Lecture 5: Symmetry

1
Lecture 5 Symmetry
What is symmetry? What is a symmetry
operator? What is a group of symmetry
operators? What is a space group?
2
symmetry
An object or function is symmetrical if a spatial
transformation of it looks identical to the
original.
This is the original
This is rotated by 180
X
X
3
Symmetry operators
A spatial transformation can be expressed as an
operator that changes the coordinates of every
point in the object the same way. Symmetry
operators do not distort the object. The distance
between any two points is the same before and
after the symmetry operation.
Here is the operator for a 180 rotation around Z.
equivalent positions
4
3x3 Matrix multiplication
REMINDER
by the way
5
Types of symmetry operations

• Point of inversion
• mirror plane
• glide plane
• rotation (2,3,4 or 6-fold)
• screw axis
• lattice symmetry

6
point of inversion
Object
Object
7
mirror plane
Object
Object
8
glide plane
Object
Object
x
9
rotation
Object
Object
10
screw-rotation
Object
Object
1/3
11
Why proteins cannot have centric symmetry
Mirror images and points of inversion cannot be
re-created by pure rotations. Centric
operations would change the chirality of chiral
centers such as the alpha-carbon of amino acids
or the ribosal carbons of RNA or DNA.
R
H
Ca
C
N
O
12
Rotational symmetry
A 2-fold (180) rotation around the Z-axis
13
rotation
14
Rotation matrices
... the mathematical description of a rotation.
...goes here
y
..rotates by b..
(x,y)
axis of rotation
atom starts here...
(x,y)
?
r
?
x
A rotation is the addition of angles.
15
REMINDER sum of angles rules
cos (????? cos ??cos ????sin ??sin ? sin
(??????? sin ??cos ????sin ??cos ?
16
Adding angles in Cartesian space
converting internal motion to Cartesian motion
y
x rcos ? y rsin ?
(x,y)
(x,y)
?
r
?
x
y' r sin (???? ??? r(sin ??cos ????sin
??cos ?? ??? (r sin ???cos ?????r cos ???sin
? ??? y?cos ????x sin ?
x' r cos (???? ??? r(cos ??cos ????sin
??sin ?? ??? (r cos ???cos ?????r sin ??sin
? ??? x?cos ????y sin ?
in matrix notation...
rotation matrix
17
lets see that again
row times column
x' x?cos ????y sin ? (r cos ???cos
?????r sin ??sin ? ? r cos (???? ???
y' y?cos ????x sin ? ??? (r sin ???cos
?????r cos ???sin ? r sin (??????
18
Reversing the rotation
To rotate the opposite direction, flip the matrix
about the diagonal.
the transpose
inverse rotation matrix transposed rotation
matrix.
...because cosb cosb sinb sinb 1
19
A 3D rotation matrix
Is the product of 2D rotation matrices.
Rotation around y-axis
Rotation around z-axis
3D rotation
20
Example
Rotate v(1.,2.,3.) around Z by 60, then rotate
around Y by -60
21
Examples
z
90 rotation around
y
X
x
x
Y
z
y
y
Z
x
z
Helpful hint
For a R-handed rotation, the minus sine is the
one on the Right.
22
Euler angles, a b g
3D angle conventions
axis of rotation
z
x
z
Order of rotations
1
2
3
Polar angles, fyk
y
z
z
-z
-y
4
1
2
3
5
Net rotation k
23
Properties of rotation matrices
More at http//mathworld.wolfram.com/RotationMatri
x.html
24
Orthogonal matrices

• The dot-product of any row or column with itself
  is one.
• The dot-product of any row or column with a
  different row or column is zero.

