Superspace description of the homologous series Ga₂O₃(ZnO)_m

A unified description for the structures of the homologous series Ga₂O₃(ZnO)_m, gallium zinc oxide, is presented using the superspace formalism. The structures were treated as a compositely modulated structure consisting of two subsystems. One is constructed with metal ions and the other with O ions. The ideal model is given, in which the displacive modulations of ions are well described by the zigzag function with large amplitudes. Alternative settings are also proposed which are analogous to the so-called modular structures. The validity of the model has been confirmed by refinements for phases with m = 6 and m = 9 in the homologous series. A few complex phenomena in real structures are taken into account by modifying the ideal model.

Computing details

For both structures, program(s) used to refine structure: Jana2006 (Petricek, Dusek & Palatinus, 2006); software used to prepare material for publication: Jana2006 (Petricek, Dusek & Palatinus, 2006).

(m=6)

Crystal data

Ga0.25O1.25Zn0.75	F(000) = 314
Mr = 84.5	Dx = 5.68 Mg m^−3
Orthorhombic, Cmmm(00γ)0s0†	Mo Kα radiation, λ = 0.71069 Å
q = 1.12500c*‡	Cell parameters from 25 reflections
a = 3.2465 (9) Å	θ = 12.6–15.7°
b = 19.640 (5) Å	µ = 24.6 mm^−1
c = 3.0979 (8) Å	T = 298 K
V = 197.53 (9) Å^3	Irregular, colorless
Z = 8	0.12 × 0.06 × 0.02 mm

† Symmetry operations: (1) x₁, x₂, x₃, x₄; (2) −x₁, x₂, x₃, x₄; (3) x₁, −x₂, x₃, x₄+1/2; (4) −x₁, −x₂, x₃, x₄+1/2; (5) −x₁, −x₂, −x₃, −x₄; (6) x₁, −x₂, −x₃, −x₄; (7) −x₁, x₂, −x₃, −x₄+1/2; (8) x₁, x₂, −x₃, −x₄+1/2; (9) x₁+1/2, x₂+1/2, x₃, x₄; (10) −x₁+1/2, x₂+1/2, x₃, x₄; (11) x₁+1/2, −x₂+1/2, x₃, x₄+1/2; (12) −x₁+1/2, −x₂+1/2, x₃, x₄+1/2; (13) −x₁+1/2, −x₂+1/2, −x₃, −x₄; (14) x₁+1/2, −x₂+1/2, −x₃, −x₄; (15) −x₁+1/2, x₂+1/2, −x₃, −x₄+1/2; (16) x₁+1/2, x₂+1/2, −x₃, −x₄+1/2.

‡ ;

Data collection

Rigaku AFC-7R diffractometer	1349 reflections with I > 2σ(I)
Radiation source: Rotating anode	Rint = 0.122
Graphite monochromator	θmax = 45.0°, θmin = 2.7°
ω scans	h = 0→6
Absorption correction: ψ scan Absorption correction given as ψ scan	k = 0→38
Tmin = 0.18, Tmax = 0.61	l = −7→7
7287 measured reflections	3 standard reflections every 200 reflections
3699 independent reflections	intensity decay: 0.0%

Refinement

Refinement on F^2	4 constraints
R[F^2 > 2σ(F^2)] = 0.044	Weighting scheme based on measured s.u.'s w = 1/(σ^2(I) + 0.0001I^2)
wR(F^2) = 0.099	(Δ/σ)max = 0.017
S = 1.03	Δρmax = 2.29 e Å^−3
3699 reflections	Δρmin = −2.27 e Å^−3
115 parameters	Extinction correction: B-C type 1 Gaussian isotropic (Becker & Coppens, 1974)
0 restraints	Extinction coefficient: 404 (14)

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

	x	y	z	Uiso*/Ueq	Occ. (<1)
M1	0	0.38426 (4)	0	0.01344 (16)	0.75
M1g	0	0.38426 (4)	0	0.01344 (16)	0.25
M1a	0	−0.10985 (19)	0.049 (5)	0.034 (2)	0.1769 (16)
M1ag	0	−0.10985 (19)	0.049 (5)	0.034 (2)	0.0590 (5)
M1b	0.5	−0.13516 (11)	0	0.0113 (6)	0.396 (3)
M1bg	0.5	−0.13516 (11)	0	0.0113 (6)	0.1320 (11)
M1c	0	0.87414 (6)	0	0.0089 (3)	0.75
M1cg	0	0.87414 (6)	0	0.0089 (3)	0.25
M2	0	0.88428 (2)	0.5	0.01097 (10)	0.75
M2g	0	0.88428 (2)	0.5	0.01097 (10)	0.25
O1	0	0.63435 (13)	0.25	0.0155 (6)
O2	0	0.13556 (14)	0.75	0.0254 (9)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
M1	0.0106 (3)	0.0081 (2)	0.0216 (3)	0	0	0
M1g	0.0106 (3)	0.0081 (2)	0.0216 (3)	0	0	0
M1a	0.068 (3)	0.0092 (13)	0.024 (7)	0	0	−0.004 (2)
M1ag	0.068 (3)	0.0092 (13)	0.024 (7)	0	0	−0.004 (2)
M1b	0.0178 (12)	0.0078 (9)	0.0083 (9)	0	0	0
M1bg	0.0178 (12)	0.0078 (9)	0.0083 (9)	0	0	0
M1c	0.0084 (5)	0.0076 (5)	0.0107 (5)	0	0	0
M1cg	0.0084 (5)	0.0076 (5)	0.0107 (5)	0	0	0
M2	0.00950 (19)	0.00838 (16)	0.01504 (19)	0	0	0
M2g	0.00950 (19)	0.00838 (16)	0.01504 (19)	0	0	0
O1	0.0143 (9)	0.0115 (11)	0.0207 (12)	0	0	0
O2	0.0312 (17)	0.0163 (13)	0.0287 (16)	0	0	0

Bond lengths (Å)

	Average	Minimum	Maximum
M1—O1^i	44.91 (6)	2.135 (10)	11.748 (7)

Symmetry code: (i) x₁, x₂, x₃−1, x₄.

