1 Crystalline State
1.1 Classification of Degree of Periodicity
Solids are classified according to regularity and structure of their building blocks, typically atoms and can be the following:
1. Amorphous – No periodic structure at all. All constituent atoms are disordered. An example of this category is glass.
2. Crystal – Periodical arrangement of constituent species. Perfect array of atoms. Examples are diamonds, metals and salt.
3. Polycrystalline – Many small regions of single crystals with various orientations connected by grain boundaries.
To describe crystals, we use the concepts of lattice and basis:
1. Lattice – A collection of mathematical points in a periodic arrangement.
- periodically spaced
- infinite in extent
- all points are identically surrounded
- mathematically described by translation vector
2. Basis – Set of atoms attached to each lattice point. These are the actual physical things that make up the material.
1.2 Translation Vectors
Definition – A lattice can be mathematically described by a set of primative translation vectors denoted by a1, a2, and a3. These are vectors such that the atomic arrangement looks the same in every respect when viewed from one point r as when viewed from the point:
r’=r+n1 a1+n2 a2+n3 a3 where n1, n2, n3 are integers
A lattice translation operation is defined as the displacement of a crystal by a crystal translation vector
T=n1 a1+n2 a2+n3 a3 where n1, n2, n3 are integers
1.3 Primitive Lattice Cell
The parallelepiped defined by a1, a2, and a3 is called a primitive lattice cell and is a minimum volume cell and has only one lattice point per unit cell.
There is not a unique primitive cell because the primitive lattice vectors themselves are not unique. One commonly used method to define the primitive cell is by constructing a Wigner-Seitz cell. Primitive unit cells will have only one lattice point per cell. Other commonly used unit cells such as the body centered cubic (BCC), face centered cubic (FCC), and hexagonal conventional unit cells have more than one lattice point per cell.
Crystal (a) is a simple cubic, crystal (b) is a body centered cubic and crystal (c) is a face centered cubic. Two additional commonly encountered structures are called the zinc-blend and diamond structures. Examples of zinc-blend and diamond structures are GaAs and Si respectively and are shown below.
1.4 Miller Indices
The use of Miller indices for describing planes and directions within a crystal lattice is very common. To obtain the Miller indices describing a plane, you use the following four-step procedure given by Pierret and is as follows:
1. Set up coordinate axes along the edges of the unit cell and then note where the plane to be indexed intercepts the axes. Then divide each intercept value by the unit cell length along the respective coordinate axis. Record the resulting normalized intercept sent in the order x,y,z.
2. Invert the intercept values (i.e. 1/intercepts)
3. Using an appropriate multiplier, convert the 1/intercept set to the smallest possible set of whole numbers
4. Enclose the whole-number set in curvilinear brackets
The Miller indices for a direction for cubic crystals turn out to be the same as the Miller indices for the plane normal to the direction. For noncubic crystals, the procedure is more lengthy but just a straightforward calculation where you find the components of the direction vector in terms of the basis vectors followed by multiplying these components by a whole number to get the smallest set of whole numbers possible.
There exists the following notation:
(hkl) Crystal Plane
{hkl} Equivalent Planes
[hkl] Crystal Direction
<hkl> Equivalent Directions