Bragg's Law Calculator

Solve for the diffraction angle, wavelength, or interplanar spacing in X-ray diffraction using Bragg's law (nλ = 2d·sinθ).

Diffraction Setup

Diffraction order (n) Integer reflection order — usually 1 for the strongest peak.

Wavelength (λ), nm Incident X-ray wavelength. Cu Kα radiation ≈ 0.154 nm (1.54 Å).

Interplanar spacing (d), nm Distance between adjacent crystal lattice planes.

Diffraction angle (θ), degrees Angle between the incident beam and the lattice plane (not the 2θ detector angle).

Result

—

Enter wavelength and interplanar spacing above to compute the diffraction angle.

4 min read 3 steps 7 terms 3 examples 6 FAQs nλ = 2d·sinθ

📋

Walk-through

How to Use This Calculator

3 steps

Pick what you want to solve for

Choose one of the three tabs above the calculator — Diffraction Angle, Interplanar Spacing, or Wavelength. The tab you pick hides that field as an input and computes it instead; the other two quantities plus the diffraction order (n) stay editable.

Enter the known quantities

Fill in the diffraction order (n, usually 1 for the strongest peak) plus whichever of wavelength (λ), interplanar spacing (d), and diffraction angle (θ) are still shown as inputs. The result updates instantly as you type.

Read the result

The result card shows the solved quantity plus a summary of the other values used. The interpretation line spells out the full relationship (nλ = 2d·sinθ) with your numbers plugged in, so you can double-check the math by hand.

⚡

Reference

Formula & Methodology

1 formula

Bragg's Law

nλ = 2d·sinθ

n is the diffraction order (a positive integer), λ is the wavelength of the incident radiation (nm), d is the spacing between adjacent crystal lattice planes (nm), and θ is the angle between the incident beam and the lattice plane (degrees) — not the 2θ angle read off an XRD detector. Solved for each variable: θ = asin(nλ / 2d), λ = 2d·sinθ / n, and d = nλ / (2·sinθ).

📖

Glossary

Key Terms Explained

7 terms

Bragg's law The condition nλ = 2d·sinθ under which X-rays (or other short-wavelength waves) scattered by parallel planes of atoms in a crystal interfere constructively, producing a detectable diffraction peak. Derived independently by William Henry Bragg and William Lawrence Bragg in 1913.

X-ray diffraction (XRD) An analytical technique that fires X-rays at a crystalline sample and measures the angles and intensities of the diffracted beams to determine the sample's atomic structure, phase composition, and crystallite size.

Interplanar spacing (d) The perpendicular distance between adjacent, parallel planes of atoms in a crystal lattice, indexed by Miller indices (hkl). Typical values for common crystals fall between about 0.1 and 1 nm.

Diffraction order (n) A positive integer (1, 2, 3, ...) representing how many whole wavelengths of path difference separate waves reflecting off successive lattice planes. First-order (n = 1) reflections are the strongest and most commonly used.

Crystallography The scientific study of the arrangement of atoms in crystalline solids, largely enabled by X-ray diffraction techniques built on Bragg's law.

Constructive interference The reinforcement of waves that are in phase with one another, producing a wave of greater amplitude. Bragg's law identifies exactly the angles at which reflected X-ray waves stay in phase and interfere constructively.

XRD pattern A plot of diffracted X-ray intensity versus scattering angle (typically 2θ) produced by an X-ray diffractometer. Each peak in the pattern corresponds to a set of lattice planes satisfying Bragg's law.

👥

Scenarios

Real-World Examples

3 worked examples

🔬

Materials scientist

Reading an XRD peak

Wavelength (λ) 0.154 nm (Cu Kα) Interplanar spacing (d) 0.314 nm Order (n) 1

With Cu Kα radiation (λ ≈ 0.154 nm) diffracting off planes spaced 0.314 nm apart at first order, Bragg's law gives sinθ = nλ/2d = 0.154/0.628 ≈ 0.245, so θ ≈ 14.19°. A detector recording 2θ would show a peak near 28.4° — decreasing the d-spacing would push that peak to a larger angle.

🧪

Lab technician

Solving for spacing from a measured peak

Wavelength (λ) 0.154 nm Diffraction angle (θ) 14.19° Order (n) 1

If an XRD scan shows a first-order peak at θ = 14.19° using the same Cu Kα source, solving d = nλ/(2·sinθ) recovers d ≈ 0.314 nm — matching the forward calculation above and confirming the same crystal plane spacing.

📈

Student checking higher orders

Comparing diffraction order

Wavelength (λ) 0.154 nm Interplanar spacing (d) 0.314 nm Order (n) 2

Doubling the order to n = 2 with the same λ and d gives sinθ = 2(0.154)/0.628 ≈ 0.490, so θ ≈ 29.4° — a noticeably larger angle for the second-order reflection of the same lattice planes. Pushing n high enough (so that nλ exceeds 2d) makes sinθ exceed 1, which the calculator flags as no diffraction possible at that order.

