# 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.

- The plane intercepts at
**3a**_{1}, 2a_{2}, 2a_{3} - Reciprocals:
**1/3, 1/2, 1/2** - The smallest three integers having the same ratio:
**2,3,3** - 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 d_{hkl}

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 2a _{1}, 2a_{2}, 2a_{3}**

**Reciprocals:**

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