The modulated average structure of mullite

Homogeneous and inclusion-free single crystals of 2:1 mullite (Al_4.8Si_1.2O_9.6) grown by the Czochralski technique were examined by X-ray and neutron diffraction methods. The observed diffuse scattering together with the pattern of satellite reflections confirm previously published data and are thus inherent features of the mullite structure. The ideal composition was closely met as confirmed by microprobe analysis (Al_4.82 (3)Si_1.18 (1)O_9.59 (5)) and by average structure refinements. 8 (5) to 20 (13)% of the available Si was found in the T* position of the tetrahedra triclusters. The strong tendencey for disorder in mullite may be understood from considerations of hypothetical superstructures which would have to be n-fivefold with respect to the three-dimensional average unit cell of 2:1 mullite and n-fourfold in case of 3:2 mullite. In any of these the possible arrangements of the vacancies and of the tetrahedral units would inevitably be unfavorable. Three directions of incommensurate modulations were determined: q₁ = [0.3137 (2) 0 ½], q₂ = [0 0.4021 (5) 0.1834 (2)] and q₃ = [0 0.4009 (5) −0.1834 (2)]. The one-dimensional incommensurately modulated crystal structure associated with q₁ was refined for the first time using the superspace approach. The modulation is dominated by harmonic occupational modulations of the atoms in the di- and the triclusters of the tetrahedral units in mullite. The modulation amplitudes are small and the harmonic character implies that the modulated structure still represents an average structure in the overall disordered arrangement of the vacancies and of the tetrahedral structural units. In other words, when projecting the local assemblies at the scale of a few tens of average mullite cells into cells determined by either one of the modulation vectors q₁, q₂ or q₃ a weak average modulation results with slightly varying average occupation factors for the tetrahedral units. As a result, the real structure of mullite is locally ordered (as previously known), but on the long-range its average is not completely disordered, the modulated structure of mullite may be denoted the true `average structure of mullite'.

Computing details

(I)

Crystal data

Al4.8O9.6Si1.2	Z = 1
Mr = 316.8	F(000) = 156
Orthorhombic, Pbam(α01/2)0ss†	Dx = 3.117 Mg m^−3
q = 0.313700a* + 0.500000c*	Mo Kα radiation, λ = 0.71073 Å
a = 7.5911 Å	µ = 1.06 mm^−1
b = 7.6924 Å	T = 293 K
c = 2.8899 Å	Stump wedge, colorless
V = 168.75 Å^3	0.36 × 0.28 × 0.21 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

Bruker CCD diffractometer	1234 independent reflections
Radiation source: X-ray tube	772 reflections with I > 3σ(I)
Graphite monochromator	Rint = 0.028
integration technique scans	θmax = 36.6°, θmin = 4.0°
Absorption correction: empirical (using intensity measurements)	h = −12→12
Tmin = 0.741, Tmax = 0.831	k = −12→12
8049 measured reflections	l = −5→5

Refinement

Refinement on F	48 constraints
R[F^2 > 2σ(F^2)] = 0.028	Weighting scheme based on measured s.u.'s w = 1/(σ^2(F) + 0.0001F^2)
wR(F^2) = 0.057	(Δ/σ)max = 0.032
S = 3.01	Δρmax = 0.39 e Å^−3
1234 reflections	Δρmin = −0.50 e Å^−3
106 parameters	Extinction correction: B-C type 1 Gaussian isotropic (Becker & Coppens, 1974)
0 restraints	Extinction coefficient: 2000 (200)

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

	x	y	z	Uiso*/Ueq	Occ. (<1)
Al1	0	0	0	0.00670 (13)
Al2	0.14887 (5)	0.34028 (4)	0.5	0.00720 (13)	0.5
Si2	0.14887 (5)	0.34028 (4)	0.5	0.00720 (13)	0.2963
Al3	0.2626 (2)	0.2055 (2)	0.5	0.0087 (4)	0.2037
Si3	0.2626 (2)	0.2055 (2)	0.5	0.0087 (4)	0
O1	0.35820 (9)	0.42264 (10)	0.5	0.0132 (2)
O2	0.12772 (10)	0.21868 (11)	0	0.0134 (2)
O3	0	0.5	0.5	0.0198 (16)	0.3889
O4	0.4523 (8)	0.0457 (11)	0.5	0.0119 (17)	0.2037

