Table 6.1.1.7

Maslen, E. N.; Fox, A. G.; O'Keefe, M. A.

doi:10.1107/97809553602060000600

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. C. ch. 6.1, p. 584

Table 6.1.1.7

E. N. Maslen,^e ^‡ A. G. Fox^b and M. A. O'Keefe^c

Table 6.1.1.7 | top | pdf |
Indices allowed by the site symmetry for the real form of the spherical harmonics $[Y_{lmp(\theta,\varphi)}]$ ; λ, μ and j are integers such that l, m ≥ 0; (−)ⁿ implies p = − for n odd and p = + for n even

Site symmetry	Coordinate axes	Indices
1	Any	All
$[\bar1]$	Any	$[(2\lambda,m,p)]$
2	$[2\parallel x]$	$[(l,m,(-)^{l-m})]$
	$[2\parallel y]$	$[(l,m,(-)^{l})]$
	$[2\parallel z]$	$[(l,2\mu,p)]$
m	$[m\,\bot\, x]$	$[(l,m,(-)^{m})]$
	$[m\,\bot\, y]$
	$[m\,\bot\, z]$
	$[2\parallel x,m\,\bot\,x]$	$[(2\lambda ,m,(-)^{m})]$
	$[2\parallel y,m\,\bot\,y]$	$[(2\lambda,m,+)]$
	$[2\parallel z,m\,\bot\,z]$	$[(2\lambda,2\mu,p)]$
222	$[2\parallel z,2\parallel y]$	$[(l,2\mu,(-)^l)]$
	$[2\parallel x,m\,\bot\,z]$
	$[2\parallel y,m\,\bot\,z]$
	$[2\parallel z,m\,\bot\,y]$	$[(l,2\mu,+)]$
	$[m\,\bot\,z,m\,\bot\,y,m\,\bot\,z]$	$[(2\lambda,2\mu,+)]$
4	$[4\parallel z]$	$[(l,4\mu,p)]$
$[\bar4]$	$[\bar4\parallel z]$
	$[4\parallel z,m\,\bot\,z]$	$[(2\lambda,4\mu,p)]$
422	$[4\parallel z,2\parallel y]$	$[(l,4\mu,(-)^l)]$
	$[4\parallel z,m\,\bot\,y]$	$[(l,4\mu,+)]$
$[\bar42m]$	$[\bar4\parallel z,2\parallel x]$
$[\bar42m]$	$[\bar4\parallel z,m\,\bot\,y]$
	$[4\parallel z,m\,\bot\,z,m\,\bot\,x]$	$[(2\lambda,4\mu,+)]$
3	$[3\parallel z]$	$[(l,3\mu,p)]$
$[\bar3]$	$[\bar3\parallel z]$	$[(2\lambda,3\mu,p)]$
32	$[3\parallel z,2\parallel y]$	$[(l,3\mu,(-)^l)]$
32	$[3\parallel z,2\parallel x]$	$[(l,3\mu,(-)^{l-m})]$
	$[3\parallel z,m\,\bot\,y]$	$[(l,3\mu,+)]$
	$[3\parallel z,m\,\bot\,x]$	$[(l,3\mu,(-)^m)]$
$[\bar3m]$	$[\bar3\parallel z,m\,\bot\,y]$	$[(2\lambda,3\mu,+)]$
$[\bar3m]$	$[\bar3\parallel z,m\,\bot\,x]$	$[(2\lambda,3\mu,(-)^m)]$
6	$[6\parallel z]$	$[(l,6\mu,p)]$
$[\bar6]$	$[\bar6\parallel z]$	$[(m+2j,3\mu,p)]$
	$[6\parallel z,m\,\bot\,z]$	$[(2\lambda,6\mu,p)]$
622	$[6\parallel z,2\parallel y]$	$[(l,6\mu,(-)^l)]$
	$[6\parallel z,m\,\bot\,y]$	$[(l,6\mu,+)]$
$[\bar6m2]$	$[\bar6\parallel z,m\,\bot\,y]$	$[(m+2j,3\mu,+)]$
$[\bar6m2]$	$[\bar6\parallel z,m\,\bot\,x]$	$[(m+2j,3\mu,(-)^l)]$
	$[6\parallel z,m\,\bot\,z,m\,\bot\,y]$	$[(2\lambda,6\mu,+)]$