Penta­terbium lithium tris­tannide, Tb₅LiSn₃

Abstract

The new ternary phase penta­terbium lithium tris­tannide, Tb₅LiSn₃, crystallizes in the hexa­gonal Hf₅CuSn₃ structure type, which is a ‘filled’ version of the binary RE ₅Sn₃ phases (Mn₅Si₃-type) (RE is rare earth). The asymmetric unit contains two Tb sites (site symmetries 3.2 and m2m), one Li site (site symmetry .m) and one Sn site (site symmetry m2m). The 14-vertex Frank–Kasper polyhedra are typical for Li and Tb atoms. The environment of the Sn atom is a pseudo-Frank–Kasper polyhedron with a coordination number of 13 for the tin atom. One of the Tb atoms is enclosed in a 17-vertex polyhedron. The metallic type of bonding was indicated by an analysis of the inter­atomic distances.

Related literature

For the Hf₅CuSn₃ structure type, see: Rieger & Parthé (1965 ▶). For related structures, see: Pavlyuk & Bodak (1992a ▶,b ▶); Pavlyuk et al. (1989 ▶, 1991 ▶, 1993 ▶). For the magnetic properties of related compounds, see: Tran et al. (2008 ▶).

Experimental

Crystal data

• Tb₅LiSn₃

• M _r = 1157.72

• Hexagonal,

• a = 9.0122 (14) Å

• c = 6.5744 (13) Å

• V = 462.4 (2) ų

• Z = 2

• Mo Kα radiation

• μ = 45.56 mm⁻¹

• T = 293 K

• 0.07 × 0.05 × 0.03 mm

Data collection

• Oxford Diffraction Xcalibur3 CCD diffractometer

• Absorption correction: analytical (CrysAlis RED; Oxford Diffraction, 2008 ▶) T _min = 0.322, T _max = 0.657

• 1907 measured reflections

• 216 independent reflections

• 207 reflections with I > 2σ(I)

• R _int = 0.021

Refinement

• R[F ² > 2σ(F ²)] = 0.021

• wR(F ²) = 0.066

• S = 1.33

• 216 reflections

• 14 parameters

• Δρ_max = 1.08 e Å⁻³

• Δρ_min = −1.47 e Å⁻³

Data collection: CrysAlis CCD (Oxford Diffraction, 2008 ▶); cell refinement: CrysAlis CCD; data reduction: CrysAlis RED (Oxford Diffraction, 2008 ▶); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008 ▶); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008 ▶); molecular graphics: DIAMOND (Brandenburg, 2006 ▶); software used to prepare material for publication: SHELXL97.

Supplementary Material

Additional supplementary materials: crystallographic information; 3D view; checkCIF report

supplementary crystallographic information

Comment

The RE₅TM₃ (RE - rare earth, T - Cu, Ag and M - Sn, Pb) ternary stannides crystallize in a hexagonal Hf₅CuSn₃ (superstructure to Ti₅Ga₄-type) with space group P6₃/mcm (Rieger and Parthé, 1965). These intermetallic compounds are characterized by two different sites for the RE atoms located at 4 d and 6 g, respectively. The Sn or Pb atoms are located at the next 6 g site and the transitions atoms occupy 2 b site.The RE₅TM₃ intermetallics are 'filled' version of the binary RE₅M₃ phases which crystallize in Mn₅Si₃ structure type. It is also possible, that the transition metals fill the octahedral voids.

For the Ce based compounds, Ce₅TM₃, investigated by (Tran et al., 2008) are found multiple magnetic phase transitions at low temperatures and discussed the role of f-spd hybridization on the evolution of heavy-fermion behaviour.

We detected the new ternary compound during the systematic study of ternary alloys of Tb—Li—Sn system from the concentration region with low content of lithium. The powder diffraction pattern of this compound is similar to the powder pattern of the RE₅Sn₃ (RE - rare-earth metals) binary phases, but has some differences. So we decided to further study this phase using single-crystal method. Obtained single-crystal data show that the title compound crystallizes with the hexagonal space group P6₃/mcm as a Hf₅CuSn₃ type. The projection of the unit cell and coordination polyhedra of the atoms are shown in Fig. 1. The distribution of tin and lithium atoms in three-dimensional-nets consisted of Tb atoms are shown in Fig. 2.

