Crystallography Calculators
Free crystallography calculators: Bragg's law diffraction angle, d-spacing from Miller indices, and unit cell volume for diffraction and materials science.
Crystallography Calculators - X-Ray and Neutron Diffraction
Crystallography studies how atoms pack into repeating, periodic lattices, and how that periodic structure diffracts X-rays or neutrons to reveal atomic-scale detail. These calculators cover the three foundational relationships behind diffraction analysis and crystal structure determination: Bragg’s law, interplanar d-spacing from Miller indices, and unit cell volume.
Three Crystallography Calculators
The Bragg Law Calculator solves nλ = 2d sinθ for the Bragg angle, d-spacing, or wavelength, the fundamental condition for constructive interference in X-ray and neutron diffraction. The d-Spacing from Miller Indices Calculator finds the interplanar spacing d from Miller indices (h,k,l) for cubic, tetragonal, and orthorhombic crystal systems, the direct input Bragg’s law needs to predict a diffraction pattern. The Unit Cell Volume Calculator finds the volume of any unit cell, cubic through fully triclinic, from its edge lengths a, b, c and angles α, β, γ using a single general formula that reduces to V = a³ for the cubic case and to the full trigonometric expression for the least symmetric triclinic case.
Who Uses These Calculators
Materials science and condensed matter physics students use these tools to work through powder XRD problem sets, converting between diffraction angle, d-spacing, and Miller indices. Crystallographers and X-ray diffraction lab technicians use the Bragg Law Calculator for quick peak-position sanity checks before running full Rietveld refinement software. Solid-state chemistry students use the Unit Cell Volume Calculator to compute theoretical density (mass of unit cell contents divided by unit cell volume) as a standard lab exercise.
Constants Behind Crystallography
Most crystallography calculations reduce to trigonometry and geometry applied to a repeating lattice. The standard X-ray wavelength for laboratory diffractometers is Cu-Kα radiation at λ ≈ 1.5406 Å, and typical interplanar spacings in real crystals (silicon, quartz, common oxides) fall in the 1 to 5 Å range, which is exactly why diffraction angles are measurable with laboratory equipment rather than requiring exotic instrumentation.
Frequently Asked Questions
What is Bragg's law used for?
Bragg's law (nλ = 2d sinθ) is the fundamental condition for constructive interference of X-rays or neutrons scattered by the periodic planes of a crystal lattice. It underlies X-ray diffraction (XRD), neutron diffraction, and every technique used to determine crystal structure from a diffraction pattern. The Bragg Law Calculator solves it for any of the three unknowns.
What are Miller indices?
Miller indices (h,k,l) are a set of three integers that uniquely identify a family of crystal planes and their orientation within the unit cell. Every d-spacing calculation starts from a chosen set of Miller indices, and the relationship between them depends on the crystal system, cubic, tetragonal, orthorhombic, and beyond. The d-Spacing from Miller Indices Calculator covers the three most common cases.
How do I find the unit cell volume of a triclinic crystal?
For the lowest-symmetry triclinic system, where none of the three edge lengths or three angles are equal or 90°, the volume formula is V = abc·√(1 − cos²α − cos²β − cos²γ + 2cosα·cosβ·cosγ). Higher-symmetry systems are special cases of this same formula: it reduces to V = a³ for cubic (a=b=c, α=β=γ=90°) and V = abc for orthorhombic (α=β=γ=90°, a≠b≠c). The Unit Cell Volume Calculator applies the general formula automatically for any crystal system.
What X-ray wavelength should I use in the Bragg law calculator?
Laboratory X-ray diffractometers almost always use copper anode tubes, giving Cu-Kα radiation at λ ≈ 1.5406 Å (the Kα1 line; some instruments report the weighted Kα1/Kα2 average of 1.5418 Å). Other common sources include Mo-Kα (0.7107 Å, used for single-crystal work) and Co-Kα (1.7889 Å, preferred for iron-containing samples to avoid fluorescence). Enter the correct source wavelength into the Bragg Law Calculator to get an accurate d-spacing from a measured 2θ angle.