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Acta Cryst. (2009). E65, i88    [ doi:10.1107/S1600536809046431 ]

Octarubidium di-[mu]-sulfato-[kappa]4O:O'-bis[cis-dioxido-cis-disulfatotungstate(VI)]

K. Ståhl and R. W. Berg

Abstract top

The title compound, Rb8[W2O4(SO4)6], was precipitated from a melt of tungsten(VI) oxide and rubidium sulfate in rubidium disulfate. The unit cell contains two discrete [{WVIO2(SO4)2}2([mu]-SO4)2]8- units connected by Rb-O coordination. The W atom is octahedrally surrounded by two oxide ligands, two terminal sulfate ligands and two bridging sulfate groups. One Rb atom is coordinated by eight O atoms, whereas the three other Rb atoms are coordinated by nine O atoms from sulfate and oxide groups, leading to distorted [RbOx] polyhedra.

Comment top

Tungsten trioxide is known as a highly inert solid, practically insoluble in acids. It was discovered that WO3 can be dissolved in considerable amounts in acidic sulfate melts at high temperatures (Schäffer and Berg, 2005 and Berg et al., 2006). When therefore WO3, Rb2SO4 and Rb2S2O7 are mixed in varying molar amounts in sealed ampoules and heated for equilibration in a rocking furnace at appr. 600 °C for appr. 1 hr, clear melts are formed (Rb2S2O7 is hygroscopic and needs to be handled in a dry box). It has been shown that [WO2]2+ ions formed are solvated by SO42- ions. The stoichiometry of the reaction have been determined fairly accurately as 1:1:1, or 2WO3 + 2M2SO4 + 2M2S2O7 → M8[(WO2)2(µ-SO4)2(SO4)4] for the case of M = K, Schäffer and Berg, 2005. Here we have studied the case of M = Rb and have found closely analogous results for single crystals of the compound, 1WO3:1Rb2S2O7:1Rb2SO4. The main difference for M = Rb as compared to M = K is as expected in the M - O coordination: Rb1/K1, CN = 8/8, <M - O> = 2.999 (1)/2.822 (1); Rb2/K2, CN = 9/7, <M - O> = 3.042 (1)/2.825 (1); Rb3/K3, CN = 9/8, <M - O> = 3.029 (1)/2.909 (1); Rb4/K4, CN = 9/9, <M - O> = 3.079 (1)/2.945 (1). As the oxo-sulfatotungstate groups are extended in the b-direction, the major changes in the unit cell axes are observered in the a- and c- directions.

Related literature top

For methods used in the synthesis, see: Berg et al. (2006); Borup et al. (1990); Nørbygaard et al. (1998). For the crystal structure of the potassium analog, see: Schäffer & Berg (2005).

Experimental top

The crystals were grown from a melt of 20.6 mol % tungsten trioxide and 19.9 mol % rubidium sulfate in rubidium disulfate, using the methods described by Borup et al. (1990), Nørbygaard et al. (1998) and Berg et al. (2006).

Computing details top

Data collection: SMART (Bruker, 2002); cell refinement: SAINT-Plus (Bruker, 2002); data reduction: SAINT-Plus (Bruker, 2002) and SADABS (Sheldrick, 2002); program(s) used to solve structure: SHELXTL (Sheldrick, 2008); program(s) used to refine structure: SHELXTL (Sheldrick, 2008); molecular graphics: ORTEP-3 (Farrugia, 1997) and ATOMS (Dowty, 2000); software used to prepare material for publication: SHELXTL (Sheldrick, 2008).

