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inorganic compounds
Crystals of KAlSiO4-O1 (potassium aluminium silicate) were synthesized using a flux method and analysed utilizing single-crystal X-ray diffraction and electron microprobe analysis. Both methods confirm that the crystals are nonstoichiometric according to K1-xAl1-xSi1+xO4 with x = 0.04 (1). KAlSiO4-O1 is closely related to the stuffed derivatives of tridymite, although the topology of the Si/Al-ordered framework is different. Six-membered rings of UUDDUD and UUUDDD (U = up and D = down; ratio 2:1) configurations are present in layers parallel to the ab plane. In contrast, the framework of tridymite exhibits UDUDUD rings. The crystals are affected by inversion, pseudo-orthorhombic and pseudo-hexagonal twinning.
Supporting information
Crystallographic Information File (CIF) https://doi.org/10.1107/S0108270113006069/qs3022sup1.cif | |
Structure factor file (CIF format) https://doi.org/10.1107/S0108270113006069/qs3022Isup2.hkl |
Computing details top
Data collection: APEX2 (Bruker, 2011); cell refinement: SAINT (Bruker, 2011); data reduction: SAINT (Bruker, 2011); program(s) used to solve structure: SIR97 (Altomare et al., 1999); program(s) used to refine structure: JANA2006 (Petříček et al., 2006); molecular graphics: DIAMOND (Brandenburg & Putz, 2005); software used to prepare material for publication: publCIF (Westrip, 2010).
Potassium aluminium silicate top
Crystal data top
KAlSiO4 | F(000) = 936 |
Mr = 158.2 | Dx = 2.597 Mg m−3 |
Monoclinic, P21 | Mo Kα radiation, λ = 0.71073 Å |
Hall symbol: P 2yb | Cell parameters from 31861 reflections |
a = 15.6553 (2) Å | θ = 2.4–47.5° |
b = 9.0565 (1) Å | µ = 1.7 mm−1 |
c = 8.5568 (1) Å | T = 293 K |
β = 90.017 (1)° | Plate, colourless |
V = 1213.20 (2) Å3 | 0.5 × 0.4 × 0.1 mm |
Z = 12 |
Data collection top
Bruker APEXII area-detector diffractometer | 20167 independent reflections |
Radiation source: X-ray tube | 18282 reflections with I > 3σ(I) |
Graphite monochromator | Rint = 0.018 |
ω scans | θmax = 47.5°, θmin = 2.4° |
Absorption correction: multi-scan (SADABS; Bruker, 2011) | h = −25→32 |
Tmin = 0.604, Tmax = 0.749 | k = −18→18 |
35192 measured reflections | l = −17→17 |
Refinement top
Refinement on F | 1 constraint |
R[F > 3σ(F)] = 0.040 | Weighting scheme based on measured s.u.'s w = 1/(σ2(F) + 0.000625F2) |
wR(F) = 0.055 | (Δ/σ)max = 0.017 |
S = 1.44 | Δρmax = 1.82 e Å−3 |
20167 reflections | Δρmin = −0.63 e Å−3 |
