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13‐11‐2012
1
Fundamentals of Raman spectroscopy
Part1
Cees Otto
Vibrational Spectroscopy
IR absorptionspectroscopy
Light scattering
StokesRaman scattering
anti‐StokesRaman scattering
1‐photoneffect
2‐photoneffect
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2
Nanometers, wavenumbers and relative wavenumbers
Example: 500 nm corresponds to 20000 cm-1
Example: A Raman band at 1020 cm‐1 and excited with a laser wavelength of 500 nm scatters light at a wavelength of 527 nm.
A vibration that absorbs light at 1020 cm‐1 absorbs light in the infrared at a wavelength of 9.8 µm (Check everything!)
Relative wavenumbers:
Absolute wavenumbers:
The harmonic oscillator
m1 m2
k
k
2
1
The frequency of the vibration depends on the “spring”Constant, k, and the reduced mass, µ, of the atoms involved in the motion.
The frequency is expressed in cm-1
1 cm-1 ~ 30 GigaHz
Motions of light atoms (C-H, N-H, O-H) have high frequencies: ~ 2700-4000 cm-1
Motions of heavy atoms (S-S- bridges) have low frequencies: < 600 cm-1
Motions involving C, N, O can be anywhere between 600 - 2700 cm-1
Tables relate chemical groups with vibration frequencies are available in the literature
21
111
mm
The Schrödinger equation for the harmonic oscillator describes atomic vibrations in molecules, which are also calledmolecular vibrations:
For a di‐atomic molecule:
scmccmHz /~ 1
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Raman spectroscopy
1: Band positions2: Band width 3: Amplitude of a band
(related to Raman cross section)
A comparison between cross sections
Electronic (UV-Vis) Absorption spectroscopy: 10-20 m2
Fluorescence spectroscopy: Q x 10-20 m2
Vibrational (IR) absorption spectroscopy: 10-23 m2
Resonance Raman spectroscopy: 10-29 m2
Non-resonant Raman spectroscopy: 10-33 m2
Surface Enhanced Raman Scattering: 10-? m2
Surface enhanced Raman scattering cross sections vary widely in literature reports.There seems to be consensus developing that estimates the SERS cross sectionsbetween 6 to 8 orders of magnitude larger than the “normal” non-resonant and resonant Raman cross sections.
So:Surface Enhanced Raman Scattering: 10-21 to 10-27 m2
Reported literature values are even as high as 10-16 m2. This requires further research.
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Fluorescence “cross section”
Fluorescence imaging
detdet EQA
PNn flmolabsfl
22010 mabs
A(rea): 2.4*10‐13 m2
Edet : 6 ‐ 13 %
molfl Nn 5det 10
Intensity: ~107 W/m2
Flux density: ~1026 1/(m2s)
exc = 457 nm.128 x 128 pixelsimage time : 8 secondspower: 5 microwatt
Raman cross section
Raman imaging
detdet 63 EA
PNmn molRR
.)(10 233 resnonmR
A(rea): 2.4*10‐13 m2
Edet : 6 ‐ 13 %
Intensity: ~1011 W/m2
Flux density: ~1030 1/(m2s)
molR Nmn 4det 1063
~ 10 photons/day.mol.vib
exc = 647 nm.64 x 64 pixelsimage time : 1800 secondspower: 60 miliwatt
Status:2003
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5
The harmonic oscillator
In non‐linear molecules with “n” atoms 3n‐6 vibrations exist.Each vibration has a distinct force constant and reduced mass and, therefore, frequency. Raman spectra contain information on the vibration frequencies. Chemical analysis is possible because the frequencies are unique for molecular groups.
The parabolic potentialfor a classical oscillation
021 hvEv
01 hEEE vv
The energy states of a QM harmonic oscillator form a “ladder”
Transitions can only occurbetween consecutivestates: 1
From Atkins
Band positions
The wave functions:
021 hvEv
k
2
10
01 hEEE vv The energy difference between adjacent levels is:
The eigenvalues:
xkF Hooke’s law: with “k”, the force constant.“x” is the displacement from the equilibrium position
The relation between “force” and “potential energy” is:x
VF
Follows: 221 xkV
The S. eq. for the harmonic oscillator becomes:
xExxkx
x
2
21
2
22
2
.....,3,2,1,02
2
2
ve
xHNx
x
vvv
41
2
km
a parabolic potential
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The wavefunctions of the harmonic oscillator
.....,3,2,1,02
2
2
ve
xHNx
x
vvv
with Hv the Hermite polynomials (See page 340, table 12.1).α is proportional to the steepness of the parabolic function, meaninglarge α shallow potential well, small α steep potential well.
