Crystal structure of a new tripotassium hexa­nickel iron hexa­phosphate

K₃Ni₆Fe(PO₄)₆ has been synthesized by solid-state reaction and structurally characterized by single-crystal X-ray diffraction. Its structure is built up by [PO₄] tetra­hedra and [FeO₆] and [NiO₆] octa­hedra linked to each other by edge or corner sharing, leading to a three-dimensional framework delimiting tunnels along the [100] direction in which the K⁺ cations are localized.

Abstract

A new potassium-nickel iron phosphate, K₃Ni₆Fe(PO₄)₆, has been synthesized by solid-state reaction and structurally characterized by single-crystal X-ray diffraction and qualitative energy dispersive X-ray spectroscopy (EDS) analysis. The structure is built up by [FeO₆], [PO₄], and [NiO₆] coordination polyhedra, which are linked to each other by edge and corner sharing to form zigzag layers parallel to the ab plane. These layers are inter­connected by [PO₄] tetra­hedra and [NiO₆] octa­hedra via common corners, leading to a three-dimensional framework delimiting large channels running along the [100] direction in which the K⁺ cations are localized.

Chemical context  

Iron-based phosphates are widely studied materials today. They present a promising field for various applications such as electronics (Saw et al., 2014 ▸), ferroelectrics (Lazoryak et al., 2004 ▸), magnetic materials (Hatert et al., 2004 ▸; Essehli et al., 2015 ▸) and catalytic processes (Moffat, 1978 ▸). The introduction of alkali metals into these phosphates materials can be of great inter­est to improve the ion-conduction properties for applications in rechargeable alkaline batteries (La Parola et al., 2018 ▸; Orikasa et al., 2016 ▸). The present work is part of our activity devoted particularly to the investigation of new materials based on phosphates belonging to the A ₂O–MO–Fe₂O₃–P₂O₅ (A = an alkali metal; M = divalent cation) quaternary system, which could have inter­esting ionic conductivity or magnetic proprieties. We report herein on the synthesis and structural characterization by single crystal X-ray diffraction of a new potassium nickel iron phosphate with formula K₃Ni₆Fe(PO₄)₆.

Structural commentary  

The asymmetric unit of the title compound, K₃Ni₆Fe(PO₄)₆, consists of two [NiO₆] octa­hedra, one [FeO₆] octa­hedron, two [PO₄] tetra­hedra, and three K atoms, as shown in Fig. 1 ▸. One Ni²⁺, Fe³⁺, P⁵⁺, two K⁺ cations and two of the seven oxygen atoms lie on special positions. The Ni2 atom occupies Wyckoff position 4g (2), the Fe atom is localized on the 2a (2/m) Wyckoff position, P2, K1, K3, O6 and O7 are positioned on 4i (m) sites. The octa­hedral coordination sphere of the nickel(II) cation is more distorted than that of the iron(III) atom, with average <Ni—O> distances of 2.066 and 2.119 Å for Ni1 and Ni2, respectively. The mean <P—O> distance in the two PO₄ tetra­hedra is equal to 1.547 Å for P1 and 1.543 Å for P2. The Fe atoms are coordinated octa­hedrally with an average <Fe—O> distance of 2.038 Å. The structure of the title compound is built up from two types of nickel sites and one iron site, each with an octa­hedral coordination environment, [Ni1O₆], [Ni2O₆] and [FeO₆], besides two independent phosphor tetra­hedra [P1O₄] and [P2O₄]. Edge-sharing [Ni2O₆] octa­hedra build up a dimeric [Ni2₂O₁₀] unit. Two [P2O₆] octa­hedra are connected to the [Ni2₂O₁₀] dimer by sharing edges to form an [Ni(2)₂P(2)₂O₁₂] unit, which alternates with an [FeO₆] octa­hedron to establish an infinite chain along the [100] direction (Fig. 2 ▸). In addition, the association between the [P1O₄] tetra­hedra and the [Ni1O₆] octa­hedra by means of edge-sharing allows the formation of a zigzag chain running parallel to the [100] direction. Each of the P1O₄ tetra­hedra and Ni1O₆ octa­hedra, both belonging to the same layer, share vertices with Ni1O₆ and P1O₄, respectively, of the adjacent one (Fig. 3 ▸). The two types of chain linkages lead to the formation of layers parallel to the ab plane (Fig. 4 ▸). One vertex of an Ni1O₆ octa­hedron belonging to one layer is shared with a P1O₄ vertex of the neighbouring layer. This configuration leads to a three-dimensional centrosymmetric framework, delimiting hexa­gonal tunnels along the [100] direction, in which the K⁺ cations are located (Fig. 5 ▸). The potassium cations are distributed over three independent crystallographic positions with partial occupancies

Figure 1.

