1

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

'

1 0 1 1 1

r a r r a

Space Group: translations and point Group

' with traslation rotation matrix

( 1 No rotation). Th

Gro

e o

up elements:

pe (ration denoted b | )y :

r r a a

a

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

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

Space GroupPoint Group is a quotient Group: Point Group

Translation Group

2

'

1 0 1 1 1

r a r r a

Multiplication ( | )( | ) ( | )

(never Abelian) It is called semidirect product.

b a b a

( )

10 1 1 1

r ab r a b r a b

1 1 1 1 1

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

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

here 1 no rotation.

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

a b

b a b a

w

b a b a a

3

1

1

1 1 1 1 1

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

( | ) ( | ). Inserting the inver

Clas

se,

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

Th e rotations must be conjugated, i.e. same

conjugation is

angle

ses:

.

b a b a

a b a

b a b a b a a

1 1 1 1

1 1

( | ) :first rotate then translate

( | ) ( ):first translate back then rotate back

( | ) ( | ) ( )

a r r a

a r r a r a

a a r r a a r

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

The translations make an invariant subgroup.

b

That is, if =1 (translation) conjugation with any

element gives a translation:

translations are an invariant subgroup

(rotations are non-invariant subgroup (conjugation

does not change angle but may add translation)

Shift of origin

Consider the operation ' with ' .

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

' becomes ' ( ) : ' '

same rotation, .but a S

r r r r a

r r

r s br r a

a b b

s b s b ar s b

hifting the origin changes the translation.

Consider the Space Group operation ( |a), ' : '

When is it a pure rotation around some center site?

r r r r a

Pure rotations

1

1 1a b b a b a b

One wants b such that =0a b b

O

The shift of the origin is obtained by rotating the old translation, if a solution

exists. Then one can consider the Space Group operation as a pure rotation

around some origin.

When is it a pure rotation? One wants b such that =0.a b b

6

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 symmetry elements are called screw axes and glide planes .

glide: ( , )a

screw: ( , )a

7

Binary compounds with Hexagonal structure (CdS)

8

Example: Binary compounds with Hexagonal structure (CdS)

2

c

Screw axis: C6 operation and C/2 translation : ( , )screw a

9

2

c

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

10

Graphite is elemental but nonsymmorphic

C6 rotation screw axis and glide plane

screw axes and glide planes depend on special relations between the dimensions of the basis (that is, of the unit which is periodically repeated) and of the Bravais translations.

11

If solution exists, the Space Group operation is a pure rotation around -b,

the c/2 translation is not needed, and the Group is symmorphic..

1

1 a b

Recall the condition for pure rotation around -b

but if a=a there is no solution since (1- )-1 a=(1++2+3+…)a blows up

The translation cannot be removed when it is along

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

cIt is natural to ask whether one can eliminate the translation

2

from the screw axis operation by shifting the origin to some -b .

Doubt: When is a screw axis really needed?

screw operation: ( , )a

12

let us iterate ( ,a) recalling multiplica

(

tion:

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

2 2 2( | ) ( | ( 2) | ).a a a a

Let the translation can be taken 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

screw operation: ( , ), with not a lattice translation t.

How arbitrary is the choice of ?

a a

a

13

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

1

0

( | ) ( | ) ( | ), and

for some n ,( | ) (1| )

nn n k n

k

n

a a na

a na

14

C6

½ c

½ c

Glide plane: is a reflection, n=2 a=1/2 t

Kinds of lattices in 3d

Primitive (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.

17

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

4

2

3

4 23 Face centered Cubic

4axis+orthogonal plane

2axis+orthogonal plane

3 axis+inversion

hF F Om m

Cm

Cm

C

this is symmorphic,while the diamond Group is not

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

18

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/

19

CdS also has a cubic form with space group

43F m

CdS in Wurtzite crystal structure P63mc group (P=primitive, c means glide translation along c axis)

20

Representations of the Translation Group

and of the Space Group

1. .( , ) 1

Consider first the effect on plane waves,

which are eigenfunctions of all the translations.

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

1 1 1 1

1

1

1 1

denotes '

( | ) :first rotate then translate

( |

inverse

Recall Group

) ( ):firs

: ( |

t translate back then rotate back

elements:

) ( | )

( | |

|

(

( )

)

a

r r a

a r r a

a r r a r a

a

a

a

a

1

-1 1 1 Since f(r), Rf(r)=f(R ), ( | )f(r

) ( )

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

r r a a r

r a a

In terms of Bloch functions, (α,a )ψn(k ,r ) yields a linear combination of

ψn’ (αk ,r ), where n → n’ because in general Point Group operations mix degenerate bands.

. ( ).( )( , ) ik r i k r aa e e

21

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

so we may write

labels a representation of translation Group, basis=plane waves.

Such representations are mixed by the Space Group.

k

Star of k

is the set , int Group .

High symmetry have smaller sets

k po

k

Star of k: subspace which is a basis set for a

representation of all T and R 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.

The star of some special k may comprise just that k.

22

Example: square lattice

Special Points: , ,

Special Lines: ,Z,

M X

4v

4v

2v

is invariant under

is invariant under since theother corners areconnected to M byG

X is invariant under

C

M C

C

is invariant under ,

is invariant under ,

is invariant under ,sinceit takes toequivalent points.

d

y

xZ

2vthis is C

4vthis is C

just a reflection

Special Points: , , ,

Special Lines: ,Z, , ,

M X R

S T

, are invariant under

, are invariant under 4 /

hR O

X M mmm

2

, invariant under 2 ,

, invariant under 4 ,

has 2 mirror planes and .

S are mm

T are mm

Z C

29

In general one may have a set of wave functions

at each k, so a basis for the Space Group must

comprise all of them

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.

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. They 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 T is no symmetry, but T times a one-step translation is:

T can only be a symmetry if there are no spins and no currents.

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

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

way. 30

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

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

G=H+aH, with a H, aG.

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.

31

yT i K

From Tinkham