III Crystal Symmetry
3-1 Symmetry elements
(1) Rotation symmetry
• two fold (diad ) 2
• three fold (triad ) 3
• Four fold (tetrad ) 4
• Six fold (hexad ) 6
(2) Reflection (mirror) symmetry m
LH
RH
mirror
(3) Inversion symmetry (center of symmetry) 1
RH
LH
(4) Rotation-Inversion axis
360o
= Rotate by
，then invert.
𝑛
(a) one fold rotation inversion (1)
(b) two fold rotation inversion (2)
= mirror symmetry (m)
(c) inversion triad (3)
3
= Octahedral site in
an octahedron
(d) inversion tetrad (4)
x
= tetrahedral site in a
tetrahedron
(e) inversion hexad (6)
x
Hexagonal close-packed (hcp) lattice
3-2. Fourteen Bravais lattice structures
3-2-1. 1-D lattice
3 types of symmetry can be arranged in a
1-D lattice
(1) mirror symmetry (m)
(2) 2-fold rotation
(2)
(3) center of symmetry ( 1 )
Proof: There are only 1, 2, 3, 4, and 6 fold
rotation symmetries for crystal with
translational symmetry.
graphically
The space cannot
be filled!

T
Start with the translation
A
lattice
point
Add a rotation
lattice point

A
lattice point

pT
T: scalar
T
T

A

T

T
A translation vector connecting two
lattice
 points! It must be some integer
of T or we contradicted the basic
Assumption of our construction.
p: integer
Therefore,  is not arbitrary! The basic constrain has to be met!
B’
B
b
T
T
T
A

T
tcos
A’
T

tcos
T
To be consistent with the
original translation t:
b  pT
b  T  2T cos   pT  p  1  2 cos 
2 cos   1  p  M
p
M
4 -3
3 -2
2 -1
1
0
0
1
-1 2
cos

-1.5
-1
-0.5
0
0.5
1
--

2/3
/2
/3
0
p must be integer
M must be integer
n (= 2/)
b
-2
3
4
6
 (1)
M >2 or
M<-2:
no solution
-3T
2T
T Allowable rotational
0 symmetries are 1, 2,
3, 4 and 6.
-T
Look at the
 case
 of p = 2
n = 3; 3-fold
pT  2T

=


T1  T2

T2
 
T1 T2  120o

T1
120o
angle
3-fold lattice.
Look at the
case of p = 1

n = 4; 4-fold
pT  T

 = 90o


T1  T2

T2

T1
4-fold lattice.
 
T1 T2  90o
Look at the
 case
 of p = 0
n = 6; 6-fold
pT  0T

T2

=


T1  T2

T1
60o
 
T1 T2  60o
Exactly the same as 3-fold lattice.
Look at the case of p = 3
n = 2; 2-fold


pT  3T
Look at the case
 of p = -1
pT  1T
1
2
n = 1; 1-fold
1-fold
2-fold
3-fold
4-fold
6-fold
Parallelogram
 


T1  T2
T1 T2  general
Hexagonal
Net
 


T1  T2
T1 T2  120
Square

Net 
T1  T2
o

T1 T2  90o
Can accommodate
1- and 2-fold
rotational symmetries
Can accommodate
3- and 6-fold
rotational symmetries
Can accommodate
4-fold rotational
Symmetry!
These are the lattices obtained by combining
rotation and translation symmetries?
How about combining mirror and translation
Symmetries?
Combine mirror line with translation:

T2

T1
constrain
m
m
Unless
Or
0.5T


T1  T2
 
T1 T2  90o
Primitive cell
centered rectangular
Rectangular
5 lattices in 2D
(1) Parallelogram
 (Oblique)



T1  T2
T1 T2  general
(2) Hexagonal



T1  T2
(3) Square


T1  T2

T1 T2  120o
 
T1 T2  90o
(4) Centered rectangular
 


T1  T2
T1 T2  90o
(5) Rectangular 


T1  T2

T1 T2  90o
Double cell (2 lattice points)
Primitive cell
Symmetry elements in 2D lattice
Rectangular = center rectangular?
3-2-2. 2-D lattice
Two ways to repeat 1-D  2D
(1) maintain 1-D symmetry
(2) destroy 1-D symmetry
m 2 1
(a) Rectangular lattice (𝑎 ≠ 𝑏; 𝛾 = 90o)
Maintain mirror symmetry
(𝑎 ≠ 𝑏; 𝛾 = 90o)
m
(b) Center Rectangular lattice (𝑎 ≠ 𝑏; 𝛾 = 90o)
Maintain mirror symmetry
m
(𝑎 ≠ 𝑏; 𝛾 = 90o)
Rhombus cell
(Primitive unit cell)
𝑎 = 𝑏; 𝛾 ≠ 90o
(c) Parallelogram lattice (𝑎 ≠ 𝑏; 𝛾 ≠ 90o)
Destroy mirror symmetry
(d) Square lattice (𝑎 = 𝑏; 𝛾 = 90o)
b
a
(e) hexagonal lattice (𝑎 = 𝑏; 𝛾 = 120o)
3-2-3. 3-D lattice: 7 systems, 14 Bravais lattices
Starting from parallelogram lattice
(𝑎 ≠ 𝑏; 𝛾 ≠ 90o)
(1)Triclinic system
c