41
2
km
The normalization constant Nv is:
21
21
!2
1
vN
vv Fig. 8.19 The graph of the Gaussian function
Fig. 8.20 the wavefunction and It’s square for the v=0
Fig. 8.21 the wavefunction and It’s square for the v=1
From Atkins From Atkins
Hermite polynomials
.....,3,2,1,02
2
2
ve
xHNx
x
vvv
The wave functions for the harmonic oscillator look as follows:
The polynomial functions are a power series of the argument and take the following form (with y=x/α):
x
Hv
512016032
4124816
3128
224
12
01
35
24
3
2
vyyyyH
vyyyH
vyyyH
vyyH
vyyH
vyH
v
v
v
v
v
v
See page 302-306 for more details.
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Some properties of the harmonic oscillator
km
vx
xv
212
0
x=0 is the equilibrium position of two nuclei connected with a “spring”.The expectation value for x and x2, respectively and is:x 2x
Interpretation:The average value of the position is equal to the equilibrium constant, so the oscillator is equally likely to be found on either side of the equilibrium position.The mean square displacement is proportionalto the vibrational quantum number.
Fig. 8.23 p. 304The probability distribution for the first 5 states of a harmonic oscillator and the state v=20. Note how the regions of highest probability move towards the turning points of the classical motion as “v” increases.
From Atkins
Molecular vibrations in DNA measured with Raman spectroscopy
Poly(dG-dC).poly(dG-dC): C) Z-form: Watson Crick LH-helixB) B-form: Watson-Crick RH-helixA) Hoogsteen basepairing at pH=4.3
400 800 1200 1600
596
118
0
C 1579
1536
1483
1426
1356
1211
1245
1290
13
16
1264
810
793
1093
782
685
625
Ram
an in
tens
ity
wavenumber / cm-1
597
1177
B
1317
1240
1577
1530
1489
1420
1362
1334
1218
129312
59
1093
830
783
681
1181
A
1578
1538
1488
1424
1359
1328
1259
1094
824
786
675
616
Changes in physico-chemicalParameters (pH, T, etc.) givesrise to profound changes in the Raman spectrum.
G.M.J. Segers‐Nolten, N.M. Sijtsema, C. Otto, Biochemistry 36(43), 13241‐13247 (1997)
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Band positions
xkxV
F
i
ii
k
21
For 3n‐6 vibrations
The force constant, ki, are obtained from :2
2
xV
k
Substituting Hooke’s law in the Newton equation:
Hooke’s law
xktx
2
2
And solving the equation of motion results in:
tAtx i2sin withi
ii
k
21
How does this work out for N atoms?
Band Positions
xy
z
xy
z
xy
z
The dynamic model for molecular vibrations may start from N individual atoms. Each of these atoms has a (x,y,z) Cartesian coordinate system attached. Displacement of each atom along any of its coordinates may affect other atoms
along their (x,y,z) coordinates through the spring constant. For a three‐atomic molecule nine (9) degrees of freedom exist. Three degrees of freedom are connected to translations of the centerof mass of the molecule (translationsof the molecule),Three degrees of freedom are connectedto the rotation of the molecule around the center of mass (rotations of the molecule).
3N‐6 degrees of freedom are connected to internal motion of atoms with respect to the center of mass. These are the molecular vibrations. In a linear molecule 3N‐5 vibrations exist.The description of vibrations in cartesian coordinates of the atoms is not chemicallyintuitive. A coordinate systems based on internal coordinates is more suited.
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Band positionsLet us first look at a description in Cartesian coordinates. First substitute the solution to the Newton equation back into it. We obtain:
N atoms have 3N degrees of freedom and 3N equations of motionresult. The motion of each atom in the x‐, y‐ and z‐direction is drivenback to equilibrium by a spring constant and by coupling, via springconstants, to each of the other degrees of freedom. For a three‐atomicmolecule we obtain 9 coupled equations of motion:
xkx 224
333
333
333
232
232
232
131
131
131
3322
313
313
313
212
212
212
111
111
111
1122
313
313
313
212
212
212
111
111
111
1122
313
313
313
212
212
212
111
111
111
1122
4
4
4
4
zkykxkzkykxkzkykxkz
zkykxkzkykxkzkykxkz
zkykxkzkykxkzkykxky
zkykxkzkykxkzkykxkx
zzzyzxzzzyzxzzzyzx
zzzyzxzzzyzxzzzyzx
yzyyyxyzyyyxyzyyyx
xzxyxxxzxyxxxzxyxx
Band positionsThe notation is much more transparent in matrix form using mass‐weighted coordinates and force constants:
The equations become:
21
1212~
mm
kk xy
xy iii xmx ~
4
=⋅ ⋅ ⋅⋅ ⋅ ⋅
⋅ ⋅ ⋅⋅ ⋅ ⋅
⋅ ⋅ ⋅⋅ ⋅ ⋅
⋅ ⋅ ⋅⋅ ⋅ ⋅
⋅ ⋅ ⋅⋅ ⋅ ⋅
⋅ ⋅ ⋅⋅ ⋅ ⋅
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Band positions
ξ
Introducing the following shorthand notation for the Cartesian coordinates of the atoms:
the equation becomes:
This is an eigenvalue equation and can be solved by standard means once one knows the matrix of forceconstants.
