Crystals are three-dimensional symmetric arrangements of atoms, molecules or ions. Such arrangements repeat themselves at regular intervals keeping the same relative orientation to one another. This is a unique property of crystalline materials which are specific to different crystalline compounds irrespective of the source of origin be it natural or synthetic.
If you consider each atom, molecule or ion as a point then such an arrangement is in translational symmetry and the outline of such arrangement is called a crystal lattice .A unit cell comprising of single type of atoms is monoatomic whereas one comprising of more than one type of atoms is called polyatomic cell.
Crystals can be considered as planes joining groups of atoms with fixed distances and angles .These dimensional constants are characteristic features of different crystalline materials.
A crystal system is a group of crystal structures used to describe the axial arrangement of crystals. There are seven basic crystal shapes
This arrangement consists of three axis perpendicular to each other with all sides equal in length. The cubic system has a lattice point at each of its eight corners and has six faces.
The hexagonal arrangement has four axes. Three of these are horizontal at 120° to each other and the fourth axes is perpendicular to the three horizontal axes. It comprises of eight faces
A tetragonal system has a square base and top like in cubic arrangement but has an extended vertical height. It has three axes at 90° to each other and a total of six faces
The rhombohedral is similar in shape to a cube but is inclined in one direction.Its three axes are perpendicular to each other with two horizontal and one vertical. It has six faces.
Orthogonal crystals consist of three axes perpendicular to each but of different lengths.It has six faces.
Monoclinic crystal has three unequal axes. The front face axes are oblique to each other and the third axes is perpendicular to the other two. The system has six faces
The structure has three unequal crystallographic axes which intersect one another obliquely. It has six faces.
Bravais in 1848 postulated that seven crystal systems can exist in 14 distinct types of configurations. The unit cells of Bravais lattice are
Cubic – 3(simple cubic, body centred cubic, face centred cubic)
Tetragonal – 2(simple, body centred )
Orthorhombic – 4 (simple, body centred, base centred, face centred)
Rhombohedral – 1(simple)
Monoclinic – 2(simple, base centred)
Triclinic – 1 (simple)
The atomic orientations in crystals are responsible for their shapes. It often becomes necessary to define different planes within a lattice mathematically. Physical properties of materials such as electrical conductivity, thermal conductivity, deformation under loads,etc are dependent on orientations in some crystals. Such behaviour is referred to as anisotropy.
To understand Miller induces it is important to understand the commonly used expressions
x, y, z are axes passing through origin
a,b,c are unit cell lengths along the three axes
Miller induces express planes as (hkl) where h, k and l are integers.
Gen Convention for assigning Miller indices:
- Determine the intersection of the plane along the three axes-a,b and c.
- Suppose a plane intersects x axis at a/2, y axis at the end and c axis at c/3. These are expressed as 1/2,1and 1/3.
- The reciprocals of these become 2,1, 3.
- The Miller indice of this plane is expressed as (213), ie, in brackets and without commas.
In a cubic system planes having same indices regardless of order and sign are equivalent but the same will not be true for other geometries.