More on symmetry. Learning Outcomes: By the end of this section you should: have consolidated your knowledge of point groups and be able to draw stereograms be able to derive equivalent positions for mirrors, and certain rotations, rotoinversions, glides and screw axes
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Learning Outcomes:
By the end of this section you should:
In the lectures we have discussed point symmetry:
Below
Back to stereograms and point symmetryExample: 2fold rotation perpendicular to plane (2)
Example: 2fold rotation perpendicular to mirror (2/m)
Example: 3 perpendicular 2fold rotations (222)
A rotation followed by an inversion through the origin (in this case the centre of the stereogram)
Example: “bar 4” = inversion tetrad
More examples in sheet.
When the object under study lies on a symmetry element mm2 example
General positions
Special positions
Equivalent positions
b
x, y, z
a
In terms of axes…Again, from workshop:
(x’ y’ z’)
r’
r
(x y z)
b
a
General conventionor r’ = Rr
R represents the matrix of the point operation
For nonorthogonal, high symmetry axes, it becomes more complex, in terms of deriving from a figure. 3fold example:
b
a
Centrosymmetric – have a centre of symmetry
Enantiomorphic – opposite, like a hand and its mirror
*  polar, or pyroelectric, point groups
These involve a point operation R (rotation, mirror, rotoinversion) followed by a translation
Can be described by the Seitz operator:
e.g.
a
,
c
,
Glide planesThe simplest glide planes are those that act along an axis, a b or c
Thus the translation is ½ way along the cell followed by a reflection (which changes the handedness: )
Here the a glide plane is perpendicular to the caxisThis gives symmetry operator ½+x, y, z.
n glide = Diagonal glide
Here the translation vector has components in two (or sometimes three) directions
a
+
+
,

b
+
+
So for example the translations would be (a b)/2
Special circumstances for cubic & tetragonal
+
+
,

b
+
+
n glideHere the glide plane is in the plane xy (perpendicular to c)
Symmetry operator ½+x, ½+y, z
d glide = Diamond glide
Here the translation vector has components in two (or sometimes three) directions
,
,


a
+
+
,

+
,

,

b
+
+
So for example the translations would be (a b)/4
Special circumstances for cubic & tetragonal
,


a
+
+
,

+
,

,

b
+
+
d glideHere the glide plane is in the plane xy (perpendicular to c)
Symmetry operator ¼+x, ¼+y, z
Studied (briefly) in the workshop
Combinations of point symmetry and glide planes
E. S. Fedorov (1881)
Build up from one point:
Rotation followed by a translation
Notation is nx where n is the simple rotation, as before
x indicates translation as a fraction x/n along the axis
/2
2 rotation axis
21 screw axis
If we build up into 3d we go from point to plane to space groups
From the 32 point groups and the different Bravais lattices, we can get 73 space groups which involve ONLY rotations, reflection and rotoinversions.
Nonsymmorphic space groups involve translational elements (screw axes and glide planes).
There are 157 nonsymmorphic space groups
230 space groups in total!
Systematic absences in (hkl) reflections Bravais lattices
e.g. Reflection conditions h+k+l = 2n Body centred
Example:
0kl – glide plane is perpendicular to a
if k=2n b glide
if l = 2n c clide
if k+1 = 2n n glide
P21/c : note glide plane shifted to y=¼ because convention “likes” inversions at origin
Equivalent positions:
Taken from last example
If the general equivalent positions are:
Handout of Structure and Space group