389 parameters | Absolute structure: Flack (1983), with 8056 Friedel pairs |
0 restraints | Absolute structure parameter: 0.32 (2) |
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
x | y | z | Uiso*/Ueq | Occ. (<1) | |
K1 | 0.29498 (3) | −0.03227 (6) | 0.24888 (11) | 0.02487 (11) | 0.983 (2) |
K2 | 0.20979 (3) | 0.02900 (5) | 0.74823 (9) | 0.02285 (10) | 0.989 (2) |
K3 | −0.06023 (4) | −0.00379 (5) | 0.75172 (11) | 0.02664 (13) | 0.977 (2) |
K4 | −0.10788 (4) | −0.01824 (8) | 0.25179 (14) | 0.02846 (15) | 0.892 (2) |
K5 | −0.39711 (3) | 0.01452 (5) | 0.75160 (9) | 0.01702 (8) | 0.999 (2) |
K6 | −0.45263 (3) | 0.00776 (5) | 0.25163 (11) | 0.01884 (9) | 0.909 (2) |
Si1 | 0.27484 (5) | −0.34492 (7) | 0.55560 (7) | 0.00732 (11) | |
Al1 | 0.27580 (5) | −0.34424 (9) | −0.06512 (8) | 0.00781 (14) | |
Al2 | 0.11335 (5) | −0.16717 (9) | 0.43553 (8) | 0.00704 (13) | |
Si2 | 0.11319 (5) | −0.16915 (8) | 0.05446 (7) | 0.00750 (12) | |
Si3 | −0.05474 (4) | −0.33463 (8) | 0.55244 (7) | 0.00713 (12) | |
Al3 | −0.05485 (5) | −0.33616 (9) | −0.06213 (9) | 0.00798 (14) | |
Al4 | −0.23183 (5) | −0.17634 (9) | 0.56483 (8) | 0.00775 (14) | |
Si4 | −0.23077 (5) | −0.17781 (8) | −0.05826 (7) | 0.00794 (12) | |
Si5 | −0.40085 (5) | −0.32250 (8) | 0.44168 (8) | 0.00776 (12) | |
Al5 | −0.40016 (6) | −0.32430 (10) | 0.06499 (8) | 0.00892 (15) | |
Al6 | −0.56359 (5) | −0.15606 (9) | 0.56259 (9) | 0.00766 (14) | |
Si6 | −0.56392 (5) | −0.15599 (8) | −0.05692 (7) | 0.00753 (12) | |
O1 | 0.20725 (15) | −0.2119 (2) | 0.5355 (2) | 0.0154 (4) | |
O2 | 0.19964 (15) | −0.2072 (3) | −0.0378 (3) | 0.0175 (4) | |
O3 | 0.03264 (13) | −0.2972 (3) | 0.4640 (3) | 0.0161 (4) | |
O4 | 0.04267 (14) | −0.2927 (3) | 0.0266 (3) | 0.0156 (4) | |
O5 | −0.12681 (16) | −0.2121 (3) | 0.5055 (2) | 0.0157 (5) | |
O6 | −0.13258 (16) | −0.2082 (3) | −0.0074 (2) | 0.0146 (4) | |
O7 | −0.30109 (13) | −0.2885 (3) | 0.4590 (2) | 0.0151 (4) | |
O8 | −0.29270 (13) | −0.2808 (3) | 0.0456 (2) | 0.0151 (4) | |
O9 | −0.45595 (14) | −0.1986 (3) | 0.5303 (3) | 0.0153 (4) | |
O10 | −0.46470 (13) | −0.1926 (3) | −0.0242 (3) | 0.0145 (4) | |
O11 | −0.63190 (16) | −0.2905 (3) | 0.4879 (2) | 0.0144 (4) | |
O12 | −0.62761 (15) | −0.2811 (3) | 0.0140 (2) | 0.0130 (4) | |
O13 | 0.08350 (18) | 0.0042 (2) | 0.5084 (3) | 0.0164 (5) | |
O14 | 0.08040 (17) | −0.0099 (2) | −0.0098 (2) | 0.0140 (5) | |
O15 | −0.24615 (15) | 0.0117 (2) | 0.5420 (2) | 0.0155 (4) | |
