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Bragg's Law Calculator

Calculate X-ray diffraction angles, d-spacing between crystal lattice planes, and wavelength using Bragg's Law: nλ = 2d·sin(θ). Includes Miller index d-spacing formulas, common crystal reference table, and SAXS vs WAXS guidance.

Reverse-engineer the microscopic geometric spacing of crystal lattices by analyzing macroscopic angle wave reflections.

Usually 1 for highest intensity peak

nm
nm

Crystalline Analysis

Resolved Diffraction Angle (θ)

22.6437
° (Degrees)

Calculated Balance State

nλ = 2d • sin(θ)

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What is Seeing Atoms with X-Rays?

Bragg's Law is the foundational equation of X-Ray Crystallography (the technique famously used by Rosalind Franklin to discover the double-helix shape of DNA). Because the spacing between atoms in a crystal is roughly the same size as X-ray wavelengths, shooting X-rays at a crystal causes the waves to bounce off different layers of atoms and interfere. By measuring the angle of the brightest reflection, you can perfectly calculate exactly how far apart the microscopic atoms are.

Mathematical Foundation

nλ=2dsin⁡(θ)n\lambda = 2d \sin(\theta)
nn= The diffraction order (integer 1, 2, 3...). Represents which specific bright spot harmonic you are measuring.
λ\lambda= Wavelength of the incident photon beam (usually highly calibrated X-rays).
dd= Interplanar distance. The true physical distance between successive layers of atoms in the crystal lattice.
θ\theta= The specific grazing angle at which the reflected waves perfectly construct and combine to form a massive bright spot.

Laws & Principles

  • Constructive Interference: At random angles, the X-ray bouncing off the top layer of atoms and the X-ray bouncing off the bottom layer are out of phase. They collide and cancel out to pitch-black invisible zero. But at exactly angle $\theta$, the second wave has traveled exactly one full wavelength further ($n\lambda$), meaning the crests align perfectly, flashing a brilliant bright spot on the detector.
  • The Sine Limit Lockout: Because $\sin(\theta)$ can never mathematically exceed 1.0, diffraction is entirely impossible if the wavelength chosen $\lambda$ is more than twice as large as the atomic spacing $d$. Shooting radio waves at a crystal yields nothing; the wave simply ignores the lattice completely.

Step-by-Step Example Walkthrough

" An engineer shoots an X-ray (wavelength 0.154 nm) at a mystery metal. The detector registers the first major bright spot (n=1) at an angle of exactly 22.6 degrees. What is the distance between the atoms? "

  1. 1. Ensure n=1, λ=0.154, θ=22.6.
  2. 2. Analyze formula: d = (n × λ) / (2 × sin(θ)).
  3. 3. Calculate sine: sin(22.6°) ≈ 0.384.
  4. 4. Denominator: 2 × 0.384 = 0.768.
  5. 5. Division: (1 × 0.154) / 0.768
Final Result: The math concludes the atomic spacing d is precisely 0.200 nanometers. They can cross-reference 0.2nm with a metal database to confirm the mystery material is likely Iron or a similar transition alloy.

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
nλ=2dsin⁡(θ)n\lambda = 2d\sin(\theta)
nn= 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).
λ\lambda= 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.
dd= 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.
θ\theta= 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
d{hkl}=ah2+k2+l2d_{\{hkl\}} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}
aa= 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).
h,k,lh, k, l= 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. "

  1. Convert 2θ to Bragg angle: θ = 28.44° / 2 = 14.22°
  2. Apply Bragg's Law (n=1): d = λ / (2·sinθ) = 1.5406 / (2 × sin(14.22°))
  3. sin(14.22°) = 0.2457
  4. d = 1.5406 / (2 × 0.2457) = 1.5406 / 0.4914 = 3.135 Å
  5. 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.

Quick Answer: How do you use Bragg’s Law?

nλ = 2d·sin(θ) — solve for any variable: d = nλ / (2 sinθ), θ = arcsin(nλ / 2d), λ = 2d sinθ / n. Critical: XRD instruments report 2θ — always divide by 2 before using in Bragg’s Law. Most lab XRD uses Cu Kα = 1.5406 Å. Example: Si(111) peak at 2θ = 28.44°, Cu Kα: d = 1.5406 / (2 × sin(14.22°)) = 3.135 Å. Unit cell: a = d × √3 = 5.431 Å (matches known silicon a = 5.4309 Å).

