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, roto- inversions, glides and screw axes • understand and be able to use matrices for different symmetry elements • be familiar with the basics of space groups and know the difference between symmorphic & non-symmorphic
Transcript
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, roto-inversions, glides and screw axes• understand and be able to use matrices for different
symmetry elements• be familiar with the basics of space groups and know
the difference between symmorphic & non-symmorphic
The story so far…
In the lectures we have discussed point symmetry:
• Rotations
• Mirrors
In the workshops we have looked at plane symmetry which involves translation = ua + vb + wc
•Glides
•Screw axes
Back to stereograms and point symmetry
Example: 2-fold rotation perpendicular to plane (2)
Above Below
More examples
Example: 2-fold rotation in plane (2)
Example: mirror in plane (m)
Combinations
Example: 2-fold rotation perpendicular to mirror (2/m)
Example: 3 perpendicular 2-fold rotations (222)
Roto-Inversions
A rotation followed by an inversion through the origin (in this case the centre of the stereogram)
Example: “bar 4” = inversion tetrad
44
4
More examples in sheet.
Special positions
When the object under study lies on a symmetry element mm2 example
General positions
Special positions
Equivalent positions
In terms of axes…
Again, from workshop:• Take a point at (x y z)• Simple mirror in bc plane
x, y, z
-x, y, z
a
b
General convention
• Right hand rule• (x y z) (x’ y’ z’)
a
b(x y z)
(x’ y’ z’)
rr’
c
z
y
x
ccc
bbb
aaa
z
y
x
333231
232221
131211
'
'
'
or r’ = Rr
R represents the matrix of the point operation
Back to the mirror…
• Take a point at (x y z)• Simple mirror in bc plane
z
y
x
z
y
x
z
y
x
100
010
001
'
'
'
100
010
001
100mx, y, z
-x, y, z
a
b
Other examples
z
x
y
z
y
x
z
y
x
100
001
010
'
'
'
4
Left as an example to show with a diagram.
roto-inversion around z
z
y
x
z
x
y
z
y
x
100
001
010
'
'
'
z
x
y
z
y
x
z
y
x
100
001
010
'
'
'
100
001
010
0014
More complex cases
For non-orthogonal, high symmetry axes, it becomes more complex, in terms of deriving from a figure. 3-fold example:
z
yx
y
z
y
x
z
y
x
100
011
010
'
'
'
z
x
xy
z
yx
y
z
y
x
100
011
010
'
'
'
a
b
3-fold and 6-fold
It is “obvious” that 62 and 64 are equivalent to 3 and 32, respectively.
z
x
yx
z
y
x
z
y
x
100
001
011
'
'
'
etc.
32 crystallographic point groups
• display all possibilities for the symmetry of space-filling shapes
• form the basis (with Bravais lattices) of space groups
Enantiomorphic Centrosymmetric
Triclinic 1 *
Monoclinic 2 * 2/m m *
Orthorhombic 222 mmm mm2 *
Tetragonal 4 * 422 4/m 4/mmm 4mm * 2m
Trigonal 3 * 32 3m *
Hexagonal 6 * 622 6/m 6/mmm 6mm * 2m
Cubic 23 432 m m m 3m
1
3
33
4
4
4
6 6
Enantiomorphic Centrosymmetric
Triclinic 1 *
Monoclinic 2 * 2/m m *
Orthorhombic 222 mmm mm2 *
Tetragonal 4 * 422 4/m 4/mmm 4mm * 2m
Trigonal 3 * 32 3m *
Hexagonal 6 * 622 6/m 6/mmm 6mm * 2m
Cubic 23 432 m m m 3m
32 crystallographic point groups
• Centrosymmetric – have a centre of symmetry• Enantiomorphic – opposite, like a hand and its mirror • * - polar, or pyroelectric, point groups
1
3
33
4
4
4
6 6
Space operations
These involve a point operation R (rotation, mirror, roto-inversion) followed by a translation
Can be described by the Seitz operator:
RrrR |
e.g.
r0,0,|0012 21
Glide planes
The 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 c-axis This gives symmetry operator ½+x, y, -z.
z
y
x
z
y
x
r,,|m2
12
1
21
0
0
100
010
001
00001
a
c ,
,
,
n glide
n glide = Diagonal glideHere the translation vector has components in two (or
sometimes three) directions
So for example the translations would be (a b)/2
Special circumstances for cubic & tetragonal
a
b
-
+
+ +,
+
n glide
Here the glide plane is in the plane xy (perpendicular to c)
Symmetry operator ½+x, ½+y, -z
z
y
x
z
y
x
r,,|m 21
21
21
21
21
21
0100
010
001
0001
a
b-
+
+ +
,+
d glide
d glide = Diamond glideHere the translation vector has components in two (or
sometimes three) directions
So for example the translations would be (a b)/4
Special circumstances for cubic & tetragonal
a
b
-
+,
+
++
-,+
-,-,-
,
d glide
Here the glide plane is in the plane xy (perpendicular to c)
a
b-
+
,
+
++
-,
+
-
,
-,
-
,
z
y
x
z
y
x
rm 41
41
41
41
41
41
0100
010
001
0,,|001
Symmetry operator ¼+x, ¼+y, -z
17 Plane groups
Studied (briefly) in the workshop
Combinations of point symmetry and glide planes E. S. Fedorov (1881)
Another example
Build up from one point:
Screw axes
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
21 screw axis2 rotation axis
Screw axes - examples
Note e.g. 31 and 32 give different handedness
Looking down from above
Example
• P42 (tetragonal) – any additional symmetry?
Matrix
4 fold rotation and translation of ½ unit cell
z
x
y
z
y
x
r
21
21
21 0
0
100
001
010
,0,0|0014
Carry this on….
Symmorphic Space Groups
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.
Non-symmorphic space groups involve translational elements (screw axes and glide planes).There are 157 non-symmorphic space groups
230 space groups in total!
Example of Symmorphic Space group
Example of Symmorphic Space group
Systematic Absences #2
Systematic absences in (hkl) reflections Bravais lattices
e.g. Reflection conditions h+k+l = 2n Body centred
Similarly glide & screw axes associated with other absences:
• 0kl, h0l, hk0 absences = glide planes
• h00, 0k0, 00l absences = screw axes
Example:0kl – glide plane is perpendicular to a
if k=2n b glideif l = 2n c clideif k+1 = 2n n glide
Space Group example
• P2/c
zyxzyxzyxzyx ,,,,,, ,2
1,
2
1
Equivalent positions:
Space Group example
P21/c : note glide plane shifted to y=¼ because convention “likes” inversions at origin
2
1
2
1,
2
1
2
1, ,,,,,, zyxzyxzyxzyx
Equivalent positions:
Special positions
Taken from last example
If the general equivalent positions are:
2
1
2
1,
2
1
2
1, ,,,,,, zyxzyxzyxzyx
Special positions are at:
• ½,0,½ ½,½,0
• 0,0,½ 0,½,0
• ½,0,0 ½,½, ½
• 0,0,0 0,½,½
Space groups…
• Allow us to fully describe a crystal structure with the minimum number of atomic positions
• Describe the full symmetry of a crystal structure• Restrict macroscopic properties (see symmetry
workshop) – e.g. BaTiO3
• Allow us to understand relationships between similar crystal structures and understand polymorphic transitions
Example: YBCO
Handout of Structure and Space group• Most atoms lie on special positions
