Penkvilksite-2O: Na₂TiSi₄O₁₁·2H₂O

The crystal structure of synthetic penkvilksite-2O, disodium titanium tetra­silicate dihydrate, Na₂TiSi₄O₁₁·2H₂O, a microporous titanosilicate, confirms the major features of a previous model that had been obtained by order-disorder (OD) theory from the known structure of penkvilksite-1M. An important difference from the previous model involves the hydrogen bonding of the water mol­ecule which, on the basis of a Raman spectrum and the finding of only one of the two H atoms, is proposed to be disordered about a fixed O-H direction. The structure of penkvilksite-2O is based on (100) silicate layers linked by isolated TiO₆ octa­hedra to form a heteropolyhedral framework. The layer is strongly corrugated, based on inter­laced spiral chains, and is crossed by two different channels that have an effective channel width of about 3 Å.

S1. Comment

In recent years, titanium silicates have attracted broad attention from materials scientists because they possess properties that are useful for technological applications, e.g. in catalysis and ion exchange. The heteropolyhedral structure type of penkvilksite, Na₂TiSi₄O₁₁.2H₂O, which is, at the same time, layered and microporous, has been a model for the synthesis of several isotypic structures (Lin et al.,1997; Liu et al., 1997; Lin & Rocha, 2004, 2005, 2006). Compared with microporous monopolyhedral frameworks, like those of zeolites, the presence in the framework of higher coordination polyhedra makes the microporous heteropolyhedral structures more suitable for tuning properties via chemical composition [cf. Rocha & Lin, 2005; a recent review of microporous mineral phases is reported by Ferraris & Merlino (2005)]. Some synthetic compounds have been tested for sorption of radioactive elements (Attar & Dyer, 2001; Koudsi & Dyer, 2001) and luminescence due to the demonstrated incorporation of rare earth elements into the structure (Lin et al., 2006).

Two minerals with the same ideal chemical composition Na₂TiSi₈O₁₁.2H₂O, but crystallographically different, were reported from the alkaline massifs of Lovozero and Khibiny (Kola Peninsula, Russia) and Mont Saint-Hilaire (Canada). Merlino et al. (1994) showed that the Khibiny sample, first studied by them, and the Lovozero sample, used by Bussen et al. (1975) to define the mineral species, represent two different polytypes. These polytypes, according to Merlino et al. (1994), are: penkvilksite-1M (monoclinic P2₁/c; a = 8.956, b = 8.727, c = 7.387 Å, β = 112.74°) and penkvilksite-2O (orthorhombic Pnca; a = 16.3721, b = 8.7492, c = 7.4020 Å). The same authors solved the crystal structure of penkvilksite-1M from single-crystal X-ray diffraction data and, applying the order–disorder (OD) theory [Dornberger-Schiff, 1964; for a recent review of the theory see Ferraris et al. (2008)], obtained a model of the structure of penkvilksite-2O and refined it by the Rietveld method using X-ray powder diffraction data obtained from a sample that also contained 5% of penkvilksite-1M. According to Merlino et al. (1994), penkvilksite-1M and penkvilksite-2O represent two out of four maximum degree of order (MDO) polytypes predicted by the OD theory; the other two MDO polytypes are one that is monoclinic 2M (space group I2/c) and another, again orthorhombic 2O, but with space group Pmcn. The occurrence of these further two polytypes was considered improbable because the H₂O molecules would be too close each other, according to the structural models derived from the two known polytypes. However, Sheriff & Zhou (2004) tentatively assigned to 2M an unknown penkvilksite polytype found by an NMR spectroscopic study of a synthetic sample.

