Symmetry Operations

brief remark about the general role of symmetry in modern physics

V(X)

x

V(X1)

X1

V(X2)

X2

x

dVF

dx

change of momentumx

x

dpF

dt

V(X)

x

x

dVF 0

dx

conservation of momentum

xdp0

dt

X1 X2

xp const.

V(X1) = V(X2)

translationalsymmetry

Emmy Noether 1918: Symmetry in nature

conservation law1882 in Erlangen, Bavaria, Germany

1935 in Bryn Mawr, Pennsylvania, USA

Breaking the symmetry with magnetic field

Hamiltonian invariant with respect to rotation

Example for symmetry in QM

angular momentum conservedJ good quantum number

B=0 B0

mJ=-1

mJ=0

mJ=+1

EProton and Neutron 2 states of one particle

breaking the Isospin symmetry

Magnetic phase transition

T>TC T<TC

Zeeman splitting

called basis

Symmetry in perfect single crystals

ideally perfect single crystal

infinite three-dimensional repetition of identical building blocks

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

square unit cell hexagonal unit cell

there is often more than one reasonable choice of a repeat unit (or unit cell)

most obvious symmetry of crystalline solid

Translational symmetry

3D crystalline solid 3 translational basis vectors a, b, c

translational operation T=n1a+n2b+n3c where n1, n2, n3

arbitrary integers

-connects positions with identical atomic environments

ab n1=2

n2=1

-by parallel extensions the basis vectors form a parallelepiped, the unit cell, of volume V=a(bxc)

concept of translational invariance is more general

physical property at r (e.g.,electron density) is also found at r’=r+T

Set of operations T=n1a+n2b+n3c

r’

defines

space lattice or Bravais lattice

purely geometrical concept

+ =

lattice basis crystal structure

r

lattice and translational vectors a, b,c are primitive if every point r’ equivalent to r      

identical atomic arrangementis created by T according to r’=r+T

r

x

y

x

y rr’=r+0.5 a4

No integer!

no primitive translationvector

no primitive unit cell

Primitive basis: minimum number of atoms in the primitive (smallest) unit cell which issufficient to characterize crystal structure

r’=r+ a2

2 important examples for primitive and non primitive unit cells

face centered cubic

body centered cubic

a1=(½, ½,-½) a2=(-½, ½,½) a3=(½,- ½,½)

a1=a(½, ½,0) a2=a(0, ½,½) a3=a(½,0,½)

Primitive cell: rhombohedronprimitive 1 2 3V a a a 3

conventional

1 1a V

4 4=

1atom/Vprimitive4 atoms/Vconventinal

primitiveV 3conventional

1 1a V

2 2

1atom/Vprimitive 2 atoms/Vconventinal

Lattice Symmetry

Symmetry of the basis point group symmetry

has to be consistent with symmetry of Bravais lattice

Limitation of possible structures

Operations (in addition to translation) which leave the crystal lattice invariant

No change of thecrystal after symmetry

operation

• Reflection at a plane

(point group of the basis must be a point group of the lattice)

• Rotation about an axis H2o

NH3

SF5 Cl

Cr(C6H6)2

= 2 -fold rotation axis2

2

n

2= n -fold rotation axis

Click for more animations and details about point group theory

• point inversion

)z,y,x( )z,y,x(

• Glide = reflection + translation

• Screw = rotation + translation

Notation for the symmetry operations

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

* rotation by 2/n degrees + reflection through plane perpendicular to rotation

axis

*

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

a1 2Row A (m-1) a m

α α

Row B

X1’ m’

If rotation by α is a symmetry operation 1’ and m’ positions of atoms in row B

X=p a

p integer!

= (m-1)a – 2a + 2a cos α = (m-3)a + 2a cos α

1cos

2

3 mpcos

p-m integer 1

p-m cos

-1 1 0/2π1

2=1-fold

-2 1/2 π/3 62 / =6-fold

-3 0 π/2 42 / =4-fold

-4 -1/2 2π/3 32 / =3-fold

-5 -1 π 22 / =2-fold

order ofrotation

Plane lattices and their symmetries

5 two-dimensional lattice types

Point-group symmetry

of lattice: 2

2mm

2mm

4mm

6 mm

10 types of point groups (1, 1m, 2, 2mm,3, 3mm, 4, 4mm, 6, 6mm)possible basis:

Combination of point groups and translational symmetry 17 space groupsin 2D

Crystal=lattice+basis may have lower symmetry

Three-dimensional crystal systems

oblique lattice in 2D triclinic lattice in 3D

,cba

Special relations between axes and angles 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