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The prevalent polymorph of the energetic material 5-nitro-2,4-dihydro-1,2,4,-triazol-3-one, α-NTO, crystallizes as a four-component twin with triclinic symmetry (space group
). All crystals under investigation were fourlings,
i.e. they contained each of the four possible twin components. Complete data sets were collected for two crystals, one with a predominant amount of one individual component (55%) and one with approximately equal volumes of each component. In both cases the fourling components are related by the twofold axes inherent in the holohedral symmetry of a pseudo-orthorhombic superlattice with
ao =
at,
bo =
bt and
co =
at +
bt + 2
ct. The triclinic unit cell contains four crystallographically independent planar molecules in the asymmetric unit, each of which forms a hydrogen-bonded flat chain parallel to
at. Pairs of chains are combined into planar ribbons by additional hydrogen bonds. Thus, two independent ribbons extend parallel to
at, creating a dihedral angle of ∼ 70°. The origin of the twinning is derived from consideration of the crystal packing and the hydrogen-bonding scheme.
Supporting information
CCDC reference: 1243661
Program(s) used to solve structure: SHELXS97 (Sheldrick, 1990); program(s) used to refine structure: SHELXL97 (Sheldrick, 1997).
5-nitro-2,4-dihydro-3
H-1,2,4,-triazol-3-one
top
Crystal data top
C2H2N4O3 | Z = 8 |
Mr = 130.08 | F(000) = 528 |
Triclinic, P1 | Dx = 1.916 Mg m−3 |
a = 5.1233 (8) Å | Mo Kα radiation, λ = 0.71069 Å |
b = 10.3140 (16) Å | Cell parameters from 3399 reflections |
c = 17.998 (3) Å | θ = 2.6–28.2° |
α = 106.610 (2)° | µ = 0.18 mm−1 |
β = 97.810 (2)° | T = 293 K |
γ = 90.130 (2)° | Parallelpiped, colorless |
V = 902.1 (2) Å3 | 0.34 × 0.26 × 0.20 mm |
Data collection top
Bruker platform with 1K CCD diffractometer | 8966 independent reflections |
Radiation source: normal-focus sealed tube | 8202 reflections with I > 2σ(I) |
Graphite monochromator | Rint = 0.000 |
ω–scans | θmax = 28.7°, θmin = 2.1° |
Absorption correction: multi-scan SADABS;Sheldrick1996;Blessing,ActaC.1995,A51,33 | h = −6→6 |
Tmin = 0.851, Tmax = 1.000 | k = −13→13 |
8966 measured reflections | l = −24→23 |
Refinement top
Refinement on F2 | Primary atom site location: structure-invariant direct methods |
Least-squares matrix: full | Secondary atom site location: difference Fourier map |
R[F2 > 2σ(F2)] = 0.043 | Hydrogen site location: inferred from neighbouring sites |
wR(F2) = 0.115 | H-atom parameters constrained |
