1. NOTE: Symbolism In Schönflies notation, what does the symbol S2 mean?
Point Group of
the Space Group
Crystal
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 1
Improper Rotation
Sn = h  Cn (S3, S4, S5, …)
Inversion (x,y,z)  (–x,–y,–z)
i = S2
Mirror plane  Principal Axis
h = S1
/m (n/m is the designator: 4/m)
Mirror plane  Principal Axis
v , d (= S1)
m =
NOTE:
S2 = h  C2 x y ,
(z)
In Schönflies notation, what does the symbol S2 mean?
(x,y,z)
S2 = inversion
C2 rotation followed by sh  C2 axis
In International notation, what does the symbol mean? x y
(z)
2-fold (C2) rotation followed by inversion ( )
(x,y,z) ,
Why are the symbols S2 and not used?

2. 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
Lattice Types
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)
 = angle between b and c
 = angle between a and c
 = angle between a and b c a b

3. “Centered Lattices” 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 F
Trigonal
One 3-fold axis
a = b = c;  =  = 
a = b  c;  =  = 90,  = 120 R
Hexagonal
One 6-fold axis (c-axis)
(rhombohedral)
“Centered Lattices”
 = angle between b and c
 = angle between a and c
 = angle between a and b ? c a b I F C B A
Body-
(All) Face-
Base-

4. Symbolism: Point Groups
Schönflies
Notation
Type
Symbol
Features
Uniaxial n
Single rotation axis Cn
nh
+ mirror plane  Cn axis
nv
+ n mirror planes || Cn axis
Low
Symmetry 1
Asymmetric (NO symmetry) s
Mirror plane only i
Inversion center only
Dihedral n
Rotation axis Cn + n C2 axes  Cn axis
nd
nh
Polyhedral
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

5. Symbolism: Crystallographic Point Groups
Allowed Rotations = C1 C2 C3 C4 C6
32 Point Groups
Crystal
System
Schönflies
Symbol
International Symbol
Order /
Inversion?
Full
Abbrev.
Directions
Triclinic 1 1
1 / No
(Holohedral) i
2 / Yes
Monoclinic s
1m1 or 11m m
[010] or [001]
2 / No 2
121 or 112 2
2h
12/m1 or 112/m
2/m
4 / Yes
Orthorhombic
2v
2mm
[100][010][001]
4 / No 2
222
2h
2/m 2/m 2/m
mmm
8 / Yes
Tetragonal 4 4
[001]{100}{110} 4
4h
4/m
2d
8 / No
4v
4mm 4
422
4h
4/m 2/m 2/m
4/mmm
16 / Yes
Yes:
Laue Groups b c
m2m or mm2 a b c c a b
a+b
a–b

6. Symbolism: Crystallographic Point Groups (cont.)
System
Schönflies
Symbol
International Symbol
Order / Inversion?
Full
Abbrev.
Directions
Trigonal 3 3
[001]{100}{210}
3 / No 6
6 / Yes
3v
3m or 3m1
6 / No 3
32 or 321
(Holohedral)
3d
12 / Yes
Hexagonal 6 6
3h
6h
6/m
3h
12 / No
6v
6mm 6
622 62
6h
6/m 2/m 2/m
6/mmm
24 / Yes
Cubic T 23
{100}{111}{110} Th Td
24 / No O
432 Oh
48 / Yes c a b
31m
312 c a b a b c