Geos 306, Fall 2012, Lecture 9
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 shortranged arrangements of atoms as
found in crystals. We have focused on the bonding of atoms, and coordinated polyhedra. Now we will examine the longranged arrangements.

Most crystals are composed of a limited and simple number of components,
E.g. SiO_{2}, Mg_{2}SiO_{4}.
This is because a crystal is composed of a small number of atoms in a minimum energy arrangement that displays 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, SiO_{2}. Using Pauling's rules
we find that each Si, at ambient conditions, is coordinated to 4 O's, and each O is coordinated
to 2 Si's. Therefore, the possible numbers of different crystalline phases
of SiO_{2} are limited by the ways in which we can pack these cornerlinked
tetrahedra together. Presently, about 50 different phases of silica have been discovered. Most are synthetic, but a few are minerals.
Here are three examples that are minerals.
 Compounds with the same chemical formula but with different crystal structures are called polymorphs. We can also say that they are different phases.
So, for instance, the minerals quartz,
cristobalite,
and tridymite are polymorphs of SiO_{2} because each has a distinct crystal structure.
It would be incorrect nomenclature to say "the polymorphs of quartz", instead we would say "the polymorphs of silica or SiO_{2}."
Each of these minerals is considered to be a phase of SiO_{2}. Each has its own structural motif, a pattern of atomic level symmetry and structure, that is unique.

But there are in the order of 10^{24} atoms in a crystal. So the
patterns
of bonding must repeat over and over again.

Therefore, in a 3 dimensional crystalline solid, a particular volume of space has been defined as the unit cell.
The unit cell is a parallelpiped with a shape that is characteristic of the crystal and it contains an integral number of chemical formulas.
 Ideally, the contents of each unit cell are exactly identical, and are repeated over and over again, by translation in three dimensions.
Because the unit cell contains an integral number of chemical formulas, then it contains an integral number of atoms,
and thus the property of translational periodicity ensures that the chemical formula of a mineral will be made up of integers, like SiO_{2}.
In quartz there are 3 Si and 6 O atoms in the unit cell, Si_{3}O_{6}. We write the formula as SiO_{2}
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, Si_{4}O_{8}.
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 SiO_{2}. The formula must charge balance,
or else the mineral would have an electrical charge that would be huge because of the 10^{24} atoms in a crystal.
It is not, however, the different amounts of SiO_{2} in each unit cell that distinguishes one polymorph from the other. It is that the atoms are arranged differently.

Different minerals with different compositions can have the same crystal structure.
For instance, calcite CaCO_{3} has the same structure as rhodochrosite MnCO_{3}.
We say that these minerals are isostructural.
Isostructural minerals frequently share similar properties, and, though not always, there frequently is a chemical solidsolution between them.
The unit cells of isostructural minerals have the same shapes, but may differ in size.
The unit cell
 RenéJust Haüy introducted the concept of "molécules intégrantes" (i.e. the modern day unitcell) in his treatise "Traité de Minéralogie" (1801). He noticed how calcite cleaves into rhombohedral shapes, and that all of the various crystal shapes seen in calcite could be built of such units.

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
∠ bc = α, ∠ ac = β, and ∠ ab = γ.
Note that the boldfaced letters, a, b, c, represent vectors,
while the normal letters, a, b, c represent lengths of the vectors.

This is similar to a 3dimensional Cartesian coordinate system. With a 3dimensional Cartesian system 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 correspond to the directions and lengths of the periodicity exhibited in the crystal.

The shape of the unit cell is related to the symmetry of the crystal and defines six crystal systems.
System

a

b

c

∠ bc = α

∠ ac = β

∠ ab = γ

cubic

a

a

a

90

90

90

hexagonal

a

a

c

90

90

120

tetragonal

a

a

c

90

90

90

orthorhombic

a

b

c

90

90

90

monoclinic

a

b

c

90

β

90

triclinic

a

b

c

α

β

γ


The study of crystals is simplified by the fact that all we really need to do
is study and understand the unit cell and its contents. What makes this so appealing is that we can measure these things.
We measure the chemistry with an electron microprobe,
and we measure the unit cell parametes and atomic positions with Xray diffraction.

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 of unit cells 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,
that provides a way to describe 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 halfway 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 zcoordinate (1 decimal place
is good enough) beside the atom. Try to draw the bonds.
atom 
x

y

z

Si 
0.4697

0

0

Si 
0

0.4697

2/3

Si 
0.5303

0.5303

1/3

O 
0.4133

0.2672

0.1188

O 
0.2672

0.4133

0.5479

O 
0.7328

0.1461

0.7855

O 
0.5867

0.8539

0.2145

O 
0.8539

0.5867

0.4521

O 
0.1461

0.7328

0.8812

Zones

A zone is a direction in a crystal that can be defined by three integers,
[uvw], or as a vector, ua+vb+wc

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], [110], 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.

The diffraction of Xrays is described relative to the crystal planes, so we will discuss planes in greater detail in the next few lectures.
Reading:
Wenk and Bulakh, Chapter 3
Klein, Chapter 5
Nesse, Chapter 2