# Raman spectroscopic evidence for anharmonic phonon lifetimes and blueshifts in 1D structures

TABLE OF CONTENTS

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TITLE PAGE................................................................................................................... i

ABSTRACT...................................................................................................................... iii

DEDICATION................................................................................................................ iv

ACKNOWLEDGEMENTS......................................................................................... v

LIST OF TABLES........................................................................................................... viii

LIST OF FIGURES........................................................................................................ ix

CHAPTER

1. ANHARMONICITY IN MATERIALS................................................... 1

Introduction............................................................................................ 1 Experimental observations of anharmonicity.................................... 5

2. RAMAN SPECTROSCOPY...................................................................... 13

Introduction............................................................................................ 13 Quantum Theory of Raman Scattering............................................... 19 Selection Rules................................................................................. 25 Raman Instrumentation......................................................................... 26 Renishaw 1000 Raman Microscope.............................................. 27 Triax 550 Raman Spectrometer..................................................... 29

3. BLUESHIFTED RAMAN SCATTERING IN GALLIUM OXIDE NANOWIRES........................................... 32

Introduction............................................................................................ 32 Nanowire Synthesis................................................................................ 34 Growth Mechanism......................................................................... 35 Electron Microscopy of β-Ga 2 O 3 nanowires...................................... 37 Raman Modes of β-Ga 2 O 3 .................................................................... 39 FTIR Spectroscopy................................................................................ 49 Raman Peak Shifts due to Quantum Confinement........................... 51

x Table of Contents (Continued)

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LDA Model of the Peak Frequencies of β-Ga 2 O 3 ............................ 58

4. ANHARMONIC PHONON LIFETIMES IN CARBON NANOTUBES............................................................. 68

Introduction............................................................................................ 68 Raman Scattering in Carbon Nanotubes............................................ 71 Phonon Decay and Lifetime.......................................................... 75 Nanotube Synthesis................................................................................ 82 Electron Microscopy of Suspended SWNTs..................................... 84 Raman Spectroscopy of Suspended SWNTs..................................... 85

5. SUMMARY.................................................................................................... 105

Conclusions............................................................................................. 105 Future Work............................................................................................ 107

APPENDIX ............................................................................................................ 111

REFERENCES ............................................................................................................ 113

LIST OF TABLES

Table Page

1 Comparison of Calculated Raman Mode Frequencies With Those Measured in Bulk β-Ga 2 O 3 ................................................ 66

2 Estimated Internal Strains............................................................................... 67

3 Raman mode frequencies and frequency shifts in β-Ga 2 O 3

nanowires with the [ 40 1 ] and [110] growth directions. Overall, excellent agreement between the observed and calculated shifts is seen for all mode frequencies except the two marked with an *........................................................................ 68

xiii

LIST OF FIGURES

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1.1 Potential Energy versus Displacement Diagram for a Simple Harmonic Oscillator..................................................................................... 2

1.2 Potential Energy versus Position Diagram for a Harmonic and Anharmonic Oscillator.......................................................................... 5

1.3 This The calculated contribution of confinement (open circles) and anharmonic (solid circles) effects to the linewidth of the E g mode (143 cm -1 ) in TiO 2 nanoparticles at different temperatures............................................................................... 6

1.4 Figure 1.4: Experimental (dots) and calculated (lines) data showing the behavior of the Raman modes of TiO 2 with increasing uniaxial stress along the a axis................................................... 7

1.5 Raman spectra of β-Ga 2 O 3 nanorods (top trace) synthesized in a RF-induction furnace and β-Ga 2 O 3 powder (bottom trace)................................................................................................................ 8

1.6 Raman spectra of β-Ga 2 O 3 nanowires produced by arc- discharge and β-Ga 2 O 3 powder................................................................... 9

1.7 (a) Schematic diagram showing the set-up for performing spectroscopy on suspended SWNTs. A voltage is applied to the substrate with respect to the tip, and the current flowing from the substrate through the SWNT to the tip is measured. (b), STM image of a nanotube crossing a trench. Scale bar, 25 nm. The apparent width of the 2-nm -diameter tube is enlarged by tip convolution. (c), High- resolution image of the suspended portion of the SWNT showing atomic resolution. Scale bar, 2 nm............................................... 10

2.1 Rayleigh and Raman scattered light off a sample excited with the incident monochromatic light............................................................... 14

2.2 Energy diagram comparing Rayleigh, Stokes and anti-Stokes Raman scattering light. ∆E and ∆E’ denote the incident and scattered photon energies respectively....................................................... 15

xiv List of Figures (Continued)

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2.3 Schematic layout of the Renishaw 1000 Raman microscope......................... 27

2.4 Layout of Raman spectroscopy system used for analyzing anharmonic phonon lifetimes in suspended single-walled carbon nanotubes as described in Chapter 4............................................. 32

3.1 Schematic of the microwave plasma CVD reactor used to synthesize gallium oxide nanowires............................................................. 35

3.2 Schematic for the growth process of gallium oxide nanowires..................... 36

3.3 Raman spectrum of a gallium–coated substrate in which the entire gallium pool did not react with oxygen in a non- optimized experiment.................................................................................... 38

3.4 SEM micrograph of β-Ga 2 O 3 nanowires grown from a large molten gallium droplet using a microwave plasma mediated technique........................................................................................ 39

