The δ-phase of SrTeO₃ at 780 K

Abstract

As part of a structural investigation of strontium tellurate(IV) (STO), SrTeO₃, with particular emphasis on the crystal chemistry and phase transitions, the structure of the δ-phase has been determined at 780 K using a single-crystal analysis. Both structural and non-linear optical measurements indicate that STO undergoes a γ→δ second-order ferroelectric phase transition at 633 K from the C2 (γ) to the C2/m (δ) modification. Systematic differences between the similar γ- and δ-phase structures were determined and it was found that this phase transformation can be described by a displacive mechanism.

Related literature

Single crystals of strontium tellurate(IV) (STO) were prepared by Sadovskaya (1984 ▶). Structural phase transitions of STO have been studied by X-ray powder diffraction by Ismailzade et al. (1979 ▶) and Simon et al. (1979 ▶), neutron powder diffraction studies have been conducted by Dityatiev et al. (2006 ▶) and second harmonic generation studies by Libertz & Sadovskaya (1980 ▶). The temperature dependence of the physical properties of STO was analysed by Yamada & Iwasaki (1972 ▶, 1973 ▶), Yamada (1975 ▶) and Kudzin et al. (1988 ▶). For related literature, see: Antonenko et al. (1982 ▶); Avramenko et al. (1984 ▶); Kudzin et al. (1982 ▶); Zavodnik et al. (2007a ▶,b ▶,c ▶).

Experimental

Crystal data

• SrTeO₃

• M _r = 263.22

• Monoclinic,

• a = 28.438 (6) Å

• b = 5.950 (1) Å

• c = 15.550 (3) Å

• β = 122.45 (3)°

• V = 2220.3 (8) ų

• Z = 24

• Mo Kα radiation

• μ = 22.11 mm⁻¹

• T = 780 (2) K

• 0.13 × 0.10 × 0.04 mm

Data collection

• Enraf–Nonius CAD-4 diffractometer

• Absorption correction: analytical (Alcock, 1970 ▶) T _min = 0.169, T _max = 0.475

• 1681 measured reflections

• 1611 independent reflections

• 538 reflections with I > 2σ(I)

• R _int = 0.062

• θ_max = 22.5°

• 3 standard reflections frequency: 60 min intensity decay: none

Refinement

• R[F ² > 2σ(F ²)] = 0.036

• wR(F ²) = 0.095

• S = 0.78

• 1611 reflections

• 118 parameters

• Δρ_max = 1.22 e Å⁻³

• Δρ_min = −1.12 e Å⁻³

Data collection: CAD-4-PC (Enraf–Nonius, 1993 ▶); cell refinement: CAD-4-PC; data reduction: CAD-4-PC; program(s) used to solve structure: SHELXS97 (Sheldrick, 2008 ▶); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008 ▶); molecular graphics: DIAMOND (Brandenburg, 2005 ▶); software used to prepare material for publication: CIFTAB97 (Sheldrick, 2008 ▶) and SHELXL97.

Supplementary Material

Additional supplementary materials: crystallographic information; 3D view; checkCIF report

