Incommensurate structures and radiation damage in Rb2V3O8 and K2V3O8 mixed-valence vanadate fresnoites
Authors:Grzechnik, Andrzej; Petricekc, Vaclav; Chernyshov, Dmitry; McMonagle, Charles; Geise, Tobias; Shahed, Hend; Friese, Karen
Journal:Acta Cryst. B 79 104-113 (2023)
DOI:https://doi.org/10.1107/S2052520623000999
B-IncStrDB ID: z034DL74QTM Entry date: 2023-02-08 Last revision: 2024-01-02
I-5D
Chemical data
Structural Formula Sum: K2 O8 V3 [ Help ]
Formula weight: 359 Da [ Help ]
Crystallographic data and experimental details
Crystal system: tetragonal [ Help ]
Space group name (H-M): P 4 b m [ Help ]
Symmetry operations of the superspace group: (Show/hide table) [ Help ]
| Operation code | Operation in algebraic form |
|---|---|
| 1 | x1,x2,x3,x4,x5 |
| 2 | -x1,-x2,x3,x3-x4,x3-x5 |
| 3 | -x2,x1,x3,x3-x5+1/2,x4+1/2 |
| 4 | x2,-x1,x3,x5+1/2,x3-x4+1/2 |
| 5 | x1+1/2,-x2+1/2,x3,x3-x5,x3-x4 |
| 6 | -x1+1/2,x2+1/2,x3,x5,x4 |
| 7 | -x2+1/2,-x1+1/2,x3,x3-x4+1/2,x5+1/2 |
| 8 | x2+1/2,x1+1/2,x3,x4+1/2,x3-x5+1/2 |
a: 8.941(2) Å [ Help ]
b: 8.941(2) Å [ Help ]
c: 5.223(2) Å [ Help ]
α: 90 ° [ Help ]
β: 90 ° [ Help ]
γ: 90 ° [ Help ]
Volume: 417.6(2) Å3 [ Help ]
Modulation dimension: 2 [ Help ]
Measured independent wave vectors: (Show/hide table) [ Help ]
| Wave vector id | q_x | q_y | q_z |
|---|---|---|---|
| 1 | 0.313(2) | 0.313(2) | 0.499(1) |
| 2 | -0.313(2) | 0.313(2) | 0.499(1) |
Z: 2 [ Help ]
Cell determination reflection Nb.: 1101 [ Help ]
θ(min) for cell determination: 3.1320 ° [ Help ]
θ(max) for cell determination: 25.9320 ° [ Help ]
Cell measurement temperature: 293 K [ Help ]
μ: 15.503 mm-1 [ Help ]
Absorption correction type: multi-scan [ Help ]
Absorption correction remarks: CrysAlisPro 1.171.41.115a (Rigaku Oxford Diffraction, 2021) Empirical absorption correction using spherical harmonics, implemented in SCALE3 ABSPACK scaling algorithm. [ Help ]
Minimum transmission factor: 0.75180 [ Help ]
Maximum transmission factor: 1.00000 [ Help ]
Refinement details
Total nb. of reflections: 2043 [ Help ]
Nb. of observed reflections: 1240 [ Help ]
Intense reflections threshold: I>3σ(I) [ Help ]
Refinement based on: F [ Help ]
R(obs): 0.0156 [ Help ]
wR(obs): 0.0193 [ Help ]
R(all): 0.0207 [ Help ]
wR(all): 0.0200 [ Help ]
S(all): 0.91 [ Help ]
S(obs): 1.14 [ Help ]
Nb. of reflections: 2043 [ Help ]
Nb. of parameters: 96 [ Help ]
Number of restraints: 0 [ Help ]
Number of constraints: 2 [ Help ]
Weighting scheme: sigma [ Help ]
Weighting scheme remarks: w=1/(σ2(F)+0.0001F2) [ Help ]
Δ/σ(max): 0.0456 [ Help ]
Δ/σ(mean): 0.0022 [ Help ]
Δρ(max): 0.21 e_Å-3 [ Help ]
Δρ(min): -0.41 e_Å-3 [ Help ]
Extinction method: B-C type 1 Gaussian isotropic (Becker & Coppens, 1974) [ Help ]
Extinction coefficient: 2050(80) [ Help ]