Figures top
[Figure 1] Fig. 1. The asymmetric unit of (I) showing 50% probability displacement ellipsoids and the atomic numbering. O6iv was added to complete the S1 coordination.
[Figure 2] Fig. 2. The crystal packing of (I) viewed down the a axis. W atoms are in the centers of the octahedra and S atoms are in the centers of the tetrahedra. Rb are shown as large and O as small spheres.
Octarubidium di-µ-sulfato-κ4O:O'-bis[cis-dioxido-cis- disulfatotungstate(VI)] top
Crystal data top
Rb8[W2O4(SO4)6]F(000) = 1528
Mr = 1691.86Dx = 3.869 Mg m−3
Monoclinic, P21/nMo Kα radiation, λ = 0.71073 Å
Hall symbol: -P 2ynCell parameters from 6979 reflections
a = 9.6405 (5) Åθ = 2.4–28.0°
b = 13.9890 (7) ŵ = 21.77 mm−1
c = 10.7692 (5) ÅT = 120 K
β = 90.4720 (10)°Irregular, colourless
V = 1452.30 (12) Å30.45 × 0.20 × 0.05 mm
Z = 2
Data collection top
Bruker SMART APEX
diffractometer		3496 independent reflections
Radiation source: fine-focus sealed tube3360 reflections with I > 2σ(I)
graphiteRint = 0.036
ω scan, frame data integrationθmax = 28.0°, θmin = 2.4°
Absorption correction: multi-scan
(SADABS; Sheldrick, 2002)		h = −12→12
Tmin = 0.077, Tmax = 0.59k = −18→18
18895 measured reflectionsl = −14→14
Refinement top
Refinement on F2Primary atom site location: structure-invariant direct methods
Least-squares matrix: fullSecondary atom site location: difference Fourier map
R[F2 > 2σ(F2)] = 0.020 w = 1/[σ2(Fo2) + (0.0287P)2 + 1.1918P]
where P = (Fo2 + 2Fc2)/3
wR(F2) = 0.052(Δ/σ)max = 0.001
S = 1.11Δρmax = 1.35 e Å−3
3496 reflectionsΔρmin = −1.51 e Å−3
200 parametersExtinction correction: SHELXTL (Sheldrick, 2008), Fc*=kFc[1+0.001xFc2λ3/sin(2θ)]-1/4
0 restraintsExtinction coefficient: 0.00175 (10)
Crystal data top
Rb8[W2O4(SO4)6]V = 1452.30 (12) Å3
Mr = 1691.86Z = 2
Monoclinic, P21/nMo Kα radiation
a = 9.6405 (5) ŵ = 21.77 mm−1
b = 13.9890 (7) ÅT = 120 K
c = 10.7692 (5) Å0.45 × 0.20 × 0.05 mm
β = 90.4720 (10)°
Data collection top
Bruker SMART APEX
diffractometer		3496 independent reflections
Absorption correction: multi-scan
(SADABS; Sheldrick, 2002)		3360 reflections with I > 2σ(I)
Tmin = 0.077, Tmax = 0.59Rint = 0.036
18895 measured reflectionsθmax = 28.0°
Refinement top
R[F2 > 2σ(F2)] = 0.020200 parameters
wR(F2) = 0.0520 restraints
S = 1.11Δρmax = 1.35 e Å−3
3496 reflectionsΔρmin = −1.51 e Å−3
Special details top

Experimental. Oxford Cryosystem liquid nitrogen cryostream cooler

Geometry. All e.s.d.'s 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.

Refinement. A series of identical frames was collected twice during the experiment to monitor decay. No decay was detected and decay correction was not applied. Systematic conditions suggested the unambiguous space group. The structure was solved by direct methods (Sheldrick, 2008). The space group choice was confirmed by successful convergence of the full-matrix least-squares refinement on F2 (Sheldrick, 2008). 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 > 2σ(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. The highest peak in the final difference Fourier map was 0.85Å and the deepest hole was 0.78Å from W1. The final map had no other significant features. A final analysis of variance between observed and calculated structure factors showed no dependence on amplitude or resolution.

