Transcript Slide 1

Symmetry Operations
brief remark about the general role of symmetry in modern physics
V(X)
V(X)
dV
Fx  
dx
V(X1)
V(X2)
Fx  
dV
0
dx
conservation of momentum
change of momentum
dp x
 Fx
dt
dp x
0
dt
px  const.
V(X1) = V(X2)
X1X2
x
X1
translational
symmetry
X2
Emmy Noether 1918: Symmetry in nature
conservation law
1882 in Erlangen, Bavaria, Germany
1935 in Bryn Mawr, Pennsylvania, USA
x
Example for symmetry in QM
Hamiltonian invariant with respect to rotation
Breaking the symmetry with magnetic field
E
mJ=-1
mJ=0
angular momentum conserved
J good quantum number
Zeeman splitting
Proton and Neutron 2 states of one particle
mJ=+1
B=0
B0
breaking the Isospin symmetry
Magnetic phase transition
T>TC
T<TC
Symmetry in perfect single crystals
ideally perfect single crystal
infinite three-dimensional repetition of identical building blocks
called basis
basis
single atom
simple molecule
very complex molecular structure
Quantity of matter contained in the unit cell
Volume of space (parallelepiped)
fills all of space by translation of
discrete distances
Example: crystal from
hexagonal unit cell
square unit cell
there is often more than one reasonable choice of a repeat unit (or unit cell)
most obvious symmetry
of crystalline solid
n2=1
3D crystalline solid
n1=2
b
a
Translational symmetry
3 translational basis vectors a, b, c
-by parallel extensions the basis vectors form a parallelepiped,
the unit cell, of volume V=a(bxc)
translational operation
T=n1a+n2b+n3c where n1, n2, n3
arbitrary integers
-connects positions with identical atomic environments
concept of translational invariance is more general
is also found at r’=r+T
physical property at r (e.g.,electron density)
Set of operations
T=n1a+n2b+n3c defines
r’
r
space lattice
or
Bravais lattice
purely geometrical concept
lattice
crystal structure
basis
+
=
lattice and translational vectors a, b,c are primitive if every point r’ equivalent to r
is created by T according to r’=r+T
y
r
identical atomic arrangement
r’=r+ a2
no primitive
unit cell
x
Primitive basis: minimum number of atoms
in the primitive (smallest)
unit cell which is
sufficient to characterize
crystal structure
y
r
r’=r+0.5 a4
x
No integer!
no primitive
translation
vector
2 important examples for primitive and non primitive unit cells
face centered cubic
1atom/Vprimitive
4 atoms/Vconventinal
a1=a(½, ½,0) a2=a(0, ½,½) a3=a(½,0,½)
Primitive cell: rhombohedron Vprimitive  a1  a2  a3
=
body centered cubic
1atom/Vprimitive
Vprimitive
2 atoms/Vconventinal
1 3 1
 a  Vconventional
2
2
a1=(½, ½,-½) a2=(-½, ½,½) a3=(½,- ½,½)
1 3 1
a  Vconventional
4
4
Lattice Symmetry
Symmetry of the basis point group symmetry
Limitation of possible structures
has to be consistent with symmetry of Bravais lattice
(point group of the basis must be a point group of the lattice)
No change of the
crystal after symmetry
operation
Operations (in addition to translation) which leave the crystal lattice invariant
• Reflection at a plane
• Rotation about an axis
2
n
H2o
2
2
= 2 -fold rotation axis
= n -fold rotation axis
NH3
SF5 Cl
Cr(C6H6)2
Click for more animations and details about point group theory
• point inversion
( x, y, z)
(x,y,z)
• Glide
= reflection + translation
• Screw
= rotation + translation
Notation for the symmetry operations
*
* rotation by 2/n degrees + reflection through plane perpendicular to rotation axis
Origin of the Symbols after Schönflies:
E:identity from the German Einheit =unity
Cn :Rotation (clockwise) through an angle 2π/n, with n integer
: mirror plane from the German Spiegel=mirror
h :horizontal mirror plane, perpendicular to the axis of highest symmetry
v :vertical mirror plane, passing through the axis with the highest symmetry
n-fold rotations with n=1, 2, 3,4 and 6 are the only rotation symmetries
consistent with translational symmetry
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Intuitive example: pentagon
Two-dimensional crystal with lattice constant a in horizontal direction
Row A
1
a
2
(m-1)
α
Row B
m
a
α
m’
1’
X
If rotation by α is a symmetry operation
X=p a
p integer!
cos   1
p-m integer  1
1’ and m’ positions of atoms in row B
= (m-1)a – 2a + 2a cos α
3pm
cos  
2
= (m-3)a + 2a cos α
order of
p-m cos 

rotation
2
-1
1
0/2π
=1-fold
1
-2
1/2 π/3 2  / 6 =6-fold
-3
-4
-5
0
-1/2
-1
π/2 2 / 4 =4-fold
2π/3 2  / 3 =3-fold
π 2 / 2 =2-fold
Plane lattices and their symmetries
4mm
Point-group symmetry
of lattice:
2
2mm
2mm
6 mm
5 two-dimensional lattice types
Crystal=lattice+basis may have lower symmetry
possible basis: 10 types of point groups (1, 1m, 2, 2mm,3, 3mm, 4, 4mm, 6, 6mm)
Combination of point groups and translational symmetry
17 space groups
in 2D
Three-dimensional crystal systems
a  b  c,     
oblique lattice in 2D
Special relations between axes and angles
triclinic lattice in 3D
14 Bravais (or space) lattices
7 crystal systems
There are 32 point groups in 3D, each compatible with one of the 7 classes
32 point groups and compound operations applied to 14 Bravais lattices
230 space groups or structures exist
Many important solids share a few relatively simple structures