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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

Introduction to five and six dimensional modulations

Michal Dušekand VáclavPetříček

Institute of Physics ASCR

Praha, CzechRepublic

Workshop Jana2006

slide3

b*

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

slide4

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.

slide6

Superspace group symbol:

H.Stokes, B.J.Campbell, S.van Smaalen, ActaCryst. (2011). A67, 45-55    

slide7

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.

slide8

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

slide10

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)

slide11

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)

slide12

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.

slide13

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

slide14

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

slide15

Where are combined satellites?

Here we only see lambda/2 reflections

slide16

Jana indexing tool

We see both combined satellites and lambda/2 reflections