Transcript Chapter 8: Major Elements

Symmetry
Motif: the fundamental part of a symmetric
design that, when repeated, creates the whole
pattern
Operation: some act that reproduces the motif to
create the pattern
Element: an operation located at a particular
point in space
2-D Symmetry
= 360o/2 rotation
to reproduce a
motif in a
symmetrical
pattern
A Symmetrical Pattern
6
6
Symmetry Elements
1. Rotation
a. Two-fold rotation
2-D Symmetry
Symmetry Elements
Operation
1. Rotation
a. Two-fold rotation
= the symbol for a two-fold
rotation
6
Element
6
= 360o/2 rotation
to reproduce a
motif in a
symmetrical
pattern
Motif
2-D Symmetry
Symmetry Elements
1. Rotation
a. Two-fold rotation
= the symbol for a two-fold
rotation
6
second
operation
step
6
= 360o/2 rotation
to reproduce a
motif in a
symmetrical
pattern
first
operation
step
2-D Symmetry
Symmetry Elements
1. Rotation
a. Two-fold rotation
Some familiar
objects have an
intrinsic
symmetry
2-D Symmetry
Symmetry Elements
1. Rotation
a. Two-fold rotation
Some familiar
objects have an
intrinsic
symmetry
2-D Symmetry
Symmetry Elements
1. Rotation
a. Two-fold rotation
Some familiar
objects have an
intrinsic
symmetry
2-D Symmetry
Symmetry Elements
1. Rotation
a. Two-fold rotation
Some familiar
objects have an
intrinsic
symmetry
2-D Symmetry
Symmetry Elements
1. Rotation
a. Two-fold rotation
Some familiar
objects have an
intrinsic
symmetry
2-D Symmetry
Symmetry Elements
1. Rotation
a. Two-fold rotation
Some familiar
objects have an
intrinsic
symmetry
2-D Symmetry
Symmetry Elements
1. Rotation
a. Two-fold rotation
Some familiar
objects have an
intrinsic
symmetry
180o rotation makes it coincident
Second 180o brings the object
back to its original position
What’s the motif here??
2-D Symmetry
Symmetry Elements
1. Rotation
b. Three-fold rotation
= 360o/3 rotation
to reproduce a
motif in a
symmetrical pattern
2-D Symmetry
Symmetry Elements
1. Rotation
b. Three-fold rotation
step 1
= 360o/3 rotation
to reproduce a
motif in a
symmetrical pattern
step 3
step 2
2-D Symmetry
Symmetry Elements
1. Rotation
6
6
6
6
6
6
6
2-fold
6
1-fold
3-fold
Objects with symmetry:
a
identity
Z
t
4-fold
9
6-fold
d
5-fold and > 6-fold rotations will not work in combination with translations in crystals
(as we shall see later). Thus we will exclude them now.
4-fold, 2-fold, and 3-fold
rotations in a cube
Click on image to run animation
2-D Symmetry
Symmetry Elements
2. Inversion (i)
inversion is identical to 2-fold
rotation in 2-D, but is unique
in 3-D (try it with your hands)
6
6
inversion through a
center to reproduce a
motif in a symmetrical
pattern
= symbol for an
inversion center
2-D Symmetry
Symmetry Elements
3. Reflection (m)
Reflection across a
“mirror plane”
reproduces a motif
= symbol for a mirror
plane
2-D Symmetry
We now have 6 unique 2-D symmetry operations:
1 2 3 4 6 m
Rotations are congruent operations
reproductions are identical
Inversion and reflection are enantiomorphic operations
reproductions are “opposite-handed”
2-D Symmetry
Combinations of symmetry elements are also possible
To create a complete analysis of symmetry about a point in
space, we must try all possible combinations of these symmetry
elements
In the interest of clarity and ease of illustration, we continue to
consider only 2-D examples
2-D Symmetry
Try combining a 2-fold rotation axis with a mirror
2-D Symmetry
Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
(could do either step first)
2-D Symmetry
Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate (everything)
2-D Symmetry
Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate (everything)
Is that all??
2-D Symmetry
Try combining a 2-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate (everything)
No! A second mirror is required
2-D Symmetry
Try combining a 2-fold rotation axis with a mirror
The result is Point Group 2mm
“2mm” indicates 2 mirrors
The mirrors are different
(not equivalent by symmetry)
2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect
2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate 1
2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate 2
2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect
Step 2: rotate 3
2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Any other elements?
2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Any other elements?
Yes, two more mirrors
2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Any other elements?
Yes, two more mirrors
Point group name??
2-D Symmetry
Now try combining a 4-fold rotation axis with a mirror
Any other elements?
Yes, two more mirrors
Point group name??
4mm
Why not 4mmmm?
2-D Symmetry
3-fold rotation axis with a mirror creates point group 3m
Why not 3mmm?
