Transcript Document
Symbolism
Point Group of Crystal
the Space Group System
Space Group
Molecules
Solids
Symmetry Operation
Schönflies Notation
International Notation
Proper Rotation (by 2π/n)
Cn (C2, C3, C4, …)
“n” (2, 3, 4, …)
Identity
E = C1
Sn = sh Cn (S3, S4, S5, …)
Improper Rotation
1
n 1 n (3, 4, 5, )
Inversion (x,y,z) (–x,–y,–z)
i = S2
1
Mirror plane Principal Axis
sh = S1
/m (n/m is the designator: 4/m)
Mirror plane Principal Axis
sv , sd (= S1)
m= 2
S2 2
NOTE: S n n
( x, y,z )
( x, y,z )
S2 = s h C 2
,
In Schönflies notation, what does the symbol S2 mean?
S2 = inversion
2 = reflection
(x,y,z)
C2 rotation followed by sh C2 axis
In International notation, what does the symbol 2 mean?
x
2 12
( x, y,z )
y
2-fold (C2) rotation followed by inversion ( 1 )
,
Why are the symbols S2 and 2 not used?
(z)
y
x
(x,y,z)
( x, y,z )
(z)
1
Symbolism: Crystal Systems
What rotational symmetries are consistent with a lattice (translational symmetry)?
C1 C2 (2π/2) C3 (2π/3)
C4 (2π/4)
C6 (2π/6)
Crystal System
Minimum Symmetry
Primitive Unit Cell
Triclinic
None
a b c;
Monoclinic
One 2-fold axis (b-axis)
a b c; = = 90, 90
Orthorhombic
Three orthogonal 2-fold axes
a b c; = = = 90
Tetragonal
One 4-fold axis (c-axis)
a = b c; = = = 90
Cubic
Four 3-fold axes
a = b = c; = = = 90
Trigonal
One 3-fold axis
a = b = c; = =
a = b c; = = 90, = 120
Hexagonal
One 6-fold axis (c-axis)
a = b c; = = 90, = 120
c
a
b
Lattice Types
= angle between b and c
= angle between a and c
= angle between a and b
2
Symbolism: Bravais Lattices
7 Crystal Systems = 7 Primitive Lattices (Unit Cells):
P
Crystal System
Minimum Symmetry
Primitive Unit Cell
Lattice Types
Triclinic
None
a b c;
P
Monoclinic
One 2-fold axis (b-axis)
a b c; = = 90, 90
P C
Orthorhombic
Three orthogonal 2-fold axes
a b c; = = = 90
P C (A) I F
Tetragonal
One 4-fold axis (c-axis)
a = b c; = = = 90
P I
Cubic
Four 3-fold axes
a = b = c; = = = 90
P I
Trigonal
One 3-fold axis
a = b = c; = =
a = b c; = = 90, = 120
R (rhombohedral)
P
Hexagonal
One 6-fold axis (c-axis)
a = b c; = = 90, = 120
P
F
“Centered Lattices”
c
a
I
F
Body-
(All) Face-
C
B
Base-
?
b
= angle between b and c
= angle between a and c
= angle between a and b
A
3
Symbolism: Point Groups
Schönflies
Notation
Type
Symbol
Features
Uniaxial
n
Single rotation axis Cn
Low
Symmetry
Dihedral
Polyhedral
nh
+ mirror plane Cn axis
nv
+ n mirror planes || Cn axis
1
Asymmetric (NO symmetry)
s
Mirror plane only
i
Inversion center only
n
Rotation axis Cn + n C2 axes Cn axis
nd
+ n mirror planes || Cn axis
nh
+ mirror plane Cn axis
T, Th , Td
Tetrahedral; 4 C3 axes (cube body-diagonals)
O, Oh
Octahedral; 4 C3 axes + 3 C4 axes (cube faces)
I, Ih
Icosahedral; 6 C5 axes
4
Symbolism: Crystallographic Point Groups
Allowed Rotations = C1 C2 C3
Crystal
System
Schönflies
Symbol
Triclinic
(Holohedral)
Monoclinic
(Holohedral)
Orthorhombic
(Holohedral)
Tetragonal
(Holohedral)
C4
32 Point Groups
C6
International Symbol
Directions
Order /
Inversion?
Full
Abbrev.
1
1
1
1 / No
i
1
1
2 / Yes
s
1m1 or 11m
m (2)
2
121 or 112
2
2h
12/m1 or 112/m
2/m
2v
2mm
2mm
2
222
222
2h
2/m 2/m 2/m
mmm
4
4
4
4
4
4
4h
4/m
4/m
2d
42m
42m
8 / No
4v
4mm
4mm
8 / No
4
422
422
8 / No
4h
4/m 2/m 2/m
4/mmm
[010] or [001]
b
c
2 / No
Yes:
Laue Groups
2 / No
4 / Yes
[100][010][001]
a
b
c
4 / No
m2m or mm2
4 / No
8 / Yes
[001]{100}{110}
c
a
b
a+b
a–b
4 / No
4 / No
8 / Yes
4 m2
16 / Yes
5
Symbolism: Crystallographic Point Groups (cont.)
International Symbol
Crystal
System
Schönflies
Symbol
Full
Abbrev.
Directions
Trigonal
3
3
3
[001]{100}{210}
6
3
3
3v
3m or 3m1
3m or 3m1
3
32 or 321
32 or 321
3d
3 m or 3 m1
3m or 3 m1
6
6
6
3h
6
6
6h
6/m
6/m
3h
6 m2
6 m2
12 / No
6v
6mm
6mm
12 / No
6
622
62
12 / No
6h
6/m 2/m 2/m
6/mmm
24 / Yes
T
23
23
Th
2 / m3
m3
Td
43m
43m
O
432
432
24 / No
Oh
4 / m32 / m
m3m
48 / Yes
(Holohedral)
Hexagonal
(Holohedral)
Cubic
(Holohedral)
c
a
b
[001]{100}{210}
c
a
b
{100}{111}{110}
a
b
c
Order /
Inversion?
3 / No
6 / Yes
6 / No
31m
6 / No
312
12 / Yes
31m
6 / No
6 / No
12 / Yes
62m
12 / No
24 / Yes
24 / No
6