Bragg's law

X-rays interact with the atoms in a crystal.

Bragg diffraction (also referred to as the Bragg formulation of X-ray diffraction) was first proposed by Lawrence Bragg and his father, William Henry Bragg, in 1913[1] 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 these crystals, at certain specific wavelengths and incident angles, produced intense peaks of reflected radiation. The derived Bragg's law is a special interpretation of Laue diffraction, where the Braggs interpreted the constructive Laue-Bragg interference in a geometric way by reflection of waves from crystal lattice planes, such that the path-difference becomes a multiple of the incident wavelength.

According to the 2θ deviation, the phase shift causes constructive (left figure) or destructive (right figure) interferences.

Lawrence 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 interference is constructive when the phase shift is a multiple of 2π; this condition can be expressed by Bragg's law (see Bragg condition section below) and was first presented by Lawrence Bragg on 11 November 1912 to the Cambridge Philosophical Society.[2][3] 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 in the form of X-ray and neutron diffraction. Lawrence Bragg and his father, William Henry Bragg, were awarded the Nobel Prize in physics in 1915 for their work in determining crystal structures beginning with NaCl, ZnS, and diamond. They are the only father-son team to jointly win.

The concept of Bragg diffraction applies equally to neutron diffraction and approximately to electron diffraction.[4] In all these the wavelengths are comparable with inter-atomic distances (~ 150 pm) and thus are an excellent probe for this length scale. Many other types of matter waves have also been shown to diffract.

Bragg diffraction[5]^: 16 Two beams with identical wavelength and phase approach a crystalline solid and are scattered off two different atoms within it. The lower beam traverses an extra length of 2dsinθ. Constructive interference occurs when this length is equal to an integer multiple of the wavelength of the radiation.

Bragg diffraction occurs when radiation of wavelength λ comparable to atomic spacings, is scattered in a specular fashion (mirror-like reflection) by the atoms of a crystalline system, and undergoes constructive interference. For a crystalline solid, the waves are scattered from lattice planes separated by the distance d between successive layers of atoms.[6]^: 223 When the scattered waves interfere constructively they remain in phase. They are reflected only when they strike the surface at a definite angle, the glancing angle (optics) 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 being connected by the relation:[7]^: 1026

$${\displaystyle n\lambda =2d\sin \theta }$$

${\displaystyle n}$ is the diffraction order (${\displaystyle n=1}$ is first order, ${\displaystyle n=2}$ is second order,[6]^: 221 ${\displaystyle n=3}$ is third order[7]^: 1028). The effect of the constructive or destructive interference intensifies because of the cumulative effect of reflection in successive crystallographic planes (h,k,l) of the crystalline lattice (as described by Miller notation). This leads to Bragg's law, which describes the condition on θ for the constructive interference to be at its strongest:[8]

A diffraction pattern is obtained by measuring the intensity of scattered waves as a function of scattering angle. Strong intensities known as Bragg peaks are obtained in the diffraction pattern when the scattering angles satisfy Bragg condition. As mentioned in the introduction, this condition is a special case of the more general Laue equations, and the Laue equations can be shown to reduce to the Bragg condition under additional assumptions.

When X-rays are incident on an atom, they make the electronic cloud move, as does any electromagnetic wave. The movement of these charges re-radiates waves with the same frequency, blurred slightly due to a variety of effects; this phenomenon is known as Rayleigh scattering (or elastic scattering). The scattered waves can themselves be scattered but this secondary scattering is assumed to be negligible.

A similar process occurs upon scattering neutron waves from the nuclei or by a coherent spin interaction with an unpaired electron. These re-emitted wave fields interfere with each other either constructively or destructively (overlapping waves either add up together to produce stronger peaks or are subtracted from each other to some degree), producing a diffraction pattern on a detector or film. The resulting wave interference pattern is the basis of diffraction analysis. This analysis is called Bragg diffraction.

Suppose that a single monochromatic wave (of any type) is incident on aligned planes of lattice points, with separation ${\displaystyle d}$, at angle ${\displaystyle \theta }$. Points A and C are on one plane, and B is 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 along AB, then reflected along BC. This path difference is

$${\displaystyle (AB+BC)-\left(AC'\right)\,.}$$

The two separate waves will arrive at a point (infinitely displaced 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.

$${\displaystyle n\lambda =(AB+BC)-\left(AC'\right)}$$

where ${\displaystyle n}$ and ${\displaystyle \lambda }$ are an integer and the wavelength of the incident wave respectively.