Absolute structure remarks: 988 of Friedel pairs used in the refinement [ Help ]
Structural Information
Average Structure: (Show/hide table) [ Help ]
| Atom site label | Atom symbol | x | y | z | ADP type | Uiso/equiv | Symmetry multiplicity | Occupancy | Coords from (d)iffraction or (c)alculated | Coords restraints or constraints | Disordered cluster | Disordered group |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| K | K | 0.331293(17) | 0.168707(17) | 0 | Uani | 0.01578(7) | 4 | 1 | d | . | . | . |
| V1 | V | 0.5 | 0.5 | 0.46108(4) | Uani | 0.00764(6) | 2 | 1 | d | . | . | . |
| V2 | V | 0.134016(12) | 0.365984(12) | 0.46737(3) | Uani | 0.00736(5) | 4 | 1 | d | . | . | . |
| O1 | O | 0.5 | 0.5 | 0.7693(2) | Uani | 0.0146(3) | 2 | 1 | d | . | . | . |
| O2 | O | 0.13020(6) | 0.36980(6) | 0.78120(18) | Uani | 0.0121(2) | 4 | 1 | d | . | . | . |
| O3 | O | 0.30794(6) | 0.41328(7) | 0.35308(12) | Uani | 0.01270(17) | 8 | 1 | d | . | . | . |
| O4 | O | 0 | 0.5 | 0.34218(19) | Uani | 0.0115(3) | 2 | 1 | d | . | . | . |
ADP components: (Show/hide table) [ Help ]
| Atom site label | Atom site symbol | U11 | U22 | U33 | U12 | U13 | U23 |
|---|---|---|---|---|---|---|---|
| K | K | 0.01796(9) | 0.01796(9) | 0.01142(15) | 0.00966(8) | 0.00010(7) | -0.00010(7) |
| V1 | V | 0.00660(8) | 0.00660(8) | 0.00972(15) | 0 | 0 | 0 |
| V2 | V | 0.00624(7) | 0.00624(7) | 0.00961(12) | -0.00011(5) | 0.00031(5) | -0.00031(5) |
| O1 | O | 0.0166(4) | 0.0166(4) | 0.0105(6) | 0 | 0 | 0 |
| O2 | O | 0.0132(3) | 0.0132(3) | 0.0098(6) | 0.0015(3) | -0.0008(2) | 0.0008(2) |
| O3 | O | 0.0096(3) | 0.0131(3) | 0.0154(3) | -0.0021(2) | 0.00142(19) | -0.0008(3) |
| O4 | O | 0.0120(3) | 0.0120(3) | 0.0106(6) | 0.0034(4) | 0 | 0 |
Fourier Wave Vectors (explicit: q_x,q_y,q_z or coefficients: q_1,q_2,...): (Show/hide table) [ Help ]
| Wave vector code | q_1 | q_2 |
|---|---|---|
| 1 | 1 | 0 |
| 2 | 0 | 1 |
Definition of the displacive (translational) Fourier series: (Show/hide table) [ Help ]
| Modulation code | Atom site label | Displacement axis | Wave vector code |
|---|---|---|---|
| Kx1 | K | x | 1 |
| Ky1 | K | y | 1 |
| Kz1 | K | z | 1 |
| Kx2 | K | x | 2 |
| Ky2 | K | y | 2 |
| Kz2 | K | z | 2 |
| V1x1 | V1 | x | 1 |
| V1y1 | V1 | y | 1 |
| V1z1 | V1 | z | 1 |
| V1x2 | V1 | x | 2 |
| V1y2 | V1 | y | 2 |
| V1z2 | V1 | z | 2 |
| V2x1 | V2 | x | 1 |
| V2y1 | V2 | y | 1 |
| V2z1 | V2 | z | 1 |
| V2x2 | V2 | x | 2 |
| V2y2 | V2 | y | 2 |
| V2z2 | V2 | z | 2 |
| O1x1 | O1 | x | 1 |
| O1y1 | O1 | y | 1 |
| O1z1 | O1 | z | 1 |
| O1x2 | O1 | x | 2 |
| O1y2 | O1 | y | 2 |
| O1z2 | O1 | z | 2 |
| O2x1 | O2 | x | 1 |
| O2y1 | O2 | y | 1 |
| O2z1 | O2 | z | 1 |
| O2x2 | O2 | x | 2 |
| O2y2 | O2 | y | 2 |
| O2z2 | O2 | z | 2 |
