Bragg's law

Bragg diffraction (also referred to as the Bragg formulation of X-ray diffraction) was first proposed by

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. 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.^[3] They are the only father-son team to jointly win.

The concept of Bragg diffraction applies equally to neutron diffraction^[4] and approximately to electron diffraction.^[5] In both cases the wavelengths are comparable with inter-atomic distances (~ 150 pm). Many other types of matter waves have also been shown to diffract,^[6][7] and also light from objected with a larger ordered structure such as opals.^[8]

Bragg condition

Bragg diffraction occurs when radiation of a

^: 1026

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

where ${\displaystyle n}$ is the

diffraction order

(${\displaystyle n=1}$ is first order, ${\displaystyle n=2}$ is second order,

${\displaystyle n=3}$

A map of the intensities of the scattered waves as a function of their angle is called a diffraction pattern. Strong intensities known as Bragg peaks are obtained in the diffraction pattern when the scattering angles satisfy Bragg condition. This is a special case of the more general Laue equations, and the Laue equations can be shown to reduce to the Bragg condition with additional assumptions.^[13]

Heuristic derivation

Suppose that a plane wave (of any type) is incident on planes of lattice points, with separation ${\displaystyle d}$, at an angle ${\displaystyle \theta }$ as shown in the Figure. 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 far from these lattice planes) with the same

, 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, from the geometry

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

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 Figure then the transition from constructive to destructive interference would be gradual as a function of angle, with gentle

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

Beyond Bragg's law

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

With X-rays the effect of having small crystals is described by the Scherrer equation.^[13][17][18] This leads to broadening of the Bragg peaks which can be used to estimate the size of the crystals.

Bragg scattering of visible light by colloids

A

Volume Bragg gratings (VBG) or

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

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.).^[23]

Selection rules and practical crystallography

The measurement of the angles can be used to determine crystal structure, see

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

where ${\displaystyle a}$ is the lattice spacing of the

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

One can derive selection rules for the

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.

See also

• 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

References