The above-mentioned MDO polytypes of penkvilksite differ only in the stacking of a basic module (Fig. 1a) with layer-group symmetry P(1)2₁/c1. This module is stacked by the action of the following symmetry operations: 2₁ (1M), n (2O, space group Pnca), 2 (2M) and m (2O, space group Pmcn); the MDO polytypes shown in parentheses and, in particular, silicate layers like that shown in Fig. 1b are thus formed [cf. Fig. 5 in Merlino et al. (1994)]. Both the 1M and 2O (Pnca) polytypes have been found several times in the syntheses of penkvilksite-type structures quoted above [1M also occurs in the mineral tumchaite, Na₂(Zr,Sn)Si₄O₁₁.2H₂O (Subbotin et al., 2000)], but the crystal structure of penkvilksite-2O has not been independently determined from diffraction data. In this paper, we describe the structure of penkvilksite-2O as obtained by single-crystal X-ray diffraction data from a synthetic crystal and discuss it further on the basis of its Raman spectrum (Fig. 2). The most important features of the structure model obtained by OD theory are confirmed, including bond lengths and angles, in spite of the fact that the XRPD [X-ray powder diffraction?] pattern used by Merlino et al. (1994) for the structure refinement contained a 5% contribution from penkvilksite-1M. However, the hydrogen-bonding scheme proposed by these authors for the H₂O molecule must be revised taking into account details of our structural study and the Raman spectrum of Fig. 2.

The structure of penkvilksite-2O consists of (100) silicate layers connected by isolated TiO₆ octahedra along [100], i.e. along the stacking direction of the module shown in Fig. 1a, such that, finally, it consists of a mixed tetrahedral–octahedral heteropolyhedral framework. The silicate layer is strongly corrugated, being based on a spiral Si1—Si2—Si1^(vii)—Si2^(vii)··· chain that runs along [001] and is linked to its two adjacent chains by sharing O6 (Fig. 1b). The layer is crossed by two types of channels that are delimited by six Si tetrahedra along [010] (Fig. 3a) and by six tetrahedra plus two Ti octahedra along [001] (Fig. 3b). However, because of the spiral form of the silicate chains, the polyhedra delimiting the channels do not form closed rings. Consequently, the effective channel width (e.c.w.), as calculated according to McCusker et al. (2003) by subtracting the ionic diameter of O^2-(2.7 Å) from the O···O distances across each channel, can only be approximate and corresponds to about 3 Å. The channels host seven-coordinate Na1 (Table 1) and, even if they show an e.c.w. smaller than the value of 3.2 Å required to classify a structure as porous (McCusker et al., 2003), allow sorption of radioactive elements (Attar & Dyer, 2001; Koudsi & Dyer, 2001), which is likely due to their open spiral development.

O7W of the water molecule establishes O···O contacts shorter than 3.1 Å as shown in Table 2. O7W···O5 and O7W···O2^ix are edges of the Na1 coordination polyhedron and could not correspond to hydrogen bonds, unless the relevant angles O7W—H1···O were very bent to avoid Na1···H1 contacts. Moreover, the angle O1^viii···O7W···O6^viii = 54.0 (1)° would require an unusually bent hydrogen bond in the case that O1^viii and O6^viii were acceptors at the same time. To propose a reasonable configuration for the H₂O molecule, one must take into account: (1) only one residual peak in the difference electron density can be attributed to a H atom (H1); (2) no trace of a second H atom (H') was found; (3) in the O—H stretching region, the Raman spectrum (Fig. 2) shows a broad, structured peak (maxima at 3410 and 3527 cm^-1; shoulders at 3354 and 3471 cm^-1) that can be fitted by four Gaussian curves centred at 3363, 3409, 3422 and 3527 cm^-1. Reasonably, the configuration around O7W, which is twice coordinated to Na1 (Table 1) and is the donor of a medium-strength hydrogen bond to O1^viii, can be represented as follows. Due to the unfavourable geometry required to establish an O7W—H'···O6^viii hydrogen bond, the O7W—H' direction is disordered (likely by rotation around O7W—H1) and may alternatively establish very bent hydrogen bonds with O2^ix, O5 and O6^viii. To note that, despite the fact that O7W···O2^ix and O7W···O5 correspond to edges of the Na1 coordination polyhedron, the angles O1^viii···O7W···O5 and O1^viii···O7W···O2^ix (Table 2) actually match the range of values expected for an acceptor–donor–acceptor angle of bent hydrogen bonds (cf. Chiari & Ferraris, 1982). In conclusion, the maxima given above for the four Gaussian curves fitting the O—H stretching range of the Raman spectrum (Fig. 2) should correspond, in the order from lower to higher frequencies, to the hydrogen bonds O7W—H'···O5, O7W—H1···O1^viii, O7W—H'···O6^viii and O7W—H'···O2^ix. Precisely [This is precisely what is found, that], the sharpest fitting peak at 3410 cm^-1 is originated by the ordered O7W—H1···O1^viii hydrogen bond, whereas the three statistically disordered O7W—H'···O hydrogen bonds contribute to the features of the Raman spectrum shown in the inset of Fig. 2.

