What is Bragg's Law: X-Ray Diffraction & Crystal Lattice Spacing?
Bragg’s Law (1913) is the fundamental equation of X-ray crystallography — the scientific technique that determined the atomic structures of DNA (1953), proteins, pharmaceuticals, and virtually every material scientists study at the atomic scale. When an X-ray beam strikes a crystalline material, it reflects off successive parallel planes of atoms. If the path length difference between waves reflecting off adjacent planes is an exact integer multiple of the X-ray wavelength, the waves interfere constructively (Bragg condition), producing a strong diffracted beam at a specific angle. By measuring these angles, scientists reverse-engineer the three-dimensional spacing and arrangement of atoms. The 1915 Nobel Prize in Physics was awarded to William Henry Bragg and William Lawrence Bragg for this discovery. Over 200 further Nobel Prizes have been enabled by crystallographic techniques built on Bragg’s Law.
Mathematical Foundation
Bragg's Law
= Diffraction order (integer: 1, 2, 3, ...). n=1 is the first-order reflection (most intense). n=2 is the second-order reflection at a different angle. In practice, most XRD analysis uses n=1. Higher-order reflections appear at larger θ angles. The maximum observable n for a given d and λ is n_max = 2d/λ (since sinθ cannot exceed 1).
= X-ray wavelength in Ångströms (Å) or nanometers (nm). Common laboratory X-ray sources: Cu Kα = 1.5406 Å (0.15406 nm) — most widely used. Mo Kα = 0.7107 Å. Co Kα = 1.7902 Å. Ag Kα = 0.5594 Å. Synchrotron sources are tunable (0.5–2.5 Å range). Wavelength must match d-spacing scale: for atomic-scale d (1–5 Å), X-ray wavelengths in the 0.5–2.5 Å range are required. Neutron diffraction also follows Bragg's Law at similar wavelengths.
= Interplanar spacing — the perpendicular distance between adjacent parallel planes of atoms in the crystal lattice, in Å or nm. Determined by crystal structure and Miller indices (hkl). For a cubic crystal: d_{hkl} = a / √(h²+k²+l²), where a is the unit cell parameter. For NaCl (a=5.64 Å, planes 002): d = 5.64/√4 = 2.82 Å. This is what crystallographers solve FOR when given λ and θ.
= Bragg angle: the angle between the incident X-ray beam and the crystal plane surface (NOT the angle from the normal). Measured in degrees or radians. θ ranges from 0° to 90° in theory; physically observable range is typically 5°–80° in laboratory XRD instruments. In XRD powder patterns, peaks are plotted against 2θ (twice the Bragg angle). A peak at 2θ=28.4° means θ=14.2°. Always divide 2θ values by 2 before using in Bragg's Law.
Miller Index d-Spacing for Cubic Crystal Systems
= Unit cell lattice parameter — the edge length of the cubic unit cell in Å. Examples: Silicon (a=5.431 Å), NaCl (a=5.640 Å), Iron α-Fe (a=2.866 Å), Aluminum (a=4.049 Å), Gold (a=4.078 Å), Diamond (a=3.567 Å).
= Miller indices — integers that define the orientation of a crystal plane. (100) plane: d = a. (110) plane: d = a/√2. (111) plane: d = a/√3. (200) plane: d = a/2. Higher Miller indices give smaller d-spacings and appear at larger 2θ angles in XRD patterns. For non-cubic systems (tetragonal, hexagonal, orthorhombic, etc.), more complex d-spacing formulas apply: for hexagonal, 1/d² = (4/3)(h²+hk+k²)/a² + l²/c².
Laws & Principles
- The path length difference condition: Bragg's Law d=nλ/2sinθ is derived from the requirement that waves reflecting off adjacent planes travel a path length difference of exactly nλ for constructive interference. The geometry gives path difference = 2d·sinθ (factor of 2 because the wave enters at angle θ, reflects, and exits at angle θ past the second plane). If the path difference is not an integer multiple of λ, the reflected waves are out of phase and cancel by destructive interference. This is why XRD diffractograms show sharp intensity peaks at specific angles against a low-intensity background — constructive interference at Bragg angles, destructive interference everywhere else.
- 2θ notation in XRD instruments: all commercial powder diffractometers scan in 2θ (twice the Bragg angle). The detector moves at 2θ while the sample rotates at θ (the θ-2θ geometry). XRD peaks are always reported and plotted at 2θ positions. CRITICAL: when using Bragg's Law with data from a diffractogram, always use θ = (reported 2θ) / 2 in the sin(θ) term. Forgetting to halve 2θ is the most common calculation error, producing d-spacings that are systematically off by a factor of sin(2θ)/sin(θ) = 2cosθ.
- SAXS vs WAXS: Small-angle X-ray scattering (SAXS) operates at 2θ < 5° (very small angles), probing large d-spacings from 1–60 nm — nanoparticle size, polymer chain spacing, protein quaternary structure. Wide-angle X-ray scattering (WAXS or conventional XRD) operates at 2θ = 5°–160°, probing atomic d-spacings of 0.5–1.5 nm — crystal planes, unit cell parameters. Electron diffraction (TEM) follows the same Bragg equation but uses electron wavelengths (λ = 0.00197 Å at 300 kV), enabling much smaller d-spacings to be resolved. Neutron diffraction is especially sensitive to light atoms (H, Li) that are weak X-ray scatterers.
- Systematic absences in XRD patterns: Not all (hkl) planes produce Bragg peaks, even when geometry is satisfied. Bravais lattice centering causes systematic absences: Face-centered cubic (FCC): reflections absent when h, k, l are mixed (not all odd or all even). Body-centered cubic (BCC): reflections absent when h+k+l is odd. These systematic absences are fingerprints of the crystal system and are used to identify unknown materials. For FCC aluminum, peaks at (100), (110), (210) — where h,k,l are mixed — are absent. Present peaks: (111), (200), (220), (311).
Step-by-Step Example Walkthrough
" An XRD scan of a silicon wafer using Cu Kα radiation (λ = 1.5406 Å) shows a strong peak at 2θ = 28.44°. Confirm it is the Si(111) reflection and calculate the unit cell parameter. "
- Convert 2θ to Bragg angle: θ = 28.44° / 2 = 14.22°
- Apply Bragg's Law (n=1): d = λ / (2·sinθ) = 1.5406 / (2 × sin(14.22°))
- sin(14.22°) = 0.2457
- d = 1.5406 / (2 × 0.2457) = 1.5406 / 0.4914 = 3.135 Å
- For Si(111): d = a / √(1²+1²+1²) = a / √3. So a = d × √3 = 3.135 × 1.7321 = 5.431 Å
Final Result: d(111) = 3.135 Å and unit cell parameter a = 5.431 Å, matching the known silicon lattice parameter (5.4309 Å) to 4 significant figures. This confirms the peak is the Si(111) reflection.