Bragg's law

Result

Bragg's law

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Statement

Diffraction reveals the spacing of crystal planes.

Why it matters

crystal-lattices-unit-cell describes a crystal as a periodic, repeating arrangement of atoms, but that description alone does not say how one could ever actually determine such an arrangement experimentally for an unknown solid. Bragg's law is the key that unlocks X-ray diffraction as a structural technique: it converts a measured set of diffraction angles into the spacing between families of crystal planes, and from a full set of such spacings, into the complete three-dimensional atomic arrangement.

Nearly every precisely known crystal structure — the very lattice geometries close-packing-efficiency describes, the ionic separations born-lande-lattice-energy's lattice-energy calculations require as direct input — was determined experimentally by exactly this technique, making Bragg's law arguably the single most consequential practical tool in the whole of solid-state and structural chemistry.

Hypotheses

The crystal is treated as a stack of parallel, evenly spaced atomic planes, each acting as a partial reflector of incident X-rays.This is a deliberate simplification: X-rays are actually scattered by individual atoms (or, more precisely, by the electron density surrounding them) throughout the crystal, not literally "reflected" by planes as if they were mirrors; the plane picture is nonetheless mathematically equivalent to, and vastly easier to reason about than, the full scattering treatment, for the specific purpose of finding the angles of constructive interference. The incident X-ray beam is monochromatic (a single, known wavelength \(\lambda\)), comparable in magnitude to the interplanar spacing \(d\).Diffraction (constructive interference producing sharp, well-defined peaks rather than a smeared continuum) specifically requires a wavelength on the same order of magnitude as the spacing being probed; X-rays, with wavelengths of order \(1\,\text{Å}\), are comparable to typical interatomic spacings, which is exactly why X-rays, rather than visible light, are the standard probe for atomic-scale crystal structure. Real crystals are not perfectly, infinitely periodic; finite crystallite size and lattice strain broaden the observed diffraction peaks away from the idealised infinitely sharp peaks Bragg's law alone predicts, and this peak broadening is itself a standard, separate diagnostic of crystallite size and strain (Scherrer's equation extends Bragg's law specifically to quantify this).

Proof

\text{Path difference between rays reflected from two adjacent planes, separated by } d\text{, at glancing angle } \theta: \quad 2d\sin\theta

Consider two parallel rays striking two adjacent atomic planes at the same glancing angle \(\theta\) (measured from the plane, not the normal, the standard crystallographic convention); simple trigonometry on the right triangle formed by the incident ray, the interplanar spacing \(d\), and the reflected ray shows the second ray travels an extra path length of exactly \(2d\sin\theta\) compared with the first. A

\text{Constructive interference requires the path difference to equal a whole number of wavelengths: } 2d\sin\theta = n\lambda

Two waves reinforce (constructive interference, producing a detectable diffracted beam) only if they remain in phase after travelling their respective paths, which requires their path difference to be an integer multiple \(n\) of the wavelength \(\lambda\); any other path difference leads to at least partial destructive interference and no observable diffracted intensity in that direction. A

\text{At any angle } \theta \text{ not satisfying Step 2, contributions from many successive planes interfere destructively and cancel.}

Because a real crystal contains not just two but very many successive, identical planes, even a small deviation from the exact Bragg angle causes the path-difference mismatch to accumulate plane after plane, driving the net diffracted intensity from the whole stack to essentially zero except extremely close to the angles satisfying Step 2 — this is why observed diffraction peaks are sharp, not broad. B

\text{Measuring } \theta \text{ for a known } \lambda \text{ and known } n \text{ gives } d \text{ directly.}

Since \(\lambda\) is fixed and known (from the X-ray source) and \(n\) is an integer identified from the pattern of observed peaks, a single measured diffraction angle \(\theta\) determines the corresponding interplanar spacing \(d\) via direct rearrangement of Step 2's condition. A

Result

n\lambda = 2d\sin\theta

Reading. A diffracted X-ray beam appears only at specific glancing angles \(\theta\), each corresponding to constructive interference from a particular family of atomic planes with spacing \(d\); measuring these angles converts directly into the spacings that define a crystal's internal structure.

Scope. Requires monochromatic radiation of wavelength comparable to \(d\) and a reasonably well-ordered, periodic crystal (Hypotheses); a full structure determination requires collecting many such spacings from many different plane orientations, then combining them systematically (indexing) into the complete three-dimensional lattice.

Corollaries & converses

Because larger \(d\) gives smaller \(\sin\theta\) for fixed \(\lambda\) and \(n\), widely spaced planes diffract at small angles and closely spaced planes diffract at large angles, an inverse relationship that must be kept in mind when interpreting a diffraction pattern intuitively.
close-packing-efficiency's geometric description of a crystal structure predicts a specific, calculable set of interplanar spacings for that structure type; comparing predicted spacings against a measured diffraction pattern is the standard way of confirming (or ruling out) a proposed structural model.
Converse: given a full set of measured diffraction angles and their associated diffracted intensities, the interplanar spacings (and, with further analysis of intensities, the actual atomic positions within the unit cell) can be reconstructed — Bragg's law run in reverse is precisely how essentially every crystal structure in the literature has been solved.

