A new rubidium beryllium borate, RbBe₄(BO₃)₃

Single crystals of a new rubdidium beryllium borate, RbBe₄(BO₃)₃, have been obtained by spontaneous nucleation from a high-temperature melt. This new ortho­rhom­bic (Pnma) structure type contains [Be₂BO₄]^- rings, made of two BeO₄ tetra­hedra and one BO₃ triangle, which constitute the basic structural units. The m plane runs through the B and one of the O atoms and intersects the ring. These rings form chains in the a direction, which are connected in the b and c directions to form zeolite-type cages in which the Rb⁺ cations are located, at sites of m symmetry.

S1. Comment

Since the discovery of β-BaB₂O₄ (BBO) as a nonlinear optical (NLO) material about 20 years ago (Chen et al., 1985), several borate NLO crystals have been developed which can sustain high laser damage (Chen et al., 1989; Wu et al., 1993; Mori et al.,1995). Based on a theoretical study, one of the present authors has proposed that beryllium borates possess the largest energy gap among all alkaline and alkaline earth borates, and hence the shortest transmission cut-off wavelength (Li, 1989). Therefore, beryllium borates are good candidates for NLO applications in the ultraviolet (UV) region. In reality, two families of crystals, ABe₂BO₃F₂ (A = Na, K, Rb and Cs) (Baydina et al., 1975; Mei et al., 1994, 1995) and M₂Be₂B₂O₇ (M = Sr and Ba) (Chen et al., 1995; Qi & Chen, 2001), have been found to crystallize in noncentrosymmetric space groups and to possess NLO properties. Among these, KBBF (KBe₂BO₃F₂) has proved to be the best material for deep-UV (DUV) applications and is the only material that can generate coherent light down to 170 nm by direct second-harmonic generation (Lu et al., 2001). Apart from these two families of compounds, relatively few synthetic beryllium borate compounds are known. The compounds MBe₂B₂O₆ (M = Sr and Ba) (Schaffers & Keszler, 1990, 1994), CaBeB₂O₅ (Schaffers & Keszler, 1993), and Li₁₄Be₅B(BO₃)₉ (Luce et al., 1994) have only been partially studied, because of the high toxicity of the Be compounds used as starting materials during their preparation. In this paper, we describe the crystal structure of a new rubidium beryllium borate, RbBe₄(BO₃)₃.

A perspective view of the RbBe₄(BO₃)₃ structure along the a direction is shown in Fig. 1. In the structure, both B atoms are coordinated to three O atoms to form planar BO₃ groups, with a mean B—O bond length of 1.372 Å and O—B—O bond angles ranging from 115.9 (2) to 123.7 (2)°. The two Be atoms are bonded to four O atoms and the BeO₄ tetrahedra are distorted, with Be—O bond lengths ranging from 1.585 (3) to 1.666 (3) Å and O—Be—O angles from 95.5 (2) to 116.3 (2)°. All O atoms except O3 are in threefold coordinated, as is typical for dense borate compounds (Parthé, 2004). Except for atom O3, the B—O bond lengths are close to their average value of 1.372 Å. The B2—O3 bond involving the two-coordinated atom O3 is the shortest [1.339 (3) Å] and is significantly shorter than the mean B—O length of 1.365 Å in the BO₃ groups of most borate compounds (Wells, 1984). The same observation applies to the Be1—O3 bond. Even shorter B—O bonds have been found in other rubidium borate compounds, such as RbLi₂Ga₂(BO₃)₃ (Kissick et al., 2000) and LiRbB₄O₇ (Ono et al., 2000). The short B2—O3 and Be1—O3 bonds reflect not only strong Be1—O3 and B2—O3 bonding, but also underbonding between the Rb and O3 atoms. The small O5—Be2—O5 bond angle [95.5 (2)°] corresponds to edge-sharing between two very distorted Be2O₄ tetrahedra.