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzUiso*/Ueq
W10.168190 (12)0.117132 (8)0.917586 (11)0.00604 (6)
Rb10.31202 (3)0.30953 (2)0.65221 (3)0.01095 (8)
Rb20.50882 (3)0.01145 (2)0.79641 (3)0.01149 (8)
Rb30.19055 (3)0.01388 (2)0.51347 (3)0.01057 (8)
Rb4−0.23085 (3)0.22589 (2)0.74755 (3)0.01173 (8)
S1−0.07712 (8)−0.01972 (5)0.80402 (7)0.00671 (15)
S20.46037 (8)0.22894 (6)0.97953 (7)0.00823 (15)
S30.03580 (8)0.34141 (5)0.94629 (7)0.00726 (15)
O10.2116 (3)0.13695 (17)0.7656 (2)0.0126 (5)
O20.2550 (3)0.01169 (17)0.9456 (2)0.0134 (5)
O30.3015 (2)0.20560 (17)0.9939 (2)0.0105 (5)
O4−0.0203 (2)0.06004 (16)0.8874 (2)0.0095 (4)
O50.0314 (2)0.23235 (16)0.9395 (2)0.0113 (5)
O60.0953 (2)0.10525 (15)1.1153 (2)0.0086 (4)
O70.1394 (3)0.37617 (16)0.8585 (2)0.0123 (5)
O80.4896 (3)0.22782 (18)0.8473 (2)0.0146 (5)
O90.5342 (3)0.1533 (2)1.0446 (3)0.0221 (6)
O10−0.1042 (3)0.37142 (17)0.9097 (2)0.0143 (5)
O110.0230 (3)−0.04158 (17)0.7090 (2)0.0118 (5)
O120.4761 (3)0.32204 (18)1.0374 (2)0.0160 (5)
O130.0695 (3)0.36869 (18)1.0740 (2)0.0168 (5)
O14−0.2088 (3)0.01462 (17)0.7559 (2)0.0121 (5)
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
W10.00608 (8)0.00633 (8)0.00575 (8)−0.00051 (4)0.00225 (5)−0.00028 (4)
Rb10.01021 (15)0.01196 (15)0.01072 (15)−0.00113 (11)0.00177 (11)0.00109 (11)
Rb20.01089 (16)0.01125 (15)0.01242 (16)−0.00146 (11)0.00470 (11)−0.00220 (11)
Rb30.01010 (15)0.01395 (15)0.00769 (15)−0.00045 (11)0.00251 (11)0.00227 (10)
Rb40.01259 (16)0.01381 (15)0.00883 (15)−0.00152 (12)0.00221 (11)−0.00042 (11)
S10.0069 (3)0.0080 (3)0.0052 (3)−0.0015 (3)0.0008 (3)0.0002 (3)
S20.0065 (3)0.0102 (4)0.0080 (4)−0.0011 (3)0.0004 (3)0.0005 (3)
S30.0086 (4)0.0070 (3)0.0062 (3)0.0008 (3)0.0016 (3)−0.0007 (3)
O10.0157 (12)0.0143 (11)0.0077 (11)−0.0046 (10)0.0025 (9)−0.0001 (9)
O20.0144 (12)0.0098 (11)0.0161 (12)0.0047 (9)0.0057 (9)0.0011 (9)
O30.0071 (11)0.0141 (11)0.0104 (11)−0.0017 (9)0.0015 (8)−0.0033 (9)
O40.0099 (11)0.0093 (10)0.0094 (11)−0.0024 (9)0.0007 (8)−0.0019 (8)
O50.0088 (11)0.0081 (10)0.0170 (12)0.0002 (9)0.0006 (9)0.0009 (9)
O60.0110 (11)0.0068 (10)0.0080 (11)−0.0025 (8)0.0025 (9)0.0006 (8)
O70.0123 (12)0.0127 (12)0.0120 (12)−0.0024 (9)0.0043 (9)0.0030 (8)