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Al1	0.0079 (2)	0.0068 (2)	0.0054 (2)	0.00073 (11)	0	0
Al2	0.0065 (2)	0.0081 (2)	0.0070 (2)	−0.00052 (10)	0	0
Si2	0.0065 (2)	0.0081 (2)	0.0070 (2)	−0.00052 (10)	0	0
Al3	0.0088 (7)	0.0090 (6)	0.0082 (6)	0.0015 (5)	0	0
Si3	0.0088 (7)	0.0090 (6)	0.0082 (6)	0.0015 (5)	0	0
O1	0.0141 (4)	0.0188 (4)	0.0066 (3)	−0.0079 (3)	0	0
O2	0.0156 (3)	0.0136 (4)	0.0109 (3)	−0.0063 (2)	0	0
O3	0.007 (3)	0.024 (4)	0.0283 (17)	−0.0002 (16)	0	0
O4	0.005 (3)	0.017 (4)	0.0138 (18)	−0.003 (2)	0	0

Bond lengths (Å)

	Average	Minimum	Maximum
Al1—O1^i	1.8977 (7)	1.8907 (8)	1.9047 (8)
Al1—O1^ii	1.8977 (7)	1.8907 (8)	1.9047 (8)
Al1—O1^iii	1.8977 (7)	1.8907 (8)	1.9047 (8)
Al1—O1^iv	1.8977 (7)	1.8907 (8)	1.9047 (8)
Al1—O2	1.9416 (8)	1.9415 (8)	1.9416 (8)
Al1—O2^v	1.9416 (8)	1.9415 (8)	1.9416 (8)
Al2—O1	1.7109 (12)	1.7033 (12)	1.7182 (12)
Al2—O2	1.7289 (8)	1.7186 (8)	1.7391 (8)
Al2—O2^vi	1.7289 (8)	1.7186 (8)	1.7391 (8)
Al2—O3	1.670 (3)	1.650 (6)	1.689 (6)
Al2—O4^vii	1.757 (12)	1.692 (15)	1.822 (15)
Al2—O4^iv	1.731 (11)	1.688 (12)	1.774 (12)
Si2—O1	1.7109 (12)	1.7033 (12)	1.7182 (12)
Si2—O2	1.7289 (8)	1.7186 (8)	1.7391 (8)
Si2—O2^vi	1.7289 (8)	1.7186 (8)	1.7391 (8)
Si2—O3	1.670 (3)	1.650 (6)	1.689 (6)
Si2—O4^vii	1.757 (12)	1.692 (15)	1.822 (15)
Si2—O4^iv	1.731 (11)	1.688 (12)	1.774 (12)
Al3—O1	1.822 (3)	1.806 (3)	1.838 (3)
Al3—O2	1.7739 (16)	1.7611 (16)	1.7865 (16)
Al3—O2^vi	1.7739 (16)	1.7611 (16)	1.7865 (16)
Al3—O4	1.894 (11)	1.847 (13)	1.940 (13)
Si3—O1	1.812 (3)	1.806 (3)	1.825 (3)
Si3—O2	1.7722 (16)	1.7611 (16)	1.7862 (16)
Si3—O2^vi	1.7722 (16)	1.7611 (16)	1.7862 (16)
Si3—O4	1.865 (11)	1.847 (13)	1.898 (13)

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+1/2, x₂+1/2, −x₃+1, −x₄+1/2.

(II)

Crystal data

Al4.78O9.61Si1.22	Z = 1
Mr = 317	F(000) = 156
Orthorhombic, Pbam	Dx = 3.141 Mg m^−3
Hall symbol: -P -2xab;-2yab;-2z	X-ray radiation, λ = 1.0408 Å
a = 7.5757 Å	µ = 0.00044 mm^−1
b = 7.6651 Å	T = 293 K
c = 2.885 Å	Almost a cube, colorless
V = 167.53 Å^3	4 × 4 × 4 mm

Data collection

Eulerian cradle diffractometer	θmax = 58.5°, θmin = 5.5°
Graphite monochromator	h = −12→9
1545 measured reflections	k = −8→12
1534 independent reflections	l = −4→4
1516 reflections with I > 3σ(I)

