Bragg laaw

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Bragg·s law:

When a beam of X-rays (wavelength ) strikes a crystal surface in which the layers of atoms or ions are

separated by a distance d , the maximum intensity of the reflected ray occurs when sin = n/2d ,

where (known as the Bragg angle) is the complement of the angle of incidence and n is an integer.The law enables the structure of many crystals to be determined. It was discovered in 1912 by Sir

William Lawrence Bragg.

In physics, Bragg's law gives the angles for coherent and incoherent scattering from a crystal lattice. When X-rays are incident onan 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 Rayleighscattering (or elastic scattering). The scattered waves can themselves be scattered but this secondary scattering is assumed tobe negligible. A similar process occurs upon scattering neutron waves from the nuclei or by a coherent spin interaction with anunpaired electron. These re-emitted wave fields interfere with each other either constructively or destructively (overlappingwaves either add together to produce stronger peaks or subtract from each other to some degree), producing a diffractionpattern on a detector or film. The resulting wave interference pattern is the basis of diffraction analysis. This analysis is

called Bragg diff raction.

Bragg diff raction (also referred to as the Bragg formulation of X-ray diffraction) was first proposed by William LawrenceBragg and William Henry Bragg in 1913 in response to their discovery that crystalline solids produced surprising patterns ofreflected X-rays (in contrast to that of, say, a l iquid). They found that these crystals, at certain specific wavelengths andincident angles, produced intense peaks of reflected radiation (known as Bragg peaks). The concept of Bragg diff raction appliesequally to neutron diffraction and electron diffraction processes.[1] Both neutron and X-ray wavelengths are comparable withinter-atomic distances (~150 pm) and thus are an excellent probe for this length scale.

W. L. Bragg explained this result by modeling the crystal as a set of discrete parallel planes separated by a constantparameter d . It was proposed that the incident X-ray radiation would produce a Bragg peak if their reflections off the variousplanes interfered constructively.

X-rays interact with the atoms in a crystal.

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

where n is an integer, is the wavelength of incident wave, d is the spacing between the planes in the atomic lattice, and isthe angle between the incident ray and the scattering planes.

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According to the 2 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 was derived by physicist Sir William Lawrence Bragg[3] in 1912 and first presented on 11 November 1912 tothe 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 LawrenceBragg 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. W. L. Bragg was 25years old, making him the youngest Nobel laureate.

Contents

y 1 The Bragg condition

y 2 Reciprocal space

y 3 Alternate derivation

y 4 Bragg scattering of visible light by colloids

y 5 Selection rules and practical crystallography

y 6See also

y 7 References

y 8 Further reading

y 9 External links

The Bragg condition

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Bragg diffraction. Two beams with identical wavelength and phase approach a crystalline solid and are scattered off two different atomswithin it. The lower beam traverses an extra length of 2d sin. Constructive interference occurs when this length is equal to an integer

multiple of the wavelength of the radiation.

Bragg diffraction occurs when electromagnetic radiation or subatomic particle waves with wavelength comparable to atomicspacings are incident upon a crystalline sample, scattered in a specular fashion by the atoms in the system, and undergoconstructive interference in accordance to Bragg's law. For a crystalline solid, the waves are scattered from lattice planesseparated by the interplanar distance d . Where the scattered waves interfere constructively; they remain in phase since the path

length of each wave is equal to an integer multiple of the wavelength. The path difference between two waves undergoingconstructive interference is given by 2d sin, where is the scattering angle. This leads to Bragg's law which describes thecondition for constructive interference from successive crystallographic planes (h,k,l)[4] of the crystalline lattice:

where n is an integer determined by the order given, and is the wavelength.[5] A diffraction pattern is obtained by measuringthe intensity of scattered waves as a function of scattering angle. Very strong intensities known as Bragg peaks are obtained in

the diffraction pattern when scattered waves satisfy the Bragg condition.

R eciprocal space

Although the misleading common opinion reigns that Bragg's Law measures atomic distances in real space, it does not.

Furthermore, the term demonstrates that it measures the number of wavelengths fitting between two rows of atoms,thus measuring reciprocal distances. Reciprocal lattice vectors describe the set of lattice planes as a normal vector to this setwith length G = 2 / d . Max von Laue had interpreted this correctly in a vector form, the Laue equation

where is a reciprocal lattice vector and and 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 this leads equivalentlyto Bragg's equation. This is simply explained by the conservation of momentum transfer. In this system the scanning variable canbe the length or the direction of the incident or exit wave vectors relating to energy- and angle-dispersive setups. The simple

relationship between diffraction angle and Q-space is then:

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

Alternate derivation

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

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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

The two separate waves will arrive at a point with the same phase, and hence undergo constructive interference, if and only ifthis path difference is equal to any integer value of the wavelength, i.e.

(C' needs to be defined)

where the same definition of n and apply as above.

Clearly,

and

from which it follows that

Putting everything together,

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which simplifies to

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 toone centimeter) in length, and which appear analogous to their atomic or molecular counterparts.[6] The periodic arrays ofspherical 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.[7][8][9]

Thus, it has been known for many years that, due to repulsive Coulombic interactions, electrically charged macromolecules inan aqueous environment can exhibit long-range crystal-like correlations with interparticle separation distances often beingconsiderably greater than the individual particle diameter. In all of these cases in nature, the same brilliant iridescence (or playof colours) 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 solid.

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:

where a is the lattice spacing of the cubic crystal, and h, k , and l are the Miller indices of the Bragg plane. Combining thisrelation with Bragg's law:

One can derive selection rules for the Miller indices for different cubic Bravais lattices; here, selection rules for several will begiven as is.

Selection rules for the Miller indices

Bravais lattice Example compounds Allowed reflections Forbidden reflections

Simple cubic N aC l , LiH , PbS , KCl Any h, k , l None

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Body-centered cubic (At RTP) Pure Fe, W , T a, Cr h + k + l even h + k + l odd

Face-centered cubic (At RTP) Pure Cu, Al , N i h, k , l all odd or all even h, k , l mixed odd and even

Diamond F.C.C. Z nS e , CuC l , AgI , Cu F , S i, Ge all: odd, or even & h+k +l = 4n above, or even &h+k +l � 4n

Triangular lattice (HCP at RTP) Pure T i, Z r , C d 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. Selection rules for other structures can bereferenced elsewhere, or derived.

Read more: http://www.answers.com/topic/bragg-s-law#ixzz1P4jZz5i9

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Bragg laaw