| O3x1 | O3 | x | 1 |
| O3y1 | O3 | y | 1 |
| O3z1 | O3 | z | 1 |
| O3x2 | O3 | x | 2 |
| O3y2 | O3 | y | 2 |
| O3z2 | O3 | z | 2 |
| O4x1 | O4 | x | 1 |
| O4y1 | O4 | y | 1 |
| O4z1 | O4 | z | 1 |
| O4x2 | O4 | x | 2 |
| O4y2 | O4 | y | 2 |
| O4z2 | O4 | z | 2 |
Displacive (translational) Fourier coefficients: (Show/hide table) [ Help ]
| Modulation code | Cosine coefficient | Sine coefficient |
|---|---|---|
| Kx1 | 0.00220(3) | -0.00079(2) |
| Ky1 | 0.00220(3) | 0.00079(2) |
| Kz1 | 0 | -0.00362(7) |
| Kx2 | -0.00189(3) | -0.00113(3) |
| Ky2 | -0.00189(3) | -0.00113(3) |
| Kz2 | 0 | 0 |
| V1x1 | -0.00196(2) | 0.000241(2) |
| V1y1 | -0.00147(2) | 0.000181(2) |
| V1z1 | -0.000237(6) | -0.00193(5) |
| V1x2 | -0.00147(2) | 0.000181(2) |
| V1y2 | 0.00196(2) | -0.000241(2) |
| V1z2 | 0.000237(6) | 0.00193(5) |
| V2x1 | -0.00186(10) | 0.001519(14) |
| V2y1 | 0.00213(10) | 0.001108(15) |
| V2z1 | -0.00128(5) | 0.000131(5) |
| V2x2 | -0.000710(15) | 0.000021(15) |
| V2y2 | -0.000710(15) | 0.000021(15) |
| V2z2 | 0 | 0 |
| O1x1 | -0.00183(6) | -0.00207(7) |
| O1y1 | -0.00114(6) | -0.00128(7) |
| O1z1 | 0.00136(14) | -0.00120(12) |
| O1x2 | -0.00114(6) | -0.00128(7) |
| O1y2 | 0.00183(6) | 0.00207(7) |
| O1z2 | -0.00136(14) | 0.00120(12) |
| O2x1 | -0.00190(6) | 0.00011(7) |
| O2y1 | -0.00047(7) | 0.00184(6) |
| O2z1 | -0.00097(12) | -0.00118(14) |
| O2x2 | 0.00047(7) | -0.00071(7) |
| O2y2 | 0.00047(7) | -0.00071(7) |
| O2z2 | 0 | 0 |
| O3x1 | -0.00234(6) | 0.00072(6) |
| O3y1 | 0.00254(6) | 0.00332(6) |
| O3z1 | -0.00215(13) | -0.00097(13) |
| O3x2 | -0.00081(6) | 0.00223(6) |
| O3y2 | 0.00084(6) | -0.00442(6) |
| O3z2 | 0.00059(14) | 0.00186(13) |
| O4x1 | -0.00198(7) | 0.00107(4) |
| O4y1 | 0.00198(7) | -0.00107(4) |
| O4z1 | 0 | 0 |
| O4x2 | -0.00384(8) | 0.00208(4) |
| O4y2 | -0.00384(8) | 0.00208(4) |
| O4z2 | 0 | 0 |
Definition of the ADP Fourier series: (Show/hide table) [ Help ]
| Modulation code | Atom site label | Tensor element | Wave vector code |
|---|---|---|---|
| KU111 | K | U11 | 1 |
| KU221 | K | U22 | 1 |
| KU331 | K | U33 | 1 |
| KU121 | K | U12 | 1 |
| KU131 | K | U13 | 1 |
| KU231 | K | U23 | 1 |
| KU112 | K | U11 | 2 |
| KU222 | K | U22 | 2 |
| KU332 | K | U33 | 2 |
| KU122 | K | U12 | 2 |
| KU132 | K | U13 | 2 |
| KU232 | K | U23 | 2 |
| V1U111 | V1 | U11 | 1 |
| V1U221 | V1 | U22 | 1 |
| V1U331 | V1 | U33 | 1 |
| V1U121 | V1 | U12 | 1 |
| V1U131 | V1 | U13 | 1 |
| V1U231 | V1 | U23 | 1 |
| V1U112 | V1 | U11 | 2 |
| V1U222 | V1 | U22 | 2 |
| V1U332 | V1 | U33 | 2 |
| V1U122 | V1 | U12 | 2 |
| V1U132 | V1 | U13 | 2 |
| V1U232 | V1 | U23 | 2 |
| V2U111 | V2 | U11 | 1 |
| V2U221 | V2 | U22 | 1 |
| V2U331 | V2 | U33 | 1 |
| V2U121 | V2 | U12 | 1 |
| V2U131 | V2 | U13 | 1 |
| V2U231 | V2 | U23 | 1 |
