(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
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 , 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:
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.
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
Then Bragg's law is simply expressed by the conservation of momentum transfer
with incident and final wave vectors ki
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
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:
where is the lattice spacing of the cubic crystal, and , , and are the Miller indices of the Bragg plane. Combining this relation with Bragg's law:
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 4n |
| Triangular lattice
|| Hexagonal close packed
|| l even, h + 2k 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
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.
- Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: Orlando, 1976).
- Nobel Prize in Physics - 1915