Transcript Chapter 3_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) 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