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diffrac_handout2 - Reciprocal Space Removed due to...

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Reciprocal Space The reflection h , k , l is generated by diffraction of the X-ray beam at the Bragg plane set h , k , l , which intersects the three edges of the unit cell at 1/ h , 1/ k and 1/ l . Sets of planes in real space (with spacing d ) correspond to points in reciprocal space (distance d* from the origin). The vector d* is perpendicular to the Bragg planes and has the length | d *| = 2sin θ / λ . A reflection is visible when the corresponding set of Bragg planes is in reflex position, that is when Bragg’s law is fulfilled. In an alternative description: a reflection is visible when the corresponding scattering vector s = d * intersects with the Ewald sphere. Removed due to copyright re strictions . Please see: Massa, Werner. Crystal Structure Determination . 2nd ed. Translated into English by R. O. Gould. New York, NY : Springer, 2004. ISBN: 3540206442.
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Reciprocal Space The reflections form a lattice in reciprocal space. Reciprocal unit cell: a * , b * , c * The dimensions and angles of the reciprocal cell are inversely proportional to the real space cell: if the unit cell doubles, the space between the X-ray reflections will be reduced by factor two. Removed due to copyright re strictions . Please see: Massa, Werner. Crystal Structure Determination . 2nd ed. Translated into English by R. O. Gould. New York, NY : Springer, 2004. ISBN: 3540206442.
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Reciprocal Space For orthorhombic tetragonal and cubic unit cells: a * = 1/ a b * = 1/ b c * = 1/ c α * = β * = γ * = α = β = γ = 90° Triclinic more complicated: α * β * γ * c* b* a* 100 110 010 011 001 111 101 000 1/ V = V * = a * b * c * [ 1 – cos 2 α * – cos 2 β * – cos 2 γ * + 2cos α *cos β *cos γ * ] ½ a = b * c *sin α * / V * and cos α = ( cos β *cos γ * – cos α * ) / ( sin β *sin γ * ) Same thing for b , c , cos β , cos γ and for a *, cos α * etc. Courtesy of George M. Sheldrick. Used with permission.
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The Reciprocal Lattice: Ewald Construction Incident beam Detector Diffracted beam hkl reflection Reciprocal lattice hkl lattice planes Crystal hkl reciprocal lattice point s s Q C O P d θ θ θ Ewald sphere with radius r = 1/ λ 2 θ
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Intensities of the Reflections With the help of Braggs law and the Ewald construction, we can calculate the place of a reflection on the detector, provided we know the unit cell dimensions. Indeed, the position of a spot is determined alone by the metric symmetry of the unit cell. The intensity of a spot,
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