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General introduction to representation theory
Laurent C. Chapon
ISIS Facility, Rutherford Appleton Laboratory, UK
Representation theory
• Method for simplifying analysis of a problem in systems possessing some degree of symmetry.
• What is allowed vs. what is not allowed
Keyword : Invariance of the physical properties under
application of symmetry operators.
Spectroscopy
• Ground state characterized by φ0
• Excited state characterized by φ1
• Operator Ο
• Transition integral :
• The integrand must be invariant under
application of all symmetry operations
∫= 10 φφ OT
φ0
φ1
Use to predict vibration spectroscopic transitions
that can be observed
IR-Raman active modes
IR active, change in dipole moment Raman active, change in polarizability
CO2
∫= 10µφφT ∫= 10αφφT
Dipole moment operator Operator for polarizability
Crystal field
↑
• Ce3+ 4f1 electronic configuration
• J=|L-S|=5/2
Cubic environmentFree ion
Ground state
multiplet ∫= 10 φφ iJT
∆J=0;+1;-1
MO-LCAO
∑=Ψr
iicφ
The molecular orbitals of polyatomic species are linear
combinations of atomic orbitals:
If the molecule has symmetry,
group theory predicts which atomic orbitals can contribute to each molecular orbital.
3 HN
a1 (2pz)
a1 (2s)
e (2px, 2py)
e
a1 2s1 - s2 - s3
- s2 + s3
-25.6 eV
-15.5 eV
-13.5 eV
s1 + s2 + s33a1
1e
2e
-17.0 eV
2a1-31.0 eV
4a1
MO-LCAO
SALCs of
H atoms
Orbitals of
Central atom
Phase transitions in solidsPhase transitions often take place between phases of different symmetry.
High symmetry phase, Group G0
Low symmetry phase, Group G1
• This is a “spontaneous” symmetry-breaking process.
• Transition are classified as either 1st order (latent heat) or 2d order (or continuous)
(T,P)
A simple example: Paramagnetic -> Ferromagnetic transition
“Time-reversal” is lost
• Symmetry under reversal
of the electric current
Landau theory
• Ordering is characterized by a function ρ(x) that changes at the transition.
•Above Tc, ρ0(x) is invariant under all operations of G0
•Below Tc, ρ1(x) is invariant under all operations of G1
• At T=Tc, all the coefficients cin vanish
)('
01 xcn i
n
i
n
i∑∑ Φ=−= ρρδρ Basis functions of irreducible
Representation of G0.
Landau theory (2)
-100 -50 0 50 100
E
η
.......))((2
1 42
0 ++−+Φ=Φ ηη CTTTa c
T>Tc
In a second order phase transition,
a single symmetry mode is involved.
...........)(),(2
'
0 ++Φ=Φ ∑ ∑n i
n
i
n cTPA
Φ is invariant under operations of G, each order of the expansion can be written
is given by some polynomal invariants of cin.
• Thermodynamic equilibrium requires that all A are >0
above Tc.
• In order to have broken symmetry, one A has to change
sign at the transition.
Outline
1. Symmetry elements and operations
2. Symmetry groups (molecules)
3. Representation of a group
4. Irreducible representations (IR)
5. Decomposition into IRs
6. Projection
7. Space groups
Inversion point i ( )1
Change coordinates of a point (x,y,z) to (-x,-y,-z)
Mirror planes
σh
σv σd
Proper rotation Cn (n)
Fe(C5H5)2
5-fold axis
Rotation axis of order n
1)-nk(1 ≤≤k
nCRotation of 2πk/n
Improper rotation Sn( )nCombination of two successive operations:
1) Rotation Cn around an axis.
2) Mirror operation in a plane perpendicular to rotation axis
S4 in tetrahedral geometry
Group structure
• Collection of elements for which an associative lawof combination is defined and such that for any pair of elements g and h, the product gh is also element of the collection
• It contains a unitary element, E, such that gE=g
• Every element g has an inverse, noted g-1 such that gg-1=E.
The order of a group is simply the number of elements in a group.
We will note the order of a group h.
