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| {{Redirect|Bragg scattering|[[wind waves]] [[radar]] [[remote sensing]]|Wave radar}}
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| In [[physics]], '''Bragg's law''' gives the angles for coherent and incoherent [[scattering]] from a crystal lattice. When [[X-ray]]s are incident on an [[atom]], they make the [[electron|electronic cloud]] move as does any [[electromagnetic wave]]. The [[Motion (physics)|movement]] of these [[electric charge|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.
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| A similar [[Process (science)|process]] occurs upon [[scattering]] neutron waves from the [[atomic nucleus|nuclei]] or by a [[Coherence (physics)|coherent]] [[Spin (physics)|spin]] interaction with an unpaired [[electron]]. These re-emitted wave fields [[Interference (wave propagation)|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. This analysis is called ''Bragg diffraction''.
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| Bragg diffraction (also referred to as the '''Bragg formulation of X-ray diffraction''') was first proposed by [[William Lawrence Bragg]] and [[William Henry Bragg]] in 1913 in response to their discovery that [[crystal]]line 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 (known as ''Bragg peaks''). The concept of Bragg diffraction applies equally to [[neutron diffraction]] and [[electron diffraction]] processes.<ref>John M. Cowley (1975) ''Diffraction physics'' (North-Holland, Amsterdam) ISBN 0-444-10791-6.</ref> Both neutron and X-ray wavelengths are comparable with inter-atomic distances (~150 pm) and thus are an excellent probe for this [[length scale]].
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| [[File:Diffusion rayleigh et diffraction svg.svg|thumb|450px|X-rays interact with the atoms in a [[crystal]].]]
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| W. L. 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 2π; this condition can be expressed by Bragg's law,<ref>See, for example, [http://www.encalc.com/?expr=n%20lambda%20%2F%20(2*sin(theta))%20in%20nanometers&var1=n&val1=1&var2=lambda&val2=620%20nm&var3=theta&val3=45%20degrees&var4=&val4= this example calculation] of interatomic spacing with Bragg's law.</ref>
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| :<math>n\lambda=2d\sin\theta\!</math>
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| where ''n'' is an integer, ''λ'' is the [[wavelength]] of incident wave, ''d'' is the spacing between the planes in the atomic lattice, and ''θ'' is the angle between the incident ray and the scattering planes. Note that moving particles, including electrons, protons and neutrons, have an associated [[De Broglie wavelength]].
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| [[Image:Braggs Law.svg|thumb|450px|According to the 2''θ'' deviation, the phase shift causes constructive (left figure) or destructive (right figure) interferences.]]
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| Bragg's Law was derived by physicist [[William Lawrence Bragg|Sir William Lawrence Bragg]]<ref>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 [http://nobelprize.org/nobel_prizes/physics/laureates/1915/present.html 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.</ref> in 1912 and first presented on 11 November 1912 to the [[Cambridge Philosophical Society]]. Although simple, Bragg's law confirmed the existence of real [[Subatomic particle|particle]]s at the atomic scale, as well as providing a powerful new tool for studying [[crystal]]s in the form of X-ray and neutron diffraction. William Lawrence Bragg and his father, [[William Henry Bragg|Sir William Henry Bragg]], were awarded the [[Nobel Prize]] in physics in 1915 for their work in determining crystal structures beginning with [[Sodium chloride|NaCl]], [[Zinc sulfide|ZnS]], and [[diamond]]. They are the only father-son team to jointly win. W. L. Bragg was 25 years old, making him the youngest Nobel laureate.
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| == Bragg condition ==
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| [[Image:BraggPlaneDiffraction.svg|thumb|400px|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 2''d''sin''θ''. Constructive interference occurs when this length is equal to an integer multiple of the wavelength of the radiation.]]
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| Bragg diffraction occurs when electromagnetic radiation or subatomic particle waves with wavelength comparable to atomic spacings are incident upon a crystalline sample, are scattered in a specular fashion by the atoms in the system, and undergo constructive interference in accordance to Bragg's law. For a crystalline solid, the waves are scattered from lattice planes separated by the interplanar distance ''d''. Where the scattered waves [[Interference (wave propagation)|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 undergoing constructive interference is given by 2''d''sin''θ'', where ''θ'' is the scattering angle. This leads to Bragg's law, which describes the condition for constructive interference from successive [[crystallographic planes]] (''h'', ''k'', and ''l'', as given in [[Miller index|Miller Notation]])<ref>{{Cite book|title=Introductory Solid State Physics|author=H. P. Myers|publisher=Taylor & Francis|year=2002|isbn=0-7484-0660-3|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref> of the crystalline lattice:
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| :<math>
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| 2 d\sin\theta = n\lambda,\!