In the RbBe₄(BO₃)₃ structure, two BeO₄ tetrahedra share one common O atom, and each of them also shares a different O atom with a BO₃ group, to form a six-membered [Be₂BO_2x3/3O_3/3O_2/2]¹⁻ ring (hereinafter referred to as [Be₂BO₄]¹⁻) (Fig. 2). These rings are repeated in the a direction to form a zigzag chain. The basic structural unit, [Be₂BO₄]¹⁻, is similar to the [Be₂BO₃F₂]¹⁻ unit in KBe₂BO₃F₂. However, the [Be₂BO₃F₂]¹⁻ rings in KBBF are coplanar with the Be—F bonds alternating up and down, whereas in the title compound, both Be—O bonds point in the same direction and bend the [Be₂BO₄]¹⁻ unit into a zigzag chain. From the study of KBBF and Sr₂Be₂B₂O₇, it is known that the [Be₂BO₄]¹⁻ unit can yield large NLO effects and short UV transmission cut-offs. If the [Be₂BO₄]_∞ chain can be arranged favourably, the resulting structure may be a good candidate for DUV NLO applications. Unfortunately, in the case of RbBe₄(BO₃)₃, adjacent chains of [Be₂BO₄]¹⁻ rings in the b direction point in exactly opposite directions and, therefore, their contributions to the NLO effect cancel out.

The [Be₂BO₄]_∞ chains in RbBe₄(BO₃)₃ are connected by O—Be2—O—Be2—O and O—B2—O bonds in the b direction, and by O—Be1—O—B2—O bonds in the c direction, to build three-dimensional zeolite-like cages 6 Å in diameter and tunnels of 5 and 4 Å along the a and b axes, respectively (Fig. 1). The Rb⁺ cations are located at the centres of the cages and are coordinated by ten O atoms with a bond valence sum (BVS; Brese & O'Keeffe, 1991) of 0.94. The BVS value for Rb and O3 (−1.94) show that they are slightly underbonded and may indicate a possible mobilization of the Rb⁺ cations in the tunnels at elevated temperature.

S2. Experimental

Single crystals of RbBe₄(BO₃)₃ were obtained from a melt in a tightly covered Pt crucible with initial materials of BeF₂ (0.14 mol), BeO (0.14 mol), Rb₂CO₃ (0.187 mol), NaF (0.3 mol) and B₂O₃ (0.4 mol). The melt was cooled slowly from 988 to 853 K at a rate of 3 K day⁻¹ and then cooled down to room temperature at a rate of 20 K h⁻¹. Block-shaped crystals of dimensions up to 8 × 4 × 4 mm³ were obtained after treating the contents of the crucible in hot dilute hydrochloric acid to remove the excess flux.

The title compound and its Cs analogue can also be synthesized by solid-state reaction, with the determined composition of RbBe₄(BO₃)₃.

S3. Refinement

The positions of the Rb atom and some of the O atoms were found in the first round of direct-phase determination. The remaining O, B and Be atoms were located in the subsequent difference Fourier syntheses. The atomic and anisotropic displacement parameters of all atoms were subjected to least-squares refinement. The final difference electron-density map shows a highest peak of 2.01 e Å⁻³ located 0.64 Å from Rb, and a deepest hole of −1.46 e Å⁻³ located 1.11 Å from Rb.

Computing details

Data collection: XSCANS (Bruker, 1997); cell refinement: XSCANS; data reduction: SHELXTL (Bruker, 1997); program(s) used to solve structure: SHELXS97 (Sheldrick, 1997); program(s) used to refine structure: SHELXL97 (Sheldrick, 1997); molecular graphics: ORTEP-3 for Windows (Farrugia, 1997); software used to prepare material for publication: enCIFer (Allen et al., 2004).

