Pedestrian Approach to Raman Scattering
Optics Group Meeting _April 2004
S. Guha
• Classical and Semiclassical Approach
(including resonance Raman scattering)
• Raman Microscopy
• Raman Imaging/mapping
• Applications (stresses in semiconductors, crystallographic orientation,
chemical composition, mineralogy, biological applications, biomedical applications,
art, jewelry, and forensic science)
• Surface-Enhanced Raman Scattering (SERS)
Later in the semester (or summer)
© S. Guha
Scattering
Liquid
Observer
Sunlight
(white)
Violet Filter
Green Filter
Raman Scattering-inelastic process
1 in 108 photons get inelastically scattered
Excitations can
be plasmons,
polaritons,
magnons,
phonons
(thermal
vibrations).
Invention of Laser 1960
(Townes & Schawlow)
CCD camera 1969
-in market since 1983
Holographic* early-mid 90’s
Notch filters
High power diode lasers
Impact on Raman spectroscopy
Inspector
Raman: fits
in your
hand!!*Lippmann discovered in 1891 and was
awarded the Nobel prize. These reflect 90%
of the laser excitation and transmits 90% of
the scattered light.
Quantum mechanically.........
Phonon is emitted
into the lattice
Phonon is absorbed from
the lattice
hωf
hωL
Rayleigh
Scattering
Stokes
Scattering
Anti-Stokes
Scattering
hωf
hωsc hωL hωsc hωschωL
Phenomenological introduction to Raman scattering
qi (incident light)
Eie-iωit
Sample
Ese-iωst
qs (scattered light)
∑ −=
β
ω
βαβα ω .),()( tiL LeEuPtM r
Incident light induces a dipole moment----
Polarizability of the medium
(electronic)
The incident frequency
(visible photons) is 2 to
3 orders of magnitude
higher than the
phonon frequencies,
hence the massive
atoms do not directly
respond to light!! The
induced dipole
moment is electronic in
origin.
Pαβ will depend upon the incident
frequency ωL. Expand Pαβ above the
equilibrium configuration u=0:
(1)
∑ +
∂
∂+=
γ
αβ
αβαβ γω
l
tlu
u
P
PuP ......);()0(),(
0
r
u(l,α) is the displacement of the lth atom in γth direction. For
simplicity assume only the fth normal mode is excited, then
( ) == tfltlu fωγχγ cos)|(; ( ) ).)(|(21; titi ff eefltlu ωωγχγ −+=
Amplitude
( ) ........)|(
)(2
1)0(),(
0
++
∂
∂+= −∑ titi
l
ff eefl
lu
P
PuP ωω
γ
αβ
αβαβ γχγω
r
(2)
(3)
( )
........
)|(
)(2
1)0()( ((
0
++
∂
∂+= −−+−− ∑∑∑ titiL
l
ti
L
fLfLL eeEfl
lu
P
eEPtM ωωωωβ
β γ
αβ
β
ω
βαβα γχγ
(4)
( ) ........)|(
)(2
1
)0()(
((
0
++
∂
∂
+=
−−+−
−
∑∑
∑
titi
L
l
ti
L
fLfL
L
eeEfl
lu
P
eEPtM
ωωωω
β
β γ
αβ
β
ω
βαβα
γχγ
Origin of inelastic scattering-----
Rayleigh
Anti Stokes Stokes
(5)
Higher-order terms in Equation (5) gives rise to higher-order
Raman scattering, where two or more phonons are involved.
∂Pαβ/∂u(lγ)|0 transforms as a third rank tensor under operation;
this leads to the result that under inversion symmetry
∂Pαβ/∂u(0γ)|0=0 at an inversion site.
For Raman scattering to occur, polarizability of the molecule
must change during a vibration.