a


1-fold rotation (1)
b
(𝑎 ≠ 𝑏 ≠ 𝑐; 𝛼 ≠ 𝛽 ≠ 𝛾 ≠ 90o)
lattice center symmetry at lattice point as
shown above which the molecule is
isotropic (1)
(2) Monoclinic system
(𝑎 ≠ 𝑏 ≠ 𝑐; 𝛼 = 𝛽 = 90o ≠ 𝛾)
one diad axis
(only one axis perpendicular to the drawing
plane maintain 2-fold symmetry in a
parallelogram lattice)
(1) Primitive monoclinic lattice (P cell)
c
b
a
(2) Base centered monoclinic lattice
c
a
b
B-face centered monoclinic lattice
The second layer coincident to the middle of
the first layer and maintain 2-fold symmetry
Note: other ways to maintain 2-fold symmetry
c
a
b
A-face centered
monoclinic lattice
If relabeling lattice coordination


a

b
b

a
A-face centered monoclinic = B-face centered
(2) Body centered monoclinic lattice
Body centered monoclinic = Base centered
monoclinic
So monoclinic has two types
1. Primitive monoclinic
2. Base centered monoclinic
(3) Orthorhombic system
c
a
b
(𝑎 ≠ 𝑏 ≠ 𝑐; 𝛼 = 𝛽 = 𝛾 = 90o)
3 -diad axes
(1) Derived from rectangular lattice
(𝑎 ≠ 𝑏; 𝛾 = 90o)
 to maintain 2 fold symmetry
The second layer superposes directly on the
first layer
(a) Primitive orthorhombic lattice
c
a
b
(b) B- face centered orthorhombic
= A -face centered orthorhombic
c
a
b
(c) Body-centered orthorhombic (I- cell)
rectangular
body-centered orthorhombic
 based centered orthorhombic
(2) Derived from centered rectangular lattice
(𝑎 ≠ 𝑏; 𝛾 = 90o)
(a) C-face centered Orthorhombic
a
c b
C- face centered orthorhombic
= B- face centered orthorhombic
(b) Face-centered Orthorhombic (F-cell)
Up & Down
Left & Right
Front & Back
Orthorhombic has 4 types
1. Primitive orthorhombic
2. Base centered orthorhombic
3. Body centered orthorhombic
4. Face centered orthorhombic
(4) Tetragonal system
c
𝛽𝛼b
a 𝛾
(𝑎 = 𝑏 ≠ 𝑐;
𝛼 = 𝛽 = 𝛾 = 90o)
One tetrad axis
starting from square lattice
(𝑎 = 𝑏; 𝛾 = 90o)
starting from square lattice (𝑎 = 𝑏; 𝛾 = 90o)
(1) maintain 4-fold symmetry
(a) Primitive tetragonal lattice
First layer
Second layer
(b) Body-centered tetragonal lattice
First layer
Second layer
Tetragonal has 2 types
1. Primitive tetragonal
2. Body centered tetragonal
(5) Hexagonal system
c
a

 b

a = b  c;
 =  = 90o;  = 120o
One hexad axis
starting from hexagonal lattice (2D)
a = b;  = 120o
(1) maintain 6-fold symmetry
Primitive hexagonal lattice
c



b
a
(2) maintain 3-fold symmetry
2/3
2/3
1/3

1/3
a = b = c;
 =  =   90o
Hexagonal has 1 types
1. Primitive hexagonal
Rhombohedral (trigonal)
2. Primitive rhombohedral (trigonal)
(6) Cubic system
c
 
a 
b
a = b = c;
 =  =  = 90o
4 triad axes ( triad axis = cube diagonal )
Cubic is a special form of Rhombohedral
lattice
Cubic system has 4 triad axes mutually
inclined along cube diagonal
 = 90o
(a) Primitive cubic
c
 
a 
b
a = b = c;
 =  =  = 90o
(b) Face centered cubic
 = 60o
a = b = c;
 =  =  = 60o
(c) body centered cubic
 = 109o
a = b = c;
 =  =  = 109o
cubic (isometric)
Special case of orthorhombic with a = b = c
Primitive (P)
Body centered (I)
Face centered (F)
Base center (C)
Tetragonal (I)?
Tetragonal (P) a = b  c
Cubic has 3 types
1. Primitive cubic (simple cubic)
2. Body centered cubic (BCC)
3. Face centered cubic (FCC)

=P
=TP
http://www.theory.
nipne.ro/~dragos/S
olid/Bravais_table.
jpg
=I