the rules of groups, there is a limited number of ways in which the 10 proper and im proper rotations can be combined to form groups, that is, there are 32 possible combinations to form groups. These are the 32 3-dimensional point groups which correspond to the 32 Crystal Classes. Each of the 32 crystal classes can be ascribed to one of the 7 crystal systems. The various crystal classes are outlined in the table below. Table 2.2. The 32 Crystal Classes.
System		Classes (Point Group)

Triclinic 1, -1 Monoclinic 2, -2 (= m), 2/m Orthorhombic 222, 2mm, 2/m2/m2/m Trigonal 3, -3, 3m, 32, -32/m Hexagonal 6, -6(= 3/m), 6/m, 6mm, 622, -6m2, 6/m2/m2/m Tetragonal 4, -4, 4/m, 4mm, 422, -42/m, 4/m2/m2/m Cubic 23, 2/m-3, -43m, 432, 4/m-32/m

Each of the 10 allowed proper and improper rotations is, by itself, one of the 32 point groups, and we have seen stereographic projections of each of these. The additional 22 point groups are gener ated by combinations of these 10 symmetry operations. These are illustrated below.

_________________________________________________________________ Monoclinic

____________________________________________________________________

Orthorhombic

____________________________________________________________________

Tetragonal

____________________________________________________________________

Tetragonal

____________________________________________________________________

Trigonal

____________________________________________________________________ Hexagonal 6mm Hexagonal 6/m Hexagonal 622

____________________________________________________________________

Hexagonal

____________________________________________________________________

Cubic

____________________________________________________________________ Cubic

____________________________________________________________________

2.11. The 14 Three-Dimensional Bravais Lattices

1. Primitive Triclinic

C-centered are no constraints on axial lengths or on b. The standard choice of axes is that b is unique. The centered cell is also unique and preserves the axis identities and has an additional lattice point at 1/2+x, 1/2+y, z, so that there is an additional lattice point per cell for a total of two, whereas the primitive cell have just one lattice point per cell (eight corners, each of which is 1/8th within the cell). Because there are several possible choices of the a and b axes, it is possible to choose an A- centered or I-centered cell. It is possible to show that centering of any other face reduces to one of these two.

4. & 5.

Primitive C-centered Primitive and C-centered orthorhombic. alpha = beta gamma = 90º. There are no constraints on axial lengths. Labeling of axes is arbitrary. Just as in the monoclinic case, the centered cell preserves the axis orthogonality and has an additional lattice point at 1/2+x, 1/2+y, z. It would also be possible to choose equivalent A or B centered cell, but conventionally c is the centered axis.

6 & 7.

I-centered F-centered centered (body-centered) cell has an additional lattice point at 1/2+x, 1/2+y, 1/2+z, for a total of two points per cell. In the F-centered cell each of the faces contains 1/2 of an additional lattice point for a total of four per cell. In each, the labeling of axes is arbitrary.

8 & 9.

Primitive I-centered Primitive and I-centered Tetragonal alpha = beta = gamma = 90 degrees. a = b; and c is unique axis with 4 or -4 symmetry. The I- centered (body-centered) cell has an additional lattice point at 1/2+x, 1/2+y, 1/2+z, for a total of two points per cell. Note that there is no C-centered tetragonal cell be cause it would be possible to choose a primitive tetragonal cell with half the volume. Similarly an F-centered cell would reduce to an I-centered cell.

10 & 11. Primitive I-centered Primitive and I-centered Cubic (Isometric) alpha = beta = gamma = 90 º a = b = c. These are similar to the above except for the additional axial length constraint. 12. F-centered Face Centered Cubic (Isometric) alpha = beta = gamma = 90º a = b = c. Each of the six faces contains a lattice point for a total of four per cell.

13. 120 120 Primitive R-centered Primitive Hexagonal

The R-centered cell contains lattice points at 1/3 and 2/3 along the body diagonal. This makes it pos sible to choose an alternative primitive cell such that alpha = beta = gamma, but not 90 or 120 degrees.

