Bragg Law Calculator
Solve nλ = 2d sinθ, the fundamental X-ray and neutron diffraction condition, for the Bragg angle, d-spacing, or wavelength.
📐 What is Bragg's Law?
Bragg's law (nλ = 2d sinθ) is the fundamental condition for constructive interference when X-rays or neutrons scatter off the regularly spaced planes of a crystal lattice. n is the diffraction order (a positive integer, usually 1), λ is the wavelength of the incident radiation, d is the spacing between adjacent crystal planes, and θ is the angle of incidence, measured from the plane itself rather than from the normal. When this exact relationship holds, waves reflected from successive planes stay in phase, reinforcing each other into a measurable diffraction peak.
Bragg's law is the working principle behind X-ray diffraction (XRD), neutron diffraction, and electron diffraction, techniques used across materials science, chemistry, and structural biology. A materials scientist identifying an unknown crystalline phase measures a series of diffraction angles and works backward through Bragg's law to find the d-spacings that fingerprint the material. A protein crystallographer solving a molecular structure relies on thousands of Bragg reflections collected from a single crystal. Semiconductor manufacturers use it to verify that silicon wafers have the correct crystal orientation.
A common point of confusion is the angle convention: θ in Bragg's law is measured from the crystal plane, not from the surface normal as in ordinary mirror-like reflection. Diffractometers also report 2θ, the angle between the incoming and outgoing beam actually swept by the detector, rather than θ itself, so both values matter in practice. Another subtlety is that not every combination of wavelength, spacing, and order has a solution, if nλ/2d exceeds 1, no real angle satisfies the equation and diffraction simply cannot occur.
This calculator solves Bragg's law for whichever quantity is unknown, the angle θ, the d-spacing d, or the wavelength λ, and shows the full working, useful for crystallography coursework, XRD data analysis, and checking diffraction geometry by hand.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Cu-Kα X-Rays off Silicon (111) Planes
Find θ: λ = 1.5406 Å (standard Cu-Kα radiation), d = 3.13 Å (close to silicon's real (111) spacing), n = 1
Example 2 — Finding d-Spacing from a Measured Angle
Find d: λ = 1.5406 Å, θ = 20°, n = 1
Example 3 — Finding Wavelength from a Known Plane
Find λ: d = 2.0 Å, θ = 25°, n = 1
❓ Frequently Asked Questions
🔗 Related Calculators
What is Bragg's law?
Bragg's law, nλ = 2d sinθ, is the condition for constructive interference when X-rays or neutrons reflect off the parallel planes of a crystal lattice. n is the diffraction order, λ is the wavelength, d is the spacing between planes, and θ is the angle of incidence measured from the plane.
How do you calculate the Bragg angle?
Rearrange Bragg's law to θ = arcsin(nλ/2d). For example, Cu-Kα radiation (λ=1.5406 Å) reflecting off planes spaced d=3.13 Å at order n=1 gives θ = arcsin(1.5406/6.26) = arcsin(0.2461) ≈ 14.25°.
Is the Bragg angle measured from the plane or the normal?
From the plane. This differs from everyday optics, where angles of incidence and reflection are measured from the surface normal. In Bragg diffraction, θ is measured from the crystal plane itself, so grazing incidence corresponds to a small θ.
What does it mean if nλ/2d is greater than 1?
It means no real solution exists for θ, since sine can never exceed 1. Physically, diffraction cannot occur for that combination of wavelength, spacing, and order, the wavelength is simply too long (or the spacing too small) to satisfy Bragg's law at all.
What is the difference between θ and 2θ?
θ is the Bragg angle used inside the formula nλ = 2d sinθ, measured between the incident beam and the crystal plane. 2θ is the angle between the incident beam and the diffracted beam, which is what a diffractometer detector actually scans and reports in a diffraction pattern.
What wavelength should I use for X-ray diffraction?
Cu-Kα radiation at λ ≈ 1.5406 Å is the most common laboratory X-ray source. Other common sources include Co-Kα (≈1.7902 Å) and Mo-Kα (≈0.7107 Å), chosen depending on the sample and the resolution needed.
What units does d-spacing use?
Angstroms (Å), where 1 Å = 0.1 nanometers = 10⁻¹⁰ meters. This is the standard unit in crystallography because typical interplanar spacings in real crystals fall in the 1 to 10 Å range, a convenient, human-scale number in angstroms.
What is the diffraction order n?
The diffraction order n is a positive integer (usually 1) representing which harmonic of the diffraction condition is satisfied. n=1 is the primary (first-order) reflection off a given set of planes; n=2, 3, ... correspond to higher-order reflections from the same planes, appearing at larger angles, as long as nλ/2d stays at or below 1.
Can I use Bragg's law for neutron diffraction, not just X-rays?
Yes. Bragg's law applies to any wave scattering off a periodic lattice, including neutrons (with wavelengths set by the neutron's de Broglie wavelength) and electrons. Neutron diffraction commonly uses wavelengths in the same 1 to 3 Å range as laboratory X-ray sources.
How accurate is this calculator?
This calculator computes θ, d, or λ using exact double-precision trigonometry (arcsine and sine), so results match textbook Bragg's law calculations to within standard floating-point precision, verified here against known real crystal examples like silicon's (111) plane.