Crystallography and Diffraction. Theory and Modern Methods of Analysis

Workshop 1

Symmetry and Space Group Determination

1.Complete the following table:

Unit cells and crystal systems for a range of inorganic solids

Formula / a (Å) / b (Å) / c (Å) /  () /  () /  () / Crystal
system
Ti5O9 / 5.57 / 7.12 / 8.86 / 97.5 / 112.3 / 108.5
Al / 4.05 / 4.05 / 4.05 / 90 / 90 / 90
Zn / 2.66 / 4.95 / hexagonal
V2O5 / 11.5 / 3.56 / 4.37 / orthorhombic
NaSb / 6.80 / 6.34 / 12.48 / 117.7 / monoclinic
Cu2O / 4.27 / cubic
TiO2 / 4.59 / 2.95 / tetragonal
CaCO3 / 6.36 / 46.1 / trigonal
Fe3C / 4.52 / 5.09 / 6.74 / 90 / 90 / 90
P4S10 / 9.07 / 9.18 / 9.19 / 92.4 / 101.2 / 110.5
Pb2O3 / 7.05 / 5.62 / 3.86 / 80.1 / 90 / monoclinic
-U / 10.59 / 10.59 / 5.63 / 90 / 90 / 90
WO3 / 7.27 / 7.50 / 3.82 / 90 / 89.9 / 90
MgO / 4.44 / cubic
Al2O3 / 5.13 / 5.13 / 5.13 / 55.3 / 55.3 / 55.3
FeCl3 / 6.76 / 53.2 / trigonal
AgNO2 / 3.53 / 5.17 / 6.17 / 90 / 90 / 90
H2O / 4.52 / 4.52 / 7.37 / 90 / 90 / 120
CaCl2 / 6.24 / 6.43 / 4.20 / orthorhombic
CaC2 / 3.87 / 6.37 / tetragonal

2. Rb2CoI4 crystallises in the monoclinic space group P21/m with cell dimensions a = 10.382, b = 8.144, c = 7.657 Å,  = 109.8.

(i) Describe each of the terms that make up the space group symbol P21/m.

(ii) What equivalent positions are generated in this space group ?

(iii) What systematic absences would be expected in the X-ray diffraction data for Rb2CoI4 ?

3.WO2Cl2 crystallises in the monoclinic crystal system. The following X-ray powder diffraction data were collected using Cu K radiation ( = 1.5418 Å).

h / k / l / d/Å / h / k / l / d/Å
2 / 0 / 0 / 6.7005 / 2 / 2 / 0 / 1.8679
1 / 1 / 0 / 3.7358 / 5 / 1 / -3 / 1.8596
0 / 0 / 2 / 3.7021 / 0 / 0 / 4 / 1.8511
2 / 0 / -2 / 3.6804 / 2 / 2 / -1 / 1.8438
1 / 1 / -1 / 3.4393 / 4 / 0 / -4 / 1.8402
4 / 0 / 0 / 3.3502 / 2 / 2 / 1 / 1.7802
1 / 1 / 1 / 3.2402 / 7 / 1 / -1 / 1.7686
3 / 1 / 0 / 2.9336 / 3 / 1 / 3 / 1.7446
2 / 0 / 2 / 2.9280 / 5 / 1 / 2 / 1.7361
3 / 1 / -1 / 2.9064 / 7 / 1 / -2 / 1.7227
4 / 0 / -2 / 2.8960 / 0 / 2 / 2 / 1.7218
1 / 1 / -2 / 2.7327 / 6 / 0 / 2 / 1.7209
3 / 1 / 1 / 2.5777 / 2 / 2 / -2 / 1.7196
1 / 1 / 2 / 2.5374 / 7 / 1 / 0 / 1.7177
3 / 1 / -2 / 2.5234 / 1 / 1 / -4 / 1.7097
5 / 1 / -1 / 2.2543 / 8 / 0 / -2 / 1.7057
6 / 0 / 0 / 2.2335 / 3 / 1 / -4 / 1.7054
4 / 0 / 2 / 2.2093 / 4 / 2 / -1 / 1.6897
5 / 1 / 0 / 2.2070 / 4 / 2 / 0 / 1.6821
6 / 0 / -2 / 2.1864 / 8 / 0 / 0 / 1.6751
1 / 1 / -3 / 2.1332 / 2 / 0 / 4 / 1.6738
3 / 1 / 2 / 2.1258 / 6 / 0 / -4 / 1.6578
5 / 1 / -2 / 2.1095 / 2 / 2 / 2 / 1.6201
3 / 1 / -3 / 2.0753 / 4 / 2 / -2 / 1.6146
5 / 1 / 1 / 1.9988 / 1 / 1 / 4 / 1.6118
1 / 1 / 3 / 1.9925 / 5 / 1 / -4 / 1.6011
0 / 2 / 0 / 1.9450 / 7 / 1 / -3 / 1.5999
2 / 0 / -4 / 1.9199 / 4 / 2 / 1 / 1.5950
0 / 2 / 1 / 1.8812 / 7 / 1 / 1 / 1.5918