The number of neighbouring atoms correlates well with the dimensions of the central atoms. The Tb atoms are enclosed in 14- and 17-vertex polyhedra. The coordination polyhedron of the Sn atom is pseudo Frank-Kasper polyhedron with CN=13. Lithium atom is surrounded by 14 neighbours atoms in the form of 14-vertex Frank-Kasper polyhedron. The shortest interatomic distances in the title compound are in the typical for intermetallic compounds ranges and indicate metallic type of bonding.

In the title compound lithium atoms occupy the same crystallographic position that the atoms of transition metal in the original structure type. The same was observed previously when we studied RELiSn₂ compounds with the CeNiSi₂ structure type (Pavlyuk et al., 1989), RELiGe with the ZrNiAl type (Pavlyuk et al., 1991 and Pavlyuk & Bodak, 1992a), RE₃Li₂Ge₃ with Hf₃Ni₂Si₃ type (Pavlyuk & Bodak, 1992b), solid solutions RLi_xCu_{2 -x}Si₂ and RLi_xCu_{2 -x}Ge₂ (Pavlyuk et al., 1993).

Experimental

Terbium, lithium and tin, all with a nominal purity more than 99.9 wt. %, were used as starting elements. First, the pieces of the pure metals with a stoichiometry Tb₅₅Li₁₀Sn₃₅ were pressed into pellet, enclosed in tantalum crucible and placed in a resistance furnace with a thermocouple controller. Heating rate from room temperature to 670 K was equal 5 K per minute. At this temperature the alloy was held over 2 d and then the temperature was increased from 670 to 1070 K over 1 h. Then the alloy was annealed at this temperature for 8 h and slowly cooled down to room temperature. After the melting and annealing procedures, the total weight loss was less than 2%. Small good quality single-crystal of the title compound was isolated from alloy.

Figures

Fig. 1.

The projection of the unit cell and coordination polyhedra of the atoms.

Fig. 2.

The distribution of tin and lithium atoms in three-dimensional-nets consisted of Tb atoms.

Crystal data

Tb5LiSn3	Dx = 8.315 Mg m−3
Mr = 1157.72	Mo Kα radiation, λ = 0.71073 Å
Hexagonal, P63/mcm	Cell parameters from 1907 reflections
Hall symbol: -P 6c 2	θ = 2.6–27.4°
a = 9.0122 (14) Å	µ = 45.56 mm−1
c = 6.5744 (13) Å	T = 293 K
V = 462.4 (2) Å3	Prism, metallic dark grey
Z = 2	0.07 × 0.05 × 0.03 mm
F(000) = 956

Data collection

Oxford Diffraction Xcalibur3 CCD diffractometer	216 independent reflections
Radiation source: fine-focus sealed tube	207 reflections with I > 2σ(I)
graphite	Rint = 0.021
Detector resolution: 0 pixels mm-1	θmax = 27.4°, θmin = 2.6°
ω scans	h = −11→11
Absorption correction: analytical (CrysAlis RED; Oxford Diffraction, 2008)	k = −11→11
Tmin = 0.322, Tmax = 0.657	l = 0→8
1907 measured reflections

Refinement

Refinement on F2	0 restraints
Least-squares matrix: full	Primary atom site location: structure-invariant direct methods
R[F2 > 2σ(F2)] = 0.021	Secondary atom site location: difference Fourier map
wR(F2) = 0.066	w = 1/[σ2(Fo2) + (0.0268P)2 + 3.977P] where P = (Fo2 + 2Fc2)/3
S = 1.33	(Δ/σ)max < 0.001
216 reflections	Δρmax = 1.08 e Å−3
14 parameters	Δρmin = −1.47 e Å−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.

