International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. B. ch. 2.1, pp. 201-202   | 1 | 2 |

Table 2.1.7.1 

U. Shmuelia* and A. J. C. Wilsonb‡

a School of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and bSt John's College, Cambridge, England
Correspondence e-mail:  ushmueli@post.tau.ac.il

Table 2.1.7.1 | top | pdf |
Some even absolute moments of the trigonometric structure factor

The symbols p, q, r and s denote the second, fourth, sixth and eighth absolute moments of the trigonometric structure factor T [equation (2.1.7.12)[link]], respectively, and the columns of the table contain (for some conciseness) [p, q, r/p] and [s/p^{2}]. The numbers in parentheses, appearing beside some space-group entries, refer to hkl subsets which are defined in the note at the end of the table. These subset references are identical with those given by Shmueli & Kaldor (1981[link], 1983[link]). The symbols q, r and s are also equivalent to [\gamma_{4}P^{2}], [\gamma_{6}P^{3}] and [\gamma_{8}P^{4}], respectively, where [\gamma_{2n}] are the normalized absolute moments given by equation (2.1.7.11)[link].

Space groups(s) p q [r/p] [s/p^{2}]
Point group: 1        
P1 1 1 1 1
Point group: [\bar{\bf 1}]        
[P\bar{1}] 2 6 10 17½
Point groups: 2, m        
All P 2 6 10 17½
All C 4 48 160 560
Point group: [{\bf 2/}{\bi m}]        
All P 4 36 100 306¼
All C 8 288 1600 9800
Point group: 222        
All P 4 28 64 169¾
All C and I 8 224 1024 5432
F222 16 1792 16384 173824
Point group: mm2        
All P 4 36 100 306¼
All A, C and I 8 288 1600 9800
Fmm2 16 2304 25600 313600
Fdd2 (1) 16 2304 25600 313600
Fdd2 (2) 16 1280 7168 43264
Point group: mmm        
All P 8 216 1000 5359[3\over8]
All C and I 16 1728 16000 171500
Fmmm 32 13824 256000 5488000
Fddd (1) 32 13824 256000 5488000
Fddd (2) 32 7680 71680 757120
Point group: 4        
[P4, P4_{2}] 4 36 100 306¼
[P4_{1}] † (3) 4 36 100 306¼
[P4_{1}] † (4) 4 20 28 42¼
[I4] 8 288 1600 9800
[I4_{1}] (5) 8 288 1600 9800
[I4_{1}] (6) 8 160 448 1352
Point group: [\bar{\bf 4}]        
[P\bar{4}] 4 28 64 169¾
[I\bar{4}] 8 224 1024 5432
Point group: [{\bf 4}/{\bi m}]        
All P 8 216 1000 5359[3\over8]
[I4/m] 16 1728 16000 171500
[I4_{1}/a] (7) 16 1728 16000 171500
[I4_{1}/a] (8) 16 960 4480 23660
Point group: 422        
[P422], [P42_{1}2], [P4_{2}22], [P4_{2}2_{1}2] 8 136 424 1682[1\over8];
[P4_{1}22],† [P4_{1}2_{1}2] † (3) 8 136 424 1682[1\over8]
[P4_{1}22],† [P4_{1}2_{1}2] † (4) 8 104 208 470[1\over8]
I422 16 1088 6784 53828
[I4_{1}22] (7) 16 1088 6784 53828
[I4_{1}22] (8) 16 832 3328 15044