| V2U112 | V2 | U11 | 2 |
| V2U222 | V2 | U22 | 2 |
| V2U332 | V2 | U33 | 2 |
| V2U122 | V2 | U12 | 2 |
| V2U132 | V2 | U13 | 2 |
| V2U232 | V2 | U23 | 2 |
| O1U111 | O1 | U11 | 1 |
| O1U221 | O1 | U22 | 1 |
| O1U331 | O1 | U33 | 1 |
| O1U121 | O1 | U12 | 1 |
| O1U131 | O1 | U13 | 1 |
| O1U231 | O1 | U23 | 1 |
| O1U112 | O1 | U11 | 2 |
| O1U222 | O1 | U22 | 2 |
| O1U332 | O1 | U33 | 2 |
| O1U122 | O1 | U12 | 2 |
| O1U132 | O1 | U13 | 2 |
| O1U232 | O1 | U23 | 2 |
| O2U111 | O2 | U11 | 1 |
| O2U221 | O2 | U22 | 1 |
| O2U331 | O2 | U33 | 1 |
| O2U121 | O2 | U12 | 1 |
| O2U131 | O2 | U13 | 1 |
| O2U231 | O2 | U23 | 1 |
| O3U111 | O3 | U11 | 1 |
| O3U221 | O3 | U22 | 1 |
| O3U331 | O3 | U33 | 1 |
| O3U121 | O3 | U12 | 1 |
| O3U131 | O3 | U13 | 1 |
| O3U231 | O3 | U23 | 1 |
| O3U112 | O3 | U11 | 2 |
| O3U222 | O3 | U22 | 2 |
| O3U332 | O3 | U33 | 2 |
| O3U122 | O3 | U12 | 2 |
| O3U132 | O3 | U13 | 2 |
| O3U232 | O3 | U23 | 2 |
ADP Fourier coefficients: (Show/hide table) [ Help ]
| Modulation code | Cosine coefficient | Sine coefficient |
|---|---|---|
| KU111 | -0.00055(13) | 0.00172(10) |
| KU221 | 0.00055(13) | 0.00172(10) |
| KU331 | 0 | 0 |
| KU121 | 0 | 0.00157(14) |
| KU131 | -0.00034(10) | 0 |
| KU231 | -0.00034(10) | 0 |
| KU112 | 0.00034(14) | 0.00026(13) |
| KU222 | -0.00034(14) | -0.00026(13) |
| KU332 | 0 | 0 |
| KU122 | 0 | 0 |
| KU132 | 0.00047(11) | 0 |
| KU232 | 0.00047(11) | 0 |
| V1U111 | 0 | 0 |
| V1U221 | 0.000010(10) | 0.00008(8) |
| V1U331 | 0 | 0 |
| V1U121 | 0 | 0 |
| V1U131 | 0 | 0 |
| V1U231 | 0 | 0 |
| V1U112 | -0.000010(10) | -0.00008(8) |
| V1U222 | 0 | 0 |
| V1U332 | 0 | 0 |
| V1U122 | 0 | 0 |
| V1U132 | 0 | 0 |
| V1U232 | 0 | 0 |
| V2U111 | -0.00001(5) | 0 |
| V2U221 | -0.00001(5) | 0.000002(11) |
| V2U331 | 0 | 0 |
| V2U121 | 0.000185 | -0.000019 |
| V2U131 | 0 | 0 |
| V2U231 | 0 | 0 |
| V2U112 | 0.00004(7) | 0.00006 |
| V2U222 | -0.00004(7) | -0.00006 |
| V2U332 | 0 | 0 |
| V2U122 | 0 | 0 |
| V2U132 | 0 | 0 |
| V2U232 | 0 | 0 |
| O1U111 | 0 | 0 |
| O1U221 | 0 | 0 |
| O1U331 | 0 | 0 |
| O1U121 | -0.0003(3) | 0.0003(2) |
| O1U131 | 0 | 0 |
| O1U231 | 0 | 0 |
| O1U112 | 0 | 0 |
| O1U222 | 0 | 0 |
| O1U332 | 0 | 0 |
| O1U122 | -0.0003(3) | 0.0003(2) |
| O1U132 | 0 | 0 |
| O1U232 | 0 | 0 |
| O2U111 | 0 | 0 |
| O2U221 | 0 | 0 |
| O2U331 | 0 | 0 |
| O2U121 | -0.0003(2) | -0.0004(3) |
| O2U131 | 0 | 0 |
| O2U231 | 0 | 0 |
| O3U111 | -0.0004(3) | -0.0003(3) |
| O3U221 | 0 | -0.0006(3) |
| O3U331 | 0 | 0 |
| O3U121 | 0.0001(2) | 0.0005(2) |
| O3U131 | 0 | 0 |
| O3U231 | 0 | 0 |
| O3U112 | 0.0004(3) | -0.0003(3) |
| O3U222 | -0.0001(3) | 0.0009(3) |
| O3U332 | 0 | 0 |
| O3U122 | -0.0001(2) | 0.0000(2) |
| O3U132 | 0 | 0 |
| O3U232 | 0 | 0 |
I-4D
Chemical data
Structural Formula Sum: K2 O8 V3 [ Help ]