S = 1.12 | w = 1/[σ2(Fo2) + (0.0554P)2 + 0.2255P] where P = (Fo2 + 2Fc2)/3 |
8966 reflections | (Δ/σ)max = 0.002 |
328 parameters | Δρmax = 0.34 e Å−3 |
0 restraints | Δρmin = −0.36 e Å−3 |
Crystal data top
C2H2N4O3 | γ = 90.130 (2)° |
Mr = 130.08 | V = 902.1 (2) Å3 |
Triclinic, P1 | Z = 8 |
a = 5.1233 (8) Å | Mo Kα radiation |
b = 10.3140 (16) Å | µ = 0.18 mm−1 |
c = 17.998 (3) Å | T = 293 K |
α = 106.610 (2)° | 0.34 × 0.26 × 0.20 mm |
β = 97.810 (2)° | |
Data collection top
Bruker platform with 1K CCD diffractometer | 8966 independent reflections |
Absorption correction: multi-scan SADABS;Sheldrick1996;Blessing,ActaC.1995,A51,33 | 8202 reflections with I > 2σ(I) |
Tmin = 0.851, Tmax = 1.000 | Rint = 0.000 |
8966 measured reflections | |
Refinement top
R[F2 > 2σ(F2)] = 0.043 | 0 restraints |
wR(F2) = 0.115 | H-atom parameters constrained |
S = 1.12 | Δρmax = 0.34 e Å−3 |
8966 reflections | Δρmin = −0.36 e Å−3 |
328 parameters | |
Special details top
Experimental. identical or equivalent reflections were not merged |
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. Using the twin laws above, the structural model was refined as a four-component twin using the procedure associated with the HKLF 5 format for merohedral twins. Fractional contributions k1, k2, k3, k4 of the twin components were refined constraining k1 = 1 - k2 - k3 - k4. An HKLF 5 format data file was created where each line contained h,k,l, Fo2, σ(Fo2), m where m is the twin component number. Thus, four lines of the data file correspond to each measured reflection: h, k, l, Fo2, σ(Fo2), −1 h, −k, −h-l, Fo2, σ(Fo2), −2 −h, k, −k-l, Fo2, σ(Fo2), −3 −h, −k, h+k+l, Fo2, σ(Fo2), 4 The volumes of the fourling domains refine to 55: 9: 20: 16. |
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top | x | y | z | Uiso*/Ueq | |
N11 | 0.3084 (4) | −0.06728 (18) | 0.89274 (10) | 0.0254 (4) | |
N21 | 0.4387 (4) | −0.1449 (2) | 0.93397 (11) | 0.0264 (4) | |
H21 | 0.3621 | −0.1907 | 0.9586 | 0.037* | |
C31 | 0.7044 (5) | −0.1426 (3) | 0.93212 (13) | 0.0264 (5) | |
O31 | 0.8741 (3) | −0.20130 (18) | 0.96269 (9) | 0.0356 (4) | |
N41 | 0.7383 (4) | −0.0578 (2) | 0.88705 (10) | 0.0260 (4) | |
H41 | 0.8842 | −0.0346 | 0.8747 | 0.037* | |
C51 | 0.4949 (5) | −0.0188 (2) | 0.86632 (11) | 0.0236 (5) | |
N51 | 0.4449 (4) | 0.0715 (2) | 0.81750 (10) | 0.0272 (5) | |
O51 | 0.6354 (4) | 0.1093 (2) | 0.79429 (11) | 0.0372 (5) | |
O61 | 0.2172 (4) | 0.1003 (2) | 0.80308 (11) | 0.0409 (5) | |
N12 | 1.0939 (4) | 0.43300 (19) | 0.89177 (10) | 0.0272 (4) | |
N22 | 1.0000 (4) | 0.3554 (2) | 0.93374 (10) | 0.0285 (5) | |
