|Figure 1. Bragg's Law reflection. The diffracted X-rays exhibit constructive interference when the distance between paths ABC and A'B'C' differs by an integer number of wavelengths (λ)|
Bragg diffraction occurs when electromagnetic radiation or subatomic particle waves with wavelength comparable to atomic spacings are incident upon a crystalline sample, are scattered in a specular fashion 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 2dsinθ, 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, and l, as given in Miller Notation) of the crystalline lattice:
where n is an integer determined by the order given, and λ is the wavelength. 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.
It should be taken into account that if only two planes of atoms were diffracting, as shown in the pictures, then the transition from constructive to destructive interference would be gradual as the angle is varied. However, since many atomic planes are interfering in real materials, very sharp peaks surrounded by mostly destructive interference result.
Alternate derivation: Suppose that a single monochromatic wave (of any type) is incident on aligned planes of lattice points, with separation , at angle . Points A and C are on one plane, and Bis on the plane below. Points ABCC' form a quadrilateral.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 respectively. This path difference is
The two separate waves will arrive at a point with the same phase, and hence undergo constructive interference, if and only if this path difference is equal to any integer value of the wavelength, i.e.
where the same definition of and apply as above.
from which it follows that
Putting everything together,
which simplifies to
which is Bragg's law.
Bragg scattering of visible light by colloids
A colloidal crystal is a highly ordered array of particles which can be formed over a very long range (from a few millimeters to one centimeter) in length, and which appearanalogous to their atomic or molecular counterparts. The periodic arrays of spherical particles make similar arrays of interstitial voids (the spaces between the particles), which act as a natural diffraction grating for visible light waves, especially when the interstitial spacing is of the same order of magnitude as the incident lightwave.
Thus, it has been known for many years that, due to repulsive Coulombic interactions, electrically charged macromolecules in an aqueous environment can exhibit long-rangecrystal-like correlations with interparticle separation distances often being considerably greater than the individual particle diameter. In all of these cases in nature, the same brilliant iridescence (or play of colours) can be attributed to the diffraction and constructive interference of visible lightwaves which satisfy Bragg’s law, in a matter analogous to the scattering of X-rays in crystalline solid.
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.
|Bravais lattice||Example compounds||Allowed reflections||Forbidden reflections|
|Simple cubic||Po||Any h, k, l||None|
|Body-centered cubic||Fe, W, Ta, Cr||h + k + l = even||h + k + l = odd|
|Face-centered cubic||Cu, Al, Ni, NaCl, LiH, PbS||h, k, l all odd or all even||h, k, l mixed odd and even|
|Diamond F.C.C.||Si, Ge||all odd, or all even with h+k+l = 4n||h, k, l mixed odd and even, or all even with h+k+l ≠ 4n|
|Triangular lattice||Ti, Zr, Cd, Be||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. KCl exhibits a fcc cubic structure. However, the K+ and the Cl− ion have the same number of electrons and are quite close in size, so that the diffraction pattern becomes essentially the same as for a simple cubic structure with half the lattice parameter. Selection rules for other structures can be referenced elsewhere, or derived.
Applications of Braggs Law:
- In X-ray diffraction (XRD) the interplanar spacing (d-spacing) of a crystal is used for identification and characterization purposes. In this case, the wavelength of the incident X-ray is known and measurement is made of the incident angle (Θ) at which constructive interference occurs. Solving Bragg's Equation gives the d-spacing between the crystal lattice planes of atoms that produce the constructive interference. A given unknown crystal is expected to have many rational planes of atoms in its structure; therefore, the collection of "reflections" of all the planes can be used to uniquely identify an unknown crystal. In general, crystals with high symmetry (e.g. isometric system) tend to have relatively few atomic planes, whereas crystals with low symmetry (in the triclinic or monoclinic systems) tend to have a large number of possible atomic planes in their structures.
- In the case of wavelength dispersive spectrometry (WDS) or X-ray fluorescence spectroscopy (XRF), crystals of known d-spacings are used as analyzing crystals in the spectrometer. Because the position of the sample and the detector is fixed in these applications, the angular position of the reflecting crystal is changed in accordance with Bragg's Law so that a particular wavelength of interest can be directed to a detector for quantitative analysis. Every element in the Periodic Table has a discrete energy difference between the orbital "shells" (e.g. K, L, M), such that every element will produce X-rays of a fixed wavelength. Therefore, by using a spectrometer crystal (with fixed d-spacing of the crystal) and positioning the crystal at a unique and fixed angle (Θ), it is possible to detect and quantify elements of interest based on the characteristic X-ray wavelengths produced by each element.