Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)INTERNATIONAL SCHOOL14th to 16th JUNE, NANCY (FRANCE)RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE
Theory of Raman SpectroscopyFernando Rull & Valentín García Baonza
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
MOLECULAR SYMMETRY CLASSIFICATION (POINT GROUP)
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
CHARACTER TABLES
C2V E C2 σσσσv (xz) σσσσ’v (yz)
A1 1 1 1 1
A2 1 1 -1 -1
B1 1 -1 1 -1
B2 1 -1 -1 1Irreducible Representation for B2
Point Group Symmetry Operations
Irreducible Representation (ΓΓΓΓi )
Order C2v (h) = 4 (1 E, 1 C2, 1 σv, 1 σ’ v)
Character: χχχχ
Classes
In vibrational spectroscopy we are very concerned with the symmetry of each normal
mode of vibration in a molecule.
Each normal mode of vibration will (behaves like) form a basis for an irreducible
representation of the point group of the molecule.
The transformation of all the atomic displacements (coordinates) will result in what
we shall call a Reducible Representation, which should be expressed as a sum of
Irreducible Representations (Symmetry Modes)
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
CHARACTER TABLES
Simmetry of Functions
C2V E C2 σv (xz) σ’v (yz)
A1 1 1 1 1 z x2,y2,z2
A2 1 1 -1 -1 Rz xy
B1 1 -1 1 -1 x, Ry xz
B2 1 -1 -1 1 y, Rx yz
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
C2vz
y
x
z
y
x
EC2
σv (xz)σ’v (yz)
≡
orbital pz
C2V E C2 σv (xz) σ’v (yz)
A1 1 1 1 1 z x2,y2,z2
A2 1 1 -1 -1 Rz xy
B1 1 -1 1 -1 x, Ry xz
B2 1 -1 -1 1 y, Rx yz
z
y
xEσv (xz)
z
y
x≡z
y
xC2
σ’v (yz)
z
y
x
orbital px
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
C2V E C2 σv (xz) σ’v (yz)
A1 1 1 1 1 z x2,y2,z2
A2 1 1 -1 -1 Rz xy
B1 1 -1 1 -1 x, Ry xz
B2 1 -1 -1 1 y, Rx yz
y
xRn
EC2
y
x
y
xσv (xz)σ’v (yz)
y
x
C2v
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
C2V E C2 σv (xz) σ’v (yz)
A1 1 1 1 1 z x2,y2,z2
A2 1 1 -1 -1 Rz xy
B1 1 -1 1 -1 x, Ry xz
B2 1 -1 -1 1 y, Rx yz
y
xdXY
EC2
y
x
y
xσv (xz)σ’v (yz)
y
x
C2v
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
ΓVIB = A1+E+2T2
Γ = A1+E+T1+3T2
νννν1(A1) νννν2(E)
νννν3(T2) νννν4(T2)
460 cm-1 218 cm-1
762 cm-1 314 cm-1
MOLECULAR VIBRATIONS
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
ΓVIB = A1+E+2T2
Γ = A1+E+T1+3T2
νννν1(A1) νννν2(E)
νννν3(T2) νννν4(T2)
460 cm-1 218 cm-1
762 cm-1 314 cm-1
Wavenumber (cm-1) Wavenumber (cm-1)
SAMPLEPOLARIZED
LIGHT
III
I⊥=ρ
MOLECULAR VIBRATIONS AND POLARIZABILITY
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
CCl235Cl2
37
CCl435
CCl335Cl37
ISOTOPIC CONTRIBUTIONS
νννν1(A1)
460 cm-1
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
Celestine (SrSO4)
Vibrations of SO4
2- group
-Symmetry group in the free state Td
(43m)
-Symmetry domain group Cs
(m)
-Crystallographic factor group D2h
(mmm)
CRYSTAL VIBRATIONS AND POLARIZABILITY
SAMPLEPOLARIZED
LIGHT
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
Celestine (SrSO4)
Vibrations of SO4
2- group
-Symmetry group in the free state Td
(43m)
-Symmetry domain group Cs
(m)
-Crystallographic factor group D2h
(mmm)
CRYSTAL VIBRATIONS AND POLARIZABILITY
SAMPLEPOLARIZED
LIGHT
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
SYMMETRY IN CRYSTALS
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
Celestine
(SrSO4)
CRYSTAL VIBRATIONS AND RAMAN ACTIVITY
Td (43m) Cs (m) D2h (mmm)
Vibrations of SO42- group
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
Celestine (SrSO4)
CRYSTAL VIBRATIONS AND RAMAN ACTIVITY
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
CRYSTAL VIBRATIONS AND RAMAN ACTIVITY
Celestine (SrSO4)
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
CRYSTAL VIBRATIONS AND RAMAN ACTIVITY
Celestine (SrSO4)
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
Phonon confinement
PHONONS IN DIAMOND: SIZE EFFECTS
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
PHONONS IN DIAMOND: PRESSURE AND TEMPERATURE EFFECTS
0 500 1000 1500 20001270
1280
1290
1300
1310
1320
1330
1340
This work: Anastassakis et al.Borer Cui et al.Herchen and Cappelli:Zoubolis and Grimsditch:Liu et al.
