Miller indices are the universal crystallographic shorthand for identifying atomic planes (hkl) and directions [uvw] within a crystal lattice. Appearing throughout X-ray diffraction, electron backscatter diffraction (EBSD), transmission electron microscopy, and slip system analysis, Miller indices are the language of crystallography. Every peak in an X-ray diffraction pattern, every slip system formula, and every EBSD orientation map uses Miller index notation.
KEY TAKEAWAYS
- Plane indices (hkl) are derived from the reciprocal of the fractional intercepts on the three crystallographic axes.
- Direction indices [uvw] are the integer components of the direction vector in terms of unit cell dimensions.
- Negative indices are written with an overbar: (1̄10) means h=−1, k=1, l=0.
- Curly braces {hkl} denote a family of equivalent planes; angle brackets ⟨uvw⟩ denote a family of equivalent directions.
- In cubic systems ONLY, the plane (hkl) is perpendicular to the direction [hkl] — this simplification does not hold in hexagonal systems.
- Hexagonal metals use the 4-index Miller-Bravais notation (hkil) where i = −(h+k), preserving 3-fold symmetry.
- The d-spacing between (hkl) planes in cubic systems: d_hkl = a / √(h² + k² + l²) — the foundation of XRD peak indexing.
📷 IMAGE: Miller Indices Diagram: (100), (110) and (111) Planes in Cubic Unit Cell
Diagram showing the three most important crystallographic planes in a cubic unit cell: (100) cube face plane, (110) face diagonal plane, and (111) octahedral plane. These planes have successively higher atom density in FCC, with (111) being the close-packed slip plane.
Search terms: Miller indices crystal planes (100) (110) (111) cubic unit cell diagram
Source:
https://en.wikipedia.org/wiki/Miller_index
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Step-by-Step: How to Find Miller Indices of a Plane
If plane is parallel to an axis → intercept = ∞
If plane passes through origin → translate origin to adjacent cell
2. Take the RECIPROCAL of each intercept
∞ → 0 (plane parallel to that axis gives index 0)
3. Multiply by lowest common factor to get integers
4. Enclose in parentheses WITHOUT commas: (hkl)
Examples:
Intercepts x=1, y=∞, z=∞ → reciprocals 1,0,0 → (100) [cube face]
Intercepts x=1, y=1, z=1 → reciprocals 1,1,1 → (111) [octahedral plane, FCC slip plane]
Intercepts x=1, y=2, z=∞ → reciprocals 1,½,0 → ×2 → (210)
Intercepts x=1, y=−1, z=∞ → reciprocals 1,−1,0 → (11̄0) [overbar on negative index]
Important Planes in BCC and FCC Metals
| Plane (hkl) | Atom density | d-spacing (cubic) | Role in BCC | Role in FCC |
|---|---|---|---|---|
| (100) | Face — medium | a | Low-density; not slip plane | Medium; used in XRD |
| (110) | Face diagonal — high (BCC) | a/√2 | PRIMARY slip plane {110}⟨111⟩ | Used in XRD; not slip plane |
| (111) | Octahedral — highest (FCC) | a/√3 | Secondary slip; also cleavage in BCC | PRIMARY slip plane {111}⟨110⟩ |
| (112) | Complex | a/√6 | Secondary BCC slip plane | XRD identification |
| (200) | Same family as (100) | a/2 | XRD systematic absence absent in BCC | XRD first allowed peak in FCC |
Direction Indices [uvw] and Families ⟨uvw⟩
To find direction indices: draw the vector from the origin, read off x, y, z components as multiples of unit cell lengths, reduce to smallest integers, enclose in square brackets.
[110] = face diagonal (in FCC = close-packed direction)
[111] = body diagonal (in BCC = close-packed direction, slip direction)
[1̄10] = negative x, positive y, zero z
<100> = 6 cube edge directions: [100],[010],[001],[1̄00],[01̄0],[001̄]
<110> = 12 face diagonal directions (FCC slip directions)
<111> = 8 body diagonal directions (BCC slip directions)
Miller-Bravais Indices for HCP Metals (hkil)
| Miller-Bravais (hkil) | Plane Name | Significance in HCP Metals |
|---|---|---|
| (0001) | Basal plane | Primary slip plane in Mg, Zn, Co; perpendicular to c-axis |
| (101̄0) | Prismatic plane | Important secondary slip plane in Ti (enables ductility) |
| (101̄1) | Pyramidal plane | Allows ⟨c+a⟩ slip; critical for Ti polycrystal plasticity |
| [0001] | c-axis direction | Direction along which twins and ⟨c⟩ dislocations move |
| [112̄0] | a-axis direction | In-plane close-packed direction; ⟨a⟩ Burgers vector type |
📷 IMAGE: Miller Indices: Crystallographic Directions [100] [110] [111] in BCC
Principal crystallographic directions in a BCC unit cell: [100] cube edge, [110] face diagonal, [111] body diagonal (the BCC close-packed direction and slip direction). The ⟨111⟩ family of 8 body diagonals are the slip directions in all BCC metals including α-iron.
Search terms: crystallographic directions [100] [110] [111] BCC unit cell Miller indices
Source:
https://en.wikipedia.org/wiki/Miller_index#Crystallographic_direction_indices
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Frequently Asked Questions
Q: How are Miller indices used in X-ray diffraction?
A: Each XRD peak corresponds to Bragg reflection from a specific set of (hkl) planes separated by d-spacing d_hkl. Bragg’s Law: nλ = 2d sinθ. Measuring 2θ for each peak and solving for d_hkl identifies the crystal structure (cubic/HCP/BCC), lattice parameter, and specific phase. For cubic structures: d_hkl = a/√(h²+k²+l²). Peak positions follow systematic absence rules: BCC allows only h+k+l = even; FCC allows only all-odd or all-even indices. These rules immediately distinguish BCC from FCC in an X-ray pattern.
Q: What is the significance of the d-spacing in Miller indices?
A: The d-spacing d_hkl (interplanar spacing) is the perpendicular distance between adjacent parallel (hkl) planes. It determines: (1) the angle of XRD diffraction peak by Bragg’s law; (2) the elastic strain measured by XRD residual stress analysis (strain = Δd/d₀); (3) the stability of the plane against cleavage fracture (larger d-spacing = wider separation = weaker bond across plane = preferred cleavage). The (100) plane in BCC iron (d = a = 0.287 nm) is the most common cleavage plane in ferritic steel fracture at low temperatures.
Q: Why is it impossible to have a (000) Miller index?
A: The (000) plane would have infinite intercepts in all three directions — it would be a plane at the origin passing through all lattice points, which is not a specific crystallographic plane at all. Miller indices require at least one non-zero index to define a specific family of parallel planes with a finite d-spacing. By convention, the smallest possible non-zero integers are used (e.g. (100) not (200) for the cube face, unless specifically referring to the second-order diffraction from those planes in XRD, where (200) notation is conventional to indicate the second harmonic).
References
- Callister, W.D. and Rethwisch, D.G., Materials Science and Engineering. 10th ed. Wiley, 2018.
- Cullity, B.D. and Stock, S.R., Elements of X-Ray Diffraction. 3rd ed. Prentice Hall, 2001.
- Hammond, C., The Basics of Crystallography and Diffraction. 4th ed. Oxford University Press, 2015.
Related: BCC FCC HCP Crystal Structures · Slip Systems in Metals · XRD in Metallurgy
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