Strontium chromium(II) diphosphate, SrCrP₂O₇

The structure of the title compound, SrCrP₂O₇, belongs to the series of isotypic crystal structures SrMP₂O₇ (M = Cr, Mn, Fe, Co, Ni, Cu, Zn or Cd) and is closely related to [alpha]-Ca₂P₂O₇. Chromium(II) shows a 4+1 square-pyramidal coordination by oxy­gen. [CrO₅] units and the [P₂O₇] groups build a three-dimensional framework with channels along the a axis, and Sr occupies these channels. In addition to the work on SrCrP₂O₇, lattice parameters for SrMnP₂O₇ have been determined for the first time and unit-cell dimensions for SrZnP₂O₇ have been redetermined.

S1. Comment

Despite recent progress (Glaum, 1999; Glaum & Schmidt, 1997), the crystal chemistry of Cr²⁺ in a plain O coordination is still not well established. Apart from preliminary investigations (Schmidt & Glaum, 1997, 1999; Belsky et al., 1984), the spectroscopic and magnetic properties of Cr²⁺ in oxidic solids have not been studied. For this reason we have started to synthesize new phosphates of Cr²⁺ in the systems A/Cr/P/O (A = Mg, Ca, Sr or Ba). Our particular interest is focused on structures containing only one Cr²⁺ within the asymmetric unit, because these are particularly suitable for spectroscopic and magnetic measurements. The series of isotypic diphosphates SrMP₂O₇ [M = Mn (Maa\&s et al., 1999), Fe (Le Meins & Courbion, 1999), Co (Riou & Raveau, 1991), Ni (El-Bali, 1999), Cu (Moqine et al., 1993), Zn (Maa\&s, 1999; Murashova et al., 1991) or Cd (Alaoui El Belghiti et al., 1991)] with square-pyramidal [MO₅] units meets this requirement. In this paper we report on synthesis of SrCrP₂O₇ and its crystal structure refinement from X-ray single-crystal data. Lattice parameters for SrMnP₂O₇ and SrZnP₂O₇ have also been determined. \scheme

The X-ray single-crystal structure analysis of SrCrP₂O₇ shows square-pyramidal coordination for Cr²⁺ [equatorial Cr—O distances 1.995 (2)–2.109 (2), and axial Cr—O 2.362 (3) Å], similar to those observed for M²⁺ in the other members of this series. The P—O distances and O—P—O angles in SrCrP₂O₇ are in the typical ranges found for diphosphate groups (Table 2). An almost eclipsed conformation (dihedral angle 15–20°) and a P—O—P bridging angle of 128.0 (2)° are observed for the [P₂O₇] units.

Each Cr²⁺ is coordinated to the terminal O atoms of five [P₂O₇] groups. These [CrO₅] and [P₂O₇] groups are joined together to form a three-dimensional framework, with channels running along the crystallographic a and b axes (Fig. 1). The Sr atoms are located at the interceptions of these channels. Calculations according to the concept of effective coordination numbers, ECoN (Hoppe, 1979), using the program MAPLE-4 (Hübenthal, 1993) yield ECoN(Sr²⁺) = 7.88. Naïve counting of all O atoms closer to Sr²⁺ than the nearest cation leads to nine O neighbours. For O7ⁱ, showing the longest distance to Sr, a contribution δ(ECoN) = 0.098 to ECoN(Sr²⁺) has been calculated, while for O5ⁱ, with the shortest Sr—O distance, δ(ECoN) = 1.227 is obtained [symmetry code: (i) 1/2 - x, 1/2 + y, 1/2 - z].

The structure of SrCrP₂O₇ is closely related to that of α-Ca₂P₂O₇ (Calvo, 1968). Sr²⁺ occupies the position Ca(1), while Cr²⁺ is located at the position Ca(2). In contrast to Ca²⁺ (coordination number = 8, [Ca₂O₁₄] `dimers'), Cr²⁺ shows coordination by only five O atoms. This leads to isolated [CrO₅] groups.

