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

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.

Read article Similar articles

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).

Format		BIBTeX
		EndNote
		RefMan
		Refer
		Medline
		CIF
		SGML
		Text
		Plain Text

Format		BIBTeX
		EndNote
		RefMan
		Refer
		Medline
		CIF
		SGML
		Text
		Plain Text

Search IUCr Journals		doi		Advanced search
Author		volume	page