3-1 symmetry elements iii crystal symmetry (1) rotation symmetry two fold (diad ) 2 three fold...
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3-1 Symmetry elements
III Crystal Symmetry
(1) Rotation symmetry• two fold (diad ) 2• three fold (triad ) 3• Four fold (tetrad ) 4• Six fold (hexad ) 6
= Rotate by , then invert. 360o
𝑛(a) one fold rotation inversion (
(b) two fold rotation inversion ( = mirror symmetry (m)
(4) Rotation-Inversion axis
(c) inversion triad ()
3
= Octahedral site in an octahedron
(d) inversion tetrad ()
= tetrahedral site in a tetrahedron
x
(e) inversion hexad ()
Hexagonal close-packed (hcp) lattice
x
3-2-1. 1-D lattice3 types of symmetry can be arranged in a 1-D lattice
(1) mirror symmetry (m)(2) 2-fold rotation (2)(3) center of symmetry ()
3-2. Fourteen Bravais lattice structures
Proof: There are only 1, 2, 3, 4, and 6 foldrotation symmetries for crystal withtranslational symmetry.
The space cannotbe filled!
graphically
Start with the translation T
Add a rotation A
A
lattice point
lattice point
latticepoint
A T
TT
T: scalar
Tp
T
A translation vector connecting twolattice points! It must be some integerof or we contradicted the basicAssumption of our construction.
T
p: integer
Therefore, is not arbitrary! The basic constrain has to be met!
T
T T
tcos tcos
b
To be consistent with theoriginal translation t:
pTb p must be integer cos21cos2 ppTTTb
Mp 1cos2 M must be integer
p M cos n (= 2/) b-3 -1.5 -- -- ---2 -1 2 3T-1 -0.5 2/3 3 2T 0 0 /2 4 T 1 0.5 /3 6 0 2 1 0 (1) -T
M >2 orM<-2:no solution
T
T T
A A’
B’B
43210-1
Allowable rotationalsymmetries are 1, 2,3, 4 and 6.
Look at the case of p = 2
= 120o
TTp
2
1T
2T
21 TT
o21 120 TT
angle
Look at the case of p = 1
n = 3; 3-fold
n = 4; 4-fold
= 90o
TTp
21 TT
o21 90 TT
1T
2T
3-fold lattice.
4-fold lattice.
Look at the case of p = 0
= 60o
TTp
0
1T
2T 21 TT
o21 60 TT
n = 6; 6-fold
Look at the case of p = 3 n = 2; 2-fold
TTp
3
Look at the case of p = -1 n = 1; 1-fold
1 2
TTp
1
Exactly the same as 3-fold lattice.
1-fold2-fold3-fold4-fold6-fold
Parallelogram21 TT
general21 TT
Hexagonal Net21 TT
o21 201 TT
Can accommodate1- and 2-foldrotational symmetries
Can accommodate3- and 6-foldrotational symmetriesSquare Net
21 TT
o21 90 TT
Can accommodate4-fold rotationalSymmetry!
These are the lattices obtained by combiningrotation and translation symmetries?
How about combining mirror and translationSymmetries?
Combine mirror line with translation:
m
Unless
0.5T
centered rectangular
constrain
Or21 TT
o21 09 TT
Primitive cell
Rectangular
1T
2T
m
Parallelogram (Oblique)21 TT
general21 TT
Hexagonal21 TT
o21 201 TT
Square21 TT
o21 90 TT
(1)
(2)
(3)
21 TT
o21 09 TT
Double cell (2 lattice points)
Centered rectangular(4)
21 TT
o21 09 TT
Primitive cell
Rectangular(5)
5 lattices in 2D
(b) Center Rectangular lattice (; 90o)
Maintain mirror symmetry m
(; 90o)Rhombus cell
(Primitive unit cell); 90o
3-2-3. 3-D lattice: 7 systems, 14 Bravais lattices
Starting from parallelogram lattice(; 90o)
(1)Triclinic system 1-fold rotation (1)
b
a
c
(; 90o)
one diad axis (only one axis perpendicular to the drawing plane maintain 2-fold symmetry in a parallelogram lattice)
(2) Monoclinic system
(; )
(1) Primitive monoclinic lattice (P cell)
ab
c
(2) Base centered monoclinic lattice
c
ba
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
a
c
bA-face centeredmonoclinic lattice
If relabeling lattice coordination
A-face centered monoclinic = B-face centered
b
a b
a
(1) Derived from rectangular lattice(; 90o)
to maintain 2 fold symmetryThe second layer superposes directly on the first layer
(a) Primitive orthorhombic lattice
ab
c
(b) B- face centered orthorhombic = A -face centered orthorhombic
c
a b
(c) Body-centered orthorhombic (I- cell)
(2) Derived from centered rectangular lattice
(; 90o)
(a) C-face centered Orthorhombic
abc
C- face centered orthorhombic= B- face centered orthorhombic
Orthorhombic has 4 types1. Primitive orthorhombic 2. Base centered orthorhombic3. Body centered orthorhombic4. Face centered orthorhombic
starting from square lattice (; )
(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 types1. Primitive tetragonal 2. Body centered tetragonal
(5) Hexagonal system
a = b c; = = 90o; = 120o
b
a
c
One hexad axis
starting from hexagonal lattice (2D)
a = b; = 120o
(1) maintain 6-fold symmetry
Primitive hexagonal lattice
b
a
c
(2) maintain 3-fold symmetry
ab
c
1/32/3
1/3
2/3
a = b = c; = = 90o
Hexagonal has 1 types1. Primitive hexagonal
Rhombohedral (trigonal)2. Primitive rhombohedral (trigonal)
(6) Cubic system
4 triad axes ( triad axis = cube diagonal )
Cubic is a special form of Rhombohedral latticeCubic system has 4 triad axes mutually inclined along cube diagonal
ba
c
a = b = c;
= = = 90o
(a) Primitive cubic = 90o
ba
c
a = b = c;
= = = 90o
(b) Face centered cubic = 60o
a = b = c; = = = 60o
cubic (isometric)
Primitive (P)Body centered (I)Face centered (F)Base center (C)
Special case of orthorhombic with a = b = c
a = b cTetragonal (P)
Tetragonal (I)?
Cubic has 3 types1. Primitive cubic (simple cubic) 2. Body centered cubic (BCC)3. Face centered cubic (FCC)
http://www.theory.nipne.ro/~dragos/Solid/Bravais_table.jpg
= P = I
= T P