Multiplication tableFour different operations:
• E
• σ(xz)• σ(yz)• C2(z)
EC2(z)σ(xz)σ(yz)σ(yz)
C2(z)Eσ(yz)σ(xz)σ(xz)
σ(xz)σ(yz)EC2(z)C2(z)
σ(yz)σ(xz)C2(z)EE
σ(yz)σ(xz)C2(z)E
Classes
Symmetry operations:
2
31
1
3
1
1
321
2
3
1
3 ,,,,,
CC
CCE
vv
vvv
=− σσ
σσσ
Similarity transform: gxxh 1−=
The set of elements that are all conjugate to one another is called a (conjugacy) class.
g and h are conjugate
Determination of G
Linear?
2 Cn
n>2?i?
i?
i?
C5
Select
Cn of highest n
nC2⊥⊥⊥⊥Cn?
Cn?
σσσσh?
nnnnσσσσd?
σσσσh?
nnnnσσσσv?
σσσσ?
S2n?
D∞h C∞v
TdOhIh
Dnh
Dnd Dn
Cnh
Cnv
S2n Cn
Ci C1
Cs
y
y
y
y
y
y
y
y
y
yy
y
y y
• 1 C5
• 5 C2
• 5 σv• 1 σh
Flow chart (Cotton)
Representation of G
A group G is represented in a vector space E, of dimension n,
if we form an homomorphism D from G to GLn(E) :
( ) 11
11
−=∈∀
=
=∈∀
∈∈∀
D(g)) G, D(g g
)D(
D(g)D(g') G, D(gg') g,g'
(E) GLD(g) G, g g
-
na
Matrices
• If a basis of E is chosen, then we can write D(g) as n by n matrices.
• We will note Dαβ(g) the matrix elements (line α, row β)
....
....
....
....
β
α
Character
....
....
....
.... The trace (sum of diagonal elements)
is noted χ.
(g)Dχ(g)α
αα∑=
Important reminder :
D’=P-1DP
Matrices that are conjugate to one another
have the same trace.
Example : H2O modes
A symmetry operation produces linear transformations in the vector space E.
H2O- E
=
O1z
O1y
O1x
H2z
H2y
H2x
H1z
H1y
H1x
100000000
010000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
O1z'
O1y'
O1x'
H2z'
H2y'
H2x'
H1z'
H1y'
H1x'
χ(E)=9
H2O- C2-axis
−
−
−
−
−
=
O1z
O1y
O1x
H2z
H2y
H2x
H1z
H1y
H1x
100000000
010000000
001000000
000000100
000000010
000000001
000100000
000010000
000001-000
O1z'
O1y'
O1x'
H2z'
H2y'
H2x'
H1z'
H1y'
H1x'
χ(C2)=-1
H2O-σ(xz)
−
−
−
=
O1z
O1y
O1x
H2z
H2y
H2x
H1z
H1y
H1x
100000000
010000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
O1z'
O1y'
O1x'
H2z'
H2y'
H2x'
H1z'
H1y'
H1x'
χ(σ(xz))=3
Matrix multiplication
−
−
−
=
−
−
−
−
−
−
−
−
−
100000000
010000000
001000000
000000100
000000010
000000001
000100000
000010000
000001000
100000000
010000000
001000000
000000100
000000010
000000001
000100000
000010000
000001000
100000000
010000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
C2(z)σ(xz) σ(yz)x =
Irreducible representations (IRs)
D is a representation of a group in a space E.
D is reducible if it leaves at least one subspace of E invariant, otherwise the representation is irreducible.
∑⊕
⊕== ......21 EEEE i
Every element of E can be written in one and only one way
as a sum of elements of Ei.
IRs
• In matrix terms:
A representation is reducible if one can find a similarity transformation (change of basis) that send all the matrices D(g) to the same block-diagonal form.
0
0
Properties of block diagonal matrices
0
0
0
0x
D1 D2
Corresponding blocks are multiplied separately.
=
0
0
IRs• In a finite group, there is a limited number of IRs.
( )∑ =i
i hl2
0-12E
-111A2
111A1
3σv2C3EC3v
Character tables• In a finite group, there is a limited number of IRs.
• IRs are described in character tables:
A table that list the symmetry operations horizontally, IRs labels vertically and corresponding characters.
