The complex phosphate K_0.92In_0.46Nb_0.54OPO₄: a new ­representative of the KTiOPO₄ family

Aliovalent K_0.92In_0.46Nb_0.54OPO₄ (KINP, potassium indium niobium oxide phosphate) features a chain structure involving corner-sharing [MO₆] octa­hedra (M = In/Nb), which allows this compound to be recognized as a member of the KTiOPO₄ (KTP) family. However, its crystallization in the polar space group P4₁ belonging to the 4/m Laue group generates its own subclass of KTP-related compounds. The unit cell contains 18 symmetry-independent atoms occupying general positions. The cis-cis principle of octa­hedral arrangement is observed for each of the separate [MO₆]_[infinity] chains. The observed variations in the cis-cis linkage are critically important for nonlinear optical properties and distinguish the present compound from other KTP-related structures. The anionic framework adopts one-dimensional tunnels running orthogonal to the ab plane. The K⁺ ions are arranged in the [001] direction at a distance of c. Merohedral twinning was detected during the structure refinement.

S1. Comment

Compounds related to the KTiOPO₄ (KTP) structure have received steady attention since its outstanding nonlinear optical properties were reported (Zumsteg et al., 1976). Its metal–oxygen chain structure is constructed on a special cis–trans principle which, in combination with a strong distortion of the [TiO₆] octahedra, provokes a significant non-linear effect (Bierlein, 1989). Previously, it was shown that KTiOPO₄ can serve as a model for the exploration of structure–property relationships by means of aliovalent substitution of K, Ti or P atoms (Stucky et al., 1989). Among the range of KTP family compounds, the non-linear polarizabilities of Nb^V– and Ta^V-doped KTiOPO₄ crystals are characterized by a high β_parallel:β_{perpendicular} ratio, and the non-linear response of Nb(Ta)—O bonds has been described in terms of a bond-additivity model (Bergman & Crane, 1975). [Original meaning not clear - please check rephrasing] These compounds may serve as ferroelastic-based multipurpose laser sources for scientific instruments, optical communications, optical signal processing, data storage etc. (Canalias, 2005).

Replacement of Ti^IV atoms by Nb^V is limited to 15 mol%, due to the structure transformations (Alekseeva et al., 2001) that accompany the decrease in the framework anionic charge and consequent lesser quantity of K⁺ cations relative to the parent KTiOPO₄. Therefore, two necessary conditions need to be addressed for the design of NLO materials, namely the retention of the KTP structure and the prevalence of Nb^V in the composition of the target compound. This work continues the systematic study of the aliovalent combination approach [M^III + M^V] towards KTP-related structures and we report here the structure of a mixed In^III/Nb^V-substituted phase of formula K_0.92In_0.46Nb_0.54OPO₄ (hereinafter denoted KINP). A brief discussion of its merohedral twinning is also included.

Despite our expectations, the structure of KINP is essentially different from earlier reported structures of KM^III_0.5Nb^V_0.5OPO₄ phases with M = V, Cr or Fe (Rangan et al., 1998; Babaryk et al., 2006). The overwhelming majority of previously reported KTP family compounds containing three- and five-valent cations belong to the orthorhombic crystal system (mmm Laue group) and crystallize in either Pna2₁ (noncentrosymmetric) or Pnan (centrosymmetric) space groups, which are related by a ferroelectric–paraelectric phase transition (Belokoneva et al., 1990, 1993; Stefanovich et al., 1996). The precedent for increasing lattice symmetry has been observed for KMg_0.333Nb_0.667OPO₄ (KMNP; McCarron & Calabrese, 1993) and KNi_0.5W_0.5OPO₄ (KNiWP; Peuchert et al., 1995). These compounds belong to the tetragonal crystal system (4/mmm Laue group). By contrast, the KINP crystal structure displays 4/m Laue symmetry, unlike all previously published structures of the KTP family. This change in crystal symmetry may be associated with a gradual increase of octahedrally coordinated metal ion radii from 0.76–0.79 Å (Cr–Fe) to 0.83–0.94 Å (Mg–In).

