Miller Indices Calculator
Calculate interplanar distances and analyze crystal plane orientations
Miller Indices Calculator
Cubic unit cell lattice parameter in Angstroms
📐 Crystallographic Analysis
Plane Type: Prismatic (parallel to yz)
Planar Density: 0.078595 atoms/Ų
Formula: d = a/√(h² + k² + l²)
Calculation: 3.567/√(1² + 0² + 0²) = 3.5670 Å
🔬 X-ray Diffraction Analysis
Bragg's Law: nλ = 2d sin(θ)
Calculation: θ = arcsin(1.5406/(2 × 3.5670)) = 12.47°
🧊 Crystal Structure Properties
👥 Equivalent Planes in Family {100}
Multiplicity: 6 equivalent planes
Family notation: Curly braces {100} denote the complete family of equivalent planes
📊 Geometric Analysis
Normal Vector: [1, 0, 0]
Magnitude: √(1² + 0² + 0²) = 1.000
Example: Miller Indices for Diamond
Given Parameters
Material: Diamond (C)
Lattice constant: 3.567 Å
Miller indices: (111)
Crystal system: Diamond Cubic
Calculation
Formula: d₁₁₁ = a/√(h² + k² + l²)
Substitution: d₁₁₁ = 3.567/√(1² + 1² + 1²)
Calculation: d₁₁₁ = 3.567/√3 = 3.567/1.732
Result: d₁₁₁ = 2.059 Å
Notation Guide
(hkl)
Individual plane
{hkl}
Family of equivalent planes
[hkl]
Direction vector
⟨hkl⟩
Family of directions
h̄kl
Negative index (bar notation)
Common d-spacings
Crystal Systems
Cubic
a = b = c, α = β = γ = 90°
Tetragonal
a = b ≠ c, α = β = γ = 90°
Orthorhombic
a ≠ b ≠ c, α = β = γ = 90°
Hexagonal
a = b ≠ c, α = β = 90°, γ = 120°
Calculation Tips
Miller indices are reciprocals of plane intercepts
d-spacing formula: d = a/√(h² + k² + l²) for cubic
Higher indices mean smaller d-spacings
Structure factor rules determine allowed reflections
Bragg's law: nλ = 2d sin(θ) for X-ray diffraction