(m=9)

Crystal data

Ga0.1818O1.0909Zn0.8181	F(000) = 311
Mr = 83.6	Dx = 5.68 Mg m^−3
Orthorhombic, Cmcm(00γ)000†	Mo Kα radiation, λ = 0.71069 Å
q = 1.09091c*‡	Cell parameters from 25 reflections
a = 3.2494 (14) Å	θ = 15.0–16.8°
b = 19.722 (6) Å	µ = 24.7 mm^−1
c = 3.0513 (11) Å	T = 298 K
V = 195.54 (12) Å^3	Plate, colorless
Z = 8	0.20 × 0.12 × 0.02 mm

† Symmetry operations: (1) x₁, x₂, x₃, x₄; (2) −x₁, −x₂, x₃+1/2, x₄; (3) −x₁, x₂, −x₃+1/2, −x₄; (4) x₁, −x₂, −x₃, −x₄; (5) −x₁, −x₂, −x₃, −x₄; (6) x₁, x₂, −x₃+1/2, −x₄; (7) x₁, −x₂, x₃+1/2, x₄; (8) −x₁, x₂, x₃, x₄; (9) x₁+1/2, x₂+1/2, x₃, x₄; (10) −x₁+1/2, −x₂+1/2, x₃+1/2, x₄; (11) −x₁+1/2, x₂+1/2, −x₃+1/2, −x₄; (12) x₁+1/2, −x₂+1/2, −x₃, −x₄; (13) −x₁+1/2, −x₂+1/2, −x₃, −x₄; (14) x₁+1/2, x₂+1/2, −x₃+1/2, −x₄; (15) x₁+1/2, −x₂+1/2, x₃+1/2, x₄; (16) −x₁+1/2, x₂+1/2, x₃, x₄.

‡ ;

Data collection

Rigaku AFC-7R diffractometer	993 reflections with I > 2σ(I)
Radiation source: Rotating anode	Rint = 0.048
Graphite monochromator	θmax = 30.1°, θmin = 2.8°
ω scans	h = 0→4
Absorption correction: analytical Tompa Analytical	k = −27→0
Tmin = 0.034, Tmax = 0.417	l = −7→7
3662 measured reflections	3 standard reflections every 200 reflections
1862 independent reflections	intensity decay: 0.0%

Refinement

Refinement on F^2	5 constraints
R[F^2 > 2σ(F^2)] = 0.034	Weighting scheme based on measured s.u.'s w = 1/(σ^2(I) + 0.0001I^2)
wR(F^2) = 0.068	(Δ/σ)max = 0.017
S = 1.35	Δρmax = 1.60 e Å^−3
1862 reflections	Δρmin = −1.21 e Å^−3
149 parameters	Extinction correction: B-C type 1 Gaussian isotropic (Becker & Coppens, 1974)
0 restraints	Extinction coefficient: 590 (20)

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

	x	y	z	Uiso*/Ueq	Occ. (<1)
M1	0	0.63438 (3)	0.25	0.01132 (14)	0.8181
M1g	0	0.63438 (3)	0.25	0.01132 (14)	0.1919
M1a	0	−0.04159 (19)	0.25	0.0422 (12)	0.456 (4)
M1ag	0	−0.04159 (19)	0.25	0.0422 (12)	0.1013 (8)
M1b	0.5	−0.06856 (18)	0.25	0.0154 (9)	0.362 (4)
M1bg	0.5	−0.06856 (18)	0.25	0.0154 (9)	0.0805 (8)
M2	0	0.13443 (2)	0.75	0.01253 (12)	0.8181
M2g	0	0.13443 (2)	0.75	0.01253 (12)	0.1919
O1	0	0.38420 (13)	0	0.0173 (7)
O2	0	0.88545 (15)	0.5	0.0217 (8)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
M1	0.00793 (18)	0.0099 (3)	0.0161 (2)	0	0	0
M1g	0.00793 (18)	0.0099 (3)	0.0161 (2)	0	0	0
M1a	0.076 (3)	0.0118 (17)	0.0391 (19)	0	0	0
M1ag	0.076 (3)	0.0118 (17)	0.0391 (19)	0	0	0
M1b	0.0270 (16)	0.0081 (16)	0.0113 (14)	0	0	0
M1bg	0.0270 (16)	0.0081 (16)	0.0113 (14)	0	0	0
M2	0.00863 (16)	0.0098 (3)	0.0192 (2)	0	0	0
M2g	0.00863 (16)	0.0098 (3)	0.0192 (2)	0	0	0
O1	0.0109 (10)	0.0158 (14)	0.0251 (13)	0	0	0
O2	0.0241 (14)	0.0134 (13)	0.0276 (14)	0	0	0

Bond lengths (Å)

	Average	Minimum	Maximum
M1—O1	53.5 (4)	1.91 (4)	11.75 (3)