Helical screw type magnetic structure of the multiferroic CaMn7O12 with low Cu-doping

Slawiński, W.; Przenioslo, R.; Sosnowska, I.; Petříček, V.

doi:10.1107/S0108768112008038

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Helical screw type magnetic structure of the multiferroic CaMn₇O₁₂ with low Cu-doping

W. Sławiński, R. Przeniosło, I. Sosnowska and V. Petříček

The modulated crystal structure and modulated magnetic ordering of the multiferroic CaCu_xMn_7−xO₁₂ is studied by analysing neutron and synchrotron-radiation (SR) powder diffraction data with a model based on the magnetic superspace group R31′(00γ)ts. Both atomic position modulations and magnetic modulations are described with the modulation vector (0, 0, q). The magnetic ordering is a screw-type circular helix where the magnetic moments are perpendicular to the c direction. The temperature dependence of the modulation vector length and the ordered magnetic moments of Mn³⁺ and Mn⁴⁺ ions is given between T = 50 K and the Néel temperature T_N ≃ 90 K. The atomic position modulation length L_p and the magnetic modulation length L_m fulfil the relation L_m = 2L_p at all temperatures between 50 K and T_N.

Keywords: modulated structure; magnetic ordering; multiferroic; powder diffraction; transition metal oxides.

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Supporting information

	Crystallographic Information File (CIF) https://doi.org/10.1107/S0108768112008038/pz5097sup1.cif Contains datablocks global, I, II, III, IV, V, VI
	Rietveld powder data file (CIF format) https://doi.org/10.1107/S0108768112008038/pz5097Isup2.rtv Contains datablock I

B-IncStrDB reference: 6302EkCcoh

Computing details top

Data collection: KM4B8 (Kuma Diffraction Ltd., 1996) for (I), (II). Cell refinement: KM4B8 (Kuma Diffraction Ltd., 1996) for (I), (II). Data reduction: Jana2006 (Petricek, Dusek & Palatinus, 2006) for (I), (II). Program(s) used to solve structure: SIR97 (Altomare et al., 1997) for (I), (II). Program(s) used to refine structure: Jana2006 (Petricek, Dusek & Palatinus, 2006) for (I), (II). Molecular graphics: Diamond 3 (Brandenburg, 1999) for (I), (II). Software used to prepare material for publication: Jana2006 (Petricek, Dusek & Palatinus, 2006) for (I), (II).

Figures top

(I) top

Crystal data top

CaMn₇O₁₂	Z = 3
M_r = 616.6	F(000) = 873
Trigonal, R3(00γ)t†	D_x = 5.125 Mg m⁻³
q = 0.921268c*	X-ray radiation, λ = 0.39996 Å
a = 10.443658 (10) Å	T = 70 K
c = 6.343562 (7) Å	?, ? × ? × ? mm
V = 599.20 (1) Å³

Data collection top

2θ_min = 2.001°, 2θ_max = 37.199°, 2θ_step = 0.001°

Refinement top

R_p = 0.151	59 parameters
R_wp = 0.232	0 restraints
R_exp = 0.127	0 constraints
χ² = NOT FOUND	Weighting scheme based on measured s.u.'s
47540 data points	(Δ/σ)_max = 0.529
Excluded region(s): from 0.4 to 2.000, from 37.2 to 47.987	Background function: Manual background combined with 10 Legendre polynoms
Profile function: Pseudo-Voigt	Preferred orientation correction: none

Crystal data top

CaMn₇O₁₂	V = 599.20 (1) Å³
M_r = 616.6	Z = 3
Trigonal, R3(00γ)t†	X-ray radiation, λ = 0.39996 Å
q = 0.921268c*	T = 70 K
a = 10.443658 (10) Å	?, ? × ? × ? mm
c = 6.343562 (7) Å

Data collection top

2θ_min = 2.001°, 2θ_max = 37.199°, 2θ_step = 0.001°

Refinement top

R_p = 0.151	47540 data points
R_wp = 0.232	59 parameters
R_exp = 0.127	0 restraints
χ² = NOT FOUND	(Δ/σ)_max = 0.529

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
Ca	0	0	0	0.0035 (6)*
Mn1	0.5	0	0	0.00089 (8)*
Mn2	0.5	0.5	0.5	0.00089 (8)*
Mn3	0	0	0.5	0.00089 (8)*
O1_1	0.2236 (3)	0.2740 (3)	0.0814 (4)	0.00120 (19)*
O1_2	−0.2236 (3)	−0.2740 (3)	−0.0814 (4)	0.00120 (19)*
O2_1	0.3416 (3)	0.5218 (3)	0.3407 (4)	0.00120 (19)*
O2_2	−0.3416 (3)	−0.5218 (3)	−0.3407 (4)	0.00120 (19)*

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
?	?	?	?	?	?	?