Refinement

Refinement on F	16 constraints
R[F^2 > 2σ(F^2)] = 0.050	Weighting scheme based on measured s.u.'s w = 1/(σ^2(F) + 0.0004F^2)
wR(F^2) = 0.067	(Δ/σ)max = 0.007
S = 2.68	Δρmax = 1.12 e Å^−3
1534 reflections	Δρmin = −0.69 e Å^−3
47 parameters	Extinction correction: B-C type 1 Lorentzian anisotropic (Becker & Coppens, 1975)
0 restraints	Extinction coefficient: -0.014(5), -0.026(2), -0.0078(8), -0.008(2), -0.001(1), -0.0009(7)

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

	x	y	z	Uiso*/Ueq	Occ. (<1)
Al1	0	0	0	0.0045 (3)
Al2	0.14897 (9)	0.34022 (13)	0.5	0.0049 (3)	0.56 (5)
Si2	0.14897 (9)	0.34022 (13)	0.5	0.0049 (3)	0.24 (5)
Al3	0.2619 (4)	0.2066 (4)	0.5	0.0054 (9)	0.13 (5)
Si3	0.2619 (4)	0.2066 (4)	0.5	0.0054 (9)	0.06 (5)
O1	0.35845 (5)	0.42238 (8)	0.5	0.0111 (2)
O2	0.12734 (5)	0.21852 (7)	0	0.0116 (2)
O3	0.5	0	0.5	0.0194 (16)	0.42 (2)
O4	0.4496 (11)	0.0518 (9)	0.5	0.0072 (9)	0.195 (8)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Al1	0.0056 (4)	0.0065 (4)	0.0013 (5)	0.0011 (3)	0	0
Al2	0.0038 (4)	0.0081 (5)	0.0029 (5)	−0.0013 (2)	0	0
Si2	0.0038 (4)	0.0081 (5)	0.0029 (5)	−0.0013 (2)	0	0
Al3	0.0016 (14)	0.0091 (13)	0.0055 (19)	0.0014 (10)	0	0
Si3	0.0016 (14)	0.0091 (13)	0.0055 (19)	0.0014 (10)	0	0
O1	0.0116 (4)	0.0199 (4)	0.0018 (4)	−0.00831 (17)	0	0
O2	0.0131 (3)	0.0138 (4)	0.0078 (4)	−0.00698 (15)	0	0
O3	0.018 (3)	0.018 (4)	0.0224 (13)	−0.006 (3)	0	0
O4	0.0094 (18)	0.0047 (15)	0.0073 (12)	−0.0002 (11)	0	0

Bond lengths (Å)

Al1—O1^i	1.8933 (3)	Si2—O3^vi	1.790 (3)
Al1—O1^ii	1.8933 (3)	Si2—O3^iv	1.705 (3)
Al1—O1^iii	1.8933 (3)	Si2—O4	1.7243 (6)
Al1—O1^iv	1.8933 (3)	Si2—O4^vii	1.7243 (6)
Al1—O4	1.9331 (5)	Al3—O1	1.808 (4)
Al1—O4^v	1.9331 (5)	Al3—O1^ii	2.362 (4)
Al2—Al3	1.334 (3)	Al3—O2	2.398 (3)
Al2—Si3	1.334 (3)	Al3—O3	1.879 (4)
Al2—O1	1.7078 (9)	Al3—O4	1.7699 (17)
Al2—O2^vi	1.6680 (9)	Al3—O4^vii	1.7699 (17)
Al2—O3^vi	1.790 (3)	Si3—O1	1.808 (4)
Al2—O3^iv	1.705 (3)	Si3—O1^ii	2.362 (4)
Al2—O4	1.7243 (6)	Si3—O2	2.398 (3)
Al2—O4^vii	1.7243 (6)	Si3—O3	1.879 (4)
Si2—Al3	1.334 (3)	Si3—O4	1.7699 (17)
Si2—Si3	1.334 (3)	Si3—O4^vii	1.7699 (17)
Si2—O1	1.7078 (9)	O1—O1^viii	2.4525 (7)
Si2—O2^vi	1.6680 (9)	O3—O3^ix	1.045 (4)

Symmetry codes: (i) −x+1/2, y−1/2, −z; (ii) −x+1/2, y−1/2, −z+1; (iii) x−1/2, −y+1/2, −z; (iv) x−1/2, −y+1/2, −z+1; (v) −x, −y, z; (vi) −x+1/2, y+1/2, −z+1; (vii) x, y, z+1; (viii) −x+1, −y+1, z; (ix) −x+1, −y, z.