Formula weight: 359 Da [ Help ]
Crystallographic data and experimental details
Crystal system: orthorhombic [ Help ]
Superspace group name: Cmm2(0β1/2)s00 [ 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,x3-x4 |
| 3 | x1,-x2+1/2,x3,x3-x4+1/2 |
| 4 | -x1,x2+1/2,x3,x4+1/2 |
| 5 | x1+1/2,x2+1/2,x3,x4 |
| 6 | -x1+1/2,-x2+1/2,x3,x3-x4 |
| 7 | x1+1/2,-x2,x3,x3-x4+1/2 |
| 8 | -x1+1/2,x2,x3,x4+1/2 |
a: 12.645(2) Å [ Help ]
b: 12.645(2) Å [ Help ]
c: 5.223(2) Å [ Help ]
α: 90 ° [ Help ]
β: 90 ° [ Help ]
γ: 90 ° [ Help ]
Volume: 835.2(4) Å3 [ Help ]
Modulation dimension: 1 [ Help ]
Measured independent wave vectors: (Show/hide table) [ Help ]
| Wave vector id | q_x | q_y | q_z |
|---|---|---|---|
| 1 | 0.000000 | 0.626000 | 0.500000 |
Z: 16 [ Help ]
Cell measurement temperature: 100 K [ Help ]
μ: 15.503 mm-1 [ Help ]
Refinement details
Total nb. of reflections: 3790 [ Help ]
Nb. of observed reflections: 1929 [ Help ]
Intense reflections threshold: I>3σ(I) [ Help ]
Refinement based on: F [ Help ]
R(obs): 0.0158 [ Help ]
wR(obs): 0.0198 [ Help ]
R(all): 0.0240 [ Help ]
wR(all): 0.0209 [ Help ]
S(all): 0.87 [ Help ]
S(obs): 1.17 [ Help ]
Nb. of reflections: 3790 [ Help ]
Nb. of parameters: 144 [ Help ]
Number of restraints: 0 [ Help ]
Number of constraints: 2 [ Help ]
Weighting scheme: sigma [ Help ]
Weighting scheme remarks: w=1/(σ2(F)+0.0001F2) [ Help ]
Δ/σ(max): 0.0470 [ Help ]
Δ/σ(mean): 0.0047 [ Help ]
Δρ(max): 0.25 e_Å-3 [ Help ]
Δρ(min): -0.35 e_Å-3 [ Help ]
Extinction method: B-C type 1 Gaussian isotropic (Becker & Coppens, 1974) [ Help ]
Extinction coefficient: 2140(60) [ Help ]
Absolute structure remarks: 1139 of Friedel pairs used in the refinement [ Help ]
Structural Information
Average Structure: (Show/hide table) [ Help ]
| Atom site label | Atom symbol | x | y | z | ADP type | Uiso/equiv | Symmetry multiplicity | Occupancy | Coords from (d)iffraction or (c)alculated | Coords restraints or constraints | Disordered cluster | Disordered group |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| K1_1 | K | 0.08170(8) | 0.25 | 0 | Uani | 0.0154(3) | 4 | 1 | d | . | . | . |
| K1_2 | K | -0.25 | 0.08088(8) | -0.00083(8) | Uani | 0.0162(3) | 4 | 1 | d | . | . | . |
| V1 | V | 0 | 0.5 | 0.46096(4) | Uani | 0.0076(2) | 4 | 1 | d | . | . | . |
| V2_1 | V | -0.11599(6) | 0.25 | 0.46650(12) | Uani | 0.0068(2) | 4 | 1 | d | . | . | . |
| V2_2 | V | -0.25 | -0.11593(6) | 0.46772(12) | Uani | 0.0080(2) | 4 | 1 | d | . | . | . |
| O1 | O | 0 | 0.5 | 0.7696(2) | Uani | 0.0148(10) | 4 | 1 | d | . | . | . |
| O2_1 | O | -0.1192(2) | 0.25 | 0.7778(5) | Uani | 0.0104(11) | 4 | 1 | d | . | . | . |
| O2_2 | O | -0.25 | -0.1206(2) | 0.7852(5) | Uani | 0.0123(12) | 4 | 1 | d | . | . | . |
