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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. E. ch. 1.2, pp. 19-20   | 1 | 2 |

Section 1.2.15.1.1. Blocks I and IIa

V. Kopskýa and D. B. Litvinb*

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

1.2.15.1.1. Blocks I and IIa

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In blocks I and IIa, every maximal subgroup S of a subperiodic group G is listed with the following information:[\displaylines{\quad[i]\quad\hbox{HMS1}\quad(\hbox{HMS2})\quad\hbox{Sequence of numbers}\hfill}]The symbols have the following meaning:

  • [i]: index of S in G.

  • HMS1: short Hermann–Mauguin symbol of S, referred to the coordinate system and setting of G; this symbol may be unconventional.

  • (HMS2): conventional short Hermann–Mauguin symbol of S, given only if HMS1 is not in conventional short form.

  • Sequence of numbers: coordinate triplets of G retained in S. The numbers refer to the numbering scheme of the coordinate triplets of the general position. For the centred layer groups the following abbreviations are used:

    • Block I (all translations retained). Number +: coordinate triplet given by Number, plus that obtained by adding the centring translation (1/2, 1/2, 0) of G. (Numbers) +: the same as above, but applied to all Numbers between parentheses.

    • Block IIa (not all translations retained). Number + (1/2, 1/2, 0): coordinate triplet obtained by adding the translation (1/2, 1/2, 0) to the triplet given by Number. (Numbers) + (1/2, 1/2, 0): the same as above, but applied to all Numbers between parentheses.

Examples

  • (1) G: Layer group c211 (L10)[\matrix{{\bf I}\hfill & [2]\hfill & c1\;(p1)\hfill &1+\hfill\cr {\bf IIa}\hfill & [2]\hfill &p2_111\hfill &1;2+(1/2, 1/2, 0)\hfill\cr & [2]\hfill & p211\hfill & 1;2\hfill}]where the numbers have the following meaning:[\matrix{1+\hfill &x,y,z \quad x+1/2,y+1/2,z\hfill\cr 1;2\hfill &x,y,z\quad x,\bar{y}, \bar{z}\hfill\cr 1; 2 +\hfill &x,y,z\quad x+1/2,\bar{y}+1/2,\bar{z}\hfill \cr}]

  • (2) G: Rod group [{\scr p}422] (R30)[\matrix{{\bf I}\hfill & [2]\hfill & {\scr p}411\;({\scr p}4)\hfill & 1;2;3;4\cr &[2]\hfill & {\scr p}221\;({\scr p}222)\hfill & 1;2;5;6\hfill\cr& [2]\hfill &{\scr p}212\;({\scr p}222)\hfill& 1;2;7;8\hfill}]

The HMS1 symbol in each of the three subgroups S is given in the tetragonal coordinate system of the group G. In the first case, [{\scr p}411] is not the conventional short Hermann–Mauguin symbol and a second conventional symbol [{\scr p}4] is given. In the latter two cases, since the subgroups are orthorhombic rod groups, a second conventional symbol of the subgroup in an orthorhombic coordinate system is given.








































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