It is interesting to note that the [CrO₅] groups are aligned with their long axis almost parallel to the crystallographic a axis (ϕ = 8.42°). Fig. 2 gives a summary of the lattice parameters observed for the whole series SrMP₂O₇ with M = Cr—Zn and Cd. It is evident that the particular stereochemistry of Cr²⁺ and Cu²⁺, which is due to the Jahn-Teller-effect, leads to an expansion of the crystallographic a axis compared with the other members of the series. At the same time, shrinking of the b and c axes is observed for SrCrP₂O₇ and SrCuP₂O₇. SrFeP₂O₇, containing Fe²⁺ which is subject to a second-order Jahn-Teller distortion, shows similar behaviour, although to a far smaller extent.

In addition to the characterization of SrCrP₂O₇, two more mixed diphosphates, SrZnP₂O₇ (Maa\&s, 1999; Murashova et al., 1991) and SrMnP₂O₇, have been synthesized and their lattice parameters have been refined. The Mn compound has been obtained for the first time. Their unit-cell dimensions fit well into the whole series obtained for SrMP₂O₇ (M = Cr—Zn and Cd). The unit cell volumes directly follow the development of the ionic radii of divalent 3 d metals as described in textbooks. The variation of the three axes, however, shows the anisotropy in bonding due to the particular number of antibonding d-electrons.

S2. Experimental

Equimolar mixtures of the starting materials, Cr₂P₂O₇ (Glaum et al., 1991) and Sr₂P₂O₇ [obtained from SrCO₃ and (NH₄)₂HPO₄ by slowly increasing the temperature up to 1273 K], were sealed in evacuated silica ampoules (l ~10 cm, d ~1.6 cm) with iodine (100 mg) and CrP (5 mg) as mineralizers and heated at 1223 K for 10 d. The addition of CrP was intended to ensure reducing conditions. Furthermore, we have observed an improved recrystallization via the gas phase, and even chemical vapour transport reactions for many phosphates, using iodine as a mineraliser/transport agent in combination with reducing agents such as metal, phosphorus or metal phosphide (Glaum et al., 1991; Glaum, 1999). The reaction products were washed with dilute NaOH and water and dried at 393 K. The experiments led to light blue powders of SrCrP₂O₇ which always contained small amounts of well recrystallized CrP and Cr₂P₂O₇ as by-products. Occasionally, growth of prismatic crystals of SrCrP₂O₇ with an edge-length of up 1 mm was observed. Chemical vapour transport experiments in a temperature gradient (1323 → 1223 K), aimed at purification and crystallization of SrCrP₂O₇, led to decomposition of the mixed phosphate. Cr₂P₂O₇ has been deposited at the lower temperature region, while Sr₂P₂O₇ remained at the higher temperature zone. SrCrP₂O₇ is remarkably stable in air and against mild oxidizing agents. By heating SrCO₃ with MnCO₃ or ZnO and stoichiometric amounts of (NH₄)₂HPO₄ in air slowly to 1073 K, single phase powders of the isotypic diphosphates SrMnP₂O₇ and SrZnP₂O₇ have also been synthesized (Maa\&s et al., 1999; Maa\&s, 1999).

S3. Refinement

The Guinier photograph of SrCrP₂O₇ was indexed on the basis of the pattern of SrCoP₂O₇ (Riou & Raveau, 1991). The structure refinement of SrCrP₂O₇ from X-ray single-crystal data, using SHELXL97 (Sheldrick, 1997) and the coordinates of SrCoP₂O₇ (Riou & Raveau, 1991) as starting parameters, proceeded in a straightforward manner. The crystal shape was optimized by minimizing the internal R value of the whole data set using the program HABITUS (Herrendorf, 1993). The HABITUS so derived was used for the numerical absorption correction. The lattice parameters of SrMnP₂O₇ and SrZnP₂O₇ were also determined from Guinier photographs [Cu Kα, α-SiO₂ as internal standard, SrMnP₂O₇: a = 5.378 (2), b = 8.424 (2) and c = 12.770 (3) Å, and β = 90.24 (3)°; SrZnP₂O₇: a = 5.309 (2), b = 8.225 (2) and c = 12.749 (6) Å, and β = 90.22 (3)°]. Lattice parameters obtained for SrZnP₂O₇ have lower standard deviations and deviate somewhat from the values reported earlier by Murashova et al. (1991).