Mol­ecular structure of the title compound with the atom-labelling scheme. Displacement ellipsoids are drawn at the 50% probability level. Symmetry codes: (i) −x + , −y + , −z + 1; (ii) −x + 1, y, −z + 1; (iii) x, y, z − 1; (iv) −x + 1, y, −z + 2; (v) x + , −y + , z + 1; (vi) x + , y + , z + 1; (vii) −x + 1, −y + 1, −z; (viii) x, −y + 1, z − 1; (ix) −x + 1, −y + 1, −z + 1; (x) x, −y + 1, z; (xi) x, y, z + 1; (xii) −x + , −y + , −z; (xiii) −x, −y + 1, −z; (xiv) x, −y + 2, z.

Figure 2.

A chain formed by sharing edges and corners of [Ni2₂O₁₀] dimers, [P2O₄] tetra­hedra and [FeO₆] octa­hedra along the [100] direction

Figure 3.

Corner- and edge-sharing [P1O₄] tetra­hedra and [Ni1O₆] octa­hedra forming a zigzag shape chain running parallel to [100]

Figure 4.

View along the c axis of corner- and edge-sharing [PO₄] tetra­hedra and [NiO₆] octa­hedra forming a layer parallel to the ab plane.

Figure 5.

Polyhedral representation of the crystal structure of K₃Ni₆Fe(PO₄)₆ showing large tunnels running along the [100] direction that contain the K⁺ cations.

Database survey  

The investigated compound is a new member of the β-xenophyllite family that includes Na₄Ni₇(PO₄)₆ (Moring & Kostiner, 1986 ▸), Na₄Co₇(PO₄)₆ (Kobashi et al., 1998 ▸), K₄Ni₇(AsO₄)₆ (Ben Smail et al., 1999 ▸), Na₄Co_5.63Al_0.91(AsO₄)₆ (Marzouki et al., 2010 ▸), Na₄Li_0.62Co_5.67Al_0.71(AsO₄)₆ (Marzouki et al., 2013 ▸), Ag₄Co₇(AsO₄)₆ (Marzouki et al., 2014 ▸) and Na₄Co₇(AsO₄)₆ (Ben Smida et al., 2016 ▸). The phosphates of these compounds crystallize in the non-centrosymmetric Cm space group while the arsenates adopt the C2/m space group.

Synthesis and crystallization  

Single crystals of K₃Ni₆Fe(PO₄)₆ were prepared by solid-state reaction in air. A mixture of K₂CO₃, Ni(NO₃)₂·6H₂O, Fe(NO₃)₃·9H₂O and H₃PO₄ (85 wt.%) reagents with a K:Ni:Fe:P molar ratio of 2:2:1:3 was dissolved in 50 mL of distilled water. The resulting solution was stirred without heating for 24 h and was subsequently evaporated to dryness at 343 K. The obtained dry residue was progressively heated in a platinum crucible up to 673 K in order to eliminate volatile products. In a second step, the powder was homogenized in an agate mortar and then progressively heated to 1303 K. Kept at this temperature for 2 h, the reaction mixture then underwent slow cooling at a rate of 5 Kh⁻¹ to 1103 K and then to room temperature with the furnace inertia. After washing with distilled water, the obtained crystals were brown with block-type shape. A qualitative EDX analysis (energy dispersive X-ray spectroscopy) detected the presence of the expected chemical elements corresponding to K, Ni, Fe, P and O atoms (see Fig. 6).

Refinement  

Crystal data, data collection and structure refinement details of K₃Ni₆Fe(PO₄)₆ are summarized in Table 1 ▸. The highest peak and the deepest hole in the final Fourier map are at 0.71 and 0.59 Å, respectively, from atom K2.