Fails without

Illuminate the crystal with polychromatic radiation without accounting for it: multiple wavelengths satisfy Step 2's condition at different angles for the same spacing \(d\), smearing or multiplying the observed diffraction peaks unless the wavelength is properly selected or resolved (Hypotheses, second point).
Assume perfect, infinite periodicity for a real, finite or strained crystal: finite crystallite size and lattice strain broaden the observed peaks away from the idealised, infinitely sharp peaks Bragg's law alone predicts (Hypotheses, third point), a broadening that must be accounted for separately rather than treated as an error in the underlying law.

Common errors

Measuring \(\theta\) from the surface normal rather than from the plane itself, the opposite of the crystallographic convention used in the Result (unlike the optical convention for ordinary mirror reflection).
Forgetting that a given interplanar spacing \(d\) can satisfy Step 2 at multiple angles, corresponding to different integer values of \(n\) (higher-order reflections), not only the smallest angle observed.
Assuming a single measured diffraction angle is sufficient to determine a crystal's full three-dimensional structure, when in fact many spacings from many differently oriented plane families are required (Result's Scope).
Confusing constructive interference between successive planes (this result, giving the diffraction angles) with the separate question of diffracted intensity, which depends additionally on what atoms occupy which positions within each unit cell.

Discussion

William Henry Bragg and his son William Lawrence Bragg developed this relationship in 1913, shortly after Max von Laue's earlier demonstration that crystals diffract X-rays at all; the Braggs went on to use the law to solve some of the very first crystal structures determined by X-ray diffraction, work recognised by the 1915 Nobel Prize in Physics, shared between father and son.

Common misconception: that Bragg's law describes X-rays literally bouncing off atomic planes the way light reflects off a mirror. As the Hypotheses note, this "reflection" picture is a convenient geometric equivalent to the true physical process, which is scattering from the electron density of individual atoms throughout the crystal; the plane-reflection picture correctly predicts the angles of constructive interference without needing to invoke the full, more complex scattering treatment.

Worked examples

\lambda = 1.54\,\text{Å (Cu K}_\alpha\text{ radiation)}, \qquad \theta = 13.7^\circ, \qquad n=1

A crystal is illuminated with a common laboratory X-ray source of well-known wavelength, and a first-order diffraction peak is observed at the stated glancing angle; the interplanar spacing follows directly from rearranging the Result. A

d = \frac{n\lambda}{2\sin\theta} = \frac{(1)(1.54)}{2\sin(13.7^\circ)} \approx \frac{1.54}{2(0.2368)} \approx 3.25\,\text{Å}

Substituting the measured angle and known wavelength gives an interplanar spacing of roughly \(3.25\,\text{Å}\), a typical magnitude for spacings between densely packed planes in many common ionic and metallic crystal structures. A

d \approx 3.25\,\text{Å}

Reading. One measured diffraction angle, together with the known X-ray wavelength, gives the spacing of one particular family of crystal planes directly.

Scope. A complete structure determination repeats this measurement across the full pattern of observed diffraction peaks, then combines the resulting spacings systematically to reconstruct the unit cell.

Problems

Using Cu K\(_\alpha\) radiation (\(\lambda=1.54\,\text{Å}\)), a crystal shows a first-order diffraction peak at \(\theta=10.4^\circ\). Find the corresponding interplanar spacing \(d\).
Solution
\(d=n\lambda/(2\sin\theta) = 1.54/(2\sin10.4^\circ) = 1.54/(2\times0.1805)\approx4.27\,\text{Å}\).
For the same plane spacing found in Problem 1, at what angle would the second-order (\(n=2\)) reflection appear?
Solution
Rearranging for \(\theta\): \(\sin\theta = n\lambda/(2d) = (2)(1.54)/(2\times4.27) = 0.3606\), giving \(\theta\approx21.1^\circ\). The second-order reflection appears at a larger angle than the first-order reflection from the identical plane spacing, consistent with the Result.
Two different families of planes in the same crystal, with spacings \(d_1=3.0\,\text{Å}\) and \(d_2=1.5\,\text{Å}\), are illuminated with the same wavelength. Which family diffracts at the larger first-order angle, and why?
Solution
The Result gives \(\sin\theta = n\lambda/(2d)\), which is inversely proportional to \(d\); the more closely spaced family (\(d_2=1.5\,\text{Å}\), half of \(d_1\)) therefore diffracts at the larger angle, since a smaller \(d\) requires a larger \(\sin\theta\) to keep the path-difference condition of Step 2 satisfied for the same \(n\) and \(\lambda\).