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Sahithyan's S1
Sahithyan's S1 — Properties of Materials

Miller Indices

Miller Indices in Simple Cubic

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)(1,1,0) - Atom/Vertex
  • [110][110] - Direction, no commas
  • <110>\text{<}110\text{>} - Family of directions
  • (100)(100) - Plane, no commas
  • {100}\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: 66
  • [100],[010],[001],[1ˉ00],[01ˉ0],[001ˉ][100],[010],[001],[\bar{1}00],[0\bar{1}0],[00\bar{1}]

<110>

  • No of planes: 1212
  • [011],[011ˉ],[01ˉ1],[01ˉ1ˉ],[101],[101ˉ],[1ˉ01],[1ˉ01ˉ],[110],[11ˉ0],[1ˉ10],[1ˉ1ˉ0][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: 88
  • [111],[111ˉ],[11ˉ1],[1ˉ11],[1ˉ1ˉ1],[1ˉ11ˉ],[11ˉ1ˉ],[1ˉ1ˉ1ˉ][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][132], for example:

Take the point (1,3,2)(1,3,2). Divide by the highest number (33, in this case) to bring the point inside the unit cell. The resulting point will be (13,1,23)(\frac{1}{3},1,\frac{2}{3}). The direction is given by vector from (0,0,0)(0,0,0) to the resulting point.

Close packed direction

All neighbour atoms in a direction touch each other. For example: (110)(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 {100}\set{100}
  • No of planes: 66
  • (100),(010),(001),(1ˉ00),(01ˉ0),(001ˉ)(100),(010),(001),(\bar{1}00),(0\bar{1}0),(00\bar{1})

110

  • Denotes as {110}\set{110}
  • No of planes: 1212
  • (011),(011ˉ),(01ˉ1),(01ˉ1ˉ),(101),(101ˉ),(1ˉ01),(1ˉ01ˉ),(110),(11ˉ0),(1ˉ10),(1ˉ1ˉ0)(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 {111}\set{111}
  • No of planes: 88
  • (111),(111ˉ),(11ˉ1),(1ˉ11),(1ˉ1ˉ1),(1ˉ11ˉ),(11ˉ1ˉ),(1ˉ1ˉ1ˉ)(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)(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.