In conclusion, our single-crystal diffraction and Raman study improves – particularly for the description of hydrogen bonding and the microporous structure – the structural characterization of penkvilksite-2O, whose structure type was shown in several reports to be technologically important for its exchange and photoluminescent properties. At the same time, aside from details of the hydrogen bonding, the present results confirm the structure model deduced from OD theory (Merlino et al., 1994), thus providing a further example of the power of this theory in modelling unknown polytypes (cf. Ferraris et al., 2008).

Related literature

For related literature, see: Attar & Dyer (2001); Bussen et al. (1975); Chiari & Ferraris (1982); Dornberger-Schiff (1964); Ferraris & Gula (2005); Ferraris & Merlino (2005); Ferraris et al. (2008); Koudsi & Dyer (2001); Lin & Rocha (2004, 2005, 2006); Lin et al. (1997); Lin, Rainho, Rocha & Carlos (2006); Liu et al. (1997); McCusker et al. (2003); Merlino et al. (1994); Rocha & Lin (2005); Sheriff & Zhou (2004); Subbotin et al. (2000).

S2. Experimental

Crystals of penkvilksite-2O (Fig. 4) were obtained as a product of hydrothermal runs directed at the synthesis of rhodesite-type microporous silicates (cf. Ferraris & Gula, 2005) using a mixture with composition 1.10SiO₂ : 0.065TiO₂: 0.220SrO: 0.725Na₂O: 0.20K₂O. A gel with this composition was prepared by adding solutions of titanium isopropoxide in ethanol and, subsequently, of strontium nitrate, under vigorous stirring, to a strongly alkaline gel obtained by dissolving fumed silica in NaOH and KOH solution. Static crystallization was carried out in a 25 ml Teflon-lined stainless steel autoclave at 503 K, under autogenous pressure, for a period of 12 d.

An X-ray powder diffraction pattern collected on a Siemens D5000 diffractometer using Cu Kα radiation revealed the presence of quite pure penkvilskite-2O with minor ETS-4. Examination by scanning electron microscopy (SEM; Stereoscan S360 Cambridge electron microscope equipped with an Energy 200 Oxford Instruments EDS apparatus) was carried out and a secondary electron image of (100) tabular penkvilksite-2O is shown in Fig. 4.

The Raman spectrum reported in Fig. 2 was collected using a LabRam HR800 micro-Raman spectrometer (Jobin Yvon, equipped with a HeNe laser at an excitation wavelength of 562 nm, a CCD detector and an Olympus BX41 optical microscope).

S3. Refinement

Only one H atom is reported for the H₂O molecule because the second one is disordered, as discussed in the text. The coordinates of this atom are derived from the difference electron density map and were not refined. Consequently, the fixed position of this H atom does not contribute to the calculated s.u.s of the related bond lengths and angles.

Computing details

Data collection: SMART (Bruker, 2004); cell refinement: SMART (Bruker, 2004); data reduction: SAINT-Plus (Bruker, 2004) and SADABS (Sheldrick, 2002); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: ATOMS 6.2 (Dowty, 2002); software used to prepare material for publication: SHELXL97 (Sheldrick, 2008).

