phasing_handout2 - Symmetry in Reciprocal Space The...

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Symmetry in Reciprocal Space The diffraction pattern is always centrosymmetric (at least in good approximation). Friedel’s law: I hkl = I - h - k - l . Fourfold symmetry in the diffraction pattern corresponds to a fourfold axis in the space group ( 4, 4, 4 1 , 4 2 or 4 3 ), threefold to a threefold, etc. If you take away the translational part of the space group symmetry and add an inversion center, you end up with the Laue group. The Laue group describes the symmetry of the diffraction pattern. The Laue symmetry can be lower than the metric symmetry of the unit cell, but never higher. That means: A monoclinic crystal with β = 90° is still monoclinic. The diffraction pattern from such a crystal will have monoclinic symmetry, even though the metric symmetry of the unit cell looks orthorhombic. There are 11 Laue groups: -1, 2/m, mmm, 4/m, 4/mmm, -3, -3/m, 6/m, 6/mmm, m3, m3m
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Laue Symmetry 432, -43m, m3m m3m 23, m3 m3 Cubic 622, 6mm, -6m2, 6/mmm 6/mmm 6, -6, 6/m 6/m Hexagonal 32, 3m, -3m -3/m 3, -3 -3 Trigonal/ Rhombohedral 422, 4mm, -42m, 4/mmm 4/mmm 4, -4, 4/m 4/m Tetragonal 222, mm2, mmm mmm Orthorhombic 2, m, 2/m 2/m Monoclinic 1, -1 -1 Triclinic Point Group Laue Group Crystal System
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Space Group Determination The first step in the determination of a crystal structure is the determination of the unit cell from the diffraction pattern. Second step: Space group determination. From the symmetry of the diffraction pattern, we can determine the Laue group, which narrows down the choice quite considerably. Usually the Laue group and the metric symmetry of the unit cell match. The <| E 2 -1 |> statistics, can give us an idea, whether the space group is centrosymmetric or acentric. Even thought the diffraction pattern is always centrosymmetric, the intensity distribution across the reciprocal space is much more even for a centrosymmetric space group. From systematic absences, we can determine the lattice type as well as screw axes and glide planes. This is usually enough to narrow down the choice to a very short list.
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E 2 -1 Statistics We measure intensities I IF 2 F : structure factors Normalized structure factors E: E 2 = F 2 /< F 2 >< F 2 >: mean value for reflections at same resolution < E 2 > = 1 < | E 2 -1 | > = 0.736 for non-centrosymmetric structures 0.968 for centrosymmetric structures Heavy atoms on special positions and twinning tend to lower this value. Pseudo translational symmetry tend to increase this value.
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E 2 -1 Statistics < | E 2 -1 | > = 0.736 for non-centrosymmetric structures 0.968 for centrosymmetric structures <| E 2 –1|> = 0.968 <| E 2 –1|> = 0.736 2 kl projection of the reflections of a structure in the space group P- 1 . 2 kl projection of the reflections of a structure in the space group P 1 . Courtesy of George M. Sheldrick. Used with permission.
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Systematic Absences Lattice centering and symmetry elements with translation (glide planes and screw axes) cause certain reflections to have zero intensity in the diffraction pattern. If, e.g. , all reflections 0, k , 0 with odd values for k are absent, we know that we have a 2 1 axis along b .
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phasing_handout2 - Symmetry in Reciprocal Space The...

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