d-Spacing from Miller Indices Calculator
Find the interplanar spacing d from Miller indices (h,k,l) for cubic, tetragonal, and orthorhombic crystal systems.
🧊 What is d-Spacing from Miller Indices?
The d-spacing from Miller indices calculator finds the interplanar spacing d, the perpendicular distance between a set of parallel crystal planes, from the plane's Miller indices (h,k,l) and the crystal's lattice parameters. Every crystal structure is built from a repeating unit cell, and every diffraction measurement (Bragg's law, X-ray diffraction, neutron diffraction) needs this exact d-spacing as an input, so calculating it correctly from lattice geometry is a foundational step in crystallography.
This calculator covers the three most common crystal systems in increasing order of generality. Cubic crystals (silicon, diamond, many metals and salts) have the simplest geometry, one lattice parameter a shared equally in all three directions. Tetragonal crystals (rutile TiO₂, many oxide ceramics) stretch or compress one axis relative to the other two, needing a separate parameter c. Orthorhombic crystals (many minerals, some organic crystals) have three fully independent edge lengths a, b, and c, all still meeting at 90° angles.
A common misconception is that Miller indices behave like ordinary coordinates. They don't, they describe an entire family of parallel planes, not a single point, and only their squares appear in the d-spacing formulas, so the sign of h, k, or l never changes the calculated spacing. The one invalid input is (0,0,0), which does not correspond to any physical plane at all.
This calculator is useful for materials science and crystallography students checking a d-spacing by hand, for anyone indexing an X-ray diffraction pattern, and as a companion to the Bragg Law Calculator, which takes this same d value as a direct input.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Silicon's (111) Plane, Cubic
Cubic: a = 5.43 Å (silicon's real lattice parameter), (h,k,l) = (1,1,1)
Example 2 — Rutile TiO₂'s (110) Plane, Tetragonal
Tetragonal: a = 4.59 Å, c = 2.96 Å (real rutile TiO₂ parameters), (h,k,l) = (1,1,0)
Example 3 — An Orthorhombic Crystal's (211) Plane
Orthorhombic: a = 5.0 Å, b = 6.0 Å, c = 7.0 Å, (h,k,l) = (2,1,1)
❓ Frequently Asked Questions
🔗 Related Calculators
What is the d-spacing in crystallography?
The d-spacing (or interplanar spacing) is the perpendicular distance between adjacent, parallel planes of atoms in a crystal lattice, identified by a set of Miller indices (h,k,l). It directly sets the diffraction angle for those planes through Bragg's law, nλ = 2d sinθ.
What are Miller indices?
Miller indices (h,k,l) are three integers that identify a family of parallel crystal planes and their orientation relative to the unit cell axes. They can be positive, negative, or zero, but cannot all be zero simultaneously since (0,0,0) does not correspond to any real plane.
How do you calculate d-spacing for a cubic crystal?
Use 1/d² = (h²+k²+l²)/a², so d = a/√(h²+k²+l²). For silicon (a=5.43 Å) and the (111) plane, d = 5.43/√3 ≈ 3.135 Å, matching the real measured spacing for silicon's most prominent diffraction peak.
How is the tetragonal formula different from cubic?
Tetragonal crystals have two equal edge lengths and one different (a=b≠c), so the formula splits the in-plane and out-of-plane contributions: 1/d² = (h²+k²)/a² + l²/c². Setting a=c recovers the cubic formula exactly.
How is the orthorhombic formula different from tetragonal?
Orthorhombic crystals have three independent edge lengths (a≠b≠c, all angles 90°), so each Miller index gets its own lattice parameter: 1/d² = h²/a² + k²/b² + l²/c². Setting a=b recovers the tetragonal formula, and setting a=b=c recovers the cubic formula.
Can Miller indices be negative?
Yes. Negative indices (conventionally written with a bar over the number, such as 1̄) represent planes on the opposite side of the origin. Since only the squares of h, k, and l appear in every d-spacing formula, the sign does not affect the calculated spacing.
Why can't h, k, and l all be zero?
The indices (0,0,0) do not correspond to any physical plane, there is no set of parallel planes with that orientation, so the d-spacing formula is undefined (it would require dividing by zero). This calculator blocks that input combination with a clear error.
What is the reciprocal spacing 1/d used for?
1/d is the magnitude of the reciprocal lattice vector for that set of planes (up to a factor of 2π, depending on convention). It appears directly in the scattering vector formula q = 2π/d used throughout diffraction physics and reciprocal-space analysis.
What units does this calculator use?
Angstroms (Å) for all lattice parameters (a, b, c) and for the resulting d-spacing, the standard unit in crystallography since 1 Å = 0.1 nanometers conveniently matches the scale of real interatomic spacings.
How does this relate to the Bragg Law Calculator?
The d-spacing found here is exactly the d value used in Bragg's law, nλ = 2d sinθ. Compute d from Miller indices and lattice parameters here, then plug it into the Bragg Law Calculator to find the diffraction angle θ for a given wavelength.