O80.0156 (12)0.0178 (12)0.0105 (12)−0.0028 (10)0.0057 (9)−0.0012 (9)
O90.0171 (13)0.0202 (14)0.0288 (15)0.0015 (11)−0.0074 (11)0.0102 (11)
O100.0099 (12)0.0123 (11)0.0207 (14)0.0039 (9)0.0018 (10)−0.0009 (9)
O110.0142 (12)0.0130 (11)0.0084 (11)−0.0017 (9)0.0049 (8)−0.0008 (9)
O120.0153 (12)0.0157 (12)0.0170 (13)−0.0054 (10)0.0020 (10)−0.0047 (10)
O130.0250 (15)0.0173 (12)0.0079 (12)0.0004 (11)−0.0014 (10)−0.0039 (9)
O140.0110 (12)0.0151 (12)0.0102 (11)0.0011 (9)−0.0017 (9)0.0014 (9)
Geometric parameters (Å, °) top
W1—O11.716 (2)Rb3—O12i3.101 (2)
W1—O21.721 (2)Rb3—O11vii3.170 (2)
W1—O31.960 (2)Rb3—O12v3.173 (3)
W1—O42.009 (2)Rb3—O13.220 (2)
W1—O52.097 (2)Rb3—O13v3.225 (3)
W1—O62.254 (2)Rb4—O8viii2.910 (3)
Rb1—O12.877 (2)Rb4—O3i2.914 (2)
Rb1—O82.931 (2)Rb4—O102.941 (3)
Rb1—O72.939 (3)Rb4—O142.964 (2)
Rb1—O9i2.955 (3)Rb4—O13i2.981 (3)
Rb1—O6ii3.009 (2)Rb4—O12i3.103 (3)
Rb1—O11iii3.010 (2)Rb4—O6i3.221 (2)
Rb1—O2iii3.084 (2)Rb4—O53.253 (2)
Rb1—O5ii3.185 (2)Rb4—O43.423 (2)
Rb2—O14iv2.761 (3)S1—O111.445 (2)
Rb2—O7v2.893 (2)S1—O141.449 (2)
Rb2—O9vi2.903 (3)S1—O6ix1.490 (2)
Rb2—O22.939 (3)S1—O41.531 (2)
Rb2—O13ii2.986 (2)S2—O121.451 (3)
Rb2—O83.082 (3)S2—O91.452 (3)
Rb2—O10v3.106 (3)S2—O81.454 (3)
Rb2—O93.337 (3)S2—O31.575 (2)
Rb2—O13.374 (3)S3—O131.461 (2)
Rb3—O112.776 (2)S3—O71.464 (3)
Rb3—O10ii2.788 (3)S3—O101.464 (3)
Rb3—O7v2.874 (2)S3—O51.528 (2)
Rb3—O14vii2.935 (2)
O1—W1—O2100.59 (12)O7v—Rb3—O12i146.54 (7)
O1—W1—O397.67 (11)O14vii—Rb3—O12i103.19 (7)
O2—W1—O398.71 (11)O11—Rb3—O11vii103.26 (6)
O1—W1—O497.90 (10)O10ii—Rb3—O11vii94.90 (7)
O2—W1—O497.20 (11)O7v—Rb3—O11vii143.97 (6)
O3—W1—O4155.24 (9)O14vii—Rb3—O11vii46.63 (6)
O1—W1—O598.15 (11)O12i—Rb3—O11vii62.82 (6)
O2—W1—O5160.75 (11)O11—Rb3—O12v66.26 (7)
O3—W1—O583.02 (9)O10ii—Rb3—O12v141.04 (7)
O4—W1—O575.81 (9)O7v—Rb3—O12v78.58 (6)
O1—W1—O6173.62 (10)O14vii—Rb3—O12v75.47 (7)
O2—W1—O685.75 (10)O12i—Rb3—O12v107.66 (6)
O3—W1—O681.90 (9)O11vii—Rb3—O12v69.40 (6)
O4—W1—O680.50 (9)O11—Rb3—O162.79 (7)
O5—W1—O675.48 (9)O10ii—Rb3—O189.42 (7)
O1—Rb1—O864.22 (7)O7v—Rb3—O185.54 (6)
O1—Rb1—O775.57 (7)O14vii—Rb3—O1154.44 (7)
O8—Rb1—O784.99 (7)O12i—Rb3—O164.64 (6)
O1—Rb1—O9i90.40 (8)O11vii—Rb3—O1127.46 (6)
O8—Rb1—O9i150.64 (8)O12v—Rb3—O1128.76 (6)