Figures

	Fig. 1. (a) View along [010] of a module of the penkvilksite structure consisting of isolated Ti1 octahedra (blue) and [001] chains of Si tetrahedra (yellow); different stackings of this module along [100] lead to the polytypes discussed in the text. (b) (100) silicate layer formed by the [001] chains: clockwise and counterclockwise chains are differently coloured. Orange and cyan circles represent, respectively, Na1 and the O atoms of H2O.
	Fig. 2. Raman spectrum of penkvilksite-2O; the region of the O—H stretching is enlarged in the inset and the fitting curve (red), sum of four Gaussian curves (green), is shown.
	Fig. 3. Penkvilksite-2O seen (a) along [010], showing the channels delimited by Si tetrahedra (yellow) only, and (b) along [001], showing the channels delimited by Si tetrahedra and Ti1 octahedra (blue). Orange and cyan circles represent, respectively, Na1 and the O atoms of H2O.
	Fig. 4. SEM image showing (100) tabular crystals of synthetic penkvilksite-2O.

disodium titanium tetrasilicate dihydrate

Crystal data

Na2TiSi4O11·2H2O	Dx = 2.638 Mg m^−3
Mr = 418.10	Mo Kα radiation, λ = 0.71073 Å
Orthorhombic, Pnca	Cell parameters from 1306 reflections
a = 16.320 (2) Å	θ = 2.5–28.3°
b = 8.7378 (9) Å	µ = 1.43 mm^−1
c = 7.3854 (8) Å	T = 295 K
V = 1053.2 (2) Å^3	Tabular, colourless
Z = 4	0.10 × 0.09 × 0.02 mm
F(000) = 832

Data collection

Bruker SMART APEX diffractometer	1306 independent reflections
Radiation source: fine-focus sealed tube	1109 reflections with I > 2σ(I)
Graphite monochromator	Rint = 0.082
ϕ and ω scans	θmax = 28.3°, θmin = 2.5°
Absorption correction: multi-scan (SADABS; Sheldrick, 2002)	h = −21→21
Tmin = 0.871, Tmax = 0.972	k = −11→11
13075 measured reflections	l = −9→9

Refinement

Refinement on F^2	Secondary atom site location: difference Fourier map
Least-squares matrix: full	Hydrogen site location: inferred from neighbouring sites
R[F^2 > 2σ(F^2)] = 0.076	H-atom parameters not refined
wR(F^2) = 0.161	w = 1/[σ^2(Fo^2) + (0.0457P)^2 + 14.7729P] where P = (Fo^2 + 2Fc^2)/3
S = 1.23	(Δ/σ)max < 0.001
1306 reflections	Δρmax = 1.13 e Å^−3
94 parameters	Δρmin = −1.13 e Å^−3
0 restraints	Extinction correction: SHELXL97 (Sheldrick, 2008), Fc^*=kFc[1+0.001xFc^2λ^3/sin(2θ)]^-1/4
Primary atom site location: structure-invariant direct methods	Extinction coefficient: 0.0028 (13)

Special details

Geometry. All esds (except the esd in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell esds are taken into account individually in the estimation of esds in distances, angles and torsion angles; correlations between esds in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell esds is used for estimating esds 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
Ti1	0.0000	0.5000	0.0000	0.0080 (4)
Si1	0.07996 (10)	0.16049 (18)	0.1236 (2)	0.0078 (4)
Si2	0.15690 (10)	0.44096 (19)	0.2605 (2)	0.0078 (4)
Na1	0.0970 (2)	0.6916 (3)	0.6282 (5)	0.0294 (8)
O1	0.1583 (3)	0.2605 (5)	0.1991 (6)	0.0112 (9)
O2	0.0519 (3)	0.0382 (5)	0.2726 (6)	0.0117 (9)
O3	0.0097 (3)	0.2794 (5)	0.0611 (6)	0.0116 (9)
O4	0.1146 (3)	0.0513 (5)	−0.0410 (6)	0.0121 (9)
O5	0.1063 (3)	0.5405 (5)	0.1149 (6)	0.0110 (9)
O6	0.2500	0.5000	0.2702 (8)	0.0130 (12)
O7W	0.1913 (4)	0.6844 (8)	−0.1457 (9)	0.0410 (17)
H1	0.2575	0.7027	−0.1631	0.070*