(II) top

Crystal data top

CaMn₇O₁₂	Z = 3
M_r = 616.6	F(000) = 873
Trigonal, R3(00γ)0†	D_x = 5.125 Mg m⁻³
q = 0.921372c*	X-ray radiation, λ = 0.39996 Å
a = 10.443650 (11) Å	T = 70 K
c = 6.343562 (8) Å	?, ? × ? × ? mm
V = 599.20 (1) Å³

† Symmetry operations: (1) x₁, x₂, x₃, x₄; (2) −x₂, x₁−x₂, x₃, x₄; (3) −x₁+x₂, −x₁, x₃, x₄; (4) −x₁, −x₂, −x₃, −x₄; (5) x₂, −x₁+x₂, −x₃, −x₄; (6) x₁−x₂, x₁, −x₃, −x₄; (7) x₁+2/3, x₂+1/3, x₃+1/3, x₄; (8) −x₂+2/3, x₁−x₂+1/3, x₃+1/3, x₄; (9) −x₁+x₂+2/3, −x₁+1/3, x₃+1/3, x₄; (10) −x₁+2/3, −x₂+1/3, −x₃+1/3, −x₄; (11) x₂+2/3, −x₁+x₂+1/3, −x₃+1/3, −x₄; (12) x₁−x₂+2/3, x₁+1/3, −x₃+1/3, −x₄; (13) x₁+1/3, x₂+2/3, x₃+2/3, x₄; (14) −x₂+1/3, x₁−x₂+2/3, x₃+2/3, x₄; (15) −x₁+x₂+1/3, −x₁+2/3, x₃+2/3, x₄; (16) −x₁+1/3, −x₂+2/3, −x₃+2/3, −x₄; (17) x₂+1/3, −x₁+x₂+2/3, −x₃+2/3, −x₄; (18) x₁−x₂+1/3, x₁+2/3, −x₃+2/3, −x₄.

Data collection top

2θ_min = 2.001°, 2θ_max = 37.199°, 2θ_step = 0.001°

Refinement top

R_p = 0.164	56 parameters
R_wp = 0.239	0 restraints
R_exp = 0.127	0 constraints
χ² = NOT FOUND	Weighting scheme based on measured s.u.'s
47540 data points	(Δ/σ)_max = 2.263
Excluded region(s): from 0.4 to 2.000, from 37.2 to 47.987	Background function: Manual background combined with 10 Legendre polynoms
Profile function: Pseudo-Voigt	Preferred orientation correction: none

Crystal data top

CaMn₇O₁₂	V = 599.20 (1) Å³
M_r = 616.6	Z = 3
Trigonal, R3(00γ)0†	X-ray radiation, λ = 0.39996 Å
q = 0.921372c*	T = 70 K
a = 10.443650 (11) Å	?, ? × ? × ? mm
c = 6.343562 (8) Å

Data collection top

2θ_min = 2.001°, 2θ_max = 37.199°, 2θ_step = 0.001°

Refinement top

R_p = 0.164	47540 data points
R_wp = 0.239	56 parameters
R_exp = 0.127	0 restraints
χ² = NOT FOUND	(Δ/σ)_max = 2.263

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
Ca	0	0	0	0.0035 (5)*
Mn1	0.5	0	0	0.00042 (9)*
Mn2	0.5	0.5	0.5	0.00042 (9)*
Mn3	0	0	0.5	0.00042 (9)*
O1	0.2236 (3)	0.2741 (3)	0.0816 (4)	0.0006 (3)*
O2	0.3420 (3)	0.5221 (3)	0.3415 (4)	0.0006 (3)*

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
?	?	?	?	?	?	?

