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# Bragg diffraction

Bragg diffraction (also referred to as the Bragg formulation of X-ray 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. The concept of Bragg diffraction applies equally to neutron diffraction and electron diffraction processes.

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.

## The Bragg Condition

Bragg diffraction occurs when electromagnetic radiation or subatomic particle waves with wavelength comparable to atomic spacings, are incident upon a crystalline sample, scattered by the atoms in the system and undergo constructive interference in accordance to Bragg's law. For a crystalline solid, the waves are scattered from lattice planes separated by the interplanar distance d. Where the scattered waves interfere constructively they remain in phase since the path length of each wave is equal to an integer multiple of the wavelength. The path difference between two waves undergoing constructive interference is given by $2dsintheta$, where θ is the scattering angle. This leads to Bragg's law which describes the condition for constructive interference from successive crystallographic planes (h,k,l) of the crystalline lattice:

$2 dsintheta = nlambda .$

A diffraction pattern is obtained by measuring the intensity of scattered waves as a function of scattering angle. Very strong intensities known as Bragg peaks are obtained in the diffraction pattern when scattered waves satisfy the Bragg condition.

## Reciprocal space

More elegant is the description in reciprocal space. Reciprocal lattice vectors describe the set of lattice planes as a normal vector to this plane with length $G = 2pi / d .$ Then Bragg's law is simply expressed by the conservation of momentum transfer $\left(G = k_f - k_i\right)$ with incident and final wave vectors ki and kf of identical length. The precedent relation is also called Laue diffraction and not only gives the absolute value, but a full vectorial description of the phenomenon. The scanning variable can be the length or the direction of the incident or exit wave vectors relating to energy- and angle-dispersive setups. The simple relationship between diffraction angle and Q-space is then
$Q = frac \left\{4 pi sin left \left(theta right \right)\right\}\left\{lambda\right\} .$

## Selection rules and practical crystallography

Bragg's law, as stated above, can be used to obtain the lattice spacing of a particular cubic system through the following relation:

$d = frac\left\{a\right\}\left\{ sqrt\left\{h^2 + k^2 + l^2\right\}\right\}$

where $a$ is the lattice spacing of the cubic crystal, and $h$, $k$, and $l$ are the Miller indices of the Bragg plane. Combining this relation with Bragg's law:

$left\left(frac\left\{ lambda \right\}\left\{ 2a \right\} right\right)^2 = frac\left\{ sin ^2 theta \right\}\left\{ h^2 + k^2 + l^2 \right\}.$

One can derive selection rules for the Miller indices for different cubic Bravais lattices; here, selection rules for several will be given as is.

Selection rules for the Miller indices
Bravais lattice Example compounds Allowed reflections Forbidden reflections
Simple cubic Simple cubic Any h, k, l None
Body-centered cubic Body-centered cubic h + k + l even h + k + l odd
Face-centered cubic Gold, NaCl, Zinc blende h, k, l all odd or all even h, k, l mixed odd and even
Diamond F.C.C. Diamond, Si, Ge all: odd, or even & h+k+l = 4n above, or even & h+k+l $neq$ 4n
Triangular lattice Hexagonal close packed l even, h + 2k $neq$ 3n h + 2k = 3n for odd l

These selection rules can be used for any crystal with the given crystal structure. Selection rules for other structures can be referenced elsewhere, or derived.

## Nobel Prize for Bragg diffraction

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. Thomson and George Thomson, Hans von Euler-Chelpin and Ulf von Euler, and Arthur and Roger Kornberg, who were all awarded the prize for separate contributions.

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

## References

• Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: Orlando, 1976).