International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. D. ch. 1.9, pp. 235-237

Table 1.9.3.5 

W. F. Kuhsa*

a GZG Abt. Kristallographie, Goldschmidtstrasse 1, 37077 Göttingen, Germany
Correspondence e-mail: wkuhs1@gwdg.de

Table 1.9.3.5 | top | pdf |
Symmetry restrictions on coefficients in fifth-rank symmetric polar tensors

(a) A–K.

Cross-reference No. of independent coefficients Symbols and coefficient indices
A B C D E F G H I J K
1 2 3 1 1 1 1 1 1 1 1
1 2 3 1 1 1 1 1 1 1 1
1 2 3 1 1 1 1 1 2 2 2
1 2 3 1 1 2 2 3 2 2 3
1 2 3 2 3 2 3 3 2 3 3
E0 0 0 0 0 0 0 0 0 0 0 0 0
E1 1 0 0 0 0 0 0 G 0 0 0 0
E2 1 0 0 0 0 0 0 G 0 0 0 0
E3 1 0 0 0 0 0 0 G 0 0 0 0
E4 1 0 0 0 0 0 0 0 0 0 0 0
E5 2 0 0 0 D 0 D 0 0 D 0 K
E6 2 A −A 0 A/2 0 A/10 0 H −A/10 0 H/2
E7 2 0 0 0 0 0 0 G 0 0 0 0
E8 2 0 0 0 0 E 0 0 0 0 0 0
E9 2 0 0 0 0 0 0 G 0 0 0 0
E10 2 0 0 0 D 0 0 0 0 I 0 0
E11 2 0 0 0 0 0 0 G 0 0 0 0
E12 2 0 0 0 0 0 F 0 −F 0 0 0
E13 2 0 0 0 D −D F 0 −F −F 0 0
E14 2 0 0 0 D D F 0 −F −F 0 0
E15 2 0 0 0 D D F 0 −F F 0 0
E16 2 0 0 0 D −D F 0 −F F 0 0
E17 3 0 0 C 0 E 0 E/2 0 0 E/2 0
E18 3 0 0 0 0 0 0 G 0 0 0 0
E19 3 0 0 0 0 E 0 G 0 0 0 0
E20 3 0 0 0 D 0 0 G 0 I 0 0
E21 3 0 0 0 0 0 F 0 −F 0 0 0
E22 3 0 0 0 0 0 0 G 0 0 2G 0
E23 3 0 0 0 0 0 0 G 0 0 G 0
E24 4 A −A 0 D 0 (1)† 0 H (3)† 0 K
E25 4 0 0 C 0 E 0 0 0 0 J 0
E26 4 0 B 0 D 0 0 0 0 I 0 K
E27 4 A 0 0 0 0 F 0 F 0 0 0
E28 4 0 0 0 0 E 0 G 0 0 0 0
E29 4 0 0 0 D 0 0 G 0 I 0 0
E30 4 0 0 0 0 0 F G −F 0 0 0
E31 5 0 0 C 0 E 0 G 0 0 J 0
E32 5 0 B 0 D 0 0 G 0 I 0 K
E33 5 A 0 0 0 0 F 0 F 0 0 0
E34 5 A A A D D F G F F J J
E35 5 A A −A D −D F G F F J −J
E36 5 A −A A D −D F G F −F J −J
E37 5 A −A −A D D F G F −F J J
E38 5 A −A C A/2 E A/10 E/2 H −A/10 E/2 H/2
E39 5 0 0 C D E D E/2 0 D E/2 K
E40 6 0 0 C 0 E 0 0 0 0 J 0
E41 6 0 0 C 0 E 0 G 0 0 J 0
E42 6 0 B 0 D 0 0 0 0 I 0 K
E43 6 0 B 0 D 0 0 G 0 I 0 K
E44 6 A 0 0 0 0 F 0 H 0 0 0
E45 6 A 0 0 0 0 F G H 0 0 0
E46 6 A −A 0 D 0 F 0 H F 0 K
E47 6 A −A 0 D 0 F 0 H −F 0 K
E48 6 A 0 A 0 E F 0 H 0 J 0
E49 6 A 0 −A 0 E F 0 H 0 G 0
E50 6 0 B B D D 0 0 0 I J J
E51 6 0 B −B D −D 0 0 0 I J −J
E52 6 0 0 C 0 E 0 G 0 0 J 0
E53 6 0 0 C 0 E 0 E/2 0 0 J 0
E54 6 A 0 0 D 0 F 0 H (4)† 0 K
E55 6 A B 0 A/2 0 F 0 H (5)† 0 H/2
E56 6 A B 0 D 0 (2)† 0 H (6)† 0 K
E57 6 0 B 0 D 0 D 0 0 I 0 K
E58 7 A A A D E F G H H J J
E59 7 A A −A D E F G H H J −J
E60 7 A −A A D E F G H −H J −J
E61 7 A −A −A D E F G H −H J J
E62 7 A −A C D E (1)† E/2 H (3)† E/2 K
E63 9 0 0 C 0 E 0 G 0 0 J 0
E64 9 0 B 0 D 0 0 G 0 I 0 K
E65 9 A 0 0 0 0 F G H 0 0 0
E66 9 A A 0 D E F G H F 0 K
E67 9 A −A 0 D E F G H −F 0 K
E68 9 A 0 A D E F G H I J 0
E69 9 A 0 −A D E F G H I J 0
E70 9 0 B B D D F 0 −F I J J
E71 9 0 B −B D −D F 0 −F I J −J
E72 9 A 0 0 D E F G H (4)† 2G K
E73 9 A B 0 A/2 0 F G H (5)† G H/2
E74 9 A B 0 D E (2)† G H (6)† 2G K
E75 9 0 B 0 D 0 D G 0 I G K
E76 12 A B 0 D 0 F 0 H I 0 K
E77 12 A 0 C 0 E F 0 H 0 J 0
E78 12 0 B C D E 0 0 0 I J K
E79 12 A −A C D E F G H −F J K
E80 12 A A C D E F G H F J K
E81 12 A B −A D E F G H I J K
E82 12 A B A D E F G H I J K
E83 12 A B −B D −D F G F I J −J
E84 12 A B B D D F G F I J J
E85 12 A B C D E (2)† G H (6)† J K
E86 12 0 B C D E D E/2 0 I J K
E87 12 A 0 C D E F G H (4)† J K
E88 12 A B C A/2 E F E/2 H (5)† J H/2
E89 21 A B C D E F G H I J K

