1. In Schönflies notation, what does the symbol S2 mean?
Symbolism
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 = x y
In Schönflies notation, what does the symbol S2 mean?
(x,y,z)
In International notation, what does the symbol mean? x y
(x,y,z)

2. Symbolism: Crystal Systems
What rotational symmetries are consistent with a lattice (translational symmetry)?
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. “Centering?” Symbolism: Bravais Lattices
7 Crystal Systems = 7 Primitive Lattices (Unit Cells):
Including Centering: 14 Bravais Lattices
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)
“Centering?”
 = angle between b and c
 = angle between a and c
 = angle between a and b c a b

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

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

7. 4 / m m m Symbolism: Symmetry Operations (4h = 4/mmm = 4/m 2/m 2/m) y
Schönflies
International
Coordinates
(1) E 1
x, y, z (𝑥,𝑦,𝑧)
(9) i
–x, –y, –z ( 𝑥 , 𝑦 , 𝑧 )
(2)
C2 = C42
2 0,0,z
–x, –y, z ( 𝑥 , 𝑦 ,𝑧)
(10) σh
m x,y,0
x, y, –z (𝑥,𝑦, 𝑧 )
(3) C4
4+ 0,0,z
–y, x, z ( 𝑦 ,𝑥,𝑧)
(11)
S43
,0,z; 0,0,0
y, –x, –z (𝑦, 𝑥 , 𝑧 )
(4)
C43
4– 0,0,z
y, –x, z (𝑦, 𝑥 ,𝑧)
(12) S4
4 − 0,0,z; 0,0,0
–y, x, –z ( 𝑦 ,𝑥, 𝑧 )
(5)
C2
2 0,y,0
–x, y, –z ( 𝑥 ,𝑦, 𝑧 )
(13) σv
m x,0,z
x, –y, z (𝑥, 𝑦 ,𝑧)
(6)
2 x,0,0
x, –y, –z (𝑥, 𝑦 , 𝑧 )
(14)
m 0,y,z
–x, y, z ( 𝑥 ,𝑦,𝑧)
(7)
C2
2 x,x,0
y, x, –z (𝑦,𝑥, 𝑧 )
(15) σd
m x, 𝑥 ,z
–y, –x, z ( 𝑦 , 𝑥 ,𝑧)
(8)
2 x, 𝑥 ,0
–y, –x, –z ( 𝑦 , 𝑥 , 𝑧 )
(16)
m x,x,z
y, x, z (𝑦,𝑥,𝑧)
4 / m m m x y +

8. Symmorphic Space Groups
Point Group =
Space Group =
Symmorphic Space Groups (73):
Nonsymmorphic Space Groups (157):

9. BaFe2As2 Symmorphic Space Groups Space Group: I4/mmm Asymmetric Unit:
Ba (2a): 0 0 0
Fe (4d): ½ 0 ¼
As (4e):
BaFe2As2

10. Symmorphic Space Groups
What is the empirical formula of this compound?
How many formula units are in the unit cell?
Asymmetric Unit:
Dy (4e)
Dy (4d)
Ni (8j)
B (16l)
B (16m)