Point group: 4mm        
All P 8 168 640 2970[5\over8]
I4mm, I4cm 16 1344 10240 95060
[I4_{1}md, I4_{1}cd] (7) 16 1344 10240 95060
[I4_{1}md, I4_{1}cd] (8) 16 832 3328 15188
Point groups: [\bar{\bf 4}{\bf 2}{\bi m},\bar{\bf 4}{\bi m}{\bf 2}]        
All P 8 136 424 1682[1\over8]
[I\bar{4}m2, I\bar{4}2m, I\bar{4}c2] 16 1088 6784 53828
[I\bar{4}2d] (5) 16 1088 6784 53828
[I\bar{4}2d] (6) 16 832 3328 15044
Point group: 4/ mmm        
All P 16 1008 6400 51985[15\over16]
[I4/mmm], [I4/mcm] 32 8064 102400 1663550
[I4_{1}/amd, I4_{1}/acd] (5) 32 8064 102400 1663550
[I4_{1}/amd, I4_{1}/acd] (6) 32 4992 33280 265790
Point group: 3        
All P and R 3 15 31 71
Point group: [\bar{\bf 3}]        
All P and R 6 90 310 1242½
Point group: 32        
All P and R 6 66 166 508½
Point group: 3 m        
P3m1, P31m, R3m 6 66 178 604½
P3c1, P31c, (3); R3c (1) 6 66 178 604½
P3c1, P31c, (4); R3c (2) 6 66 154 412½
Point group: [\bar{\bf 3}{\bi m}]        
[P\bar{3}1m, P\bar{3}m1, R\bar{3}m] 12 396 1780 10578¾
[P\bar{3}1c, P\bar{3}c1] (3); 12 396 1780 10578¾
 [R\bar{3}c] (1)        
[P\bar{3}1c, P\bar{3}c1] (4); 12 396 1540 7218¾
 [R\bar{3}c] (2)        
Point group: 6        
P6 6 90 340 1522½
[P6_{1}] † (9) 6 90 340 1522½
[P6_{1}] † (10) 6 54 91 161½
[P6_{1}] † (11) 6 54 97 193½
[P6_{1}] † (12) 6 90 280 962½
[P6_{2}] † (13) 6 90 340 1522½
[P6_{2}] † (14) 6 54 97 193½
[P6_{3}] (3) 6 90 340 1522½
[P6_{3}] (4) 6 90 280 962½
Point group: [\bar{\bf 6}]        
[P\bar{6}] 6 90 310 1242½
Point group: [{\bf 6/{\bi m}}]        
[P6/m] 12 540 3400 26643¾
[P6_{3}/m] (3) 12 540 3400 26643¾
[P6_{3}/m] (4) 12 540 2800 16843¾
Point group: 622        
P622 12 324 1150 5506¼
[P6_{1}22] † (9) 12 324 1150 5506¼
[P6_{1}22] † (10) 12 252 577 1537¾
[P6_{1}22] † (11) 12 252 583 1601¾
[P6_{1}22] † (12) 12 324 1090 4746¼
[P6_{2}22] † (13) 12 324 1150 5506¼
[P6_{2}22] † (14) 12 252 583 1601¾
[P6_{3}22] (3) 12 324 1150 5506¼
[P6_{3}22] (4) 12 324 1090 4746¼
Point group: 6 mm        
P6mm 12 396 1930 12818¾
P6cc (3) 12 396 1930 12818¾
P6cc (4) 12 396 1450 6098¾
[P6_{3}cm, P6_{3}mc] (3) 12 396 1930 12818¾
[P6_{3}cm, P6_{3}mc] (4) 12 396 1630 8338¾
Point groups: [\bar{\bf 6}{\bi m}{\bf 2}, \bar{\bf 6}{\bf 2}{\bi m}]        
[P\bar{6}m2, P\bar{6}2m] 12 396 1780 10578¾
[P\bar{6}c2, P\bar{6}2c] (3) 12 396 1780 10578¾
[P\bar{6}c2, P\bar{6}2c] (4) 12 396 1540 7218¾
Point group: 6/mmm        
P6/mmm 24 2376 19300 224328[1\over8]
P6/mcc (3) 24 2376 19300 224328[1\over8]
P6/mcc (4) 24 2376 14500 106728[1\over8]
P6/mcm, P6/mmc (3) 24 2376 19300 224328[1\over8]
[P6/mcm], [P6/mmc] (4) 24 2376 16300 145928[1\over8]