Common Crystal d-Spacings & XRD Peak Positions (Cu Kα = 1.5406 Å)

Reference table for identifying unknown crystal phases from diffractogram peak positions. The 2θ values assume Cu Kα radiation and n=1. These are the most intense (lowest-order) reflections for each material.

Material Crystal System Plane (hkl) d-spacing (Å) 2θ (Cu Kα)
Silicon (Si)Cubic (FCC diamond)(111)3.13528.44°
Aluminum (Al)Cubic (FCC)(111)2.33838.47°
Iron α-FeCubic (BCC)(110)2.02744.67°
Gold (Au)Cubic (FCC)(111)2.35538.18°
NaCl (Halite)Cubic (FCC)(200)2.82031.70°
Calcite (CaCO&sub3;)Trigonal / Rhombohedral(104)3.03529.41°
Quartz (SiO&sub2;)Hexagonal(100)4.25520.86°
Copper (Cu)Cubic (FCC)(111)2.08743.30°
All 2θ values computed using nλ = 2d sinθ with λ = 1.5406 Å (Cu Kα1). For Mo Kα (λ = 0.7107 Å), angles are approximately half as large. Peaks shift with temperature due to thermal expansion of lattice parameter a. d-spacings from ICDD PDF database and published crystallographic data.

Pro Tips & Common Bragg’s Law Mistakes

Do This

  • ✓Always verify that your X-ray wavelength and d-spacing units match — both must be in Ångströms OR both in nm, never mixed. The most common unit in XRD is the Ångström (1 Å = 0.1 nm = 10⊃⁻¹º m). Cu Kα is 1.5406 Å = 0.15406 nm. A silicon (111) d-spacing is 3.135 Å = 0.3135 nm. If you enter λ in nm and d in Å (or vice versa), your calculated 2θ will be wrong by a factor of 10 in the sin argument. Always confirm: nλ and 2d must have the same unit for the ratio to yield a dimensionless sin(θ). Running a quick sanity check against a known peak (e.g., silicon 28.44°, quartz 20.86°) before analyzing unknown peaks is standard laboratory practice.
  • ✓Use silicon or LaB&sub6; as an internal standard to correct for instrumental aberrations (zero-angle offset, sample displacement) before drawing conclusions from measured 2θ peaks. Laboratory diffractometers have systematic errors: the sample may not be on the true focal axis (sample displacement error), or the goniometer’s zero may be off by 0.01–0.10°. For precision lattice parameter determination, these errors matter. Adding NIST-certified silicon powder (SRM 640) or lanthanum hexaboride (SRM 660) as an internal standard to your sample provides a built-in ruler: shift all measured 2θ peaks by the offset observed in the standard’s peaks. Typical silicon (111) peak at 28.444° ± 0.001° is the reference. If your instrument measures it at 28.40°, apply +0.044° correction to all your sample peaks.

Avoid This

  • ✗Don't use the full 2θ value from a diffractogram directly in Bragg’s Law — divide by 2 first to get the Bragg angle θ. XRD instruments scan and report in 2θ (the total scattering angle, which is twice the actual incidence angle). The Bragg equation uses θ (the half-angle). Example error: a peak at 2θ=44.67° (iron 110). Using 44.67° directly: d = 1.5406 / (2 × sin(44.67°)) = 1.5406 / (2 × 0.7030) = 1.095 Å (WRONG). Using θ=22.33°: d = 1.5406 / (2 × sin(22.33°)) = 1.5406 / (2 × 0.3798) = 2.027 Å (CORRECT, matches α-Fe d&sub1;&sub1;&sub0;). This 2θ/θ confusion is the single most common error in student XRD calculations.
  • ✗Don't expect Bragg’s Law alone to give crystal structure — it only gives d-spacings; you still need systematic absence analysis and peak intensities to determine the full structure. Bragg’s Law calculates d-spacings from peak positions, but d-spacings alone cannot distinguish all crystal structures. Two different materials may have very similar or accidentally coincident d-spacings. Full structure determination requires: (1) d-spacing (position), (2) relative peak intensities (structure factor F, which depends on atom types and positions within the unit cell), (3) systematic absence patterns (which reflections are missing, indicating lattice centering and space group symmetry). The Rietveld refinement method fits observed XRD patterns to calculated patterns from a proposed structural model, refining atomic positions, site occupancies, and thermal displacement parameters. Bragg’s Law provides the backbone; full crystallographic analysis layered on top determines actual atomic coordinates.

