# Miller Indices Calculator

Created by Rahul Dhari
Reviewed by Steven Wooding
Last updated: Dec 30, 2022

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 (cf. atom calculator) 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 the 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:

$\quad d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$

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:

1. Locate the x, y, and z axes intercept for the crystal face.
2. Find the reciprocal of the intercept to obtain the Miller indices.
3. Enter the lattice constant for the cubic cell.
4. Fill in the Miller indices, $(hkl)$.
5. 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. You can also explore our cubic cell calculator or lattice energy calculator for more information on the topic.

## Example: Using the Miller indices calculator

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

1. Enter the lattice constant for the cubic cell as $2\r{\text{ A}}$.
2. Fill in the Miller indices as 2, 0 and 1, respectively.
3. The interplanar distance, $d_{hkl}$ is:
$\qquad \scriptsize d_{hkl} = \frac{2}{\sqrt{2^2 + 0^2 + 1^2}} = 0.8944\r{\text{ A}}$

## 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:

1. Find the x, y, and z intercepts for the face.
2. 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.

Rahul Dhari
Material
Element / compound
Carbon (diamond)
Lattice constant, A
Å
Miller indices
Miller index, h
Miller index, k
Miller index, l
Result
Interplanar distance, d
Å
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