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
Tb1	0.25088 (10)	0.0000	0.2500	0.0479 (3)
Tb2	0.3333	0.6667	0.0000	0.0535 (3)
Sn3	0.60694 (14)	0.0000	0.2500	0.0493 (4)
Li4	0.0000	0.0000	0.0000	0.055 (13)

Atomic displacement parameters (Ų)

	U11	U22	U33	U12	U13	U23
Tb1	0.0478 (4)	0.0481 (5)	0.0479 (5)	0.0241 (3)	0.000	0.000
Tb2	0.0536 (4)	0.0536 (4)	0.0532 (6)	0.0268 (2)	0.000	0.000
Sn3	0.0491 (5)	0.0493 (7)	0.0496 (6)	0.0247 (4)	0.000	0.000
Li4	0.07 (2)	0.07 (2)	0.03 (2)	0.033 (11)	0.000	0.000

Geometric parameters (Å, °)

Tb1—Li4	2.7952 (8)	Tb2—Tb1xii	3.8093 (7)
Tb1—Li4i	2.7952 (8)	Tb2—Tb1xi	3.8093 (7)
Tb1—Sn3ii	3.1066 (11)	Tb2—Tb1xiii	3.8093 (8)
Tb1—Sn3iii	3.1066 (11)	Tb2—Tb1ix	3.8093 (7)
Tb1—Sn3	3.2090 (16)	Sn3—Tb1xvii	3.1066 (11)
Tb1—Sn3iv	3.5281 (8)	Sn3—Tb1xviii	3.1066 (11)
Tb1—Sn3v	3.5281 (8)	Sn3—Tb2vii	3.2247 (6)
Tb1—Tb2vi	3.8093 (7)	Sn3—Tb2vi	3.2247 (6)
Tb1—Tb2vii	3.8093 (7)	Sn3—Tb2ix	3.2247 (6)
Tb1—Tb2viii	3.8093 (7)	Sn3—Tb2viii	3.2247 (6)
Tb1—Tb2ix	3.8093 (7)	Sn3—Tb1iv	3.5281 (8)
Tb1—Tb1x	3.9161 (17)	Sn3—Tb1v	3.5281 (8)
Tb2—Sn3xi	3.2247 (6)	Li4—Tb1xix	2.7952 (8)
Tb2—Sn3xii	3.2247 (6)	Li4—Tb1xx	2.7952 (8)
Tb2—Sn3xiii	3.2247 (6)	Li4—Tb1xiii	2.7952 (8)
Tb2—Sn3ix	3.2247 (6)	Li4—Tb1x	2.7952 (8)
Tb2—Sn3xiv	3.2247 (6)	Li4—Tb1xiv	2.7952 (8)
Tb2—Sn3iii	3.2247 (6)	Li4—Li4i	3.2872 (6)
Tb2—Tb2xv	3.2872 (6)	Li4—Li4xxi	3.2872 (6)
Tb2—Tb2xvi	3.2872 (6)
Li4—Tb1—Li4i	72.03 (3)	Sn3xiv—Tb2—Tb1xii	144.94 (2)
Li4—Tb1—Sn3ii	82.67 (2)	Sn3iii—Tb2—Tb1xii	123.785 (13)
Li4i—Tb1—Sn3ii	82.67 (2)	Tb2xv—Tb2—Tb1xii	64.439 (7)
Li4—Tb1—Sn3iii	82.67 (2)	Tb2xvi—Tb2—Tb1xii	115.561 (7)
Li4i—Tb1—Sn3iii	82.67 (2)	Sn3xi—Tb2—Tb1xi	53.50 (2)
Sn3ii—Tb1—Sn3iii	161.86 (5)	Sn3xii—Tb2—Tb1xi	110.39 (2)
