14.22378
14.22378
°
28.447561
°
3.135
Å
1.5406
Å
0.24571
1.5406
Å
14.22378
14.22378
°
28.447561
°
3.135
Å
1.5406
Å
0.24571
1.5406
Å
The Bragg's Law Calculator solves the fundamental equation of X-ray diffraction for any unknown variable: diffraction angle (θ), interplanar spacing (d), or wavelength (λ). Bragg's law describes the condition for constructive interference when electromagnetic radiation scatters from a periodic crystal lattice, forming the theoretical foundation of X-ray crystallography.
Discovered by William Lawrence Bragg and William Henry Bragg in 1913, this relationship earned them the Nobel Prize in Physics in 1915. It remains the most important equation in crystallography, enabling determination of crystal structures, lattice parameters, phase identification, and residual stress measurements across materials science, chemistry, geology, and biology.
Bragg's law relates the wavelength of incident radiation to the diffraction angle and interplanar spacing:
$$n\lambda = 2d\sin\theta$$
Where $$n$$ is the order of diffraction (positive integer), $$\lambda$$ is the wavelength of the incident radiation, $$d$$ is the spacing between parallel crystal planes, and $$\theta$$ is the angle of incidence (measured from the plane surface, not the normal).
Solving for each variable:
$$\theta = \arcsin\left(\frac{n\lambda}{2d}\right)$$
$$d = \frac{n\lambda}{2\sin\theta}$$
$$\lambda = \frac{2d\sin\theta}{n}$$
The condition $$n\lambda/(2d) \leq 1$$ must hold for a real solution. When this ratio exceeds 1, diffraction is geometrically impossible for that combination of parameters. In practice, XRD patterns plot intensity versus 2θ (the angle between incident and diffracted beams), so the 2θ value is also provided.
Each diffraction peak in an XRD pattern corresponds to a specific set of crystal planes satisfying Bragg's law. Peaks at low 2θ correspond to large d-spacings (low-index planes), while high-angle peaks correspond to small d-spacings. Cu Kα radiation (λ = 1.5406 Å) is the most common laboratory X-ray source. For synchrotron sources, the wavelength is tunable, and for neutron diffraction, λ depends on neutron energy.
Inputs
Results
θ = arcsin(1 × 1.5406 / (2 × 3.135)) = arcsin(0.2457) = 14.22°, 2θ = 28.44°
Inputs
Results
d = 1 × 1.5406 / (2 × sin(21.5°)) = 1.5406 / 0.7330 = 2.102 Å
Bragg's law (nλ = 2d sinθ) describes the condition for constructive interference of X-rays scattered by crystal planes. When the path difference between rays reflected from adjacent planes equals a whole number of wavelengths, the scattered waves reinforce each other, producing a diffraction peak at that specific angle.
θ is the angle between the incident beam and the crystal plane (Bragg angle). 2θ is the angle between the incident and diffracted beams, which is what XRD instruments actually measure. The detector scans in 2θ because the diffracted beam exits at angle θ on the other side of the plane normal, making the total deflection 2θ.
The most common is Cu Kα radiation at 1.5406 Å (actually a doublet: Kα1 = 1.5405 Å, Kα2 = 1.5443 Å). Other common sources: Mo Kα = 0.7107 Å (used for single-crystal work), Co Kα = 1.7890 Å (preferred for iron-containing samples), and Cr Kα = 2.2897 Å. Synchrotrons provide tunable wavelengths.
The order n is a positive integer representing the number of wavelengths in the path difference. First order (n=1) is the strongest. Higher orders appear at larger angles and are weaker. In practice, nth-order reflections from (hkl) planes are equivalent to first-order reflections from (nh, nk, nl) planes, so n is usually set to 1.
When nλ/(2d) > 1, there is no real value of θ that satisfies the equation, because sin θ cannot exceed 1. This occurs when the wavelength is too large relative to the d-spacing. This is why X-rays (λ ≈ 1 Å) are needed to diffract from crystal planes (d ≈ 1–10 Å), while visible light (λ ≈ 5000 Å) cannot.
With careful experimental technique, d-spacings can be measured to ±0.001 Å or better. High-precision lattice parameter determination requires calibration with internal standards, temperature control, and correction for systematic errors like sample displacement, zero shift, and beam divergence.
Yes. Electron diffraction in transmission electron microscopy (TEM) follows Bragg's law with the electron wavelength given by the de Broglie equation: λ = h/p. At 200 keV, λ ≈ 0.025 Å, much shorter than X-rays, allowing diffraction from much smaller d-spacings and enabling imaging of crystal defects.
For thin film XRD, the same Bragg's law applies but with specialized geometries. Grazing incidence XRD (GIXRD) uses very small incident angles to enhance surface sensitivity. Rocking curves measure peak width at fixed 2θ to assess crystallographic quality. Reciprocal space mapping provides complete strain and composition information.
Bragg's law and Laue conditions are equivalent descriptions of diffraction. Laue conditions require that the scattering vector equals a reciprocal lattice vector: k' - k = G. This is satisfied when Bragg's law holds for a set of planes. The Ewald sphere construction geometrically connects both formulations in reciprocal space.
Bragg's law determines d-spacings and thus lattice parameters, but not the complete crystal structure. Full structure determination requires measuring peak intensities (related to structure factors), which encode atomic positions, thermal vibrations, and site occupancies. Methods like Rietveld refinement fit a structural model to the complete diffraction pattern.
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