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Ti1	0.0120 (7)	0.0076 (6)	0.0045 (6)	0.0008 (6)	−0.0004 (6)	−0.0005 (6)
Si1	0.0131 (8)	0.0069 (7)	0.0035 (7)	−0.0007 (6)	0.0001 (6)	−0.0007 (6)
Si2	0.0112 (8)	0.0090 (7)	0.0032 (7)	0.0000 (6)	0.0000 (6)	0.0001 (6)
Na1	0.045 (2)	0.0103 (12)	0.0324 (18)	−0.0055 (13)	−0.0193 (16)	0.0045 (12)
O1	0.016 (2)	0.0088 (19)	0.009 (2)	0.0043 (18)	−0.0036 (17)	0.0010 (16)
O2	0.016 (2)	0.011 (2)	0.007 (2)	0.0024 (17)	0.0011 (17)	0.0023 (16)
O3	0.013 (2)	0.011 (2)	0.011 (2)	0.0042 (17)	−0.0031 (17)	0.0018 (16)
O4	0.019 (2)	0.011 (2)	0.007 (2)	0.0017 (18)	0.0064 (17)	−0.0017 (16)
O5	0.018 (2)	0.0075 (19)	0.0078 (19)	−0.0030 (16)	−0.0011 (17)	0.0016 (16)
O6	0.011 (3)	0.017 (3)	0.011 (3)	−0.001 (2)	0.000	0.000
O7W	0.031 (3)	0.056 (4)	0.036 (4)	−0.015 (3)	−0.007 (3)	0.027 (3)

Geometric parameters (Å, º)