H22 | 1.1004 | 0.3106 | 0.9588 | 0.037* | |
C32 | 0.7344 (5) | 0.3559 (2) | 0.93208 (13) | 0.0247 (5) | |
O32 | 0.5925 (4) | 0.29716 (17) | 0.96306 (9) | 0.0349 (4) | |
N42 | 0.6593 (4) | 0.4398 (2) | 0.88654 (10) | 0.0256 (4) | |
H42 | 0.5014 | 0.4622 | 0.8741 | 0.037* | |
C52 | 0.8799 (5) | 0.4801 (2) | 0.86520 (11) | 0.0228 (5) | |
N52 | 0.8821 (4) | 0.5705 (2) | 0.81713 (11) | 0.0266 (4) | |
O52 | 0.6643 (4) | 0.6049 (2) | 0.79464 (11) | 0.0383 (5) | |
O62 | 1.0904 (4) | 0.6061 (2) | 0.80328 (12) | 0.0443 (5) | |
N13 | −0.0965 (4) | 0.04197 (19) | 0.60938 (10) | 0.0262 (4) | |
N23 | −0.0007 (4) | −0.07541 (19) | 0.56712 (11) | 0.0263 (4) | |
H23 | −0.0993 | −0.1459 | 0.5415 | 0.037* | |
C33 | 0.2647 (5) | −0.0701 (2) | 0.56948 (13) | 0.0251 (5) | |
O33 | 0.4089 (4) | −0.15743 (16) | 0.53844 (9) | 0.0365 (4) | |
N43 | 0.3376 (4) | 0.0603 (2) | 0.61626 (11) | 0.0258 (4) | |
H43 | 0.4944 | 0.0968 | 0.6293 | 0.037* | |
C53 | 0.1129 (4) | 0.1182 (2) | 0.63718 (12) | 0.0211 (4) | |
N53 | 0.1048 (4) | 0.2559 (2) | 0.68601 (11) | 0.0266 (4) | |
O53 | 0.3155 (4) | 0.31856 (19) | 0.70961 (11) | 0.0388 (5) | |
O63 | −0.1129 (4) | 0.2971 (2) | 0.69924 (11) | 0.0422 (5) | |
N14 | 0.7013 (4) | 0.53429 (19) | 0.60426 (10) | 0.0274 (4) | |
N24 | 0.5640 (4) | 0.41422 (19) | 0.56377 (11) | 0.0259 (4) | |
H24 | 0.6379 | 0.3433 | 0.5389 | 0.037* | |
C34 | 0.3013 (5) | 0.4189 (2) | 0.56700 (13) | 0.0269 (5) | |
O34 | 0.1287 (4) | 0.33027 (16) | 0.53764 (9) | 0.0374 (4) | |
N44 | 0.2754 (4) | 0.5511 (2) | 0.61176 (11) | 0.0262 (4) | |
H44 | 0.1314 | 0.5879 | 0.6246 | 0.037* | |
C54 | 0.5181 (5) | 0.6103 (2) | 0.63109 (12) | 0.0248 (5) | |
N54 | 0.5689 (4) | 0.7486 (2) | 0.67853 (11) | 0.0271 (4) | |
O54 | 0.3796 (4) | 0.80735 (19) | 0.70440 (11) | 0.0392 (5) | |
O64 | 0.7956 (4) | 0.7932 (2) | 0.68975 (11) | 0.0423 (5) | |
Atomic displacement parameters (Å2) top | U11 | U22 | U33 | U12 | U13 | U23 |
N11 | 0.0224 (11) | 0.0280 (9) | 0.0272 (9) | 0.0030 (9) | 0.0016 (8) | 0.0113 (7) |
N21 | 0.0257 (11) | 0.0273 (10) | 0.0315 (9) | 0.0035 (8) | 0.0042 (8) | 0.0169 (8) |
C31 | 0.0257 (13) | 0.0275 (12) | 0.0282 (11) | 0.0016 (10) | 0.0032 (9) | 0.0115 (10) |
O31 | 0.0268 (10) | 0.0470 (10) | 0.0422 (9) | 0.0105 (8) | 0.0023 (7) | 0.0286 (7) |
N41 | 0.0203 (10) | 0.0313 (10) | 0.0302 (9) | 0.0023 (8) | 0.0013 (8) | 0.0163 (8) |
C51 | 0.0238 (12) | 0.0222 (10) | 0.0251 (10) | 0.0014 (9) | 0.0007 (9) | 0.0085 (8) |
N51 | 0.0334 (12) | 0.0247 (10) | 0.0244 (9) | 0.0037 (9) | 0.0019 (8) | 0.0094 (7) |
