Chemistry 4000
3.
Symmetry and the Crystalline Solid State
Lecture 3
Using stereographic projections to describe symmetry elements
In this course, we are much more interested in the symmetry properties of certain regions of space rather than specific
molecules. Often they are just of points (x,y,z) in a 3-D lattice. This is often done using stereographic representations. In
this nomenclature, a circle is drawn, which is a cross-section at the greatest diameter of a sphere. A general point (x,y,z) is
indicated by a point, or an X. It is located above the plane containing the circle. The reflection of the point to (x,y,–z) is
indicated by an open circle. The following diagram illustrates the principle for several types of symmetry operations:
For further explanations of this important concept, please read the extract from West’s book that is provided at the end of
this set of notes. It is essential that all members of this class thoroughly understand the notation and symbolism of
stereographic representations, which is explained in the article. I encourage you to practice the use of these stereographic
projections by making models of the compounds he discusses and sketching out your own stereographic projections. (West
pp. 188-193).
4.
Point symmetry groups
In any real molecule, or in any repeating unit in a crystal, which is the more restrictive case we are now considering, certain
symmetry operations occur together. These natural sets of symmetry operations, or more specifically the symmetry elements
w.r.t. which the operations are carried out, are called point symmetry groups. They are important to us for several reasons:
(i)
Describe the full symmetry properties of real objects, and hence mathematically defines their shapes (or at the least,
restricts their possible shapes within defined categories.)
(ii)
Affects the locations molecules or ions can have in crystalline solids
(iii)
Governs the basic repeating units of a crystal structure, the so-called unit cell.
Because of the restrictions on the orders of the rotation and inversion axes, there is a finite limit to the allowed symmetry
point groups for crystallographic purposes. There are indeed no more and no less than 32 crystallographic point groups.
In the HM system, these are named by a combination of (some of) the symmetry elements which they contain. These are
called the "generating elements". The combination of these elements creates the remainder, so therefore they do not
necessarily need to be listed (for the simplest systems all may be listed; for complex symmetry only a few are). A / symbol
means that the following plane is perpendicular to the trailing axis, e.g. in 2/m the mirror plane in perpendicular to the
two-fold axis (i.e. it is σh) whereas in 2mm, the mirror planes are parallel to the two-fold axis (i.e. they are σv). I do not
wish to go on at length about how these names are constituted. If you need to know all the symmetry elements in a
particular point group, it is probably easiest to convert to the Schönflies notation and look them up in a set of character
tables. (For example, in the back of Shriver, Atkins, “Inorganic Chemistry”.)
In the Table below are listed the 32 crystallographic point groups. We will be using the HM notation, but I have also given
you the corresponding Schönflies symbol for each point group. Please make a point of learning the HM names thoroughly.
The meaning of the Crystal System sorting system will be discussed next. Suffice it to say that there is always a relationship
between the internal symmetry of the contents of a crystal and its outward, overall shape and symmetry. This relationship
may not be simple, but is constrained by the laws of mathematics to certain allowed values.
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Chemistry 4000
Symmetry and the Crystalline Solid State
Crystal System
Lecture 3
The 32 Crystallographic Point Groups in Hermann-Mauguin notation (Scönflies symbol)
Triclinic
1
(C1)
1
(Ci)
Monoclinic
2
(C2)
m
(Cs)
2/m
(C2h)
Orthorhombic
222
(D2)
mm2
(C2v)
mmm
(D2h)
Tetragonal
4
(C4)
4
(S4)
4/m
(C4h)
4mm
(C4v)
4 2m
(D2d)
4/mmm (D4h)
3
(C3)
3
(S6)
32
3m
(D3d)
6
(C6)
6
(C3h)
6/m
6mm
(C6v)
6 m2
(D3h)
6/mmm (D6h)
23
(T)
m3
(Th)
432
m3m
(Oh)
Trigonal
Hexagonal
Cubic
5.
422
(D4)
(D3)
3m
(C3v)
(C6h)
622
(D6)
4 3m
(Td)
(O)
Stereographic representation of point groups
Crystallographers like to use the stereographic projection diagrams introduced in subsection (3) to list the point groups.
These little diagrams contain all the symmetry elements of the point groups. I am including two different sets of diagrams
of these stereographic notations on the subsequent full pages. The first contains only a “top view”, i.e. it is simply the
projection diagram. The second diagram, which covers two pages, includes both the projection and a “side-on” view. The
meaning of the side-on views is explained quite well by Smart and Moore, and I encourage you to look at this diagram
while reading the relevant sections of Chapter 1 of the text. Note that the projection diagrams are very simple for 1 and 1 ,
but extremely complex for the cubic groups. Hence the simple diagrams for cubic symmetry are split into two, one
containing only the spots, the other the location of the axes. For a full description of the notation, refer to the reading
assigned from West.
6.
The seven crystal systems
We now consider a very important aspect of a crystal lattice: the
unit cell. The unit cell is the smallest repeating unit which shows
the full symmetry of the crystal structure.
Look at the crystal model of NaCl shown at right. (Later on we will
worry about choosing the "right" unit cell for a specific structure.)
For now notice that the cube containing the three ions Cl–Na–Cl
along each edge is the repeating unit in the crystal. There are
smaller units, but they do not have the full symmetry of the overall
structure. For example, the small cube containing a Cl at one
corner and a Na at the other. This piece does not in fact have cubic
symmetry because the identities of the ions are different on the
eight corner sites.
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Chemistry 4000
Fall 2007
Symmetry and the Crystalline Solid State
Lecture 3
Page 3-3
Chemistry 4000
Symmetry and the Crystalline Solid State
Lecture 3
Symmetries of the 32 Crystallographic Point Groups
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Chemistry 4000
Symmetry and the Crystalline Solid State
Lecture 3
Symmetries of the 32 Crystallographic Point Groups, Continued
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Chemistry 4000
Symmetry and the Crystalline Solid State
Lecture 3
In general, there are seven possible unit cell symmetries which are capable of existence in crystal structures. These are
called the seven crystal systems. These crystal systems are shown in the following diagram.
These are of course the seven categories used in the Table above to group the crystallographic point groups.
The distinguishing features of these crystals systems are the relative edge lengths and inter-axial angles. The cubic class has
the highest symmetry. It has all edge lengths the same, and all inter-axial angles as right angles. The triclinic class has the
lowest symmetry. It has all three edge lengths different from each other, and all three inter-axial angles different and non90°. Note that crystallographers love simplicity; whenever two edge lengths are identical, i.e. a = b, then both edges are
expressed by a. Hence the only parameter needed to characterize a cubic unit cell is the value of a. In the tetragonal
system, we need only specify a and c. Monoclinic, as the name suggests, has one non-90° angle, and by convention this is
always taken to be the angle β between a and c. I urge you to become totally familiar with the contents of the above figure.
These angle and axis relationships are absolutely fundamental to understanding crystal structures!
The crystal systems as illustrated above are also known as the seven primitive cells from among the 14 conventional Bravais
lattices, about which we will learn more soon. Please take time to examine the models of these cells provided.
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