2.12. The 230 Three-Dimensional Space Groups

2.12.1. Screw Axes. In addition to glide planes we have a new symmetry operation - the screw axis. A screw axis involves both rotation and translation. The symbol Nm indicates a rotation of 360/N º plus a translation m/N times the repeat (cell edge) in the direction of the rotation axis.

A. when they are parallel to c and as when they lie in the plane of the page.

B. axis or screw triad. 31 = 120º rotation + 1/3 cell translation 32 = 120º rotation + 2/3 cell translation 31 and 32 are right and left handed enantiomorphs of each other and so can only occur in acentric trigonal space groups.

C. 43 42 41, 42, and 43 - Screw tetrads occur in the tetragonal and cubic systems

41 = 90º rotation + 1/4 cell translation 42 = 90º rotation + 2/4 cell translation 43 = 90º rotation + 3/4 cell translation 41 and 43 are right and left handed enantiomorphs of each other and so can only occur in acentric space groups.

D. 61 62 64 65 agonal system.

61 = 60º  rotation + 1/6 cell translation
62 = 60º  rotation + 2/6 cell translation
63 = 60º  rotation + 3/6 cell translation
64 = 60º  rotation + 4/6 cell translation
65 = 60º  rotation + 5/6 cell translation

2.12.2. Glide Planes. Glide planes are reflections plus translations. In space groups, glide planes can have several orientations and are noted according to their orientation.

A c-glide is a reflection plus a translation of 1/2 cell along the c-axis. They can be perpendicular to a or b. In symmetry diagrams, in the standard orientation with a vertical, b horizontal and c normal to the page, they are denoted as row of dots:

of 1/2 cell along the b-axis. It can be perpendicular to a or c.

An a-glide is a reflection plus a translation of 1/2 cell along the a-axis.

An n-glide is a reflection plus a translation of 1/2 cell along both the axes in the plane of reflection.

There is also a d-glide (a diamond glide), but we will not encounter it in rigorous fashion. It occurs in both the cu bic and the tetragonal systems with its glide plane in clined to a major axis.

Space group symbols are called Hermann-Mauguin symbols and are entirely consistent with our point group and plane group notations. The first symbol refers to the type of lattice (space or Bravais Lattice). In the case of protoenstatite, the P refers to primitive orthorhombic. The three symbols which follow refer to symmetry operations which are normal (for plane operations, i.e glide operations) to or parallel (rotation operations) to the major axes a, b, c, respectively. For Pbcn we have a b-glide normal to the a-axis, a c -glide normal to the b-axis and an n-glide normal to the c-axis. When we refer to a glide operation as a c-glide, the c denotes the translation direction (e.g. 1/2 cell along c), and the position refers to its orientation, as perpendicular to a, b, or c. .

2.13.3. Example: Pbcn

At right is asymmetry diagram for space group Pbcn showing the location and orientation of the symmetry symbols. In these diagrams a is horizontal, b vertical, and c normal to the page, so that the symmetry diagram is a c-axis projection of the symmetry operations of the unit cell.

The full Hermann-Mauguin symbol for the space group symbol Pbcn is P21/b 2/c 21/n.

Inversions are marked with a small circle. Mirrors are marked with a heavy solid line when perpendicular to the page and an arrow (2 barbs) when in the plane of the page. Screw and rotational axes have symbols as described above.

In space group Pbcn, it is important to understand how the symmetry elements affect the location and point symmetry of the atoms that compose the structure. If we place an atom at a random location in the unit cell de noted by x, y, z where x, y, and z are in fractions of the unit cell edges, then the symmetry operations of the space group will generate exactly equivalent atoms at the following coordinates.

These locations are shown in the diagram at right. Thus for any atom at a random location in the cell (called the general position) there are eight equivalent positions (atoms) in the unit cell. Because the general position does not lie on a symmetry operation, the point symmetry or symmetry of the site is always 1, i.e. no symmetry. However, some atoms may lie directly on a symmetry operation such as mirrors, two-folds, or inversions where the symmetry operation repeats the atom onto itself giving a redundant position. These are special positions, and have point symmetries greater than one, and will have fewer equivalents in the cell. If an atom in Pbcn lies on the inversion at 0,0,0, it can be seen by substituting into the equivalence equations above that there will only be four such positions: (0,0,0); (0, 0, 1/2);a (1/2, 1/2, 1/2); and (1/2, 1/2, 0) rather than eight. This site will have point symmetry -1.