(III) top

Crystal data top

CaMn₆MnO₁₂	Z = 6
M_r = 671.6	F(000) = 873
Trigonal, R31(00γ)ts†	D_x = 11.157 Mg m⁻³
q = 1.039135c*	Neutron radiation, λ = 1.59493 Å
a = 10.4452 (2) Å	T = 60 K
c = 6.3453 (2) Å	?, ? × ? × ? mm
V = 599.54 (2) Å³

Data collection top

D2B
diffractometer

2θ_min = 5.05°, 2θ_max = 159.85°, 2θ_step = 0.05°

Refinement top

R_p = 0.051	52 parameters
R_wp = 0.069	0 restraints
R_exp = 0.046	5 constraints
χ² = NOT FOUND	Weighting scheme based on measured s.u.'s
3134 data points	(Δ/σ)_max = 0.028
Excluded region(s): from 3.197 to 5.000, from 149 to 153.000	Background function: 30 Legendre polynoms
Profile function: Pseudo-Voigt	Preferred orientation correction: none

Crystal data top

CaMn₆MnO₁₂	V = 599.54 (2) Å³
M_r = 671.6	Z = 6
Trigonal, R31(00γ)ts†	Neutron radiation, λ = 1.59493 Å
q = 1.039135c*	T = 60 K
a = 10.4452 (2) Å	?, ? × ? × ? mm
c = 6.3453 (2) Å

Data collection top

D2B
diffractometer

2θ_min = 5.05°, 2θ_max = 159.85°, 2θ_step = 0.05°

Refinement top

R_p = 0.051	3134 data points
R_wp = 0.069	52 parameters
R_exp = 0.046	0 restraints
χ² = NOT FOUND

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
Ca1	0	0	0	0.0199 (9)*
Mn1	0.5	0	0	0.0183 (3)*
Mn2	0.5	0.5	0.5	0.0183 (3)*
Mn3	0	0	0.5	0.0183 (3)*
O1	0.67557 (18)	0.18888 (15)	0.0073 (3)	0.02106 (15)*
O1'	−0.67557 (18)	−0.18888 (15)	−0.0073 (3)	0.02106 (15)*
O2	0.61602 (19)	0.5561 (2)	0.25311 (18)	0.02106 (15)*
O2'	−0.61602 (19)	−0.5561 (2)	−0.25311 (18)	0.02106 (15)*

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
?	?	?	?	?	?	?

(IV) top

Crystal data top

CaMn₆MnO₁₂	Z = 6
M_r = 576.6	F(000) = 873
Trigonal, R31(00γ)ts†	D_x = 9.599 Mg m⁻³
q = 1.040311c*	? radiation, λ = 2.41835 Å
a = 10.4372 (11) Å	T = 60 K
c = 6.3414 (9) Å	?, ? × ? × ? mm
V = 598.25 (12) Å³

Data collection top

2θ_min = 10.049°, 2θ_max = 139.99°

Refinement top

R_p = 0.007	28 parameters
R_wp = 0.010	0 restraints
R_exp = 0.007	5 constraints
χ² = ?	Weighting scheme based on measured s.u.'s
1534 data points	(Δ/σ)_max = 0.044
Excluded region(s): from 1.153 to 10.000, from 140 to 154.616	Background function: 15 Legendre polynoms
Profile function: Gaussian	Preferred orientation correction: none

Crystal data top

CaMn₆MnO₁₂	V = 598.25 (12) Å³
M_r = 576.6	Z = 6
Trigonal, R31(00γ)ts†	? radiation, λ = 2.41835 Å
q = 1.040311c*	T = 60 K
a = 10.4372 (11) Å	?, ? × ? × ? mm
c = 6.3414 (9) Å

Data collection top

2θ_min = 10.049°, 2θ_max = 139.99°

Refinement top

R_p = 0.007	1534 data points
R_wp = 0.010	28 parameters
R_exp = 0.007	0 restraints
χ² = ?

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
Ca	0	0	0	0.022 (6)*
Mn1	0.5	0	0	0.048 (3)*
Mn2	0.5	0.5	0.5	0.048 (3)*
Mn3	0	0	0.5	0.048 (3)*
O1	0.675633	0.188816	0.00749	0.060 (2)*
O1'	−0.675633	−0.188816	−0.00749	0.060 (2)*
O2	0.616083	0.556128	0.253145	0.060 (2)*
O2'	−0.616083	−0.556128	−0.253145	0.060 (2)*

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
?	?	?	?	?	?	?

