Bragg's law

In physics, Bragg's law, or Wulff–Bragg's condition in post-Soviet countries, a special case of Laue diffraction, gives the angles for coherent and incoherent scattering from a crystal lattice. 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.

History

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 William 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 concept of Bragg diffraction applies equally to neutron diffraction and electron diffraction processes.^[2] Both neutron and X-ray wavelengths are comparable with inter-atomic distances (~ 150 pm) and thus are an excellent probe for this length scale.

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

Sir William 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 2 $π$ ; this condition can be expressed by Bragg's law (see Bragg condition section below) and was first presented by Sir William Lawrence Bragg on 11 November 1912 to the Cambridge Philosophical Society.^[3]^[4] 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. William Lawrence Bragg and his father, Sir 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. William Lawrence Bragg was 25 years old, making him then, the youngest physics Nobel laureate.

Bragg condition

Bragg diffraction. 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, with wavelength comparable to atomic spacings, is scattered in a specular fashion 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 interplanar distance d. When the scattered waves interfere constructively, they remain in phase since the difference between the path lengths of the two waves is equal to an integer multiple of the wavelength. The path difference between two waves undergoing interference is given by 2dsinθ, where θ is the scattering angle (see figure on the right). The effect of the constructive or destructive interference intensifies because of the cumulative effect of reflection in successive crystallographic planes 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:^[5]

2d\sin \theta =n\lambda \,,

where n is a positive integer and λ is the wavelength of incident wave. Note that moving particles, including electrons, protons and neutrons, have an associated wavelength called de Broglie 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 at the points where the scattering angles satisfy Bragg condition.

Heuristic derivation

Suppose that a single monochromatic wave (of any type) is incident on aligned planes of lattice points, with separation $d$ , at angle $\theta$ . 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, then reflected, along AB and BC respectively. This path difference is

(AB+BC)-(AC')\,.

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.

(AB+BC)-(AC')=n\lambda \,,

where the same definition of $n$ and $\lambda$ apply as above.

Therefore,

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

from which it follows that

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,

n\lambda ={\frac {2d}{\sin \theta }}(1-\cos ^{2}\theta )={\frac {2d}{\sin \theta }}\sin ^{2}\theta \,,

which simplifies to

n\lambda =2d\sin \theta \,,

which is Bragg's law.

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 interfering in real materials, very sharp peaks surrounded by mostly destructive interference result.^[6]

Bragg scattering of visible light by colloids

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.^[7] 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.^[8]^[9]^[10] 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.

Volume Bragg gratings

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.^[11] Bragg's law (adapted for volume hologram) dictates which wavelength will be diffracted:^[12]

2n\Lambda \sin(\theta +\phi )=\lambda _{B}\,,

where n is a positive integer, λ_B the diffracted wavelength, Λ the step 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.).

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:

d={\frac {a}{{\sqrt {h^{2}+k^{2}+l^{2}}}}}\,,

where $a$ 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:

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

Bravais lattices	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. Lattice spacing for the other crystal systems can be found here.

References

↑ Bragg, W.H.; Bragg, W.L. (1913). "The Reflexion of X-rays by Crystals". Proc R. Soc. Lond. A. 88 (605): 428–38. Bibcode:1913RSPSA..88..428B. doi:10.1098/rspa.1913.0040. (Free access)
↑ John M. Cowley (1975) Diffraction physics (North-Holland, Amsterdam) ISBN 0-444-10791-6.
↑ See, for example, this example calculation Archived July 10, 2011, at the Wayback Machine. of interatomic spacing with Bragg's law.
↑ There are some sources, like the Academic American Encyclopedia, that attribute the discovery of the law to both W.L Bragg and his father W.H. Bragg, but the official Nobel Prize site and the biographies written about him ("Light Is a Messenger: The Life and Science of William Lawrence Bragg", Graeme K. Hunter, 2004 and "Great Solid State Physicists of the 20th Century", Julio Antonio Gonzalo, Carmen Aragó López) make a clear statement that William Lawrence Bragg alone derived the law.
↑ H. P. Myers (2002). Introductory Solid State Physics. Taylor & Francis. ISBN 0-7484-0660-3.
↑ x-ray diffraction, Bragg's law and Laue equation on electrons.wikidot.com.
↑ Pieranski, P (1983). "Colloidal Crystals". Contemporary Physics. 24: 25. Bibcode:1983ConPh..24...25P. doi:10.1080/00107518308227471.
↑ Hiltner, PA; IM Krieger (1969). "Diffraction of Light by Ordered Suspensions". Journal of Physical Chemistry. 73: 2306. doi:10.1021/j100727a049.
↑ Aksay, IA (1984). "Microstructural Control through Colloidal Consolidation". Proceedings of the American Ceramic Society. 9: 94.
↑ Luck, W. et al., Ber. Busenges Phys. Chem. , Vol. 67, p.84 (1963).
↑ Barden, S.C.; Williams, J.B.; Arns, J.A.; Colburn, W.S. (2000). "Tunable Gratings: Imaging the Universe in 3-D with Volume-Phase Holographic Gratings (Review)". ASP Conf. Ser. 195: 552.
↑ C. Kress, Bernard (2009). Applied Digital Optics : From Micro-optics to Nanophotonics. ISBN 978-0-470-02263-4.

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