Table 1. Experimental details.

Crystal data
Chemical formula	K3Ni6Fe(PO4)6
M r	1095.23
Crystal system, space group	Monoclinic, C2/m
Temperature (K)	296
a, b, c (Å)	10.6853 (4), 14.1009 (5), 6.5481 (2)
β (°)	103.842 (1)
V (Å3)	957.97 (6)
Z	2
Radiation type	Mo Kα
μ (mm−1)	7.79
Crystal size (mm)	0.36 × 0.27 × 0.20
Data collection
Diffractometer	Bruker D8 VENTURE Super DUO
Absorption correction	Multi-scan (SADABS; Krause et al., 2015 ▸)
T min, T max	0.638, 0.746
No. of measured, independent and observed [I > 2σ(I)] reflections	13870, 1981, 1891
R int	0.021
(sin θ/λ)max (Å−1)	0.781
Refinement
R[F 2 > 2σ(F 2)], wR(F 2), S	0.024, 0.062, 1.07
No. of reflections	1981
No. of parameters	112
Δρmax, Δρmin (e Å−3)	2.34, −1.16

Computer programs: APEX3 and SAINT-Plus (Bruker, 2016 ▸), SHELXT2014 (Sheldrick, 2015a ▸), SHELXL2016 (Sheldrick, 2015b ▸), ORTEP-3 for Windows (Farrugia, 2012 ▸), DIAMOND (Brandenburg, 2006 ▸) and publCIF (Westrip, 2010 ▸).

Supplementary Material

CCDC reference: 1898755

Additional supporting information: crystallographic information; 3D view; checkCIF report

supplementary crystallographic information

Crystal data

K3Ni6Fe(PO4)6	F(000) = 1066
Mr = 1095.23	Dx = 3.797 Mg m−3
Monoclinic, C2/m	Mo Kα radiation, λ = 0.71073 Å
a = 10.6853 (4) Å	Cell parameters from 1981 reflections
b = 14.1009 (5) Å	θ = 2.4–33.7°
c = 6.5481 (2) Å	µ = 7.79 mm−1
β = 103.842 (1)°	T = 296 K
V = 957.97 (6) Å3	Block, brown
Z = 2	0.36 × 0.27 × 0.20 mm

Data collection

Bruker D8 VENTURE Super DUO diffractometer	1981 independent reflections
Radiation source: INCOATEC IµS micro-focus source	1891 reflections with I > 2σ(I)
HELIOS mirror optics monochromator	Rint = 0.021
Detector resolution: 10.4167 pixels mm-1	θmax = 33.7°, θmin = 2.4°
φ and ω scans	h = −14→16
Absorption correction: multi-scan (SADABS; Krause et al., 2015)	k = −22→22
Tmin = 0.638, Tmax = 0.746	l = −10→10
13870 measured reflections

Refinement

Refinement on F2	112 parameters
Least-squares matrix: full	0 restraints
R[F2 > 2σ(F2)] = 0.024	w = 1/[σ2(Fo2) + (0.0262P)2 + 6.1957P] where P = (Fo2 + 2Fc2)/3
wR(F2) = 0.062	(Δ/σ)max = 0.001
S = 1.07	Δρmax = 2.34 e Å−3
1981 reflections	Δρmin = −1.16 e Å−3

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.

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

	x	y	z	Uiso*/Ueq	Occ. (<1)
Ni1	0.33034 (2)	0.69342 (2)	0.18429 (4)	0.00599 (6)
Ni2	0.500000	0.87799 (2)	1.000000	0.00683 (7)
Fe1	0.500000	0.500000	0.000000	0.00327 (9)
K1	0.40983 (18)	0.500000	0.4730 (2)	0.0200 (3)	0.472
K2	0.38224 (13)	0.90311 (11)	0.48618 (19)	0.0242 (3)	0.417
K3	0.1849 (7)	0.500000	0.4893 (8)	0.0560 (19)	0.192
P1	0.41063 (4)	0.69522 (3)	0.72442 (7)	0.00485 (8)
P2	0.19569 (6)	0.500000	−0.01061 (10)	0.00557 (11)
O1	0.31025 (13)	0.70594 (10)	0.8587 (2)	0.0072 (2)
O2	0.34525 (14)	0.68172 (11)	0.4959 (2)	0.0102 (2)
O3	0.50488 (13)	0.60977 (10)	0.7970 (2)	0.0069 (2)
O4	0.50525 (13)	0.78185 (10)	0.7692 (2)	0.0082 (2)
O5	0.19430 (13)	0.58983 (10)	0.1265 (2)	0.0085 (2)
O6	0.06136 (18)	0.500000	−0.1681 (3)	0.0087 (3)
O7	0.30728 (19)	0.500000	−0.1148 (3)	0.0103 (3)

