en>Daddi Moussa Ider Abdallah |
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| [[File:Vonlaue.png|thumb|300px|Ray diagram of Von Laue formulation]]
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| In [[physics]], a '''Bragg plane''' is a [[Plane (geometry)|plane]] in [[reciprocal space]] which bisects one reciprocal lattice vector <math>\mathbf{K}</math>.<ref>{{Cite book
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| | last1 = Ashcroft | first1 = Neil W.
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| | last2 = Mermin | first2 = David
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| | title = Solid State Physics
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| | publisher = Brooks Cole
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| | edition = 1
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| | date = January 2, 1976
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| | pages = 96–100
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| | isbn = 0-03-083993-9}}</ref> It is relevant to define this plane as part of the definition of the Von Laue condition for [[Interference (wave propagation)|diffraction peaks]] in [[X-ray_crystallography|x-ray diffraction crystallography]].
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| Considering the diagram at right, the arriving [[x-ray]] [[plane wave]] is defined by:
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| :<math>e^{i\mathbf{k}\cdot\mathbf{r}}=\cos {(\mathbf{k}\cdot\mathbf{r})} +i\sin {(\mathbf{k}\cdot\mathbf{r})}</math>
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| Where <math>\mathbf{k}</math> is the incident wave vector given by:
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| :<math>\mathbf{k}=\frac{2\pi}{\lambda}\hat n</math>
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| where <math>\lambda</math> is the [[wavelength]] of the incident [[photon]]. While the [[Bragg's law|Bragg formulation]] assumes a unique choice of direct lattice planes and [[specular reflection]] of the incident X-rays, the Von Laue formula only assumes monochromatic light and that each scattering center acts as a source of secondary wavelets as described by the [[Huygens principle]]. Each scattered wave contributes to a new plane wave given by:
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| :<math>\mathbf{k^\prime}=\frac{2\pi}{\lambda}\hat n^\prime</math>
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| The condition for constructive interference in the <math>\hat n^\prime</math> direction is that the path difference between the photons is an integer multiple (m) of their wavelength. We know then that for constructive interference we have:
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| :<math>|\mathbf{d}|\cos{\theta}+|\mathbf{d}|\cos{\theta^\prime}=\mathbf{d}\cdot(\hat n-\hat n^\prime)=m\lambda</math>
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| where <math>m\in\mathbb{Z}</math>. Multiplying the above by <math>2\pi/\lambda</math> we formulate the condition in terms of the wave vectors <math>\mathbf{k}</math> and <math>\mathbf{k^\prime}</math>:
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| :<math>\mathbf{d}\cdot(\mathbf{k}-\mathbf{k^\prime})=2\pi m</math> | |
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| [[File:Bragg plane_illustration.png|thumb|300px|The Bragg plane in blue, with its associated reciprocal lattice vector K.]] | |
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| Now consider that a crystal is an array of scattering centres, each at a point in the [[Bravais lattice]]. We can set one of the scattering centres as the origin of an array. Since the lattice points are displaced by the Bravais lattice vectors <math>\mathbf{R}</math>, scattered waves interfere constructively when the above condition holds simultaneously for all values of <math>\mathbf{R}</math> which are Bravais lattice vectors, the condition then becomes:
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| :<math>\mathbf{R}\cdot(\mathbf{k}-\mathbf{k^\prime})=2\pi m</math>
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| An equivalent statement (see [[Reciprocal_lattice#Mathematical_description|mathematical description of the reciprocal lattice]]) is to say that:
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| :<math>e^{i(\mathbf{k}-\mathbf{k^\prime})\cdot\mathbf{R}}=1</math>
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| By comparing this equation with the definition of a reciprocal lattice vector, we see that constructive interference occurs if <math>\mathbf{K}=\mathbf{k}-\mathbf{k^\prime}</math> is a vector of the reciprocal lattice. We notice that <math>\mathbf{k}</math> and <math>\mathbf{k^\prime}</math> have the same magnitude, we can restate the Von Laue formulation as requiring that the tip of incident wave vector <math>\mathbf{k}</math> must lie in the plane that is a perpendicular bisector of the reciprocal lattice vector <math>\mathbf{K}</math>. This reciprocal space plane is the ''Bragg plane''.
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| ==References==
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| {{reflist}}
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| ==See also==
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| * [[X-ray crystallography]]
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| * [[Reciprocal lattice]]
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| * [[Bravais lattice]]
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| * [[Powder diffraction]]
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| * [[Kikuchi line]]
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| * [[Brillouin zone]]
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| [[Category:Crystallography]]
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| [[Category:Geometry]]
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| [[Category:Fourier analysis]]
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| [[Category:Lattice points]]
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| [[Category:Diffraction]]
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