1. MODULE 2 - Introduction to Basic Crystallography
Crystal Forms
Bravais Lattices
Crystal system
Miller Indices
Crystallographic Direction
Zone Axis
Zones in the EBSP
Indexing
Euler Angle
Euler Space - ODF's
Orientation Matrix
Ideal Orientation Nomenclature
Misorientation
Describing Planes and Directions
Determining the Orientation
Describing Orientations
Crystallography is the general term applied to describe the study of crystalline materials. In nature, materials may exist in an amorphous state, i.e., having no discernible order in the arrangement of the atoms making up the whole, or alternatively possess some degree of crystallinity. When a material is crystalline the atoms form into regular arrays, the smallest component of which can be considered as a 'unit cell'. This unit cell is dictated by the crystal system which varies from one material to another, and also as a result of the thermodynamic history of the material. Thus different crystal forms are possible for a material and the types of atom present in the unit cell lead to differing unit cell dimensions and angles.
Crystalline materials can be in the form of single crystal, whereby the material has a uniform crystal structure throughout the material. Examples of such materials include silicon and germanium wafers used in the semiconductor industry. Single crystal germanium is used as a calibration material for EBSD as a single crystal is convenient for this purpose.
Most metals and alloys form polycrystalline structures (although under special circumstances single crystal metals are possible). In a polycrystalline material, regions with the same crystal structure exist in close proximity to make up the whole. These regions of discrete crystalline structure are referred to as grains and the interface between neighbouring grains referred to as grain boundaries. The orientation of individual grains is important, as is the misorientation between grains in governing the physical properties of the material, along with other parameters such as texture i.e. preferred orientation, which may or may not be present.
Describing Misorientations

2. Introduction to Basic Crystallography - Bravais Lattices
There are 14 Bravais Lattices:
Note:
Body Centered Cubic (BCC) and
Face Centred Cubic (FCC) forms
In 1835 Frankenheim proposed that there were 15 space lattices. Unfortunately two that he described were in fact identical and…
In 1848 Bravais pointed out Frankenheim’s error and correctly described the 14 space lattices which are now called Bravais Lattices.
Bravais Lattices take into account the possible variations within a given unit cell type. For example, a cubic unit cell can be simple cubic, body centred cubic (BCC) which has an atom at the centre of the unit cell, or face centered cubic (FCC) with an atom located in each face. Similar variants exist for the other types as shown.
These 14 Bravais Lattices can be derived to classify all crystals into 7 systems.
From these 7 crystal systems are derived

3. Introduction to Basic Crystallography - Crystal System
These terms describe the geometry of the unit cell - the structure that is repeated throughout the crystal.
There are seven Crystal systems:
1. Cubic
2. Hexagonal
3. Trigonal
4. Tetragonal
5. Orthorhombic
6. Monoclinic
7. Triclinic
These are ordered with respect to symmetry, i.e. Cubic is the most symmetrical with all lengths equal and all angles equal. Conversely, Triclinic is the least symmetrical with all sides of different length, and all angles different.
As can be seen by the table, these two extreme cases bound a range of variants in between.
The lengths of the sides of the unit cell are shown below as a, b and c. The corresponding angles are shown as alpha, beta and gamma.

4. Introduction to Basic Crystallography - Miller Indices
For a cubic example, Miller indices can be derived to describe a plane
Consider a cubic unit cell with sides a, b, c, with an origin as shown
William Miller , devised a method for describing crystallographic planes in numerical (integer) form. He described a method for deriving indices, and notation which became widely adopted. The method can be used to describe planes and crystallograophic directions.
100. This describes the plane of the cube face, as shown on the left. This plane is 1 from the origin along the 'a' axis, and can be considered to extend for infinity in other directions. By taking a reciprocal of infinity, the value 0 is abtained and hence the orientation is quoted as '100'.
110. This describes the diagonal plane shown in the center diagram. This plane is 1 from the origin along the 'a' axis, 1 from the origin along the 'b' axis and extends for infinity with respect to the 'c' axis. Therefore the orientation is quotes as '110'.
111. This describes the plane shown on the right. This plane lies the distance 1 along each of the axes a, b, and c. Thus it is described as '111'.
All other orientations are determined in the same manner.
Miller indices are usually displayed in brackets. These brackets are used as a way of showing whether the indices relate to a plane, a family of planes (i.e. all those of the same type), a direction or family of directions. Therefore the type of brackets signify whether the indices describe a plane or a direction.
Notation: 'hkl' are frequently used to denote a plane, and 'uvw' are used to denote a direction. These terms are substituted by miller indices when a plane or direction is being described. Therefore:
(hkl) is used to denote the indices of a plane. In the above examples, the indices could be quoted as (100), (110) and (111) to refer to the specific planes shown.
{hkl} is used to denote the indices of a family of planes, i.e. all those of the same type. Therefore in the above example, if {100}, {110} or {111} were quoted, then the reader would know that the indices refer to all equivalent planes of that type.
[uvw] is used to denote the indices of a specific direction.
<uvw> is used to denote the indices of a family of directions.

5. Introduction to Basic Crystallography - Crystallographic Direction
A crystallographic direction describes the intersection of specific faces or lattice planes
Miller Indices can be used to describe directions
A crystallographic direction in a crystal describes the intersection of specific faces or lattice planes. A crystallographic direction is described in the same manner as crystallographic planes, using Miller indices.
For example, in a cubic system, the crystallographic directions lie perpendicular to planes. These also happen to have the same numerical indices (Miller indices) of the plane that the direction lies parallel to.
Specific directions written as Miller indices are shown in brackets ’[ ]' so that the term can be discriminated from the indices that describe a plane.
Families of directions are shown in brackets: ’< >'.
For a cubic material the plane and the normal to the plane have the same indices

6. Introduction to Basic Crystallography - Zone Axis
‘Zone’: faces or planes in a crystal with parallel intersections.
Zone Axis: The common direction of the intersections
Christian Samuel Weiss , formulated the concept of a ‘Zone’. The concept was originally conceived with respect to a direction of prominent crystal growth. A 'Zone' was defined as a collection of crystal faces parallel to a line - the Zone Axis.
‘Zone’: faces or planes in a crystal with parallel intersections.
Zone Axis: The common direction of the intersections
In an Electron Backscatter Diffraction Pattern (EBSP), the intersection of Kikuchi bands correspond to a Zone Axis in the crystal, and all Kikuchi bands with one intersection belong to one zone. A Zone Axis is a crystallographic direction and is expressed by [uvw].