Figures

	Fig. 1. The crystal structure of RbBe4(BO3)3, viewed along the a axis.
	Fig. 2. The [Be2BO2x3/3O3/3O2/2]^1− unit in RbBe4(BO3)3. [Symmetry codes: (vii) 1/2 + x, y, 1/2 − z; (viii) x, 1/2 − y, z; (ix) 3/2 − x, 1 − y, −1/2 + z; (xii) 3/2 − x, −1/2 + y, −1/2 + z.]

rubidium tetraberyllium tris(borate)

Crystal data

RbBe4(BO3)3	F(000) = 560
Mr = 297.94	Dx = 2.789 Mg m^−3
Orthorhombic, Pnma	Mo Kα radiation, λ = 0.71073 Å
Hall symbol: -P 2ac 2n	Cell parameters from 44 reflections
a = 8.496 (4) Å	θ = 5.7–15.9°
b = 13.274 (6) Å	µ = 7.01 mm^−1
c = 6.292 (4) Å	T = 295 K
V = 709.7 (6) Å^3	Block, colourless
Z = 4	0.4 × 0.4 × 0.2 mm

Data collection

Bruker P4 diffractometer	1156 reflections with I > 2σ(I)
Radiation source: fine-focus sealed tube	Rint = 0.043
Graphite monochromator	θmax = 32.5°, θmin = 3.1°
ω scans	h = −12→12
Absorption correction: empirical (using intensity measurements) ψ scan (Reference)	k = −20→20
Tmin = 0.088, Tmax = 0.246	l = −9→9
2586 measured reflections	3 standard reflections every 97 reflections
1324 independent reflections	intensity decay: none

Refinement

Refinement on F^2	Primary atom site location: structure-invariant direct methods
Least-squares matrix: full	Secondary atom site location: difference Fourier map
R[F^2 > 2σ(F^2)] = 0.044	w = 1/[σ^2(Fo^2) + (0.0701P)^2 + 0.2245P] where P = (Fo^2 + 2Fc^2)/3
wR(F^2) = 0.112	(Δ/σ)max < 0.001
S = 1.06	Δρmax = 2.01 e Å^−3
1324 reflections	Δρmin = −1.46 e Å^−3
83 parameters	Extinction correction: SHELXL97 (Sheldrick, 1997), Fc^*=kFc[1+0.001xFc^2λ^3/sin(2θ)]^-1/4
0 restraints	Extinction coefficient: 0.083 (6)

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
Rb	0.56713 (4)	0.2500	0.67089 (5)	0.02922 (18)
B1	0.4493 (4)	0.2500	0.1890 (5)	0.0159 (5)
B2	0.6761 (3)	0.53556 (16)	0.3063 (4)	0.0171 (4)
Be1	0.7205 (3)	0.35637 (19)	0.1461 (4)	0.0184 (5)
Be2	0.4369 (3)	0.44254 (19)	0.0938 (4)	0.0176 (5)
O1	0.2972 (2)	0.2500	0.2605 (3)	0.0174 (4)
O2	0.52638 (19)	0.33876 (11)	0.1526 (2)	0.0185 (3)
O3	0.71178 (18)	0.61848 (11)	0.4175 (2)	0.0202 (3)
O4	0.76103 (19)	0.44568 (11)	0.3193 (2)	0.0190 (3)
O5	0.55048 (18)	0.53595 (12)	0.1638 (2)	0.0170 (3)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Rb	0.0289 (2)	0.0342 (2)	0.0246 (2)	0.000	−0.00395 (11)	0.000
B1	0.0157 (11)	0.0191 (12)	0.0129 (12)	0.000	−0.0017 (10)	0.000
B2	0.0173 (9)	0.0200 (8)	0.0140 (8)	0.0006 (7)	0.0001 (7)	0.0004 (7)
Be1	0.0174 (10)	0.0185 (10)	0.0193 (11)	−0.0004 (8)	−0.0001 (9)	0.0000 (8)
Be2	0.0180 (11)	0.0190 (10)	0.0157 (12)	0.0000 (7)	0.0005 (8)	−0.0008 (8)
O1	0.0161 (8)	0.0179 (8)	0.0183 (9)	0.000	0.0032 (7)	0.000
O2	0.0159 (6)	0.0179 (6)	0.0216 (7)	0.0000 (5)	0.0002 (5)	0.0009 (5)
O3	0.0219 (6)	0.0194 (6)	0.0193 (7)	0.0006 (5)	−0.0038 (6)	−0.0025 (5)
O4	0.0179 (6)	0.0191 (6)	0.0201 (7)	0.0016 (5)	−0.0032 (5)	−0.0018 (5)
O5	0.0175 (6)	0.0198 (6)	0.0136 (7)	0.0001 (5)	−0.0017 (4)	0.0001 (4)

