Block IIb

Kopskyacute, V.; Litvin, D. B.

doi:10.1107/97809553602060000647

International
Tables for
Crystallography
Volume E
Subperiodic groups
Edited by V. Kopský and D. B. Litvin

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. E. ch. 1.2, p. 20 | 1 | 2 |

Section 1.2.15.1.2. Block IIb

V. Kopský^a and D. B. Litvin^b ^*

^a Department of Physics, University of the South Pacific, Suva, Fiji, and Institute of Physics, The Academy of Sciences of the Czech Republic, Na Slovance 2, PO Box 24, 180 40 Prague 8, Czech Republic, and ^bDepartment of Physics, Penn State Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610-6009, USA
Correspondence e-mail: u3c@psu.edu

1.2.15.1.2. Block IIb

| top | pdf |

Whereas in blocks I and IIa every maximal subgroup S of G is listed, this is no longer the case for the entries of block IIb. The information given in this block is $[\displaylines{\quad[i]\quad\hbox{HMS1}\quad(\hbox{Vectors})\quad(\hbox{HMS2})\hfill}]$

The symbols have the following meaning:

[i]: index of S in G.
HMS1: Hermann–Mauguin symbol of S, referred to the coordinate system and setting of G; this symbol may be unconventional.
(Vectors): basis vectors of S in terms of the basis vectors of G. No relations are given for basis vectors which are unchanged.
(HMS2): conventional short Hermann–Mauguin symbol, given only if HMS1 is not in conventional short form.

Examples

(1) G: Rod group $[{\scr p}222]$ (R13) $[\displaylines{\quad{\bf IIb}\quad[2]\; {\scr p}222_{1}\;({\bf c}'=2{\bf c})\hfill}]$ There are two subgroups which obey the same basis-vector relation. Apart from the translations of the enlarged cell, the generators of the subgroups, referred to the basis vectors of the enlarged cell, are $[\matrix{x,y,z\hfill&x,\bar{y},\bar{z}+1/2\hfill&\bar{x},y,\bar{z}\hfill\cr x,y,z\hfill&x,\bar{y},\bar{z}\hfill&\bar{x},y,\bar{z}+1/2.\hfill}]$
(2) G: Layer group pm2₁b (L28) $[\displaylines{\quad{\bf IIb}\quad[2]\; pm2_{1}n\;({\bf a}'=2{\bf a})\hfill}]$ This entry represents two subgroups whose generators, apart from the translations of the enlarged cell, are $[\matrix{x,y,z\hfill&\bar{x}+1/2,y,z\hfill&\bar{x},y+1/2,\bar{z}\hfill\cr x,y,z\hfill&\bar{x},y,z\hfill &\bar{x}+1/2,y+1/2,\bar{z}.\hfill}]$ The difference between the two subgroups represented by the one entry is due to the different sets of symmetry operations of G which are retained in S. This can also be expressed as different conventional origins of S with respect to G: the two subgroups in the first example above are related by a translation c/4 of the origin, and the two subgroups in the second example by a/4.

References

International Tables for Crystallography (2006). Vol. E. ch. 1.2, p. 20