Computing details

Data collection: IPDS Software (Stoe & Cie, 1997); cell refinement: IPDS Software; data reduction: IPDS Software; program(s) used to solve structure: details?; program(s) used to refine structure: SHELXL97; molecular graphics: ATOMS for Windows (Dowty, 1998); software used to prepare material for publication: SHELXL97.

Figures

	Fig. 1. The crystal structure of SrCrP2O7 projected along the a and b axes. Light grey shading [PO4], dark grey shading [CrO5] and black circles Sr.
	Fig. 2. Unit-cell parameters of the series of diphosphates SrMP2O7 (M = Cr—Zn and Cd).

strontium chromium(II) diphosphate

Crystal data

SrCrP2O7	F(000) = 592
Mr = 313.56	unit cell parameters from Guinier photograph
Monoclinic, P21/n	Dx = 3.679 Mg m^−3
Hall symbol: -P 2yn	Mo Kα radiation, λ = 0.71073 Å
a = 5.422 (2) Å	Cell parameters from 36 reflections
b = 8.3254 (19) Å	θ = 6.3–26.1°
c = 12.542 (4) Å	µ = 11.89 mm^−1
β = 90.39 (3)°	T = 293 K
V = 566.1 (3) Å^3	Prismatic, light blue
Z = 4	0.34 × 0.10 × 0.09 mm

Data collection

Stoe IPDS diffractometer	1260 independent reflections
Radiation source: fine-focus sealed tube	1178 reflections with I > 2σ(I)
Graphite monochromator	Rint = 0.032
Detector resolution: 6.66 pixels mm^-1	θmax = 28.2°, θmin = 2.9°
integrated intensities from imaging plate scans	h = −6→6
Absorption correction: numerical (HABITUS; Herrendorf, 1993)	k = −11→11
Tmin = 0.259, Tmax = 0.366	l = −16→16
4907 measured reflections

Refinement

Refinement on F^2	0 restraints
Least-squares matrix: full	w = 1/[σ^2(Fo^2) + (0.0555P)^2 + 0.2729P] where P = (Fo^2 + 2Fc^2)/3
R[F^2 > 2σ(F^2)] = 0.027	(Δ/σ)max < 0.001
wR(F^2) = 0.076	Δρmax = 0.78 e Å^−3
S = 1.10	Δρmin = −0.74 e Å^−3
1260 reflections	Extinction correction: SHELXL97 (Sheldrick, 1997), Fc^*=kFc[1+0.001xFc^2λ^3/sin(2θ)]^-1/4
101 parameters	Extinction coefficient: 0.093 (4)