Geometric parameters (Å, º)

Rb—O3^i	2.9954 (18)	Be1—O1^vii	1.662 (3)
Rb—O3^ii	2.9954 (17)	Be1—O2	1.667 (3)
Rb—O3^iii	2.9974 (18)	Be1—Be2	2.687 (4)
Rb—O3^iv	2.9974 (18)	Be1—Be2^vii	2.715 (4)
Rb—O5^ii	3.187 (2)	Be1—Rb^ix	3.554 (3)
Rb—O5^i	3.187 (2)	Be1—Rb^vii	3.827 (3)
Rb—B1	3.193 (4)	Be2—O4^xi	1.592 (3)
Rb—O2^v	3.271 (2)	Be2—O2	1.616 (3)
Rb—O2^vi	3.271 (2)	Be2—O5	1.632 (3)
Rb—O1^vii	3.345 (3)	Be2—O5^xii	1.649 (3)
Rb—B1^v	3.411 (4)	Be2—Be2^xii	2.206 (5)
Rb—O1	3.454 (3)	Be2—Be1^xi	2.715 (4)
B1—O2	1.367 (2)	Be2—Rb^ix	3.852 (3)
B1—O2^viii	1.367 (2)	O1—Be1^xiii	1.662 (3)
B1—O1	1.369 (4)	O1—Be1^xi	1.662 (3)
B1—Rb^ix	3.411 (4)	O1—Rb^xi	3.345 (3)
B2—O3	1.339 (3)	O2—Rb^ix	3.271 (2)
B2—O5	1.394 (3)	O3—Be1^iii	1.585 (3)
B2—O4	1.397 (3)	O3—Rb^ii	2.9954 (18)
B2—Rb^ii	3.520 (2)	O3—Rb^x	2.9974 (18)
B2—Rb^x	3.686 (2)	O4—Be2^vii	1.592 (3)
Be1—O3^x	1.585 (3)	O5—Be2^xii	1.649 (3)
Be1—O4	1.647 (3)	O5—Rb^ii	3.187 (2)
O3^i—Rb—O3^ii	71.30 (7)	O4—Be1—Be2	87.83 (14)
O3^i—Rb—O3^iii	158.65 (3)	O1^vii—Be1—Be2	139.13 (17)
O3^ii—Rb—O3^iii	104.62 (5)	O2—Be1—Be2	34.45 (9)
O3^i—Rb—O3^iv	104.62 (5)	O3^x—Be1—Be2^vii	102.28 (15)
O3^ii—Rb—O3^iv	158.65 (3)	O4—Be1—Be2^vii	32.43 (10)
O3^iii—Rb—O3^iv	71.25 (6)	O1^vii—Be1—Be2^vii	83.07 (14)
O3^i—Rb—O5^ii	109.41 (5)	O2—Be1—Be2^vii	135.61 (16)
O3^ii—Rb—O5^ii	44.99 (4)	Be2—Be1—Be2^vii	120.12 (13)
O3^iii—Rb—O5^ii	60.54 (5)	O3^x—Be1—Rb^ix	56.85 (10)
O3^iv—Rb—O5^ii	123.15 (5)	O4—Be1—Rb^ix	156.85 (14)
O3^i—Rb—O5^i	44.99 (4)	O1^vii—Be1—Rb^ix	95.94 (13)
O3^ii—Rb—O5^i	109.41 (5)	O2—Be1—Rb^ix	66.55 (10)
O3^iii—Rb—O5^i	123.15 (5)	Be2—Be1—Rb^ix	74.77 (8)
O3^iv—Rb—O5^i	60.54 (5)	Be2^vii—Be1—Rb^ix	157.38 (11)
O5^ii—Rb—O5^i	126.17 (6)	O3^x—Be1—Rb	170.47 (14)
O3^i—Rb—B1	64.90 (5)	O4—Be1—Rb	76.42 (11)
O3^ii—Rb—B1	64.90 (5)	O1^vii—Be1—Rb	60.94 (11)
O3^iii—Rb—B1	133.47 (5)	O2—Be1—Rb	65.74 (10)
O3^iv—Rb—B1	133.47 (5)	Be2—Be1—Rb	87.56 (9)