Li4—Tb1—Sn3	143.985 (13)	Sn3xiii—Tb2—Tb1xi	59.520 (16)
Li4i—Tb1—Sn3	143.985 (13)	Sn3ix—Tb2—Tb1xi	51.60 (2)
Sn3ii—Tb1—Sn3	99.07 (2)	Sn3xiv—Tb2—Tb1xi	123.785 (13)
Sn3iii—Tb1—Sn3	99.07 (2)	Sn3iii—Tb2—Tb1xi	144.94 (2)
Li4—Tb1—Sn3iv	147.31 (3)	Tb2xv—Tb2—Tb1xi	115.561 (7)
Li4i—Tb1—Sn3iv	75.28 (2)	Tb2xvi—Tb2—Tb1xi	64.439 (7)
Sn3ii—Tb1—Sn3iv	93.283 (5)	Tb1xii—Tb2—Tb1xi	63.16 (2)
Sn3iii—Tb1—Sn3iv	93.283 (5)	Sn3xi—Tb2—Tb1xiii	59.520 (16)
Sn3—Tb1—Sn3iv	68.70 (3)	Sn3xii—Tb2—Tb1xiii	123.785 (13)
Li4—Tb1—Sn3v	75.28 (2)	Sn3xiii—Tb2—Tb1xiii	53.50 (2)
Li4i—Tb1—Sn3v	147.31 (3)	Sn3ix—Tb2—Tb1xiii	144.94 (2)
Sn3ii—Tb1—Sn3v	93.283 (5)	Sn3xiv—Tb2—Tb1xiii	110.39 (2)
Sn3iii—Tb1—Sn3v	93.283 (5)	Sn3iii—Tb2—Tb1xiii	51.60 (2)
Sn3—Tb1—Sn3v	68.70 (3)	Tb2xv—Tb2—Tb1xiii	64.439 (7)
Sn3iv—Tb1—Sn3v	137.41 (5)	Tb2xvi—Tb2—Tb1xiii	115.561 (7)
Li4—Tb1—Tb2vi	102.887 (9)	Tb1xii—Tb2—Tb1xiii	102.753 (8)
Li4i—Tb1—Tb2vi	136.924 (8)	Tb1xi—Tb2—Tb1xiii	93.848 (16)
Sn3ii—Tb1—Tb2vi	54.444 (14)	Sn3xi—Tb2—Tb1ix	123.785 (13)
Sn3iii—Tb1—Tb2vi	140.12 (2)	Sn3xii—Tb2—Tb1ix	59.520 (16)
Sn3—Tb1—Tb2vi	53.886 (12)	Sn3xiii—Tb2—Tb1ix	144.94 (2)
Sn3iv—Tb1—Tb2vi	100.83 (2)	Sn3ix—Tb2—Tb1ix	53.50 (2)
Sn3v—Tb1—Tb2vi	51.970 (13)	Sn3xiv—Tb2—Tb1ix	51.60 (2)
Li4—Tb1—Tb2vii	136.924 (8)	Sn3iii—Tb2—Tb1ix	110.39 (2)
Li4i—Tb1—Tb2vii	102.887 (9)	Tb2xv—Tb2—Tb1ix	115.561 (7)
Sn3ii—Tb1—Tb2vii	140.12 (2)	Tb2xvi—Tb2—Tb1ix	64.439 (7)
Sn3iii—Tb1—Tb2vii	54.444 (14)	Tb1xii—Tb2—Tb1ix	93.848 (16)
Sn3—Tb1—Tb2vii	53.886 (12)	Tb1xi—Tb2—Tb1ix	102.753 (8)
Sn3iv—Tb1—Tb2vii	51.970 (13)	Tb1xiii—Tb2—Tb1ix	160.55 (2)
Sn3v—Tb1—Tb2vii	100.83 (2)	Tb1xvii—Sn3—Tb1xviii	78.14 (5)
Tb2vi—Tb1—Tb2vii	107.77 (2)	Tb1xvii—Sn3—Tb1	140.93 (2)
Li4—Tb1—Tb2viii	136.924 (8)	Tb1xviii—Sn3—Tb1	140.93 (2)