examples
a11a11a21a21a31a31 1 a12a11a22a21a32a31 0
(row 1 row 1) (row 2 row 1)
When the dot-product of any two vectors is zero,
the vectors are orthogonal (90 apart).
25
2-fold rotation
R
2-fold symbol
P2
R
Equivalent positions in fractional
coordinates x,y,z -x,-y,z
180 rotation. Called a 2-fold because doing it
twice brings you back to where you started.
26
3-fold rotation
R
R
3-fold symbol
P3
R
In fractional coordinates
Equivalent positions x,y,z -y,x-y,z -xy,-x,z
27
4-fold rotation
R
R
R
P4
R
4-fold symbol
In fractional coordinates (same as orthogonal
coords)
Equivalent positions x,y,z -x, -y,z-y,
x,z y,-x,z
28
6-fold rotation
R
R
R
R
R
P6
R
6-fold symbol
In fractional coordinates
Equivalent positions x,y,z -y,x-y,z -xy,-x,z-x,
-y,z y,-xy,z x-y,x,z
29
In class exercise rotating a point
(a)
Choose a point r(0.1,0.2,0.3) orthogonal
coordinates Rotate the point by 30 in x. Then
rotate it by -90 in y. What are the new
coordinates?
(b)
Choose a point r(0.1,0.2,0.3) fractional
coordinates Multiply by the symmetry operator
What are the new fractional coordinates?
30
No 5-fold symmetry in crystals??
A crystal lattice must be space-filling and
periodic. This Penrose tile pattern is
spacefilling but not periodic. Look for
translational symmetry in this image. Is there
any?
31
Quasicrystals Space-filling, 5-fold symmetry,
but no lattice
32
Poliovirus has 5-fold and 3-fold point-group
symmetry.
33
Screw symmetry
21
31
61
62
Example 6-fold in the projection. Screw moves up
and to the right 4/6 units.
32
41
64
63
42
65
43
Equivalent positions are related by rotation AND
translation
34
Screw axes
R
A screw rotation is a rotation of 2p/n plus a
translation along the axis of rotation by 1/n
(right-handed screw) or -1/n (left-handed screw).
R
P31
R
R
Symbol for 3-fold screw
A right-handed 3-fold screw
35
translational symmetry
The crystal lattice is an example of
translational symmetry. Equivalent positions are
(x,y,z) and (x1,y1,z1), in fractional
coordinates. Space groups that have no other
translational symmetry operations are called
primitive. Space group letter P Space groups
have letters indicating the type of translational
symmetry C (centered) F (face-centered) I
(body-centered)
36
Centered lattices
Centered C Translational symmetry operator
(1/2,1/2,0)
This is face-centered but only on one face.
37
Centered lattices
Face-centered F Translational symmetry
operators(1/2,1/2,0),(0,1/2,1/2),(1/2,0,1/2)
38
Centered lattices
Body-centered I Translational symmetry
operator(1/2,1/2,1/2)
39
systematic absences
In the h direction, in this precession photo of
the 0-layer in l, only the even h reflections are
visible. What happened to the odd-h reflections?
Systematic absences are the result of screw
symmetry.
40
Systematic absences
R
If the molecules along a screw axis are collapsed
to the axis, we see a repeating pattern of length
1/n times the unit cell length.
R
P31
R
R
Symbol for 3-fold screw
41
Space groups
A symmetry group is a set of symmetry operators
that is closed, meaning any two operations,
applied in succession, create a third operation
that is part of the group. A space group is a
symmetry group that includes lattice symmetry
operators.
All space groups implicitly include lattice
operators (1,1,1)
42
The International Tables for Crystallography
Equivalent positions (x, y, z), (-x,-y,z1/2)
43
Group theory
A group is a closed set of operators. Try
applying two operators in succession. the result
is another operator
Equivs (x, y, z), (-x,-y,z1/2)
44
A cubic space group
x,y,z -x1/2,-y,z1/2 -x,y1/2,z1/2 x1/2,-y1/2,
-z z,x,y z1/2,-x1/2,-y -z1/2,-x,y1/2 -z,x1/2,
-y1/2 y,z,x -y,z1/2,-x1/2 y1/2,-z1/2,-x -y1/
2,-z,x1/2
45
www-sphys.unil.ch/escher/
In class exercise Find equivalent positions
In class exercise Use the Escher web sketch
applet to find the equivalent positions for cm,
p4mm, and p6.
Draw a dot at fractional coordinates (0.1, 0.2,
0.3)What are the fractional coordinates of the
equivalent positions? Write the 2D symmetry
operators (matrix and vector)
46
Point group symbols, etc.
47
Finding symmetry in an image
48
Plane groups
p1
49
(No Transcript)
50
(No Transcript)
51
(No Transcript)
52
(No Transcript)
53
(No Transcript)
54
(No Transcript)
55
(No Transcript)
56
(No Transcript)
57
(No Transcript)
58
(No Transcript)
59
(No Transcript)
60
(No Transcript)
61
(No Transcript)
62
(No Transcript)