O51 | 0.0393 (11) | 0.0392 (11) | 0.0410 (9) | −0.0009 (9) | 0.0078 (8) | 0.0229 (8) |
O61 | 0.0323 (12) | 0.0486 (12) | 0.0479 (10) | 0.0149 (9) | 0.0014 (8) | 0.0256 (9) |
N12 | 0.0242 (11) | 0.0303 (10) | 0.0282 (9) | 0.0033 (9) | 0.0033 (8) | 0.0103 (7) |
N22 | 0.0287 (12) | 0.0298 (11) | 0.0318 (10) | 0.0034 (9) | 0.0026 (8) | 0.0173 (8) |
C32 | 0.0211 (12) | 0.0242 (12) | 0.0298 (11) | 0.0024 (9) | 0.0004 (8) | 0.0112 (9) |
O32 | 0.0280 (10) | 0.0425 (10) | 0.0432 (9) | −0.0002 (8) | 0.0041 (8) | 0.0270 (7) |
N42 | 0.0223 (11) | 0.0282 (10) | 0.0309 (9) | 0.0026 (8) | 0.0029 (8) | 0.0160 (8) |
C52 | 0.0203 (11) | 0.0248 (11) | 0.0239 (10) | 0.0021 (9) | 0.0009 (8) | 0.0088 (8) |
N52 | 0.0280 (11) | 0.0269 (10) | 0.0266 (10) | −0.0005 (8) | 0.0048 (8) | 0.0100 (8) |
O52 | 0.0382 (11) | 0.0437 (12) | 0.0390 (9) | 0.0058 (9) | −0.0018 (8) | 0.0250 (9) |
O62 | 0.0413 (12) | 0.0430 (11) | 0.0581 (11) | 0.0001 (9) | 0.0166 (9) | 0.0256 (9) |
N13 | 0.0210 (11) | 0.0246 (9) | 0.0323 (10) | 0.0035 (8) | 0.0046 (8) | 0.0069 (7) |
N23 | 0.0261 (11) | 0.0170 (9) | 0.0325 (10) | −0.0018 (8) | 0.0014 (8) | 0.0034 (7) |
C33 | 0.0308 (14) | 0.0173 (11) | 0.0261 (11) | 0.0033 (9) | 0.0044 (9) | 0.0044 (9) |
O33 | 0.0342 (10) | 0.0237 (8) | 0.0447 (9) | 0.0066 (8) | 0.0063 (8) | −0.0015 (7) |
N43 | 0.0160 (10) | 0.0210 (9) | 0.0367 (10) | 0.0024 (8) | 0.0021 (8) | 0.0033 (8) |
C53 | 0.0169 (11) | 0.0190 (10) | 0.0274 (10) | 0.0044 (8) | 0.0034 (8) | 0.0067 (8) |
N53 | 0.0283 (11) | 0.0205 (10) | 0.0304 (10) | 0.0052 (8) | 0.0042 (8) | 0.0059 (8) |
O53 | 0.0334 (11) | 0.0272 (9) | 0.0503 (10) | −0.0039 (8) | 0.0012 (8) | 0.0049 (8) |
O63 | 0.0355 (11) | 0.0350 (10) | 0.0512 (10) | 0.0125 (9) | 0.0132 (8) | 0.0013 (8) |
N14 | 0.0259 (11) | 0.0227 (9) | 0.0336 (10) | 0.0023 (8) | 0.0046 (8) | 0.0079 (7) |
N24 | 0.0256 (11) | 0.0172 (9) | 0.0318 (9) | 0.0027 (8) | 0.0063 (7) | 0.0012 (7) |
C34 | 0.0315 (14) | 0.0221 (12) | 0.0248 (11) | 0.0033 (10) | 0.0061 (9) | 0.0023 (9) |
O34 | 0.0319 (10) | 0.0236 (8) | 0.0485 (10) | −0.0058 (8) | 0.0046 (8) | −0.0018 (7) |
N44 | 0.0200 (10) | 0.0206 (9) | 0.0352 (10) | 0.0036 (8) | 0.0068 (8) | 0.0021 (7) |
C54 | 0.0267 (13) | 0.0218 (11) | 0.0248 (10) | 0.0015 (10) | 0.0054 (9) | 0.0042 (8) |
N54 | 0.0310 (12) | 0.0185 (9) | 0.0309 (10) | −0.0006 (8) | 0.0013 (8) | 0.0072 (8) |
O54 | 0.0375 (11) | 0.0264 (10) | 0.0493 (10) | 0.0061 (8) | 0.0122 (8) | 0.0012 (8) |
O64 | 0.0322 (11) | 0.0337 (10) | 0.0574 (11) | −0.0140 (9) | 0.0002 (9) | 0.0106 (9) |
Geometric parameters (Å, º) top
N11—C51 | 1.283 (3) | N13—C53 | 1.283 (3) |