Intrinsic contributionThis work: Zoubolis and Grimsditch:
Diamond
ω (
cm-1)
T (K)
mass reduced :
c21
constant force :21
Energy Harmonic 2
µµπ
ω K
K
Kr
=
=
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
T
j
T
j
j
jVV
V
∂
∂−=
∂
∂−=
ln
lnνν
νγ
T
jT
T
j
j
Tj
pB
p
B
∂
∂=
∂
∂=
νν
νγ
ln
T
TV
pVB
∂∂−=
FreeMolecule
Molecular Solid
Molecular Solid at High Pressure
Internal Modes(HIGH Frequency)
ν (cm-1)
ν (cm-1)
ν (cm-1)
Davydov Splittings
External Modes(LOW Frequency)
Elementary vibrational model for Molecular Crystals
Chattopadhyay et al.(PRB, 1981)
GRÜNEISEN PARAMETERS AND DAVIDOV SPLITTINGS
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
Two level Perturbation ModelTwo level Perturbation Model
δ = ν+−ν−R= I+ / I-
W: Fermi Coupling Parameter
W = δ (R)1/2 / (R+1)
δ0 = [δ2 -4W2]1/2ν+
0 = ν+−1/2[δ−δ0]
ν−0 = ν−+1/2[δ−δ0]
Exact Degeneration:
δδδδ0 is zeroδδδδ is minimumR =1
2
2/10
2/10
2/10
2/10
2/10
2/10
)()(
)()(
+−−−++=
δδδδδδδδ
R
RR
FERMI RESONANCE
CO2
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
FERMI RESONANCE
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
04
04
04
044
03
03
03
033
02
02
02
022
01
01
01
011
CCBBAAFF
CCBBAAFF
CCBBAAFF
CCBBAAFF
cccc
cccc
cccc
cccc
φφφφφφφφ
φφφφφφφφ
+++=Φ
+++=Φ
+++=Φ
+++=Φ
0
)(
)(
)(
)(
0
0
0
0
=
−−
−−
EEWWW
WEEWW
WWEEW
WWWEE
CBCACFC
BCBABFB
ACABAFA
FCFBFAF
2800 2900 3000 3100
νOV2
νOV1
νAS
(CH3)
FR(ν−)
FR(ν+)
Inte
nsity
(ar
b. u
nits
)
P (GPa)
0.5
1.6
1.9
1.0
RP
Raman Shift (cm-1)
FERMI RESONANCE
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)INTERNATIONAL SCHOOL14th to 16th JUNE, NANCY (FRANCE)RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE
Theory of Raman SpectroscopyFernando Rull & Valentín García Baonza
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
Q3
Q2
Q1
NORMAL MODES IN WATER MOLECULE
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
x1
x3 x2
y1
z1
y3y2
z2z3
C2v E C2 σxz σyz
A1 1 1 1 1 z x2, y2, z2
A2 1 1 -1 -1 Rz xy
B1 1 -1 1 -1 x, Ry xz
B2 1 -1 -1 1 y, Rx yz
=
⋅
3
3
3
2
2
2
1
1
1
3
3
3
2
2
2
1
1
1
100000000
010000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
−−
−−
−−
=
⋅
−−
−−
−−
2
2
2
3
3
3
1
1
1
3
3
3
2
2
2
1
1
1
000100000
000010000
000001000
100000000
010000000
001000000
000000100
000000010
000000001
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
−
−
−
=
⋅
−
−
−
2
2
2
3
3
3
1
1
1
3
3
3
2
2
2
1
1
1
000100000
000010000
000001000
100000000
010000000
001000000
000000100
000000010
000000001
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
−
−
−
=
⋅
−
−
−
3
3
3
2
2
2
1
1
1
3
3
3
2
2
2
1
1
1
100000000
010000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
z
y
x
E, χ(E)=9 C 2, χ(C 2)=-1
σv , χ(σv) = 1 σ'v , χ(σ'v) = 3
Γred = (9)E + (-1)C2 + (1)σv + (3)σv’
Γred = 3 A1 + 1 A2 + 2 B1 + 3 B2
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
x1
x3 x2
y1
z1
y3y2
z2z3
C2v E C2 σxz σyz
A1 1 1 1 1 z x2, y2, z2
A2 1 1 -1 -1 Rz xy
B1 1 -1 1 -1 x, Ry xz
B2 1 -1 -1 1 y, Rx yz
x
z
y
R z
R x
R y
Γred = (9)E + (-1)C2 + (1)σv + (3)σv’
Γred = 3 A1 + 1 A2 + 2 B1 + 3 B2
Γtran = A1 + B1 + B2
Γrot = A2 + B1 + B2
ΓVib= 2 A1 + B2
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
x1
x3 x2
y1
z1
y3y2
z2z3
x
z
y
R z
R x
R y
Γred = (9)E + (-1)C2 + (1)σv + (3)σv’
Γred = 3 A1 + 1 A2 + 2 B1 + 3 B2
Γtran = A1 + B1 + B2
Γrot = A2 + B1 + B2
ΓVib= 2 A1 + B2
Theory of Raman Spectroscopy - Fernando Rull & Valentín García Baonza
INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)
RAMAN ACTIVITY
Simmetry of Functions
C2V E C2 σv (xz) σ’v (yz)
A1 1 1 1 1 z x2,y2,z2
A2 1 1 -1 -1 Rz xy
B1 1 -1 1 -1 x, Ry xz
B2 1 -1 -1 1 y, Rx yz
ΓVib= 2 A1 + B2
00uv ≠ΨΨ αi
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