Basic Concepts of Crystallography

Crystal Symmetry

Here, the key aspects of crystal symmetry are introduced, leading on from the concepts relating to Crystal and Lattices that were introduced on the previous tab. The crystal symmetry is an important concept for electron backscatter diffraction since a knowledge of a particular phase’s crystal symmetry will influence how we might display and interpret data collected using EBSD.

Point groups

A crystallographic point group is a group of symmetry elements that operate on a 3-dimensional lattice to leave it unchanged. The symmetry elements are those of rotation (1, 2, 3, 4 and 6 fold axes) and rotation followed by inversion. For example, rotation about a 3-fold symmetry axis leaves space unchanged for every rotation of 120 degrees. The inversion symmetry elements are shown by the notation -1, -2, -3, -4, -6. -2 is equivalent to a mirror plane perpendicular to the rotation axis. The symmetry elements combine to form 32 point groups.

View along a <111> direction in a face centred cubic lattice showing the 3-fold symmetry. The lattice is unchanged for every rotation of 120 degrees about the <111> axis.

Crystal systems

The point groups can be assigned to seven crystal systems, each with a symmetry element in common. A lattice with a unit cell exhibiting the required symmetry can then be assigned to each crystal system as shown in the following table. The terms a, b and c and α, β and γ refer to the unit cell dimensions (the lattice parameters and inter-axis angles, respectively). The unit cell is primitive because it has only one lattice point per cell (which is shared by the neighbouring cells).

Adapted from "Essentials of Crystallography" McKie D and McKie C, Blackwell Scientific Publications (1986)

Bravais lattices

Each of the lattices described above is based on a primitive unit cell. It is also possible to define lattices based on non-primitive unit cells which are consistent with the crystal symmetry in a system. These are the fourteen Bravais Lattices which are described in the following table.

System	Lattice Symbol	Conventional Unit Cell	Number of lattice points
Triclinic	P	a ≠ b ≠ c; α ≠ β ≠ γ	1
Monoclinic	P	a ≠ b ≠ c; α = γ = 90°, β > 90°	1
Monoclinic	C	a ≠ b ≠ c; α = γ = 90°, β > 90°	2
Orthorhombic	P	a ≠ b ≠ c; α = β = γ = 90°	1
Orthorhombic	C	a ≠ b ≠ c; α = β = γ = 90°	2
Orthorhombic	I	a ≠ b ≠ c; α = β = γ = 90°	2
Orthorhombic	F	a ≠ b ≠ c; α = β = γ = 90°	4
Tetragonal	P	a = b ≠ c; α = β = γ = 90°	1
Tetragonal	I	a = b ≠ c; α = β = γ = 90°	2
Cubic	P	a = b = c; α = β = γ = 90°	1
Cubic	I	a = b = c; α = β = γ = 90°	2
Cubic	F	a = b = c; α = β = γ = 90°	4
Trigonal	R	a = b ≠ c; α = β = 90°, γ = 120° or         a = b = c; α = β = γ < 120°, ≠ 90°	3
Hexagonal	P	a = b ≠ c; α = β = 90°, γ = 120°	1

Adapted from "Essentials of Crystallography" McKie D and McKie C, Blackwell Scientific Publications (1986)

The fourteen Bravais lattices

Directions, Planes and Zone Axes

Here we introduce the concepts relating to the labelling and representation of crystallographic planes and directions. This is usually done using the Miller or Miller-Bravais notations, which are described below.

These are essential for the effective recording of electron backscatter diffraction (EBSD) data in terms of textures and boundaries (rotation axes or boundary planes), as well as for a deeper understanding of the crystallographic principles of the EBSD technique.

Directions

Here we show how to describe a crystal direction using the Miller index notation. Consider a crystal lattice with unit cell edges a, b and c. A crystal direction [uvw] is parallel to the direction joining the origin of the crystal lattice with the point with coordinates (ua, vb, wc), as shown in the images below.

Illustration of the crystal direction [uvw]. The direction shown here is [422].

This shows how the [100], [110] and [111] directions are defined. The directions are perpendicular to the planes with same Miller indices only in materials with cubic symmetry.

Planes

Here we show how to describe a crystal plane using the Miller index notation. A plane with Miller indices (hkl) passes through the three points (a/h,0,0), (0, b/k,0) and (0,0, c/l) on the edges of the unit cell. The set of parallel lattice planes passes through all similar points in the lattice. The plane d-spacing is the perpendicular distance from the origin to the closest plane and also the perpendicular distance between successive planes. In materials with cubic symmetry the crystal direction [uvw] and the normal to the plane (uvw) are parallel.

Crystal Planes 1: The plane with Miller indices (hkl) makes intercepts a/h, b/k and c/l on the edges of the unit cell. The plane shown here is (221) because the intercept in the a direction is ½a, in the b direction ½b and in c direction c.

Crystal Planes 2: these schematic images show how the (100), (110) and (111) planes are defined.

Crystal Planes 3: a set of parallel (221) planes intersecting the edges of the unit cells. The d-spacing, d for these planes is the perpendicular distance between successive planes or equivalently the perpendicular distance from the origin to the closest plane.

Symmetrically related planes and directions

The family of symmetrically related directions [uvw] is shown by the notation <uvw> and the family of symmetrically related planes (hkl) is shown by the notation {hkl}.

Miller-Bravais Indices

It is sometimes easier to define the unit cell of some crystals in a hexagonal reference frame (i.e. hexagonal and rhombohedral/trigonal crystal systems). In these cases there are, in fact, 4 axes in the unit cell – one along the long axis of the hexagonal cylinder (i.e. the 6-fold rotation axis) which is usually referred to as the “c” axis and 3 “a” axes in the basal plane, at 120° intervals.

For such systems the crystal directions and planes can be defined using a 4-digit notation, known as the Miller-Bravais notation. Calculating the Miller-Bravais indices for a plane is done in the same way as with the 3-digit Miller indices.

It is possible to use the Miller indices notation (3 digits) for Hexagonal unit cells, since the 3rd a-axis (a3) is symmetrically equivalent to the first 2. The Miller indices <UVW> for a direction can be converted into Miller-Bravais notation <uvtw> using these equations:

u=(2U-V)/3

v=(2V-U)/3

t=-(U+V)/3

w=W

Tip: it is useful to know that, when using the Miller-Bravais notation (hkil), the sum of the first 3 indices h+k+i = 0.