O16 | −0.25041 (15) | −0.0039 (2) | −0.0347 (2) | 0.0156 (4) | |
O17 | −0.58120 (16) | 0.0177 (2) | 0.4779 (2) | 0.0130 (4) | |
O18 | −0.58343 (15) | 0.0032 (2) | 0.0224 (2) | 0.0123 (4) | |
O19 | 0.28296 (9) | −0.38970 (16) | 0.73883 (19) | 0.0142 (3) | |
O20 | 0.13679 (9) | −0.15920 (17) | 0.23818 (17) | 0.0148 (3) | |
O21 | −0.03799 (10) | −0.33272 (17) | 0.73902 (19) | 0.0163 (3) | |
O22 | −0.24583 (10) | −0.22158 (15) | 0.7594 (2) | 0.0161 (3) | |
O23 | −0.42963 (10) | −0.32083 (17) | 0.26136 (19) | 0.0145 (3) | |
O24 | −0.58104 (10) | −0.14600 (18) | 0.75997 (19) | 0.0169 (3) |
Atomic displacement parameters (Å2) top
U11 | U22 | U33 | U12 | U13 | U23 | |
K1 | 0.01753 (17) | 0.0341 (2) | 0.02296 (18) | −0.00664 (16) | 0.0021 (3) | −0.0003 (3) |
K2 | 0.0292 (2) | 0.02448 (19) | 0.01488 (14) | −0.01146 (15) | −0.0006 (3) | 0.0001 (2) |
K3 | 0.0468 (3) | 0.01568 (17) | 0.01747 (16) | −0.00974 (18) | −0.0015 (4) | 0.0015 (2) |
K4 | 0.0249 (2) | 0.0347 (3) | 0.0257 (2) | −0.0078 (2) | −0.0011 (4) | 0.0004 (4) |
K5 | 0.02033 (16) | 0.01732 (14) | 0.01341 (12) | −0.00304 (12) | −0.0017 (2) | 0.0004 (2) |
K6 | 0.01846 (17) | 0.02073 (17) | 0.01732 (15) | −0.00470 (14) | −0.0019 (3) | −0.0007 (2) |
Si1 | 0.0082 (2) | 0.00609 (19) | 0.00766 (17) | −0.00071 (19) | −0.00123 (17) | −0.00159 (18) |
Al1 | 0.0070 (3) | 0.0087 (2) | 0.0078 (2) | −0.0001 (2) | 0.00074 (19) | −0.0004 (2) |
Al2 | 0.0068 (3) | 0.0068 (2) | 0.00751 (19) | 0.0001 (2) | 0.00026 (19) | −0.0001 (2) |
Si2 | 0.0075 (2) | 0.0075 (2) | 0.00753 (17) | 0.00045 (19) | −0.00011 (17) | −0.00042 (19) |
Si3 | 0.0067 (2) | 0.0070 (2) | 0.00768 (17) | −0.00021 (19) | 0.00115 (16) | −0.00074 (19) |
Al3 | 0.0072 (3) | 0.0081 (3) | 0.0087 (2) | −0.0007 (2) | 0.0001 (2) | −0.0012 (2) |
Al4 | 0.0069 (3) | 0.0078 (2) | 0.0086 (2) | 0.0014 (2) | −0.0010 (2) | 0.0013 (2) |
Si4 | 0.0081 (2) | 0.0077 (2) | 0.00804 (17) | 0.0002 (2) | 0.00054 (18) | 0.00114 (18) |
Si5 | 0.0069 (2) | 0.0073 (2) | 0.00906 (18) | −0.0006 (2) | −0.00029 (17) | 0.00030 (19) |
Al5 | 0.0095 (3) | 0.0090 (3) | 0.0083 (2) | −0.0008 (3) | −0.0002 (2) | 0.0029 (2) |
Al6 | 0.0081 (3) | 0.0066 (2) | 0.0083 (2) | 0.0002 (2) | −0.0005 (2) | 0.0007 (2) |
Si6 | 0.0077 (2) | 0.0080 (2) | 0.00692 (17) | −0.00015 (19) | −0.00028 (18) | 0.00125 (19) |
O1 | 0.0138 (7) | 0.0122 (6) | 0.0203 (7) | 0.0025 (6) | −0.0071 (6) | −0.0010 (6) |
O2 | 0.0134 (7) | 0.0203 (8) | 0.0187 (6) | 0.0101 (7) | 0.0064 (6) | 0.0051 (6) |