The solution contains the 3N frequencies associated with the 3N degrees of freedom.The 6 degrees of freedom associated with translations and rotations will have a frequencyclose to “ zero”. The frequencies of the 3N‐6 (or 3N‐5 for a linear molecule) will be different from “zero” and can be associated with the frequencies of internal modes of freedom, the molecular vibartions, of a molecule.The amplitude of each of the atoms in each of the vibrations can also be obtained.This is a classical model for the nuclei.Real nuclei are not “point‐like” but need to be described by a vibrational wave function,which reflects the probability to find an atom at a certain position. The classical approachgives very reasonable results for the frequencies in spite of this approximation.
k 224
Internal CoordinatesInternal coordinates are defined according to “common sense” or chemical intuition. Severaltypes of internal coordinates can be defined. In the three‐atomic molecule below, two typesoccur, namely bond stretch motions and a bending motion. Two stretch motions can be
distinguished, one for each bond between the atoms. The angle between the two OH‐groups varies around the equilibrium value θ. This angle defines a bending coordinate.
For more complicated molecules other internal coordinates can be convenientlydefined as well. The most common ones arelisted below:
• Bond stretch coordinates• In plane Bending coordinates• Angle between a bond and a plane defined by two bonds• Torsion coordinate
Other coordinates may be defined whenever a molecule “suggests” this.
θ
S1
S1 S2
S2
O
H H
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Normal coordinates
θ
S1
S1 S2
S2
The internal coordinates do not form a symmetry optimized set of motions for the nuclei. For instance in the example below for “water”, it is hard to imagine that the oxygen atom willsimultaneously move to the left and to the right. The symmetry optimized motions of the
of the nuclei are the normal coordinates.One normal coordinate is associated with each normal mode of motion.
The normal modes form an orthogonal system of motions. They are linear combinations of the internal coordinates (and also of the Cartesian coordinates).
Properties of Normal Modes are:• Each normal normal mode behaves like a harmonic oscillator• A normal mode is a concerted motion of many atoms• A normal mode does not translate or rotate the molecule• All atoms involved in a normal mode pass simultaneously
through the equilibrium position• Normal modes are independent, that is they form an
orthonormal set, that is they do not interact.
Example of Normal ModesNormal modes can be constructed from the internal coordinates by forming symmetry‐adapted linear combinations. For a three‐atomic system this is rather straight forward, asfollows:
θ
The length of the arrows represent the amplitude of the motion of the atom in the normal mode.
In the drawings the amplitude is greatly exaggerated. The amplitude of atomic vibrations in molecules is typically of the order of 1% of the inter‐atomic distance. The inter‐atomic distance is typically 0.1 nm.
The amplitude of the vibrations is therefore typically 1 pm (1 picometer). Light atoms, like H,have somewhat larger amplitudes.
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Raman spectrum of water
OH
sym
met
ric
stre
tch
OH
ant
i-sy
mm
etri
cst
retc
h
OH
ben
dθ
Raman spectra of different compounds
OH
sym
met
ric
stre
tch
OH
ant
i-sy
mm
etri
cst
retc
h
OH
ben
d
Raman spectra of different compounds are different. A compound can be determined from the measured Raman spectrum. Raman spectroscopy is an analytical chemical technique!!In mixtures of compounds the Raman spectra are partially overlapping. Complicated Raman spectra result. The spectrum of the mixture can be viewed as a “pattern”. Raman pattern recognition is an important approach for medical applications concerning cell and tissue Raman spectroscopy. Multivariate analysis methods provide powerful tools to perform pattern recognition in Raman spectra.
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Raman cross section
The previous “classical” introduction explains qualitatively all aspects but does not explain quantitative aspects of Raman scattering.
A quantum-mechanical derivation of the molecular polarizability gives the right treatment.
We will not do the derivation, but the result of the treatment provides theexpression for the Raman polarizability for each normal mode (and for the polarizability in general).
The formula explain also intensities of light scattering for normal modes
and is helpful to understand the difference in non-electronic resonanceRaman scattering and electronic resonance Raman scattering.