O7—Rb1—O9i73.96 (7)O11—Rb3—O13v117.97 (7)
O1—Rb1—O6ii134.22 (7)O10ii—Rb3—O13v74.48 (7)
O8—Rb1—O6ii73.97 (7)O7v—Rb3—O13v46.39 (7)
O7—Rb1—O6ii119.62 (6)O14vii—Rb3—O13v64.96 (6)
O9i—Rb1—O6ii134.35 (7)O12i—Rb3—O13v166.10 (7)
O1—Rb1—O11iii123.25 (6)O11vii—Rb3—O13v108.61 (6)
O8—Rb1—O11iii67.10 (7)O12v—Rb3—O13v77.31 (7)
O7—Rb1—O11iii72.87 (7)O1—Rb3—O13v122.86 (6)
O9i—Rb1—O11iii123.13 (7)O8viii—Rb4—O3i116.73 (7)
O6ii—Rb1—O11iii46.76 (6)O8viii—Rb4—O1098.95 (7)
O1—Rb1—O2iii147.59 (7)O3i—Rb4—O10106.38 (7)
O8—Rb1—O2iii136.20 (7)O8viii—Rb4—O1493.69 (7)
O7—Rb1—O2iii81.33 (6)O3i—Rb4—O14110.37 (7)
O9i—Rb1—O2iii61.21 (7)O10—Rb4—O14130.00 (7)
O6ii—Rb1—O2iii77.25 (6)O8viii—Rb4—O13i68.94 (7)
O11iii—Rb1—O2iii69.12 (6)O3i—Rb4—O13i68.39 (7)
O1—Rb1—O5ii112.24 (7)O10—Rb4—O13i160.41 (7)
O8—Rb1—O5ii93.25 (6)O14—Rb4—O13i67.81 (7)
O7—Rb1—O5ii170.30 (6)O8viii—Rb4—O12i150.99 (7)
O9i—Rb1—O5ii110.80 (7)O3i—Rb4—O12i46.59 (6)
O6ii—Rb1—O5ii50.90 (6)O10—Rb4—O12i108.45 (7)
O11iii—Rb1—O5ii97.65 (6)O14—Rb4—O12i76.14 (7)
O2iii—Rb1—O5ii93.51 (6)O13i—Rb4—O12i82.09 (7)
O14iv—Rb2—O7v113.73 (7)O8viii—Rb4—O6i71.11 (7)
O14iv—Rb2—O9vi104.59 (7)O3i—Rb4—O6i53.40 (6)
O7v—Rb2—O9vi75.43 (7)O10—Rb4—O6i88.10 (6)
O14iv—Rb2—O2155.92 (7)O14—Rb4—O6i141.38 (6)
O7v—Rb2—O284.64 (7)O13i—Rb4—O6i73.59 (6)
O9vi—Rb2—O263.56 (7)O12i—Rb4—O6i99.87 (6)
O14iv—Rb2—O13ii70.38 (7)O8viii—Rb4—O5118.84 (7)
O7v—Rb2—O13ii90.22 (7)O3i—Rb4—O5119.85 (6)
O9vi—Rb2—O13ii161.61 (8)O10—Rb4—O544.51 (6)
O2—Rb2—O13ii127.42 (7)O14—Rb4—O587.33 (6)
O14iv—Rb2—O894.19 (7)O13i—Rb4—O5154.80 (6)
O7v—Rb2—O8135.64 (7)O12i—Rb4—O588.19 (6)
O9vi—Rb2—O8131.74 (8)O6i—Rb4—O5131.24 (6)
O2—Rb2—O881.38 (7)O8viii—Rb4—O4113.04 (6)
O13ii—Rb2—O866.65 (7)O3i—Rb4—O4124.58 (6)
O14iv—Rb2—O10v66.37 (7)O10—Rb4—O488.06 (6)
O7v—Rb2—O10v47.37 (7)O14—Rb4—O443.02 (6)
O9vi—Rb2—O10v88.01 (8)O13i—Rb4—O4110.63 (6)
O2—Rb2—O10v130.12 (7)O12i—Rb4—O477.99 (6)
O13ii—Rb2—O10v73.72 (7)O6i—Rb4—O4174.76 (6)
O8—Rb2—O10v139.93 (7)O5—Rb4—O444.32 (5)
O14iv—Rb2—O992.92 (7)O11—S1—O14113.92 (14)
O7v—Rb2—O9152.12 (7)O11—S1—O6ix108.97 (14)
O9vi—Rb2—O990.56 (7)O14—S1—O6ix111.57 (14)
O2—Rb2—O967.50 (7)O11—S1—O4109.38 (14)
O13ii—Rb2—O9107.16 (7)O14—S1—O4106.03 (14)
O8—Rb2—O943.73 (6)O6ix—S1—O4106.66 (13)
