Lesson 2, Topic 11
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Planes and Directions

Abdulaziz July 26, 2020
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Miller Indices for Crystal Planes

Steps:
1) Check where plane intercepts each axis. Express as n times the unit vector length.
2) Invert the intercepts:1 /intercept
3) Convert the 1/intercept set into smallest possible set of whole numbers by choosing a suitable multiplier.
4) Enclose in curvilinear brackets.

  1. The plane intercepts at 3a1, 2a2, 2a3
  2. Reciprocals: 1/3, 1/2, 1/2
  3. The smallest three integers having the same ratio: 2,3,3
  4. The indices of the plane is (233)—(hkl) plane notation.

Common errors:
1) Forget to invert the intercepts
2) Forget to enclose in curvilinear brackets.

Indexing Rules:

  • For intercept at infinity, the corresponding index is zero.
  • If a plane cuts an axis on the negative side of the origin, the corresponding index is negative, indicated by placing a minus sign above the index:
  • If the plane passes through the origin of coordinates, the origin of coordinates must be moved to a lattice point not on the plane to be indexed.
  • All planes which fold into each other upon application of crystalsymmetry operations* cannot be distinguished from each other by any physical measurement and therefore are said to be “equivalent”.
    A group of equivalent planes is denoted by braces around the indices.
  • For example
  • are equivalent planes and are collectively denoted as planes.
    {100} planes.

Example: Planes

Distance between adjacent (hkl) planes

In cubic crystal structures, the interplanar spacing between two closest parallel planes with the same Miller indices is designated dhkl

where a is the lattice constant.

d= distance from a selected origin containing one plane and another parallel plane with the same indices which is closest to it.

Example 1:

Example 2:

The plane intercepts at 2a1, 2a2, 2a3

  • Reciprocals:
    1/2, 1/2, 1/2, the indices of the plane are (111)