Bragg's Law Calculator
Bragg's Law calculator solves nλ = 2d·sin θ for lattice-plane spacing, diffraction angle 2θ, or X-ray wavelength. Enter any three of the four quantities and get the fourth instantly.
Solve Bragg's Law for Spacing, Angle or Wavelength
Bragg's Law Formula
Where:
- \(n\) — reflection order (a whole number, 1, 2, 3, …)
- \(\lambda\) — wavelength of the incident X-rays (Å, nm or pm)
- \(d\) — spacing between adjacent lattice planes (same units as \(\lambda\))
- \(\theta\) — glancing (Bragg) angle, half of the diffractometer angle \(2\theta\)
The law states that constructive interference — and therefore a diffraction peak — occurs only when the path difference between X-rays reflected from successive planes equals a whole number of wavelengths. Note that diffractometers report \(2\theta\), the angle between the incident and diffracted beams, so \(\theta = 2\theta/2\) must be used in the formula.
Bragg's Law – Calculation Example
Example: a powder diffraction pattern is recorded with copper Kα radiation (λ = 1.5406 Å). A first-order reflection (n = 1) appears at 2θ = 30°. Find the spacing of the planes that produced it.
\[ \theta = \frac{2\theta}{2} = \frac{30°}{2} = 15° \]
\[ d = \frac{n\lambda}{2\sin\theta} = \frac{1 \times 1.5406}{2 \times \sin 15°} = \frac{1.5406}{2 \times 0.2588} = \frac{1.5406}{0.5176} \]
\[ d \approx 2.9758\ \text{Å} \]
Result: the lattice-plane spacing is d ≈ 2.976 Å (about 0.2976 nm or 297.6 pm).
How to Use the Calculator
- Open the Solve for dropdown and choose whether you want the lattice-plane spacing (d), the diffraction angle (2θ), or the X-ray wavelength (λ).
- Enter the Wavelength (λ) of the radiation and pick its unit (Å, nm or pm). For copper Kα radiation, keep the default 1.5406 Å.
- Enter the Diffraction angle (2θ) in degrees, exactly as reported by the diffractometer — the calculator halves it for you.
- Set the Reflection order (n); use 1 unless you are analyzing a higher-order reflection.
- If you are solving for the angle or the wavelength, also fill in the Plane spacing (d) with its unit.
- The results appear immediately: the requested quantity, the half-angle θ, sin θ, and a step-by-step substitution showing the check against Bragg's law.
What Is Bragg's Law?
Bragg's law, formulated by William Henry Bragg and his son William Lawrence Bragg in 1913, is the fundamental equation of X-ray diffraction. When X-rays strike a crystal, they scatter off the regularly arranged atoms. Rays reflected from parallel lattice planes travel different distances: the ray bouncing off a deeper plane travels an extra \(2d\sin\theta\) compared with the ray from the plane above. If this extra path equals a whole number of wavelengths (\(n\lambda\)), the reflected waves arrive in phase and reinforce each other, producing a sharp diffraction peak. If not, they interfere destructively and almost no intensity is observed.
The law is the bridge between a diffraction pattern and the atomic structure of a material. Every peak position in a powder pattern corresponds to one family of lattice planes satisfying \(n\lambda = 2d\sin\theta\). By measuring the angles of the peaks and knowing the wavelength, crystallographers reconstruct the lattice spacings and, from them, the unit cell and the arrangement of atoms. This is how crystal structures of minerals, metals, pharmaceuticals and proteins were — and still are — determined.
Interpreting the result is straightforward: a peak at a large angle corresponds to a small d-spacing (closely spaced planes), and vice versa. If the calculator reports an error when solving for the angle, the combination \(n\lambda > 2d\) would require \(\sin\theta > 1\), which is physically impossible — that reflection simply cannot occur for the given wavelength and spacing.
Assumptions, Limitations and Practical Tips
- Use θ, not 2θ. This is the most common mistake. Diffractometers scan and report 2θ; the formula requires the half-angle. This calculator accepts 2θ directly and does the halving internally.
- Units must match. The wavelength and the spacing can be in any consistent unit; the calculator converts Å, nm and pm automatically, so you can mix them freely in the two unit selectors.
- Higher orders. A second-order reflection (n = 2) from a set of planes appears at exactly the same angle as a first-order reflection from planes with half the spacing, which is why n = 1 with the "effective" d is the usual convention in powder diffraction indexing.
- Idealized model. The law assumes perfectly parallel, equally spaced planes and elastic, specular reflection. Real crystals have strain, size broadening, thermal vibrations and defects, which broaden and weaken peaks but do not shift the Bragg positions.