(V) top

Crystal data top

CaCu_0.113Mn_5.887MnO₁₂	Z = 6
M_r = 617.6	F(000) = 874
Trigonal, R31(00γ)ts†	D_x = 10.294 Mg m⁻³
q = 1.047854c*	? radiation, λ = 2.41835 Å
a = 10.4312 (13) Å	T = 60 K
c = 6.3417 (9) Å	?, ? × ? × ? mm
V = 597.59 (14) Å³

Data collection top

2θ_min = 10.05°, 2θ_max = 139.991°

Refinement top

R_p = 0.028	36 parameters
R_wp = 0.039	0 restraints
R_exp = 0.007	5 constraints
χ² = ?	Weighting scheme based on measured s.u.'s
1534 data points	(Δ/σ)_max = 0.042
Excluded region(s): from 1.154 to 10.000, from 140 to 154.617	Background function: 15 Legendre polynoms
Profile function: Gaussian	Preferred orientation correction: none

Crystal data top

CaCu_0.113Mn_5.887MnO₁₂	V = 597.59 (14) Å³
M_r = 617.6	Z = 6
Trigonal, R31(00γ)ts†	? radiation, λ = 2.41835 Å
q = 1.047854c*	T = 60 K
a = 10.4312 (13) Å	?, ? × ? × ? mm
c = 6.3417 (9) Å

Data collection top

2θ_min = 10.05°, 2θ_max = 139.991°

Refinement top

R_p = 0.028	1534 data points
R_wp = 0.039	36 parameters
R_exp = 0.007	0 restraints
χ² = ?

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq	Occ. (<1)
Ca	0	0	0	0.052 (3)*
Mn1	0.5	0	0	0.0489 (18)*	0.9623
Cu	0.5	0	0	0.0489 (18)*	0.0377
Mn2	0.5	0.5	0.5	0.0489 (18)*
Mn3	0	0	0.5	0.0489 (18)*
O1	0.2232 (5)	0.2754 (4)	0.0807 (7)	0.0518 (9)*
O1'	−0.2232 (5)	−0.2754 (4)	−0.0807 (7)	0.0518 (9)*
O2	0.3433 (5)	0.5257 (5)	0.3400 (10)	0.0518 (9)*
O2'	−0.3433 (5)	−0.5257 (5)	−0.3400 (10)	0.0518 (9)*

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
?	?	?	?	?	?	?

(VI) top

Crystal data top

CaCu_0.246Mn_5.754MnO₁₂	Z = 6
M_r = 618.8	F(000) = 876
Trigonal, R31(00γ)ts†	D_x = 10.381 Mg m⁻³
q = 1.052835c*	? radiation, λ = 2.41835 Å
a = 10.403 (2) Å	T = 60 K
c = 6.3340 (14) Å	?, ? × ? × ? mm
V = 593.7 (2) Å³

Data collection top

2θ_min = 10.05°, 2θ_max = 139.991°

Refinement top

R_p = 0.025	35 parameters
R_wp = 0.037	0 restraints
R_exp = 0.007	5 constraints
χ² = ?	Weighting scheme based on measured s.u.'s
1536 data points	(Δ/σ)_max = 0.051
Excluded region(s): from 1.154 to 10.000, from 52.015 to 55.384, from 140 to 154.826	Background function: 20 Legendre polynoms
Profile function: Gaussian	Preferred orientation correction: none

Crystal data top

CaCu_0.246Mn_5.754MnO₁₂	V = 593.7 (2) Å³
M_r = 618.8	Z = 6
Trigonal, R31(00γ)ts†	? radiation, λ = 2.41835 Å
q = 1.052835c*	T = 60 K
a = 10.403 (2) Å	?, ? × ? × ? mm
c = 6.3340 (14) Å

Data collection top

2θ_min = 10.05°, 2θ_max = 139.991°

Refinement top

R_p = 0.025	1536 data points
R_wp = 0.037	35 parameters
R_exp = 0.007	0 restraints
χ² = ?

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq	Occ. (<1)
Ca	0	0	0	0.051 (4)*
Mn1	0.5	0	0	0.049 (2)*	0.918
Cu	0.5	0	0	0.049 (2)*	0.082
Mn2	0.5	0.5	0.5	0.049 (2)*
Mn3	0	0	0.5	0.049 (2)*
O1	0.223239	0.275435	0.080719	0.0543 (11)*
O1'	−0.223239	−0.275435	−0.080719	0.0543 (11)*
O2	0.3433	0.525721	0.340079	0.0543 (11)*
O2'	−0.3433	−0.525721	−0.340079	0.0543 (11)*

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
?	?	?	?	?	?	?

