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Chapter 8
Aims & Objectives
The aim of this chapter is to teach you about point-group symmetry. This chapter
follows on from the section on 3-D symmetry
elements. I hope that by now concepts concerning symmetry elements have had
time to sink in and that you are ready to move further with the concept of
point-group symmetry. There are two reasons why you need to understand
point group symmetry: Firstly, the symmetry of three-dimensional diffraction
data is determined by the symmetry of the origin in reciprocal space.
Secondly, the atoms/ions in a crystal structure may be sited on
so-called special positions, which have
a particular point-group symmetry. The relevance of the latter will
become more apparent in later sections of this course.
Specific learning objectives for this section are given below:
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Before starting with the concept of how certain symmetry elements can
interact together at a particular point in space, we start with the concept
of how symmetry in 3-dimensions is projected onto a plane as in a
paper drawing. The concept of the
simple flat projection will be exploited in
the later section on space group symmetry, while the concept of the
sterographic projection will be used
in this section as it is the best method for
showing symmetry elements that through a single point in space.
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We then explain the concept
of a group since we shall be using
this word a lot in the context of both point groups and space groups.
The definition of a point group is important, but you should also be aware
of the concepts given here concerning symmetry group in general.
We then move on to some practical examples provided by simple
chemical molecules of molecular point
symmetry. These demonstrate how rotation and
rotary-inversion axes may be combined together to form some of the
crystallographic point groups. We also point out that some molecular
point groups do not have crystallographic symmetry.
-
Having been introduced to various example of point-group symmetry,
we then proceed to a formal table of the complete 32 crystallographic point
groups and how they may be classified according to the seven crystal
systems. This is followed by an explanation of how the symmetry of a water
molecule, mm2, relates to a sterographic
point-group diagram of this
symmetry. Do look at the diagrams for the low-symmetry point
groups (triclinic, monoclinic, and orthorhombic crystal systems)
and make sure you understand the various possibilities for the combination
of mirror planes, twofold rotation axes, and a point of inversion.
If you have time, you may want to contrast them with some of the higher-symmetry
diagrams that are provided as reference material.
-
We then move on to the subject of
diffraction symmetry and
reflection multiplicity.
At this point, it is important to learn about some of the key differences
between powder and single-crystal diffraction data.
The next two pages deal with a few other sub-groups of the
32 crystallographic point groups, namely the
polar point groups and the
enantiomorphic point groups.
A knowledge of these is useful when refining certain types of
crystal structure. This section ends with some comments on the
determination of point-group symmetry.
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Finally, it is strongly emphasised (again) that you can look at
the high-symmetry examples of point-group symmetry in the various tables,
but that you are not expected to remember the details about them
(especially the stereographic diagrams) for examination purposes.
They are given (a) so that the tables are complete;
(b) to complement, and to give an appreciation of, the low-symmetry
point groups; and (c) to act as reference information should you need to
refer to them as some future date.
However, where they are discussed in the text, you should understand any
of the concepts involved.
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