Ordered vacancy distribution in 2/1 mullite: a superspace model

A mullite single crystal with composition Al_4.84Si_1.16O_9.58 (2) exhibiting sharp satellite reflections was investigated by means of X-ray diffraction. For the refinement of a superspace model in the superspace group Pbam(α0½)0ss different scale factors for main and satellite reflections were used in order to describe an ordered mullite structure embedded in a disordered polymorph. The ordered fraction of the mullite sample exhibits a completely ordered vacancy distribution and can be described as a block structure of vacancy blocks (VBs) that alternate with vacancy-free blocks (VFBs) along a and c. The incommensurate nature of mullite originates from a modulation of the block size, which depends on the composition. The displacive modulation is analyzed with respect to the vacancy distribution and a possible Al/Si ordering scheme is derived, although the measurement itself is not sensitive to the Al/Si distribution. An idealized, commensurate approximation for 2/1 mullite is also presented. Comparison of the ordered superspace model with different preceding models reconciles many key investigations of the last decades with partly contradicting conclusions, where mullite was usually treated as either ordered or disordered instead of considering simultaneously different states of order.

Computing details

Data collection: STOE X-AREA for (I). Cell refinement: STOE X-AREA for (I). Data reduction: STOE X-AREA for (I). Program(s) used to solve structure: superflip for (I). Program(s) used to refine structure: Jana 2006 for (I). Molecular graphics: VESTA3 for (I).

(I)

Crystal data

2.1(Al2O3)·(SiO2)	F(000) = 156
Mr = 316.4	Dx = 3.130 Mg m^−3
Orthorhombic, Pbam(α01/2)0ss†	Mo Kα radiation, λ = 0.71069 Å
q = 0.300100a* + 0.500000c*	Cell parameters from 2808 reflections
a = 7.5817 (6) Å	θ = 3.8–39.5°
b = 7.6752 (7) Å	µ = 1.06 mm^−1
c = 2.8843 (3) Å	T = 293 K
V = 167.84 (3) Å^3	Irregularly shaped block, colorless
Z = 1	0.06 × 0.05 × 0.03 × 0.03 (radius) mm

† Symmetry operations: (1) x₁, x₂, x₃, x₄; (2) −x₁, −x₂, x₃, x₃−x₄+1/2; (3) −x₁+1/2, x₂+1/2, −x₃, −x₄+1/2; (4) x₁+1/2, −x₂+1/2, −x₃, −x₃+x₄; (5) −x₁, −x₂, −x₃, −x₄; (6) x₁, x₂, −x₃, −x₃+x₄+1/2; (7) x₁+1/2, −x₂+1/2, x₃, x₄+1/2; (8) −x₁+1/2, x₂+1/2, x₃, x₃−x₄.

Data collection

STOE Stadi Vari diffractometer	864 independent reflections
Radiation source: X-ray tube	516 reflections with I > 3σ(I)
Detector resolution: 5.8 pixels mm^-1	Rint = 0.051
integration technique, STOE X–Area scans	θmax = 39.5°, θmin = 3.8°
Absorption correction: numerical STOE X-Red32	h = −6→13
Tmin = 0.965, Tmax = 0.981	k = −13→11
3095 measured reflections	l = −4→5

Refinement

Refinement on F^2	0 restraints
R[F^2 > 2σ(F^2)] = 0.035	12 constraints
wR(F^2) = 0.112	Weighting scheme based on measured s.u.'s w = 1/(σ^2(I) + 0.0049000001I^2)
S = 0.98	(Δ/σ)max = 0.001
864 reflections	Δρmax = 1.18 e Å^−3
52 parameters	Δρmin = −1.02 e Å^−3

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq	Occ. (<1)
Al1	0	0	0	0.00645 (18)	0.25
Al2	0.1492 (2)	0.3407 (2)	0.5	0.00716 (16)	0.317127
Si2	0.1492 (2)	0.3407 (2)	0.5	0.00716 (16)	0.182873
Al3	0.2629 (3)	0.2050 (3)	0.5	0.0068 (4)	0.5
O1	0.35872 (15)	0.42222 (18)	0.5	0.0082 (6)	0.5
O2	0.12751 (16)	0.21796 (17)	0	0.0132 (4)	0.5
O3a	0	0.5	0.5	0.0195 (18)	0.25
O3b	0	0.5	0.5	0.0195 (18)	0.5
O4	0.4514 (9)	0.0492 (8)	0.5	0.0112 (14)	0.5

