Bragg's law

In many areas of science, Bragg's law, Wulff–Bragg's condition, or Laue–Bragg interference are a special case of Laue diffraction, giving the angles for coherent scattering of waves from a large crystal lattice.

This law was initially formulated for X-rays, but it also applies to all types of matter waves including neutron and electron waves if there are a large number of atoms, as well as visible light with artificial periodic microscale lattices.

They found that these crystals, at certain specific wavelengths and incident angles, produced intense peaks of reflected radiation.

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

The interference is constructive when the phase difference between the wave reflected off different atomic planes is a multiple of 2π; this condition (see Bragg condition section below) was first presented by Lawrence Bragg on 11 November 1912 to the Cambridge Philosophical Society.

[2] Although simple, Bragg's law confirmed the existence of real particles at the atomic scale, as well as providing a powerful new tool for studying crystals.

Many other types of matter waves have also been shown to diffract,[6][7] and also light from objects with a larger ordered structure such as opals.

[8] Bragg diffraction occurs when radiation of a wavelength λ comparable to atomic spacings is scattered in a specular fashion (mirror-like reflection) by planes of atoms in a crystalline material, and undergoes constructive interference.

[10] When the scattered waves are incident at a specific angle, they remain in phase and constructively interfere.

The glancing angle θ (see figure on the right, and note that this differs from the convention in Snell's law where θ is measured from the surface normal), the wavelength λ, and the "grating constant" d of the crystal are connected by the relation:[11]: 1026

This equation, Bragg's law, describes the condition on θ for constructive interference.

[12] A map of the intensities of the scattered waves as a function of their angle is called a diffraction pattern.

[13] In Bragg's original paper he describes his approach as a Huygens' construction for a reflected wave.

The two separate waves will arrive at a point (infinitely far from these lattice planes) 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.

Because the scattering of X-rays and neutrons is relatively weak, in many cases quite large crystals with sizes of 100 nm or more are used.

In contrast, electrons interact thousands of times more strongly with solids than X-rays,[5] and also lose energy (inelastic scattering).

Typical diffraction patterns, for instance the Figure, show spots for different directions (plane waves) of the electrons leaving a crystal.

A colloidal crystal is a highly ordered array of particles that forms over a long range (from a few millimeters to one centimeter in length); colloidal crystals have appearance and properties roughly analogous to their atomic or molecular counterparts.

[8] It has been known for many years that, due to repulsive Coulombic interactions, electrically charged macromolecules in an aqueous environment can exhibit long-range crystal-like correlations, with interparticle separation distances often being considerably greater than the individual particle diameter.

Periodic arrays of spherical particles give rise to interstitial voids (the spaces between the particles), which act as a natural diffraction grating for visible light waves, when the interstitial spacing is of the same order of magnitude as the incident lightwave.

[21][22][23] In these cases brilliant iridescence (or play of colours) is attributed to the diffraction and constructive interference of visible lightwaves according to Bragg's law, in a matter analogous to the scattering of X-rays in crystalline solid.

The effects occur at visible wavelengths because the interplanar spacing d is much larger than for true crystals.

Depending on the orientation of the refractive index modulation, VBG can be used either to transmit or reflect a small bandwidth of wavelengths.

[24] Bragg's law (adapted for volume hologram) dictates which wavelength will be diffracted:[25]

where m is the Bragg order (a positive integer), λB the diffracted wavelength, Λ the fringe spacing of the grating, θ the angle between the incident beam and the normal (N) of the entrance surface and φ the angle between the normal and the grating vector (KG).

The output wavelength can be tuned over a few hundred nanometers by changing the incident angle (θ).

VBG are being used to produce widely tunable laser source or perform global hyperspectral imagery (see Photon etc.).

[5][13] As a simple example, Bragg's law, as stated above, can be used to obtain the lattice spacing of a particular cubic system through the following relation:

is the lattice spacing of the cubic crystal, and h, k, and ℓ are the Miller indices of the Bragg plane.

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.