Geos 306, Fall 2007, Lecture 8, Crystallography I

Geos 306, Fall 2007, Lecture 8
Crystallography - Unit Cells

Definition of crystal and mineral

W&B define a crystal as: “A homogeneous chemical compound with a regular and periodic arrangement of atoms.”
W&B define a mineral as: “A mineral is a naturally occurring chemical compound. Most minerals are crystalline.”
Klein defines a mineral as: “A mineral is a naturally occurring homogeneous solid with a definite (but generally not fixed) chemical composition and a highly ordered atomic arrangement. It is usually formed by inorganic processes.”

Periodicity

So far we have examined the short-ranged arrangements of atoms as found in crystals. Now we will examine the long-ranged arrangements.
Most crystals are composed of a limited and simple number of components, E.g. SiO₂, Mg₂SiO₄. These components are charge balanced.
This is because a crystal is a structure with a minimum energy arrangement of atoms that has translational periodicity.
As a consequence, and making use of the concept of coordinated polyhedra, there are a limited number of ways to pack the atoms together to form a solid. For instance, silica, SiO₂. Using Pauling's rules we find that each Si is coordinated to 4 O's, and each O is coordinated to 2 Si's. Therefore, the possible numbers of different crystalline phases of SiO₂ are limited by the ways in which we can pack these corner-linked tetrahedra together. Presently, we have found about 50 different phases. Here are three examples.

Different crystal structures that are found for a single chemical compound are called polymorphs. So, for instance, the crystal structures of quartz, cristobalite, and tridymite are polymorphs of SiO₂. Each of these minerals is considered to be a phase of SiO₂.
But there are in the order of 10²⁴ atoms in a crystal. So the patterns of bonding must repeat over and over again.
Therefore, in a 3 dimensional solid, we have defined a volume of space that is known as the unit cell. The unit cell is a parallelpiped with a shape that is characteristic of the crystal.
The contents of the unit cell are repeated over and over again, by translation in three directions. There are an integral number of atoms in the unit cell, and thus the property of translational periodicity ensures that the chemical formula of a mineral will be made up of integers, like SiO₂. In quartz there are 3 Si and 6 O atoms in the unit cell, Si₃O₆. We write the formula as SiO₂ because the Si and O atoms are found in that ratio. In a unit cell of cristobalite there are 4 Si and 8 O atoms, Si₄O₈. In the unit cell of a tridymite crystal that was recovered from a meteorite, there were 12 Si and 24 O atoms. In each of these three examples the ratio of Si to O provided a formula that is SiO₂. The formula must charge balance, or else the mineral would have an electrical charge that would be huge because of the 10²⁴ atoms in a crystal.

The unit cell

The shape of the unit cell is defined by 6 parameters, a, b, c, a, b, g. We choose a corner to be the origin, and the three edges define the vectors a, b, c of lengths a, b, c. These edges need not be 90° from each other so a, b and g define the angles between the edges, with Ð bc = a, Ð ac = b, and Ð ab = g. Note that the bold-faced letters, a, b, c, represent vectors, while the normal letters, a, b, c represent lengths of the vectors.
This is similar to a 3-dimension Cartesian coordinate system. With Cartesian systems you have three axes, all of length equal to 1, and they are each 90° from the others. So, in the Cartesian system, a = b = c = 1, α = β = γ = 90°. In crystals we have a more general system, chosen to mimic the symmetry of the crystal.
The shape of the unit cell is related to the symmetry of the crystal. In fact, often it can be considered to define it.

So, in order to study and understand a crystal, all we really need to do is study and understand the unit cell and its contents.
For instance, the compressibility of a mineral is obtained simply by finding out how its unit cell volume decreases with pressure. Furthermore, the compressibility of a mineral is directly related to how fast seismic waves can propagate through the the crystal. So we model the seismic properties of the earth's deep interior by examining the unit cell parameters of minerals at high pressure.
Examples that can be viewed with XtalDraw: cubic, hexagonal, tetragonal, orthorhombic, monoclinic, and triclinic.
Examples of crystal structures of minerals with different symmetry that can be viewed with XtalDraw: cubic, hexagonal, tetragonal, orthorhombic, monoclinic, and triclinic.

Atomic coordinate systems

We define a certain corner of the unit cell to be the origin. This is the corner that defines the angles a, b, and g. The vectors a, b, c define the basis of our coordinate system, called the direct basis, or real space.
Atoms are located within the unit cell as fractional coordinates, [xyz]. The beginning of a, b and c are at a coordinate value of 0, defining a point called the origin at [0,0,0]. The end points of a, b and c are at the coordinate value of 1. A point that is half-way along vector a is at coordinate [1/2 0 0].
E.g., quartz, which has hexagonal symmetry and cell parameters a = 4.9137 Å, c = 5.4047 Å. Construct a diagram of the crystal structure of quartz using a unit cell template. Render the atoms as circles or spheres, with Si smaller than O. Put the z-coordinate (1 decimal place is good enough) beside the atom. Try to draw the bonds.

Zones

A zone is a direction in a crystal that can be defined by three integers, [uvw].
Every edge of a crystal can be described as a zone.
They are simply vectors that are useful in defining the direction in which you are viewing a crystal.
Eg, find the directions [100], [010], [-1-10], and [210] in quartz.

Planes

Any 3 points define a plane. Three corners of any unit cell define a lattice plane. All faces of any crystal are parallel to such planes. The faces that are observed, in fact, are the slowest growing planes in a crystal. E.g., cube or octahedron.
Each plane has an identifying set of integer coordinates that define the given plane, (hkl). (Notice that these brackets are round ( ), while the brackets for a zone are square [ ].) These coordinates represent the vector that is perpendicular to the plane and are given in a coordinate system that is referred to as reciprocal space since it intercepts the x,y, and z axes at a/h, b/k, and c/l.

Reading:
Wenk and Bulakh, Chapter 3
Klein, Chapter 5
Nesse, Chapter 2