The unit cell of KINP (Fig. 1) contains the atomic set (all atoms are in general positions) corresponding to KTiOPO₄. The In(Nb)2 atoms are located in a significantly distorted octahedral environment, with four In(Nb)2—O bonds of typical length (Table 1) close to the sum of the corresponding ionic radii and two bonds that are much shorter. For comparison of the local geometry of the octahedra, the bond-valence sum (BVS) method was used (Reference?). The BV parameter was additionally weighted to correct its value for competitive occupation of the octahedral position by the following equation:

BV = BV_M1q_M1 + BV_M2q_M2,

where BV is the bond valence, q is the site occupancy, and M1 and M2 are In or Nb, respectively.

BV parameter values for the shortened bonds are 1.254 and 1.214, while the shortest bonds of the [TiO₆] octahedra in the KTP structure correspond to bond-valence values of 1.300 and 1.231 (Brown & Altermatt, 1985; Brese & O'Keeffe, 1991; Tytko et al., 1999). The [In(Nb)1O₆] octahedron adopts a comparatively undistorted configuration (Table 1). The In atoms selectively occupy the In(Nb)1 position [qIn = 0.8307 (12) and qNb = 0.1693 (12)], while In(Nb)2 are preferentially occupied by Nb [qIn = 0.086 (3) and qNb = 0.914 (3)]. Based on the distribution of the metal atoms over the two positions, it is clear that the degree of distortion of the octahedron depends on the nature of the metal which is in preference at the site.

Sequential alternation of two regular In(Nb)1—O6 and In(Nb)1—O10 bonds and two shorter In(Nb)2—O6 and In(Nb)2—O10 bonds forms an infinite helical chain stretching along the [001] direction, in contrast with the reference KTP structure where the Ti atoms are coplanar. Owing to this circumstance, the only possible means of chain linkage is a cis–cis type (Fig. 2a). This type of chain was reported earlier for orthorhombic γ-NaTiOPO₄ (Nagornyi et al., 1990), and tetragonal KMg_0.333Nb_0.667OPO₄ (McCarron & Calabrese, 1993) and KNi_0.5W_0.5OPO₄ (Peuchert et al., 1995). Each KINP octahedron is bonded to two neighbours via oxygen vertices along the O6—O10 edge, whereas in KTiOPO₄, the octahedra are additionally linked via apical vertices, giving rise to trans fragments (Fig. 2b). In KINP, the formation of the cis–cis chain is essentially [Is this in the sense of crucial or the sense of fundamental?] associated with the connectivity of Nb^V atoms. However, differences in the local geometry of the octahedra distinguish the cis–cis linkages of the tetragonal aliovalent analogues. For KMNP, only shortened bonds [Nb—O = 1.980 (1) Å] participate in the construction of the chain and hence no alternation is observed. In KNiWP, both Ni and W atoms possess twofold axis symmetry requiring the equal approach of the W atom to two Ni atoms.

The symmetry relations within the [In(Nb)O₆]_∞ chains determine the polarity along the chains, which is critically important for an NLO medium (Shaldin et al., 2006). The In(Nb)—O bonds that are not involved in the formation of the chains complete the coordination of the In(Nb) atoms through the two independent PO₄ groups. In turn, each PO₄ tetrahedron bonds to the two closest [In(Nb)O₆] octahedra in adjacent chains, giving a framework structure (Fig. 3).

An anionic sublattice contains one-dimensional hollow channels running orthogonal to the ab plane (Fig. 4). These channels are populated by the K atoms, which are separated by a 1/4 translation along c. Both K1 and K2 are coordinated by eight O atoms, with K1—O distances of 2.674 (5)–3.306 (5) Å and K2—O distances of 2.655 (5)–3.194 (4) Å.