N11—N21 | 1.358 (3) | N13—N23 | 1.365 (3) |
N21—C31 | 1.367 (3) | N23—C33 | 1.355 (3) |
N21—H21 | 0.8600 | N23—H23 | 0.8600 |
C31—O31 | 1.220 (3) | C33—O33 | 1.223 (3) |
C31—N41 | 1.377 (3) | C33—N43 | 1.386 (3) |
N41—H41 | 0.8600 | N43—C53 | 1.348 (3) |
N41—C51 | 1.348 (3) | N43—H43 | 0.8600 |
C51—N51 | 1.454 (3) | C53—N53 | 1.444 (3) |
N51—O51 | 1.217 (3) | N53—O53 | 1.216 (3) |
N51—O61 | 1.217 (3) | N53—O63 | 1.224 (3) |
N12—C52 | 1.288 (3) | N14—C54 | 1.280 (3) |
N12—N22 | 1.373 (3) | N14—N24 | 1.378 (3) |
N22—C32 | 1.357 (3) | N24—C34 | 1.356 (3) |
N22—H22 | 0.8600 | N24—H24 | 0.8600 |
C32—O32 | 1.228 (3) | C34—O34 | 1.220 (3) |
C32—N42 | 1.374 (3) | C34—N44 | 1.389 (3) |
N42—C52 | 1.343 (3) | N44—C54 | 1.341 (3) |
N42—H42 | 0.8600 | N44—H44 | 0.8600 |
C52—N52 | 1.443 (3) | C54—N54 | 1.440 (3) |
N52—O62 | 1.208 (3) | N54—O64 | 1.217 (3) |
N52—O52 | 1.223 (3) | N54—O54 | 1.220 (3) |
| | | |
C51—N11—N21 | 102.69 (18) | C53—N13—N23 | 102.64 (19) |
N11—N21—C31 | 112.8 (2) | C33—N23—N13 | 113.47 (19) |
N11—N21—H21 | 123.6 | C33—N23—H23 | 123.3 |
C31—N21—H21 | 123.6 | N13—N23—H23 | 123.3 |
O31—C31—N21 | 128.8 (2) | O33—C33—N23 | 129.4 (2) |
O31—C31—N41 | 127.4 (2) | O33—C33—N43 | 127.5 (2) |
N21—C31—N41 | 103.7 (2) | N23—C33—N43 | 103.10 (19) |
C51—N41—C31 | 105.5 (2) | C53—N43—C33 | 106.10 (19) |
C51—N41—H41 | 127.2 | C53—N43—H43 | 127.0 |
C31—N41—H41 | 127.2 | C33—N43—H43 | 127.0 |
N11—C51—N41 | 115.2 (2) | N13—C53—N43 | 114.70 (19) |
N11—C51—N51 | 121.9 (2) | N13—C53—N53 | 122.04 (19) |
N41—C51—N51 | 122.8 (2) | N43—C53—N53 | 123.3 (2) |
O51—N51—O61 | 126.5 (2) | O53—N53—O63 | 126.9 (2) |
O51—N51—C51 | 116.5 (2) | O53—N53—C53 | 116.43 (19) |
O61—N51—C51 | 117.0 (2) | O63—N53—C53 | 116.7 (2) |
C52—N12—N22 | 101.7 (2) | C54—N14—N24 | 102.31 (19) |
C32—N22—N12 | 113.9 (2) | C34—N24—N14 | 113.47 (19) |
C32—N22—H22 | 123.0 | C34—N24—H24 | 123.3 |
N12—N22—H22 | 123.0 | N14—N24—H24 | 123.3 |
O32—C32—N22 | 129.6 (2) | O34—C34—N24 | 129.2 (2) |
O32—C32—N42 | 127.7 (2) | O34—C34—N44 | 128.2 (2) |
N22—C32—N42 | 102.7 (2) | N24—C34—N44 | 102.6 (2) |
C52—N42—C32 | 106.9 (2) | C54—N44—C34 | 106.8 (2) |
C52—N42—H42 | 126.6 | C54—N44—H44 | 126.6 |
C32—N42—H42 | 126.6 | C34—N44—H44 | 126.6 |
N12—C52—N42 | 114.8 (2) | N14—C54—N44 | 114.8 (2) |
N12—C52—N52 | 121.7 (2) | N14—C54—N54 | 122.6 (2) |
N42—C52—N52 | 123.5 (2) | N44—C54—N54 | 122.6 (2) |
O62—N52—O52 | 126.1 (2) | O64—N54—O54 | 127.2 (2) |
O62—N52—C52 | 119.2 (2) | O64—N54—C54 | 116.8 (2) |
O52—N52—C52 | 114.6 (2) | O54—N54—C54 | 116.0 (2) |
Hydrogen-bond geometry (Å, º) top
