Theory of Raman Spectroscopygeoraman2014.wustl.edu/previous/2012/georaman10.uhp-nancy.fr... ·...

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


INTERNATIONAL SCHOOL - RAMAN SPECTROSCOPY APPLIED TO EARTH SCIENCES AND CULTURAL HERITAGE 14th to 16th JUNE, NANCY (FRANCE)

MOLECULAR SYMMETRY CLASSIFICATION (POINT GROUP)



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)



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



C2vz

y

x

z

y

x

EC2

σv (xz)σ’v (yz)

≡

orbital pz


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




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




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



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



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



CCl235Cl2

37

CCl435

CCl335Cl37

ISOTOPIC CONTRIBUTIONS

νννν1(A1)

460 cm-1



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



SYMMETRY IN CRYSTALS



Celestine

(SrSO4)

CRYSTAL VIBRATIONS AND RAMAN ACTIVITY

Td (43m) Cs (m) D2h (mmm)

Vibrations of SO42- group



Celestine (SrSO4)




Phonon confinement

PHONONS IN DIAMOND: SIZE EFFECTS



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

=

=



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



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



FERMI RESONANCE



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


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



Q3

Q2

Q1

NORMAL MODES IN WATER MOLECULE



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



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



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



RAMAN ACTIVITY

Simmetry of Functions


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



PROPIEDADES DE SÓLIDOS CRISTALINOS: INTERPRETACIÓN

RAMAN ACTIVITY IN CRISTALS

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