(i) Use the data above to determine the space group of WO2Cl2.

(ii) What are the equivalent positions for this space group?

4.MgO possesses the cubic NaCl structure. The standard X-ray powder

diffraction pattern for MgO is as follows:

h k l / 2() / d-spacing (Å)
1 1 1 / 36.91
2 0 0 / 42.88
2 2 0 / 62.27
3 1 1 / 74.61
2 2 2 / 78.61

Using Bragg's law, complete the table above (assume CuK radiation, =1.5418Å). Prove that the structure is cubic by calculating the unit cell dimensions a, b and c, assuming that the angles are orthogonal and that the following relationship holds.

5. For each of the following X-ray powder diffraction data sets determine the probable

space group (or groups if there is more than one option). Where necessary use the determinative tables to aid you.

(i)YBa2CuO5 orthorhombic

h / k / l / d/Å / h / k / l / d/Å
1 / 1 / 0 / 6.1547 / 1 / 3 / 2 / 2.2072
0 / 2 / 0 / 6.0905 / 3 / 0 / 1 / 2.1917
1 / 2 / 0 / 4.6315 / 2 / 1 / 2 / 2.1805
1 / 0 / 1 / 4.4325 / 3 / 1 / 1 / 2.1571
1 / 1 / 1 / 4.1653 / 2 / 4 / 1 / 2.1432
0 / 2 / 1 / 4.1453 / 1 / 5 / 1 / 2.1350
1 / 2 / 1 / 3.5839 / 2 / 2 / 2 / 2.0827
2 / 0 / 0 / 3.5660 / 0 / 4 / 2 / 2.0726
1 / 3 / 0 / 3.5286 / 3 / 2 / 1 / 2.0623
2 / 1 / 0 / 3.4224 / 3 / 3 / 0 / 2.0516
2 / 2 / 0 / 3.0773 / 0 / 6 / 0 / 2.0302
0 / 4 / 0 / 3.0452 / 2 / 5 / 0 / 2.0116
1 / 3 / 1 / 2.9940 / 1 / 4 / 2 / 1.9903
2 / 1 / 1 / 2.9283 / 1 / 6 / 0 / 1.9526
0 / 0 / 2 / 2.8290 / 2 / 3 / 2 / 1.9453
1 / 4 / 0 / 2.8006 / 3 / 3 / 1 / 1.9287
2 / 2 / 1 / 2.7033 / 0 / 6 / 1 / 1.9109
0 / 4 / 1 / 2.6815 / 2 / 5 / 1 / 1.8954
2 / 3 / 0 / 2.6794 / 3 / 4 / 0 / 1.8739
1 / 1 / 2 / 2.5705 / 1 / 6 / 1 / 1.8458
0 / 2 / 2 / 2.5657 / 1 / 0 / 3 / 1.8233
1 / 4 / 1 / 2.5100 / 1 / 1 / 3 / 1.8032
2 / 3 / 1 / 2.4216 / 0 / 2 / 3 / 1.8016
1 / 2 / 2 / 2.4143 / 3 / 1 / 2 / 1.8000
3 / 1 / 0 / 2.3333 / 2 / 4 / 2 / 1.7919
2 / 4 / 0 / 2.3158 / 1 / 5 / 2 / 1.7871
1 / 5 / 0 / 2.3054 / 4 / 0 / 0 / 1.7830
2 / 0 / 2 / 2.2163 / 3 / 4 / 1 / 1.7789
3 / 2 / 0 / 2.2146 / 2 / 6 / 0 / 1.7643

