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Symmetry Operations

brief remark about the general role of symmetry

in modern physics

conservation of momentum

change of momentum

V(X1) V(X2)

translational symmetry

Emmy Noether 1918 Symmetry in nature

conservation law

1882 in Erlangen, Bavaria, Germany

1935 in Bryn Mawr, Pennsylvania, USA

Example for symmetry in QM

angular momentum conserved J good quantum number

Hamiltonian invariant with respect to rotation

Zeeman splitting

Breaking the symmetry with magnetic field

Proton and Neutron 2 states of one particle

breaking the Isospin symmetry

Magnetic phase transition

Symmetry in perfect single crystals

ideally perfect single crystal

infinite

three-dimensional

repetition

of

identical building blocks

basis

very complex molecular structure

Volume of space (parallelepiped) fills all of

space by translation of discrete distances

Quantity of matter contained in the unit cell

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

Translational symmetry

n21

3D crystalline solid

3 translational basis vectors a, b, c

n12

-by parallel extensions the basis vectors form a

parallelepiped, the unit cell, of volume Va(bxc)

translational operation

-connects positions with identical atomic

environments

concept of translational invariance is more

general

physical property at r (e.g.,electron density)

is also found at

rrT

Set of operations

Tn1an2bn3c

defines

r

space lattice

or

Bravais lattice

purely geometrical concept

lattice and translational vectors a, b,c are

primitive if every point r equivalent to r

is created by

T

according to

rrT

Primitive basis

minimum number of atoms in the primitive

(smallest) unit cell which is sufficient to

characterize crystal structure

No integer!

2 important examples for primitive and non

primitive unit cells

face centered cubic

1atom/Vprimitive

4 atoms/Vconventinal

Primitive cell rhombohedron

body centered cubic

1atom/Vprimitive

2 atoms/Vconventinal

Lattice Symmetry

Symmetry of the basis

point group symmetry

has to be consistent with

symmetry of Bravais lattice

No change of the crystal after symmetry operation

(point group of the basis must be a point group

of the lattice)

Operations (in addition to translation) which

leave the crystal lattice invariant

- Reflection at a plane

- Rotation about an axis

H2o

2 -fold rotation axis

NH3

SF5 Cl

Cr(C6H6)2

Click for more animations and details about point

group theory

- point inversion

- Glide

reflection translation

rotation translation

- Screw

Notation for the symmetry operations

rotation by 2?/n degrees reflection through

plane perpendicular to rotation axis

Origin of the Symbols after Schönflies

Eidentity from the German Einheit unity

Cn Rotation (clockwise) through an angle 2p/n,

with n integer

? mirror plane from the German Spiegelmirror

?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 n1, 2, 3,4 and 6 are the

only rotation symmetries consistent with

translational symmetry

!

?

?

?

?

?

?

?

Intuitive example pentagon

Two-dimensional crystal with lattice constant a

in horizontal direction

1

a

2

(m-1)

Row A

Row B

m

1

If rotation by a is a symmetry operation

1

and

m

positions of atoms in row B

(m-1)a 2a

2a cos a

p-m

-1

1

0/2p

-2

1/2

p/3

-3

0

p/2

-4

-1/2

2p/3

-5

-1

p

Plane lattices and their symmetries

4mm

Point-group symmetry of lattice 2

2mm

2mm

6 mm

5 two-dimensional lattice types

Crystallatticebasis may have lower symmetry

10 types of point groups (1, 1m, 2, 2mm,3, 3mm,

4, 4mm, 6, 6mm)

possible basis

Combination of point groups and translational

symmetry

Three-dimensional crystal systems

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

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