Selection rules for Raman spectra
+
Polarization of a diatomic
molecule in an electric field
+δ
-δ
–
hν Mx
My
Mz
Pxx Pxy Pxz
Pyx Pyy Pyz
Pzx Pzy Pzz
Ex
Ey
Ez
=
Polarizability tensor
O
O
O C OC O O C O
O C OC O O C O
O C OO
C
O O C O
+q q = 0 -q
ν1- Raman
active
ν2
ν3
P/1
Plot
Infrared absorption and Raman scattering
0 50 100 150 200 250 300
Frequency (cm-1)
IR
Raman
-300 -200 -100 0 100 200 300
Raman shift (cm-1)
)exp(
TkI
I
Bstokes
antiStokes ωh−=
Energy units for optical spectroscopy
hω=hck=2πhcλ
k ≡ wave vector; 1/λ = ν/c = wavenumber
1/λ ∝ energy
1 eV = 8067.5 cm-1
30 GHz = 1 cm-1
Typical energies: 100 -10,000 cm-1 (1012-1014 Hz)
Conservation rules for a Raman process
fSL
fSL
kkk
r
h
r
h
r
h
hhh
+=
+= ωωω
Gkk f
rrr =+∆
where G is the reciprocal lattice
vector and Ls kkk
rrr −=∆
Ls
LfLs
kk
rr =∴
≈+= ωωωω
15103.1
4880
22 −×≈°== cmAk LL
π
λ
πr
For visible light
181022 −×≈= cm
a
G πr
1510,0 −== cmkandG f
rrHence,
Raman
Brillouin
(cm-1)
qphonon
ωf
π
0.5 nm
optic
acoustical
Silicon and GaAs
Selection Rules
In centrosymmetric crystals, the vibrational modes must have
either even (gerade) or odd parity (ungerade) under inversion.
Odd parity modes: IR active
Even parity modes: Raman active
Diamond structure with
inversion symmetry.
Noncentrosymmetric
zinc blende structure
Phonon dispersion curves in Si and GaAs
LO: LONGITUDINAL OPTICAL
PHONON
TO: TRANSVERSE OPTICAL PHONON
LA: LONGITUDINAL ACOUSTIC
PHONON
TA: TRANSVERSE ACOUSTIC
PHONON
TO: Raman
active (not IR
active)
TO modes:
Both Raman
and IR active
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C60
Raman
IR
What determines Raman intensities?
For a classical dipole moment M(t) located at the origin and oscillating at
frequency ω, the magnitude of the Poynting vector associated with
radiation polarized along a unit vector n at position R is given by
∑=
αβ
βαβαπ
ω )()(
4 32
4
tMtMnn
cR
S
∑=
αβ
βαβαπ
ω )()(
4 32
4
tMtMnn
cR
S
Average of S is (Use Eq. (1)):
[ ][ ]
)(
)()(
**
**
βλαγ
γλ
βλαγλγ
ω
βλ
ω
βλ
γλ
ω
αγ
ω
αγλγβα
PPPPEE
ePePePePEEtMtM
LL
titititi
LL
∑
∑
+=
++= −−
System is at the origin,
subjected to ω; for
short time intervals
the system’s config. u
is fixed, and just the
electronic system
responds.
(6)
(7)
( )∑ +=
αβγλ
λγβλαγβλαγβαπ
ω
LL EEPPPPnncR
S **32
4
4
For the radiated power per unit solid angle, one needs to multiply
the above equation by R2dΩ
( )∑=
αβγλ
λγβλαγβα ωωπ
ω
LL EEuPuPnnc
uI ),(),(
2
)( *3
4
(8)
In Raman experiments we watch for times long compared to
vibrational period, so one must take a thermal average of the
above expression over the vibrational levels:
,),(),(
2
)( *3
4 ∑=
αβγλ
λγβλαγβα ωωπ
ω
LL EEuPuPnnc
uI (9)
.|),(|''|),(|1
'
* νωννων βλ
νν
αγ
β ν uPuPe
Z
E
vib
∑ −=
TkB/1=β
The Raman tensor is defined as……
)(|),(|''|),(|1)( '
'
*
ννβλ
νν
αγ
β
αβγλ ωωδνωννωνω ν −= ∑ − uPuPeZi Evib
Zvib = partition functionα,β = for incident light and γ,λ=for scattered light
ν= initial vibrational state
ν’=final vibrational state
(10)
∫ ∞+∞− −−=− dte it )(' '21)( ννωωνν πωωδ
Use:
∫ ∑∞+∞− −−=
'
* |),(|''|),(|1
2
1)( '
νν
βλ
ω
αγ
βω
αβγλ νωννωνπω
ννν uPeuPe
Z
dtei itE
vib
ti
(11)
A thermal average in the above equation means that the system is
free to vibrate
For any vibrational state----
,νν νEH =
.// νν ν hh tiEiHt ee =
(12)
Using the Heisenberg representation of the operator
hh //)( iHtiHt AeetA −=
∫ ∞+∞− −= νωννωνπω βλαγωαβγλ |),(|''|),,(|21)( * uPtuPdtei ti (13)
Eq. (13) is the common starting point for Raman calculations. It
expresses the Raman scattering tensor as the Fourier transform of the
time correlation function.
