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bragg re-flection

Bragg's law

In physics, 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 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.

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 to 2π; this condition can be expressed by Bragg's law:

nlambda=2dcdotsintheta ,

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.

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 2 theta 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

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 paths respectively. This path difference is:

(AB+BC) - (AC') ,
If this path difference is equal to any integer value of the wavelength then the two separate waves will arrive at a point with the same phase, and hence undergo constructive interference. Expressed mathematically:
(AB+BC) - (AC') = nlambda ,
Where the same definition of n and λ apply from the article above
Using the Pythagorean theorem it is easily shown that:
AB=frac{d}{sintheta}, and BC=frac{d}{sintheta}, and AC=frac{2d}{tantheta},
also it can be shown that:
AC'=ACcdotcostheta=frac{2d}{tantheta}costheta,
Putting everything together and using known identities for sinusoidal functions:
nlambda=frac{2d}{sintheta}-frac{2d}{tantheta}costheta=frac{2d}{sintheta}(1-cos^2theta)=frac{2d}{sintheta}sin^2theta
Which simplifies to:
nlambda=2dcdotsintheta ,
yielding Bragg's law.

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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