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Al1	0.0084 (2)	0.0065 (2)	0.0044 (4)	0.00029 (17)	0	0
Al2	0.0065 (3)	0.0071 (3)	0.0079 (2)	−0.0014 (2)	0	0
Si2	0.0065 (3)	0.0071 (3)	0.0079 (2)	−0.0014 (2)	0	0
Al3	0.0059 (7)	0.0078 (8)	0.0066 (8)	0.0010 (6)	0	0
O1	0.0113 (8)	0.0073 (15)	0.0059 (4)	−0.0011 (9)	0	0
O2	0.0164 (5)	0.0141 (4)	0.0091 (9)	−0.0069 (4)	0	0
O3a	0.018 (3)	0.015 (3)	0.025 (3)	0.002 (2)	0	0
O3b	0.018 (3)	0.015 (3)	0.025 (3)	0.002 (2)	0	0
O4	0.012 (3)	0.008 (3)	0.015 (2)	0.0014 (18)	0	0

Geometric parameters (Å, º)

	Average	Minimum	Maximum
Al1—O1^i	1.897 (6)	1.868 (7)	1.930 (7)
Al1—O1^ii	1.897 (6)	1.868 (7)	1.930 (7)
Al1—O1^iii	1.897 (6)	1.868 (7)	1.930 (7)
Al1—O1^iv	1.897 (6)	1.868 (7)	1.930 (7)
Al1—O2	1.9332 (13)	1.9322 (13)	1.9342 (13)
Al1—O2^v	1.9332 (13)	1.9322 (13)	1.9342 (13)
Al2—O1	1.708 (10)	1.691 (10)	1.732 (10)
Al2—O2	1.725 (8)	1.680 (8)	1.783 (8)
Al2—O2^vi	1.725 (8)	1.680 (8)	1.783 (8)
Al2—O3a	1.683 (6)	1.681 (6)	1.684 (6)
Al2—O3b	1.665 (6)	1.657 (6)	1.672 (6)
Al2—O3b^vii	1.653 (5)	1.649 (5)	1.659 (5)
Al2—O4^viii	1.769 (9)	1.752 (9)	1.786 (9)
Al2—O4^iv	1.721 (8)	1.713 (9)	1.729 (9)
Al3—O1	1.729 (9)	1.718 (9)	1.758 (9)
Al3—O2	1.788 (7)	1.748 (7)	1.825 (7)
Al3—O2^vi	1.788 (7)	1.748 (7)	1.825 (7)
Al3—O4	1.864 (7)	1.864 (7)	1.864 (7)
O1^i—Al1—O1^ii	99.4 (3)	99.2 (3)	99.6 (3)
O1^i—Al1—O1^iii	80.7 (3)	77.5 (4)	84.4 (4)
O1^i—Al1—O1^iv	175.5 (4)	172.9 (4)	180.0 (5)
O1^i—Al1—O2	89.5 (3)	85.0 (3)	94.0 (3)
O1^i—Al1—O2^v	90.5 (3)	87.4 (3)	93.6 (3)
O1^ii—Al1—O1^iii	175.5 (4)	172.9 (4)	180.0 (5)
O1^ii—Al1—O1^iv	80.7 (3)	77.5 (4)	84.4 (4)
O1^ii—Al1—O2	89.4 (3)	85.0 (3)	94.0 (3)
O1^ii—Al1—O2^v	90.6 (3)	87.4 (3)	93.6 (3)
O1^iii—Al1—O1^iv	99.4 (3)	99.2 (3)	99.6 (3)
O1^iii—Al1—O2	90.6 (3)	87.4 (3)	93.6 (3)
O1^iii—Al1—O2^v	89.4 (3)	85.0 (3)	94.0 (3)
O1^iv—Al1—O2	90.5 (3)	87.4 (3)	93.6 (3)
O1^iv—Al1—O2^v	89.5 (3)	85.0 (3)	94.0 (3)
O2—Al1—O2^v	178.6 (4)	177.7 (5)	180.0 (5)
O1—Al2—O2	107.4 (3)	103.4 (3)	110.3 (3)
O1—Al2—O2^vi	107.4 (3)	103.4 (3)	110.3 (3)
O1—Al2—O3a	104.5 (5)	104.1 (4)	105.1 (5)