supplementary crystallographic information

Comment

The origin of unusual ferroelectric properties and several phase transitions between α–β (350 K), β–γ (590 K) and γ–δ (760 K) polymorphs of STO were hotly debated long time (Yamada & Iwasaki, 1973; Yamada, 1975; Simon et al., 1979; Ismailzade et al., 1979; Libertz & Sadovskaya, 1980, Antonenko et al., 1982; Kudzin et al., 1988) but the detailed structure data are lacking. Recently, in our several papers of the present series (Zavodnik et al., 2007a,b,c,) the structures of α (T < 363 K), β (363 K < T < 563 K) and γ (563 K < T <633 K) phases STO were reported. The purpose of the present communication is to report on the structure of δ-phase and clarify the nature of γ–δ ferroelectric phase transition at 633 K. There is a number of experimental studies of dielectric, elastic, piezoelectric and optical properties near well known reversible γ–δ phase transition at 633 K (Ismailzade et al., 1979; Libertz & Sadovskaya, 1980; Kudzin et al., 1982, 1988; Sadovskaya, 1984; Antonenko et al., 1982). All the measured constants exhibit significant changes but the lack of thermal hysteresis or phase coexistence at this transition is indicative of a second order transformation. Around 633 K the SHG signal vanishes indicating that the δ-structure is centrosymmetrical. No success was obtained in earlier attempts to determine the δ phase structure using X-ray and neutron powder diffraction studies (Simon et al., 1979; Ismailzade et al., 1979; Dityatiev et al., 2006). The structure of δ-phase STO forms a three-dimensional lattice consisting types of irregular n-vertex SrOn (n = 6, 7, 8) polyhedra sharing corners or faces and TeO₃ pyramidal units which share edges with Sr-polyhedra but are not connected to each other. The projection along the b axis (Fig. 1) shows two sorts of tunnels running along that direction. Te⁴⁺ cations are located inside the tunnels of different sizes and shapes which represent the required space for the lone-electron pairs within the structure. From a comparison of atomic coordinates of comparable atoms in γ and δ phases the atomic polar displacements required to achieve centrosymmetry were determined. The structures of these polymorphs are similar and the phase transformation can be realised by the orientation and tilts of the TeO₃ pyramids and also by the variation in n-vertex SrO_n polyhedra without a serious changing the building structural blocks. The Te6—O31 bond length is located at distance greater than 2.8 Å and does not contribute to the first coordination sphere of Te⁴⁺. Probably, γ–δ phase transition in STO can be described by displacive mechanism rather than by order–disorder model. The structure–property correlation in STO is in progress and will be reported later.

Experimental

The single crystals of STO were grown by Czochralski technique as described earlier (Libertz & Sadovskaya, 1980; Avramenko et al., 1984). The products were characterized in a scanning electron microscope (Jeol 820) with an energy-dispersive spectrometer (LINK AN10000), confirming the presence and stoichiometry of Sr and Te. SHG measurements showed that there is a symmetry centre in δ-phase (which is stable above 633 K) in a full agreement with the results (Libertz & Sadovskaya, 1980).

Refinement

The atomic coordinates of all Sr and Te cations in γ-phase were used as starting parameters. The O atoms were localized by difference Fourier maps. The selection of space group C2/m for description of crystal structure of δ-phase STO was based on the experimental data of second harmonic generation (SHG) obtained on tested single crystals. The temperature dependence of SHG signal confirms that the structure of δ-phase STO is centrosymmetric. Precise X-ray diffraction study of single crystals at high temperatures is a challenging task because there is usually only a small number of measured X-ray reflections in the data and they cover a rather limited range of sinθ/λ. At 780 K it was impossible to registrate any reflections with sinθ/λ > 0.54. The thermal vibration parameters for oxygen anions were very high and strongly anisotropic. It was difficult to use an anisotropic approximation in this high-temperature refinement because the ratio of statistically reliable reflections to a number of refined parameters was very far from an optimal value. The positive definite refinements with anisotropic atomic displacement parameters were impossible for O atoms at 780 K. It was a main reason why the oxygen atoms were refined isotropically. A special attention must be given to the accuracy of interatomic distances of Te—O which are not rather similar as in the case of α, β and γ-phases (Zavodnik et al., 2007a,b,c). But all these Te—O bond lengths can be found acceptable if we take into account the standard deviation. The highest residual electron density peak is located 0.87 Å from atom Sr1 and the deepest hole is located 0.12 Å from atom Te4. Several atoms (Sr6, O12, O22 and O52) have increased isotropic atomic displacement parameters. These atoms are located inside significant voids which are larger than the voids for the rest of the atoms. The same peculiarity was also observed for the α- β and γ-STO structures.

Figures

Fig. 1.

The crystal structure of δ-SrTeO3 at 780 K. The sequence of Sr polyhedra are presented, Te cations occupy two different kinds of voids in a three-dimensional lattice.

Crystal data

SrTeO3	F000 = 2736
Mr = 263.22	Dx = 4.725 Mg m−3
Monoclinic, C2/m	Mo Kα radiation λ = 0.71073 Å
Hall symbol: -C 2y	Cell parameters from 24 reflections
a = 28.438 (6) Å	θ = 12.3–13.8º
b = 5.950 (1) Å	µ = 22.11 mm−1
c = 15.550 (3) Å	T = 780 (2) K
β = 122.45 (3)º	Triangular prism, colourless
V = 2220.3 (8) Å3	0.13 × 0.10 × 0.04 mm
Z = 24

