# Miller Indices Calculator

The Miller indices calculator determines the **interplanar distance for cubic crystal systems**. It was **introduced in 1839 by a British mineralogist named Prof. William Hallowes Miller**. Miller indices are part of **X-ray crystallography**, which are useful to study the **optical properties of atoms and molecules using their crystal structures**.

In addition to the optical properties, it is also helpful to **observe dislocations during the plastic deformations in an atomic structure**, i.e., the strain at the atomic level. The knowledge is crucial in **nanofabrication or machining nano wafers**. There are different kinds of crystal lattices; this calculator focuses on cubic structures in particular.

The topic is a fundamental concept in material sciences that has implications in various fields. This article will help you understand the definition of Miller indices and steps on how to calculate Miller indices.

## What are Miller indices?

Miller indices are the lynchpin in a notation system that denotes the family of lattice planes for a given Bravais lattice. The notation is written as $hkl$ but has several variations based on what it represents. For instance:

Notation | Represents |
---|---|

$(h,k,l)$ | Point |

$[hkl]$ | Direction |

$\text{<}hkl\text{>}$ | Family of directions |

$(hkl)$ | Plane |

${hkl}$ | Family of planes |

The Miller indices represent the **reciprocal of the intercepts a plane makes with each axis**. In a 3-dimensional plane, consider a crystal **face parallel to the XY plane**. The intercepts for the crystal face with the axes are $(\infin, \infin, 1)$. Therefore, the **Miller indices for the face are $(1/\infin, 1/\infin, 1) = (100)$**.

If the intercepts are fractions, the indices are **converted to integers by multiplying by their common denominator**. The **negative intercepts** are represented with a **bar** on the index.

Using the Miller indices and the lattice constant, you can determine the **interplanar distance** for the unit cell. Mathematically, this is:

where:

- $h,k,l$ – Miller indices;
- $a$ – Lattice constant; and
- $d_{hkl}$ – Interplanar distance.

## How to calculate Miller indices for planes

To calculate Miller indices using this calculator:

**Locate the x, y, and z axes intercept**for the crystal face.- Find the
**reciprocal of the intercept**to obtain the Miller indices. - Enter the
**lattice constant**for the cubic cell. - Fill in the
**Miller indices**, $(hkl)$. - The Miller indices calculator will return the interplanar distance for the cubic cell.

**Lattice constants**

The list of compounds and elements contains the lattice constant you can directly use upon selecting.

## Example: Using the Miller indices calculator

Find the interplanar distance for a cubic unit cell having Miller indices (201) and lattice constant $2\r{\text{ A}}$.

- Enter the
**lattice constant**for the cubic cell as $2\r{\text{ A}}$. - Fill in the
**Miller indices**as 2, 0 and 1, respectively. - The
**interplanar distance**, $d_{hkl}$ is:

## FAQ

### What do you mean by Miller indices?

The Miller indices are the **notation to depict a plane or a family of planes in a two or three-dimensional cartesian coordinate space**. You can also represent a **point using the Miller indices**. The parenthesis enclosure explains what the indices denote. For instance, if the **Miller indices are enclosed in ( ), e.g. (1,0,0) separated by commas, it represents a point**. The index inside

`( )`

**without commas represent a plane**such as

**(100)**or

**(001)**.

### How do you calculate Miller indices?

To determine the Miller indices:

**Find**the**x, y, and z intercepts**for the face.**Calculate the reciprocal**of the intercepts to obtain the Miller indices.

In the case of the Miller indices being fractions, multiply the index with the common denominator to convert them into integers.

### What are different types of crystal systems?

There are **seven different types of crystal systems** which are **triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic**. Each system has a different set of symmetry properties, such as the cubic system having four threefold axes of rotation. Some of the cubic crystal system elements are **nickel, copper, sodium chloride, silver, and gold**.

### What are some applications of Miller indices?

The **applications of Miller indices** include the field of **X-ray crystallography**, the study of material deformation in the plastic region considering the strains at the atomic level otherwise known as **dislocations**. Other areas include **diffraction, surface tension, nanofabrication, and machining**.