Miller Indices

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
• $\text{<}110\text{>}$ - Family of directions
• $(100)$ - Plane, no commas
• $\set{100}$ - 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: $6$
• $[100],[010],[001],[\bar{1}00],[0\bar{1}0],[00\bar{1}]$

<110>

• No of planes: $12$
• $[011],[01\bar{1}],[0\bar{1}1],[0\bar{1}\bar{1}],[101],[10\bar{1}],[\bar{1}01],[\bar{1}0\bar{1}],[110],[1\bar{1}0],[\bar{1}10],[\bar{1}\bar{1}0]$

<111>

• No of planes: $8$
• $[111],[11\bar{1}],[1\bar{1}1],[\bar{1}11],[\bar{1}\bar{1}1],[\bar{1}1\bar{1}],[1\bar{1}\bar{1}],[\bar{1}\bar{1}\bar{1}]$

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 $(\frac{1}{3},1,\frac{2}{3})$. 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. $\infty$ if parallel.
• Find the reciprocals.

Plane families

100

• Denotes as $\set{100}$
• No of planes: $6$
• $(100),(010),(001),(\bar{1}00),(0\bar{1}0),(00\bar{1})$

110

• Denotes as $\set{110}$
• No of planes: $12$
• $(011),(01\bar{1}),(0\bar{1}1),(0\bar{1}\bar{1}),(101),(10\bar{1}),(\bar{1}01),(\bar{1}0\bar{1}),(110),(1\bar{1}0),(\bar{1}10),(\bar{1}\bar{1}0)$

111

• Denotes as $\set{111}$
• No of planes: $8$
• $(111),(11\bar{1}),(1\bar{1}1),(\bar{1}11),(\bar{1}\bar{1}1),(\bar{1}1\bar{1}),(1\bar{1}\bar{1}),(\bar{1}\bar{1}\bar{1})$

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. $\infty$ 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.