Data collection

Enraf–Nonius CAD-4 with high-temperature device diffractometer	Rint = 0.062
Radiation source: fine-focus sealed tube	θmax = 22.5º
Monochromator: β-filter	θmin = 1.6º
T = 780(2) K	h = −30→25
ω/2θ scans	k = 0→6
Absorption correction: analytical(Alcock, 1970)	l = 0→16
Tmin = 0.169, Tmax = 0.475	3 standard reflections
1681 measured reflections	every 60 min
1611 independent reflections	intensity decay: none
538 reflections with I > 2σ(I)

Refinement

Refinement on F2	Secondary atom site location: difference Fourier map
Least-squares matrix: full	w = 1/[σ2(Fo2) + (0.0453P)2] where P = (Fo2 + 2Fc2)/3
R[F2 > 2σ(F2)] = 0.036	(Δ/σ)max = 0.001
wR(F2) = 0.095	Δρmax = 1.22 e Å−3
S = 0.78	Δρmin = −1.12 e Å−3
1611 reflections	Extinction correction: SHELXL97 (Sheldrick, 2008), Fc*=kFc[1+0.001xFc2λ3/sin(2θ)]-1/4
118 parameters	Extinction coefficient: 0.00009 (2)
Primary atom site location: isomorphous structure methods

Special details

Geometry. All e.s.d.'s (except the e.s.d. in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell e.s.d.'s are taken into account individually in the estimation of e.s.d.'s in distances, angles and torsion angles; correlations between e.s.d.'s in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell e.s.d.'s is used for estimating e.s.d.'s involving l.s. planes.

Refinement. Refinement of F2 against ALL reflections. The weighted R-factor wR and goodness of fit S are based on F2, conventional R-factors R are based on F, with F set to zero for negative F2. The threshold expression of F2 > σ(F2) is used only for calculating R-factors(gt) etc. and is not relevant to the choice of reflections for refinement. R-factors based on F2 are statistically about twice as large as those based on F, and R- factors based on ALL data will be even larger.

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

	x	y	z	Uiso*/Ueq
Te1	−0.01963 (9)	0.5000	0.1468 (2)	0.0530 (8)
Te2	0.49464 (8)	0.5000	0.65585 (18)	0.0423 (7)
Te3	0.11483 (9)	0.0000	0.2806 (2)	0.0396 (6)
Te4	0.35697 (9)	0.0000	0.2329 (2)	0.0566 (8)
Te5	0.14971 (9)	1.0000	−0.00115 (19)	0.0548 (8)
Te6	0.26235 (9)	0.5000	0.41676 (19)	0.0345 (7)
Sr1	0.12254 (12)	0.5000	0.4219 (3)	0.0458 (10)
Sr2	0.24767 (12)	0.5000	0.1092 (3)	0.0437 (10)
Sr3	0.24223 (13)	0.0000	0.2740 (3)	0.0507 (11)
Sr4	0.37724 (11)	0.5000	0.3964 (3)	0.0479 (11)
Sr5	0.12358 (13)	0.5000	0.1520 (3)	0.0501 (11)
Sr6	0.0000	0.0000	0.0000	0.070 (2)
Sr7	0.5000	1.0000	0.5000	0.0588 (18)
O11	0.0554 (10)	0.5000	0.223 (2)	0.097 (9)*
O12	−0.0384 (15)	0.317 (7)	0.050 (3)	0.277 (19)*
O21	0.4492 (5)	0.724 (3)	0.5733 (11)	0.071 (5)*
O22	0.5456 (15)	0.5000	0.629 (3)	0.158 (14)*
O31	0.1641 (7)	0.231 (4)	0.3268 (14)	0.100 (6)*
O32	0.0981 (11)	0.0000	0.148 (2)	0.100 (9)*
O41	0.3148 (5)	−0.225 (3)	0.2374 (10)	0.071 (5)*
O42	0.4029 (12)	0.0000	0.373 (2)	0.112 (9)*
O51	0.1756 (6)	0.761 (3)	0.0923 (11)	0.078 (5)*
O52	0.2055 (17)	1.0000	−0.017 (4)	0.197 (18)*
O61	0.2317 (7)	0.5000	0.2770 (15)	0.056 (6)*
O62	0.3118 (5)	0.737 (3)	0.4416 (11)	0.066 (5)*

Atomic displacement parameters (Ų)