O10v—Rb2—O9158.07 (7)O12—S2—O9113.40 (16)
O14iv—Rb2—O1144.45 (6)O12—S2—O8114.19 (15)
O7v—Rb2—O182.45 (6)O9—S2—O8111.50 (16)
O9vi—Rb2—O1110.23 (7)O12—S2—O3104.03 (14)
O2—Rb2—O148.93 (6)O9—S2—O3105.97 (15)
O13ii—Rb2—O178.50 (7)O8—S2—O3106.88 (13)
O8—Rb2—O156.93 (6)O13—S3—O7111.90 (16)
O10v—Rb2—O1120.93 (6)O13—S3—O10112.06 (16)
O9—Rb2—O179.98 (6)O7—S3—O10111.32 (15)
O11—Rb3—O10ii152.16 (7)O13—S3—O5108.18 (14)
O11—Rb3—O7v77.41 (7)O7—S3—O5108.64 (14)
O10ii—Rb3—O7v100.02 (7)O10—S3—O5104.36 (14)
O11—Rb3—O14vii138.78 (7)S2—O3—W1136.45 (14)
O10ii—Rb3—O14vii68.51 (7)S1—O4—W1134.71 (14)
O7v—Rb3—O14vii110.14 (7)S3—O5—W1139.07 (14)
O11—Rb3—O12i75.60 (7)S1ix—O6—W1130.47 (13)
O10ii—Rb3—O12i94.76 (7)
Symmetry codes: (i) x−1/2, −y+1/2, z−1/2; (ii) x+1/2, −y+1/2, z−1/2; (iii) −x+1/2, y+1/2, −z+3/2; (iv) x+1, y, z; (v) −x+1/2, y−1/2, −z+3/2; (vi) −x+1, −y, −z+2; (vii) −x, −y, −z+1; (viii) x−1, y, z; (ix) −x, −y, −z+2.
Table 1
Selected geometric parameters (Å) top
W1—O11.716 (2)Rb3—O12i3.101 (2)
W1—O21.721 (2)Rb3—O11vii3.170 (2)
W1—O31.960 (2)Rb3—O12v3.173 (3)
W1—O42.009 (2)Rb3—O13.220 (2)
W1—O52.097 (2)Rb3—O13v3.225 (3)
W1—O62.254 (2)Rb4—O8viii2.910 (3)
Rb1—O12.877 (2)Rb4—O3i2.914 (2)
Rb1—O82.931 (2)Rb4—O102.941 (3)
Rb1—O72.939 (3)Rb4—O142.964 (2)
Rb1—O9i2.955 (3)Rb4—O13i2.981 (3)
Rb1—O6ii3.009 (2)Rb4—O12i3.103 (3)
Rb1—O11iii3.010 (2)Rb4—O6i3.221 (2)
Rb1—O2iii3.084 (2)Rb4—O53.253 (2)
Rb1—O5ii3.185 (2)Rb4—O43.423 (2)
Rb2—O14iv2.761 (3)S1—O111.445 (2)
Rb2—O7v2.893 (2)S1—O141.449 (2)
Rb2—O9vi2.903 (3)S1—O6ix1.490 (2)
Rb2—O22.939 (3)S1—O41.531 (2)
Rb2—O13ii2.986 (2)S2—O121.451 (3)
Rb2—O83.082 (3)S2—O91.452 (3)
Rb2—O10v3.106 (3)S2—O81.454 (3)
Rb2—O93.337 (3)S2—O31.575 (2)
Rb2—O13.374 (3)S3—O131.461 (2)
Rb3—O112.776 (2)S3—O71.464 (3)
Rb3—O10ii2.788 (3)S3—O101.464 (3)
Rb3—O7v2.874 (2)S3—O51.528 (2)
Rb3—O14vii2.935 (2)
Symmetry codes: (i) x−1/2, −y+1/2, z−1/2; (ii) x+1/2, −y+1/2, z−1/2; (iii) −x+1/2, y+1/2, −z+3/2; (iv) x+1, y, z; (v) −x+1/2, y−1/2, −z+3/2; (vi) −x+1, −y, −z+2; (vii) −x, −y, −z+1; (viii) x−1, y, z; (ix) −x, −y, −z+2.
Acknowledgements top

The authors thank Astrid Schöneberg and Bodil Holten for technical assistance.

references
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