O3 | 0.0103 (6) | 0.0169 (7) | 0.0210 (7) | −0.0041 (6) | 0.0060 (5) | −0.0037 (6) |
O4 | 0.0108 (6) | 0.0140 (7) | 0.0220 (7) | −0.0073 (6) | −0.0065 (5) | 0.0021 (6) |
O5 | 0.0102 (7) | 0.0165 (8) | 0.0205 (8) | 0.0035 (7) | 0.0037 (5) | 0.0021 (6) |
O6 | 0.0095 (7) | 0.0139 (8) | 0.0205 (7) | 0.0027 (6) | 0.0016 (5) | −0.0054 (5) |
O7 | 0.0081 (6) | 0.0197 (7) | 0.0176 (6) | −0.0009 (6) | −0.0008 (5) | −0.0060 (6) |
O8 | 0.0070 (5) | 0.0203 (7) | 0.0179 (6) | −0.0060 (5) | −0.0012 (5) | 0.0078 (6) |
O9 | 0.0104 (6) | 0.0141 (7) | 0.0216 (7) | 0.0013 (6) | 0.0010 (5) | −0.0059 (6) |
O10 | 0.0085 (6) | 0.0140 (7) | 0.0211 (6) | 0.0028 (5) | −0.0017 (5) | 0.0064 (6) |
O11 | 0.0141 (8) | 0.0144 (7) | 0.0148 (6) | −0.0040 (6) | 0.0057 (5) | −0.0038 (5) |
O12 | 0.0121 (7) | 0.0141 (7) | 0.0128 (5) | −0.0072 (6) | 0.0005 (5) | 0.0051 (5) |
O13 | 0.0211 (11) | 0.0082 (7) | 0.0198 (8) | 0.0058 (7) | −0.0010 (6) | −0.0011 (5) |
O14 | 0.0204 (10) | 0.0102 (7) | 0.0114 (6) | 0.0060 (6) | −0.0004 (5) | −0.0008 (4) |
O15 | 0.0201 (8) | 0.0088 (6) | 0.0175 (6) | 0.0030 (5) | −0.0037 (6) | 0.0027 (5) |
O16 | 0.0209 (9) | 0.0079 (5) | 0.0179 (6) | 0.0051 (5) | −0.0006 (6) | −0.0026 (5) |
O17 | 0.0205 (9) | 0.0069 (5) | 0.0116 (6) | 0.0022 (5) | 0.0008 (5) | −0.0006 (4) |
O18 | 0.0201 (9) | 0.0066 (6) | 0.0101 (5) | 0.0011 (5) | 0.0018 (5) | −0.0008 (4) |
O19 | 0.0172 (5) | 0.0164 (5) | 0.0088 (4) | 0.0015 (4) | 0.0005 (5) | −0.0026 (5) |
O20 | 0.0160 (5) | 0.0208 (5) | 0.0074 (4) | −0.0019 (4) | −0.0007 (4) | −0.0002 (5) |
O21 | 0.0195 (5) | 0.0209 (6) | 0.0086 (5) | −0.0009 (5) | 0.0010 (5) | −0.0004 (5) |
O22 | 0.0249 (6) | 0.0140 (4) | 0.0095 (4) | −0.0011 (4) | 0.0007 (6) | 0.0013 (5) |
O23 | 0.0172 (5) | 0.0174 (5) | 0.0088 (4) | −0.0012 (4) | −0.0009 (5) | 0.0007 (5) |
O24 | 0.0241 (6) | 0.0193 (5) | 0.0073 (4) | −0.0027 (5) | −0.0007 (5) | 0.0006 (5) |
Geometric parameters (Å, º) top
K1—O17i | 2.793 (2) | Al2—O13 | 1.736 (2) |
K1—O18i | 2.736 (2) | Al2—O20 | 1.7298 (16) |
K1—O20 | 2.7318 (15) | Si2—O2 | 1.604 (2) |
K1—O22ii | 2.9179 (15) | Si2—O4 | 1.590 (2) |
K2—O1 | 2.841 (2) | Si2—O14 | 1.627 (2) |
K2—O2iii | 2.820 (2) | Si2—O20 | 1.6173 (16) |
K2—O7ii | 2.814 (2) | Si3—O3 | 1.600 (2) |
K2—O8ii | 2.786 (2) | Si3—O5 | 1.633 (3) |
K2—O13 | 2.858 (3) | Si3—O13v | 1.614 (2) |
K2—O14iii | 2.918 (2) | Si3—O21 | 1.6179 (18) |
K3—O3ii | 2.664 (2) | Al3—O4 | 1.750 (2) |
K3—O4ii | 2.707 (2) | Al3—O6 | 1.744 (3) |
K3—O6iii | 2.993 (2) | Al3—O14vi | 1.736 (2) |