2-D Symmetry
6-fold rotation axis with a mirror creates point group 6mm
2-D Symmetry
All other combinations are either:
Incompatible
(2 + 2 cannot be done in 2-D)
Redundant with others already tried
m + m  2mm because creates 2-fold
This is the same as 2 + m  2mm
2-D Symmetry
The original 6 elements plus the 4 combinations creates
10 possible 2-D Point Groups:
1 2 3 4 6 m 2mm 3m 4mm 6mm
Any 2-D pattern of objects surrounding a point must
conform to one of these groups
3-D Symmetry
New 3-D Symmetry Elements
4. Rotoinversion
a. 1-fold rotoinversion ( 1 )
3-D Symmetry
New 3-D Symmetry Elements
4. Rotoinversion
a. 1-fold rotoinversion ( 1 )
Step 1: rotate 360/1
(identity)
3-D Symmetry
New 3-D Symmetry Elements
4. Rotoinversion
a. 1-fold rotoinversion ( 1 )
Step 1: rotate 360/1
(identity)
Step 2: invert
This is the same as i, so not a new
operation
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
b. 2-fold rotoinversion ( 2 )
Step 1: rotate 360/2
Note: this is a temporary
step, the intermediate
motif element does not
exist in the final pattern
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
b. 2-fold rotoinversion ( 2 )
Step 1: rotate 360/2
Step 2: invert
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
b. 2-fold rotoinversion ( 2 )
The result:
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
b. 2-fold rotoinversion ( 2 )
This is the same as m, so not
a new operation
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
1
Step 1: rotate 360o/3
Again, this is a
temporary step, the
intermediate motif
element does not exist
in the final pattern
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Step 2: invert through
center
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
1
Completion of the first
sequence
2
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Rotate another 360/3
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Invert through center
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
3
1
Complete second step to
create face 3
2
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Third step creates face 4
(3  (1)  4)
3
1
4
2
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Fourth step creates face
5 (4  (2)  5)
1
5
2
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
Fifth step creates face 6
(5  (3)  6)
Sixth step returns to face 1
5
1
6
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
c. 3-fold rotoinversion ( 3 )
3
This is unique
5
1
4
6
2
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
2: Invert
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
1: Rotate 360/4
2: Invert
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
4: Invert
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
3: Rotate 360/4
4: Invert
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
5: Rotate 360/4
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
5: Rotate 360/4
6: Invert
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
This is also a unique operation
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
d. 4-fold rotoinversion ( 4 )
A more fundamental
representative of the
pattern
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
Begin with this framework:
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
1
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
1
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
1
e. 6-fold rotoinversion ( 6 )
2
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
1
e. 6-fold rotoinversion ( 6 )
2
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
1
e. 6-fold rotoinversion ( 6 )
3
2
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
1
e. 6-fold rotoinversion ( 6 )
3
2
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
1
e. 6-fold rotoinversion ( 6 )
3
2
4
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
1
e. 6-fold rotoinversion ( 6 )
3
2
4
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
1
e. 6-fold rotoinversion ( 6 )
3
5
2
4
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
1
e. 6-fold rotoinversion ( 6 )
3
5
2
4
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
1
e. 6-fold rotoinversion ( 6 )
3
5
2
6
4
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
Note: this is the same as a 3-fold
rotation axis perpendicular to a
mirror plane
Top View
(combinations of elements follows)
3-D Symmetry
New Symmetry Elements
4. Rotoinversion
e. 6-fold rotoinversion ( 6 )
A simpler pattern
Top View
3-D Symmetry
We now have 10 unique 3-D symmetry operations:
1 2 3 4 6 i m 3 4 6
Combinations of these elements are also possible
A complete analysis of symmetry about a point in space requires
that we try all possible combinations of these symmetry elements
3-D Symmetry
3-D symmetry element combinations
a. Rotation axis parallel to a mirror
Same as 2-D
2 || m = 2mm
3 || m = 3m, also 4mm, 6mm
b. Rotation axis  mirror
2  m = 2/m
3  m = 3/m, also 4/m, 6/m
c. Most other rotations + m are impossible
2-fold axis at odd angle to mirror?
Some cases at 45o or 30o are possible, as we shall see
3-D Symmetry
3-D symmetry element combinations
d. Combinations of rotations
2 + 2 at 90o  222 (third 2 required from
combination)
4 + 2 at 90o  422 ( “
“
“
)
6 + 2 at 90o  622 ( “
“
“
)
3-D Symmetry
As in 2-D, the number of possible combinations is
limited only by incompatibility and redundancy
There are only 22 possible unique 3-D combinations,
when combined with the 10 original 3-D elements
yields the 32 3-D Point Groups
3-D Symmetry
But it soon gets hard to
visualize (or at least
portray 3-D on paper)
Fig. 5.18 of Klein (2002) Manual of
Mineral Science, John Wiley and
Sons
3-D Symmetry
The 32 3-D Point Groups
Every 3-D pattern must conform to one of them.
This includes every crystal, and every point within
a crystal
Increasing Rotational Symmetry
Rotation axis only
1
2
3
4
6
1 (= i )
2 (= m)
3
4
6 (= 3/m)
Combination of rotation axes
222
32
422
622
One rotation axis  mirror
2/m
3/m (= 6)
4/m
6/m
One rotation axis || mirror
2mm
3m
4mm
6mm
3 2/m
4 2/m
6 2/m
2/m 2/m 2/m
4/m 2/m 2/m
6/m 2/m 2/m
23
432
4/m 3 2/m
2/m 3
43m
Rotoinversion axis only
Rotoinversion with rotation and mirror
Three rotation axes and  mirrors
Additional Isometric patterns
Table 5.1 of Klein (2002) Manual of Mineral Science, John Wiley and Sons
3-D Symmetry
The 32 3-D Point Groups
Regrouped by Crystal System
(more later when we consider translations)
Crystal System
No Center
Center
1
1
Monoclinic
2, 2 (= m)
2/m
Orthorhombic
222, 2mm
2/m 2/m 2/m
Tetragonal
4, 4, 422, 4mm, 42m
4/m, 4/m 2/m 2/m
Hexagonal
3, 32, 3m
3, 3 2/m
6, 6, 622, 6mm, 62m
6/m, 6/m 2/m 2/m
23, 432, 43m
2/m 3, 4/m 3 2/m
Triclinic
Isometric
Table 5.3 of Klein (2002) Manual of Mineral Science, John Wiley and Sons
3-D Symmetry
The 32 3-D Point Groups
After Bloss, Crystallography and
Crystal Chemistry. © MSA