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
Sr1	0.28539 (6)	0.34270 (3)	0.27851 (2)	0.0090 (2)
Cr1	0.79177 (11)	0.14493 (6)	0.11053 (4)	0.0080 (2)
P1	0.7538 (2)	0.53916 (9)	0.16150 (6)	0.0069 (2)
P2	0.3140 (2)	0.19796 (9)	0.98505 (6)	0.0069 (2)
O1	0.6791 (5)	0.3652 (3)	0.1493 (2)	0.0113 (5)
O2	0.6620 (5)	0.4009 (3)	0.4052 (2)	0.0110 (5)
O3	0.9351 (5)	0.1222 (3)	0.2666 (2)	0.0100 (5)
O4	0.7676 (5)	0.1166 (3)	0.4543 (2)	0.0109 (5)
O5	0.4865 (5)	0.0633 (3)	0.2980 (2)	0.0132 (5)
O6	0.0897 (5)	0.3351 (3)	0.4716 (2)	0.0103 (5)
O7	0.2224 (5)	0.1818 (3)	0.0979 (2)	0.0113 (5)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Sr1	0.0061 (2)	0.0107 (2)	0.0102 (2)	−0.00009 (9)	−0.00037 (12)	0.00059 (9)
Cr1	0.0075 (4)	0.0078 (3)	0.0086 (3)	0.0010 (2)	−0.0021 (2)	−0.0004 (2)
P1	0.0050 (5)	0.0077 (4)	0.0080 (3)	0.0002 (3)	−0.0004 (3)	−0.0001 (2)
P2	0.0057 (5)	0.0085 (4)	0.0064 (4)	0.0002 (3)	−0.0001 (3)	0.0004 (3)
O1	0.0123 (15)	0.0079 (10)	0.0137 (10)	0.0000 (8)	−0.0006 (9)	−0.0003 (8)
O2	0.0127 (14)	0.0089 (10)	0.0113 (9)	−0.0001 (8)	−0.0038 (8)	0.0006 (8)
O3	0.0089 (13)	0.0124 (10)	0.0087 (9)	−0.0023 (8)	0.0004 (8)	0.0004 (8)
O4	0.0128 (14)	0.0101 (10)	0.0099 (10)	−0.0003 (8)	0.0008 (9)	0.0008 (8)
O5	0.0070 (14)	0.0155 (11)	0.0170 (10)	−0.0001 (8)	−0.0032 (9)	−0.0006 (8)
O6	0.0054 (15)	0.0139 (11)	0.0116 (10)	−0.0011 (8)	0.0002 (8)	−0.0008 (8)
O7	0.0099 (14)	0.0169 (11)	0.0071 (10)	−0.0013 (9)	0.0010 (8)	−0.0003 (8)

Geometric parameters (Å, º)