(b) L–V.

Cross-reference No. of independent coefficients Symbols and coefficient indices
L M N P Q R S T U V
1 1 1 1 1 1 2 2 2 2
1 2 2 2 2 3 2 2 2 3
3 2 2 2 3 3 2 2 3 3
3 2 2 3 3 3 2 3 3 3
3 2 3 3 3 3 3 3 3 3
E0 0 0 0 0 0 0 0 0 0 0 0
E1 1 0 0 G 0 G 0 0 0 0 0
E2 1 0 0 −G 0 0 0 0 0 0 0
E3 1 0 0 0 0 −G 0 0 0 0 0
E4 1 0 0 N 0 −N 0 0 0 0 0
E5 2 0 D 0 K 0 0 0 0 0 0
E6 2 0 −A/2 0 −H/2 0 0 0 0 −H 0
E7 2 0 0 G 0 Q 0 0 0 0 0
E8 2 L 0 0 0 0 0 −E 0 −L 0
E9 2 0 0 N 0 G 0 0 0 0 0
E10 2 0 0 0 0 0 0 0 −I 0 −D
E11 2 0 0 N 0 N 0 0 0 0 0
E12 2 0 M 0 0 0 −M 0 0 0 0
E13 2 F −D 0 0 0 D D F −F −D
E14 2 −F −D 0 0 0 D −D F F −D
E15 2 F D 0 0 0 −D −D −F −F −D
E16 2 −F D 0 0 0 −D D −F F −D
E17 3 L 0 E/2 0 L/2 0 E 0 L 0
E18 3 0 0 N 0 Q 0 0 0 0 0
E19 3 L 0 −G 0 0 0 −E 0 −L 0
E20 3 0 0 0 0 −G 0 0 −I 0 −D
E21 3 0 M N 0 −N −M 0 0 0 0
E22 3 L 0 2G 0 L 0 S 0 0 0
E23 3 0 0 N 0 Q 0 (14)† 0 Q 0
E24 4 0 (7)† 0 (13)† 0 0 0 −H 0 0
E25 4 L 0 0 0 0 0 E 0 L 0
E26 4 0 0 0 0 0 0 0 I 0 D
E27 4 0 M 0 P 0 M 0 0 0 0
E28 4 L 0 G 0 Q 0 −E 0 −L 0
E29 4 0 0 N 0 G 0 0 −I 0 −D
E30 4 0 M N 0 N −M 0 0 0 0
E31 5 L 0 −G 0 0 0 E 0 L 0
E32 5 0 0 0 0 −G 0 0 I 0 D
E33 5 0 M N P −N M 0 0 0 0
E34 5 F D G J G D D F F D
E35 5 −F D G −J G D −D F −F D
E36 5 F −D G J G −D −D −−F F D
E37 5 −F −D G −J G −D D −F −F D
E38 5 L −A/2 E/2 −H/2 L/2 0 E −H L 0
E39 5 L D E/2 K L/2 0 E 0 L 0
E40 6 L 0 0 0 0 0 S 0 U 0
E41 6 L 0 G 0 Q 0 E 0 L 0
E42 6 0 0 0 0 0 0 0 T 0 V
E43 6 0 0 N 0 G 0 0 I 0 D
E44 6 0 M 0 P 0 R 0 0 0 0
E45 6 0 M N P N M 0 0 0 0
E46 6 0 D 0 K 0 R 0 H 0 R
E47 6 0 −D 0 −K 0 R 0 −H 0 −R
E48 6 H M 0 J 0 E M 0 F 0
