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Modulated crystal structure of incommensurate low tridymite

Authors:

Graetsch, Heribert A.

Journal:

Acta Crystallographica, Section B 65 543-550 (2009)

DOI:

https://doi.org/10.1107/S0108768109029127

B-IncStrDB ID: 1kFTvnwy0qb Entry date: 2022-01-23 Last revision: 2022-01-24

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Chemical data

Common Name: tridymite [ Help ]

Structural Formula Sum: O2 Si1 [ Help ]

Formula weight: 60.1 Da [ Help ]

Crystallographic data and experimental details

Crystal system: monoclinic [ Help ]

Superspace group name: Cc(α0γ)0 [ Help ]

Symmetry operations of the superspace group: [ Help ]

Operation code Operation in algebraic form
1 x1,x2,x3,x4
2 x1,-x2,x3+1/2,x4
3 x1+1/2,x2+1/2,x3,x4
4 x1+1/2,-x2+1/2,x3+1/2,x4

a: 5.0049(12) Å [ Help ]

b: 8.6044(18) Å [ Help ]

c: 8.214(3) Å [ Help ]

α: 90 ° [ Help ]

β: 91.54(3) ° [ Help ]

γ: 90 ° [ Help ]

Volume: 354.60(2) Å3 [ Help ]

Z: 8 [ Help ]

Cell measurement temperature: 294 K [ Help ]

Modulation dimension: 1 [ Help ]

Measured independent wave vectors: [ Help ]

Wave vector id q_x q_y q_z
1 0.6543(7) 0.00000 -0.4946(9)

μ: 0.846 mm-1 [ Help ]

Absorption correction type: none [ Help ]

Minimum transmission factor: 1 [ Help ]

Maximum transmission factor: 1 [ Help ]

Refinement details

Total nb. of reflections: 7927 [ Help ]

Nb. of observed reflections: 3679 [ Help ]

Intense reflections threshold: I>3σ(I) [ Help ]

Refinement based on: F [ Help ]

R(all): 0.0755 [ Help ]

R(obs): 0.0303 [ Help ]

wR(obs): 0.0295 [ Help ]

wR(all): 0.0359 [ Help ]

S(all): 0.99 [ Help ]

Nb. of reflections: 7927 [ Help ]

Nb. of parameters: 293 [ Help ]

Number of restraints: 20 [ Help ]

Number of constraints: 0 [ Help ]

Weighting scheme: sigma [ Help ]

Weighting scheme remarks: w=1/(σ2(F)+0.0001F2) [ Help ]

Δ/σ(max): 0.0078 [ Help ]

Δ/σ(mean): 0.0013 [ Help ]

Δρ(max): 0.36 e_Å-3 [ Help ]

Δρ(min): -0.34 e_Å-3 [ Help ]

Extinction method: B-C type 1 Gaussian isotropic (Becker & Coppens, 1974) [ Help ]

Extinction coefficient: 464E1(9) [ Help ]

Structural Information

Average Structure: [ Help ]

Atom symbol Atom site label x y z Uiso/equiv ADP type Coords from (d)iffraction or (c)alculated Coords restraints or constraints Occupancy Disordered cluster Disordered group
Si Si1 0.089528 0.30758(3) 0.21154 0.01252(8) Uani d . 1 . .
Si Si2 0.09056(9) 0.64149(3) 0.33670(5) 0.01242(8) Uani d . 1 . .
O O1 0.30010(19) 0.75222(11) 0.25668(13) 0.0247(3) Uani d . 1 . .
O O2 0.16026(19) 0.46589(9) 0.30179(12) 0.0224(3) Uani d . 1 . .
O O3 0.30187(19) 0.18034(9) 0.26440(14) 0.0200(3) Uani d . 1 . .
O O4 0.0976(3) 0.33257(12) 0.02435(14) 0.0288(3) Uani d . 1 . .

ADP components: [ Help ]

Atom site label U11 U22 U33 U12 U13 U23 Atom site symbol
Si1 0.01151(15) 0.01154(13) 0.01448(13) 0.00027(12) -0.00007(11) 0.00135(11) Si
Si2 0.01143(15) 0.01149(13) 0.01434(13) 0.00041(12) 0.00025(11) -0.00112(11) Si
O1 0.0143(5) 0.0234(5) 0.0365(5) -0.0017(4) 0.0047(4) 0.0077(4) O
O2 0.0215(5) 0.0136(4) 0.0318(5) 0.0032(3) -0.0065(4) -0.0056(4) O
O3 0.0124(4) 0.0158(4) 0.0318(6) 0.0013(3) -0.0027(4) 0.0003(4) O
O4 0.0364(6) 0.0348(6) 0.0152(3) -0.0017(6) 0.0010(4) 0.0053(4) O

Fourier Wave Vectors (explicit: q_x,q_y,q_z or coefficients: q_1,q_2,...): [ Help ]