K4—O5 | 2.808 (2) | Al3—O21vii | 1.7222 (18) |
K4—O6 | 2.833 (2) | Al4—O5 | 1.751 (3) |
K4—O19ii | 2.9790 (16) | Al4—O7 | 1.740 (2) |
K4—O21ii | 2.8361 (17) | Al4—O15 | 1.728 (2) |
K5—O9 | 2.856 (2) | Al4—O22 | 1.7284 (18) |
K5—O10iii | 2.885 (2) | Si4—O6 | 1.621 (3) |
K5—O11iv | 2.743 (2) | Si4—O8 | 1.612 (2) |
K5—O12iv | 2.756 (2) | Si4—O16 | 1.618 (2) |
K5—O15 | 2.967 (2) | Si4—O22vii | 1.6272 (18) |
K5—O16iii | 2.940 (2) | Si5—O7 | 1.599 (2) |
K6—O10 | 2.983 (2) | Si5—O9 | 1.606 (2) |
K6—O17 | 2.795 (2) | Si5—O17viii | 1.627 (2) |
K6—O18 | 2.835 (2) | Si5—O23 | 1.6073 (17) |
K6—O19ii | 2.8152 (15) | Al5—O8 | 1.736 (2) |
K6—O23 | 2.9987 (16) | Al5—O10 | 1.739 (2) |
Si1—O1 | 1.613 (2) | Al5—O18ix | 1.751 (2) |
Si1—O11i | 1.646 (3) | Al5—O23 | 1.7429 (17) |
Si1—O15v | 1.608 (2) | Al6—O9 | 1.751 (2) |
Si1—O19 | 1.6243 (17) | Al6—O11 | 1.742 (3) |
Al1—O2 | 1.737 (3) | Al6—O17 | 1.754 (2) |
Al1—O12i | 1.752 (2) | Al6—O24 | 1.7134 (18) |
Al1—O16vi | 1.725 (2) | Si6—O10 | 1.613 (2) |
Al1—O19vii | 1.7311 (18) | Si6—O12 | 1.627 (2) |
Al2—O1 | 1.748 (2) | Si6—O18 | 1.622 (2) |
Al2—O3 | 1.744 (2) | Si6—O24vii | 1.5921 (17) |
O17i—K1—O18i | 89.72 (7) | O12i—Al1—O16vi | 106.32 (11) |
O17i—K1—O20 | 136.11 (6) | O12i—Al1—O19vii | 113.32 (9) |
O17i—K1—O22ii | 92.51 (6) | O16vi—Al1—O19vii | 107.18 (9) |
O18i—K1—O20 | 131.04 (6) | O1—Al2—O3 | 112.61 (11) |
O18i—K1—O22ii | 93.05 (5) | O1—Al2—O13 | 104.91 (12) |
O20—K1—O22ii | 99.55 (5) | O1—Al2—O20 | 107.99 (9) |
O1—K2—O2iii | 80.38 (6) | O3—Al2—O13 | 110.98 (12) |
O1—K2—O7ii | 93.11 (7) | O3—Al2—O20 | 108.58 (10) |
O1—K2—O8ii | 152.46 (7) | O13—Al2—O20 | 111.75 (10) |
O1—K2—O13 | 58.00 (7) | O2—Si2—O4 | 111.16 (12) |
O1—K2—O14iii | 110.63 (6) | O2—Si2—O14 | 106.87 (13) |
O2iii—K2—O7ii | 152.03 (7) | O2—Si2—O20 | 107.33 (10) |
O2iii—K2—O8ii | 94.84 (7) | O4—Si2—O14 | 110.74 (13) |
O2iii—K2—O13 | 111.55 (7) | O4—Si2—O20 | 110.09 (10) |
O2iii—K2—O14iii | 53.74 (7) | O14—Si2—O20 | 110.56 (10) |
O7ii—K2—O8ii | 78.40 (6) | O3—Si3—O5 | 109.30 (12) |
O7ii—K2—O13 | 86.86 (7) | O3—Si3—O13v | 106.12 (13) |
O7ii—K2—O14iii | 150.53 (7) | O3—Si3—O21 | 109.02 (10) |
O8ii—K2—O13 | 145.61 (7) | O5—Si3—O13v | 110.03 (13) |
O8ii—K2—O14iii | 87.07 (6) | O5—Si3—O21 | 110.34 (10) |
O13—K2—O14iii | 91.13 (7) | O13v—Si3—O21 | 111.91 (10) |
O3ii—K3—O4ii | 88.46 (7) | O4—Al3—O6 | 110.04 (12) |
O3ii—K3—O6iii | 166.68 (7) | O4—Al3—O14vi | 104.55 (12) |
O4ii—K3—O6iii | 89.58 (7) | O4—Al3—O21vii | 106.89 (10) |