O5^ii—Rb—B1	102.23 (4)	Be2^vii—Be1—Rb	82.28 (9)
O5^i—Rb—B1	102.23 (4)	Rb^ix—Be1—Rb	117.08 (8)
O3^i—Rb—O2^v	107.35 (4)	O3^x—Be1—Rb^vii	47.57 (9)
O3^ii—Rb—O2^v	83.01 (4)	O4—Be1—Rb^vii	116.73 (13)
O3^iii—Rb—O2^v	51.44 (4)	O1^vii—Be1—Rb^vii	64.48 (11)
O3^iv—Rb—O2^v	78.25 (5)	O2—Be1—Rb^vii	136.29 (13)
O5^ii—Rb—O2^v	48.93 (4)	Be2—Be1—Rb^vii	141.55 (11)
O5^i—Rb—O2^v	89.16 (5)	Be2^vii—Be1—Rb^vii	87.05 (9)
B1—Rb—O2^v	147.88 (5)	Rb^ix—Be1—Rb^vii	72.37 (6)
O3^i—Rb—O2^vi	83.01 (4)	Rb—Be1—Rb^vii	125.21 (7)
O3^ii—Rb—O2^vi	107.35 (4)	O4^xi—Be2—O2	112.63 (17)
O3^iii—Rb—O2^vi	78.25 (5)	O4^xi—Be2—O5	116.26 (17)
O3^iv—Rb—O2^vi	51.44 (4)	O2—Be2—O5	107.93 (16)
O5^ii—Rb—O2^vi	89.16 (5)	O4^xi—Be2—O5^xii	113.20 (17)
O5^i—Rb—O2^vi	48.93 (4)	O2—Be2—O5^xii	110.03 (16)
B1—Rb—O2^vi	147.88 (5)	O5—Be2—O5^xii	95.48 (15)
O2^v—Rb—O2^vi	42.23 (6)	O4^xi—Be2—Be2^xii	128.4 (2)
O3^i—Rb—O1^vii	108.16 (5)	O2—Be2—Be2^xii	118.9 (2)
O3^ii—Rb—O1^vii	108.16 (5)	O5—Be2—Be2^xii	48.07 (11)
O3^iii—Rb—O1^vii	93.09 (5)	O5^xii—Be2—Be2^xii	47.41 (12)
O3^iv—Rb—O1^vii	93.09 (5)	O4^xi—Be2—Be1	144.16 (17)
O5^ii—Rb—O1^vii	116.63 (3)	O2—Be2—Be1	35.67 (10)
O5^i—Rb—O1^vii	116.63 (3)	O5—Be2—Be1	76.13 (12)
B1—Rb—O1^vii	54.02 (7)	O5^xii—Be2—Be1	97.82 (13)
O2^v—Rb—O1^vii	144.49 (4)	Be2^xii—Be2—Be1	85.66 (15)
O2^vi—Rb—O1^vii	144.49 (4)	O4^xi—Be2—Be1^xi	33.69 (10)
O3^i—Rb—B1^v	86.87 (6)	O2—Be2—Be1^xi	79.69 (13)
O3^ii—Rb—B1^v	86.87 (6)	O5—Be2—Be1^xi	123.91 (15)
O3^iii—Rb—B1^v	71.88 (6)	O5^xii—Be2—Be1^xi	135.19 (15)
O3^iv—Rb—B1^v	71.88 (6)	Be2^xii—Be2—Be1^xi	160.53 (19)
O5^ii—Rb—B1^v	66.16 (3)	Be1—Be2—Be1^xi	110.73 (12)
O5^i—Rb—B1^v	66.16 (3)	O4^xi—Be2—Rb^ix	121.62 (13)
B1—Rb—B1^v	144.67 (10)	O2—Be2—Rb^ix	57.17 (10)
O2^v—Rb—B1^v	23.49 (4)	O5—Be2—Rb^ix	121.40 (13)
O2^vi—Rb—B1^v	23.49 (4)	O5^xii—Be2—Rb^ix	54.37 (9)
O1^vii—Rb—B1^v	161.31 (6)	Be2^xii—Be2—Rb^ix	87.13 (14)