Frequently Asked Questions

Why do X-ray diffractograms plot 2θ instead of θ?

In a θ–2θ diffractometer (Bragg-Brentano geometry), the X-ray source is fixed, the sample rotates at angle θ from horizontal, and the detector is positioned at 2θ from the incident beam on the other side. When the sample angle is θ, the detector angle is automatically 2θ (the total deflection of the beam). Recording as 2θ is purely instrumental convention — the detector positions are measured as 2θ, so that’s what instruments record. To use Bragg’s Law (which contains sinθ, not sin2θ): always halve the reported 2θ value. For a reported peak at 2θ = 38.47°, the Bragg angle is θ = 19.235°, and d = 1.5406 / (2 × sin(19.235°)) = 2.338 Å (aluminum 111 plane). The 2θ convention is universal across all commercial powder diffractometers.

What are Miller indices and how do they relate to d-spacing?

Miller indices (h, k, l) are the reciprocals of the fractional intercepts of a crystal plane with the unit cell axes, rounded to integers. The (100) plane is parallel to b and c axes and intersects only the a axis. The (111) plane cuts all three axes at 1. Key d-spacing patterns for cubic crystals (d = a/√(h²+k²+l²)): (100): d=a; (110): d=a/√2=0.707a; (111): d=a/√3=0.577a; (200): d=a/2=0.500a; (220): d=a/√8=0.354a. Higher Miller index planes have smaller d-spacings and appear at larger 2θ angles. Systematic absences occur when the structure factor for a given (hkl) plane is zero due to destructive interference from atoms in centered positions: FCC lattices only show peaks for h,k,l all odd or all even; BCC lattices only show peaks for h+k+l = even. These patterns allow crystallographers to identify the lattice type from the absence pattern alone.

How was Bragg’s Law used to discover the structure of DNA?

In 1952, Rosalind Franklin and Raymond Gosling at King’s College London captured “Photo 51” — an X-ray diffraction image of DNA fiber in B-form. The pattern showed: a large X-shaped pattern (characteristic of a helical structure, arising from diffraction by repeating diagonal planes of the helix) and specific d-spacings: the meridional reflection at d ≈ 3.4 Å (corresponding to the base-pair stacking distance, 2θ ≈ 26.5° with Cu Kα) and the layer line spacing revealing a helix pitch of 34 Å per complete turn. The absence of a reflection at the 4th layer line indicated the helix contained 10 base pairs per turn with the phosphate backbone on the outside. Watson and Crick used Franklin’s d-spacing data (combined with Chargaff’s base-pairing ratios) to fit a structural model that placed AT and GC base pairs in the interior, yielding the double helix. The 1962 Nobel Prize was awarded to Watson, Crick, and Wilkins (Franklin had died in 1958).

What is the difference between SAXS and WAXS?

SAXS (Small-Angle X-ray Scattering): 2θ < 5°, probes d-spacings of 1–60 nm. Used for: nanoparticle size and shape, polymer chain conformation and spacing, protein quaternary structure (size, shape in solution), lamellar repeat spacing in block copolymers. Bragg’s Law applies: at 2θ=1° with Cu Kα, d = 1.5406/(2 × sin(0.5°)) ≈ 88 Å = 8.8 nm. WAXS (Wide-Angle X-ray Scattering) / Conventional XRD: 2θ = 5°–160°, probes d-spacings of 0.5–15 Å. Used for: crystal identification, unit cell parameters, crystallite size (Scherrer equation: size = Kλ/(FWHM × cosθ)), phase quantification (Rietveld), residual stress analysis. Why different setups: SAXS requires a much longer sample-to-detector distance (often 1–10 m) and a high-brilliance source to separate the very small angular differences near the direct beam. WAXS uses a compact goniometer at standard laboratory scale. Synchrotron facilities offer simultaneous SAXS/WAXS measurements for complete structural characterization from nano to atomic scale.

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