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

International Tables for Crystallography (2006). Vol. B. ch. 1.2, p. 15

Table 1.2.7.2 

P. Coppensa*

aDepartment of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260-3000, USA
Correspondence e-mail: coppens@acsu.buffalo.edu

Table 1.2.7.2| top | pdf |
Index-picking rules of site-symmetric spherical harmonics (Kara & Kurki-Suonio, 1981[link])

λ, μ and j are integers.

SymmetryChoice of coordinate axesIndices of allowed [y_{lmp}], [d_{lmp}]
1 Any [{\rm All}\;(l, m, \pm)]
[\bar{1}] Any [(2\lambda, m, \pm)]
2 [2\!\!\parallel\!\!z] [(l, 2\mu, \pm)]
m [m\perp z] [(l, l-2j, \pm)]
[2/m] [2\!\!\parallel\!\! z, m\perp z] [(2\lambda, 2\mu, \pm)]
222 [2\!\!\parallel\!\! z, 2\!\!\parallel\!\! y] [(2\lambda, 2\mu, +)], [(2\lambda + 1, 2\mu, -)]
mm2 [2\!\!\parallel\!\! z, m\perp y] [(l, 2\mu, +)]
mmm [m\perp z, m\perp y, m\perp x] [(2\lambda, 2\mu, +)]
4 [4\!\!\parallel\!\! z] [(l, 4\mu, \pm)]
[\bar{4}] [\bar{4}\!\!\parallel\!\! z] [(2\lambda, 4\mu, \pm)], [(2\lambda + 1, 4\mu + 2, \pm)]
[4/m] [4\!\!\parallel\!\! z, m\perp z] [(2\lambda, 4\mu, \pm)]
422 [4\!\!\parallel\!\! z, 2\!\!\parallel\!\! y] [(2\lambda, 4\mu, +)], [(2\lambda + 1, 4\mu, -)]
4mm [4\!\!\parallel\!\! z, m\perp y] [(l, 4\mu, +)]
[\bar{4}]2m [\bar{4}\!\!\parallel\!\! z, 2\!\!\parallel\!\! x] [(2\lambda, 4\mu, +)], [(2\lambda + 1, 4\mu + 2, -)]
  [m\perp y] [(2\lambda, 4\mu, +)], [(2\lambda + 1, 4\mu + 2, +)]
[4/mmm] [4\!\!\parallel\!\! z, m\perp z, m\perp x] [(2\lambda, 4\mu, +)]
3 [3\!\!\parallel\!\! z] [(l, 3\mu, \pm)]
[\bar{3}] [\bar{3}\!\!\parallel\!\! z] [(2\lambda, 3\mu, \pm)]
32 [3\!\!\parallel\!\! z, 2\!\!\parallel\!\! y] [(2\lambda, 3\mu, +), (2\lambda + 1, 3\mu, -)]
  [2\!\!\parallel\!\! x] [(3\mu + 2j, 3\mu, +)],
    [(3\mu + 2j + 1, 3\mu, -)]
3m [3\!\!\parallel\!\! z, m \perp y] [(l, 3\mu, +)]
  [m \perp x] [(l, 6\mu, +), (l, 6\mu + 3, -)]
[\bar{3}m] [\bar{3}\!\!\parallel\!\! z, m \perp y] [(2\lambda, 3\mu, +)]
  [m \perp x] [(2\lambda, 6\mu, +), (2\lambda, 6\mu + 3, -)]
6 [6\!\!\parallel\!\! z] [(l, 6\mu, \pm)]
[\bar{6}] [\bar{6}\!\!\parallel\!\! z] [(2\lambda, 6\mu, \pm), (2\lambda + 1, 6\mu + 3, \pm)]
[6/m] [6\!\!\parallel\!\! z, m \perp z] [(2\lambda, 6\mu, \pm)]
622 [6\!\!\parallel\!\! z, 2\!\!\parallel\!\! y] [(2\lambda, 6\mu, +), (2\lambda + 1, 6\mu, -)]
6mm [6\!\!\parallel\!\! z, m\!\!\parallel\!\! y] [(l, 6\mu, +)]
[\bar{6}m2] [\bar{6}\!\!\parallel\!\! z, m \perp y] [(2\lambda, 6\mu, +), (2\lambda + 1, 6\mu + 3, +)]
  [m \perp x] [(2\lambda, 6\mu, +), (2\lambda + 1, 6\mu + 3, -)]
[6/mmm] [6\!\!\parallel\!\! z, m \perp z, m \perp y] [(2\lambda, 6\mu, +)]