| O3_1 | O | -0.05297(16) | 0.36020(15) | 0.3528(4) | Uani | 0.0145(6) | 8 | 1 | d | . | . | . |
| O3_2 | O | -0.36105(14) | -0.05249(14) | 0.3527(4) | Uani | 0.0113(6) | 8 | 1 | d | . | . | . |
| O4_1 | O | -0.25 | 0.25 | 0.3459(7) | Uani | 0.0107(11) | 2 | 1 | d | . | . | . |
| O4_2 | O | -0.25 | -0.25 | 0.3379(8) | Uani | 0.0119(12) | 2 | 1 | d | . | . | . |
ADP components: (Show/hide table) [ Help ]
| Atom site label | Atom site symbol | U11 | U22 | U33 | U12 | U13 | U23 |
|---|---|---|---|---|---|---|---|
| K1_1 | K | 0.0092(6) | 0.0244(5) | 0.0128(6) | 0 | -0.0005(4) | 0 |
| K1_2 | K | 0.0310(6) | 0.0072(6) | 0.0103(6) | 0 | 0 | 0.0006(4) |
| V1 | V | 0.0069(5) | 0.0064(5) | 0.00968(13) | -0.0007(3) | 0 | 0 |
| V2_1 | V | 0.0077(4) | 0.0051(4) | 0.0075(4) | 0 | 0.0015(4) | 0 |
| V2_2 | V | 0.0077(4) | 0.0053(4) | 0.0111(4) | 0 | 0 | -0.0007(4) |
| O1 | O | 0.024(2) | 0.0102(19) | 0.0103(6) | -0.0023(14) | 0 | 0 |
| O2_1 | O | 0.012(2) | 0.0160(19) | 0.0033(18) | 0 | -0.0001(13) | 0 |
| O2_2 | O | 0.013(2) | 0.010(2) | 0.014(2) | 0 | 0 | -0.0019(13) |
| O3_1 | O | 0.0189(12) | 0.0106(10) | 0.0141(9) | 0.0010(7) | 0.0044(11) | 0.0014(8) |
| O3_2 | O | 0.0076(10) | 0.0102(10) | 0.0162(9) | 0.0039(6) | 0.0000(8) | -0.0007(9) |
| O4_1 | O | 0.016(2) | 0.016(2) | 0.0006(17) | 0 | 0 | 0 |
| O4_2 | O | 0.013(2) | 0.0015(18) | 0.021(2) | 0 | 0 | 0 |
Fourier Wave Vectors (explicit: q_x,q_y,q_z or coefficients: q_1,q_2,...): (Show/hide table) [ Help ]
| Wave vector code | q_1 |
|---|---|
| 1 | 1 |
Definition of the displacive (translational) Fourier series: (Show/hide table) [ Help ]
| Modulation code | Atom site label | Displacement axis | Wave vector code |
|---|---|---|---|
| K1_1x1 | K1_1 | x | 1 |
| K1_1y1 | K1_1 | y | 1 |
| K1_1z1 | K1_1 | z | 1 |
| K1_2x1 | K1_2 | x | 1 |
| K1_2y1 | K1_2 | y | 1 |
| K1_2z1 | K1_2 | z | 1 |
| V1x1 | V1 | x | 1 |
| V1y1 | V1 | y | 1 |
| V1z1 | V1 | z | 1 |
| V2_1x1 | V2_1 | x | 1 |
| V2_1y1 | V2_1 | y | 1 |
| V2_1z1 | V2_1 | z | 1 |
| V2_2x1 | V2_2 | x | 1 |
| V2_2y1 | V2_2 | y | 1 |
| V2_2z1 | V2_2 | z | 1 |
| O1x1 | O1 | x | 1 |
| O1y1 | O1 | y | 1 |
| O1z1 | O1 | z | 1 |
| O2_1x1 | O2_1 | x | 1 |
| O2_1y1 | O2_1 | y | 1 |
| O2_1z1 | O2_1 | z | 1 |
| O2_2x1 | O2_2 | x | 1 |
| O2_2y1 | O2_2 | y | 1 |
| O2_2z1 | O2_2 | z | 1 |
| O3_1x1 | O3_1 | x | 1 |
| O3_1y1 | O3_1 | y | 1 |
| O3_1z1 | O3_1 | z | 1 |
| O3_2x1 | O3_2 | x | 1 |
| O3_2y1 | O3_2 | y | 1 |
| O3_2z1 | O3_2 | z | 1 |
| O4_1x1 | O4_1 | x | 1 |
| O4_1y1 | O4_1 | y | 1 |
| O4_1z1 | O4_1 | z | 1 |
| O4_2x1 | O4_2 | x | 1 |
| O4_2y1 | O4_2 | y | 1 |
| O4_2z1 | O4_2 | z | 1 |
Displacive (translational) Fourier coefficients: (Show/hide table) [ Help ]
| Modulation code | Cosine coefficient | Sine coefficient |
|---|---|---|