0-12E
-111A2
111A1
3σv2C3EC3v
Conjugacy classes
Great Orthogonality theorem
''''
)*()( ββααβααβ δδδ ij
ji
j
Gg
i
ll
hgDgD =∑
∈
• For two given IRs Di and Dj, of dimension li and lj respectively.
( )
0)()(
)(
2
=
=
∑
∑
∈
∈
Gg
ji
Gg
i
gg
hg
χχ
χ
GOT
D is a reducible representation.
The number of times that a representation i appears in a decomposition is :
)(*)(1
ggh
nGg
ii χχ∑∈
=
0-12E
103D
-111A2
111A1
3σv2C3EC3v
nA1=1/6(3*1+2*1*0+3*1*1)=1
nA2=1/6(3*1+2*1*0+3*-1*1)=0
nE =1/6(3*1+2*-1*0+3*0*1)=1
Projection
ggDPGg
ˆ)(ˆ *∑∈
= λµνλ
• Project a vector of the vector space into the space of the IR to find the
symmetry adapted vectors.
* indicates complex conjugate
Integrals
∫∞
∞−
dxxf )(
y
x
f(x)
y
x
f(x)
Space group Symmetry operations
• Use the Seitz notation αααα|tαααα
−α rotational part (proper or improper)
− tα translational part
α|τα β|τβ=αβ|αtβ+tα
Space group: infinite number of symmetry operations
Group of translation T
........ 200|1 |1 010|1 100|1 000|1 tΤ
•Infinite abelian group•Infinite number of irreducible representations, and consists of the complex root of unity.Basis are Bloch functions.
ΓK …………………………….
……
e-ikt
)().()()(|1
on) translatilattice a is ( )()(
).()(
)( reetrutrrt
trutru
erur
kikttrik
k
kk
kk
ikr
k
k
Φ=−=−Φ=Φ
=+
=Φ
−−
Space group
Consider a symmetry element g=h|t and a Bloch-function Φ’:
(r)hkΦΦ
uhkikuikhkuikh
k
k
k
ikr
k
k
erthereth
ruhth
rthuu
rth
erur
function bloch a is '
')(|)(|
)(|1|
)(||1'|1
)(|'
)()(
)(
1
11
φφφ
φ
φφ
φφ
φ
−−−
−
===
=
=
=
=
−−
Little group Gk
• By applying the rotational part of the symmetry elements of the paramagnetic group, one founds a set of k vectors, known as the “star of k”
• Two vectors k1 and k2 are equivalent if they equal or related by a reciprocal lattice vector.
• In the general case, if all vectors k1, k2,……ki in the star are not equivalent, the functions Φki are linearly independent.
• The group generated from the point group operations that leave k invariant elements + translations is called the group of the propagation vector k or little group and noted Gk.
• In Gk, the functions Φki are not all linearly independent, and the representation is reducible.
IRs of Gk
Tabulated (Kovalev tables) or calculable
for all space group and all k vectors for
finite sets of point group elements h
Despite the infinite number of
atomic positions in a crystal
symmetry elements in a space group
…a representation theory of space groups is feasible using Bloch
functions associated to k points of the reciprocal space. This means
that the group properties can be given by matrices of finite
dimensions for the
- Reducible (physical) representations can be constructed on the
space of the components of a set of generated points in the zero cell.
- Irreducible representations of the Group of vector k are
constructed from a finite set of elements of the zero-block.
