Miller Indices Calculator — d-Spacing & Bragg Angle (Cubic)
Enter the cubic lattice parameter a and Miller indices (h k l) to get the interplanar spacing d and the expected XRD Bragg angle 2θ. Defaults to nickel (a = 3.52 Å) and Cu Kα radiation.
Å
Å
Spacing between adjacent (hkl) planes — cubic crystal only
Formula for cubic interplanar spacing
Substitute values
Evaluate denominator
Interplanar spacing
- 1
Sum of squared indices
1² + 1² + 1² = 3 - 2
Square root of sum
√3 = 1.732051 - 3
Interplanar spacing d
3.52 ÷ 1.732051 = 2.0323d = a / √(h² + k² + l²) for cubic crystals.
How does this calculator work?
d = a / √(h²+k²+l²) for cubic crystals, where a is the lattice parameter and (h,k,l) are the Miller indices. Bragg's law (2d sin θ = λ) gives the XRD peak position. Enter a, h, k, l, and X-ray wavelength to get d in Angstroms and the expected 2θ diffraction angle.
Formula
How this is calculated
Miller indices (h, k, l) label families of parallel planes in a crystal lattice. They are defined as the smallest integers proportional to the reciprocals of the fractional intercepts the planes make on the three crystallographic axes. In a cubic system, where all axes are equal and mutually perpendicular, the interplanar spacing simplifies to d = a / √(h² + k² + l²), where a is the unit-cell edge length in Angstroms. Larger indices mean more closely-spaced planes and smaller d.
Bragg's law (2d sin θ = λ, n = 1 for first-order) relates d to the angle θ at which constructive interference (a diffraction peak) occurs for a given X-ray wavelength λ. Cu Kα radiation (λ = 1.5406 Å) is the most common laboratory source; Mo Kα (0.7107 Å) and Co Kα (1.7902 Å) are also widely used. If λ > 2d, the equation has no solution and no diffraction peak is geometrically possible.
This formula applies only to cubic (isometric) crystals such as NaCl (a = 5.64 Å), diamond (3.57 Å), iron α-Fe (2.87 Å, BCC), or nickel (3.52 Å, FCC). For tetragonal, hexagonal, orthorhombic, monoclinic, and triclinic systems, the d-spacing formula includes additional lattice parameters and is system-specific.
Frequently asked questions
Miller indices (h, k, l) describe a family of parallel, equally-spaced crystal planes. For example, (100) cuts only the a-axis; (111) cuts all three axes at equal fractional distance; (200) planes are twice as dense as (100), so d₂₀₀ = d₁₀₀/2. Negative indices are written with an overbar (e.g. h̄ = −h) and entered as negative numbers here.
In a cubic system all three lattice parameters (a = b = c) and all angles (α = β = γ = 90°) are equal, which greatly simplifies the metric tensor. Other crystal systems require separate formulas: tetragonal needs a and c; hexagonal needs a, c, and a cross-term; orthorhombic needs a, b, and c separately.
Copper Kα is the characteristic X-ray emitted when an electron drops into the K-shell of a copper anode. Its wavelength (1.5406 Å) is comparable to interatomic spacings in crystals, making it ideal for powder X-ray diffraction (XRD). The 2θ values it produces for common metals fall conveniently in the 20°–90° range most diffractometers cover.
Also known as
TG we-Calculate Editorial Team. (2026). Miller Indices Calculator — d-Spacing & Bragg Angle (Cubic) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/miller-indices-calculator
TG we-Calculate Editorial Team. "Miller Indices Calculator — d-Spacing & Bragg Angle (Cubic)." TG we-Calculate. 2026. https://we-calculate.com/calculator/miller-indices-calculator.
TG we-Calculate Editorial Team, "Miller Indices Calculator — d-Spacing & Bragg Angle (Cubic)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/miller-indices-calculator
@misc{wecalculate_miller_indices_calculator, title = {Miller Indices Calculator — d-Spacing & Bragg Angle (Cubic)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/miller-indices-calculator}}, year = {2026}, note = {TG we-Calculate} }
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