O1—Al2—O3b	112.3 (4)	109.5 (4)	115.1 (5)
O1—Al2—O3b^vii	116.2 (4)	113.9 (4)	118.0 (5)
O1—Al2—O4^viii	91.8 (5)	88.2 (5)	96.0 (5)
O1—Al2—O4^iv	123.8 (5)	121.8 (5)	126.9 (5)
O2—Al2—O2^vi	113.0 (4)	107.4 (4)	118.2 (4)
O2—Al2—O3a	109.6 (2)	109.6 (3)	109.6 (2)
O2—Al2—O3b	109.6 (2)	109.56 (18)	109.6 (2)
O2—Al2—O3b^vii	109.42 (19)	109.41 (19)	109.5 (2)
O2—Al2—O4^viii	117.4 (3)	116.9 (3)	117.7 (2)
O2—Al2—O4^iv	101.4 (2)	101.1 (2)	101.5 (2)
O2^vi—Al2—O3a	109.6 (2)	109.6 (3)	109.6 (2)
O2^vi—Al2—O3b	109.6 (2)	109.56 (18)	109.6 (2)
O2^vi—Al2—O3b^vii	109.42 (19)	109.41 (19)	109.5 (2)
O2^vi—Al2—O4^viii	117.4 (3)	116.9 (3)	117.7 (2)
O2^vi—Al2—O4^iv	101.4 (2)	101.1 (2)	101.5 (2)
O1—Al3—O2	102.54 (18)	101.84 (18)	102.81 (19)
O1—Al3—O2^vi	102.54 (18)	101.84 (18)	102.81 (19)
O1—Al3—O4	102.2 (4)	101.9 (4)	103.1 (4)
O2—Al3—O2^vi	109.6 (3)	107.7 (3)	111.2 (2)
O2—Al3—O4	118.44 (17)	117.81 (16)	119.15 (17)
O2^vi—Al3—O4	118.44 (17)	117.81 (16)	119.15 (17)
Al1^ix—O1—Al1^viii	98.9 (4)	95.5 (4)	102.7 (4)
Al1^ix—O1—Al2	129.7 (2)	127.5 (2)	131.7 (2)
Al1^ix—O1—Al3	126.3 (3)	125.0 (3)	126.9 (3)
Al1^viii—O1—Al2	129.7 (2)	127.5 (2)	131.7 (2)
Al1^viii—O1—Al3	126.3 (3)	125.0 (3)	126.9 (3)
Al1—O2—Al2^x	121.0 (4)	116.0 (4)	126.6 (4)
Al1—O2—Al2	120.9 (4)	116.0 (4)	126.6 (4)
Al1—O2—Al3^x	104.7 (3)	103.1 (3)	106.1 (3)
Al1—O2—Al3	104.7 (3)	103.1 (3)	106.1 (3)
Al2^x—O2—Al2	112.8 (2)	112.8 (2)	112.9 (2)
Al2^x—O2—Al3	131.1 (2)	130.2 (2)	131.85 (19)
Al2—O2—Al3^x	131.1 (2)	130.2 (2)	131.85 (19)
Al2—O3a—Al2^vii	179.5 (3)	179.2 (3)	180.0 (5)
Al2—O3b—Al2^vii	177.0 (2)	176.6 (2)	177.8 (3)
Al2^ii—O4—Al2^xi	147.5 (5)	146.2 (5)	148.1 (5)
Al2^ii—O4—Al3	103.5 (4)	102.9 (4)	104.3 (4)
Al2^xi—O4—Al3	109.1 (4)	108.9 (4)	109.5 (4)

Symmetry codes: (i) −x₁+1/2, x₂−1/2, −x₃, −x₄+1/2; (ii) −x₁+1/2, x₂−1/2, −x₃+1, −x₄+1/2; (iii) x₁−1/2, −x₂+1/2, −x₃, −x₃+x₄; (iv) x₁−1/2, −x₂+1/2, −x₃+1, −x₃+x₄; (v) −x₁, −x₂, x₃, x₃−x₄+1/2; (vi) x₁, x₂, x₃+1, x₄; (vii) −x₁, −x₂+1, x₃, x₃−x₄+1/2; (viii) −x₁+1/2, x₂+1/2, −x₃+1, −x₄+1/2; (ix) −x₁+1/2, x₂+1/2, −x₃, −x₄+1/2; (x) x₁, x₂, x₃−1, x₄; (xi) x₁+1/2, −x₂+1/2, −x₃+1, −x₃+x₄.