Sr1—O5^i	2.539 (3)	P1—O4^iii	1.593 (2)
Sr1—O5	2.580 (3)	P1—Sr1^vi	3.6164 (14)
Sr1—O2	2.624 (2)	P1—Sr1^iii	3.6289 (12)
Sr1—O3^ii	2.645 (2)	P2—O7^vii	1.509 (2)
Sr1—O6	2.651 (3)	P2—O6^viii	1.530 (3)
Sr1—O7	2.652 (2)	P2—O2^ix	1.533 (2)
Sr1—O1	2.695 (3)	P2—O4^ix	1.611 (3)
Sr1—O3^iii	2.836 (3)	P2—Sr1^viii	3.667 (2)
Sr1—O7^i	3.221 (2)	O2—P2^v	1.533 (2)
Sr1—P1	3.3660 (13)	O2—Cr1^iii	2.057 (2)
Sr1—P1^ii	3.6164 (14)	O3—P1^iv	1.534 (3)
Sr1—P1^iv	3.6289 (12)	O3—Sr1^vi	2.645 (2)
Cr1—O1	1.994 (2)	O3—Sr1^iv	2.836 (3)
Cr1—O2^iv	2.057 (2)	O4—P1^iv	1.593 (2)
Cr1—O6^v	2.058 (3)	O4—P2^v	1.611 (3)
Cr1—O3	2.109 (2)	O5—P1^iv	1.507 (3)
Cr1—O7^vi	2.361 (3)	O5—Sr1^x	2.539 (3)
Cr1—Sr1^iv	3.6715 (10)	O6—P2^xi	1.530 (3)
Cr1—Sr1^vi	3.7731 (13)	O6—Cr1^ix	2.058 (3)
P1—O5^iii	1.507 (3)	O7—P2^xii	1.509 (2)
P1—O1	1.511 (2)	O7—Cr1^ii	2.361 (3)
P1—O3^iii	1.534 (3)	O7—Sr1^x	3.221 (2)
O5^i—Sr1—O5	158.73 (5)	O3—Cr1—O7^vi	73.48 (9)
O5^i—Sr1—O2	122.87 (8)	O1—Cr1—Sr1^iv	136.83 (7)
O5—Sr1—O2	77.43 (7)	O2^iv—Cr1—Sr1^iv	44.20 (6)
O5^i—Sr1—O3^ii	93.78 (8)	O6^v—Cr1—Sr1^iv	134.71 (7)
O5—Sr1—O3^ii	71.48 (8)	O3—Cr1—Sr1^iv	50.32 (7)
O2—Sr1—O3^ii	135.89 (7)	O7^vi—Cr1—Sr1^iv	59.96 (6)
O5^i—Sr1—O6	97.33 (8)	O1—Cr1—Sr1^vi	71.32 (8)
O5—Sr1—O6	93.64 (7)	O2^iv—Cr1—Sr1^vi	113.46 (7)
O2—Sr1—O6	76.47 (8)	O6^v—Cr1—Sr1^vi	143.98 (7)
O3^ii—Sr1—O6	75.15 (7)	O3—Cr1—Sr1^vi	42.75 (7)
O5^i—Sr1—O7	88.36 (8)	O7^vi—Cr1—Sr1^vi	44.18 (6)
O5—Sr1—O7	71.18 (7)	Sr1^iv—Cr1—Sr1^vi	69.43 (2)
O2—Sr1—O7	134.88 (8)	O1—Cr1—Sr1	41.50 (7)
O3^ii—Sr1—O7	60.84 (8)	O2^iv—Cr1—Sr1	124.31 (7)
O6—Sr1—O7	135.91 (8)	O6^v—Cr1—Sr1	92.94 (8)
O5^i—Sr1—O1	100.52 (8)	O3—Cr1—Sr1	77.88 (7)
O5—Sr1—O1	77.51 (8)	O7^vi—Cr1—Sr1	133.76 (6)
O2—Sr1—O1	74.62 (8)	Sr1^iv—Cr1—Sr1	122.17 (2)
O3^ii—Sr1—O1	125.84 (7)	Sr1^vi—Cr1—Sr1	90.78 (3)
O6—Sr1—O1	150.96 (8)	O5^iii—P1—O1	114.28 (15)
O7—Sr1—O1	67.64 (8)	O5^iii—P1—O3^iii	111.58 (14)
O5^i—Sr1—O3^iii	68.95 (8)	O1—P1—O3^iii	108.15 (15)
O5—Sr1—O3^iii	122.18 (8)	O5^iii—P1—O4^iii	108.39 (14)
O2—Sr1—O3^iii	63.54 (7)	O1—P1—O4^iii	106.12 (14)
O3^ii—Sr1—O3^iii	160.51 (4)	O3^iii—P1—O4^iii	108.01 (14)
O6—Sr1—O3^iii	114.79 (7)	O5^iii—P1—Sr1	128.61 (10)
O7—Sr1—O3^iii	108.06 (7)	O1—P1—Sr1	51.38 (10)
O1—Sr1—O3^iii	52.88 (7)	O3^iii—P1—Sr1	56.88 (10)
O5^i—Sr1—O7^i	62.52 (7)	O4^iii—P1—Sr1	122.93 (10)
O5—Sr1—O7^i	138.72 (7)	O5^iii—P1—Sr1^vi	35.25 (10)