Therefore,

$${\displaystyle AB=BC={\frac {d}{\sin \theta }}{\text{ and }}AC={\frac {2d}{\tan \theta }}\,,}$$

from which it follows that

$${\displaystyle AC'=AC\cdot \cos \theta ={\frac {2d}{\tan \theta }}\cos \theta =\left({\frac {2d}{\sin \theta }}\cos \theta \right)\cos \theta ={\frac {2d}{\sin \theta }}\cos ^{2}\theta \,.}$$

Putting everything together,

$${\displaystyle n\lambda ={\frac {2d}{\sin \theta }}-{\frac {2d}{\sin \theta }}\cos ^{2}\theta ={\frac {2d}{\sin \theta }}\left(1-\cos ^{2}\theta \right)={\frac {2d}{\sin \theta }}\sin ^{2}\theta }$$

which simplifies to ${\displaystyle n\lambda =2d\sin \theta \,,}$ which is Bragg's law shown above.

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 a function of angle, with gentle maxima at the Bragg angles. However, since many atomic planes are participating interference in most real materials, very sharp peaks surrounded by mostly destructive interference result.[9]

A rigorous derivation from the more general Laue equations is available (see page: Laue equations).

Typical selected area electron diffraction pattern. Each spot corresponds to a different diffracted direction.

The Bragg condition is correct for very large crystals. 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. While there can be additional effects due to crystal defects, these are often quite small. In contrast, electrons interact thousands of times more strongly with solids than X-rays,[4] and also lose energy (inelastic scattering).[10] Therefore samples used in transmission electron diffraction are much thinner. Typical diffraction patterns, for instance the Figure, show spots for different directions (plane waves) of the electrons leaving a crystal. The angles that Bragg's law predicts are still approximately right, but in general there is a lattice of spots which are close to projections of the reciprocal lattice that is at right angles to the direction of the electron beam. (In contrast, Bragg's law predicts that only one or perhaps two would be present, not simultaneously tens to hundreds.) With low-energy electron diffraction where the electron energies are typically 30-1000 electron volts, the result is similar with the electrons reflected back from a surface.[11] Also similar is reflection high-energy electron diffraction which typically leads to rings of diffraction spots.[12]

With X-rays the effect of having small crystals is described by the Scherrer equation.[13][14][15] This leads to broadening of the Bragg peaks which can be used to estimate the size of the crystals.

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.[16] 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.[17][18][19] In these cases in nature, 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 separation parameter d is much larger than for true crystals. Precious opal is one example of colloidal crystals producing striking optical effects.

Volume Bragg gratings (VBG) or volume holographic gratings (VHG) consist of a volume where there is a periodic change in the refractive index. Depending on the orientation of the modulation of the refractive index, VBG can be used either to transmit or reflect a small bandwidth of wavelengths.[20] Bragg's law (adapted for volume hologram) dictates which wavelength will be diffracted:[21]

$${\displaystyle 2\Lambda \sin(\theta +\varphi )=m\lambda _{B}\,,}$$

where m is the Bragg order (a positive integer), λ_B 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 (K_G). Radiation that does not match Bragg's law will pass through the VBG undiffracted. 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.).

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

$${\displaystyle d={\frac {a}{\sqrt {h^{2}+k^{2}+\ell ^{2}}}}\,,}$$

where ${\displaystyle a}$ 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 gives:

$${\displaystyle \left({\frac {\lambda }{2a}}\right)^{2}=\left({\frac {\lambda }{2d}}\right)^{2}{\frac {1}{h^{2}+k^{2}+\ell ^{2}}}}$$

One can derive selection rules for the Miller indices for 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. KCl has a face-centered cubic Bravais lattice. 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. Lattice spacing for the other crystal systems can be found here.

• Bragg plane
• Crystal lattice
• Diffraction
• Distributed Bragg reflector
  • Fiber Bragg grating
• Dynamical theory of diffraction
• Electron diffraction
• Georg Wulff
• Henderson limit
• Laue conditions
• Powder diffraction
• Radar angels
• Structure factor
• X-ray crystallography

• Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: Orlando, 1976).
• Bragg W (1913). "The Diffraction of Short Electromagnetic Waves by a Crystal". Proceedings of the Cambridge Philosophical Society. 17: 43–57.

• Nobel Prize in Physics – 1915
• https://web.archive.org/web/20110608141639/http://www.physics.uoguelph.ca/~detong/phys3510_4500/xray.pdf
• Learning crystallography