Download Space Group: translations and point Group

Space Group: translations and point Group
Fyodorov and Schönflies in 1891 listed the 230 space Groups in 3d
Group elements: r '   r  a with a traslation  rotation matrix
(  1  No rotation). The operation denoted by ( | a ) :
Useful faithful matrix representation (different matrices for different operations):
 r ' 
 
1 0
a  r    r  a 
   

1  1   1 
Point Group is a quotient Group: Point Group 
The direct product would be
Space Group
Translation Group
  | b   | a     | b  a .
Bad!
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 r ' 
 
1 0


0
a  r    r  a 
   

1  1   1 
b    r  a    ( r  a )  b    r   a  b 




1  1  
1
1
 

Multiplication (  | b )( | a )  (  | b   a )
(never Abelian) It is called semidirect product.
Inverse of ( | a ) must be
(  | b ) such that
(  | b )( | a )  (  | b   a )  (1| 0)
where 1  no rotation.
   1 (1| b   1a )  (1| 0)  (  | b )  ( | a ) 1  ( 1 |  1a )
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( | a ) r   r  a :first rotate then translate
( | a ) 1 r   1r   1a   1 (r  a ):first translate back then rotate back
( | a ) 1 ( | a ) r   1 ( r  a  a )  r
Classes: conjugation is
(  | b )  ( | a ) 1 (  | b )( | a )
 ( | a ) 1 (  | b   a ). Inserting the inverse,
(  | b )  ( 1 |  1a ) 1 (  | b   a )  ( 1 |  1 (b   a  a )).
Th e rotations must be conjugated, i.e. same angle.
If  =1, (  | b ) is a translation and the conjugate is a translation.
The translations make an invariant subgroup.
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)
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Shift of origin
Consider the operation r  r ' with r '   r  a.
Shifting the origin to -b, the operation r  r ' must be rewritten s  s' :
 r  s b
 r '   r  a becomes s ' b   ( s  b)  a :

r '  s ' b
same rotation,but a  a  b   b. Shifting the origin changes the translation.
Pure rotations
Consider the Space Group operation ( |a), r  r ' :
When is it a pure rotation around some center site?
r '  r  a
When is it a pure rotation? One wants b such that a  b   b=0.
O
One wants b such that a  b  b=0
a  b   b  a    1 b 
 1
1
ab
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 .
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 .
screw: ( , a )
glide: ( , a )
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Binary compounds with Hexagonal structure (CdS)
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Example: Binary compounds with Hexagonal structure (CdS)
c
2
Screw axis: C6 operation and C/2 translation
screw : ( , a)
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c
2
glide plane:reflection and c/2 translation
glide: ( , a )
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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.
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screw operation: ( , a)
Doubt: When is a screw axis really needed?
c
translation
2
from the screw axis operation by shifting the origin to some -b .
It is natural to ask whether one can eliminate the
Recall the condition for pure
rotation around -b
  1
1
ab
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..
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.
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Let the translation can be taken parallel to the rotation axis,  a=a
 screw operation: ( , a ), with a not a lattice translation t.
How arbitrary is the choice of a ?
na  t
Now we show that for some integer n
(t= Bravais lattice translation)
Proof: Since  belongs to the point group  n = 1 for some n;
let us iterate ( ,a) recalling multiplication:
(  | b )( | a )  (  | b   a )
 ( | a )  ( | a   a )  ( | 2a ).
2
2
2
n 1
( | a )  ( |   a )  ( | na ), and
n
n
k
n
k 0
for some n ,( | a )  (1| na )
n
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n 1
( | a ) n  ( n |   k a )  ( n | na ), and
k 0
for some n ,( | a )  (1| na )
n
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
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Glide plane:  is a reflection, n=2
a=1/2 t
C6
½c

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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.
International notation
(International Tables for X-Ray
Crystallography (1952)
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
Screw axis with translation ¼ Bravais vector
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Screw axis with translation 2/4 Bravais vector
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Screw axis with translation ¾ Bravais vector
Screw axis with translation = Bravais vector
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44
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International Notation: Group symbols are lists of elements
Example and comparison with Schoenflies notation:
F
4 2
3
m m
F  Face centered Cubic Oh
4
 C4 axis+orthogonal plane
m
2
 C2 axis+orthogonal plane
m
3  C3 axis+inversion
this is symmorphic,while the diamond Group is not
41 2
F 3
F  Face centered Cubic Oh
d m
41
1
 C4 screw axis with
t translation+glide plane
d
4
2
 C2 axis+orthogonal plane
m
3  C3 axis+inversion
Tables readily available for purchase on internet
http://it.iucr.org/
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CdS in Wurtzite crystal structure P63mc
along c axis)
group (P=primitive, c means glide translation
CdS also has a cubic form with space group
F 43m
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Representations of the Translation Group
and of the Space Group
Recall Group elements:
( | a ) denotes r '   r  a
inverse: ( | a) 1  ( 1 |  1a )
( | a ) r   r  a :first rotate then translate
( | a ) 1 r   1r   1a   1 (r  a ):first translate back then rotate back
( | a ) 1 ( | a ) r   1 ( r  a  a )  r
Since f(r),
Rf(r)=f(R -1r ), ( | a)f(r)  f(( | a) 1 r)=f( 1 (r-a))
Consider first the effect on plane waves,
which are eigenfunctions of all the translations.
( , a )e
ik .r
e
ik .( , a )1 r
 exp[ik . 1 (r  a )]
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Rotating two vectors by the same angle the scalar product does not change;
so we may write
( , a)e
ik .r
e
i ( k ).( r  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.
k labels a representation of translation Group, basis=plane waves.
Such representations are mixed by the Space Group.
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Star of k


is the set  k ,   po int Group .
High symmetry k have smaller sets
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: , M , X
Special Lines: ,Z,
 is invariant under C4v
M is invariant under C4v since the other corners are connected to M by G
X is invariant under C2v
 is invariant under  d ,
 is invariant under  y ,
Z is invariant under  x ,sinceit takes to equivalent points.
this is C4v
this is C2v
just a reflection
Special Points: , M , X , R
Special Lines: ,Z,, S , T
 , R are invariant under Oh
X , M are invariant under 4 / mmm
 , S are invariant under 2mm,
 , T are invariant under 4mm,
Z has 2 mirror planes and C2 .
Define: Group of the wave vector or Little
Group
is the Subgroup Gk  G which consists of the operations (a, )
such that :  k = k + G.
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.
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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.
Лев Васи́льевич Шу́бников
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.
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G point group , H subgroup having index 2, that is,
G=H+aH, with a  H, aG.
Then the magnetic Group is G’=H+TaH.
T  i y K
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.
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
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From Tinkham