S2. Experimental

Crystal growth experiments for K_0.92In_0.46Nb_0.54OPO₄ were carried out by a self-flux method in the melted system K₂O–In₂O₃–Nb₂O₅–P₂O₅. The initial mixture of KPO₃ (analytically pure), K₄P₂O₇ (analytically pure), In₂O₃ (extra pure) and Nb₂O₅ (extra pure) was melted at 1373 (20) K and exposed at this temperature for 1–2 h. The melt was stirred once or twice every 30 s using a platinum stirring rod during the exposure period. The melt was cooled at a rate of 20 K h¹ in the range 1033–1093 K [Not clear - what was the final temperature?]. Biaxial prismatic crystals were separated from the flux by decantation at the final crystallization temperature and were washed with dilute H₃PO₄ solution. The quantities of In₂O₃ and Nb₂O₅ were equimolar and they constituted 10–15% weight of the initial mixture. Suitable values of the K₂O:P₂O₅ ratio for the synthesis vary in the range 1.4–1.5. However, the pure compound could only be obtained when the ratio was equal to 1.5. Decreasing this ratio provokes langbeinite-type tetrahedral and octahedral crystalline impurities (Zatovsky et al., 2006). Precise unit-cell parameters were determined using an automated Siemens D500 powder diffractometer operating in a Bragg–Brentano (θ/2θ) geometry (Cu Kα radiation, λ = 1.54184 Å; curved graphite monochromator on the counter arm; step size 0.021; scanning rate 10 s per point). The In and Nb contents were determined using X-ray fluorescence analysis (Philips PW1400 spectrometer). Additional analysis of the elements K, In, Nb and P was performed by energy dispersive spectroscopy using a Link Isis analyser mounted on a Philips XL 30 FEG scanning electron microscope.

S3. Refinement

The structure twinning was initially indicated by a low value of E²-1. The space group was chosen among those in the 4/m Laue class, despite the small difference in R_int values compared with space groups belonging to the 4/mmm Laue class that served as an additional reason to consider twinning by merohedry. All attempts to solve the structure in space groups of higher symmetry failed. The TWINLAW (Schlessman & Litvin, 1995) analysis of the data is in good agreement with that of the TWINROTMAT subroutine implemented in the PLATON software package (Spek, 2003). The twinning corresponds to a twofold axis rotation along the [110] direction, which is a recognized phenomenon for tetragonal structures (Parsons, 2003). The cause of the twinning might be a phase transition, as mentioned for aliovalent KTP analogues (Peuchert et al., 1997), accompanied with loss of symmetry elements (Parsons, 2003).

Competitive occupancies of In and Nb atoms were refined according to chemical analysis results using free variables with linear restraints (SHELXL97; Sheldrick, 1997).

Computing details

Data collection: CrysAlis CCD (Oxford Diffraction, 2005); cell refinement: CrysAlis CCD (Oxford Diffraction, 2005); data reduction: CrysAlis RED (Oxford Diffraction, 2005); program(s) used to solve structure: SHELXS97 (Sheldrick, 1997); program(s) used to refine structure: SHELXL97 (Sheldrick, 1997); molecular graphics: DIAMOND (Brandenburg, 1999); software used to prepare material for publication: WinGX (Farrugia, 1999).

Figures

	Fig. 1. A connected set of numbered atoms, showing displacement ellipsoids at the 70% probability level. M is In or Nb. [Symmetry codes: (i) 1 − y, x, 1/4 + z; (ii) −x, 1 − y, 1/2 − z; (iii) −1 + y, 1 − x, −1/4 + z; (iv) −x, 1 − y, 1/2 + z; (v) −1 + y, 1 − x, 3/4 − z.]
	Fig. 2. (a) The cis-linkage of [In(Nb)O6] octahedra in KINP arranged into the helical chain. (b) The alternating cis and trans fragments within the single [TiO6]∞ chain in KTP (Tordjman et al., 1974). All Ti atoms lie in the same plane.
	Fig. 3. The architecture of the anionic sublattice {[In0.46Nb0.54OPO4]^−}∞ in the best-view projection. [PO4] tetrahedra and [In(Nb)O6] octahedra are shaded dark grey and light grey, respectively.
	Fig. 4. A view, along the c axis, of the square tunnels that are occupied by K^+ cations (not shown).