D—H···A | D—H | H···A | D···A | D—H···A |
N21—H21···O32i | 0.86 | 2.01 | 2.769 (2) | 147 |
N21—H21···O31ii | 0.86 | 2.51 | 3.089 (3) | 125 |
N41—H41···N11iii | 0.86 | 2.20 | 2.912 (3) | 141 |
N22—H22···O31iv | 0.86 | 2.04 | 2.791 (3) | 146 |
N22—H22···O32iii | 0.86 | 2.52 | 3.100 (3) | 126 |
N42—H42···N12ii | 0.86 | 2.19 | 2.913 (3) | 142 |
N23—H23···O34v | 0.86 | 2.01 | 2.778 (2) | 147 |
N23—H23···O33ii | 0.86 | 2.51 | 3.073 (3) | 123 |
N43—H43···N13iii | 0.86 | 2.22 | 2.923 (3) | 139 |
N24—H24···O33vi | 0.86 | 2.01 | 2.776 (2) | 147 |
N24—H24···O34iii | 0.86 | 2.52 | 3.081 (3) | 124 |
N44—H44···N14ii | 0.86 | 2.23 | 2.928 (3) | 139 |
Symmetry codes: (i) −x+1, −y, −z+2; (ii) x−1, y, z; (iii) x+1, y, z; (iv) −x+2, −y, −z+2; (v) −x, −y, −z+1; (vi) −x+1, −y, −z+1. |
Experimental details
Crystal data |
Chemical formula | C2H2N4O3 |
Mr | 130.08 |
Crystal system, space group | Triclinic, P1 |
Temperature (K) | 293 |
a, b, c (Å) | 5.1233 (8), 10.3140 (16), 17.998 (3) |
α, β, γ (°) | 106.610 (2), 97.810 (2), 90.130 (2) |
V (Å3) | 902.1 (2) |
Z | 8 |
Radiation type | Mo Kα |
µ (mm−1) | 0.18 |
Crystal size (mm) | 0.34 × 0.26 × 0.20 |
|
Data collection |
Diffractometer | Bruker platform with 1K CCD diffractometer |
Absorption correction | Multi-scan SADABS;Sheldrick1996;Blessing,ActaC.1995,A51,33 |
Tmin, Tmax | 0.851, 1.000 |
No. of measured, independent and observed [I > 2σ(I)] reflections | 8966, 8966, 8202 |
Rint | 0.000 |
(sin θ/λ)max (Å−1) | 0.676 |
|
Refinement |
R[F2 > 2σ(F2)], wR(F2), S | 0.043, 0.115, 1.12 |
No. of reflections | 8966 |
No. of parameters | 328 |
H-atom treatment | H-atom parameters constrained |
Δρmax, Δρmin (e Å−3) | 0.34, −0.36 |
Hydrogen-bond geometry (Å, º) top
D—H···A | D—H | H···A | D···A | D—H···A |
N21—H21···O32i | 0.86 | 2.01 | 2.769 (2) | 146.5 |
N21—H21···O31ii | 0.86 | 2.51 | 3.089 (3) | 125.0 |
N41—H41···N11iii | 0.86 | 2.20 | 2.912 (3) | 140.6 |
N22—H22···O31iv | 0.86 | 2.04 | 2.791 (3) | 146.1 |
N22—H22···O32iii | 0.86 | 2.52 | 3.100 (3) | 125.8 |
N42—H42···N12ii | 0.86 | 2.19 | 2.913 (3) | 141.9 |
N23—H23···O34v | 0.86 | 2.01 | 2.778 (2) | 147.4 |
N23—H23···O33ii | 0.86 | 2.51 | 3.073 (3) | 123.4 |
N43—H43···N13iii | 0.86 | 2.22 | 2.923 (3) | 138.7 |
N24—H24···O33vi | 0.86 | 2.01 | 2.776 (2) | 147.4 |
N24—H24···O34iii | 0.86 | 2.52 | 3.081 (3) | 123.5 |
N44—H44···N14ii | 0.86 | 2.23 | 2.928 (3) | 138.8 |
Symmetry codes: (i) −x+1, −y, −z+2; (ii) x−1, y, z; (iii) x+1, y, z; (iv) −x+2, −y, −z+2; (v) −x, −y, −z+1; (vi) −x+1, −y, −z+1. |
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