E49 6 −H M 0 −J 0 −E −M 0 −F 0
E50 6 I 0 0 0 0 0 S T T S
E51 6 −I 0 0 0 0 0 S T −T −S
E52 6 L 0 (10)† 0 Q 0 (15)† 0 2Q 0
E53 6 L 0 (11)† 0 L/2 0 S 0 U 0
E54 6 0 (4)† 0 K 0 R 0 0 0 0
E55 6 0 (8)† 0 P 0 R 0 (16)† 0 R/2
E56 6 0 B/2 0 (12)† 0 R 0 (17)† 0 2R
E57 6 0 (9)† 0 K 0 0 0 T 0 V
E58 7 F E G J G D D F H E
E59 7 −F −E G −J G D −D F −H −E
E60 7 F E G J G −D −D −F H −E
E61 7 −F −E G −J G −D D −F −H E
E62 7 L (7)† E/2 (13)† L/2 0 E −H L 0
E63 9 L 0 N 0 Q 0 S 0 U 0
E64 9 0 0 N 0 Q 0 0 T 0 V
E65 9 0 M N P Q R 0 0 0 0
E66 9 L D −G K 0 R −E H −L R
E67 9 L −D −G −K 0 R −E −H −L −R
E68 9 H M 0 J −G E M −I F −D
E69 9 −H M 0 −J −G −E −M −I −F −D
E70 9 I M N 0 −N −M S T T S
E71 9 −I M N 0 −N −M S T −T −S
E72 9 L (4)† 2G K L R 0 0 0 0
E73 9 0 (8)† N P Q R (14)† (16)† Q R/2
E74 9 L B/2 2G (12)† L R 0 (17)† 0 2R
E75 9 0 (9)† N K Q 0 (14)† T Q V
E76 12 0 M 0 P 0 R 0 T 0 V
E77 12 L M 0 P 0 R S 0 U 0
E78 12 L 0 0 0 0 0 S T U V
E79 12 L −D G −K Q R E −H L −R
E80 12 L D G J Q R E H L R
E81 12 −H M N −J G −E −M I −F D
E82 12 H M N J G E M I F D
E83 12 −I M N P N M S T −T −S
E84 12 I M N P N M S T T S
E85 12 L B/2 (10)† (12)† N M (15)† (17)† 2N 2M
E86 12 L 2I (11)† K L/2 0 S T U V
E87 12 L (4)† (10)† K N M (15)† 0 2N 0
E88 12 L (8)† (11)† P L/2 R S (16)† U U/2
E89 21 L M N P Q R S T U V
† (1) −2A/5 + D; (2) −3A/5 + B/10 + 3D/2; (3) −3A/5 + D; (4) −D + 2F; (5) −A/4 + 3F/2; (6) −2A/5 + B/5 + D; (7) −A + D; (8) −A/5 + 2B/5 + F; (9) −D + 2I; (10) −2G + 3J; (11) −E/4 + 3J/2; (12) −2H + 3K; (13) −H + K; (14) −G + 2N; (15) −4G + 6J; (16) −H/4 + 3P/2; (17) −4H + 6K.