Experimental details

	(I)	(II)	(III)	(IV)
Crystal data
Chemical formula	CaMn₇O₁₂	CaMn₇O₁₂	CaMn₆MnO₁₂	CaMn₆MnO₁₂
M_r	616.6	616.6	671.6	576.6
Crystal system, space group	Trigonal, R3(00γ)t†	Trigonal, R3(00γ)0‡	Trigonal, R31(00γ)ts§	Trigonal, R31(00γ)ts††
Temperature (K)	70	70	60	60
Wave vectors	q = 0.921268c*	q = 0.921372c*	q = 1.039135c*	q = 1.040311c*
a, c (Å)	10.443658 (10), 6.343562 (7)	10.443650 (11), 6.343562 (8)	10.4452 (2), 6.3453 (2)	10.4372 (11), 6.3414 (9)
V (Å³)	599.20 (1)	599.20 (1)	599.54 (2)	598.25 (12)
Z	3	3	6	6
Radiation type	X-ray, λ = 0.39996 Å	X-ray, λ = 0.39996 Å	Neutron, λ = 1.59493 Å	?, λ = 2.41835 Å
Specimen shape, size (mm)	?, ? × ? × ?	?, ? × ? × ?	?, ? × ? × ?	?, ? × ? × ?

Data collection
Diffractometer	?	?	D2B diffractometer	?
Specimen mounting	?	?	?	?
Data collection mode	?	?	?	?
Scan method	?	?	?	?
2θ values (°)	2θ_min = 2.001 2θ_max = 37.199 2θ_step = 0.001	2θ_min = 2.001 2θ_max = 37.199 2θ_step = 0.001	2θ_min = 5.05 2θ_max = 159.85 2θ_step = 0.05	2θ_min = 10.049 2θ_max = 139.99 2θ_step = ?

Refinement
R factors and goodness of fit	R_p = 0.151, R_wp = 0.232, R_exp = 0.127, χ² = NOT FOUND	R_p = 0.164, R_wp = 0.239, R_exp = 0.127, χ² = NOT FOUND	R_p = 0.051, R_wp = 0.069, R_exp = 0.046, χ² = NOT FOUND	R_p = 0.007, R_wp = 0.010, R_exp = 0.007, χ² = ?
No. of data points	47540	47540	3134	1534
No. of parameters	59	56	52	28
(Δ/σ)_max	0.529	2.263	0.028	0.044

	(V)	(VI)
Crystal data
Chemical formula	CaCu_0.113Mn_5.887MnO₁₂	CaCu_0.246Mn_5.754MnO₁₂
M_r	617.6	618.8
Crystal system, space group	Trigonal, R31(00γ)ts‡‡	Trigonal, R31(00γ)ts§§
Temperature (K)	60	60
Wave vectors	q = 1.047854c*	q = 1.052835c*
a, c (Å)	10.4312 (13), 6.3417 (9)	10.403 (2), 6.3340 (14)
V (Å³)	597.59 (14)	593.7 (2)
Z	6	6
Radiation type	?, λ = 2.41835 Å	?, λ = 2.41835 Å
Specimen shape, size (mm)	?, ? × ? × ?	?, ? × ? × ?

Data collection
Diffractometer	?	?
Specimen mounting	?	?
Data collection mode	?	?
Scan method	?	?
2θ values (°)	2θ_min = 10.05 2θ_max = 139.991 2θ_step = ?	2θ_min = 10.05 2θ_max = 139.991 2θ_step = ?

Refinement
R factors and goodness of fit	R_p = 0.028, R_wp = 0.039, R_exp = 0.007, χ² = ?	R_p = 0.025, R_wp = 0.037, R_exp = 0.007, χ² = ?
No. of data points	1534	1536
No. of parameters	36	35
(Δ/σ)_max	0.042	0.051

† Symmetry operations: (1) x₁, x₂, x₃, x₄; (2) −x₂, x₁−x₂, x₃, x₄+1/3; (3) −x₁+x₂, −x₁, x₃, x₄+2/3; (4) x₁+2/3, x₂+1/3, x₃+1/3, x₄; (5) −x₂+2/3, x₁−x₂+1/3, x₃+1/3, x₄+1/3; (6) −x₁+x₂+2/3, −x₁+1/3, x₃+1/3, x₄+2/3; (7) x₁+1/3, x₂+2/3, x₃+2/3, x₄; (8) −x₂+1/3, x₁−x₂+2/3, x₃+2/3, x₄+1/3; (9) −x₁+x₂+1/3, −x₁+2/3, x₃+2/3, x₄+2/3.