(ii)BICOVOX.10 Tetragonal

h / k / l / d/Å / h / k / l / d/Å
0 / 0 / 2 / 7.7125 / 3 / 0 / 1 / 1.3008
0 / 0 / 4 / 3.8563 / 0 / 0 / 12 / 1.2854
1 / 0 / 1 / 3.7960 / 3 / 0 / 3 / 1.2653
1 / 0 / 3 / 3.1155 / 3 / 1 / 0 / 1.2385
1 / 1 / 0 / 2.7693 / 2 / 1 / 9 / 1.2250
1 / 1 / 2 / 2.6064 / 3 / 1 / 2 / 1.2228
0 / 0 / 6 / 2.5708 / 2 / 2 / 6 / 1.2191
1 / 0 / 5 / 2.4234 / 2 / 0 / 10 / 1.2117
1 / 1 / 4 / 2.2494 / 3 / 0 / 5 / 1.2023
2 / 0 / 0 / 1.9582 / 3 / 1 / 4 / 1.1792
0 / 0 / 8 / 1.9281 / 1 / 1 / 12 / 1.1659
1 / 0 / 7 / 1.9205 / 1 / 0 / 13 / 1.1356
2 / 0 / 2 / 1.8980 / 2 / 2 / 8 / 1.1247
1 / 1 / 6 / 1.8841 / 3 / 0 / 7 / 1.1232
2 / 0 / 4 / 1.7460 / 3 / 1 / 6 / 1.1158
2 / 1 / 1 / 1.7403 / 0 / 0 / 14 / 1.1018
2 / 1 / 3 / 1.6579 / 2 / 1 / 11 / 1.0947
1 / 1 / 8 / 1.5824 / 3 / 2 / 1 / 1.0835
1 / 0 / 9 / 1.5701 / 2 / 0 / 12 / 1.0746
2 / 0 / 6 / 1.5578 / 3 / 2 / 3 / 1.0628
0 / 0 / 10 / 1.5425 / 3 / 1 / 8 / 1.0420
2 / 1 / 5 / 1.5231 / 3 / 0 / 9 / 1.0385
2 / 2 / 0 / 1.3847 / 2 / 2 / 10 / 1.0304
2 / 0 / 8 / 1.3739 / 3 / 2 / 5 / 1.0246
2 / 1 / 7 / 1.3711 / 1 / 1 / 14 / 1.0237
2 / 2 / 2 / 1.3629 / 1 / 0 / 15 / 0.9946
1 / 1 / 10 / 1.3476 / 2 / 1 / 13 / 0.9823
1 / 0 / 11 / 1.3202 / 4 / 0 / 0 / 0.9791
2 / 2 / 4 / 1.3032 / 3 / 2 / 7 / 0.9743