| K1_1x1 | 0 | -0.00116(4) |
| K1_1y1 | 0.00302(4) | 0 |
| K1_1z1 | 0 | -0.00512(10) |
| K1_2x1 | -0.0028 | 0.00162(4) |
| K1_2y1 | 0 | 0 |
| K1_2z1 | 0 | 0 |
| V1x1 | -0.00034(2) | 0.000042(2) |
| V1y1 | -0.00239(2) | 0.000295(2) |
| V1z1 | -0.000336(8) | -0.00273(7) |
| V2_1x1 | -0.00290(2) | 0.000306(2) |
| V2_1y1 | 0.000193(2) | 0.00183(2) |
| V2_1z1 | -0.00185(7) | 0.000195(8) |
| V2_2x1 | 0.00101(2) | -0.00020(2) |
| V2_2y1 | 0 | 0 |
| V2_2z1 | 0 | 0 |
| O1x1 | -0.00052(7) | -0.00058(8) |
| O1y1 | -0.00206(6) | -0.00233(7) |
| O1z1 | 0.00196(18) | -0.00173(16) |
| O2_1x1 | -0.00102(6) | -0.00121(7) |
| O2_1y1 | -0.00163(7) | 0.00136(6) |
| O2_1z1 | -0.00135(16) | -0.00161(19) |
| O2_2x1 | 0.00110(9) | -0.00043(9) |
| O2_2y1 | 0 | 0 |
| O2_2z1 | 0 | 0 |
| O3_1x1 | -0.00359(6) | -0.00183(6) |
| O3_1y1 | 0.00011(6) | 0.00283(6) |
| O3_1z1 | -0.00301(18) | -0.00128(19) |
| O3_2x1 | -0.00125(6) | -0.00089(5) |
| O3_2y1 | -0.00433(6) | -0.00186(6) |
| O3_2z1 | -0.00140(19) | -0.00210(19) |
| O4_1x1 | -0.00299(11) | 0.00157(6) |
| O4_1y1 | 0 | 0 |
| O4_1z1 | 0 | 0 |
| O4_2x1 | 0.00538(12) | -0.00301(7) |
| O4_2y1 | 0 | 0 |
| O4_2z1 | 0 | 0 |
Definition of the ADP Fourier series: (Show/hide table) [ Help ]
| Modulation code | Atom site label | Tensor element | Wave vector code |
|---|---|---|---|
| K1_1U111 | K1_1 | U11 | 1 |
| K1_1U221 | K1_1 | U22 | 1 |
| K1_1U331 | K1_1 | U33 | 1 |
| K1_1U121 | K1_1 | U12 | 1 |
| K1_1U131 | K1_1 | U13 | 1 |
| K1_1U231 | K1_1 | U23 | 1 |
| K1_2U111 | K1_2 | U11 | 1 |
| K1_2U221 | K1_2 | U22 | 1 |
| K1_2U331 | K1_2 | U33 | 1 |
| K1_2U121 | K1_2 | U12 | 1 |
| K1_2U131 | K1_2 | U13 | 1 |
| K1_2U231 | K1_2 | U23 | 1 |
| V1U111 | V1 | U11 | 1 |
| V1U221 | V1 | U22 | 1 |
| V1U331 | V1 | U33 | 1 |
| V1U121 | V1 | U12 | 1 |
| V1U131 | V1 | U13 | 1 |
| V1U231 | V1 | U23 | 1 |
| V2_1U111 | V2_1 | U11 | 1 |
| V2_1U221 | V2_1 | U22 | 1 |
| V2_1U331 | V2_1 | U33 | 1 |
| V2_1U121 | V2_1 | U12 | 1 |
| V2_1U131 | V2_1 | U13 | 1 |
| V2_1U231 | V2_1 | U23 | 1 |
| V2_2U111 | V2_2 | U11 | 1 |
| V2_2U221 | V2_2 | U22 | 1 |
| V2_2U331 | V2_2 | U33 | 1 |
| V2_2U121 | V2_2 | U12 | 1 |
| V2_2U131 | V2_2 | U13 | 1 |
| V2_2U231 | V2_2 | U23 | 1 |
| O1U111 | O1 | U11 | 1 |
| O1U221 | O1 | U22 | 1 |
| O1U331 | O1 | U33 | 1 |
| O1U121 | O1 | U12 | 1 |
| O1U131 | O1 | U13 | 1 |
| O1U231 | O1 | U23 | 1 |
| O2_1U111 | O2_1 | U11 | 1 |
| O2_1U221 | O2_1 | U22 | 1 |
| O2_1U331 | O2_1 | U33 | 1 |
| O2_1U121 | O2_1 | U12 | 1 |
| O2_1U131 | O2_1 | U13 | 1 |
| O2_1U231 | O2_1 | U23 | 1 |
| O2_2U111 | O2_2 | U11 | 1 |
| O2_2U221 | O2_2 | U22 | 1 |
| O2_2U331 | O2_2 | U33 | 1 |
| O2_2U121 | O2_2 | U12 | 1 |
| O2_2U131 | O2_2 | U13 | 1 |
| O2_2U231 | O2_2 | U23 | 1 |
| O3_1U111 | O3_1 | U11 | 1 |
| O3_1U221 | O3_1 | U22 | 1 |
