Miller Indices Calculator
Our miller indices calculator helps determine crystallographic plane indices, interplanar spacing, and related parameters for various crystal systems.
If you have intercepts at 2, 3, 4 along the x, y, z axes in a cubic crystal with lattice parameter 3.615Å, the calculator process these parameters to determine the Miller indices (6,4,3) and related crystallographic information.
How to Use this miller indices calculator
- Choose between indices, spacing, or angle calculations
- Select cubic, tetragonal, orthorhombic, or hexagonal
- Input intercepts or indices and lattice parameters
The calculator will compute the reciprocals of the intercepts, normalize them to integers, and display the Miller indices.
Miller Indices Formula
Indices from Intercepts Formula:
(h,k,l) = (1/x, 1/y, 1/z) × LCMInterplanar Spacing Formula (Cubic):
d = a/√(h² + k² + l²)Bragg Angle Formula:
θ = arcsin(λ/(2d))How to Calculate Miller Indices
Determine the intercepts where the plane crosses each crystallographic axis
Transform any negative intercepts using a bar notation (overline)
Calculate reciprocals of these values
Eliminate fractions to obtain the smallest whole numbers
Present the final numbers in parentheses (hkl)
What are Typical Parameters?
For different systems, typical features include:
- Cubic: All angles = 90°, a = b = c
- Tetragonal: All angles = 90°, a = b ≠ c
- Orthorhombic: All angles = 90°, a ≠ b ≠ c
- Hexagonal: α = β = 90°, γ = 120°, a = b ≠ c
Given a plane with intercepts (1,-2,4)
- List intercepts: 1, -2, 4
- Calculate reciprocals: 1, -1/2, 1/4
- Multiply by 4 to clear fractions: 4, -2, 1
- Final Miller indices: (4̄21)
For intercepts (2,3,4), calculations would be:
(h,k,l) = (1/2, 1/3, 1/4) × 12 = (6,4,3)Miller Indices Examples
- Parallel to xy-plane: Intercepts: (∞,∞,1), Reciprocals: (0,0,1), Miller indices: (001)
- Diagonal plane: Intercepts: (1,1,1), Reciprocals: (1,1,1), Miller indices: (111)
- Complex intersection: Intercepts: (2,-3,6), Reciprocals: (1/2,-1/3,1/6), Multiply by 6: (3,-2,1), Miller indices: (3̄21)
- Plane parallel to z-axis: Intercepts: (2,3,∞), Reciprocals: (1/2,1/3,0), Multiply by 6: (3,2,0), Miller indices: (320)
- Negative intercepts: Intercepts: (-1,2,-2), Reciprocals: (-1,1/2,-1/2), Multiply by 2: (-2,1,-1), Miller indices: (2̄11̄)
What is Miller Indices?
Miller indices is a crystallographic notation system that describes planes and directions in crystal lattices using three integers (h,k,l). These numerical indicators denote the reciprocals of the fractional intercepts that a plane makes with the crystallographic axes. The significance of Miller indices lies in their ability to efficiently communicate complex spatial relationships within crystal structures.