O5—K4—O6 | 102.19 (7) | O6—Al3—O14vi | 110.25 (12) |
O5—K4—O19ii | 97.25 (6) | O6—Al3—O21vii | 111.11 (9) |
O5—K4—O21ii | 115.74 (7) | O14vi—Al3—O21vii | 113.71 (9) |
O6—K4—O19ii | 97.64 (6) | O5—Al4—O7 | 109.03 (11) |
O6—K4—O21ii | 119.41 (7) | O5—Al4—O15 | 105.75 (12) |
O19ii—K4—O21ii | 120.57 (5) | O5—Al4—O22 | 110.78 (9) |
O9—K5—O10iii | 83.29 (6) | O7—Al4—O15 | 115.81 (11) |
O9—K5—O11iv | 89.60 (7) | O7—Al4—O22 | 106.49 (9) |
O9—K5—O12iv | 168.91 (7) | O15—Al4—O22 | 109.01 (9) |
O9—K5—O15 | 81.39 (6) | O6—Si4—O8 | 108.92 (11) |
O9—K5—O16iii | 128.74 (6) | O6—Si4—O16 | 108.18 (13) |
O10iii—K5—O11iv | 167.52 (7) | O6—Si4—O22vii | 110.67 (10) |
O10iii—K5—O12iv | 90.23 (7) | O8—Si4—O16 | 112.34 (12) |
O10iii—K5—O15 | 133.53 (6) | O8—Si4—O22vii | 107.51 (10) |
O10iii—K5—O16iii | 80.57 (6) | O16—Si4—O22vii | 109.23 (9) |
O11iv—K5—O12iv | 95.07 (7) | O7—Si5—O9 | 110.26 (12) |
O11iv—K5—O15 | 54.69 (7) | O7—Si5—O17viii | 107.52 (12) |
O11iv—K5—O16iii | 111.82 (7) | O7—Si5—O23 | 111.13 (10) |
O12iv—K5—O15 | 109.51 (6) | O9—Si5—O17viii | 109.22 (12) |
O12iv—K5—O16iii | 58.39 (6) | O9—Si5—O23 | 107.24 (10) |
O15—K5—O16iii | 75.72 (6) | O17viii—Si5—O23 | 111.47 (10) |
O10—K6—O17 | 121.48 (6) | O8—Al5—O10 | 111.43 (11) |
O10—K6—O18 | 53.01 (6) | O8—Al5—O18ix | 107.70 (12) |
O10—K6—O19ii | 106.45 (6) | O8—Al5—O23 | 110.16 (9) |
O10—K6—O23 | 54.96 (5) | O10—Al5—O18ix | 109.84 (11) |
O17—K6—O18 | 87.68 (6) | O10—Al5—O23 | 104.86 (10) |
O17—K6—O19ii | 130.47 (6) | O18ix—Al5—O23 | 112.89 (9) |
O17—K6—O23 | 95.69 (5) | O9—Al6—O11 | 112.28 (12) |
O18—K6—O19ii | 134.92 (6) | O9—Al6—O17 | 106.46 (12) |
O18—K6—O23 | 95.24 (5) | O9—Al6—O24 | 108.72 (10) |
O19ii—K6—O23 | 102.31 (4) | O11—Al6—O17 | 112.29 (11) |
O1—Si1—O11i | 108.71 (12) | O11—Al6—O24 | 107.50 (10) |
O1—Si1—O15v | 111.41 (12) | O17—Al6—O24 | 109.54 (9) |
O1—Si1—O19 | 109.91 (10) | O10—Si6—O12 | 112.47 (12) |
O11i—Si1—O15v | 107.85 (12) | O10—Si6—O18 | 106.95 (12) |
O11i—Si1—O19 | 110.19 (10) | O10—Si6—O24vii | 110.16 (10) |
O15v—Si1—O19 | 108.75 (9) | O12—Si6—O18 | 110.33 (12) |
O2—Al1—O12i | 107.89 (12) | O12—Si6—O24vii | 107.67 (10) |
O2—Al1—O16vi | 111.96 (11) | O18—Si6—O24vii | 109.25 (10) |
O2—Al1—O19vii | 110.18 (10) |
Symmetry codes: (i) x+1, y, z; (ii) −x, y+1/2, −z+1; (iii) x, y, z+1; (iv) −x−1, y+1/2, −z+1; (v) −x, y−1/2, −z+1; (vi) −x, y−1/2, −z; (vii) x, y, z−1; (viii) −x−1, y−1/2, −z+1; (ix) −x−1, y−1/2, −z. |
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