diffrac_handout2 - Reciprocal Space The reflection h , k ,...

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Unformatted text preview: 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 / . A reflection is visible when the corresponding set of Bragg planes is in reflex position, that is when Braggs 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. 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. Reciprocal Space For orthorhombic tetragonal and cubic unit cells: a * = 1/ a b * = 1/ b c * = 1/ c * = * = * = = = = 90 Triclinic more complicated: * * * c* b* a* 100 110 010 011 001 111 101 000 1/ V = V * = a * b * c * [ 1 cos 2 * cos 2 * cos 2 * + 2cos *cos *cos * ] a = b * c *sin * / V * and cos = ( cos *cos * cos * ) / ( sin *sin * ) Same thing for b , c , cos , cos and for a *, cos * etc. Courtesy of George M. Sheldrick. Used with permission. 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 Ewald sphere with radius r = 1/ 2 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 the unit cell dimensions....
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This note was uploaded on 11/27/2011 for the course CHEMICAL E 20.410j taught by Professor Rogerd.kamm during the Spring '03 term at MIT.

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diffrac_handout2 - Reciprocal Space The reflection h , k ,...

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