
Any vertex can be chosen as the origin.
Notation
- Minus noted with a bar
- Addition and subtraction is carried out like vectors
- (1,1,0) - Atom/Vertex
- [110] - Direction, no commas
- <110> - Family of directions
- (100) - Plane, no commas
- {100} - Family of planes
- Always will be whole numbers. Fractions must be multiplied by LCM.
Direction
Equivalent directions are grouped into a family.
Direction families
<100>
- No of planes: 6
- [100],[010],[001],[1ˉ00],[01ˉ0],[001ˉ]
<110>
- No of planes: 12
- [011],[011ˉ],[01ˉ1],[01ˉ1ˉ],[101],[101ˉ],[1ˉ01],[1ˉ01ˉ],[110],[11ˉ0],[1ˉ10],[1ˉ1ˉ0]
<111>
- No of planes: 8
- [111],[111ˉ],[11ˉ1],[1ˉ11],[1ˉ1ˉ1],[1ˉ11ˉ],[11ˉ1ˉ],[1ˉ1ˉ1ˉ]
The above are the common directions. There are other directions as well.
Show the direction
To show the direction [132], for example:
Take the point (1,3,2). Divide by the highest number (3, in this case) to
bring the point inside the unit cell. The resulting point will be
(31,1,32). The direction is given by vector from (0,0,0)
to the resulting point.
Close packed direction
All neighbour atoms in a direction touch each other. For example: (110) of
fcc.
Plane
Steps
- If sitting on any axes, move the origin.
- Find the intercepts. ∞ if parallel.
- Find the reciprocals.
Plane families
100
- Denotes as {100}
- No of planes: 6
- (100),(010),(001),(1ˉ00),(01ˉ0),(001ˉ)
110
- Denotes as {110}
- No of planes: 12
- (011),(011ˉ),(01ˉ1),(01ˉ1ˉ),(101),(101ˉ),(1ˉ01),(1ˉ01ˉ),(110),(11ˉ0),(1ˉ10),(1ˉ1ˉ0)
111
- Denotes as {111}
- No of planes: 8
- (111),(111ˉ),(11ˉ1),(1ˉ11),(1ˉ1ˉ1),(1ˉ11ˉ),(11ˉ1ˉ),(1ˉ1ˉ1ˉ)
The above are the common planes. There are other planes as well.
Show the plane
- Divide by the smallest non-zero number.
- Find the reciprocals. ∞ means parallel to the axis.
Close packed plane
All neighbour atoms in a crystal plane touch each other. For example: (111) of
fcc.
Planar Density / Aerial Density
Number of atoms in a unit area in a specific plane. Differs between different
planes in a single crystal structure.