(II)

Crystal data

(Al2O8)·(Si0.8O0.4)·(Al0.4Si0.4O0.4)·(Al2.4O0.8)	c = 5.7686 (3) Å
Orthorhombic, Bb21m	V = 3356.8 (4) Å^3
a = 75.8170 (6) Å	Z = 10
b = 7.6752 (7) Å

Data collection

h = →	l = →
k = →

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
Al11	0.025000	0.500000	0.245409	0.006463*
Al12	0.075000	0.000000	0.259987	0.006463*
Al13	0.125000	0.500000	0.242850	0.006463*
Al14	0.175000	0.000000	0.248418	0.006463*
Al15	0.225000	0.500000	0.259009	0.006463*
Al21	0.010357	0.836560	0.000000	0.007123*
Al22	0.040259	0.156979	0.000000	0.007123*
Al23	0.109786	0.841865	0.000000	0.007123*
Al24	0.139725	0.163262	0.000000	0.007123*
Al25	0.160411	0.663215	0.000000	0.007123*
Al26	0.190271	0.343947	0.000000	0.007123*
Al27	0.259860	0.659408	0.000000	0.007123*
Al28	0.289826	0.337300	0.000000	0.007123*
Al29	0.389624	0.339473	0.000000	0.007123*
Al210	0.460422	0.661683	0.000000	0.007123*
Si21	0.310434	0.837391	0.000000	0.007123*
Si22	0.340249	0.155447	0.000000	0.007123*
Si23	0.359888	0.655400	0.000000	0.007123*
Si24	0.409956	0.839326	0.000000	0.007123*
Si25	0.439936	0.161809	0.000000	0.007123*
Si26	0.490194	0.344778	0.000000	0.007123*
Al31	0.048714	0.795047	0.000000	0.006733*
Al32	0.101286	0.204953	0.000000	0.006733*
Al33	0.198714	0.704953	0.000000	0.006733*
Al34	0.251286	0.295047	0.000000	0.006733*
O111	0.010083	0.428592	0.000000	0.008203*
O112	0.038006	0.596214	0.000000	0.008203*
O113	0.488339	0.915842	0.000000	0.008203*
O114	0.459750	0.059352	0.000000	0.008203*
O121	0.060678	0.066846	0.000000	0.008203*
O122	0.089657	0.937509	0.000000	0.008203*
O123	0.438934	0.588720	0.000000	0.008203*
O124	0.411402	0.406925	0.000000	0.008203*
O131	0.111889	0.402985	0.000000	0.008203*
O132	0.139627	0.577329	0.000000	0.008203*
O133	0.390145	0.941449	0.000000	0.008203*
O134	0.361371	0.078237	0.000000	0.008203*
O141	0.159870	0.066112	0.000000	0.008203*
O142	0.188011	0.906391	0.000000	0.008203*
O143	0.338126	0.589454	0.000000	0.008203*
O144	0.309756	0.438043	0.000000	0.008203*
O151	0.211033	0.427728	0.000000	0.008203*
O152	0.239941	0.559633	0.000000	0.008203*
O153	0.289288	0.916706	0.000000	0.008203*
O154	0.261685	0.095933	0.000000	0.008203*
O211	0.012249	0.717963	0.260814	0.013163*
O212	0.037751	0.282037	0.236036	0.013163*
O221	0.062249	0.782037	0.251251	0.013163*
O222	0.087751	0.217963	0.255602	0.013163*
O231	0.112249	0.717963	0.237715	0.013163*
O232	0.137751	0.282037	0.257379	0.013163*
O241	0.162249	0.782037	0.263191	0.013163*
O242	0.187751	0.217963	0.235723	0.013163*
O251	0.212249	0.717963	0.246778	0.013163*
O252	0.237751	0.282037	0.259404	0.013163*
O31	0.325000	0.000000	0.000000	0.019453*
O32	0.375000	0.500000	0.000000	0.019453*
O33	0.425000	0.000000	0.000000	0.019453*
O34	0.475000	0.500000	0.000000	0.019453*
O41	0.029857	0.950792	0.000000	0.011233*
O42	0.120143	0.049208	0.000000	0.011233*
O43	0.179857	0.549208	0.000000	0.011233*
O44	0.270143	0.450792	0.000000	0.011233*

