Bragg diffraction (also referred to as the Bragg formulation of X-ray diffraction) was first proposed by
William Lawrence Braggand William Henry Braggin 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 diffractionand electron diffractionprocesses [John M. Cowley (1975) "Diffraction physcis" (North-Holland, Amsterdam) ISBN 0 444 10791 6] .
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
integermultiple 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 lawwhich describes the condition for constructive interference from successive crystallographic planes("h,k,l") [citation|title=Introductory Solid State Physics|author=H.P.Myers|publisher=Taylor & Francis|year=2002|isbn=0-7484-0660-3] of the crystalline lattice:
: [citation|url=http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/bragg.html|title=Bragg's Law|author=Carl.R. Nave|publisher=HyperPhysics, Georgia State University|accessdate=2008-07-19]
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 transferwith incident and final wave vectors ki and kf of identical length. The precedent relation is also called Laue diffractionand 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 systemthrough the following relation:
where is the lattice spacing of the
cubic crystal, and , , and are the Miller indicesof the Bragg plane. Combining this relation with Bragg's law:
One can derive selection rules for the
Miller indicesfor different cubic Bravais lattices; here, selection rules for several will be given as is.
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
William Henry Braggand William Lawrence Braggwere awarded the Nobel Prizefor 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. Thomsonand 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.
Distributed Bragg reflector
Fiber Bragg grating
Photonic crystal fiber
* Neil W. Ashcroft and N. David Mermin, "Solid State Physics" (Harcourt: Orlando, 1976).
* [http://nobelprize.org/physics/laureates/1915/index.html Nobel Prize in Physics - 1915]
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