	U11	U22	U33	U12	U13	U23
Te1	0.0488 (12)	0.0602 (19)	0.0407 (15)	0.000	0.0179 (12)	0.000
Te2	0.0397 (13)	0.0482 (17)	0.0380 (14)	0.000	0.0202 (11)	0.000
Te3	0.0456 (12)	0.0372 (15)	0.0406 (14)	0.000	0.0262 (11)	0.000
Te4	0.0529 (14)	0.0416 (18)	0.084 (2)	0.000	0.0420 (14)	0.000
Te5	0.0457 (13)	0.0585 (19)	0.0394 (14)	0.000	0.0090 (12)	0.000
Te6	0.0346 (11)	0.0316 (15)	0.0364 (14)	0.000	0.0184 (10)	0.000
Sr1	0.0464 (18)	0.030 (2)	0.045 (2)	0.000	0.0134 (16)	0.000
Sr2	0.0412 (18)	0.037 (2)	0.044 (2)	0.000	0.0167 (17)	0.000
Sr3	0.057 (2)	0.047 (3)	0.039 (2)	0.000	0.0199 (18)	0.000
Sr4	0.0411 (18)	0.049 (3)	0.051 (2)	0.000	0.0233 (18)	0.000
Sr5	0.0465 (18)	0.055 (3)	0.046 (2)	0.000	0.0234 (17)	0.000
Sr6	0.052 (3)	0.071 (5)	0.054 (4)	0.000	0.007 (3)	0.000
Sr7	0.032 (2)	0.079 (5)	0.056 (4)	0.000	0.018 (2)	0.000

Geometric parameters (Å, °)