Orthogonalization procedures explained previously can be
employed to construct symmetry adapted functions
Symmetry analysisExample 1
• Space group P4mm, k=0, Magnetic site 2c
1
1
1
1
1
1
1x
1y1z+
2x
2y2z+
1|000
−
−
−
−
1
1
1
1
1
1
2z|000
1x
1y1z+
2x
2y
2z+
−
−
1
1
1
1
1
1
4z+|000
1x
1y
1z+
2x
2y
2z+
−
−
1
1
1
1
1
1
4z-|000
1x
1y
1z+
2x
2y
2z+
−
−
−
−
1
1
1
1
1
1
mx0z|000
2x
2y
2z-
1y
1x
1z-
−
−
−
−
1
1
1
1
1
1
m0yz|000
2x
2y2z-
1y
1x
1z-
−
−
1
1
1
1
1
1
mx-xz|000
1x
1y
1z-
2x
2y
2z-
mxxz|000
1x
1y
1z-
2y
2x 2z-
−
−
−
−
−
−
1
1
1
1
1
1
IRs
Irs/SO 1|000 2_00z|000 4+_00z|000 4-_00z|000 m_x0z|000 m_0yz|000 m_x-xz|000 m_xxz|000
--------------------------------------------------------------------------------------------------------------------------------------------------------------
ΓΓΓΓ1 1 1 1 1 1 1 1 1
ΓΓΓΓ2 1 1 1 1 -1 -1 -1 -1
ΓΓΓΓ3 1 1 -1 -1 1 1 -1 -1
ΓΓΓΓ4 1 1 -1 -1 -1 -1 1 1
ΓΓΓΓ5 1 0 -1 0 i 0 -i 0 0 1 0 -1 0 -i 0 i
0 1 0 -1 0 -i 0 i 1 0 -1 0 i 0 -i 0
χ(Γχ(Γχ(Γχ(Γ) 6 ) 6 ) 6 ) 6 −−−−2 0 0 2 0 0 2 0 0 2 0 0 −−−−2 2 2 2 −−−−2222 0 00 00 00 0
Decomposition into IRs
2)0000020200002226(8
1)(
1)1010121210101216(8
1)(
0)1010121210101216(8
1)(
1)1010121210101216(8
1)(
0)1010121210101216(8
1)(
5
4
3
2
1
=×+×+×−×−×+×+−×−×=Γ
=×+×+−×−−×−−×+−×+×−×=Γ
=−×+−×+×−×−−×+−×+×−×=Γ
=−×+−×+−×−−×−×+×+×−×=Γ
=×+×+×−×−×+×+×−×=Γ
η
η
η
η
η
542 2Γ⊕Γ⊕Γ=Γ
Projection onto Γ2
)2|1(|42|2|1|1|2|2|1|1|1|
02|2|1|1|2|2|1|1|1|
02|2|1|1|2|2|1|1|1|
>+>>=+>+>+>+>+>+>+>>=
>=+>−>+>−>+>−>−>>=
>=+>−>−>+>−>+>−>>=
zzzzzzzzzzzP
xxyyxxyyyP
yyxxyyxxxP
1z+
2z+
Shubnikov notation P4m’m’
Projection onto Γ4
)2|1(|42|2|1|1|2|2|1|1|1|
02|2|1|1||2|21|1|1|
02|2|1|1|2|2|1|1|1|
>−>>=−>−>+>+>−>−>+>>=
>=−>+>+>−>−>+>−>>=
>=−>+>−>+>+>−>−>>=
zzzzzzzzzzzP
xxyyxxyyyP
yyxxyyxxxP
1z+
2z-
Shubnikov notation P4’mm’
))(2|1(|21|
))(2|1(|21|
matrices theof elements (2,2) theusing Projection
))(1|2(|21|1|2|2|2|
))(1|2(|21|1|2|2|2|
02|2|1|1|1|
))(2|1(|22|2|1|1|1|
))(2|1(|22|2|1|1|1|
matrices theof elements (1,1) theusing Projection
4
3
1
2
2
1
φ
φ
φ
φ
φ
φ
>−>>=
>+>>=
−>+>>=+>+>+>>=
>−>>=−>−>+>>=
>=+>−>−>>=
>+>>=+>+>+>>=
>−>>=−>−>+>>=
xiyyP
yixxP
ixiyxixiyyyP
iyixyiyixxxP
zizizzzP
xiyxixiyyyP
yixyiyixxxP
Projection onto Γ5
The magnetic modes can be any linear combinationsof φ1, φ2, φ3, φ4
Symmetry analysisExample 2
• Space group P21/m, k=(0,δ,0) Magnetic site 4f
1
2
3
4
Little Group GK
Operation of the point group on the propagation vector• Identity k=(0,δ,0)• 2-fold axis k=(0,δ,0)• Inversion -k=(0,-δ,0)• Mirror -k=(0,-δ,0)
• Only 1|000 and 2y|0½0 belong to GK
• 4f sites are split into two orbits : (1,2) and (3,4) since no operations of Gk transform sites of the first orbit into that of the second orbit
IRs of Gk------------------------------------------
Irs/SO 1|000 2y|0½0
------------------------------------------
ΓΓΓΓ1 1 eiπδ
ΓΓΓΓ2 1 - eiπδ
Perform representation analysis for the first orbit. For identity, it is trivial.
