1

Useful faithful matrix representation (different matrices for different operations):

'

1 0 1 1 1

r a r r a

Space Group G: translations and point Group

'

= traslation, = rotation matrix

( 1 No rotation). The

G= space Group

operation is d

with elements:

( | ) :

Pure translation (1|

enoted b

)

y

r a

a

a

r

a

Fyodorov and Schönflies in 1891 listed the 230 space Groups in 3d

Set of Translations T =(0, ) is a subgroup H;

A traslation observed from a rotated-translated reference

is still a translation.

Therefore, the subgroup H is invariant (more about that

a a

below).

2

The direct product would be | | | . Bad!b a b a

Space GroupPoint Group is a quotient Group: Point Group

Translation Group

, , right coset ( set of

Translations T =(0

space group operations

, ) are a subgroup H G; it is invariant (see

followed by a translation)

below)

(

a a

Hg hg h H g G

Cosets are labelled by , and a multiplication of cosets may

be defined (mult

Also, , l

iplic

eft coset)

= r

ating rotations, regardless of translatio

otation mat

ns

r

)

i

.

x

gH gh h H

But G is not a direct product of H and the point Group!

Structure of the Space Group G

3

'Consider the operation ;

1 0 1 1 1

multiplication by another one gives:

r a r r a

Multiplication rule is ( | )( | ) ( | )

It is called semidirect product. (never Abelian)

b a b a

( )

10 1 1 1

r ab r a b r a b

1

1 1 1

1 1

Inverse of ( | ) must be ( | ) such that

( | )( | ) ( | ) (1| 0)

here 1 no rotation.

Evidently, . Then must be such that

(1| ) (1| 0)

( | ) ( | )

a b

b a b a

w

b

b a b

a

a

a

4

1

1

1 1 1 1 1

( | ) ( | ) ( | )( | )

( | ) ( | ). Inserting the invers

conjugation

e,

( | ) ( | ) ( | ) ( | ( )).

( | ) ( | ) if the r

Clas

o

ses of s

t

ipace Group

atio

G s

:

n

s

b a b a

a b a

b a b a b a a

b c

and are conjugated, i.e. same angle.

If =1, ( | ) is a translation and the conjugate is a translation.

The translations make a class (angle=0) and are a subgroup.

This confirms that translations ma invariant ke an subgr .oup

b

(rotations are non-invariant subgroup (conjugation

does not change angle but may add translation).

Multiplication rule is ( | )( | ) ( | )

It is called semidirect product. (never Abelian)

b a b a

1 1 1( | )( | ) ( | ) ( | ) ( ) |b a aa ab

Shift of originConsider the Space Group operation ' with ' .

Shifting the origin to -b, the operation ' must be rewritten s s' :

' becomes ' ( ) : ' '

same rotation,but

r r r r a

r r

r s br r a s b s b a

r s b

. Shifting the origin changes the translation.

Can we get a pure rotation? This requires

a

0.

a b b

a b b

The condition for Pure rotations

One wants that a translation b H exists such that =0.a b b

1

1 1a b b a b a b

O

The shift b of the origin such that the operation is a rotation is obtained by

rotating the old translation a. If a solution exists, then one can consider the

Space Group operation as a pure rotation around some origin.

One wants that a translation b H exists such that =0.a b b

but if a=a there is no solution since (1- )-1 a=(1++2+3+…)a blows up. We cannot eliminate a translation which is parallel to the

rotation axis, to obtain a pure rotation.

7

A space Group generated by the Bravais

translations and the point Group is said

symmorphic

The symmorphic Groups have only the rotations of the point Group and the translations of the Bravais lattice;

nonsymmorphic Groups have extra symmetryelements are called screw axes and glide planes .

glide: ( , )a

screw: ( , )awhere belongs to the point group

8

Binary compounds with

Hexagonal structure (CdS)

9

Consecutive planes are different

2

c

Screw axis: C6 operation and C/2 translation : ( , )screw a The translation cannot be removed when it is along

the rotation axis, Then, it is a real screw axis.

10

2

c

glide plane:reflection and c/2 translation glide: ( , )a

11

let us iterate ( ,a) recalling multiplica

(

tion:

| )( | ) ( ) | b a b a

2 2

2 2

( | ) ( | ) but sin ,

( | ) ( |

2 ).

cea a a a a

a a

The translation cannot be removed since it is parallel to the rotation axis,

a=a

na tNow we show that for some integer n

(t= Bravais lattice translation)

Proof: Since belongs to the point group n = 1 for some n;

1

0

( | ) ( | ) ( | ), and

for some n ,( | ) (1| )

nn n k n

k

n

a a na

a na

with not a lattice translation t. How arbitrary is ?a a

12

Iterating the screw axis operation must eventually give a pure translation

na t

Example

screw-axis with an angle α = π/2, n=4 can have a translation a equal to

1/4, 2/4 o 3/4 of a Bravais vector.

Example:

for glide plane n=2 (twice a glide plane operation is a Bravais translation). This is shown in the next slide.

13

C6

½ c

Glide plane: is a reflection,

n=2 a=1/2 t

Going up by c/2 and making is a glide

plane symmetry operation; the square takes

up by c, a pure translation.