Te1—O12	1.70 (5)	Sr3—O62iv	2.759 (16)
Te1—O11	1.80 (2)	Sr3—O41	2.763 (16)
Te2—O22	1.71 (4)	Sr3—O51iv	2.801 (16)
Te2—O21	1.820 (16)	Sr3—O61	2.993 (2)
Te3—O31	1.82 (2)	Sr3—O31	3.08 (2)
Te3—O32	1.85 (3)	Sr4—O22v	2.43 (4)
Te4—O41	1.822 (17)	Sr4—O41iii	2.690 (16)
Te4—O42	1.84 (3)	Sr4—O62	2.712 (15)
Te5—O52	1.73 (4)	Sr4—O21	2.736 (15)
Te5—O51	1.879 (18)	Sr4—O42	3.132 (10)
Te6—O61	1.86 (2)	Sr5—O61	2.612 (18)
Te6—O62	1.880 (16)	Sr5—O51	2.635 (18)
Sr1—O62i	2.487 (15)	Sr5—O11	2.69 (3)
Sr1—O11	2.62 (3)	Sr5—O31	2.811 (19)
Sr1—O21i	2.651 (17)	Sr5—O12vi	2.95 (4)
Sr1—O31	2.83 (2)	Sr5—O32	3.054 (6)
Sr2—O52ii	2.42 (5)	Sr6—O32	2.49 (3)
Sr2—O51	2.469 (17)	Sr6—O12	2.50 (5)
Sr2—O41iii	2.493 (16)	Sr7—O42vii	2.38 (3)
Sr2—O61	2.88 (2)	Sr7—O21	2.802 (17)
O12—Te1—O12viii	79 (3)	O41iii—Sr4—O62	110.4 (5)
O12—Te1—O11	106.2 (14)	O41vii—Sr4—O62	73.4 (5)
O22—Te2—O21	101.7 (10)	O62viii—Sr4—O62	62.7 (8)
O21viii—Te2—O21	94.1 (10)	O22v—Sr4—O21	84.9 (8)
O31—Te3—O31iii	98.7 (12)	O41iii—Sr4—O21	171.5 (5)
O31—Te3—O32	97.1 (8)	O41vii—Sr4—O21	113.3 (6)
O41—Te4—O41iii	94.4 (11)	O62viii—Sr4—O21	103.9 (5)
O41—Te4—O42	91.1 (8)	O62—Sr4—O21	74.5 (4)
O52—Te5—O51	95.8 (11)	O22v—Sr4—O21viii	84.9 (8)
O51ix—Te5—O51	98.4 (10)	O41iii—Sr4—O21viii	113.3 (6)
O61—Te6—O62	94.0 (6)	O41vii—Sr4—O21viii	171.5 (5)
O62—Te6—O62viii	97.3 (9)	O62viii—Sr4—O21viii	74.5 (4)
O62x—Sr1—O62i	77.9 (8)	O62—Sr4—O21viii	103.9 (5)
O62i—Sr1—O11	138.1 (5)	O21—Sr4—O21viii	58.3 (7)
O62x—Sr1—O21i	127.0 (5)	O22v—Sr4—O42vii	72.1 (5)
O11—Sr1—O21i	87.1 (6)	O41iii—Sr4—O42vii	123.4 (7)
O62x—Sr1—O21x	79.8 (5)	O41vii—Sr4—O42vii	52.7 (6)
O21i—Sr1—O21x	76.6 (7)	O62viii—Sr4—O42vii	139.5 (6)
O62x—Sr1—O31viii	76.0 (5)	O62—Sr4—O42vii	76.8 (7)
O62i—Sr1—O31viii	117.9 (5)	O21—Sr4—O42vii	63.9 (7)
O11—Sr1—O31viii	68.1 (6)	O21viii—Sr4—O42vii	119.0 (7)
O21i—Sr1—O31viii	155.2 (5)	O22v—Sr4—O42	72.1 (5)
O21x—Sr1—O31viii	102.0 (5)	O41iii—Sr4—O42	52.7 (6)
O62x—Sr1—O31	117.9 (5)	O41vii—Sr4—O42	123.4 (7)
O62i—Sr1—O31	76.0 (5)	O62viii—Sr4—O42	76.8 (7)
O11—Sr1—O31	68.1 (6)	O62—Sr4—O42	139.5 (6)
O21i—Sr1—O31	102.0 (5)	O21—Sr4—O42	119.0 (7)
O21x—Sr1—O31	155.2 (5)	O21viii—Sr4—O42	63.9 (7)
O31viii—Sr1—O31	68.7 (8)	O42vii—Sr4—O42	143.6 (11)
O52ii—Sr2—O51	128.6 (7)	O61—Sr5—O51	66.6 (5)
O51viii—Sr2—O51	77.9 (8)	O51—Sr5—O51viii	72.2 (8)
O52ii—Sr2—O41vii	92.5 (8)	O61—Sr5—O11	120.9 (7)
O51viii—Sr2—O41vii	137.2 (5)	O51—Sr5—O11	143.9 (4)
O51—Sr2—O41vii	84.7 (6)	O61—Sr5—O31viii	64.7 (5)
O41vii—Sr2—O41iii	82.1 (8)	O51—Sr5—O31viii	89.4 (6)
O52ii—Sr2—O61	160.0 (10)	O51viii—Sr5—O31viii	131.2 (5)
O51—Sr2—O61	64.6 (5)	O11—Sr5—O31viii	67.5 (6)
O41vii—Sr2—O61	72.6 (4)	O61—Sr5—O31	64.7 (5)
O62viii—Sr3—O62iv	69.0 (7)	O51—Sr5—O31	131.2 (5)
O62viii—Sr3—O41iii	71.6 (4)	O51viii—Sr5—O31	89.4 (6)
