Geos 306, Lecture 10
Crystallography II
Lattices

Suppose we define an origin, O, in a crystal that has the basis vectors
{a, b, c}. The location of the origin is arbitrary.
Then O + a is another point that is translationally equivalent to O.
Being translationally equivalent means that everything about the two points is identical in every
respect except that they are distinct points, located in different places. But if you were tiny,
the size of an atom,
and were situated at O, or at O+a, you could not tell the difference.

The points O+b and O+c are also equivalent to O and
O+a.

In fact, all the points, O+ua+vb+wc, {where u,v,w are integers},
are translationally equivalent. The set of all these points is called a lattice.

The faces of a crystal are parallel to planes of lattice points, while the edges of a crystal are parallel to
lines of lattice points.

We can say that translation maps a lattice into selfcoincidence. This means that if
we translated the lattice by, say, a, then it would lie exactly on top of itself
in such a way that you could not tell that the lattice had been moved.
 A lattice that we may all be familar with is a
rose garden lattice.
 Here is an example of the lattice for an orthorhombic olivine, looking down c, with b horizontal.
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. Any 3 (not colinear) lattice points 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.

General equation of a plane:
hx+ky+lz = n.
The points [xyz] are on the plane defined by (hkl) if they satisfy the equation. The variable n, is an integer.
When n = 0, then the plane passes through the origin.
When n = 1 then this is the next plane from the origin, n = 2 is the second plane, etc.

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 the plane (hkl) 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.

Eg, find the planes (100), (010), (110), and (210) in quartz.
Symmetry

The operations that map an object into selfcoincidence are called symmetry operations.
A crystal is solid matter that possesses atomic scale translational symmetry.
Aside from the liquid core and a small amount of molten mantle, most of the Earth is crystalline.
Rotations

Along with translational symmetry, there are also other symmetry operations found in crystals that map a lattice into self coincidence,
for instance, rotations. In the illustration of the lattice above, not only is
translation a symmetry operation but so are rotations of 0° = 360°, 60°,
120°, and 180°.
This does not mean that all crystals have these rotational symmetries in them; some display only rotations of
180°, like gypsum, while some have both 120°
and 180° rotations, like quartz. The feldspar, albite, does not have rotational symmetry.

Not all possible rotations can exist in a crystal, for instance, we do not see 7fold rotations.
It is only the rotations that can map a lattice into selfcoincidence, those
with turn angles of 0, 60, 90, 120, 180°, that are possible.
We call these symmetry operations 1fold, 6fold, 4fold, 3fold and 2fold rotations,
respectively.
The name comes from the fact that a nfold rotation has a turn angle of
(360/n)°.

For a nice derivation of the permissable lattice rotations,
see attached pdf, copied from W&B.

These rotation are defined such that a counterclockwise rotation is considered positive
(righthand rule). A clockwise rotation is designated as an inverse nfold rotation and is
denoted n^{1}. This is because an nfold rotation followed by an inverse nfold
rotation takes you back to where you started, i.e. n n^{1} = 1.

In the lattice illustrated above, not only is there a 6fold rotation axis, but there are also
2fold and 3fold rotation axes. The set of all symmetry operations that map an
object into selfcoincidence, while not moving the origin, is called its point group.
The translation symmetry operation is not an element of a point group, because it does not leave
any point fixed.

All crystals can be characterized by the symmetry in their point group, for instance,
the crystal form, because planes of lattice points define faces of a crystal.
The symmetry of the point group is present in the symmetry of all of the physical properties
of a crystal. For instance, the optical properties of a crystal match the point group symmetry.
So we will spend some time examining the point group symmetries of crystals.

Based upon the 1, 2, 3, 4, and 6fold rotations described above, we can define 5 different
point groups, generated by 1fold, 2fold, 3fold, 4fold, and 6fold rotations:
1 = {1}
2 = {1,2}
3 = {1,3,3^{1}}
4 = {1,4,2,4^{1}}
6 = {1,6,3,2,3^{1},6^{1}}
Inversion

The only symmetry rotations that map a crystal lattice into selfcoincidence
are 1, 2, 3, 4, and 6fold rotations, as shown above, because they preserve the
translational symmetry of the lattice.

There is yet another type of symmetry operation that maps a lattice onto itself besides the translations and rotations,
and it is called an inversion. The symbol for an inversion is i. If O marks the origin, then an
inversion takes any point, P, and maps it through the origin to a new point, P', that is exactly the
same distance away from the origin, (i.e. OP = OP') such that POP' are on the same line.

In a mathematical sense, the inversion takes a point located at {x,y,z} and maps it to {x,y,z}.
Thus an object has an inversion as a symmetry element if every point located at some {x,y,z} is equivalent to another one located at {x,y,z}.
 Things that do not display an inversion center are said to be chiral. Life is chiral, and the impact of
this symmetry affects all sorts of aspects of life. For instance, here is a review of chiral drugs and how they interface with our bodies.
see attached pdf
RotoInversion

In addition to translations, rotations and inversions, there is one final type of symmetry element called a rotoinversion.
This symmetry element is a hybrid mixture of a rotation that is followed by an inversion.

A well known example of a rotoinversion is a mirror, m. This is made from a 2fold rotation followed
by an inversion.

Based upon the rotations described above, and the rotoinversion hybrid just described, we can
define some new symmetry elements: (Note that the line is supposed to represent a bar over the number)
1i = i
2i = m
3i = `3
3^{1}i = `3^{1}
4i = `4
4^{1}i = `4^{1}
6i = `6
6^{1}i = `6^{1}
Point Groups
 The symmetry of every mineral can be defined by the set of all symmetry elements that map it into selfcoincidence.
There are 32 possible sets of symmetry elements, known as the crystallographic point groups.
Here is a table that summarizes the crystallographic point groups.
 The symmetry of a mineral also defines the constraints on the cell parameters,
so the point group is also related to crystal system.
G

GUGi

HUG\Hi

System

1

`1



Triclinic

2

2/m

m

Monoclinic

3

`3



Hexagonal

4

4/m

`4

Tetragonal

6

6/m

`6

Hexagonal

222

2/m2/m2/m

2mm

Orthorhombic

322

`32/m2/m

3mm

Hexagonal

422

4/m2/m2/m

4mm and `42m

Tetragonal

622

6/m2/m2/m

6mm and `62m

Hexagonal

332

`3`3 2/m



Cubic

432

4/m`3 2/m

`4 3 m

Cubic

Reading:
Wenk and Bulakh, Chapter 4
Klein, Chapter 5
Nesse, Chapter 2