In the crystal structure of proto enstatite (MgSiO3), there are two different Mg atom sites, both lying on the two-fold axis (point symmetry 2). Mg1 has fractional coordinates 0, 0.087, 3/4 and Mg2 is at (0, 0.262, 1/4). There are thus four Mg1 and four Mg2 atoms in each unit cell. The Si atom lies in a general position at (0.298, 0.098, 0.074). Substituting into the equivalence equations there are thus eight Si atoms per cell. There are three different oxygen sites in the cell, O1, O2, and O3, each of which is in a general position. O1 is at (0.123, 0.094, 0.084); O2 at (0.376, 0.257, 0.072); and O3 at (0.361, 0.983, 0.302). There are thus eight O1 atoms, eight O2 atoms, and eight O3 atoms in each cell, for a total of 24 oxygens per cell. There are then 24 oxygens, eight silicons, and eight magnesiums per cell giving us eight formula units (MgSiO3) per cell. The Z-number is thus 8. The group of unique atoms (Mg1, Mg2, Si, O1, O2, O3) is called the asymmetric unit. In the laboratory, we will calculate the positions of all the atoms of the unit cell, and plot their positions giving us a drawing of the crystal structure. As you can imagine, this computation is somewhat tedious, so we have several computer programs for drawing crystal structures.

The equivalent positions in each of the 230 space groups are given in International Tables for Crystallography, Vol I or Vol A (on reserve in the library, or in the X-ray Lab). In order to draw the structure, we will need the unit cell parameters (a, b, c, alpha, beta, gamma), the symmerty equations from the International Tables, and the fractional coordinates of the atoms in the asymmetric unit. To find the cell parameters and atom coordinates for your mineral, you will need to find a study of the structure in the mineralogical literature. Most commonly, these are published as research articles in scholarly journals, such as American Mineralogist, Mineralogical Magazine, European Journal of Mineralogy, Zeitschrift fuer Kristallographie, Acta Crystallographica, or in summary publications such as Structure Reports or Wyckoff's Crystal Structures. One very good place to start looking for structural data for any given mineral is in Hoelzel's Systematics of Minerals which is usually in the X-ray Lab. (The library has a copy.)

In drawing crystal structures, particularly of oxygen minerals, we typically think of a cation (positively charged ion (metal) as being surrounded by a coordination polyhedron in which the corners represent the oxygen atoms. This is because the crystals commonly have only one type of anion (negative ion), oxygen, and several types of cation, Na+, Mg2+, Al3+, Si4+, etc., and the coordination polyherda are commonly highly regular tetrahedra or octahedra. A tetrahedron is a polyhedron with four faces. It also has four corners which represent our oxygen ions. An octahedron is a polyhdron with eight faces, but only six corners reprsenting nearest-neighbor oxygens.

Tetrahedron is a polyhedron with four faces and four corners. It represents coordination number four. Octahedron is a polyhedron with eight faces and six corners. It represents cations in coordination number six.

These polyhedra are then used to help visualize complex crystal structures such as those illustrated below.

Polyhedral representation of the crystal structure of biotite mica. The structure is made up of a layer of octahedrally coordinated Mg and Fe cations with a layer of tetrahedrally coordinated Al and Si cations above and below. The circles indicate the positions of K atoms that hold the layers together. The chemical bonds holding the K atoms in place are very weak giving rise to the perfect cleavage of the micas.

Further Reading and References

Bloss, F.D. (1971) Crystallography and Crystal Chemistry, An Introduction. Holt Rinehart Winston 545 pp.

McKie, D. and McKie, C (1986) Essentials of Crystallography. Blackwell 437 pp.


GEOL 3010 Syllabus

Chapter 1 Mineralogy Notes

Chapter 3 Mineralogy Notes

Mineral Structures and Properties Data Base

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