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Al11	0.00845	0.00652	0.00442	−0.00030	0.00000	0.00000
Al12	0.00845	0.00652	0.00442	0.00030	0.00000	0.00000
Al13	0.00845	0.00652	0.00442	−0.00030	0.00000	0.00000
Al14	0.00845	0.00652	0.00442	0.00030	0.00000	0.00000
Al15	0.00845	0.00652	0.00442	−0.00030	0.00000	0.00000
Al21	0.00641	0.00706	0.00790	0.00139	0.00000	0.00000
Al22	0.00641	0.00706	0.00790	0.00139	0.00000	0.00000
Al23	0.00641	0.00706	0.00790	0.00139	0.00000	0.00000
Al24	0.00641	0.00706	0.00790	0.00139	0.00000	0.00000
Al25	0.00641	0.00706	0.00790	−0.00139	0.00000	0.00000
Al26	0.00641	0.00706	0.00790	−0.00139	0.00000	0.00000
Al27	0.00641	0.00706	0.00790	−0.00139	0.00000	0.00000
Al28	0.00641	0.00706	0.00790	−0.00139	0.00000	0.00000
Al29	0.00641	0.00706	0.00790	−0.00139	0.00000	0.00000
Al210	0.00641	0.00706	0.00790	−0.00139	0.00000	0.00000
Si21	0.00641	0.00706	0.00790	0.00139	0.00000	0.00000
Si22	0.00641	0.00706	0.00790	0.00139	0.00000	0.00000
Si23	0.00641	0.00706	0.00790	−0.00139	0.00000	0.00000
Si24	0.00641	0.00706	0.00790	0.00139	0.00000	0.00000
Si25	0.00641	0.00706	0.00790	0.00139	0.00000	0.00000
Si26	0.00641	0.00706	0.00790	−0.00139	0.00000	0.00000
Al31	0.00582	0.00778	0.00660	0.00097	0.00000	0.00000
Al32	0.00582	0.00778	0.00660	0.00097	0.00000	0.00000
Al33	0.00582	0.00778	0.00660	−0.00097	0.00000	0.00000
Al34	0.00582	0.00778	0.00660	−0.00097	0.00000	0.00000
O111	0.01136	0.00730	0.00595	−0.00112	0.00000	0.00000
O112	0.01136	0.00730	0.00595	−0.00112	0.00000	0.00000
O113	0.01136	0.00730	0.00595	0.00112	0.00000	0.00000
O114	0.01136	0.00730	0.00595	0.00112	0.00000	0.00000
O121	0.01136	0.00730	0.00595	0.00112	0.00000	0.00000
O122	0.01136	0.00730	0.00595	0.00112	0.00000	0.00000
O123	0.01136	0.00730	0.00595	−0.00112	0.00000	0.00000
O124	0.01136	0.00730	0.00595	−0.00112	0.00000	0.00000
O131	0.01136	0.00730	0.00595	−0.00112	0.00000	0.00000
O132	0.01136	0.00730	0.00595	−0.00112	0.00000	0.00000
O133	0.01136	0.00730	0.00595	0.00112	0.00000	0.00000
O134	0.01136	0.00730	0.00595	0.00112	0.00000	0.00000
O141	0.01136	0.00730	0.00595	0.00112	0.00000	0.00000
O142	0.01136	0.00730	0.00595	0.00112	0.00000	0.00000
O143	0.01136	0.00730	0.00595	−0.00112	0.00000	0.00000
O144	0.01136	0.00730	0.00595	−0.00112	0.00000	0.00000
O151	0.01136	0.00730	0.00595	−0.00112	0.00000	0.00000
O152	0.01136	0.00730	0.00595	−0.00112	0.00000	0.00000
O153	0.01136	0.00730	0.00595	0.00112	0.00000	0.00000
O154	0.01136	0.00730	0.00595	0.00112	0.00000	0.00000
O211	0.01631	0.01407	0.00911	0.00693	0.00000	0.00000
O212	0.01631	0.01407	0.00911	0.00693	0.00000	0.00000
O221	0.01631	0.01407	0.00911	−0.00693	0.00000	0.00000
O222	0.01631	0.01407	0.00911	−0.00693	0.00000	0.00000
O231	0.01631	0.01407	0.00911	0.00693	0.00000	0.00000
O232	0.01631	0.01407	0.00911	0.00693	0.00000	0.00000
O241	0.01631	0.01407	0.00911	−0.00693	0.00000	0.00000
O242	0.01631	0.01407	0.00911	−0.00693	0.00000	0.00000
O251	0.01631	0.01407	0.00911	0.00693	0.00000	0.00000
O252	0.01631	0.01407	0.00911	0.00693	0.00000	0.00000
O31	0.01835	0.01503	0.02498	−0.00171	0.00000	0.00000
O32	0.01835	0.01503	0.02498	0.00171	0.00000	0.00000
O33	0.01835	0.01503	0.02498	−0.00171	0.00000	0.00000
O34	0.01835	0.01503	0.02498	0.00171	0.00000	0.00000
O41	0.01165	0.00752	0.01453	0.00144	0.00000	0.00000
O42	0.01165	0.00752	0.01453	0.00144	0.00000	0.00000
O43	0.01165	0.00752	0.01453	−0.00144	0.00000	0.00000
O44	0.01165	0.00752	0.01453	−0.00144	0.00000	0.00000