Potassium indium niobium oxide phosphate

Crystal data

K0.92In0.46Nb0.54OPO4	Dx = 3.489 Mg m^−3
Mr = 249.76	Mo Kα radiation, λ = 0.71073 Å
Tetragonal, P41	Cell parameters from 590 reflections
Hall symbol: P 4w	θ = 15–25°
a = 9.3091 (3) Å	µ = 4.71 mm^−1
c = 10.9744 (4) Å	T = 293 K
V = 951.03 (6) Å^3	Prism, colourless
Z = 8	0.15 × 0.1 × 0.1 mm
F(000) = 936.7

Data collection

Oxford Diffraction Xcalibur-3 diffractometer	2769 independent reflections
Radiation source: sealed tube	2713 reflections with I > 2σ(I)
Graphite monochromator	Rint = 0
ϕ and ω scans	θmax = 30°, θmin = 2.9°
Absorption correction: multi-scan (Blessing, 1995)	h = −8→13
Tmin = 0.596, Tmax = 0.615	k = −8→13
2769 measured reflections	l = −15→15

Refinement

Refinement on F^2	w = 1/[σ^2(Fo^2) + (0.0299P)^2 + 2.4076P] where P = (Fo^2 + 2Fc^2)/3
Least-squares matrix: full	(Δ/σ)max = 0.001
R[F^2 > 2σ(F^2)] = 0.020	Δρmax = 1.5 e Å^−3
wR(F^2) = 0.053	Δρmin = −0.94 e Å^−3
S = 1.06	Extinction correction: SHELXL97 (Sheldrick, 1997), Fc^*=kFc[1+0.001xFc^2λ^3/sin(2θ)]^-1/4
2769 reflections	Extinction coefficient: 0.0026 (2)
153 parameters	Absolute structure: Flack (1983), with how many Friedel pairs?
4 restraints	Absolute structure parameter: −0.03 (3)

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.

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

	x	y	z	Uiso*/Ueq	Occ. (<1)
In1	0.14136 (3)	0.63012 (3)	0.27353 (4)	0.00766 (7)	0.8307 (12)
Nb1	0.14136 (3)	0.63012 (3)	0.27353 (4)	0.00766 (7)	0.1693 (12)
In2	−0.10603 (3)	0.61473 (3)	0.52301 (4)	0.00815 (7)	0.086 (3)
Nb2	−0.10603 (3)	0.61473 (3)	0.52301 (4)	0.00815 (7)	0.914 (3)
P1	0.17878 (10)	0.83201 (9)	0.52243 (11)	0.00695 (17)
P2	0.32683 (10)	0.67317 (10)	0.01576 (10)	0.00644 (17)
K1	0.2747 (2)	0.51622 (14)	0.72389 (14)	0.0414 (5)	0.971 (5)
K2	−0.00476 (15)	0.77528 (18)	−0.02728 (15)	0.0314 (4)	0.862 (5)
O1	0.0198 (3)	0.7907 (3)	0.5436 (3)	0.0115 (6)
O2	0.2662 (5)	0.7827 (4)	0.6332 (3)	0.0120 (7)
O3	0.2428 (5)	0.7361 (5)	0.1228 (3)	0.0132 (7)
O4	0.2402 (5)	0.7544 (5)	0.4094 (3)	0.0132 (8)
O5	0.2779 (5)	0.7538 (5)	−0.1002 (3)	0.0128 (8)
O6	0.0059 (4)	0.5361 (5)	0.4031 (3)	0.0137 (8)
O7	0.4883 (3)	0.6807 (3)	0.0362 (3)	0.0126 (6)
O8	0.2848 (3)	0.5140 (3)	−0.0045 (3)	0.0121 (6)
O9	0.1893 (3)	0.9934 (3)	0.5030 (3)	0.0123 (6)
O10	−0.0365 (5)	0.4979 (4)	0.6452 (3)	0.0116 (8)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
In1	0.01004 (13)	0.00723 (12)	0.00570 (10)	0.00272 (8)	0.00034 (14)	−0.00078 (14)
Nb1	0.01004 (13)	0.00723 (12)	0.00570 (10)	0.00272 (8)	0.00034 (14)	−0.00078 (14)
In2	0.01012 (15)	0.00895 (14)	0.00539 (12)	−0.00411 (9)	0.00021 (18)	0.0008 (2)
Nb2	0.01012 (15)	0.00895 (14)	0.00539 (12)	−0.00411 (9)	0.00021 (18)	0.0008 (2)
P1	0.0075 (4)	0.0069 (4)	0.0064 (4)	−0.0008 (3)	−0.0012 (4)	−0.0006 (5)
P2	0.0058 (4)	0.0059 (4)	0.0076 (5)	−0.0027 (3)	0.0011 (4)	0.0007 (4)
K1	0.0803 (13)	0.0144 (6)	0.0295 (7)	−0.0164 (6)	−0.0114 (7)	0.0037 (5)
K2	0.0133 (6)	0.0462 (9)	0.0346 (8)	0.0116 (6)	−0.0069 (5)	−0.0095 (7)
O1	0.0065 (13)	0.0151 (14)	0.0128 (17)	−0.0052 (10)	0.0015 (11)	−0.0001 (12)
O2	0.0180 (18)	0.0073 (16)	0.0106 (13)	0.0006 (12)	−0.0058 (13)	0.0009 (11)
O3	0.0142 (17)	0.0120 (17)	0.0133 (16)	0.0016 (14)	0.0038 (13)	−0.0018 (13)
O4	0.0108 (16)	0.0193 (18)	0.0095 (15)	−0.0014 (14)	−0.0008 (13)	−0.0073 (13)
O5	0.0117 (17)	0.0155 (17)	0.0111 (15)	−0.0010 (13)	−0.0005 (12)	0.0050 (13)
O6	0.018 (2)	0.0136 (17)	0.0099 (18)	−0.0001 (10)	0.0049 (13)	−0.0008 (13)
O7	0.0080 (12)	0.0154 (13)	0.0144 (16)	−0.0040 (10)	−0.0015 (13)	−0.0003 (13)
O8	0.0142 (14)	0.0074 (13)	0.0147 (17)	−0.0008 (10)	0.0034 (11)	−0.0005 (10)
O9	0.0146 (13)	0.0061 (12)	0.0162 (17)	−0.0035 (10)	0.0008 (12)	0.0028 (11)
O10	0.0116 (16)	0.0107 (18)	0.0124 (18)	0.0001 (9)	0.0001 (12)	0.0027 (13)

