Superspace description of the homologous series Ga₄Ti_m-4O_2m-2 with the crystallographic shear structure based on that of rutile

A generalized approach is proposed for the structure description of the homologous series Ga₄Ti_m-4O_2m-2 with crystallographic shear (CS) structures based on that of rutile. A (3 + 1)-dimensional model for an ideal CS structure is built up in connection with the CS operation in three-dimensional space. Its structural parameters are determined by m, the shear plane and the shear vector. The Rietveld fitting of the X-ray diffraction profile of Ga₄Ti₁₃O₃₂ (m = 17) was successfully carried out using the ideal CS structure as an initial model, where modulation functions of atomic positions are inherently discontinuous and sawtooth-like. The deviation of a real structure from the ideal one was described by a few additional Fourier terms. This method was confirmed to be efficient for reducing the number of structural parameters required in the refinement in comparison to the conventional three-dimensional description.

Computing details

Program(s) used to refine structure: (Jana2000; Petricek and Dusek, 2000); software used to prepare material for publication: (Jana2000; Petricek and Dusek, 2000).

Figures

(I)

Crystal data

Ga0.444O3.556Ti1.444	V = 57.64 Å^3
Mr = 157	Z = 1
Monoclinic, P2/m(α1/2γ)00†	F(000) = 74
q = 0.11111a* + -0.05556c*	Dx = 4.52 Mg m^−3
a = 4.0904 Å	Synchrotron radiation, λ = 0.8 Å
b = 2.9755 Å	T = 295 K
c = 4.7359 Å	?, ? × ? × ? mm
β = 90.2935°

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

Data collection

2θmin = 4.004°, 2θmax = 79.994°, 2θstep = −1°

Refinement

Rp = 0.065	Profile function: Pseudo-Voigt
Rwp = 0.094	78 parameters
Rexp = 0.023	Weighting scheme based on measured s.u.'s
R(F) = 0.041	(Δ/σ)max = 9.498
χ^2 = NOT FOUND	Background function: 20 Legendre polynoms
18999 data points	Preferred orientation correction: March & Dollase

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

	x	y	z	Uiso*/Ueq	Occ. (<1)
M1a	0	0	0	0.0093 (4)*	0.965 (5)
M1'	0.1363 (5)	0	0.1050 (4)	0.0118 (6)*
M2a	0.5	0.5	0.5	0.0095 (4)*	0.919 (6)
M2'a	0.6738 (6)	0.5	0.5838 (5)	0.0045 (7)*	0.63 (2)
O1	0.301 (7)	0	0.311 (5)	0.0041 (6)*
O1'	0.055 (2)	0	0.267 (2)	0.002 (2)*
O1"	0.456 (3)	0	0.390 (2)	0.013 (3)*
O2	0.219 (7)	0.5	−0.178 (5)	0.0080 (7)*
O2'	0.003 (3)	0.5	−0.291 (2)	0.009 (3)*
O2"	0.415 (3)	0.5	−0.076 (2)	0.019 (3)*

Bond lengths (Å)

	Average	Minimum	Maximum
M1a—O1	1.96 (4)	1.95 (3)	1.98 (3)
M1a—O1^i	1.96 (4)	1.95 (3)	1.98 (3)
M1a—O1"	1.89 (2)	1.89 (2)	1.89 (2)
M1a—O1"^i	1.89 (2)	1.89 (2)	1.89 (2)
M1a—O2^ii	1.95 (2)	1.92 (2)	1.96 (2)
M1a—O2	1.95 (2)	1.92 (2)	1.96 (2)
M1a—O2^i	1.95 (2)	1.92 (2)	1.96 (2)
M1a—O2^iii	1.95 (2)	1.92 (2)	1.96 (2)
M1a—O2"^ii	1.963 (13)	1.963 (13)	1.963 (13)
M1a—O2"	1.963 (13)	1.963 (13)	1.963 (13)
M1a—O2"^i	1.963 (13)	1.963 (13)	1.963 (13)
M1a—O2"^iii	1.963 (13)	1.963 (13)	1.963 (13)
M2a—O1	1.96 (3)	1.94 (3)	1.97 (2)
M2a—O1^iv	1.96 (3)	1.94 (3)	1.97 (2)
M2a—O1^v	1.96 (3)	1.94 (3)	1.97 (2)
M2a—O1^vi	1.96 (3)	1.94 (3)	1.97 (2)
M2a—O2^vii	1.98 (4)	1.92 (4)	2.02 (4)
M2a—O2^viii	1.98 (4)	1.92 (4)	2.02 (4)
M2a—O2'^vii	2.02 (3)	2.02 (3)	2.02 (3)
M2a—O2'^viii	2.02 (3)	2.02 (3)	2.02 (3)
M2'a—O1'^ix	2.170 (10)	2.170 (10)	2.170 (10)
M2'a—O1'^v	1.985 (7)	1.985 (7)	1.985 (7)
M2'a—O1'^vi	1.985 (7)	1.985 (7)	1.985 (7)
M2'a—O1"	1.961 (7)	1.961 (7)	1.961 (7)
M2'a—O1"^iv	1.961 (7)	1.961 (7)	1.961 (7)
M2'a—O2^vii	1.93 (3)	1.93 (3)	1.93 (3)

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