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Introduction to five and six dimensional modulations

Introduction to five and six dimensional modulations . Michal Dušek and V áclav Petříček Institute of Physics ASCR Praha, Czech Republic. Workshop Jana2006. Five - dimensional st ructures . - α. b *. - α. α. q 2. q 1. α. a *.

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Introduction to five and six dimensional modulations

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  1. Introduction to five and six dimensional modulations Michal Dušekand VáclavPetříček Institute of Physics ASCR Praha, CzechRepublic Workshop Jana2006

  2. Five-dimensional structures

  3. -α b* -α α q2 q1 α a* Example: derivation of rotation parts from the symbol and modulation vector 3+2 dimensional structure Space group P-42m q1 = (α, α,0), q2 = (-α, α,0) No systematic extinctions

  4. General symmetry element: From the basic symmetry as determined from main reflections Internal space cannot be mixed up with external one Form the metric properties (unitary conditions) → The only new information is the intrinsic part of the 4th component of the translation vector which, analogically to 3d symmetry, affects systematic extinctions of reflections (here satellites). It expressed how the modulation wave is shifted the internal space.

  5. Superspace group symbol: H.Stokes, B.J.Campbell, S.van Smaalen, ActaCryst. (2011). A67, 45-55    

  6. Translation components Jana2006 can automatically determine the 3d space group and centering in four dimensions. Test for the superspace symmetry is made just for (3+1) dimensional case.

  7. Symmetry in Melilite Natural melilite from San Venanzo, Umbria, Italy Formula: (Ca1.89Sr0.01Na0.08K0.02)(Mg0.92Al0.08)(Si1.98Al0.02)O7 Superspace group: P-421m(αα0, α-α0) + translation part Cell parameters: a=7.860 (1), c=5.024 (1) Å q vectors : 0.2815(3)(a*+b*) , 0.2815(3)(-a*+b*) l=1

  8. Symmetry in SBN Formula: Sr0.61Ba0.39Nb2O6 Superspace group: P4bm(αα½, α-α½) + translation part Cell parameters: a=12.4815 (9), c=3.9428 (6) Å q vectors : q1=(α,α,½), q2=(α,-α,½), α=0.3075(6)

  9. Cell parameters: a=12.482, c=3.943 Å P4bm(αα½, α-α½) The rational part of q-vector must be removed by doubling c. Transformation matrix on indices (row): Resulting indices: (h, k, 2l+m+n, m, n) New super space group: X4bm(αα0, α-α0) Centring: (0,0,0,1/2,1/2,1/2)

  10. Twinning can reduce dimension Hexagonal cell a=13.88 b=20.43 q1=(2/3 1/3 0) q2=(-1/3 2/3 0) (3+2)dimensional structure or a twin.

  11. Overlaps in Melilite Natural melilite from San Venanzo, Umbria, Italy Formula: (Ca1.89Sr0.01Na0.08K0.02)(Mg0.92Al0.08)(Si1.98Al0.02)O7 Superspace group: P-421m(αα0, α-α0) + translation part Cell parameters: a=7.860 (1), c=5.024 (1) Å q vectors : 0.2815(3)(a*+b*) , 0.2815(3)(-a*+b*) l=1

  12. Example: Five dimensional incommensurate structure of fresnoite L.Bindi, M.Dušek, V.Petříček, P.Bonazzi, ActaCryst. (2006). B62, 1031–1037 The sample from Museo di Storia Naturale, Sezione di Mineralogia e Litologia, Universita` di Firenze, Italy, catalogue number 44362/G originates from the sanbornite deposits of eastern Fresno County, California. Fresnoite, Ba2TiSi2O8, is structurally related to the melilite group minerals. Composition from electron microprobe: (Ba1.962Ca0.038) =Ba 2.000Ti1.000Si2.000O8.000

  13. Where are combined satellites? Here we only see lambda/2 reflections

  14. Jana indexing tool We see both combined satellites and lambda/2 reflections

  15. Verification in http://stokes.byu.edu/ssg.html by typing symmetry operators to finding tool

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