Miller Indices Calculator
Calculate miller indices with our free science calculator. Uses standard scientific formulas with unit conversions and explanations.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
1/d2 = h2/a2 + k2/b2 + l2/c2
Miller indices (hkl) are obtained by taking reciprocals of axis intercepts and clearing fractions. The d-spacing for orthorhombic and higher symmetry systems follows 1/d2 = h2/a2 + k2/b2 + l2/c2. For cubic systems, this simplifies to d = a/sqrt(h2+k2+l2). Bragg diffraction occurs at angles satisfying n*lambda = 2d*sin(theta).
Worked Examples
Example 1: Cubic (111) Plane
Problem:A plane intercepts all three axes at 1 lattice parameter each in a cubic crystal (a = 3.52 angstrom).
Solution:Intercepts: a=1, b=1, c=1\nReciprocals: 1/1, 1/1, 1/1 = 1, 1, 1\nMiller indices: (1 1 1)\nd-spacing = 3.52 / sqrt(1+1+1) = 3.52 / 1.732 = 2.033 angstrom\nBragg angle (Cu K-alpha): 2theta = 2 arcsin(1.5406 / (2 x 2.033)) = 44.5 degrees
Result:(1 1 1) | d = 2.033 angstrom | 2theta = 44.5 degrees
Example 2: Plane with Infinity Intercept
Problem:A plane intercepts a at 1, b at 2, and is parallel to c in an orthorhombic crystal.
Solution:Intercepts: a=1, b=2, c=infinity\nReciprocals: 1/1, 1/2, 1/infinity = 1, 0.5, 0\nMultiply by 2 for whole numbers: (2 1 0)\nThis plane is parallel to the c-axis\nThe family {210} has 24 equivalent planes in cubic symmetry
Result:(2 1 0) | Parallel to c-axis | Multiplicity = 24
Frequently Asked Questions
What are Miller indices and how are they determined?
Miller indices are a set of three integers (hkl) that describe the orientation of a crystallographic plane in a crystal lattice. To determine them, you first find where the plane intercepts each crystallographic axis (a, b, c) in terms of lattice parameters. Then take the reciprocals of these intercepts, and finally multiply by the smallest common factor to get whole numbers. For example, a plane intercepting at a=2, b=3, c=infinity gives reciprocals 1/2, 1/3, 0, which multiply to (3 2 0). Negative intercepts are denoted with a bar over the number. Miller indices are essential for describing crystal faces, X-ray diffraction patterns, and surface chemistry.
What is d-spacing and how is it calculated from Miller indices?
The d-spacing (interplanar spacing) is the perpendicular distance between adjacent parallel planes with the same Miller indices in a crystal. For a cubic crystal with lattice parameter a, the d-spacing is calculated as d = a / sqrt(h2 + k2 + l2). For orthorhombic crystals with parameters a, b, c: 1/d2 = h2/a2 + k2/b2 + l2/c2. The d-spacing is directly measurable through X-ray diffraction using Braggs law (n times lambda = 2d sin theta), making it one of the most important quantities in crystallography. Higher Miller indices correspond to smaller d-spacings and planes that are more closely spaced but with less atomic density.
How are Miller indices used in X-ray diffraction analysis?
In X-ray diffraction (XRD), Miller indices identify each diffraction peak by the crystal plane responsible for that reflection. According to Braggs law, constructive interference occurs when n times lambda equals 2d times sin theta, where d is the interplanar spacing determined by the Miller indices. By indexing a diffraction pattern (assigning hkl values to each peak), crystallographers can determine the crystal structure, lattice parameters, and unit cell dimensions. Systematic absences in the diffraction pattern, where certain hkl combinations produce no reflection due to the lattice type or space group symmetries, provide additional structural information about the centering and symmetry elements present.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy