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Chapter 10
Aims & Objectives
The aim of this chapter is to teach you about space-group symmetry. This
chapter follows on from the section on
point-group symmetry.
Having covered the concepts of 3-D symmetry elements and how they can interact
at at point in space to form a point group, we now move on to consider
how they can interact more generally to form space groups.
Space groups are very important to crystallographers. Firstly, they allow
crystal structrures to be described without the need to list every atom
in the unit cell. Secondly, determination of space-group symmetry
simplifies the determination of a crystal structure, whether from
single-crystal or powder diffraction data. Finally, the coordinates of
a crystal structure cannot be refined reliably without a knowledge of
the space-group symmetry. This chapter is concerned with space-group concepts:
their practical application, especially with respect to powder
diffraction data, will be dealt with in the fourth section on
symmetry.
Specific learning objectives for this section are given below:
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We begin with an introduction
to the concept of space-group symmetry as applied to the
structure of sulphur hexafluoride. We then move on directly to the
230 3-dimensional space groups.
Take your time to look through the list in order to observe how
they are classified according to crystal system, Laue class, crystal
class, and lattice centring. Since it is not practical to discuss all
of the space groups in a short crystallography course, we discuss
the frequency of space groups
as observed in the Cambridge Structural Database.
-
We then move on to the subject of
space-group diagrams,
and the concepts of special positions
and the asymmetric unit.
The concept of crystal class
is then explained with two examples. This
concept will be revisited in the later section on space-group determination,
but you should understand at this stage how the crystal class is derived
from a space-group's symmetry operators. In addition, the concept of
polar space groups is briefly mentioned at this stage.
-
We then move on to some general concepts involving
space groups belonging to the triclinic,
monoclinic,
and orthorhombic
crystal systems, using a typical example from each
to teach some of the properties of symmetry elements
in a space group. Concepts within these pages are then reinforced by the
use of a clickable examples that allow you to assess your
understanding of the space groups
P-1,
P21/c, and
P212121.
Note that the third example contrasts to the previous two exercises, in that
the example is based on a sub-group of the space group Pnma discussed
under the heading of orthorhombic space groups.
This is to give you some experience of working with the most
frequently-observed enantiomorphic space group, as well as providing you
with a simpler example of an orthorhombic space group.
-
We then move on to some general concepts involving
centred
space groups using space group
C2/c as a typical example.
The latter is also available as a clickable exercise. This may appear to be
most diffcult learning exercise in this course, but the concepts involved are
relatively simple once grasped.
-
Finally, it is strongly emphasised that you can look at
the table of space groups, but that you are not expected to remember
the details contents of it.
The list of 230 space groups is given (a) so that the table is complete;
(b) to complement, and to give an appreciation of, the symbols used
to describe the low-symmetry space groups; and (c) to act as reference
information should you need to refer to it as some future date.
At this stage, it is important that you understand the concepts involved
in space-group symmetry. Do try out the clickable exercises so that
you have confidence in working with the
three most frequently observed space groups
plus a centred space group,
namely
P-1,
P21/c,
P212121, and
C2/c.
Space groups probably form one of the most difficult subjects in
crystallography, so consult and expert if you require help in understanding
this difficult topic.
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