Point group: 23        
P23, [P2_{1}3] 12 276 760 2695¼
I23, [I2_{1}3] 24 2208 12160 86248
F23 48 17664 194560 2759936
Point group: [{\bi m}\bar{\bf 3}]        
[Pm\bar{3}, Pn\bar{3}, Pa3] 24 1800 9400 67703[1\over8]
[Im\bar{3}, Ia\bar{3}] 48 14400 150400 2166500
[Fm\bar{3}] 96 115200 2406400 69328000
[Fd\bar{3}] (1) 96 115200 2406400 69328000
[Fd\bar{3}] (2) 96 96768 1484800 28183680
Point group: 432        
[P432, P4_{2}32] 24 1272 4648 25216[7\over8]
[P4_{1}32] † (15) 24 1272 4648 25216[7\over8]
[P4_{1}32] † (16) 24 1176 3568 13916[7\over8]
[P4_{1}32] † (17) 24 1080 2776 8664[7\over8]
[P4_{1}32] † (18) 24 984 2272 6580[7\over8]
I432 48 10176 74368 806940
[I4_{1}32] (15) 48 10176 74368 806940
[I4_{1}32] (17) 48 8640 44416 277276
F432 96 81408 1189888 25822080
[F4_{1}32] (15) 96 81408 1189888 25822080
[F4_{1}32] (18) 96 62976 581632 6738816
Point group: [\bar{\bf 4}{\bf 3}{\bi m}]        
[P\bar{4}3m] 24 1272 5128 32896[7\over8]
[P\bar{4}3n] (1) 24 1272 5128 32896[7\over8]
[P\bar{4}3n] (2) 24 1272 4168 17536[7\over8]
[I\bar{4}3m] 48 10176 82048 1052700
[I\bar{4}3d] (15); (20) 48 10176 82048 1052700
[I\bar{4}3d] (15); (21) 48 10176 66688 561180
[I\bar{4}3d] (17) 48 8640 44416 277276
[F\bar{4}3m] 96 81408 1312768 33686400
[F\bar{4}3c] (15) 96 81408 1312768 33686400
[F\bar{4}3c] (18) 96 81408 1067008 17957760
Point group: [{\bi m}\bar{\bf 3}{\bi m}]        
[Pm\bar{3}m, Pn\bar{3}m] 48 8784 72160 972717[13\over16]
[Pn\bar{3}n, Pm\bar{3}n] (1) 48 8784 72160 972717[13\over16]
[Pn\bar{3}n, Pm\bar{3}n] (2) 48 8784 56800 488877[13\over16]
[Im\bar{3}m] 96 70272 1154560 31126970
[Ia\bar{3}d] (15); (20) 96 70272 1154560 31126970
[Ia\bar{3}d] (15); (21) 96 51840 432640 4497850
[Ia\bar{3}d] (17) 96 70272 908800 15644090
[Fm\bar{3}m] 192 562176 18472960 996063040
[Fm\bar{3}c] (1) 192 562176 18472960 996063040
[Fm\bar{3}c] (2) 192 562176 14540800 500610880
[Fd\bar{3}m] (1) 192 562176 18472960 996063040
[Fd\bar{3}m] (2) 192 414720 7782400 205432640
[Fd\bar{3}c] (1) 192 562176 18472960 996063040
[Fd\bar{3}c] (2) 192 414720 6799360 136619840

Note. hkl subsets: (1) [h + k + l = 2n]; (2) [h + k + l = 2n + 1]; (3) [l = 2n]; (4) [l = 2n + 1]; (5) [2h + l = 2n]; (6) [2h + l = 2n + 1]; (7) [2k + l = 2n]; (8) [2k + l = 2n + 1]; (9) [l = 6n]; (10) [l = 6n + 1, 6n + 5]; (11) [l = 6n + 2, 6n + 4]; (12) [l = 6n + 3]; (13) [l = 3n]; (14) [l = 3n + 1, 3n + 2]; (15) hkl all even; (16) only one index odd; (17) only one index even; (18) hkl all odd; (19) two indices odd; (20) [h + k + l = 4n]; (21) [h + k + l = 4n + 2].
†And the enantiomorphous space group.