O2—Sr1—O7^i	63.87 (7)	O1—P1—Sr1^vi	79.63 (10)
O3^ii—Sr1—O7^i	128.60 (7)	O3^iii—P1—Sr1^vi	119.91 (9)
O6—Sr1—O7^i	64.83 (6)	O4^iii—P1—Sr1^vi	127.31 (11)
O7—Sr1—O7^i	148.41 (7)	Sr1—P1—Sr1^vi	101.83 (3)
O1—Sr1—O7^i	103.97 (7)	O5^iii—P1—Sr1^iii	36.57 (10)
O3^iii—Sr1—O7^i	52.09 (7)	O1—P1—Sr1^iii	150.70 (11)
O5^i—Sr1—P1	85.51 (6)	O3^iii—P1—Sr1^iii	91.49 (10)
O5—Sr1—P1	99.19 (6)	O4^iii—P1—Sr1^iii	87.32 (10)
O2—Sr1—P1	65.65 (6)	Sr1—P1—Sr1^iii	140.21 (3)
O3^ii—Sr1—P1	148.89 (5)	Sr1^vi—P1—Sr1^iii	71.65 (3)
O6—Sr1—P1	135.86 (6)	O7^vii—P2—O6^viii	114.52 (15)
O7—Sr1—P1	88.06 (6)	O7^vii—P2—O2^ix	112.73 (14)
O1—Sr1—P1	25.98 (5)	O6^viii—P2—O2^ix	110.65 (14)
O3^iii—Sr1—P1	26.95 (5)	O7^vii—P2—O4^ix	104.96 (13)
O7^i—Sr1—P1	78.25 (5)	O6^viii—P2—O4^ix	107.29 (13)
O5^i—Sr1—P1^ii	20.03 (6)	O2^ix—P2—O4^ix	106.01 (12)
O5—Sr1—P1^ii	141.39 (6)	O7^vii—P2—Sr1^viii	152.83 (11)
O2—Sr1—P1^ii	140.99 (6)	O6^viii—P2—Sr1^viii	38.71 (9)
O3^ii—Sr1—P1^ii	73.80 (6)	O2^ix—P2—Sr1^viii	82.11 (10)
O6—Sr1—P1^ii	93.37 (6)	O4^ix—P2—Sr1^viii	91.60 (9)
O7—Sr1—P1^ii	77.54 (6)	P1—O1—Cr1	145.4 (2)
O1—Sr1—P1^ii	110.91 (6)	P1—O1—Sr1	102.64 (12)
O3^iii—Sr1—P1^ii	88.48 (5)	Cr1—O1—Sr1	109.13 (10)
O7^i—Sr1—P1^ii	77.64 (5)	P2^v—O2—Cr1^iii	121.85 (13)
P1—Sr1—P1^ii	101.83 (3)	P2^v—O2—Sr1	135.19 (13)
O5^i—Sr1—P1^iv	168.28 (6)	Cr1^iii—O2—Sr1	102.67 (9)
O5—Sr1—P1^iv	20.36 (6)	P1^iv—O3—Cr1	110.04 (13)
O2—Sr1—P1^iv	58.07 (5)	P1^iv—O3—Sr1^vi	139.59 (12)
O3^ii—Sr1—P1^iv	91.22 (6)	Cr1—O3—Sr1^vi	104.48 (9)
O6—Sr1—P1^iv	94.20 (6)	P1^iv—O3—Sr1^iv	96.17 (11)
O7—Sr1—P1^iv	84.82 (6)	Cr1—O3—Sr1^iv	94.75 (8)
O1—Sr1—P1^iv	68.05 (5)	Sr1^vi—O3—Sr1^iv	101.33 (9)
O3^iii—Sr1—P1^iv	104.16 (6)	P1^iv—O4—P2^v	128.1 (2)
O7^i—Sr1—P1^iv	121.46 (5)	P1^iv—O5—Sr1^x	124.72 (14)
P1—Sr1—P1^iv	84.74 (3)	P1^iv—O5—Sr1	123.07 (13)
P1^ii—Sr1—P1^iv	160.87 (2)	Sr1^x—O5—Sr1	111.86 (9)
O1—Cr1—O2^iv	165.81 (11)	P2^xi—O6—Cr1^ix	115.85 (14)
O1—Cr1—O6^v	88.28 (10)	P2^xi—O6—Sr1	120.13 (13)
O2^iv—Cr1—O6^v	93.63 (9)	Cr1^ix—O6—Sr1	123.86 (12)
O1—Cr1—O3	88.16 (10)	P2^xii—O7—Cr1^ii	113.98 (13)
O2^iv—Cr1—O3	87.46 (9)	P2^xii—O7—Sr1	135.62 (14)
O6^v—Cr1—O3	169.47 (11)	Cr1^ii—O7—Sr1	97.46 (9)
O1—Cr1—O7^vi	101.63 (10)	P2^xii—O7—Sr1^x	122.35 (12)
O2^iv—Cr1—O7^vi	90.04 (9)	Cr1^ii—O7—Sr1^x	80.65 (7)
O6^v—Cr1—O7^vi	116.97 (10)	Sr1—O7—Sr1^x	91.93 (6)

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