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| </math>
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| where ''n'' is an integer determined by the order given, and ''λ'' is the wavelength.<ref>{{Cite document|url=http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/bragg.html|title=Bragg's Law|author=Carl. R. Nave|publisher=HyperPhysics, Georgia State University|accessdate=2008-07-19|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref> 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 when scattered waves satisfy the Bragg condition.
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| It should be taken into account that 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 the angle is varied. However, since many atomic planes are interfering in real materials, very sharp peaks surrounded by mostly destructive interference result.<ref>[http://electrons.wikidot.com/x-ray-diffraction-and-bragg-s-law]</ref>
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| == Reciprocal space ==
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| Although the misleading common opinion reigns that Bragg's law measures atomic distances in real space, it does not. This first statement only seems to be true if it's further elaborated that distances measured during a Bragg experiment are inversely proportional to the distance d in the lattice diagram. Furthermore, the <math> n\lambda</math> 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 set with length <math> G = 2\pi / d .</math> [[Max von Laue]] had interpreted this correctly in a vector form, the [[Laue equation]]
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| <math> \vec{G} = \vec{k}_{f}\ - \vec{k}_{i}</math>
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| where <math>\vec{G}</math> is a reciprocal lattice vector and <math>\vec{k}_{f}</math> and <math>\vec{k}_{i}</math> are the wave vectors of the diffracted and the incident beams respectively.
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| Together with the condition for elastic scattering <math> |k_f| = |k_i|</math> and the introduction of the scattering angle <math>2 \theta</math> this leads equivalently to Bragg's equation. This is simply explained by the [[conservation of momentum]]. In this system the scanning variable can be 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:
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| :<math> Q = \frac{4 \pi \sin \left ( \theta \right )}{\lambda} </math>
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| The concept of [[reciprocal lattice]] is the [[Fourier space]] of a crystal lattice and necessary for a full mathematical description of wave mechanics.
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| == Alternate derivation ==
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| Suppose that a single [[monochromatic]] [[wave]] (of any type) is incident on aligned planes of [[Square lattice|lattice]] points, with separation <math>d</math>, at angle <math>\theta</math>. Points '''A''' and '''C''' are on one plane, and '''B''' is on the plane below. Points '''ABCC'''' form a quadrilateral.
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| <center>[[Image:Bragg's law.svg|600px]]</center>
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| There will be a path difference between the [[ray (optics)|ray]] that gets reflected along '''AC'''' and the ray that gets transmitted, then reflected, along '''AB''' and '''BC''' respectively. This path difference is
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| :<math>(AB+BC) - (AC'). \,</math>
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| The two separate waves will arrive at a point with the same [[Phase (waves)|phase]], and hence undergo [[constructive interference]], if and only if this path difference is equal to any integer value of the [[wavelength]], i.e. | |
| :<math>(AB+BC) - (AC') = n\lambda, \,</math>
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| where the same definition of <math>n</math> and <math>\lambda</math> apply as above.
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| Therefore,
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| :<math>AB=BC=\frac{d}{\sin\theta}\,</math> and <math>AC=\frac{2d}{\tan\theta}, \,</math>
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| from which it follows that
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| :<math>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. \,</math>
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| Putting everything together,
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| :<math>n\lambda=\frac{2d}{\sin\theta}(1-\cos^2\theta)=\frac{2d}{\sin\theta}\sin^2\theta,</math>
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| which simplifies to
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| :<math>n\lambda=2d\sin\theta, \,</math>
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| which is Bragg's law.
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| == Bragg scattering of visible light by colloids ==
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| A [[colloidal crystal]] is a highly [[Order (crystal lattice)|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.<ref name='Pieranski_1983'>{{Cite journal|title=Colloidal Crystals|journal=Contemporary Physics|year=1983|first=P|last=Pieranski|coauthors=|volume=24|issue=|pages=25|id= |url=|format=|doi=10.1080/00107518308227471 |bibcode = 1983ConPh..24...25P }}</ref> The periodic arrays of spherical particles make similar arrays of [[Vacancy defect|interstitial voids]] (the spaces between the particles), which act as a natural [[diffraction grating]] for [[visible spectrum|visible light waves]], especially when the interstitial spacing is of the same [[order of magnitude]] as the [[angle of incidence|incident]] lightwave.<ref name='Hiltner_1969'>{{Cite journal|title=Diffraction of Light by Ordered Suspensions|journal=Journal of Physical Chemistry|year=1969|first=PA|last=Hiltner|coauthors=IM Krieger|volume=73|issue=|pages=2306|id= |url=|format= }}</ref><ref name='Aksay_1984'>{{Cite journal|title=Microstructural Control through Colloidal Consolidation|journal=Proceedings of the American Ceramic Society|year=1984|first=IA|last=Aksay|coauthors=|volume=9|issue=|pages=94|id= |url=|format= }}</ref><ref>Luck, W. et al., Ber. Busenges Phys. Chem. , Vol. 67, p.84 (1963)</ref>
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| Thus, it has been known for many years that, due to [[Coulomb's law|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 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.