Transform to normal coordinates
∑
=
=
N
f
fdfu
3
1
.)(χr (14)
The Hamiltonian here is a harmonic oscillator problem---
f
f
ff ddH
222
2
1∑ += ω& (15)
∫ ∑
∑
∞+
∞−
−
+++
=
')(
)()0()0()0()0(
2
1)(
,
'
*
,
*
,,
**
fff
ff
f
f
ffff
ti
dtdPP
tdPPdPPPP
dtei
βλαγ
βλαγβλαγβλαγ
ω
αβγλ πω
Using the normal coordinates----
(16)
Rayleigh term
The vibrational states (phonon states) are given by
{ }
{ }'
321 ......
i
iN
n
nnnn
=′
==
ν
ν (17)
0
!
)(
00
1
11
f
n
f
f
f
ffff
ffff
n
a
n
a
nnna
nnna
+
+
=
=
−=
++=
Use the creation, annihilation
operators--- (18)
( )( )[ ] .1
2
)( ,
*
, dteenen
PP
i ti
f
ti
f
ti
f
f
ff ff ωωωβλαγ
αβγλ ωω
−
+∞
∞−
−∫∑ ++= h
One obtains----
Using the definition of the Dirac delta function..
( )( )[ ].)()(1
2
)( ,
*
,∑ −+++=
f
ffff
f
ff nn
PP
i ωωδωωδωω
βλαγ
αβγλ h
The Raman tensor is……
(19)
Stokes Anti Stokes
The intensity of Raman scattering per unit solid angle due to a
transition from a vibrational state ν to ν′
.)(
2
)( *
2,1
3
3
λγαβγλ
αβ γλ
βα ωηηπ
ωωω EEi
c
I
k
kkSL
S ∑∑∑
=
= (20)
Summation over k denotes the two
mutually perpendicular unit vectors
(perp. to the direction of scattering)
Fourier
component of
the electric
field
1
1)exp(
−
−≡
Tk
n
B
f
f
ωh
100 200 300 400 500 600 700
1000
2000
3000
4000
60000
70000
80000
λ= 785 nm
I
n
t
e
n
s
i
t
y
Raman Shift (cm-1)
λ= 514.5 nm
CdS nanocrystal
-1400 -1200 -1000 -800 -600 -400 -200 0
1000
1500
2000
2500
3000
3500
Y
A
x
i
s
T
i
t
l
e
Raman Shift (cm-1)
Anti Stokes
λ =514.5 nm
CdS nanocrystal
2nd
order
200 300 400 500 600 700
8000
10000
12000
14000
16000
18000
I
n
t
e
n
s
i
t
y
Raman Shift (cm
-1
)
CdS polycrystalline
λ =785 nm
April 04
Cds polycrystalline
sample grown using
pulsed laser deposition
Incident λ =514.5 nm
Confocal micro-
Raman system
....);(
2
1
);()0(),(
2
0
2
2
0
∑
∑
+
∂
∂+
+
∂
∂+=
γ
αβ
γ
αβ
αβαβ
γ
γω
l
l
tlu
u
P
tlu
u
P
PuP r
Hyper-Raman Effect (nonlinear Raman spectroscopy)
EM P=Earlier we used only
.....
6
1
2
1 32 +++= EEEM γβP
First and second
hyper-polarizability
CW lasers: E ~104 Vcm-1
(α>>β>>γ)
Pulsed Nd-YAG: E ~109 Vcm-1
When a sample is illuminated with a giant pulse of freq. of ωL,
the scattered radiation contains:
2 ωL(hyper-Rayleigh)
2 ωL± ωf (Stokes and anti-Stokes hyper-Raman)
Selection rules are relaxed; mode
is hyper-Raman active if the
components of the hyper-
polarizability tensor changes.
Normal Raman: 10-8 photons scattered
Hyper-Raman: 10-12 photons scattered
Examples of in vivo skin Raman spectra obtained from
various body locations of a healthy volunteer. (A) palm of
the hand; (B) fingernail; (C) surface of the forearm;
(D) volar aspect of the forearm. (Ex=785 nm)
Peak position
(cm-1)
Protein
Assignments
Lipid
Assignments Others
1745 ν(C=O)
1655 ν(C=O) Amide I
1445 δ(CH2), δ(CH3) δ(CH2) scissoring
1301 δ(CH2) twisting,wagging
1269 ν(CN),δ(NH) Amide III
1080 ν(CC) skeletal ν(CC),νs(PO2-)
1030 ν(CC) skeletal nucleic acids
1002 ν(CC) Phenyl ring
938 νCC) proline,valine
855 δ(CCH) aromatic,olefinic
polysaccar
ide
822 δ(CCH) aliphatic
Near IR Raman spectroscopy for in vivo skin measurement
From: Z. Huang et al. BC cancer Research center
Microscopic Theory of Raman Scattering
Quantum mechanical Hamiltonian of the coupled radiation field
and the scattering medium is:
electrons
ipeppeiie HHHHHHH +++++=
ions photonselectron-
ion
electron-
photon
ion-
photon
Treated exactly within
the adiabatic approx. negligible
( )( )++ +=∑ kkkk
k
p aaaakH ω2
1 Photon field is treated
exactly
Hep between the mixed electronic vibrational states is
treated by perturbation theory.
