Bragg's law

In physics, Bragg's law states that when X-rays hit 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 the 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 together to produce stronger peaks or subtract 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. Both neutron and X-ray wavelengths are comparable with inter-atomic distances (~150 pm) and thus are an excellent probe for this length scale.

The interference is constructive when the phase shift is a multiple of 2π; this condition can be expressed by Bragg's law,^[1]

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

where:

$n$ is an integer determined by the order given,
$λ$ is the wavelength of X-rays, and moving electrons, protons and neutrons,
$d$ is the spacing between the planes in the atomic lattice, and
$θ$ is the angle between the incident ray and the scattering planes.

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

Note that moving particles, including electrons, protons and neutrons, have an associated De Broglie wavelength.

Bragg's Law is the result of experiments into the diffraction of X-rays or neutrons off crystal surfaces at certain angles, derived by physicist Sir William Lawrence Bragg^[2] in 1912 and first presented on 1912-11-11 to the Cambridge Philosophical Society. 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.

1 Reciprocal space
2 Alternate derivation
3 Bragg scattering of visible light by colloids
4 See also
5 Notes
6 References

Reciprocal space

Although the misleading common opinion reigns that Bragg's Law measures atomic distances in real space, it does not. Furthermore, the $n\ \lambda$ term demonstrates that it measures the number of wavelengths fitting between two rows of atoms, thus measuring reciprocal distances. Max von Laue had interpreted this correctly in a vector form, the Laue equation

$\vec G\ =\ \vec{k_f}\ -\ \vec{k_i}$

where $\vec G$ is a reciprocal lattice vector and $\vec{k_f}$ and $\vec{k_i}$ are the wave vectors of the incident and the diffracted beams.

Together with the condition for elastic scattering $| k f | = | k i |$ and the introduction of the scattering angle $2θ$ this leads equivalently to Bragg's equation.

The concept of reciprocal lattice is the Fourier space of a crystal lattice and necessary for a full mathematical description of wave mechanics.

Alternate derivation

Suppose that a single monochromatic wave (of any type) is incident on aligned planes of lattice points, with separation $d$ , at angle $θ$ , as shown below.

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, iff 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 $λ$ apply as above.

Clearly,

$AB=BC=\frac{d}{\sin\theta}\,$ 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

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

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 appear analogous to their atomic or molecular counterparts.^[3] The periodic arrays of spherical particles make similar arrays of interstitial voids, 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.^[4]^[5]^[6]

Thus, 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. In all of these cases in nature, the same brilliant iridescence (or play of colors)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 solids.

Notes

^ See, for example, this example calculation 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.
^ Pieranski, P (1983). "Colloidal Crystals". Contemporary Physics 24: 25.
^ Hiltner, PA; IM Krieger (1969). "Diffraction of Light by Ordered Suspensions". Journal of Physical Chemistry 73: 2306.
^ 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)

References

W.L. Bragg, "The Diffraction of Short Electromagnetic Waves by a Crystal", Proceedings of the Cambridge Philosophical Society, 17 (1913), 43–57.

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