2y|0½0
• |1x> is transformed into -|2x>
• |1y> is transformed into |2y>
• |1z> is transformed into -|2z>
−
−
−
−
1
1
1
1
1
1
Decomposition into IRs
3)016(2
1)(
3)016(2
1)(
2
1
=−×+×=Γ
=×+×=Γ
πδ
πδ
η
η
i
i
e
e
Projection onto Γ1
>−>>=
>+>>=
>−>>=
−
−
−
zezzP
yeyyP
xexxP
i
i
i
2|1|1|
2|1|1|
2|1|1|
πδ
πδ
πδ
1
2
+
+
Projection onto Γ2
>+>>=
>−>>=
>+>>=
−
−
−
zezzP
yeyyP
xexxP
i
i
i
2|1|1|
2|1|1|
2|1|1|
πδ
πδ
πδ
1
2
+
-
The same can be done for the second orbit
Magnetic diffraction
L.C.Chapon
ISIS Facility, Rutherford Appleton Laboratory, UK
Outline
• Nuclear scattering
• Magnetic scattering using a non-polarized neutron beam
• Type of magnetic structures (FStudio)
• Instrumentation
Scattering cross sections
k
z
y
x
dΩ
Incident flux Φ of neutron of wavevector k.Neutron is in the initial state λAfter scattering the neutron wavevector is k’ and the neutron is in the state λ’
Partial differential cross section:
'
2
dEd
d
Ωσ Number of neutrons scattered per second
into a solid angle dΩ and with final energy between E’ and E’+dE’/(ΦdΩdE’)
Differential cross section:
Ωd
dσ Number of neutrons scattered per secondinto a solid angle dΩ/(ΦdΩ)
Scattering cross sections
)'(''2
'
''
22
2
2
EEEEkVkm
k
k
dEd
d−+−
=Ω λλδλλ
πσ
h
In the Born approximation:
Incident neutron with wavevector k and state λScattered neutron with wavevector k’ and state λ’
Elastic nuclear scattering
Hr
In the Born approximation, the scattered intensity is given by:
The interaction between the neutron and the atomic nucleus is represented by the Fermi pseudo-potential, a scalar field.
=Ωd
dσ
Elastic magnetic scattering
Hr
kr
Cross sections
××∇−=−=
×∇==
×=
2
0
2
0
ˆ.
42.
ˆ
4)(
R
RsBV
AcurlAB
R
RRA
BNn
e
σπµ
µγµµ
µπµ
qdeqsqR
Rs Rqi rrr
r rr
.ˆˆ2
1ˆ
22 ∫ ××=
××∇
π
ri
r
R
ith electron
neutron
>< σσ k|V|'k'
:element matrix theevaluate toneed wecase, magnetic In the
m
The magnetic structure we will consider must have a moment distribution that can be expanded in Fourier series.
( ) jri
kj
j
j eSfpMrrrrrr ⋅∑= κπκκ 2
)(
Unit-cell magnetic structure factor:
∑ ⋅−=
k
Rki
kjljleSmrrrr π2
.