Wave vector code q_x q_z
1 0.65390 -0.49480
2 1.30780 -0.98960
3 1.96170 -1.48440

Definition of the displacive (translational) Fourier series: [ Help ]

Modulation code Atom site label Displacement axis Wave vector code
Si1x1 Si1 x 1
Si1y1 Si1 y 1
Si1z1 Si1 z 1
Si1x2 Si1 x 2
Si1y2 Si1 y 2
Si1z2 Si1 z 2
Si2x1 Si2 x 1
Si2y1 Si2 y 1
Si2z1 Si2 z 1
Si2x2 Si2 x 2
Si2y2 Si2 y 2
Si2z2 Si2 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
O1x3 O1 x 3
O1y3 O1 y 3
O1z3 O1 z 3
O2x1 O2 x 1
O2y1 O2 y 1
O2z1 O2 z 1
O2x2 O2 x 2
O2y2 O2 y 2
O2z2 O2 z 2
O2x3 O2 x 3
O2y3 O2 y 3
O2z3 O2 z 3
O3x1 O3 x 1
O3y1 O3 y 1
O3z1 O3 z 1
O3x2 O3 x 2
O3y2 O3 y 2
O3z2 O3 z 2
O3x3 O3 x 3
O3y3 O3 y 3
O3z3 O3 z 3
O4x1 O4 x 1
O4y1 O4 y 1
O4z1 O4 z 1
O4x2 O4 x 2
O4y2 O4 y 2
O4z2 O4 z 2
O4x3 O4 x 3
O4y3 O4 y 3
O4z3 O4 z 3

Displacive (translational) Fourier coefficients: [ Help ]

Modulation code Cosine coefficient Sine coefficient
Si1x1 0.02308(9) 0.00846(13)
Si1y1 -0.00014(6) 0.01182(5)
Si1z1 0.03649(5) 0.009
Si1x2 0.00069(14) -0.00206(13)
Si1y2 0.00020(7) -0.00163(7)
Si1z2 0.00051(8) 0.00029(6)
Si2x1 0.00097(13) 0.02199(9)
Si2y1 0.01228(5) -0.00466(6)
Si2z1 0.00586(7) 0.03742(5)
Si2x2 0.00120(14) 0.00245(13)
Si2y2 -0.00117(7) -0.00054(8)
Si2z2 0.00016(8) 0.00039(6)
O1x1 0.0032(2) -0.0029(2)
O1y1 0.00471(13) -0.00704(13)
O1z1 -0.01677(15) -0.01361(15)
O1x2 -0.0020(3) -0.0023(3)
O1y2 0.00382(17) -0.0001(2)
O1z2 -0.00073(16) -0.00447(19)
O1x3 -0.0022(4) 0.0007(4)
O1y3 0.0026(3) 0.0015(3)
O1z3 -0.0002(3) 0.0012(3)
O2x1 0.0019(2) -0.0006(2)
O2y1 0.00673(12) -0.00471(12)
O2z1 0.02921(19) 0.04587(16)
O2x2 0.0012(3) 0.0014(3)
O2y2 0.00130(16) -0.00003(15)
O2z2 0.00205(18) -0.00159(19)
O2x3 0.0029(4) -0.0023(5)
O2y3 0.0002(2) -0.0013(2)
O2z3 -0.0006(3) 0.0030(3)
O3x1 0.0004(2) 0.0003(2)
O3y1 -0.00127(13) 0.00195(13)
O3z1 0.06030(19) 0.0446(2)
O3x2 0.0047(3) 0.0029(3)
O3y2 0.00516(18) 0.00144(17)
O3z2 -0.00054(18) -0.0031(2)
O3x3 -0.0031(4) 0.0005(4)
O3y3 0.0005(2) -0.0005(2)
O3z3 0.0027(3) -0.0006(3)
O4x1 0.0541(4) 0.0814(4)
O4y1 -0.0422(2) 0.0318(2)
O4z1 0.01994(16) 0.03273(14)
O4x2 0.0060(3) -0.0076(4)
O4y2 0.0044(2) 0.00059(19)
O4z2 0.00040(13) 0.00002(17)
O4x3 0.0013(4) -0.0008(5)
O4y3 -0.0011(3) 0.0014(3)
O4z3 -0.00074(18) 0.0004(2)

Definition of the ADP Fourier series: [ Help ]