- Wavelength choice matters. A shorter wavelength (e.g. Mo Kα, 0.7107 Å) reaches smaller d-spacings and compresses the pattern into lower angles; a longer wavelength (Cr Kα, 2.2910 Å) spreads it out and improves low-angle resolution.
- No absorption or intensity information. Bragg's law predicts only peak positions, not peak intensities, which depend on the arrangement of atoms within the unit cell (structure factors).
Common Laboratory X-Ray Wavelengths
| Anode | Line | λ (Å) | Typical use |
|---|---|---|---|
| Copper | Cu Kα | 1.5406 | General powder diffraction |
| Cobalt | Co Kα | 1.7902 | Iron and steel samples (less Fe fluorescence) |
| Molybdenum | Mo Kα | 0.7107 | Single crystals, small molecules |
| Chromium | Cr Kα | 2.2910 | Residual-stress analysis |
| Iron | Fe Kα | 1.9373 | Heavy-oxide and mineral samples |
X-ray diffraction is often paired with other analytical techniques to fully characterize a sample: after identifying a compound's lattice spacings here, you might confirm its molecular mass with the Mass Spectrometry m/z Calculator or check the separation quality of its components with the Chromatography Resolution Calculator.
Frequently Asked Questions
What is a typical d-spacing value for a crystal?
Most inorganic and metallic crystals have lattice-plane spacings between about 1 Å and 4 Å, while layered materials, clays and proteins can show spacings of 10–100 Å or more. There is no single "normal" value — each family of planes in the same crystal has its own d. A useful physical limit: a reflection is only possible when d ≥ nλ/2, so with Cu Kα radiation (1.5406 Å) no planes closer than about 0.77 Å can produce a first-order peak.
How do I find the d-spacing of a specific plane (hkl) in a cubic crystal?
For a cubic lattice with cell edge \(a\), the spacing of the (hkl) planes is \[ d = \frac{a}{\sqrt{h^2 + k^2 + l^2}} \] For example, if \(a = 3.615\ \text{Å}\) (copper), the (111) spacing is \(3.615/\sqrt{3} = 2.087\ \text{Å}\). Once you have d from Bragg's law, you can invert this relation to index the peak and even determine the lattice parameter from a single reflection.
Does Bragg's law work for electron or neutron diffraction too?
Yes. Bragg's law is purely geometric, so it applies to any wave that scatters from lattice planes — X-rays, electrons or neutrons. The only difference is the wavelength: electrons in an electron microscope have de Broglie wavelengths of a few picometers, and thermal neutrons around 1 Å. Because these wavelengths differ from X-ray wavelengths, the same crystal plane produces peaks at completely different angles, but the equation \(n\lambda = 2d\sin\theta\) is unchanged.
Why does my calculation give no solution or an impossible angle?
This happens when \(n\lambda > 2d\), which would force \(\sin\theta\) above 1 — mathematically and physically impossible. The reflection simply cannot occur for that combination of wavelength and spacing. You can fix it by using a shorter wavelength (e.g. Mo Kα at 0.7107 Å instead of Cu Kα), increasing d, or lowering the reflection order n. Also double-check that you entered 2θ, not θ, if the angle came from a diffractometer.
How do I convert between Ångströms, nanometers and picometers?
The conversions are simple powers of ten: \(1\ \text{Å} = 0.1\ \text{nm} = 100\ \text{pm}\). So a d-spacing of 2.976 Å equals 0.2976 nm or 297.6 pm, and a wavelength of 0.07107 nm equals 0.7107 Å. Crystallography journals traditionally use Ångströms, while some fields prefer nm or pm — this calculator converts all three automatically, so you can enter each quantity in whatever unit you have.
What is the difference between Cu Kα1 and Cu Kα2 wavelengths?
Copper Kα radiation is actually a doublet: Kα1 at 1.54056 Å and Kα2 at 1.54439 Å. The commonly used value 1.5406 Å is the Kα1 line; the weighted average of the doublet is about 1.5418 Å. At low angles the two lines overlap, but at high 2θ they split into double peaks. For routine indexing use 1.5406 Å (Kα1); if your diffractometer's software strips the Kα2 component, consistency with Kα1 is the correct choice.
Can Bragg's law tell me how strong a diffraction peak will be?
No. Bragg's law predicts only where peaks appear, not how intense they are. Peak intensities are governed by the structure factor, which depends on which atoms sit on which planes, their scattering powers and positions within the unit cell. Some allowed reflections can even be systematically absent — for example, body-centered crystals show no peaks when h + k + l is odd. For quantitative phase analysis you need structure-factor calculations or reference patterns, not Bragg's law alone.