Maximal non-isotypic non-enantiomorphic subgroups

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

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International Tables for Crystallography (2006). Vol. E. ch. 1.2, pp. 18-20 | 1 | 2 |

Section 1.2.15.1. Maximal non-isotypic non-enantiomorphic subgroups

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. Maximal non-isotypic non-enantiomorphic subgroups

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The maximal non-isotypic non-enantiomorphic subgroups S of a subperiodic group G are divided into two types:

I translationengleiche or t subgroups and
II klassengleiche or k subgroups .

Type II is subdivided again into two blocks:

IIa : the conventional cells of G and S are the same, and
IIb : the conventional cell of S is larger than that of G.

Block IIa has no entries for subperiodic groups with a primitive cell. Only in the case of the nine centred layer groups are there entries, when it contains those maximal subgroups S which have lost all the centring translations of G but none of the integral translations.

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.

1.2.15.1.2. Block IIb

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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, pp. 18-20