Atomic displacement parameters (Ų)

	U11	U22	U33	U12	U13	U23
Ni1	0.00441 (10)	0.00697 (10)	0.00638 (10)	0.00038 (7)	0.00088 (7)	−0.00015 (7)
Ni2	0.00519 (13)	0.00602 (14)	0.00944 (14)	0.000	0.00207 (10)	0.000
Fe1	0.00218 (18)	0.00293 (18)	0.00478 (19)	0.000	0.00099 (15)	0.000
K1	0.0413 (9)	0.0080 (5)	0.0103 (5)	0.000	0.0055 (5)	0.000
K2	0.0259 (6)	0.0332 (7)	0.0122 (4)	−0.0107 (5)	0.0018 (4)	0.0082 (4)
K3	0.056 (4)	0.100 (6)	0.0100 (17)	0.000	0.005 (2)	0.000
P1	0.00395 (17)	0.00570 (18)	0.00483 (18)	0.00067 (13)	0.00093 (14)	−0.00035 (13)
P2	0.0033 (2)	0.0049 (2)	0.0082 (3)	0.000	0.00071 (19)	0.000
O1	0.0055 (5)	0.0095 (5)	0.0072 (5)	0.0022 (4)	0.0028 (4)	0.0009 (4)
O2	0.0091 (6)	0.0162 (6)	0.0048 (5)	0.0011 (5)	0.0003 (4)	0.0002 (4)
O3	0.0049 (5)	0.0070 (5)	0.0089 (5)	0.0014 (4)	0.0018 (4)	0.0012 (4)
O4	0.0069 (5)	0.0072 (5)	0.0110 (6)	−0.0012 (4)	0.0031 (4)	−0.0022 (4)
O5	0.0071 (5)	0.0067 (5)	0.0120 (6)	−0.0012 (4)	0.0027 (4)	−0.0031 (4)
O6	0.0045 (7)	0.0124 (8)	0.0081 (8)	0.000	−0.0005 (6)	0.000
O7	0.0056 (7)	0.0088 (8)	0.0181 (9)	0.000	0.0060 (7)	0.000

Geometric parameters (Å, º)