Geometric parameters (Å, º)

In1—O10^i	2.087 (4)	P2—O5	1.546 (4)
In1—O6	2.093 (4)	P2—O8	1.549 (3)
In1—O4	2.100 (4)	K1—O2	2.674 (4)
In1—O7^ii	2.123 (3)	K1—O4^ii	2.679 (5)
In1—O3	2.145 (4)	K1—O8^vi	2.907 (3)
In1—O9^iii	2.185 (3)	K1—O5^vii	2.935 (4)
In2—O6	1.831 (4)	K1—O2^ii	2.960 (5)
In2—O10	1.843 (4)	K1—O8^vii	2.982 (3)
In2—O1	2.026 (3)	K1—O10	3.029 (5)
In2—O8^iv	2.073 (3)	K1—O6^iv	3.306 (5)
In2—O5^v	2.153 (4)	K2—O3^iii	2.655 (5)
In2—O2^iii	2.158 (4)	K2—O5	2.758 (4)
P1—O9	1.521 (3)	K2—O9^viii	2.775 (3)
P1—O2	1.533 (4)	K2—O3	2.856 (5)
P1—O4	1.545 (4)	K2—O6^i	2.998 (5)
P1—O1	1.547 (3)	K2—O4^iii	3.040 (5)
P2—O7	1.521 (3)	K2—O9^iii	3.094 (4)
P2—O3	1.528 (4)	K2—O10^i	3.194 (4)
O10^i—In1—O6	86.46 (9)	O4^ii—K1—O2^ii	51.94 (10)
O10^i—In1—O4	177.14 (18)	O8^vi—K1—O2^ii	126.26 (11)
O6—In1—O4	90.68 (18)	O5^vii—K1—O2^ii	101.88 (13)
O10^i—In1—O7^ii	93.09 (15)	O2—K1—O8^vii	112.29 (11)
O6—In1—O7^ii	99.54 (15)	O4^ii—K1—O8^vii	105.11 (10)
O4—In1—O7^ii	87.37 (15)	O8^vi—K1—O8^vii	84.19 (9)
O10^i—In1—O3	87.02 (18)	O5^vii—K1—O8^vii	49.28 (9)
O6—In1—O3	169.03 (18)	O2^ii—K1—O8^vii	53.83 (9)
O4—In1—O3	95.81 (12)	O2—K1—O10	85.27 (13)
O7^ii—In1—O3	89.61 (15)	O4^ii—K1—O10	76.92 (14)
O10^i—In1—O9^iii	94.31 (14)	O8^vi—K1—O10	142.40 (11)
O6—In1—O9^iii	90.67 (14)	O5^vii—K1—O10	103.86 (12)
O4—In1—O9^iii	85.72 (15)	O2^ii—K1—O10	87.22 (11)
O7^ii—In1—O9^iii	167.73 (11)	O8^vii—K1—O10	108.34 (10)
O3—In1—O9^iii	81.03 (14)	O2—K1—O6^iv	109.55 (14)
O6—In2—O10	95.00 (12)	O4^ii—K1—O6^iv	86.32 (12)
O6—In2—O1	94.26 (18)	O8^vi—K1—O6^iv	130.60 (11)
O10—In2—O1	101.12 (17)	O5^vii—K1—O6^iv	74.20 (12)
O6—In2—O8^iv	96.96 (18)	O2^ii—K1—O6^iv	52.40 (11)