Ti1—O2^i	1.911 (4)	O1—O6	2.625 (5)
Ti1—O5	1.964 (4)	O1—O4^v	2.626 (6)
Ti1—O3	1.986 (4)	O1—O3	2.637 (6)
Ti1—Na1^ii	3.265 (3)	O1—O4	2.645 (6)
Ti1—Na1^iii	3.585 (3)	O1—O2	2.662 (6)
Si1—O2	1.601 (4)	O1—O5	2.663 (6)
Si1—O3	1.615 (4)	O2—O4	2.534 (6)
Si1—O4	1.645 (4)	O2—O3	2.713 (6)
Si1—O1	1.647 (5)	O2—O5^ii	2.713 (6)
Si1—Na1^iv	3.089 (3)	O2—O3^v	2.749 (6)
Si1—Na1^ii	3.445 (4)	O2—O3^ii	2.762 (6)
Si2—O6	1.6062 (17)	O2—O5^v	2.766 (6)
Si2—O5	1.611 (5)	O2—O7W^v	3.053 (8)
Si2—O4^v	1.622 (4)	O2—O4^x	3.305 (7)
Si2—O1	1.641 (5)	O3—O4	2.733 (6)
Si2—O3	3.152 (5)	O3—O5^xi	2.784 (6)
Si2—Na1^vi	3.496 (3)	O3—O5	2.801 (6)
Na1—O7W^vii	2.273 (7)	O4—O6^xii	2.651 (5)
Na1—O5^viii	2.348 (5)	O4—O5^iv	2.668 (6)
Na1—O3^i	2.361 (5)	O5—O6	2.634 (5)
Na1—O2^v	2.389 (5)	O5—O7W	2.685 (7)
Na1—O4^v	2.480 (5)	O6—O7W^xiii	2.985 (6)
Na1—O7W^viii	2.764 (8)	O7W—O1^xiv	2.788 (7)
Na1—O3^ix	2.892 (6)	O7W—H1	1.099 (6)
Na1—O2^ix	3.466 (5)
O2^i—Ti1—O5^xi	91.12 (18)	O2^v—Na1—O2^ix	105.20 (13)
O2^i—Ti1—O5	88.88 (18)	O4^v—Na1—O2^ix	140.43 (17)
O2^i—Ti1—O3^xi	89.72 (18)	O7W^viii—Na1—O2^ix	106.16 (17)
O5—Ti1—O3^xi	89.65 (17)	O3^ix—Na1—O2^ix	49.50 (12)
O2^i—Ti1—O3	90.28 (18)	Si1^v—Na1—O2^ix	128.15 (14)
O5—Ti1—O3	90.35 (17)	Ti1^i—Na1—O2^ix	32.80 (8)
O2—Si1—O3	115.0 (2)	Si1^i—Na1—O2^ix	65.19 (10)
O2—Si1—O4	102.6 (2)	Si2—O1—Si1	126.3 (3)
O3—Si1—O4	113.9 (3)	O6—O1—O4^v	60.64 (17)
O2—Si1—O1	110.1 (2)	Si2—O1—O3	91.9 (2)
O3—Si1—O1	107.9 (2)	Si1—O1—O3	35.65 (14)
O4—Si1—O1	106.9 (2)	Si1—O2—Ti1^ii	147.9 (3)
O6—Si2—O5	109.9 (2)	Si1—O2—Na1^iv	99.6 (2)
O6—Si2—O4^v	110.4 (3)	Ti1^ii—O2—Na1^iv	112.5 (2)
O5—Si2—O4^v	111.3 (2)	Si1—O3—Ti1	138.3 (3)
O6—Si2—O1	107.90 (19)	Si1—O3—Na1^ii	118.9 (2)
O5—Si2—O1	110.0 (2)	Ti1—O3—Na1^ii	97.01 (18)
O4^v—Si2—O1	107.2 (2)	Si1—O3—Na1^ix	104.9 (2)
O7W^vii—Na1—O5^viii	90.8 (2)	Ti1—O3—Na1^ix	92.72 (17)
O7W^vii—Na1—O3^i	160.7 (3)	Na1^ii—O3—Na1^ix	93.12 (19)
O5^viii—Na1—O3^i	72.49 (17)	Si1—O3—Si2	67.26 (16)
O7W^vii—Na1—O2^v	81.8 (2)	Ti1—O3—Si2	74.47 (13)
O5^viii—Na1—O2^v	151.2 (2)	Na1^ii—O3—Si2	169.9 (2)
O3^i—Na1—O2^v	108.08 (19)	Na1^ix—O3—Si2	92.80 (14)
O7W^vii—Na1—O4^v	105.5 (2)	Si2^iv—O4—Si1	142.2 (3)
O5^viii—Na1—O4^v	145.6 (2)	Si2^iv—O4—Na1^iv	122.7 (2)
O3^i—Na1—O4^v	93.75 (18)	Si1—O4—Na1^iv	94.8 (2)
O2^v—Na1—O4^v	62.70 (15)	Si2^iv—O4—Na1^ii	103.0 (2)
O7W^vii—Na1—O7W^viii	99.9 (2)	Si1—O4—Na1^ii	67.54 (17)
O5^viii—Na1—O7W^viii	62.76 (19)	Na1^iv—O4—Na1^ii	105.17 (16)
O3^i—Na1—O7W^viii	81.38 (19)	Si2—O5—Ti1	130.1 (2)
O2^v—Na1—O7W^viii	145.9 (2)	Si2—O5—Na1^vi	122.9 (2)
O4^v—Na1—O7W^viii	84.4 (2)	Ti1—O5—Na1^vi	98.1 (2)
O7W^vii—Na1—O3^ix	80.1 (2)	Si2^xv—O6—Si2	174.9 (5)
O5^viii—Na1—O3^ix	89.14 (18)	Si2^xv—O6—Na1^vi	112.31 (12)
O3^i—Na1—O3^ix	89.86 (18)	Si2—O6—Na1^vi	66.17 (10)
O2^v—Na1—O3^ix	62.22 (16)	Na1^vi—O6—Na1^xii	148.1 (2)
O4^v—Na1—O3^ix	122.99 (18)	Na1^iii—O7W—O5	100.9 (2)
O7W^viii—Na1—O3^ix	151.89 (19)	Na1^iii—O7W—Na1^vi	98.6 (3)
O7W^vii—Na1—O2^ix	109.8 (2)	Na1^iii—O7W—H1	125.1 (4)
O5^viii—Na1—O2^ix	51.33 (15)	Na1^vi—O7W—H1	125.2 (5)
O3^i—Na1—O2^ix	52.19 (15)

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