Ni1—O2	2.0153 (14)	K1—O7xi	3.146 (3)
Ni1—O5	2.0314 (14)	K2—O4	2.631 (2)
Ni1—O1i	2.0366 (13)	K2—O6xii	2.677 (2)
Ni1—O3ii	2.0984 (14)	K2—O2i	2.737 (2)
Ni1—O1iii	2.0988 (14)	K2—O5i	2.8470 (19)
Ni1—O4ii	2.1161 (14)	K2—O4ii	2.851 (2)
Ni2—O4	2.0411 (14)	K2—O6vi	2.929 (2)
Ni2—O4iv	2.0411 (14)	K2—O1i	3.0776 (19)
Ni2—O5v	2.0928 (14)	K2—O7xii	3.086 (2)
Ni2—O5i	2.0928 (14)	K2—O2	3.149 (2)
Ni2—O6i	2.2241 (13)	K3—O7xi	2.609 (6)
Ni2—O6vi	2.2241 (13)	K3—O5	2.715 (5)
Fe1—O7	2.016 (2)	K3—O5x	2.715 (5)
Fe1—O7vii	2.016 (2)	K3—O6xi	2.862 (6)
Fe1—O3viii	2.0490 (13)	K3—O6xiii	2.949 (7)
Fe1—O3ii	2.0490 (13)	K3—O2	3.077 (4)
Fe1—O3ix	2.0490 (13)	K3—O2x	3.077 (4)
Fe1—O3iii	2.0490 (13)	P1—O2	1.5042 (15)
K1—O3	2.6263 (18)	P1—O1	1.5482 (14)
K1—O3x	2.6264 (18)	P1—O4	1.5679 (14)
K1—O2	2.6673 (16)	P1—O3	1.5694 (14)
K1—O2x	2.6673 (16)	P2—O7	1.509 (2)
K1—O3ix	2.6691 (19)	P2—O6	1.554 (2)
K1—O3ii	2.6691 (19)	P2—O5x	1.5546 (14)
K1—O5	3.091 (2)	P2—O5	1.5546 (14)
K1—O5x	3.091 (2)
O2—Ni1—O5	90.55 (6)	O3ix—K1—O5	93.66 (6)
O2—Ni1—O1i	94.23 (6)	O3ii—K1—O5	65.70 (5)
O5—Ni1—O1i	90.23 (6)	O3—K1—O5x	155.60 (8)
O2—Ni1—O3ii	91.89 (6)	O3x—K1—O5x	115.32 (5)
O5—Ni1—O3ii	99.19 (5)	O2—K1—O5x	106.12 (6)
O1i—Ni1—O3ii	168.72 (5)	O2x—K1—O5x	59.36 (5)
O2—Ni1—O1iii	178.70 (6)	O3ix—K1—O5x	65.70 (5)
O5—Ni1—O1iii	88.65 (6)	O3ii—K1—O5x	93.66 (6)
O1i—Ni1—O1iii	84.74 (6)	O5—K1—O5x	48.38 (6)
O3ii—Ni1—O1iii	89.25 (5)	O3—K1—O7xi	56.89 (5)
O2—Ni1—O4ii	92.33 (6)	O3x—K1—O7xi	56.89 (5)
O5—Ni1—O4ii	169.40 (5)	O2—K1—O7xi	78.69 (5)
O1i—Ni1—O4ii	99.72 (5)	O2x—K1—O7xi	78.69 (5)
O3ii—Ni1—O4ii	70.53 (5)	O3ix—K1—O7xi	144.33 (3)
O1iii—Ni1—O4ii	88.64 (5)	O3ii—K1—O7xi	144.33 (3)
O4—Ni2—O4iv	96.75 (8)	O5—K1—O7xi	106.17 (7)
O4—Ni2—O5v	103.69 (6)	O5x—K1—O7xi	106.17 (7)
O4iv—Ni2—O5v	92.95 (5)	O4—K2—O6xii	135.41 (8)
O4—Ni2—O5i	92.95 (5)	O4—K2—O2i	89.14 (6)
O4iv—Ni2—O5i	103.69 (6)	O6xii—K2—O2i	128.60 (7)
O5v—Ni2—O5i	154.96 (8)	O4—K2—O5i	66.22 (5)
O4—Ni2—O6i	160.71 (6)	O6xii—K2—O5i	147.28 (7)
O4iv—Ni2—O6i	94.81 (6)	O2i—K2—O5i	61.95 (5)
O5v—Ni2—O6i	91.06 (6)	O4—K2—O4ii	79.26 (7)
O5i—Ni2—O6i	69.28 (6)	O6xii—K2—O4ii	69.19 (5)
O4—Ni2—O6vi	94.80 (6)	K2xiv—K2—O4ii	126.86 (4)
O4iv—Ni2—O6vi	160.71 (6)	O2i—K2—O4ii	105.45 (6)
O5v—Ni2—O6vi	69.28 (6)	O5i—K2—O4ii	142.65 (7)