O3^i—Rb—O1	48.38 (4)	Be1—Be2—Rb^ix	62.92 (8)
O3^ii—Rb—O1	48.38 (4)	Be1^xi—Be2—Rb^ix	109.36 (10)
O3^iii—Rb—O1	143.44 (3)	B1—O1—Be1^xiii	119.09 (12)
O3^iv—Rb—O1	143.44 (3)	B1—O1—Be1^xi	119.09 (12)
O5^ii—Rb—O1	92.06 (4)	Be1^xiii—O1—Be1^xi	116.3 (2)
O5^i—Rb—O1	92.06 (4)	B1—O1—Rb^xi	106.57 (16)
B1—Rb—O1	23.35 (6)	Be1^xiii—O1—Rb^xi	93.32 (12)
O2^v—Rb—O1	128.51 (5)	Be1^xi—O1—Rb^xi	93.32 (12)
O2^vi—Rb—O1	128.51 (5)	B1—O1—Rb	67.57 (15)
O1^vii—Rb—O1	77.37 (4)	Be1^xiii—O1—Rb	89.77 (12)
B1^v—Rb—O1	121.33 (7)	Be1^xi—O1—Rb	89.77 (12)
O2—B1—O2^viii	119.0 (3)	Rb^xi—O1—Rb	174.14 (7)
O2—B1—O1	120.48 (13)	B1—O2—Be2	123.22 (18)
O2^viii—B1—O1	120.48 (13)	B1—O2—Be1	126.79 (18)
O2—B1—Rb	90.52 (15)	Be2—O2—Be1	109.88 (15)
O2^viii—B1—Rb	90.52 (15)	B1—O2—Rb^ix	84.00 (14)
O1—B1—Rb	89.08 (16)	Be2—O2—Rb^ix	98.30 (11)
O2—B1—Rb^ix	72.51 (14)	Be1—O2—Rb^ix	85.58 (11)
O2^viii—B1—Rb^ix	72.51 (14)	B1—O2—Rb	66.38 (14)
O1—B1—Rb^ix	126.25 (18)	Be2—O2—Rb	123.33 (12)
Rb—B1—Rb^ix	144.67 (10)	Be1—O2—Rb	88.41 (11)
O3—B2—O5	120.43 (19)	Rb^ix—O2—Rb	137.31 (5)
O3—B2—O4	123.68 (19)	B2—O3—Be1^iii	136.84 (17)
O5—B2—O4	115.87 (18)	B2—O3—Rb^ii	101.71 (12)
O3—B2—Rb^ii	56.42 (11)	Be1^iii—O3—Rb^ii	109.44 (12)
O5—B2—Rb^ii	64.78 (11)	B2—O3—Rb^x	110.51 (13)
O4—B2—Rb^ii	172.56 (14)	Be1^iii—O3—Rb^x	96.88 (12)
O3—B2—Rb^x	49.60 (11)	Rb^ii—O3—Rb^x	93.45 (5)
O5—B2—Rb^x	107.54 (12)	B2—O4—Be2^vii	121.84 (17)
O4—B2—Rb^x	111.55 (13)	B2—O4—Be1	117.91 (16)
Rb^ii—B2—Rb^x	74.48 (5)	Be2^vii—O4—Be1	113.88 (16)
O3^x—Be1—O4	111.92 (18)	B2—O5—Be2	128.57 (16)
O3^x—Be1—O1^vii	110.95 (18)	B2—O5—Be2^xii	125.69 (16)
O4—Be1—O1^vii	107.20 (18)	Be2—O5—Be2^xii	84.52 (15)
O3^x—Be1—O2	114.28 (18)	B2—O5—Rb^ii	91.91 (11)
O4—Be1—O2	106.95 (17)	Be2—O5—Rb^ii	125.48 (12)
O1^vii—Be1—O2	105.07 (17)	Be2^xii—O5—Rb^ii	100.77 (11)
O3^x—Be1—Be2	97.15 (15)

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