Crystal Systems
Why Crystal Systems
So many possible crystal structures, so let us divide them into groups according to unit cell configurations.
Crystallographic Points
Lattice Parameters:
The unit cell geometry is completely defined in terms of six lattice parameters (we only care about a,b,c which are lengths of each side in unit cell).
To plot:
Pretty straightforward. The position of any point located within a unit cell is specified in terms of its coordinates as fractional multiples of the unit cell edge lengths (in terms of a, b, c). Ex. 1/4 1 1/2.
Crystallographic Directions
Is a vector
Defined as a line directed between two points.
Definition
- Passes through origin
- Length of the vector project on each of the 3 axis is determined; measured in terms of the unit cell dimensions a,b,c
- These 3 numbers are reduced to smallest whole int values
- These 3 indices are inclosed in square brackets (ex. [1 1 1])
- Can be negative
Example
Procedure:
1. Calculate projections. (a/2, b, 0c)
2. Remove a,b,c. (1/2, 1, 0)
3. Reduction (1, 2, 0)
4. Enclosure ( [1 2 0] )
For reverse q's, do opposite.
Note: To convert from 3-index system to 4-index system:
Given [u',v',w'] -> [uvtw]
Planes
Specified by 3 Miller indices as (hkl).
- Any two planes parallel to each other are equivalent (same Miller indices)
Crystallographic Planes
To calculate Miller indices
1. Make sure plane doesn't pass through selected origin (either choose another origin, or translate the plane)
2. Calculate intercepts for each of the three axis.
3. Take reciprocals of these intercepts.
4. Reduce 3 numbers to smallest whole int values
5. Enclose within parentheses, like (h k l)
2. Calculate intercepts for each of the three axis.
3. Take reciprocals of these intercepts.
4. Reduce 3 numbers to smallest whole int values
5. Enclose within parentheses, like (h k l)
Linear Density
Example
1. Either translate or choose another O. We choose another O.
2. Intercepts: inf, -1b, c/2 -> inf, -1, 1/2
3. Reciprocals: 0,-1,2
5. (0 -1 2)
Planar Density
Density
Metals
Crystalline
All metals form crystalline structures. Most common: FCC, BCC, HCP. Very closely packed planes of atoms.
Ceramics
Both Crystalline and Amorphous
Ceramics can be both crystalline and amorphous.
Interatomic bonding in ceramics ranges from purely ionic to totally covalent.
Interatomic bonding in ceramics ranges from purely ionic to totally covalent.
Polymers
Vary as well
Can be completely noncrystalline or semicrystalline.
Crystal Structure
Determined by 1) charge magnitude on each ion and 2) radius of each type of ion
Importance
Important when material plastically deforms; for instance, slip occurs on the plane with the highest planar density.
X-Ray Diffraction
Important in determining how, using x-rays, atomic interplanar distances and crystal structures are deduced.
- Uses Bragg's law to specify the condition for diffraction of x-rays.
- Uses Bragg's law to specify the condition for diffraction of x-rays.