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

International Tables for Crystallography (2006). Vol. D, ch. 1.9, pp. 238-239

Table 1.9.3.6 

W. F. Kuhsa*

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

Table 1.9.3.6| top | pdf |
Symmetry restrictions on coefficients in sixth-rank symmetric polar tensors

(a) A–N.

Cross-referenceNo. of independent parametersSymbols and coefficient indices
ABCDEFGHIJKLMN
12311111111111
12311111111111
12311111111122
12311111222322
12311223223322
12323233233323
F1 3 A A A 0 0 F 0 F 0 0 0 0 F 0
F2 4 A A A 0 0 F 0 H 0 0 0 0 H 0
F3 5 A A C A/2 0 F 0 H (1) 0 H/2 0 F 0
F4 6 A A C D 0 F 0 H (2) 0 H/2 0 (5) 0
F5 6 A A C 0 0 F 0 H 0 0 0 0 F 0
F6 6 A B A 0 0 F 0 H 0 0 0 0 M 0
F7 6 A B B 0 0 F 0 F 0 0 0 0 M 0
F8 7 A A A D D F G F I J J I F J
F9 7 A A A D −D F G F I J −J −I F J
F10 7 A A A D −D F G F I J −J −I F −J
F11 7 A A A D D F G F I J J I F −J
F12 7 A A C A/2 E F E/2 H (1) E/10 H/2 I F −E/10
F13 7 A A C A/2 0 F G H (1) G H/2 0 F G
F14 8 A A C D 0 F 0 H 0 0 K 0 F 0
F15 8 A B A 0 E F 0 H 0 J 0 0 M 0
F16 8 A B B 0 0 F 0 H 0 0 0 0 M N
F17 10 A A A D E F G H I J K I H K
F18 10 A A A D E F G H I J K −I H −K
F19 10 A A A D E F G H I J K −I H K
F20 10 A A A D E F G H I J K I H −K
F21 10 A A C D E F G H (2) (4) H/2 L (5) (7)
F22 10 A B C 0 0 F 0 H 0 0 0 0 M 0
F23 10 A A C D 0 F 0 H I 0 K 0 F 0
F24 10 A B A 0 E F 0 H 0 J 0 L M 0
F25 10 A B B 0 0 F G F 0 0 0 0 M N
F26 10 A B C D 0 F 0 H (3) 0 K 0 (6) 0
F27 10 A B C A/2 0 F 0 H (1) 0 H/2 0 M 0
F28 16 A B C D 0 F 0 H I 0 K 0 M 0
F29 16 A B C 0 E F 0 H 0 J 0 L M 0
F30 16 A B C 0 0 F G H 0 0 0 0 M N
F31 16 A A C D E F G H I J K L F −J
F32 16 A A C D E F G H I J K L F J
F33 16 A B A D E F G H I J K L M N
F34 16 A B A D E F G H I J K L M N
F35 16 A B B D −D F G F I J −J −I M N
F36 16 A B B D D F G F I J J I M N
F37 16 A B C D E F G H (3) J K L (6) (8)
F38 16 A B C A/2 0 F G H (1) G H/2 0 M N
F39 16 A B C D E F G H (3) J K L (6) (9)
F40 16 A B C A/2 E F E/2 H (1) J H/2 L M (10)
F41 28 A B C D E F G H I J K L M N

(b) P–c.

Cross-referenceNo. of independent parametersSymbols and coefficient indices
PQRSTUVWXYZabc
11111111122222
11122222322223
22322223322233
23322233322333
33322333323333
33323333333333
F1 3 P 0 F 0 0 0 0 0 0 0 F 0 F 0
F2 4 P 0 F 0 0 0 0 0 0 0 F 0 H 0
F3 5 H/2 0 R A/2 0 H/2 0 R/2 0 0 H 0 R 0
F4 6 H/2 0 R (11) 0 H/2 0 R/2 0 0 H 0 R 0
F5 6 P 0 R 0 0 0 0 0 0 0 H 0 R 0
F6 6 P 0 H 0 0 0 0 0 0 0 M 0 F 0
F7 6 P 0 M 0 0 0 0 0 0 0 Z 0 Z 0
F8 7 P J F D G J J G D D F I F D
F9 7 P J F D G −J J −G −D −D F −I F −D
F10 7 P −J F D −G −J −J G −D D F I F D
F11 7 P −J F D −G J −J −G D −D F −I F −D
F12 7 H/2 I/2 R A/2 −E/2 H/2 −I/2 R/2 0 −E H −I R 0
F13 7 H/2 Q R A/2 G H/2 Q R/2 0 0 H 0 R 0
F14 8 P 0 R −D 0 −K 0 0 0 0 H 0 R 0
F15 8 P 0 H 0 0 0 −J 0 −E 0 M 0 F 0
F16 8 P −N M 0 0 0 0 0 0 Y Z 0 Z −Y
F17 10 P J F E G J K G D D F I H E
F18 10 P J F −E G −J −K −G −D −D F −I H E
F19 10 P −J F −E −G −J −K G −D D F I H −E
F20 10 P −J F E −G J K −G D −D F −I H −E
F21 10 H/2 Q R (11) (13) H/2 (18) R/2 0 −E H −L R 0
F22 10 P 0 R 0 0 0 0 0 0 0 Z 0 b 0
F23 10 P 0 R D 0 K 0 W 0 0 H 0 R 0
F24 10 P 0 H 0 T 0 J 0 E 0 M 0 F 0
F25 10 P N M 0 0 0 0 0 0 Y Z a Z Y
F26 10 P 0 R B/2 0 (16) 0 W 0 0 (22) 0 2W 0
F27 10 P 0 R (12) 0 (17) 0 R/2 0 0 Z 0 b 0
F28 16 P 0 R S 0 U 0 W 0 0 Z 0 b 0
F29 16 P 0 R 0 T 0 V 0 X 0 Z 0 b 0
F30 16 P Q R 0 0 0 0 0 0 Y Z a b c
F31 16 P Q R D −G K −Q W X −E H −L R −X
F32 16 P Q R D G K Q W X E H L R X
F33 16 P −K H S T −N J −G E −S M −I F −D
F34 16 P K H S T N J G E S M I F D
F35 16 P N M S T U −U −T −S Y Z a Z Y
F36 16 P N M S T U U T S Y Z a Z Y
F37 16 P Q R B/2 (14) (16) (19) W X (20) (22) (23) 2W 2X
F38 16 P Q R (12) (15) (17) Q R/2 0 Y Z a b c
F39 16 P Q R B/2 (9) (16) Q W X 0 (22) 0 2W 0
F40 16 P L/2 R (12) T (17) V R/2 X (21) Z (24) b X/2
F41 28 P Q R S T U V W X Y Z a b c
(1) −A/4 + F/2; (2) A/2 − 3D/2 + 3F/2; (3) B/20 − 3D/5 + 3F/2; (4) −2E/5 + G; (5) A − 2D + F; (6) B/5 − 2D/5 + F; (7) −3E/5 + G; (8) 2E − 5G + 4J; (9) −G + 2J; (10) −E/4 + 3J/2; (11) A − D; (12) A/2 − 5F/2 + 5M/2; (13) −E + G; (14) 6E − 15G + 10J; (15) −G + 2N; (16) −2K + 3P; (17) −H/4 + 3P/2; (18) −L + Q; (19) −2L + 3Q; (20) 12E − 30G + 20J; (21) E/2 − 5J/2 + 5T/2; (22) −4K + 6P; (23) −4L + 6Q; (24) −L/4 + 3V/2.