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| == Selection rules and practical crystallography ==
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| Bragg's law, as stated above, can be used to obtain the lattice spacing of a particular [[cubic system]] through the following relation:
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| : <math> d = \frac{a}{ \sqrt{h^2 + k^2 + l^2}} </math>
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| where <math>a</math> is the lattice spacing of the [[cubic crystal]], and <math>h</math>, <math>k</math>, and <math>l</math> are the [[Miller indices]] of the Bragg plane. Combining this relation with Bragg's law:
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| : <math> \left( \frac{ \lambda\ }{ 2a } \right)^2 = \frac{ \sin ^2 \theta\ }{ h^2 + k^2 + l^2 }. </math>
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| One can derive selection rules for the [[Miller indices]] for different cubic [[Bravais lattices]]; here, selection rules for several will be given as is.
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| {| class="wikitable"
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| |+ Selection rules for the Miller indices <!-- Selection Rules for the Miller Indices -->
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| ! Bravais lattice
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| ! Example compounds
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| ! Allowed reflections
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| ! Forbidden reflections
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| |-
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| | Simple cubic
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| | Po
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| | Any ''h'', ''k'', ''l''
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| | None
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| |-
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| | Body-centered cubic
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| | Fe, W, Ta, Cr
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| | ''h'' + ''k'' + ''l'' = even
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| | ''h'' + ''k'' + ''l'' = odd
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| |-
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| | Face-centered cubic
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| | Cu, Al, Ni, NaCl, LiH, PbS
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| | ''h'', ''k'', ''l'' all odd or all even
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| | ''h'', ''k'', ''l'' mixed odd and even
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| |-
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| | Diamond F.C.C.
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| | Si, Ge
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| | all odd, or all even with ''h''+''k''+''l'' = 4n
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| | ''h'', ''k'', ''l'' mixed odd and even, or all even with ''h''+''k''+''l'' ≠ 4n
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| |-
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| | [[Triangular lattice]]
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| | Ti, Zr, Cd, Be
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| | ''l'' even, ''h'' + 2''k'' ≠ 3''n''
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| | ''h'' + 2''k'' = 3''n'' for odd ''l''
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| |}
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| These selection rules can be used for any crystal with the given crystal structure. KCl exhibits a fcc cubic structure. However, the K<sup>+</sup> and the Cl<sup>−</sup> 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 [[Structure factor|derived]].
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| == See also ==
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| * [[Crystal lattice]]
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| * [[Diffraction]]
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| * [[Distributed Bragg reflector]]
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| ** [[Fiber Bragg grating]]
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| * [[Dynamical theory of diffraction]]
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| * [[Henderson limit]]
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| * [[Laue conditions]]
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| * [[Powder diffraction]]
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| * [[Structure factor]]
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| * [[William Lawrence Bragg]]
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| * [[X-ray crystallography]]
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| == References ==
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| {{reflist}}
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| == Further reading ==
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| * Neil W. Ashcroft and N. David Mermin, ''Solid State Physics'' (Harcourt: Orlando, 1976).
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| *{{cite journal | last1 = Bragg | first1 = W.L. | author-separator =, | author-name-separator= | year = 1913 | title = The Diffraction of Short Electromagnetic Waves by a Crystal | url = | journal = Proceedings of the Cambridge Philosophical Society | volume = 17 | issue = | pages = 43–57 }}
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| == External links ==
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| * [http://nobelprize.org/physics/laureates/1915/index.html Nobel Prize in Physics - 1915]
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| * http://www.citycollegiate.com/interference_braggs.htm
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| * http://www.physics.uoguelph.ca/~detong/phys3510_4500/xray.pdf
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| {{DEFAULTSORT:Bragg's Law}}
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| [[Category:Diffraction]]
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| [[Category:Neutron]]
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| [[Category:X-rays]]
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| [[Category:Condensed matter physics]]
| |
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