---
title: "Miller indices of planes and directions"
subject: "Material Science"
chapter: 2
key: "semester_3/material_science:miller-indices-planes-directions"
source: https://www.bctengineer.com/q/semester_3/material_science/miller-indices-planes-directions
site: BCT Engineering
---

# Miller indices of planes and directions

Material Science · Chapter 2: Atomic Structure, arrangement of atoms

- Asked in: year not given, 6 marks
- Video: [Working with Crystallographic Planes and Miller Indices](https://www.youtube.com/watch?v=JS9ysbgr0BE) — Scott Ramsay
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/material_science/miller-indices-planes-directions
- Whole chapter: https://www.bctengineer.com/md/semester_3/material_science/chapter-2

## Question

Explain the procedure for finding Miller indices of a crystal plane. Find the Miller indices of (a) a plane making intercepts of 2a, 3b and 6c on the three axes of a cubic cell, and (b) a plane making intercepts a/2, b and parallel to the c-axis. Also find the Miller indices of the direction joining the origin to the point (1/2, 1, 0), and of the direction from point (1, 0, 0) to point (0, 1, 1).

## Short answer

For planes, take reciprocals of intercepts in lattice constants and clear fractions: (a) (321), (b) (210). For directions, subtract tail from head coordinates and reduce to smallest integers: [120] and [̅1̅11].

## Answer

**Miller indices** $(hkl)$ are a set of three smallest integers that describe the orientation of a crystal plane.

#### Procedure for planes
1. Find the intercepts of the plane on the $x, y, z$ axes in terms of lattice constants $a, b, c$ (use $\infty$ if parallel).
2. Take the reciprocals of the intercepts.
3. Clear fractions by multiplying by the smallest common factor.
4. Write the result in round brackets without commas, $(hkl)$; a negative index has a bar over it.

#### (a) Intercepts 2, 3, 6

| Step | x | y | z |
|---|---|---|---|
| Intercepts | 2 | 3 | 6 |
| Reciprocals | 1/2 | 1/3 | 1/6 |
| Multiply by 6 | 3 | 2 | 1 |

Plane is $(321)$.

#### (b) Intercepts 1/2, 1, $\infty$

| Step | x | y | z |
|---|---|---|---|
| Intercepts | 1/2 | 1 | $\infty$ |
| Reciprocals | 2 | 1 | 0 |

Plane is $(210)$.

#### Directions
Procedure: subtract the coordinates of the tail from the head, clear fractions to the smallest integers, and write in square brackets $[uvw]$.

- Origin to $(\tfrac12, 1, 0)$: components $(\tfrac12, 1, 0)$; multiply by 2, giving $[120]$.
- $(1,0,0)$ to $(0,1,1)$: components $(0-1,\ 1-0,\ 1-0) = (-1, 1, 1)$, written $[\bar{1}11]$.

**Answer: (a) (321); (b) (210); directions [120] and $[\bar{1}11]$.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the correct order of steps for Miller indices of a plane?
   A. Reciprocals, intercepts, clear fractions
   B. Intercepts, reciprocals, clear fractions
   C. Intercepts, clear fractions, reciprocals
   D. Clear fractions, intercepts, reciprocals
   Answer: B. Find intercepts, take reciprocals, clear fractions, then write in (hkl). (Key point: Procedure)

2. Intercepts 2, 3, 6 give which plane?
   A. (236)
   B. (623)
   C. (111)
   D. (321)
   Answer: D. Reciprocals 1/2, 1/3, 1/6 times 6 give (321). (Key point: Plane (a))

3. What index results for a plane parallel to the c-axis?
   A. 0, because intercept is infinity
   B. 1
   C. Infinity
   D. Cannot be indexed
   Answer: A. The reciprocal of an infinite intercept is zero; the plane (b) is (210). (Key point: Plane parallel to an axis)

4. How is the direction from (1,0,0) to (0,1,1) found?
   A. Head minus tail: [-1 1 1], written with a bar
   B. Tail minus head: [1 -1 -1] only
   C. Add coordinates: [1 1 1]
   D. Reciprocals: [0 1 1]
   Answer: A. Subtract tail from head giving (-1,1,1) = [1bar 1 1]. (Key point: Direction from two points)

## Related in Material Science

- [BCC, FCC and HCP crystal structures](https://www.bctengineer.com/question.md?key=semester_3%2Fmaterial_science%3Abcc-fcc-hcp-structures) — Material Science, chapter 2
- [Planar and linear density of BCC iron](https://www.bctengineer.com/question.md?key=semester_3%2Fmaterial_science%3Aplanar-and-linear-density-bcc-iron) — Material Science, chapter 2

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