| O3_1U331 | O3_1 | U33 | 1 |
| O3_1U121 | O3_1 | U12 | 1 |
| O3_1U131 | O3_1 | U13 | 1 |
| O3_1U231 | O3_1 | U23 | 1 |
| O3_2U111 | O3_2 | U11 | 1 |
| O3_2U221 | O3_2 | U22 | 1 |
| O3_2U331 | O3_2 | U33 | 1 |
| O3_2U121 | O3_2 | U12 | 1 |
| O3_2U131 | O3_2 | U13 | 1 |
| O3_2U231 | O3_2 | U23 | 1 |
| O4_1U111 | O4_1 | U11 | 1 |
| O4_1U221 | O4_1 | U22 | 1 |
| O4_1U331 | O4_1 | U33 | 1 |
| O4_1U121 | O4_1 | U12 | 1 |
| O4_1U131 | O4_1 | U13 | 1 |
| O4_1U231 | O4_1 | U23 | 1 |
| O4_2U111 | O4_2 | U11 | 1 |
| O4_2U221 | O4_2 | U22 | 1 |
| O4_2U331 | O4_2 | U33 | 1 |
| O4_2U121 | O4_2 | U12 | 1 |
| O4_2U131 | O4_2 | U13 | 1 |
| O4_2U231 | O4_2 | U23 | 1 |
ADP Fourier coefficients: (Show/hide table) [ Help ]
| Modulation code | Cosine coefficient | Sine coefficient |
|---|---|---|
| K1_1U111 | 0 | 0.0004(2) |
| K1_1U221 | 0 | 0.0041(2) |
| K1_1U331 | 0 | 0 |
| K1_1U121 | -0.00087(19) | 0 |
| K1_1U131 | 0 | 0.00035(19) |
| K1_1U231 | -0.0006(2) | 0 |
| K1_2U111 | 0 | 0 |
| K1_2U221 | 0 | 0 |
| K1_2U331 | 0 | 0 |
| K1_2U121 | 0.00028(19) | 0 |
| K1_2U131 | 0 | 0 |
| K1_2U231 | 0 | 0 |
| V1U111 | 0.000006(13) | 0.00005(11) |
| V1U221 | 0 | 0 |
| V1U331 | 0 | 0 |
| V1U121 | -0.000017(13) | -0.00013(10) |
| V1U131 | 0 | 0 |
| V1U231 | 0 | 0 |
| V2_1U111 | -0.00031(15) | 0.000033(15) |
| V2_1U221 | 0.00016(12) | -0.000017(12) |
| V2_1U331 | 0 | 0 |
| V2_1U121 | 0 | 0 |
| V2_1U131 | -0.00015(15) | 0.000016(16) |
| V2_1U231 | 0.000036(17) | 0.00035(16) |
| V2_2U111 | 0 | 0 |
| V2_2U221 | 0 | 0 |
| V2_2U331 | 0 | 0 |
| V2_2U121 | -0.00014(10) | 0.00010(10) |
| V2_2U131 | 0 | 0 |
| V2_2U231 | 0 | 0 |
| O1U111 | 0.0002(4) | -0.0002(4) |
| O1U221 | 0 | 0 |
| O1U331 | 0 | 0 |
| O1U121 | -0.0004(4) | 0.0003(4) |
| O1U131 | 0 | 0 |
| O1U231 | 0.0005(4) | 0.0005(4) |
| O2_1U111 | 0.0002(4) | 0.0002(5) |
| O2_1U221 | -0.0005(4) | -0.0006(4) |
| O2_1U331 | 0 | 0 |
| O2_1U121 | 0 | 0 |
| O2_1U131 | 0 | 0 |
| O2_1U231 | -0.0006(4) | 0.0005(4) |
| O2_2U111 | 0 | 0 |
| O2_2U221 | 0 | 0 |
| O2_2U331 | 0 | 0 |
| O2_2U121 | 0.0000(5) | 0.0004(5) |
| O2_2U131 | 0 | 0 |
| O2_2U231 | 0 | 0 |
| O3_1U111 | -0.0010(4) | -0.0016(4) |
| O3_1U221 | 0 | 0.0006(4) |
| O3_1U331 | 0 | 0 |
| O3_1U121 | -0.0003(3) | 0.0002(3) |
| O3_1U131 | 0 | -0.0006(4) |
| O3_1U231 | 0 | 0 |
| O3_2U111 | -0.0006(4) | -0.0007(4) |
| O3_2U221 | 0 | -0.0002(4) |
| O3_2U331 | 0 | 0 |
| O3_2U121 | -0.0009(3) | -0.0003(3) |
| O3_2U131 | 0 | -0.0008(4) |
| O3_2U231 | 0 | -0.000625 |
| O4_1U111 | 0 | 0 |
| O4_1U221 | 0 | 0 |
| O4_1U331 | 0 | 0 |
| O4_1U121 | 0 | 0 |
| O4_1U131 | 0.0008(7) | -0.0004(4) |
| O4_1U231 | 0 | 0 |
| O4_2U111 | 0 | 0 |
| O4_2U221 | 0 | 0 |
| O4_2U331 | 0 | 0 |
| O4_2U121 | 0 | 0 |
| O4_2U131 | 0.0016(8) | -0.0009(4) |
| O4_2U231 | 0 | 0 |