Miller Indices Calculator

Calculate and Analyze Crystal Planes

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Plane Spacing Calculator

Calculate interplanar spacing and angles between crystal planes.

Plane Parameters: -

Direction Calculator

Calculate angles between directions and convert between notations.

Direction Parameters: -

Understanding Miller Indices

Basic Concepts

Miller indices notation:

  • (hkl): Single plane
  • {hkl}: Family of planes
  • [uvw]: Direction
  • <uvw>: Family of directions

Special Planes

Important crystal planes:

  • (100): Cube face
  • (110): Face diagonal
  • (111): Body diagonal
  • (hh0): Zone axis planes

Applications

Miller indices are used in:

  • X-ray diffraction
  • Crystal growth
  • Surface science
  • Materials engineering
  • Semiconductor fabrication

Advanced Topics

Additional concepts:

  • Zone Axis: Common line to planes
  • Stereographic Projection: Plane visualization
  • Weiss Zone Law: hu + kv + lw = 0
  • Reciprocal Space: Diffraction analysis

Essential Miller Indices Formulas

Plane Spacing (Cubic)

d = a/√(h² + k² + l²)

Angle Between Planes

cos φ = (h₁h₂ + k₁k₂ + l₁l₂)/√[(h₁² + k₁² + l₁²)(h₂² + k₂² + l₂²)]

Zone Equation

hu + kv + lw = 0