🔗

Reference

Cite This Calculator

APA & MLA

Use either format to cite this calculator in a paper, report, or resource list.

APA

MLA

📄

Deep Dive

Understanding Bragg's Law and X-Ray Diffraction

3 chapters · 4 min read

Bragg's law (nλ = 2d·sinθ) is the foundational equation of X-ray crystallography, explaining exactly which angles produce a detectable diffraction peak when X-rays scatter off a crystal's atomic planes. This calculator solves the equation for any one of its three variables — angle, wavelength, or spacing — given the other two plus the diffraction order.

How Bragg's law works

When X-rays strike a crystal, they scatter off successive parallel planes of atoms. Most scattered waves cancel out through destructive interference, but at specific angles the path-length difference between waves reflecting off adjacent planes equals a whole number of wavelengths — those waves stay in phase and reinforce each other, producing a measurable diffraction peak. William Lawrence Bragg and his father William Henry Bragg formalized this condition in 1913 as nλ = 2d·sinθ, where the path difference 2d·sinθ (twice the spacing times the sine of the incidence angle) must equal an integer multiple n of the wavelength λ.

Inputs and what they mean

Wavelength (λ) is the X-ray source's characteristic emission line — Cu Kα radiation at ≈0.154 nm (1.54 Å) is the most common lab source, though Mo Kα (≈0.071 nm) and synchrotron sources are also used. Interplanar spacing (d) depends on the crystal's unit cell dimensions and the specific set of lattice planes (Miller indices) being probed; it is usually on the same order of magnitude as the wavelength, which is what makes X-ray diffraction possible at all. Diffraction order (n) is almost always 1 in practice, since higher-order reflections from the same planes are typically much weaker and often indistinguishable from a different plane's first-order reflection. Diffraction angle (θ) is the angle between the incident beam and the lattice plane — note that most instruments report 2θ (the angle between the incident and diffracted beams), so halve a reported 2θ value before entering it here.

Limits and edge cases

Bragg's law has no solution when nλ/2d exceeds 1, since sine cannot exceed 1 — physically, this means the chosen order's reflection is impossible for that wavelength and spacing (the X-ray wavelength is too long, or the order too high, for the given lattice spacing). The calculator flags this case explicitly rather than returning an invalid angle. Bragg's law also treats the crystal as a simple stack of reflecting planes; it does not by itself predict peak intensity (which depends on the crystal structure factor) or account for effects like thermal vibration, crystallite size broadening, or multiple scattering — those require the fuller kinematic or dynamical theory of diffraction covered in crystallography texts.

❓

Questions

Frequently Asked Questions

6 questions

What does the Bragg's Law Calculator actually compute?+

It solves nλ = 2d·sinθ for whichever quantity you select — the diffraction angle θ, the interplanar spacing d, or the wavelength λ — given the other two variables plus the diffraction order n. This is the core equation used to interpret X-ray diffraction (XRD) patterns in crystallography and materials science.

What's the difference between θ and 2θ?+

θ is the angle between the incident X-ray beam and the crystal lattice plane, which is what Bragg's law uses directly. Most diffractometers report 2θ — the angle between the incident and diffracted beams as measured by the detector. If your instrument gives you a 2θ value, divide it by 2 before entering it as the diffraction angle here.

Why does the calculator sometimes say diffraction is not possible?+

Because sinθ can never exceed 1, Bragg's law has no valid solution when n·λ is greater than 2d. That happens when the wavelength is too long, or the chosen diffraction order too high, for the given lattice spacing. Try a lower order (often n = 1) or check that your wavelength and spacing units match (nanometres for both here).

What units does the Bragg's Law Calculator expect?+

Wavelength (λ) and interplanar spacing (d) are both entered in nanometres (nm) — 1 nm = 10 angstroms, so a common value like 1.54 Å (Cu Kα) becomes 0.154 nm. The diffraction angle (θ) is in degrees. The diffraction order (n) is a unitless positive whole number, typically 1.

Can I share my results?+

Yes. The calculator encodes your current inputs and solve mode into the page's URL as you type, so copying the address bar URL (or clicking the Share button) is enough to send a colleague the exact same scenario.

Where does the Bragg's law formula come from?+

William Henry Bragg and William Lawrence Bragg derived nλ = 2d·sinθ in 1913 to explain X-ray diffraction from crystals, work for which they jointly received the 1915 Nobel Prize in Physics. It remains the standard equation taught in crystallography and materials-science courses and used to interpret XRD instrument output today.

Download a one-page PDF of your numbers instantly. Add your email to also get our occasional calculator tips — no spam, unsubscribe anytime. Privacy.