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

13‐11‐2012

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

13‐11‐2012

3

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.

13‐11‐2012

4

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

13‐11‐2012

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

13‐11‐2012

6

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.

13‐11‐2012

7

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)

13‐11‐2012

8

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. 

13‐11‐2012

9

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

 

 

 

=⋅ ⋅ ⋅⋅ ⋅ ⋅

⋅ ⋅ ⋅⋅ ⋅ ⋅

⋅ ⋅ ⋅⋅ ⋅ ⋅

⋅ ⋅ ⋅⋅ ⋅ ⋅

⋅ ⋅ ⋅⋅ ⋅ ⋅

⋅ ⋅ ⋅⋅ ⋅ ⋅

 

 

 

13‐11‐2012

10

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

13‐11‐2012

11

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.

13‐11‐2012

12

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.

13‐11‐2012

13

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

13‐11‐2012

14

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

13‐11‐2012

15

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

13‐11‐2012

16

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.

13‐11‐2012

17

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

13‐11‐2012

18

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

13‐11‐2012

19

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

13‐11‐2012

20

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

13‐11‐2012

21

Datamining Molecular Information

Example: 12-level cluster analysis

Improving chemical resolution

END