Geometric parameters (Å, º)

Al11—Al11^i	2.9373 (3)	Al25—Al31	2.9399 (1)
Al11—Al11^ii	2.8313 (3)	Al25—O117	1.6974 (1)
Al12—Al12^i	2.9668 (3)	Al26—Al32	2.9255 (1)
Al12—Al12^ii	2.8018 (3)	Al27—Al32	2.8509 (5)
Al13—Al13^iii	2.7691 (3)	Al27—O11	1.6956 (1)
Al13—Al13^i	2.9995 (3)	Al27—O21	1.6767 (2)
Al13—Al32	2.9488 (2)	Al27—O21^i	1.6767 (2)
Al13—Al34	2.9488 (2)	Al28—Al31	2.8711 (5)
Al14—Al14^i	2.9882 (3)	Al28—O18	1.6933 (1)
Al14—Al14^ii	2.7804 (3)	Al28—O25	1.6867 (2)
Al14—Al31	2.9460 (2)	Al28—O25^i	1.6867 (2)
Al14—Al33	2.9460 (2)	Al29—Al34	2.8603 (5)
Al15—Al15^i	2.9026 (3)	Al29—O111	1.6935 (1)
Al15—Al15^ii	2.8660 (3)	Al29—O26	1.6792 (2)
Al21—Al27	2.8850 (3)	Al29—O26^i	1.6792 (2)
Al21—Al27^iv	2.8850 (3)	Al210—Al33	2.8438 (5)
Al21—O32	1.6618 (2)	Al210—O210	1.6794 (2)
Al22—Al29	2.8847 (3)	Al210—O210^i	1.6794 (2)
Al22—Al29^iv	2.8847 (3)	Si23—O33	1.6614 (2)
Al22—O34	1.6575 (2)	Si24—O34	1.6539 (2)
Al23—Al34	2.9318 (1)	Si25—O32	1.6572 (2)
Al24—Al33	2.9216 (1)	Si26—O33	1.6665 (2)
Al11^i—Al11—Al11^ii	180.0 (5)	Al22^v—Al29—Al34	90.56
Al12^i—Al12—Al12^ii	180.0 (5)	Al22^v—Al29—O111	90.96
Al13^iii—Al13—Al13^i	180.0 (5)	Al22^v—Al29—O26	144.213 (3)
Al13^iii—Al13—Al32	120.5704 (16)	Al22^v—Al29—O26^i	34.729 (3)
Al13^iii—Al13—Al34	120.5704 (16)	Al22—Al29—Al34	90.56
Al13^i—Al13—Al32	59.4296 (16)	Al22—Al29—O111	90.96
Al13^i—Al13—Al34	59.4296 (16)	Al22—Al29—O26	34.729 (3)
Al32—Al13—Al34	118.859 (3)	Al22—Al29—O26^i	144.213 (3)
Al14^i—Al14—Al14^ii	180.0 (5)	Al34—Al29—O111	51.2909 (9)
Al14^i—Al14—Al31	59.5241 (16)	Al34—Al29—O26	125.228 (3)
Al14^i—Al14—Al33	59.5240 (16)	Al34—Al29—O26^i	125.228 (3)
Al14^ii—Al14—Al31	120.4759 (16)	O111—Al29—O26	110.2250 (19)
Al14^ii—Al14—Al33	120.4760 (16)	O111—Al29—O26^i	110.2250 (19)
