1. 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. (2) Reflection (mirror) symmetry mRH LH
(3) Inversion symmetry (center of symmetry) 1 LH RH

3. (4) Rotation-Inversion axis
= Rotate by
，then invert.
(a) one fold rotation inversion ( 1 )
(b) two fold rotation inversion ( 2 )
= mirror symmetry (m)

4. (c) inversion triad ( 3 ) 3
= Octahedral site in
an octahedron

5. (d) inversion tetrad ( 4 )x
= tetrahedral site in a
tetrahedron
(e) inversion hexad ( 6 ) x
Hexagonal close-packed (hcp) lattice

6. 3-2. Fourteen Bravais lattice structures
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 )

7. Proof: There are only 1, 2, 3, 4, and 6 fold
rotation symmetries for crystal with
translational symmetry.
graphically
The space cannot
be filled!

8. Start with the translation
Add a rotation
lattice
point
lattice point  
lattice point
A translation vector connecting two lattice points! It must be some integer
of or we contradicted the basic
Assumption of our construction.
T: scalar 
p: integer
Therefore,  is not arbitrary! The basic constrain has to be met!

9. To be consistent with the original translation t: 
tcos
tcos A A’ T T
To be consistent with the
original translation t:
p must be integer
M must be integer
p M cos  n (= 2/) b
M >2 or
M<-2:
no solution 4 3 2 1 -1
 T
/ T
/ T /
 (1) T
Allowable rotational
symmetries are 1, 2,
3, 4 and 6.

10. Look at the case of p = 2 n = 3; 3-fold 3-fold lattice.
angle
 = 120o
3-fold lattice.
Look at the case of p = 1
n = 4; 4-fold 
 = 90o
4-fold lattice.

11. Exactly the same as 3-fold lattice. Look at the case of p = 3
n = 6; 6-fold 
 = 60o
Exactly the same as 3-fold lattice.
Look at the case of p = 3
n = 2; 2-fold
Look at the case of p = -1
n = 1; 1-fold 1 2

12. These are the lattices obtained by combining
Parallelogram
Can accommodate
1- and 2-fold
rotational symmetries
1-fold
2-fold
3-fold
4-fold
6-fold
Hexagonal Net
Can accommodate
3- and 6-fold
rotational symmetries
Square Net
Can accommodate
4-fold rotational
Symmetry!
These are the lattices obtained by combining
rotation and translation symmetries?
How about combining mirror and translation
Symmetries?

13. Combine mirror line with translation:
constrain m m
Unless Or
0.5T
Primitive cell
centered rectangular
Rectangular

14. 5 lattices in 2D (1) Parallelogram (Oblique) (2) Hexagonal (3) Square
(4)
Centered rectangular
Double cell (2 lattice points)
(5)
Rectangular
Primitive cell

15. Symmetry elements in 2D lattice
Rectangular = center rectangular?

16. D lattice
Two ways to repeat 1-D  2D
(1) maintain 1-D symmetry
(2) destroy 1-D symmetry m

17. (a) Rectangular lattice (𝑎≠𝑏; 𝛾= 90o)
Maintain mirror symmetry m
(𝑎≠𝑏; 𝛾= 90o)

18. (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

19. (c) Parallelogram lattice (𝑎≠𝑏; 𝛾≠ 90o)
Destroy mirror symmetry

20. (d) Square lattice (𝑎=𝑏; 𝛾= 90o)b
(e) hexagonal lattice (𝑎=𝑏; 𝛾= 120o)

21. 3-2-3. 3-D lattice: 7 systems, 14 Bravais lattices
Starting from parallelogram lattice
(𝑎≠𝑏; 𝛾≠ 90o)
(1)Triclinic system fold rotation (1) c    b a
(𝑎≠𝑏≠𝑐; 𝛼≠𝛽≠𝛾≠ 90o)

22. lattice center symmetry at lattice point as shown above which the molecule is isotropic ( 1 )

23. (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)

24. (1) Primitive monoclinic lattice (P cell)b c
(2) Base centered monoclinic lattice c b a
B-face centered monoclinic lattice

26. The second layer coincident to the middle of
the first layer and maintain 2-fold symmetry
Note: other ways to maintain 2-fold symmetry a c b
A-face centered
monoclinic lattice
If relabeling lattice coordination
A-face centered monoclinic = B-face centered

27. (2) Body centered monoclinic lattice
Body centered monoclinic = Base centered monoclinic

28. So monoclinic has two types
1. Primitive monoclinic
2. Base centered monoclinic

29. (3) Orthorhombic systema
(𝑎≠𝑏≠𝑐; 𝛼=𝛽=𝛾 = 90o)
3 -diad axes

30. (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 a b c

31. (b) B- face centered orthorhombic = A -face centered orthorhombic
(c) Body-centered orthorhombic (I- cell)

32. body-centered orthorhombic  based centered orthorhombic
rectangular
body-centered orthorhombic
 based centered orthorhombic

33. (2) Derived from centered rectangular lattice (𝑎≠𝑏; 𝛾= 90o)
(a) C-face centered Orthorhombic a b c
C- face centered orthorhombic
= B- face centered orthorhombic

34. (b) Face-centered Orthorhombic (F-cell)
Up & Down
Left & Right
Front & Back

35. Orthorhombic has 4 types
1. Primitive orthorhombic
2. Base centered orthorhombic
3. Body centered orthorhombic
4. Face centered orthorhombic

36. (4) Tetragonal system
(𝑎=𝑏≠𝑐;
𝛼=𝛽=𝛾 = 90o) c 𝛽 𝛼 b a 𝛾
One tetrad axis
starting from square lattice
(𝑎=𝑏; 𝛾 = 90o)

37. starting from square lattice (𝑎=𝑏; 𝛾 = 90o)
(1) maintain 4-fold symmetry
(a) Primitive tetragonal lattice
First layer
Second layer

38. (b) Body-centered tetragonal lattice
First layer
Second layer
Tetragonal has 2 types
1. Primitive tetragonal
2. Body centered tetragonal

39. (5) Hexagonal system
a = b  c;
 =  = 90o;  = 120o c  b 
One hexad axis  a
starting from hexagonal lattice (2D)
a = b;  = 120o

40. (1) maintain 6-fold symmetry
Primitive hexagonal lattice c  b   a
(2) maintain 3-fold symmetry
2/3 a b c  
2/3
1/3 
1/3
a = b = c;
 =  =   90o

41. Hexagonal has 1 types
1. Primitive hexagonal
Rhombohedral (trigonal)
2. Primitive rhombohedral (trigonal)

42. (6) Cubic system c
a = b = c;
 =  =  = 90o   b  a
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

43. (a) Primitive cubic
 = 90o c
a = b = c;
 =  =  = 90o   b  a
(b) Face centered cubic
 = 60o
a = b = c;
 =  =  = 60o

44. (c) body centered cubic
a = b = c;
 =  =  = 109o

45. 1. Primitive cubic (simple cubic) 2. Body centered cubic (BCC)
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)

46. http://www.theory.nipne.ro/~dragos/Solid/Bravais_table.jpg  = P = I
= T P