fyodorov and schönflies in 1891 listed the 230 space...
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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
8
Example: Binary compounds with Hexagonal structure (CdS)
2
c
Screw axis: C6 operation and C/2 translation : ( , )screw 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
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.
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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
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
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