Miller Indices Calculator
Enter the Miller indices (h, k, l) and choose a crystal system to calculate the interplanar spacing (d-spacing) between lattice planes. Optionally enter an X-ray wavelength to get the Bragg diffraction angle. Select a material preset to fill lattice constants automatically, or enter your own. Results update instantly as you type.
What are Miller indices?
Miller indices are a shorthand notation for identifying planes and directions inside a crystalline solid. A plane is described by three integers (h, k, l) that represent the reciprocals of the intercepts the plane makes with the three crystallographic axes, cleared to the smallest whole-number ratio. For example, a plane that cuts the x-axis at 1, the y-axis at 2, and is parallel to the z-axis (intercept at infinity) has reciprocals 1, 1/2, 0, which after clearing fractions gives Miller indices (2 1 0). Negative intercepts are written with an overbar over the digit: (1 0 -1) in text or with a bar above the 1. The notation was introduced by William Hallowes Miller in 1839 and is now universal across solid-state physics, mineralogy, and materials science.
How to calculate d-spacing from Miller indices
The interplanar spacing d is the perpendicular distance between adjacent parallel planes with the same (hkl) index. For a cubic crystal (a = b = c, all angles 90 degrees), the formula is simple: d = a / sqrt(h squared + k squared + l squared). For tetragonal systems (a = b, c differs), the formula becomes 1/d squared = (h squared + k squared)/a squared + l squared/c squared. Orthorhombic crystals (all three constants differ) add a b term: 1/d squared = h squared/a squared + k squared/b squared + l squared/c squared. Hexagonal and trigonal crystals use the formula 1/d squared = (4/3)(h squared + hk + k squared)/a squared + l squared/c squared. This calculator applies the correct formula automatically when you select the crystal system.
Bragg's Law and X-ray diffraction
Bragg's Law connects the d-spacing to the X-ray wavelength and the angle at which constructive interference (diffraction) occurs: n*lambda = 2*d*sin(theta), where lambda is the wavelength, theta is the Bragg angle (half the measured diffraction angle), and n is the order of diffraction (typically 1). Rearranging gives theta = arcsin(lambda / 2d). The measurable peak position in a powder diffraction pattern is 2theta. For the copper K-alpha source (lambda = 1.5406 Angstroms) most commonly used in laboratory XRD, planes with very small d-spacings may not produce a real peak because lambda / 2d would exceed 1. Molybdenum K-alpha (lambda = 0.7107 Angstroms) or synchrotron sources with shorter wavelengths can access those reflections.
Crystal systems and their lattice parameters
Seven crystal systems are recognised in crystallography, distinguished by the relationships between their three lattice constants (a, b, c) and the angles between axes (alpha, beta, gamma). This calculator covers the five most common: cubic (a = b = c, all angles 90 degrees), tetragonal (a = b, c different, all angles 90 degrees), orthorhombic (a, b, c all different, all angles 90 degrees), hexagonal (a = b, c different, alpha = beta = 90 degrees, gamma = 120 degrees), and trigonal/rhombohedral (same hexagonal convention). The material presets include lattice constants from crystallographic databases for 15 common materials including silicon, copper, iron, aluminium, and zinc. Select Custom to enter any set of lattice constants manually.
Miller index notation guide
| Notation | Meaning | Example |
|---|---|---|
| (hkl) | A specific lattice plane | (110) - a single plane |
| {hkl} | Family of equivalent planes (all symmetry-related) | {100} = (100),(010),(001) in cubic |
| [hkl] | A specific crystallographic direction | [111] - body diagonal in cubic |
| <hkl> | Family of equivalent directions | <100> = [100],[010],[001] in cubic |
| h k l with overbar | Negative index (plane intersects on negative axis side) | (1 -1 0) or written with overbar |
| (hkil) | Miller-Bravais notation for hexagonal systems (h+k+i=0) | (1 0 -1 0) for hexagonal prism face |
Standard crystallographic notation for planes, directions, and families.
Frequently asked questions
What do the numbers in Miller indices represent?
Each integer in (h, k, l) is the reciprocal of the fractional intercept that the plane makes with the corresponding crystallographic axis, scaled to the smallest whole-number ratio. A zero means the plane is parallel to that axis (intercept at infinity, reciprocal = 0). Large numbers indicate that the plane cuts the axis close to the origin, giving a high-index, densely spaced set of planes with a small d-spacing.
What is d-spacing and why does it matter?
d-spacing is the perpendicular distance between successive parallel lattice planes with the same (hkl) index. It determines which angles a given crystal diffracts X-rays according to Bragg's Law. Measured d-spacings from an XRD pattern are compared against reference databases (such as the PDF-2 or ICDD database) to identify unknown phases and confirm crystal structure.
How do I find Miller indices from intercepts?
Step 1: find where the plane intercepts each crystallographic axis, expressed as multiples of the unit cell length (e.g. the plane cuts at x = 1, y = 2, z = infinity). Step 2: take the reciprocal of each intercept (1, 1/2, 0). Step 3: multiply through by the smallest integer that clears all fractions, here 2, to get (2 1 0). Enclose in parentheses and those are the Miller indices.
What is the difference between (hkl), {hkl}, [hkl] and <hkl>?
Round parentheses (hkl) denote a specific plane; curly braces {hkl} denote the full family of planes related by the crystal's symmetry (for example {100} in a cubic crystal includes the six faces of a cube). Square brackets [hkl] denote a specific direction vector perpendicular to the (hkl) plane in cubic systems; angle brackets <hkl> denote the family of symmetry-equivalent directions.
Why does the formula change between crystal systems?
The d-spacing formula comes from the geometry of the unit cell. In a cubic cell all sides are equal and all angles are right angles, which gives the simplest form. When the cell is stretched along one axis (tetragonal) or along two axes (orthorhombic), extra terms appear to account for the different lengths. In hexagonal and trigonal systems the 120-degree angle between the a and b axes introduces the cross term hk in the formula.
What X-ray wavelength should I use for the Bragg angle calculation?
The most common laboratory source is copper K-alpha radiation at 1.5406 Angstroms. Molybdenum K-alpha (0.7107 Angstroms) is used when shorter wavelengths are needed to access small d-spacings or to reduce fluorescence from certain elements. Synchrotron sources are tunable. If your measured 2theta peaks do not match the calculated values, double-check which anode material your diffractometer uses.