(iii)ZnO Hexagonal

h / k / l / d/Å
1 / 0 / 0 / 2.8172
0 / 0 / 2 / 2.6065
1 / 0 / 1 / 2.4784
1 / 0 / 2 / 1.9132
1 / 1 / 0 / 1.6265
1 / 0 / 3 / 1.4790
2 / 0 / 0 / 1.4086
1 / 1 / 2 / 1.3799
2 / 0 / 1 / 1.3598
0 / 0 / 4 / 1.3032
2 / 0 / 2 / 1.2392
1 / 0 / 4 / 1.1828
2 / 0 / 3 / 1.0942
2 / 1 / 0 / 1.0648
2 / 1 / 1 / 1.0433
1 / 1 / 4 / 1.0170
2 / 1 / 2 / 0.9857
1 / 0 / 5 / 0.9778
2 / 0 / 4 / 0.9566
3 / 0 / 0 / 0.9391
3 / 0 / 1 / 0.9242
2 / 1 / 3 / 0.9079
3 / 0 / 2 / 0.8835
0 / 0 / 6 / 0.8688
2 / 0 / 5 / 0.8380
1 / 0 / 6 / 0.8302
3 / 0 / 3 / 0.8261
2 / 1 / 4 / 0.8246
2 / 2 / 0 / 0.8133
3 / 1 / 0 / 0.7813
2 / 2 / 2 / 0.7763
3 / 1 / 1 / 0.7727

(iv) Al2O3 Trigonal

h / k / l / d/Å / h / k / l / d/Å
0 / 1 / 2 / 3.5458 / 2 / 2 / -3 / 1.1680
1 / 0 / 4 / 2.6021 / 2 / 2 / 3 / 1.1680
1 / 1 / 0 / 2.4220 / 3 / 1 / -1 / 1.1590
0 / 0 / 6 / 2.2117 / 1 / 3 / 1 / 1.1590
1 / 1 / 3 / 2.1244 / 2 / 1 / -8 / 1.1462
1 / 1 / -3 / 2.1244 / 1 / 2 / 8 / 1.1462
2 / 0 / 2 / 2.0000 / 3 / 1 / 2 / 1.1460
0 / 2 / 4 / 1.7729 / 1 / 3 / -2 / 1.1460
1 / 1 / 6 / 1.6332 / 0 / 2 / 10 / 1.1214
1 / 1 / -6 / 1.6332 / 0 / 0 / 12 / 1.1058
2 / 1 / 1 / 1.5744 / 1 / 3 / 4 / 1.0979
1 / 2 / -1 / 1.5744 / 3 / 1 / -4 / 1.0979
0 / 1 / 8 / 1.5425 / 3 / 1 / 5 / 1.0656
1 / 2 / 2 / 1.5421 / 1 / 3 / -5 / 1.0656
2 / 1 / -2 / 1.5421 / 2 / 2 / -6 / 1.0622
1 / 2 / -4 / 1.4306 / 2 / 2 / 6 / 1.0622
2 / 1 / 4 / 1.4306 / 0 / 4 / 2 / 1.0359
3 / 0 / 0 / 1.3983 / 2 / 1 / 10 / 1.0176
2 / 1 / -5 / 1.3612 / 1 / 2 / -10 / 1.0176
1 / 2 / 5 / 1.3612 / 1 / 1 / 12 / 1.0059
2 / 0 / 8 / 1.3011 / 1 / 1 / -12 / 1.0059
1 / 0 / 10 / 1.2652 / 4 / 0 / 4 / 1.0000
1 / 1 / 9 / 1.2594 / 3 / 1 / -7 / 0.9916
1 / 1 / -9 / 1.2594 / 1 / 3 / 7 / 0.9916
1 / 2 / -7 / 1.2162 / 2 / 1 / -11 / 0.9601
2 / 1 / 7 / 1.2162 / 1 / 2 / 11 / 0.9601
2 / 2 / 0 / 1.2110 / 2 / 3 / -1 / 0.9599
0 / 3 / 6 / 1.1819 / 3 / 2 / 1 / 0.9599
3 / 0 / 6 / 1.1819 / 3 / 1 / 8 / 0.9525