Direct methods in superspace. II. The first application to an unknown incommensurate modulated structure
Authors:Shi-Bin, Xiang; Hai-Fu, Fan; Xiao-Jing, Wu; Fang-Hua, Li; Qing, P.
Journal:Acta Crystallographica A 46 929-934 (1990)
DOI:https://doi.org/10.1107/S0108767390007607
B-IncStrDB ID: 3242E4ldUl Entry date: 2010-11-08 Last revision: 2023-09-29
mod1024314151
Chemical data
Structural Formula Sum: Ba0.8 Ti8 O16 [ Help ]
Mineral Name: Ankangite [ Help ]
Crystallographic data and experimental details
a: 10.12 Å [ Help ]
b: 10.12 Å [ Help ]
γ: 90 ° [ Help ]
c: 2.96 Å [ Help ]
β: 90 ° [ Help ]
α: 90 ° [ Help ]
Modulation dimension: 1 [ Help ]
Measured independent wave vectors: (Show/hide table) [ Help ]
| Wave vector id | q_x | q_y | q_z |
|---|---|---|---|
| 1 | 0 | 0 | 0.44 |
Crystal system: tetragonal [ Help ]
Superspace group name: I4/m(00γ)00 [ Help ]
Symmetry operations of the superspace group: (Show/hide table) [ Help ]
| Operation code | Operation in algebraic form |
|---|---|
| 1 | x1,x2,x3,x4 |
| 2 | -x1,-x2,x3,x4 |
| 3 | -x2,x1,x3,x4 |
| 4 | x2,-x1,x3,x4 |
| 5 | -x1,-x2,-x3,-x4 |
| 6 | x1,x2,-x3,-x4 |
| 7 | x2,-x1,-x3,-x4 |
| 8 | -x2,x1,-x3,-x4 |
| 9 | 1/2+x1,1/2+x2,1/2+x3,x4 |
| 10 | 1/2-x1,1/2-x2,1/2+x3,x4 |
| 11 | 1/2-x2,1/2+x1,1/2+x3,x4 |
| 12 | 1/2+x2,1/2-x1,1/2+x3,x4 |
| 13 | 1/2-x1,1/2-x2,1/2-x3,-x4 |
| 14 | 1/2+x1,1/2+x2,1/2-x3,-x4 |
| 15 | 1/2+x2,1/2-x1,1/2-x3,-x4 |
| 16 | 1/2-x2,1/2+x1,1/2-x3,-x4 |
Refinement details
Nb. of observed reflections: 57 [ Help ]
R(obs): 0.190 [ Help ]
Refinement remarks: The positional modulation of Ba is given by: U3(x4) = D1[sin 2pi (x4-1/2)] + D2[sin 2pi (x4-1/2)]3 + D3[sin 2pi (x4-1/2)]5 Which is equivalent to: U3(x4) = {-(16D1+12D2+10D3) [sin 2pi x4] + (4D2+5D3) [sin 6pi x4] - D3 [sin 10pi x4]}/16 The modulation of the occupation of Ba is parameterized by a non-periodic function: P(x4)=A1*exp(-B1(x4-1/2)**2)+A2*exp(-B2(x4-1/2)**2)+C, where A1=-3.49(2), A1=4.33(2), B1=96.6(84), B2=71.5(49) and C=0.123(14) [ Help ]
Structural Information
Average Structure: (Show/hide table) [ Help ]
| Atom site label | Occupancy | Symmetry multiplicity | x | y | z | Uiso/equiv | ADP type |
|---|---|---|---|---|---|---|---|
| Ba | 0.4 | ? | 0.0 | 0.0 | 0.5 | 0.006(5) | Uiso |
| Ti | 1.0 | ? | 0.366(3) | 0.155(3) | 0.0 | 0.032(3) | Uiso |
| O1 | 1.0 | ? | 0.165(8) | 0.198(8) | 0.0 | 0.038(8) | Uiso |
| O2 | 1.0 | ? | 0.572(8) | 0.248(11) | 0.0 | 0.05(1) | Uiso |
Fourier Wave Vectors (explicit: q_x,q_y,q_z or coefficients: q_1,q_2,...): (Show/hide table) [ Help ]
| Wave vector code | q_x | q_y | q_z |
|---|---|---|---|
| 1 | 0 | 0 | 0.44 |
| 3 | 0 | 0 | 1.32 |
| 5 | 0 | 0 | 2.20 |
Definition of the displacive (translational) Fourier series: (Show/hide table) [ Help ]
| Modulation code | Atom site label | Displacement axis | Wave vector code |
|---|---|---|---|
| Baz1 | Ba | z | 1 |
| Baz3 | Ba | z | 3 |
| Baz5 | Ba | z | 5 |
Displacive (translational) Fourier coefficients: (Show/hide table) [ Help ]
| Modulation code | Sine coefficient | Cosine coefficient |
|---|---|---|
| Baz1 | -0.029(7) | 0 |
| Baz3 | 0.026(2) | 0 |
| Baz5 | 0.021(2) | 0 |