International Tables for Crystallography
Volume A1: Symmetry relations between space groups
Edited by Hans Wondratschek, Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany, and Ulrich Müller, Fachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany
First edition July 2004
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Sample pages for Volume A1 (PDF format)
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International Tables for Crystallography Volume A1, Symmetry relations between space groups, presents a systematic treatment of the maximal subgroups and minimal supergroups of
the crystallographic plane groups and space groups.
It is an extension of and a supplement to Volume A, Space-group symmetry, in which only basic data for sub-
and supergroups are provided.
Group-subgroup relations, apart from their theoretical interest, are the basis of a number
of important applications in crystallographic research:
- In solid-state phase transitions there often exists a group-subgroup relation
between the symmetry groups of the two phases. According to Landau theory, this is in fact
mandatory for displacive (continuous, second-order) phase transitions.
Group-subgroup relations are also indispensable in cases
where the symmetry groups of the two phases are not directly related but
share a common subgroup or supergroup.
- Group-subgroup relations provide a concise and powerful tool for revealing and elucidating
relations between crystal structures. They can thus help to keep up with the
ever-increasing amount of crystal-structure data. Their application requires knowledge
of the relations of the Wyckoff positions of group-subgroup related structures.
- Group-subgroup relations are of great importance in the study of twinned crystals,
domain structures and domain boundaries.
- These relations can even help
to identify errors in space-group assignment and crystal-structure determination.
- Subgroups of space groups provide a valuable approach to teaching crystallographic
Volume A1 consists of three parts:
- Part 1 presents an introduction to the theory of space groups at various
levels and with many examples. It includes a chapter on the mathematical theory of subgroups.
- Part 2 gives for each plane group and space group a complete listing of all
maximal subgroups and minimal supergroups. The treatment includes the generators of each
subgroup as well as any necessary changes of the coordinate system.
Maximal isomorphic subgroups are given in parameterized form as infinite series
because of the infinite number for each group. A special feature of the presentation is graphs
that illustrate the group-subgroup relations.
- Part 3 lists the relations between the Wyckoff positions of every space group and its
subgroups. Again, the infinite number of
maximal isomorphic subgroups of each space group are covered by parameterized series.
These data for Wyckoff positions are presented
here for the first time.
The volume is a valuable addition to the library of scientists engaged in
crystal-structure determination, crystal physics or crystal chemistry.
It is essential for those interested in phase transitions, the
systematic compilation of crystal structures, twinning phenomena and
related fields of crystallographic research.
Volume A1 has been reviewed by R. Gould (Crystallography News, No. 92, March 2005, p. 28).
Copies may be ordered directly from Springer
or from any bookseller.
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