Miller Indices
(hkl) notation for crystal planes. d = a/√(h²+k²+l²) for cubic. Bragg's Law: nλ = 2d·sin(θ). X-ray diffraction, crystal orientation.
Why This Chemistry Calculation Matters
Why: Miller indices identify crystal planes for XRD, epitaxy, surface science. d-spacing from Bragg's Law.
How: (hkl) = reciprocals of intercepts, reduced. Cubic: d = a/√(h²+k²+l²). Bragg: nλ = 2d·sin(θ).
- ●(100), (110), (111) are most common planes.
- ●Cu Kα λ = 1.5406 Å for X-ray diffraction.
- ●Silicon (111) is primary semiconductor wafer orientation.
Crystal Plane Examples
🔷 (100) Plane
Cube face - most common surface plane
🔶 (110) Plane
Diagonal plane through cube
💎 (111) Plane
Octahedral close-packed plane
📐 (200) Plane
Second-order reflection from (100)
🔺 (220) Plane
Second-order reflection from (110)
🔬 Cu Kα X-ray Diffraction
Bragg angle for Cu Kα (λ = 1.5406 Å)
📊 Intercepts to Miller Indices
Convert intercepts (2, 3, 4) to Miller indices
💻 Silicon (111) Plane
Common semiconductor crystal plane
⚙️ Iron (110) Plane
BCC metal cleavage plane
🔩 Aluminum (200) Plane
FCC metal diffraction plane
💎 Diamond (111) Plane
Diamond structure close-packed plane
🧂 NaCl (200) Plane
Rock salt structure plane
Calculate Miller Indices
For educational and informational purposes only. Verify with a qualified professional.
🔬 Chemistry Facts
(hkl) = reciprocal of intercepts, reduced to integers.
— IUCr
d = a/√(h²+k²+l²) for cubic systems.
— Crystallography
Bragg's Law: nλ = 2d·sin(θ).
— X-ray diffraction
(111) is close-packed plane in FCC.
— IUPAC
What are Miller Indices?
Miller indices (hkl) are a notation system used in crystallography to describe the orientation of crystal planes. They are three integers that represent the reciprocal of the intercepts that a plane makes with the crystal axes, reduced to the smallest integers.
h, k, l are integers representing plane orientation
Common Crystal Planes
| Plane | Miller Indices | Description | Common Uses |
|---|---|---|---|
| (100) | (100) | Cube face plane | Surface studies, Epitaxial growth |
| (110) | (110) | Diagonal plane | Cleavage planes, Slip systems |
| (111) | (111) | Octahedral plane | Close-packed planes, Diamond structure |
| (200) | (200) | Second-order (100) | X-ray diffraction, Crystal structure |
| (220) | (220) | Second-order (110) | Diffraction analysis, Texture studies |
| (210) | (210) | Mixed indices plane | Crystal orientation |
| (211) | (211) | Complex plane | Crystal defects |
| (310) | (310) | High-index plane | Surface science |
Key Concepts
Miller Indices
Three integers (hkl) that uniquely identify a crystal plane. Zero indicates the plane is parallel to that axis.
d-spacing
Interplanar spacing between parallel crystal planes. For cubic systems: d = a/√(h²+k²+l²)
Bragg's Law
nλ = 2d·sin(θ). Relates X-ray wavelength, interplanar spacing, and diffraction angle.
How to Calculate Miller Indices
Miller indices are determined from the intercepts that a crystal plane makes with the three crystallographic axes. The process involves taking reciprocals and reducing to smallest integers.
🔬 Step-by-Step Process
1. Find Intercepts
Determine where plane intersects:
a-axis: p
b-axis: q
c-axis: r
2. Take Reciprocals
h = 1/p
k = 1/q
l = 1/r
3. Clear Fractions
Multiply by common denominator
to get integers
4. Reduce to Simplest Form
Divide by GCD
Result: (hkl)
📐 Example: (111) Plane
Given: Plane intercepts axes at a=1, b=1, c=1
h = 1/1 = 1
k = 1/1 = 1
l = 1/1 = 1
Miller indices: (111)
Important Formulas
d-spacing for Cubic Systems
Where a is the lattice parameter, and h, k, l are Miller indices. This formula applies only to cubic crystal systems.
Bragg's Law
Where n is the diffraction order, λ is the X-ray wavelength, d is the interplanar spacing, and θ is the Bragg angle (half the scattering angle).
Miller Indices from Intercepts
Where p, q, r are intercepts on a, b, c axes respectively. Then reduce to smallest integers.
When to Use Miller Indices
Miller indices are essential for anyone working with crystalline materials, X-ray diffraction, or materials characterization.
X-ray Crystallography
Identify crystal structures, determine unit cell parameters, and analyze diffraction patterns.
- Single crystal XRD
- Powder diffraction
- Structure refinement
Semiconductor Physics
Understand crystal orientation, epitaxial growth, and device fabrication.
- Wafer orientation
- Epitaxial layers
- Surface preparation
Materials Science
Analyze crystal defects, texture, and mechanical properties.
- Grain orientation
- Slip systems
- Deformation modes
Practical Examples
Example: Silicon (111) Plane
Given:
- Miller indices: (111)
- Lattice parameter a = 5.431 Å
- Cubic crystal system
Solution:
d = a / √(h² + k² + l²)
d = 5.431 / √(1² + 1² + 1²)
d = 5.431 / √3
d = 3.135 Å
Example: Bragg Angle for Cu Kα Radiation
Given:
- Miller indices: (111)
- d-spacing = 3.135 Å
- Wavelength λ = 1.5406 Å (Cu Kα)
- Order n = 1
Solution:
nλ = 2d·sin(θ)
sin(θ) = nλ / (2d)
sin(θ) = 1 × 1.5406 / (2 × 3.135)
θ = arcsin(0.2457)
θ = 14.22°
Crystal Systems
Different crystal systems have different formulas for calculating d-spacing. The cubic system formula is the simplest and most commonly used.
| System | Lattice Parameters | d-spacing Formula |
|---|---|---|
| Cubic | a = b = c, α = β = γ = 90° | d = a/√(h²+k²+l²) |
| Tetragonal | a = b ≠ c, α = β = γ = 90° | More complex - requires additional parameters |
| Orthorhombic | a ≠ b ≠ c, α = β = γ = 90° | More complex - requires additional parameters |
| Hexagonal | a = b ≠ c, α = β = 90°, γ = 120° | More complex - requires additional parameters |
| Monoclinic | a ≠ b ≠ c, α = γ = 90° ≠ β | More complex - requires additional parameters |
| Triclinic | a ≠ b ≠ c, α ≠ β ≠ γ | More complex - requires additional parameters |
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