O62iv—Sr3—O41iii	103.4 (4)	O11—Sr5—O31	67.5 (6)
O41iii—Sr3—O41	57.9 (7)	O31viii—Sr5—O31	69.3 (9)
O62viii—Sr3—O51iv	173.7 (5)	O61—Sr5—O12vi	139.2 (9)
O62iv—Sr3—O51iv	114.8 (6)	O51—Sr5—O12vi	98.0 (10)
O41iii—Sr3—O51iv	102.3 (5)	O51viii—Sr5—O12vi	72.8 (9)
O41—Sr3—O51iv	73.9 (5)	O11—Sr5—O12vi	94.2 (9)
O62viii—Sr3—O51viii	114.8 (6)	O31viii—Sr5—O12vi	155.8 (9)
O62iv—Sr3—O51viii	173.7 (5)	O31—Sr5—O12vi	119.6 (10)
O41iii—Sr3—O51viii	73.9 (5)	O61—Sr5—O12xi	139.2 (9)
O41—Sr3—O51viii	102.3 (5)	O51—Sr5—O12xi	72.8 (9)
O51iv—Sr3—O51viii	61.0 (8)	O51viii—Sr5—O12xi	98.0 (10)
O62viii—Sr3—O61iv	125.1 (5)	O11—Sr5—O12xi	94.2 (9)
O62iv—Sr3—O61iv	56.6 (5)	O31viii—Sr5—O12xi	119.6 (10)
O41iii—Sr3—O61iv	125.2 (6)	O31—Sr5—O12xi	155.8 (9)
O41—Sr3—O61iv	67.3 (5)	O12vi—Sr5—O12xi	43.2 (17)
O51iv—Sr3—O61iv	59.5 (5)	O61—Sr5—O32	100.8 (5)
O51viii—Sr3—O61iv	120.1 (5)	O51—Sr5—O32	137.6 (7)
O62viii—Sr3—O61	56.6 (5)	O51viii—Sr5—O32	65.8 (6)
O62iv—Sr3—O61	125.1 (5)	O11—Sr5—O32	78.1 (5)
O41iii—Sr3—O61	67.3 (5)	O31viii—Sr5—O32	122.9 (7)
O41—Sr3—O61	125.2 (6)	O31—Sr5—O32	55.7 (7)
O51iv—Sr3—O61	120.1 (5)	O12vi—Sr5—O32	64.5 (10)
O51viii—Sr3—O61	59.5 (5)	O12xi—Sr5—O32	106.6 (11)
O61iv—Sr3—O61	167.5 (8)	O61—Sr5—O32vii	100.8 (5)
O62viii—Sr3—O31	75.4 (4)	O51—Sr5—O32vii	65.8 (6)
O62iv—Sr3—O31	104.8 (5)	O51viii—Sr5—O32vii	137.6 (7)
O41iii—Sr3—O31	124.4 (6)	O11—Sr5—O32vii	78.1 (5)
O41—Sr3—O31	176.4 (5)	O31viii—Sr5—O32vii	55.7 (7)
O51iv—Sr3—O31	107.6 (5)	O31—Sr5—O32vii	122.9 (7)
O51viii—Sr3—O31	81.2 (5)	O12vi—Sr5—O32vii	106.6 (10)
O61iv—Sr3—O31	110.4 (6)	O12xi—Sr5—O32vii	64.5 (10)
O61—Sr3—O31	57.2 (5)	O32—Sr5—O32vii	153.8 (10)
O62viii—Sr3—O31iii	104.8 (5)	O32—Sr6—O32xii	180.0 (11)
O62iv—Sr3—O31iii	75.4 (4)	O32—Sr6—O12xii	80.0 (9)
O41iii—Sr3—O31iii	176.4 (5)	O32—Sr6—O12iii	100.0 (9)
O41—Sr3—O31iii	124.4 (6)	O32xii—Sr6—O12iii	80.0 (9)
O51iv—Sr3—O31iii	81.2 (5)	O12xii—Sr6—O12iii	82 (2)
O51viii—Sr3—O31iii	107.6 (5)	O12xii—Sr6—O12vi	98 (2)
O61iv—Sr3—O31iii	57.2 (5)	O12iii—Sr6—O12vi	180 (2)
O61—Sr3—O31iii	110.4 (6)	O32—Sr6—O12	100.0 (9)
O31—Sr3—O31iii	53.2 (8)	O42v—Sr7—O42vii	180.0 (11)
O22v—Sr4—O41iii	93.3 (8)	O42v—Sr7—O21xiii	73.6 (7)
O41iii—Sr4—O41vii	75.0 (7)	O42vii—Sr7—O21xiii	106.4 (7)
O22v—Sr4—O62viii	148.0 (4)	O21xiii—Sr7—O21xiv	71.8 (7)
O41iii—Sr4—O62viii	73.4 (5)	O21xiv—Sr7—O21	108.2 (7)
O41vii—Sr4—O62viii	110.4 (5)	O21xiv—Sr7—O21ix	180.0 (5)
O22v—Sr4—O62	148.0 (4)

Symmetry codes: (i) −x+1/2, y−1/2, −z+1; (ii) −x+1/2, −y+3/2, −z; (iii) x, −y, z; (iv) x, y−1, z; (v) −x+1, −y+1, −z+1; (vi) −x, y, −z; (vii) x, y+1, z; (viii) x, −y+1, z; (ix) x, −y+2, z; (x) −x+1/2, −y+3/2, −z+1; (xi) −x, −y+1, −z; (xii) −x, −y, −z; (xiii) −x+1, −y+2, −z+1; (xiv) −x+1, y, −z+1.