Lesson 1, Topic 1
In Progress

# Planes and Directions

## 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.

## 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

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

## Example 2:

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

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