In physics, Braggs' law is the result of experiments into the diffraction of X-rays or neutrons off crystal surfaces at certain angles, derived by physicists Sir W.H. Bragg and his son Sir W.L. Bragg in 1912, and first presented on 1912-11-11 to the Cambridge Philosophical Society. Although simple, Bragg's law confirmed the existence of real particles at the atomic scale, as well as providing a powerful new tool for studying crystals in the form of X-ray and neutron diffraction. The Braggs were awarded the Nobel Prize in physics in 1915 for their work in determining crystal structures beginning with NaCl, ZnS, and diamond.
When X-rays hit an atom, they make the electronic cloud move as does any electromagnetic wave. The movement of these charges re-radiates waves with the same frequency (blurred slightly due to a variety of effects); this phenomenon is known as the Rayleigh scattering (or elastic scattering). The scattered waves can themselves be scattered but this secondary scattering is assumed to be negligible. A similar process occurs upon scattering neutron waves from the nuclei or by a coherent spin interaction with an unpaired electron. These re-emitted wave fields interfere with each other either constructively or destructively (overlapping waves either add together to produce stronger peaks or subtract from each other to some degree), producing a diffraction pattern on a detector or film. The resulting wave interference pattern is the basis of diffraction analysis. Both neutron and X-ray wavelengths are comparable with inter-atomic distances (~150 pm) and thus are an excellent probe for this length scale.
The interference is constructive when the phase shift is a multiple to 2π; this condition can be expressed by Bragg's law:
A single monochromatic wave, of any type, is incident on aligned planes of lattice points, with separation d, at angle θ, as shown below.
There will be a path difference between the 'ray' that gets reflected along AC' and the ray that gets transmitted, then reflected along AB and BC paths respectively. This path difference is:
If this path difference is equal to any integer value of the wavelength then the two separate waves will arrive at a point with the same phase, and hence undergo constructive interference. Expressed mathematically:
Where the same definition of n and λ apply from the article above
Using the Pythagorean theorem it is easily shown that:
and and
also it can be shown that:
Putting everything together and using known identities for sinusoidal functions:
Which simplifies to:
yielding Bragg's law.
References
W.L. Bragg, "The Diffraction of Short Electromagnetic Waves by a Crystal", Proceedings of the Cambridge Philosophical Society, 17 (1914), 43–57.
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