O10—In2—O8^iv	92.67 (17)	O8^vii—K1—O6^iv	55.18 (9)
O1—In2—O8^iv	161.38 (11)	O10—K1—O6^iv	53.52 (7)
O6—In2—O5^v	174.05 (17)	O3^iii—K2—O5	137.93 (14)
O10—In2—O5^v	90.94 (17)	O3^iii—K2—O9^viii	62.37 (11)
O1—In2—O5^v	84.44 (14)	O5—K2—O9^viii	133.00 (13)
O8^iv—In2—O5^v	82.86 (14)	O3^iii—K2—O3	164.42 (15)
O6—In2—O2^iii	88.14 (17)	O5—K2—O3	52.27 (10)
O10—In2—O2^iii	171.36 (19)	O9^viii—K2—O3	121.98 (12)
O1—In2—O2^iii	86.65 (15)	O3^iii—K2—O6^i	80.24 (13)
O8^iv—In2—O2^iii	78.94 (14)	O5—K2—O6^i	81.90 (13)
O5^v—In2—O2^iii	85.99 (12)	O9^viii—K2—O6^i	141.16 (11)
O9—P1—O2	111.9 (2)	O3—K2—O6^i	91.50 (12)
O9—P1—O4	109.0 (2)	O3^iii—K2—O4^iii	66.76 (9)
O2—P1—O4	107.47 (19)	O5—K2—O4^iii	153.60 (12)
O9—P1—O1	109.18 (17)	O9^viii—K2—O4^iii	59.96 (11)
O2—P1—O1	108.3 (2)	O3—K2—O4^iii	101.56 (13)
O4—P1—O1	111.0 (2)	O6^i—K2—O4^iii	97.08 (12)
O7—P2—O3	112.0 (2)	O3^iii—K2—O9^iii	114.36 (11)
O7—P2—O5	113.0 (2)	O5—K2—O9^iii	107.55 (10)
O3—P2—O5	107.2 (2)	O9^viii—K2—O9^iii	78.15 (9)
O7—P2—O8	108.35 (17)	O3—K2—O9^iii	56.27 (10)
O3—P2—O8	110.4 (2)	O6^i—K2—O9^iii	110.87 (10)
O5—P2—O8	105.8 (2)	O4^iii—K2—O9^iii	48.01 (9)
O2—K1—O4^ii	142.03 (13)	O3^iii—K2—O10^i	107.68 (14)
O2—K1—O8^vi	57.45 (11)	O5—K2—O10^i	89.96 (12)
O4^ii—K1—O8^vi	135.28 (14)	O9^viii—K2—O10^i	128.43 (12)
O2—K1—O5^vii	63.01 (9)	O3—K2—O10^i	57.28 (13)
O4^ii—K1—O5^vii	153.81 (12)	O6^i—K2—O10^i	51.77 (7)
O8^vi—K1—O5^vii	57.21 (10)	O4^iii—K2—O10^i	69.62 (12)
O2—K1—O2^ii	160.74 (14)	O9^iii—K2—O10^i	59.75 (9)

Symmetry codes: (i) −x, −y+1, z−1/2; (ii) −y+1, x, z+1/4; (iii) y−1, −x+1, z−1/4; (iv) −x, −y+1, z+1/2; (v) y−1, −x+1, z+3/4; (vi) y, −x+1, z+3/4; (vii) x, y, z+1; (viii) −x, −y+2, z−1/2.