Outline: How to tackle Hep?
• quantize the radiation field----leads to a vector potential
and electric-field operators
•include the radiation field in the electronic Hamiltonian.
•alternately one can express the electron-radiation in
terms of electric field and magnetic field operators.
•use either the A.p approach or the M.E approach
)( jjj rAepp
rrr +→
For periodic systems with extended states it is convenient to use
the A.p formalism.
Using Fermi’s golden rule, the transition rate is:
Γ = − + −∑2
2π
ε δh h
c b b a
ia bb
a b
| | | |
( ),
H H
E E
E Eint int
'1 kk nnfc −=
0'== kk nniaInitial state
Initial
matter state
Initial
photon state
Final
photon state
Final State
( )
EE
EE
M.EM.E
+
EE
M.EM.E
iLfS
Si
Lik
kSL
if
if
n
V
S
−−++−
−−
=Γ ∑∑
ωωδω
ωωω
ππ
hhh
ll
h
llh
h
l
l l
2
2
|
|22
(22)
(23)
Scattering cross section
ratio of the rate at which energy is removed from the incident beam
by the scattered beam to the rate at which energy in the beam
crosses a unit area perpendicular to its propagation direction.
σ ωω≡ =
h
h
k
k k k
c n
V
V
n c
Γ Γ ;
Spectral differential cross section:
Ωdd
d
Sω
σ2
is the rate of removal of energy from the incident beam as a result of
its scattering in volume into a solid angle element dΩ with a scattered
frequency between ωs and ωs+dωs, divided by the product of dΩdωs
with the incident beam intensity
R. Loudon, “Quantum Theory of Light” (1983)
(24)
Scattering cross section cont.
( ) ( ) .22 3
3
3
2
3 Ω=Ω→ ∫ ∫∑ ∫ ∫ ddcVddkkV SSk SSS ω
ω
ππ
Convert the summation to an integral---
( )
( ).||||+
||||
AV
2
'
'
42
3
'
2
if
Si
Li
i
SL
S
iMMf
iMMf
cdd
d
ωωδωω
ωω
ωω
ω
σ
ηη
ηη
ηη
−+
−=
Ω ∑
l
l l
ll
ll
h
Kramers-Heisenberg formula
Resonant termAnti-resonant term
(25)
Electric-dipole
-incident pol.
Electric-dipole
-scattered pol.
Resonant term:
∑ −−′l l
ll
γωω
ηη
i
iMMf
Li
||||
= Res
Expand the states as a product of the electronic and a vibrational function.
BA= Res +
For first-order Raman scattering:
A u u= − −∑ ∑M M gv ev ev gvigee eg ev gv Levη η ω ω γ' ' ,'( ) ( )
' ' | ' ' |
,0 0
B u u
u
u u
u
u
u
= − −
+ − −
∑∑ [ ( ) ( )| ' ' | ' ' | |
( )| ( )
' ' | | ' ' |
].
' '
'
' ,'
'
'
' ,
M M
gv ev ev gv
i
M M
gv ev ev gv
i
ge eg
ev gv Leve
ge eg
ev gv L
η η
η η
ω ω γ
ω ω γ
0
0
0
0
FRANCK-CONDON
HERZBERG-TELLER
(more imp. in solids with
extended wavefunctions)
0|''|'' =gvevevgv
FRANCK CONDON OVERLAP
"ν
ν
ν
'ν
0|''|'' ≠gvevevgv
A-term = 0
A-term: non-zero
(totally symmetric
modes)
Resonance Raman profile---K6C60
0
100
200
300
400
λ=775 nm
I
n
t
e
n
s
i
t
y
Raman Shift (cm
-1
)
500 cm-1
0
100
200
300
400
I
n
t
e
n
s
i
t
y
Raman Shift (cm-1)
500 cm
-1
λ=660 nm
0
100
200
300
400
λ=560 nm
I
n
t
e
n
s
i
t
y
Raman Shift (cm-1)
500 cm-1
Resonance Raman profiles---K6C60
F-C term:
( )
22
2
11
1
Γ−−+Γ−−∝ iEiE ω
β
ω
βωα hh
β1 and β1 characterize the overlap
of the vibrational wave functions.