‡ Symmetry operations: (1) x₁, x₂, x₃, x₄; (2) −x₂, x₁−x₂, x₃, x₄; (3) −x₁+x₂, −x₁, x₃, x₄; (4) −x₁, −x₂, −x₃, −x₄; (5) x₂, −x₁+x₂, −x₃, −x₄; (6) x₁−x₂, x₁, −x₃, −x₄; (7) x₁+2/3, x₂+1/3, x₃+1/3, x₄; (8) −x₂+2/3, x₁−x₂+1/3, x₃+1/3, x₄; (9) −x₁+x₂+2/3, −x₁+1/3, x₃+1/3, x₄; (10) −x₁+2/3, −x₂+1/3, −x₃+1/3, −x₄; (11) x₂+2/3, −x₁+x₂+1/3, −x₃+1/3, −x₄; (12) x₁−x₂+2/3, x₁+1/3, −x₃+1/3, −x₄; (13) x₁+1/3, x₂+2/3, x₃+2/3, x₄; (14) −x₂+1/3, x₁−x₂+2/3, x₃+2/3, x₄; (15) −x₁+x₂+1/3, −x₁+2/3, x₃+2/3, x₄; (16) −x₁+1/3, −x₂+2/3, −x₃+2/3, −x₄; (17) x₂+1/3, −x₁+x₂+2/3, −x₃+2/3, −x₄; (18) x₁−x₂+1/3, x₁+2/3, −x₃+2/3, −x₄.

§ Symmetry operations: (1) x₁, x₂, x₃, x₄; (2) −x₂, x₁−x₂, x₃, x₄+1/3; (3) −x₁+x₂, −x₁, x₃, x₄+2/3; (4) x₁+2/3, x₂+1/3, x₃+1/3, x₄; (5) −x₂+2/3, x₁−x₂+1/3, x₃+1/3, x₄+1/3; (6) −x₁+x₂+2/3, −x₁+1/3, x₃+1/3, x₄+2/3; (7) x₁+1/3, x₂+2/3, x₃+2/3, x₄; (8) −x₂+1/3, x₁−x₂+2/3, x₃+2/3, x₄+1/3; (9) −x₁+x₂+1/3, −x₁+2/3, x₃+2/3, x₄+2/3.

†† Symmetry operations: (1) x₁, x₂, x₃, x₄; (2) −x₂, x₁−x₂, x₃, x₄+1/3; (3) −x₁+x₂, −x₁, x₃, x₄+2/3; (4) x₁+2/3, x₂+1/3, x₃+1/3, x₄; (5) −x₂+2/3, x₁−x₂+1/3, x₃+1/3, x₄+1/3; (6) −x₁+x₂+2/3, −x₁+1/3, x₃+1/3, x₄+2/3; (7) x₁+1/3, x₂+2/3, x₃+2/3, x₄; (8) −x₂+1/3, x₁−x₂+2/3, x₃+2/3, x₄+1/3; (9) −x₁+x₂+1/3, −x₁+2/3, x₃+2/3, x₄+2/3.

‡‡ Symmetry operations: (1) x₁, x₂, x₃, x₄; (2) −x₂, x₁−x₂, x₃, x₄+1/3; (3) −x₁+x₂, −x₁, x₃, x₄+2/3; (4) x₁+2/3, x₂+1/3, x₃+1/3, x₄; (5) −x₂+2/3, x₁−x₂+1/3, x₃+1/3, x₄+1/3; (6) −x₁+x₂+2/3, −x₁+1/3, x₃+1/3, x₄+2/3; (7) x₁+1/3, x₂+2/3, x₃+2/3, x₄; (8) −x₂+1/3, x₁−x₂+2/3, x₃+2/3, x₄+1/3; (9) −x₁+x₂+1/3, −x₁+2/3, x₃+2/3, x₄+2/3.

§§ Symmetry operations: (1) x₁, x₂, x₃, x₄; (2) −x₂, x₁−x₂, x₃, x₄+1/3; (3) −x₁+x₂, −x₁, x₃, x₄+2/3; (4) x₁+2/3, x₂+1/3, x₃+1/3, x₄; (5) −x₂+2/3, x₁−x₂+1/3, x₃+1/3, x₄+1/3; (6) −x₁+x₂+2/3, −x₁+1/3, x₃+1/3, x₄+2/3; (7) x₁+1/3, x₂+2/3, x₃+2/3, x₄; (8) −x₂+1/3, x₁−x₂+2/3, x₃+2/3, x₄+1/3; (9) −x₁+x₂+1/3, −x₁+2/3, x₃+2/3, x₄+2/3.

Computer programs: KM4B8 (Kuma Diffraction Ltd., 1996), Jana2006 (Petricek, Dusek & Palatinus, 2006), SIR97 (Altomare et al., 1997), Diamond 3 (Brandenburg, 1999).

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