Two different planes

are superimposed

Graphite structure is not symmorphic

Kinds of lattices in 3dPrimitive (P): lattice points on the cell corners only.

Body (I): one additional lattice point at the center of the cell.

Face (F): one additional lattice point at the

center of each of the faces of the cell.

Base (A, B or C): one additional lattice point at the center of

each of one pair of the cell faces.

18

International notation

(International Tables for X-Ray

Crystallography (1952)

Screw axis with translation ¼ Bravais vector 41

Screw axis with translation 2/4 Bravais vector 42

Screw axis with translation ¾ Bravais vector 43

Screw axis with translation = Bravais vector 44

The international notation for a Space Group

starts with a letter ( P for primitive,

I for body-centered, F for face centered, R per rombohedric)

followed by a list of Group classes

Notation for screw axes

example: ( , )2

a

4

2

3

4 23 Face centered Cubic

4 axis+orthogonal plane represented by m=mirror

2axis+orthogonal plane

3 axis+inversion

hF F Om m

Cm

Cm

C

this is symmorphic,while the diamond Group is not: 41 is screw

1

14

2

3

4 23 Face centered Cubic

4 1screw axis with translation+glide plane

42

axis+orthogonal plane

3 axis+inversion

hF F Od m

C td

Cm

C

19

International Notation: Group symbols are lists of elements

Example and comparison with Schoenflies notation:

Tables readily available for purchase on internet http://it.iucr.org/

20

CdS also has a symmorphic cubic form with space group

43F m

CdS in Wurtzite crystal structure P63mc group (P=primitive, c means glide translation along c axis) with a screw axis having 3/6 of c translation

21

Representations of the Space Group

1. .( , ) 1

Applied to plane waves this is:

( , ) exp[ . ( )]ik r ik a ra e e ik r a

-1

1 1

Consider first the effect on plane waves,

which are eigenfunctions and irreps of all the translatio

Since f(r), roto-translation R Rf(r)=f(R ),

( | )f(r) f(( | ) r)=f( (r-a))

ns.

r

a a

Start from Translation Group which is an invariant subgroup

. ( ).( )

( translation

( , )

s l

Rotat

eave

io

of plane waves invarian

n

t

s rotate k.

)

ik r i k r a

pure k

a e e

22

Rotating two vectors by the same angle the scalar product does not change;

so we may write

labels a representation of translation Group,

basis=set of plane waves.

Such representations are mixed by the Space Group.

k

k

1. .( , ) 1( , ) exp[ . ( )]ik r ik a ra e e ik r a

Star of k

is the set , int Group .

High symmetry have smaller set

The star of some special k may comprise just that

s

k

k po

k

The Star of k is a subspace of k space which is a basis

set for a representation of all Translation and and

Rotations in the Space Group.

However some operations may mix k points at border of

BZ with other k points differing by reciprocal lattice vectors

G; these are equivalent and not distinct basis elements.

23

In terms of Bloch functions, the Space Group operation

(α,a )ψn(k ,r )

yields a linear combination of

ψn’ (αk ,r ), where n → n’ because in general Point Group operations mix rotated and reflected orbitals degenerate

bands.

24

Warning

Point Group operations can be represented by plane waves,

but matters are more complex with Bloch functions, which are

eigenfunctions of crystal Hamiltonian.

25

In general one may have a set of wave functions

that form the basis of a representation of the

Little group at each k . A basis for the Space

Group must comprise all of them. The Space

Group mixes them with the wave functions of the

star of k.

The set of the basis functions of a representation of

the Little Group for all the points of a star provide a

basis for a representation of the Space Group G.

Such representations can be analyzed in the

irreducible representations of the Space Group in

the usual way.

Often one can enhance the symmetry by adding P,T

Define: Group of the wave vector or Little

Group

k

is the Subgroup G G which consists of the operations (a, )

such that : k = k + G.

The Magnetic Groups

Magnetic Groups are obtained from the space groups by

adding a new generator: time reversal T. In non-magnetic

solids this is a symmetry. The Groups suitable for magnetic

solids were studied by Lev Vasilyevich Shubnikov and refered

to as color Groups.

T flips spins as well as currents. It makes a difference in magnetic

materials where equilibrium currents and magnetic moments exist. In

this chain time reversal T is no symmetry, but time reversal times a

one-step translation is:

Hamermesh (chapter 2) proves some theorems. Magnetic point Groups

can be obtained from the non-magnetic ones in most cases the following

way.

26

Лев Васи́льевич Шу́бников

In this way one finds 58 new, magnetic Groups. Including 32 point Groups the

total is 90 according to Hamermesh, 122 according to Tinkham. These can be

combined with the translation ones to form generalized space Groups.

The Magnetic Groups are 1651 in 3d

Theorem

G point group , H subgroup having index 2, that is,

G=H+aH, with a is not in H .

Then the magnetic Group is G’=H+TaH.

Along with C3v which has index 2, there is a magnetic Group where the

reflections are multipied by T. The rotation C3 cannot be multiplied by T

because otherwise the third power would give T itself as a symmetry. This

is excluded because it would reverse spins.

27

yT i K

From Tinkham

Monomer symmetries:C , ,C ,C , ......n n nh nv nS D