Modulation code Atom site label Tensor element Wave vector code
Si1U111 Si1 U11 1
Si1U221 Si1 U22 1
Si1U331 Si1 U33 1
Si1U121 Si1 U12 1
Si1U131 Si1 U13 1
Si1U231 Si1 U23 1
Si1U112 Si1 U11 2
Si1U222 Si1 U22 2
Si1U332 Si1 U33 2
Si1U122 Si1 U12 2
Si1U132 Si1 U13 2
Si1U232 Si1 U23 2
Si2U111 Si2 U11 1
Si2U221 Si2 U22 1
Si2U331 Si2 U33 1
Si2U121 Si2 U12 1
Si2U131 Si2 U13 1
Si2U231 Si2 U23 1
Si2U112 Si2 U11 2
Si2U222 Si2 U22 2
Si2U332 Si2 U33 2
Si2U122 Si2 U12 2
Si2U132 Si2 U13 2
Si2U232 Si2 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
O2U112 O2 U11 2
O2U222 O2 U22 2
O2U332 O2 U33 2
O2U122 O2 U12 2
O2U132 O2 U13 2
O2U232 O2 U23 2
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
O4U111 O4 U11 1
O4U221 O4 U22 1
O4U331 O4 U33 1
O4U121 O4 U12 1
O4U131 O4 U13 1
O4U231 O4 U23 1
O4U112 O4 U11 2
O4U222 O4 U22 2
O4U332 O4 U33 2
O4U122 O4 U12 2
O4U132 O4 U13 2
O4U232 O4 U23 2

ADP Fourier coefficients: [ Help ]

Modulation code Cosine coefficient Sine coefficient
Si1U111 -0.0001(3) -0.0003(2)
Si1U221 -0.0011(2) -0.00022(19)
Si1U331 0.0007(3) -0.0002(2)
Si1U121 -0.00042(16) 0.0002(2)
Si1U131 -0.0010(3) -0.00058(18)
Si1U231 0.00118(16) -0.00020(18)
Si1U112 0.0013(4) -0.0011(4)
Si1U222 0.0001(3) 0.0007(3)
Si1U332 -0.0031(3) -0.0023(4)
Si1U122 0.0004(3) -0.0014(2)
Si1U132 0.0010(2) 0.0002(3)
Si1U232 0.0001(2) -0.00156(19)
Si2U111 0.0004(2) 0.0006(3)
Si2U221 0.00014(19) 0.0005(3)
Si2U331 -0.0004(2) -0.0019(3)
Si2U121 -0.0002(2) 0.00070(15)
Si2U131 0.00014(18) 0.0004(3)
Si2U231 0.00030(19) 0.00048(15)
Si2U112 0.0005(3) -0.0011(4)
Si2U222 -0.0015(3) 0.0003(3)
Si2U332 0.0038(3) -0.0013(4)
Si2U122 0.0000(3) -0.0003(2)
Si2U132 -0.0002(2) -0.0004(3)
Si2U232 -0.0007(2) -0.00038(19)
O1U111 -0.0016(7) 0.0007(6)
O1U221 -0.0039(6) 0.0066(6)
O1U331 0.0013(7) -0.0023(7)
O1U121 -0.0002(5) -0.0002(5)
O1U131 -0.0020(6) 0.0036(5)
O1U231 -0.0019(5) 0.0011(5)
O1U112 0.0006(9) -0.0016(11)
O1U222 -0.0002(10) -0.0008(8)
O1U332 0.0051(9) 0.0051(8)
O1U122 0.0008(7) -0.0005(8)
O1U132 -0.0001(8) -0.0014(7)
O1U232 0.0022(7) 0.0008(7)
O2U111 -0.0019(7) 0.0022(7)
O2U221 -0.0017(5) 0.0013(6)
O2U331 0.0018(8) -0.0044(7)
O2U121 -0.0015(5) 0.0028(5)
O2U131 -0.0014(6) -0.0024(6)
O2U231 0.0025(6) -0.0015(5)
O2U112 -0.0005(10) -0.0015(10)
O2U222 -0.0010(8) -0.0028(8)
O2U332 0.0034(11) -0.0075(10)
O2U122 0.0007(7) 0.0001(7)
O2U132 -0.0001(8) 0.0058(8)
O2U232 -0.0036(7) 0.0003(7)
O3U111 0.0004(6) -0.0033(6)
O3U221 0.0000(6) 0.0002(6)
O3U331 0.0002(8) 0.0004(8)
O3U121 -0.0003(5) -0.0002(5)
O3U131 0.0006(6) 0.0012(6)
O3U231 0.0067(6) 0.0029(6)
O3U112 0.0002(9) 0.0000(9)
O3U222 -0.0004(8) 0.0002(8)
O3U332 -0.0037(10) -0.0126(11)
O3U122 0.0008(6) 0.0005(7)
O3U132 -0.0005(8) 0.0016(8)
O3U232 0.0004(7) 0.0001(8)
O4U111 0.0049(11) -0.0088(12)
O4U221 -0.0021(11) 0.0053(12)
O4U331 0.0008(7) 0.0008(7)
O4U121 -0.0003(8) -0.0020(8)
O4U131 -0.0023(7) -0.0033(7)
O4U231 -0.0006(6) 0.0015(6)
O4U112 -0.0010(14) -0.0172(12)
O4U222 -0.0062(12) 0.0090(12)
O4U332 0.0025(9) -0.0033(6)
O4U122 0.0110(9) 0.0044(10)
O4U132 0.0017(9) -0.0025(7)
O4U232 0.0013(7) 0.0008(7)