X-Ray Diffraction and the Bragg Condition
X-rays scattered by a crystal interfere at angles satisfying the Bragg condition. Reflection positions constrain atomic plane spacings, while intensities provide further structural information. Analysis of X-ray patterns helped reveal crystal structures and DNA's double helix.
What a learner can do afterwards
- Applies the Bragg condition to get a plane spacing from a measured angle
- Explains why X-rays rather than visible light are needed for atomic spacings
- Describes what a diffraction pattern reveals about the arrangement of atoms
1 · Read
X-rays scatter from atoms. For equally spaced lattice planes, rays from adjacent planes have a path difference of 2d sin(theta). Here d is plane spacing, lambda is wavelength and theta is the angle to those planes, not their normal. The planes need not be parallel to the sample surface. Constructive interference requires n lambda = 2d sin(theta), with positive integer order n. This is the Bragg condition. It permits a reflection but does not guarantee measurable intensity.
Atomic plane spacings are often around 0.1 nm. Visible wavelengths are hundreds of nanometres, thousands of times larger. For such spacings, visible light cannot satisfy n lambda = 2d sin(theta) for any positive n: even 2d is much smaller than lambda. X-ray wavelengths can satisfy it. The wavelength need not equal d. Visible light can still diffract from larger features, such as slits or grating lines.
To find spacing, rearrange to d = n lambda/(2 sin(theta)). Keep wavelength and spacing in the same units. For wavelength 0.150 nm, first order and theta = 30 degrees, sin(theta) = 0.5, so d = 0.150 nm. Some instruments report the angle between the incident and diffracted beams as 2theta. Halve that angle before using this equation.
At fixed wavelength and order, sin(theta) = n lambda/(2d). Increasing d lowers sin(theta), so theta decreases within 0 to 90 degrees. Comparing different orders or wavelengths would not isolate this spacing effect.
Bars show theta in degrees. At wavelength 0.160 nm and order 1, doubling d from 0.20 to 0.40 nm lowers theta from about 23.6 to 11.5 degrees.
Positions constrain plane spacings and lattice geometry. Intensities depend on summed scattering contributions from atoms at different positions in the repeating unit. Those contributions can cancel, so a Bragg-allowed reflection may be absent. Intensities give squared magnitudes, not phases; solving a structure therefore needs more than reading spot angles alone.
An oriented single crystal gives sharp diffraction spots in selected directions. Repeated atomic scattering adds strongly there and largely cancels between them. Finite crystals give spots some width, and background scattering may remain. A powder can give rings because its crystallites have many orientations.
For fixed d and lambda, an order is possible only if n lambda/(2d) <= 1. Equality gives theta = 90 degrees, the backscattering limit. Larger required sines have no solution. This geometric bound does not guarantee that each allowed reflection has detectable intensity.
Bars show required sin(theta) when d = 2lambda. Order 4 reaches theta = 90 degrees. Order 5 would need sin(theta) = 1.25, which is impossible.
Use n lambda = 2d sin(theta), with theta measured from the lattice planes. Positions constrain spacings; intensities add structural information but do not directly give phases.
Original lesson, questions and flashcards on the Bragg condition, written for this learning dataset.
Readings and revised flashcard text were written for this dataset. Background video references are Khan Academy's Diffraction grating and TED-Ed's How do crystals work? - Graham Baird. Video licensing terms have not been verified.
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