miller index
C2Highly Technical / Scientific
Definition
Meaning
A notation system in crystallography for identifying and describing lattice planes and directions within a crystal structure.
A triplet of integers (hkl) or a quadruplet (hkil) for hexagonal systems, derived from the reciprocals of the fractional intercepts of a plane with the crystallographic axes, used universally to specify atomic planes and directions in materials science, mineralogy, and solid-state physics.
Linguistics
Semantic Notes
Always capitalised (Miller) as it is an eponym (named after William Hallowes Miller). It refers to a specific, mathematical notation, not a general index. The term is singular but often used in the plural (Miller indices) when referring to multiple sets.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. Spelling and pronunciation are consistent across scientific English.
Connotations
Identical. Purely technical and precise.
Frequency
Identical frequency within relevant technical fields (materials science, physics, chemistry, geology). Virtually non-existent in general discourse.
Vocabulary
Collocations
Grammar
Valency Patterns
The Miller index **of** the slip plane is (110).The plane **with** Miller indices (hkl) is oriented...To index a plane **using** Miller indices.Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “N/A”
Usage
Context Usage
Business
Virtually never used.
Academic
Exclusively used in scientific papers, textbooks, and lectures in crystallography, materials science, solid-state physics, and mineralogy.
Everyday
Not used.
Technical
The primary context. Used to specify crystal planes for X-ray diffraction, electron microscopy, describing crystal growth directions, and mechanical properties like slip systems.
Examples
By Part of Speech
verb
British English
- N/A
American English
- N/A
adverb
British English
- N/A
American English
- N/A
adjective
British English
- N/A
American English
- N/A
Examples
By CEFR Level
- N/A
- N/A
- Scientists use numbers called Miller indices to describe crystal faces.
- The (100) plane is a simple example of a Miller index.
- To calculate the Miller index, one must take the reciprocal of the plane's intercepts with the crystallographic axes.
- The slip system in FCC metals is typically on the {111} plane in the <110> direction, where the curly brackets denote a family of planes with equivalent Miller indices.
Learning
Memory Aids
Mnemonic
MILLER: My Index Labels Lattice Edges Reciprocally.
Conceptual Metaphor
A CRYSTALLOGRAPHIC ADDRESS SYSTEM (like coordinates on a map, but for atomic planes inside a crystal).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid translating 'index' as 'индекс' in the general sense. It is a specific 'обозначение' or 'индекс Миллера'.
- Do not confuse with 'Miller effect' in electronics, which is unrelated.
- The word order is fixed: 'Miller index', not 'index of Miller' in normal use.
Common Mistakes
- Writing 'miller index' in lowercase.
- Using 'Miller's index' (possessive) incorrectly; the standard term is non-possessive.
- Forgetting to enclose the indices in parentheses: e.g., writing 111 instead of (111).
- Confusing plane indices (hkl) with direction indices [uvw].
Practice
Quiz
What do the parentheses ( ) around Miller indices, like (110), specifically denote?
FAQ
Frequently Asked Questions
They were introduced by the British mineralogist William Hallowes Miller in 1839.
A negative index, written with a bar over the number (e.g., h̅), indicates an intercept on the negative side of the crystallographic axis.
The principle is the same, but hexagonal and trigonal systems often use a four-index (hkil) Miller-Bravais notation for symmetry, where i = -(h+k).
Parentheses (hkl) denote a specific crystal plane. Square brackets [hkl] denote a specific crystallographic direction.