Li4i—Tb1—Tb2viii	102.887 (9)	Tb1xvii—Sn3—Tb2vii	73.951 (16)
Sn3ii—Tb1—Tb2viii	54.444 (14)	Tb1xviii—Sn3—Tb2vii	137.78 (3)
Sn3iii—Tb1—Tb2viii	140.12 (2)	Tb1—Sn3—Tb2vii	72.61 (2)
Sn3—Tb1—Tb2viii	53.886 (12)	Tb1xvii—Sn3—Tb2vi	137.78 (3)
Sn3iv—Tb1—Tb2viii	51.970 (13)	Tb1xviii—Sn3—Tb2vi	73.951 (16)
Sn3v—Tb1—Tb2viii	100.83 (2)	Tb1—Sn3—Tb2vi	72.61 (2)
Tb2vi—Tb1—Tb2viii	51.122 (14)	Tb2vii—Sn3—Tb2vi	145.22 (4)
Tb2vii—Tb1—Tb2viii	86.152 (16)	Tb1xvii—Sn3—Tb2ix	73.951 (16)
Li4—Tb1—Tb2ix	102.887 (9)	Tb1xviii—Sn3—Tb2ix	137.78 (3)
Li4i—Tb1—Tb2ix	136.924 (8)	Tb1—Sn3—Tb2ix	72.61 (2)
Sn3ii—Tb1—Tb2ix	140.12 (2)	Tb2vii—Sn3—Tb2ix	61.287 (15)
Sn3iii—Tb1—Tb2ix	54.444 (14)	Tb2vi—Sn3—Tb2ix	107.56 (2)
Sn3—Tb1—Tb2ix	53.886 (12)	Tb1xvii—Sn3—Tb2viii	137.78 (3)
Sn3iv—Tb1—Tb2ix	100.83 (2)	Tb1xviii—Sn3—Tb2viii	73.951 (16)
Sn3v—Tb1—Tb2ix	51.970 (13)	Tb1—Sn3—Tb2viii	72.61 (2)
Tb2vi—Tb1—Tb2ix	86.152 (16)	Tb2vii—Sn3—Tb2viii	107.56 (2)
Tb2vii—Tb1—Tb2ix	51.122 (14)	Tb2vi—Sn3—Tb2viii	61.287 (15)
Tb2viii—Tb1—Tb2ix	107.77 (2)	Tb2ix—Sn3—Tb2viii	145.22 (4)
Li4—Tb1—Tb1x	45.533 (9)	Tb1xvii—Sn3—Tb1iv	73.62 (2)
Li4i—Tb1—Tb1x	45.533 (9)	Tb1xviii—Sn3—Tb1iv	73.62 (2)
Sn3ii—Tb1—Tb1x	50.93 (2)	Tb1—Sn3—Tb1iv	111.30 (3)
Sn3iii—Tb1—Tb1x	110.93 (2)	Tb2vii—Sn3—Tb1iv	68.510 (11)
Sn3—Tb1—Tb1x	150.0	Tb2vi—Sn3—Tb1iv	125.693 (6)
Sn3iv—Tb1—Tb1x	108.33 (2)	Tb2ix—Sn3—Tb1iv	125.693 (6)
Sn3v—Tb1—Tb1x	108.33 (2)	Tb2viii—Sn3—Tb1iv	68.510 (11)
Tb2vi—Tb1—Tb1x	99.726 (11)	Tb1xvii—Sn3—Tb1v	73.62 (2)
Tb2vii—Tb1—Tb1x	148.420 (11)	Tb1xviii—Sn3—Tb1v	73.62 (2)
Tb2viii—Tb1—Tb1x	99.726 (11)	Tb1—Sn3—Tb1v	111.30 (3)
Tb2ix—Tb1—Tb1x	148.420 (11)	Tb2vii—Sn3—Tb1v	125.693 (6)
Sn3xi—Tb2—Sn3xii	163.38 (4)	Tb2vi—Sn3—Tb1v	68.510 (11)
Sn3xi—Tb2—Sn3xiii	72.44 (2)	Tb2ix—Sn3—Tb1v	68.510 (11)
