The condition 2d sinθ = nλ for diffraction to reinforce from crystal planes with spacing d. It is the basic equation for finding interplanar spacings from XRD peak angles.
Japanese name: ブラッグの法則
Bragg's law appears in 2 practice questions across 1 topic, including Materials characterization (SEM, TEM, XRD, XPS).
Think of a crystal as a stack of parallel atomic planes with spacing d. When X-rays of wavelength λ strike the planes at an angle θ, the path difference between X-rays reflected from adjacent planes is 2d sinθ. When this difference is an integer multiple of the wavelength, nλ, the reflections from all planes line up in phase and reinforce. This is Bragg's law, 2d sinθ = nλ. θ is the angle between the planes and the X-rays, so the angle between the incident and diffracted X-rays is 2θ, which is normally plotted on the horizontal axis in XRD.
For example, with Cu Kα radiation (λ = 0.15406 nm) and planes with d = 0.2088 nm, sinθ = 0.15406/(2 × 0.2088) ≈ 0.369, so θ ≈ 21.7° and 2θ ≈ 43.3°. Common mistakes are confusing 2θ with θ and dropping the factor of 2 in the equation. Because sinθ cannot exceed 1, planes with d < λ/2 do not diffract, and shorter wavelengths allow more peaks to be measured. Higher-order reflections (n = 2 and above) are normally treated as first-order reflections from planes with spacing d/n and labeled with indices such as (200).
For cubic crystals the interplanar spacing is d = a/√(h² + k² + l²), so combining it with Bragg's law gives sin²θ = (λ²/4a²)(h² + k² + l²), and the ratios of sin²θ for the peaks reveal the sequence of h² + k² + l². Depending on the arrangement of atoms in the unit cell, however, diffraction from some planes cancels and disappears (the extinction rules). In BCC only planes with h + k + l even appear, and in FCC only planes with h, k and l all even or all odd; from these sequences the structure can be identified and the lattice parameter determined.
Bragg's law is a condition for wave interference, so it applies not only to X-rays but also to electron and neutron diffraction. In TEM electron diffraction, the electron wavelength is very short (about 0.0025 nm at 200 kV) and θ is small, below 1°, so sinθ ≈ θ is used to find interplanar spacings from the spacing of diffraction spots. The same principle is used in EBSD in the SEM, in WDS with its analyzing crystals, and in the spectrometers of X-ray fluorescence analysis, making it a common foundation for many materials analysis techniques.
Answer: B. About 28.4°
From Bragg's law 2d sinθ = nλ (n = 1), sinθ = 0.15406/(2 × 0.3135) ≈ 0.2457, so θ ≈ 14.2° and the diffraction angle is 2θ ≈ 28.4°. XRD instruments and data normally plot 2θ, the angle between the incident and diffracted X-rays, on the horizontal axis. About 14.2° is the mistake of answering θ itself, and about 58.9° drops the 2 in Bragg's equation and uses sinθ = λ/d. The peak of silicon powder near 28.4° is strong and sharp, so it is also widely used as a standard for calibrating diffraction angles.
Answer: C. About 0.362 nm
θ = 21.65° and sinθ ≈ 0.3689, so Bragg's law gives d = λ/(2 sinθ) ≈ 0.2088 nm. In FCC the lowest-angle peak is (111), so a = d × √(1² + 1² + 1²) = 0.2088 × √3 ≈ 0.362 nm, matching the lattice parameter of copper (about 0.3615 nm). About 0.209 nm is the interplanar spacing d itself, without converting to the lattice parameter. About 0.295 nm treats the first peak as (110) and multiplies by √2, the mistake of applying the BCC extinction rules. Correct indexing of the peaks is a prerequisite for determining the lattice parameter.
θ is the angle between the crystal planes and the incident X-rays (the Bragg angle), and 2θ is the angle between the incident and diffracted X-rays (the diffraction angle). XRD data are usually plotted against 2θ, so in calculations use half of it as θ.
2d sinθ = 2λ is the same equation as a first-order reflection from planes with spacing d/2, so the second-order reflection from (100) planes is treated as a reflection from (200) planes. This way, everything can be calculated with n = 1.
For the same spacing θ becomes larger, so peaks are spread farther apart and easier to separate, but the constraint sinθ ≤ 1 reduces the number of planes that can be measured. Radiation such as Cr Kα is used for applications such as residual stress measurement.
From Bragg's law 2d sinθ = nλ (n = 1), sinθ = 0.15406/(2 × 0.3135) ≈ 0.2457, so θ ≈ 14.2° and the diffraction angle is 2θ ≈ 28.4°.
In XRD with Cu Kα radiation (wavelength 0.15406 nm), which …θ = 21.65° and sinθ ≈ 0.3689, so Bragg's law gives d = λ/(2 sinθ) ≈ 0.2088 nm.
In powder XRD of copper (FCC) with Cu Kα radiation (wavelen…XRD (X-ray diffraction) Extinction rules EDS (energy-dispersive X-ray spectroscopy) Scherrer equation SEM (scanning electron microscope) SIMS (secondary ion mass spectrometry) XPS (X-ray photoelectron spectroscopy)