The Raman scattering cross section
dd
d
A
PNn R
L
LRaman
2
420
4
4Raman
L
RvibLR
cd
d
We had: detdet 63 EA
PNmn molRR
This formula recognizes through the integral over the angular dependence that the scattering is not equal in all directions.
The angle-dependent scattering cross section is:
For one (1) vibration
The (dipolar) scattering depends on direction and this becomes:
For (3m-6)vibrations
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27
The Raman polarizability
j jfRjf
RL
jiLji
LRRaman i
ijjf
i
ijjf
Understand / Explain the formula!
Detuning
Detuning
Lji
Rjf if
j
Larger contribution Smaller contribution
28
The resonance Raman polarizability
j jfRjf
RL
jiLji
LRRaman i
ijjf
i
ijjf
The resonance Raman polarizability becomes:
Very largedetuning
j jiLji
LRRaman i
ijjf
~ 0 detuning
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The Raman spectrum of a poly‐atomic molecule: toluene
39 fundamental modes are Raman active. They are all observed.
Harmonic Oscillator selection rule: v ∆ 1
The potential energy is not parabolic
221 xkV A bond that can
be dissociated must bedescribed by a differentpotential energy.
The Morse potential energyis a useful description
The harmonic potentialis an approximation to the actual potential energy
The vibrationalenergy levels are progressively closerwhen the disscociationis approached
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Overtones and combination bands in a poly‐atomic molecule: toluene
Actually, approx. 112 modes have been observed so far.
Many more modes than 39 fundamental modes are observed.
The intermediate region improved
All these modes are overtones or combination bands.
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Raman spectroscopy
Raman spectroscopy to understand matter by probing electrons and
vibrations
Raman spectroscopy as a contrast method in
microscopy
Raman spectroscopy to determine molecular number densities
Material Science Label‐free microscopy Label‐free sensing
Raman Chemical Analysis
Raman cross section
Molecular vibrations for chemical identification
Concentration molecules of interest
Optical engineering of the optimal illumination/detection strategy
lI
LCmsn RR
263]/[#
Local field enhancement
Specificity
Sensitivity Sensitivity
Detectivity
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Hyperspectral Confocal Raman Microscopy
n
NAarcsin
2
2arcsin4
Confocal effect:Reduction of out-of-focus light Sharper imaging
High NA:Higher resolutionLarge collection efficiency
Measurement Modes
Single point measurement Confocal Raman imaging
Single spectrum N spectraN=1024, 4096, ….
Single spectrum
Rapid Scanning mode
Not representative for the whole cell Progresses to less
time consuming
Representative and fast
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Confocal Raman measurement procedure
Schematic view of the raster scanning for imaging of the cells
Data obtained :Each frame = 32 x 32 pixEach pix = 1600 data points
=> 1.64 million data points
M
N
Hyper spectral data cube
S.No Peak position (cm)-1 Peak Assignment123456789
1011121314***+
788853938
10041032109411261254130413391451166010452335590960
1070
Nucleus: O-P-O symmetric stretchTyrosine
Protein: α - helixPhenylalaninePhenylalanine
Nucleus: O-P-O symmetric stretchProtein: C-N stretch
AdenineAdenineAdenine
C-H deformationProtein: Amide 1
Collagen (Proline)Nitrogen (atmosphere)
Phosphate PO43-
Phosphate PO43-
Carbonate CO32-
Table : Raman peak assignments
Rich chemical information from cell with light
Univariate imaging
20406080100120
20
40
60
80
100
120
-20
0
20
40
60
80
100
120
140
20406080100120
20
40
60
80
100
120
-100
-50
0
50
100
150
200
250
300
20406080100120
20
40
60
80
100
120
0
50
100
150
200
250
300
20406080100120
20
40
60
80
100
120
0
200
400
600
800
1000
1200
788
cm-1
DN
A
1004
cm
-1P
heny
lala
nine
1092
cm
-1N
ucle
us
1452
cm
-1P
rote
ins
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Multivariate imaging
White light micrograph of PBL cell
Cluster level 2 Cluster level 3 Cluster level 4 Cluster level 5
600 800 1000 1200 1400 1600 1800
0
5
10
15
20
Inte
nsity
(a.
u.)
Raman Shift (cm-1)
Univariate and Multivariate Raman Imaging of Bovine Chondrocytes
White light microscopy
788 cm-1
1094 cm-1
1656 cm-1
1004 cm-1
1602 cm-1
1745 cm-1
9-level hierarchicalcluster analysis
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Datamining Molecular Information
Example: 12-level cluster analysis
Improving chemical resolution
END
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