Sn3xii—Tb2—Sn3xiii	96.334 (10)	Tb2viii—Sn3—Tb1v	125.693 (6)
Sn3xi—Tb2—Sn3ix	96.334 (10)	Tb1iv—Sn3—Tb1v	137.41 (5)
Sn3xii—Tb2—Sn3ix	72.44 (2)	Tb1xix—Li4—Tb1xx	88.933 (19)
Sn3xiii—Tb2—Sn3ix	97.06 (4)	Tb1xix—Li4—Tb1	91.067 (19)
Sn3xi—Tb2—Sn3xiv	96.334 (10)	Tb1xx—Li4—Tb1	180.0
Sn3xii—Tb2—Sn3xiv	97.06 (4)	Tb1xix—Li4—Tb1xiii	180.0
Sn3xiii—Tb2—Sn3xiv	163.38 (4)	Tb1xx—Li4—Tb1xiii	91.067 (19)
Sn3ix—Tb2—Sn3xiv	96.334 (10)	Tb1—Li4—Tb1xiii	88.933 (19)
Sn3xi—Tb2—Sn3iii	97.06 (4)	Tb1xix—Li4—Tb1x	91.067 (19)
Sn3xii—Tb2—Sn3iii	96.334 (10)	Tb1xx—Li4—Tb1x	91.067 (19)
Sn3xiii—Tb2—Sn3iii	96.334 (10)	Tb1—Li4—Tb1x	88.933 (19)
Sn3ix—Tb2—Sn3iii	163.38 (4)	Tb1xiii—Li4—Tb1x	88.933 (19)
Sn3xiv—Tb2—Sn3iii	72.44 (2)	Tb1xix—Li4—Tb1xiv	88.933 (19)
Sn3xi—Tb2—Tb2xv	120.643 (7)	Tb1xx—Li4—Tb1xiv	88.933 (19)
Sn3xii—Tb2—Tb2xv	59.357 (8)	Tb1—Li4—Tb1xiv	91.067 (19)
Sn3xiii—Tb2—Tb2xv	59.357 (7)	Tb1xiii—Li4—Tb1xiv	91.067 (19)
Sn3ix—Tb2—Tb2xv	120.643 (8)	Tb1x—Li4—Tb1xiv	180.00 (7)
Sn3xiv—Tb2—Tb2xv	120.643 (7)	Tb1xix—Li4—Li4i	126.015 (13)
Sn3iii—Tb2—Tb2xv	59.357 (7)	Tb1xx—Li4—Li4i	126.015 (13)
Sn3xi—Tb2—Tb2xvi	59.357 (7)	Tb1—Li4—Li4i	53.985 (13)
Sn3xii—Tb2—Tb2xvi	120.643 (7)	Tb1xiii—Li4—Li4i	53.985 (13)
Sn3xiii—Tb2—Tb2xvi	120.643 (8)	Tb1x—Li4—Li4i	53.985 (13)
Sn3ix—Tb2—Tb2xvi	59.357 (8)	Tb1xiv—Li4—Li4i	126.015 (13)
Sn3xiv—Tb2—Tb2xvi	59.357 (7)	Tb1xix—Li4—Li4xxi	53.985 (13)
Sn3iii—Tb2—Tb2xvi	120.643 (7)	Tb1xx—Li4—Li4xxi	53.985 (13)
Tb2xv—Tb2—Tb2xvi	180.0	Tb1—Li4—Li4xxi	126.015 (13)
Sn3xi—Tb2—Tb1xii	110.39 (2)	Tb1xiii—Li4—Li4xxi	126.015 (13)
Sn3xii—Tb2—Tb1xii	53.50 (2)	Tb1x—Li4—Li4xxi	126.015 (13)
Sn3xiii—Tb2—Tb1xii	51.60 (2)	Tb1xiv—Li4—Li4xxi	53.985 (13)
Sn3ix—Tb2—Tb1xii	59.520 (16)	Li4i—Li4—Li4xxi	180.0

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