Magnetic interaction vector
κκκκrrrrrr
××= )()( MQ
Magnetic interaction vector:
)(κrr
M
κr
)(κrr
Q
The intensity of a magnetic Bragg peak I:
)()( * κκrrrr
QQI ⋅∝
Magnetic form factor
http://neutron.ornl.gov/~zhelud/useful/formfac/index.html
In the dipole approximation:
International Tables of Crystallography, Volume C,ed. by AJC Wilson, Kluwer Ac. Pub., 1998, p. 513
><−+>=< )()2
1()()( 20 Qjg
QjQf
Notations
magneton) B(2
electron theofmoment dipole magnetic
1.913
magneton)nuclear (2
neutron theofmoment dipole magnetic
n vectorinteractio Magnetic
factor structure Magnetic
vectorscattering
B
e
N
n
ohrm
e
m
e
)(M)(Q
)(M
e
p
h
h
r
rrrr
rr
r
=
=
=
≡ ⊥
µ
µγ
µ
µ
κκ
κ
κ
Visualize magnetic structure with FStudio
Laurent C. Chapon
ISIS Facility, Rutherford Appleton Laboratory, UK
J. Rodriguez-Carvajal
ILL, France
Ions with intrinsic magnetic moments
core
Ni2+
Atoms/ions with unpaired electrons
Intra-atomic electron correlation
Hund’s rule: maximum S/J
m = gJ J (rare earths)
m = gS S (transition metals)
What is a magnetic structure?Paramagnetic state:
Snapshot of magnetic moment configuration
Jij
S Sij ij i jE J=− ⋅
0Si =
What is a magnetic structure?
Ordered state: Anti-ferromagnetic
Small fluctuations (spin waves) of the static configuration
S Sij ij i jE J=− ⋅
Jij0Si ≠
Magnetic structure:
Quasi-static configuration of magnetic moments
Types of magnetic structuresFerro Antiferro
Very often magnetic structures are complex due to :
- competing exchange interactions (i.e. RKKY)
- geometrical frustration
- competition between exchange and single ion anisotropies
-……………………..
Types of magnetic structures
“Transverse”
“Longitudinal”
Amplitude-modulated or Spin-Density Waves
Types of magnetic structures
Spiral
Cycloid
Types of magnetic structures
Conical
Shubnikov magnetic groups, are limited
to:
- Commensurate magnetic structure.
- Real representation of dimension 1.
Position of atom j in unit-cell l
is given by:
Rlj=Rl+rj where Rl is a pure
lattice translation
Formalism of prop. Vector : Basics
Rl
rj
mlj
∑ −=k
k kRSm ljlj iexp π2
∗= jj kk- SS
Necessary condition for real mlj
cb acb arRR jjjjllj zyxlll +++++=+= 321
Formalism of prop. Vector : Basics
Rl
rj
mlj
Formalism of prop. Vector : Basics
A magnetic structure is fully described by:
- Wave-vector(s) k.
- Fourier components Skj for each magnetic atom j and wave-vector k.
Skj is a complex vector (6 components) !!!
- Phase for each magnetic atom j, ΦΦΦΦkj
2k k
k
m S kR Slj j l jexp iπ= − =∑• The magnetic structure may be described within the
crystallographic unit cell
• Magnetic symmetry: conventional crystallography plus
time reversal operator: crystallographic magnetic groups
Single propagation vector
k = (0,0,0)
( ) )(2
ln
jljlj -1iexp k
k
k SkRSm =−=∑ π
REAL Fourier coefficients ≡ magnetic moments
The magnetic symmetry may also be described using
crystallographic magnetic space groups
Single propagation vector
k=1/2 H
12
2k kS uj j j jm exp( i )π φ= −
- k interior of the Brillouin zone (pair k, -k)
- Real Sk, or imaginary component in the same direction as the real
one
2 2k -km S kR S kRlj j l j lexp( i ) exp( i )π π= − +
km u kRlj j j l jm cos2 ( )π φ= +
Fourier coef. of sinusoidal structures
12
2k kS u vj uj j vj j jm im exp( i )π φ = + −
- k interior of the Brillouin zone
- Real component of Sk perpendicular to the imaginary
component
k km u kR v kRlj uj j l j vj j l jm cos2 ( ) m sin2 ( )π φ π φ= + + +
Fourier coefficients of helical structures
Centred cells!
k=(1,0,0) or (0,1,0) !!!!!
Examples. Fstudio
LATTICE P
K 0.5 0.0 0.0
SYMM x,y,z
MSYM u,v,w,0.0
MATOM Ce1 CE 0.0 0.0 0.0
SKP 1 1 2.0 0.0 0.0 0.0 0.0 0.0 0.0
Type of lattice P, C, I, F…..
Propagation vector(s)
List of symmetry operators with associated magnetic
operator
Magnetic atom
Fourier coefficients and phase
www.ill.fr/dif