The Bragg formulation of X-ray diffraction (also referred to as Bragg diffraction) was first proposed by William Lawrence Bragg and William Henry Bragg in 1913 in response to their discovery that crystalline solids produced surprising patterns of reflected X-rays (in contrast to that of, say, a liquid). They found that in these crystals, for certain specific wavelengths and incident angles, intense peaks of reflected radiation (known as Bragg peaks) were produced.

W. L. Bragg explained this result by modeling the crystal as a set of discrete parallel planes separated by a constant parameter d. It was proposed that the incident X-ray radiation would produce a Bragg peak if their reflections off the various planes interfered constructively

Mechanics

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As shown in the images on the right, a given crystal (in this case, NaCl) can be decomposed into any number of different Bragg plane configurations, due to the periodicity of the crystal lattice. The incident angle of the incoming wave and its wavelength determines which set of planes is relevant in the calculation.

As the wave enters the crystal, some portion of it will be reflected by the first layer, while the rest will continue through to the second layer, where the process continues. By the definition of constructive interference, the separately reflected waves will remain in phase if the difference in the path length of each wave is equal to an integer multiple of the wavelength.

This gives the formula for what is known as the Bragg condition or Bragg's law:

$2\; d\backslash sin\backslash theta\; =\; n\backslash lambda\backslash ,$

Waves that satisfy this condition interfere constructively and result in a reflected wave of significant intensity.

Equivalence with other formulations

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The phenomenon of crystal diffraction can also be formulated in other equivalent ways. One such example is the von Laue formulation of X-ray diffraction. In this model, the crystal is instead seen as a set of identical ions resting at the sites defined by the Bravais lattice, each of which reradiate incident radiation isotropically. The condition for constructive interference in this formulation is given by:

$\backslash vec\{R\}\backslash cdot(\backslash vec\{k\}\; -\; \backslash vec\{k\text{'}\})\; =\; 2\backslash pi\; n,$

where n is again an integer, $\backslash vec\; k$ is the wave vector describing the incoming wave, $\backslash vec\{k\}\text{'}$ is the wave vector describing the outgoing wave, and $\backslash vec\; R$ is any Bravais lattice vector. This can be equivalently stated as

$e^\{i(\backslash vec\{k\text{'}\}-\backslash vec\{k\})\backslash cdot\backslash vec\{R\}\}=1,$

or, defining $\backslash vec\; G$ to be a reciprocal lattice vector, and assuming that $|\backslash vec\; k|=|\backslash vec\{k\}\text{'}|$,

$\backslash vec\{k\}\backslash cdot\backslash vec\{G\}\; =\; \backslash frac\{1\}\{2\}\; G^2$

This final statement can be interpreted as saying that the Laue condition (for constructive interference) is satisfied if and only if the wave vector $\backslash vec\; k$ lies in a plane that is the perpendicular bisector to a reciprocal lattice vector $\backslash vec\; G$ lying at the origin of k-space. These planes are nothing other than the Bragg planes encountered earlier. The set of all Bragg planes in a crystal and the (identical) volumes they enclose define the Brillouin zones of the crystal. Thus the condition for diffraction can be equivalently stated as the requirement that the incoming wavevector (when placed at the origin of k-space) lie on a Bragg plane or on a Brillouin zone.

To further exemplify the equivalence between these two formulations (the Bragg formulation and the Van Laue formulation), note that the reciprocal lattice vector $\backslash vec\; G$ must have a magnitude which is an integer multiple of $2\backslash pi/d$, where d is again the interplanar distance (this is a consequence of the definition of the reciprocal lattice). Therefore,

$|\backslash vec\{G\}|\; =\; \backslash frac\{2\; \backslash pi\; n\}\{d\}$

Furthermore, from the results of the Van Laue formulation, we know that in the case of constructive interference, we have

$|\backslash vec\{G\}|\; =\; 2\backslash vec\{k\}\backslash cdot\backslash frac\{\backslash vec\{G\}\}\{|\backslash vec\{G\}|\}\; =\; 2|\backslash vec\{k\}|\backslash sin\; \backslash theta,$

where $\backslash theta$ is the angle between the incoming wave vector $\backslash vec\; k$ and the plane perpendicular to the reciprocal lattice vector G.

Setting these two equations equal to each other, and recognizing the magnitude of the wave vector $\backslash vec\; k$ is simply equal to $2\backslash pi/\backslash lambda$, the Bragg condition is retrieved:

$2\; d\backslash sin\backslash theta\; =\; n\backslash lambda\backslash ,$

Nobel Prize for Bragg diffraction

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In 1915, William Henry Bragg and William Lawrence Bragg were awarded the Nobel Prize for their contributions to crystal structure analysis. They were the first and (so far) the only father-son team to have jointly won the prize. Other father/son laureates include Niels and Aage Bohr, Manne and Kai Siegbahn, J.J. and George Thomson, and Hans von Euler-Chelpin and Ulf von Euler all having been awarded the prize for separate contributions.

W.L. Bragg was 25 years old at the time, making him the youngest Nobel laureate to date.

See also

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• Bragg's law
• Diffraction grating
• Diffraction
• Photonic crystal fiber

External links

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• Nobel Prize in Physics - 1915

References

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• Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: Orlando, 1976).
• http://www.citycollegiate.com/interference_braggs.htm

Condensed matter physics | Diffraction