**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.

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, -3a = bHexagonal alpha = beta = 90 gamma = 120 1, -1, 2, -2(m), 3, -3, 6, -6a = bTetragonal alpha = beta = gamma = 90 1. -1, 2, -2(m), 4, -4a = bCubic alpha = beta = gamma = 90 1. -1, 2, -2(m), 3, -3, 4, -4a = b = c

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.**

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

1-fold 360 º + ii2-fold 180 º + i -2 = m 3-fold 120 º + i -3 4-fold 90 º + i -4 6-fold 60 º + i -6

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.

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.

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

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.

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.

**2.7.1. 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.

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

____________________________________________________________________

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

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.

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

C.
4_{3}
4_{2}
4_{1}, 4_{2}, and 4_{3}
- Screw tetrads occur in the tetragonal
and cubic systems

D.
6_{1}
6_{2}
6_{4}
6_{5}
agonal system.

6_{1}= 60º rotation + 1/6 cell translation 6_{2}= 60º rotation + 2/6 cell translation 6_{3}= 60º rotation + 3/6 cell translation 6_{4}= 60º rotation + 4/6 cell translation 6_{5}= 60º rotation + 5/6 cell translation

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.

The full Hermann-Mauguin symbol for the space group symbol *Pbcn
*is *P*2_{1}/*b* 2/*c
* 2_{1}/*n.*

- (x,y,z);
- (x,-y,1/2+z);
- (-x,-y,-z);
- (-x,y,1/2-z);
- (1/2-x,1/2-y,1/2+z);
- (1/2-x,1/2+y,z);
- (1/2+x,1/2+y,1/2-z);
- (1/2+x,1/2-y,-z)

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

Mineral Structures and Properties Data Base