O5i—Ni2—O6vi	91.06 (6)	O4—K2—O6vi	68.58 (5)
O6i—Ni2—O6vi	78.66 (8)	O6xii—K2—O6vi	97.82 (7)
O7—Fe1—O7vii	180.0	O2i—K2—O6vi	126.41 (7)
O7—Fe1—O3viii	86.58 (5)	O5i—K2—O6vi	64.47 (5)
O7vii—Fe1—O3viii	93.42 (5)	O4ii—K2—O6vi	116.35 (6)
O7—Fe1—O3ii	93.42 (5)	O4—K2—O1i	109.01 (6)
O7vii—Fe1—O3ii	86.58 (5)	O6xii—K2—O1i	85.38 (5)
O3viii—Fe1—O3ii	180.00 (8)	O2i—K2—O1i	50.84 (4)
O7—Fe1—O3ix	93.42 (5)	O5i—K2—O1i	112.77 (6)
O7vii—Fe1—O3ix	86.58 (5)	O4ii—K2—O1i	64.63 (5)
O3viii—Fe1—O3ix	81.87 (8)	O6vi—K2—O1i	176.79 (6)
O3ii—Fe1—O3ix	98.13 (8)	O4—K2—O7xii	165.00 (7)
O7—Fe1—O3iii	86.58 (5)	O6xii—K2—O7xii	52.30 (6)
O7vii—Fe1—O3iii	93.42 (5)	O2i—K2—O7xii	78.76 (5)
O3viii—Fe1—O3iii	98.13 (8)	O5i—K2—O7xii	114.35 (7)
O3ii—Fe1—O3iii	81.87 (8)	O4ii—K2—O7xii	95.32 (5)
O3ix—Fe1—O3iii	180.00 (5)	O6vi—K2—O7xii	125.92 (6)
O2—P1—O1	110.92 (8)	O1i—K2—O7xii	56.34 (4)
O2—P1—O4	114.19 (8)	O4—K2—O2	52.08 (5)
O1—P1—O4	108.73 (8)	O6xii—K2—O2	125.05 (6)
O2—P1—O3	108.41 (8)	O2i—K2—O2	56.55 (6)
O1—P1—O3	112.61 (8)	O5i—K2—O2	87.27 (5)
O4—P1—O3	101.72 (8)	O4ii—K2—O2	59.33 (5)
O7—P2—O6	113.84 (12)	O6vi—K2—O2	120.57 (5)
O7—P2—O5x	112.26 (7)	O1i—K2—O2	56.94 (4)
O6—P2—O5x	104.37 (7)	O7xii—K2—O2	113.14 (5)
O7—P2—O5	112.26 (7)	O7xi—K3—O5	139.0 (2)
O6—P2—O5	104.37 (7)	O7xi—K3—O5x	139.0 (2)
O5x—P2—O5	109.14 (11)	O5—K3—O5x	55.61 (11)
O3—K1—O3x	72.22 (7)	O7xi—K3—O6xi	55.72 (11)
O3—K1—O2	56.19 (4)	O5—K3—O6xi	143.7 (2)
O3x—K1—O2	125.17 (6)	O5x—K3—O6xi	143.7 (2)
O3—K1—O2x	125.17 (6)	O7xi—K3—O6xiii	149.1 (3)
O3x—K1—O2x	56.19 (4)	O5—K3—O6xiii	65.77 (12)
O2—K1—O2x	147.77 (10)	O5x—K3—O6xiii	65.77 (12)
O3—K1—O3ix	138.56 (8)	O6xi—K3—O6xiii	93.4 (2)
O3x—K1—O3ix	93.79 (6)	O7xi—K3—O2	80.84 (14)
O2—K1—O3ix	136.75 (7)	O5—K3—O2	59.12 (9)
O2x—K1—O3ix	67.30 (4)	O5x—K3—O2	105.29 (19)
O3—K1—O3ii	93.79 (6)	O6xi—K3—O2	110.40 (11)
O3x—K1—O3ii	138.56 (8)	O6xiii—K3—O2	114.13 (11)
O2—K1—O3ii	67.30 (4)	O7xi—K3—O2x	80.84 (14)
O2x—K1—O3ii	136.75 (7)	O5—K3—O2x	105.29 (19)
O3ix—K1—O3ii	70.89 (7)	O5x—K3—O2x	59.12 (9)
O3—K1—O5	115.32 (5)	O6xi—K3—O2x	110.40 (11)
O3x—K1—O5	155.60 (8)	O6xiii—K3—O2x	114.13 (11)
O2—K1—O5	59.36 (5)	O2—K3—O2x	112.8 (2)
O2x—K1—O5	106.12 (6)

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