Al31—Al14—Al33	119.048 (3)	O26—Al29—O26^i	109.500 (6)
Al15^i—Al15—Al15^ii	180.0 (5)	Al33—Al210—O210	125.915 (3)
Al27—Al21—Al27^iv	177.4974 (2)	Al33—Al210—O210^i	125.915 (3)
Al27—Al21—O32	90.1466 (1)	O210—Al210—O210^i	108.127 (6)
Al27^iv—Al21—O32	90.1466 (1)	Al14—Al31—Al14^i	60.952 (3)
Al29—Al22—Al29^iv	178.0791 (1)	Al14—Al31—Al25	127.841 (2)
Al29—Al22—O34	90.32	Al14—Al31—Al28	68.473 (3)
Al29^iv—Al22—O34	90.32	Al14^i—Al31—Al25	127.841 (2)
Al31—Al25—O117	159.2335 (16)	Al14^i—Al31—Al28	68.473 (3)
Al21^v—Al27—Al21	177.4974 (2)	Al25—Al31—Al28	70.5809 (6)
Al21^v—Al27—Al32	90.93	Al13—Al32—Al13^i	61.141 (3)
Al21^v—Al27—O11	91.23	Al13—Al32—Al26	127.859 (2)
Al21^v—Al27—O21	143.386 (3)	Al13—Al32—Al27	68.418 (3)
Al21^v—Al27—O21^i	34.809 (3)	Al13^i—Al32—Al26	127.859 (2)
Al21—Al27—Al32	90.93	Al13^i—Al32—Al27	68.418 (3)
Al21—Al27—O11	91.23	Al26—Al32—Al27	70.7533 (6)
Al21—Al27—O21	34.809 (3)	Al14—Al33—Al14^i	60.952 (3)
Al21—Al27—O21^i	143.386 (3)	Al14—Al33—Al24	127.870 (2)
Al32—Al27—O11	52.9278 (8)	Al14—Al33—Al210	68.408 (3)
Al32—Al27—O21	125.682 (3)	Al14^i—Al33—Al24	127.870 (2)
Al32—Al27—O21^i	125.682 (3)	Al14^i—Al33—Al210	68.408 (3)
O11—Al27—O21	109.6995 (18)	Al24—Al33—Al210	70.6965 (6)
O11—Al27—O21^i	109.6995 (18)	Al13—Al34—Al13^i	61.141 (3)
O21—Al27—O21^i	108.595 (6)	Al13—Al34—Al23	127.849 (2)
Al31—Al28—O18	50.0253 (10)	Al13—Al34—Al29	68.441 (3)
Al31—Al28—O25	124.600 (3)	Al13^i—Al34—Al23	127.849 (2)
Al31—Al28—O25^i	124.600 (3)	Al13^i—Al34—Al29	68.441 (3)
O18—Al28—O25	110.374 (2)	Al23—Al34—Al29	70.7133 (6)
O18—Al28—O25^i	110.374 (2)	Al21—O32—Si25	177.65
O25—Al28—O25^i	110.743 (5)	Si23—O33—Si26	177.41
Al22^v—Al29—Al22	178.0791 (1)	Al22—O34—Si24	178.14

Symmetry codes: (i) x, y, −z; (ii) x, y, −z+1; (iii) x, y, −z−1; (iv) x, y, z+1; (v) x, y, z−1.