3.5 High resolution TEM image of an individual 13 nm thick β-Ga 2 O 3 nanowire......................................................................................... 40

3.6 Representation of the tetrahedra and octahedra which form the structure of β-Ga 2 O 3 ............................................................................... 40

3.7 Unit cell of monoclinic β-Ga 2 O 3 ........................................................................ 41

3.8 Micro-Raman spectrum of bulk β-Ga 2 O 3 ......................................................... 42

3.9 Cartesian displacements involved in the in-plane A g

vibrational mode at 111 cm -1 in β-Ga 2 O 3 The gallium and oxygen atoms are shown by the black and red dots respectively...................................................................................................... 43

3.10 Micro-Raman spectra of β-Ga 2 O 3 nanowires (top three traces) dispersed on quartz and silicon substrates collected at three different excitations showing peaks upshifted compared to bulk β-Ga 2 O 3 (bottom trace)............................................... 45

3.11 Raman spectrum of α-Ga 2 O 3 ............................................................................. 46

xv List of Figures (Continued)

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3.12 SEM image of the [40 1 ] nanowires grown by RF-induction. The inset is a high resolution TEM image from one of the wires showing twinned planes...................................................................... 49

3.13 FTIR transmittance spectra of bulk (bottom trace) and [110] β-Ga 2 O 3 nanowires (top trace).................................................................... 52

3.14 Raman spectra showing the evolution of the first-order 520 cm -1 peak of four silicon nanowire samples............................................... 53

3.15 Phonon dispersion curves for silicon. Both experimental (points) and calculated (solid lines) data are shown.................................. 56

3.16 Calculated and experimental (inelastic x-ray scattering) phonon dispersion curves for graphite....................................................... 57

3.17 Raman spectra excited using the 514 nm excitation line of an argon ion laser for crystalline graphite with differing number of graphene layers.......................................................................... 58

4.1 Calculated electron density of states for three different nanotubes. The van Hove singularity transitions are shown for both semiconducting (labeled S) and metallic nanotubes (labeled M)................................................................................... 71

4.2 Simulation of the radial breathing mode vibrations of a 0.8 nm diameter single-walled nanotube. The vertical lines over each carbon atom indicate the radial motion of the atoms......................................................................................................... 74

4.3 Raman-active normal eigevectors and frequencies for the tangential G band of a single-walled carbon nanotubes. The arrows indicate the direction of the carbon atom displacements in a unit cell for a (10,10) SWNT. A typical G band showing the G+ and G- peaks is shown on the right................ 75

4.4 Experimental points and calculated curves showing the temperature dependence of the linewidth of the 167 cm -1

band for a SWNT bundle.

The band is simulated as consisting of four close-lying Lorentzian components with Γ 0 = 0.5, 1.0, and 1.5 cm -1 .................................................................... 79

xvi List of Figures (Continued)

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4.5 Linewidth of the RBM (Γ RBM ) vs nanotube diameter (d t ) for81 metallic (solid symbols) and 89 semiconducting (open symbols) isolated SWNTs................................................................. 79

4.6 Individual single-walled nanotubes grown between pre-fabricated pillars on silicon substrates)................................................ 81

4.7 Stokes and anti-Stokes Raman spectra from an individual suspended SWNT on a silicon substrate with ω RBM

at 237 cm -1 ....................................................................................................... 82

4.8 Schematic of the chemical vapor deposition method used to synthesize suspended SWNTs................................................................ 84

4.9 SEM image showing several suspended Swnts across the corners of a trench etched into the silicon substrate. The clusters appearing at the bottom of the trench are catalyst particles and do not interfere with the Raman spectra since they do not lie in the focal plane of the focus beam of the exciting laser................................................................................................... 85

4.10 Micro-Raman spectra from an individual suspended semiconducting SWNT (left) and a metallic SWNT (right). The peak position (linewidth) are indicated in the graphs.............................................................................................................. 88

4.11 Micro-Raman spectra in the RBM range from a region on a trench containing multiple suspended SWNTs............................................. 89

4.12 The tangential bands for a semiconducting (top) and a metallic (bottom) suspended SWNT.......................................................... 91

4.13 Kataura plot between nanotube diameter and incident laser excitation energy (E L ) The vertical lines indicate the diameter distribution of the suspended SWNTs analyzed in the present study........................................................................................ 92

4.14 Deconvoluted 2 Γ (Lorentzian FWHM) values for several suspended SWNTs plotted versus ω RBM . The circled data points represent metallic SWNTs while the rest are................................. semiconducting SWNTs. Error bars from the linewidth fit are also plotted............................................................................................... 93

xvii List of Figures (Continued)

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4.15 The calculated phonon dispersion relations of a carbon nanotube (bottom), and the corresponding phonon density of states (top).................................................................................... 94

4.16 Raman spectra around the radial breathing mode frequency for two representative suspended SWNTs collected for different incident laser powers. From the lineshape of the tangential band present at ~1590 cm -1 the spectra in panels (a) and (b) are assigned to a metallic and semiconducting nanotubes, respectively.................................................... 95

4.17 Normalized linewidth versus incident total laser power P (The normalized linewidth is defined as the Lorentzian FWHM at P divided by the FWHM at P 0 = 1mW). The lines correspond to fits with eq. (4.16), in which the only adjustable parameter is the suspended length L....................................... 101

4.18 (a) Solid line: Raman phonon population decay under conditions that create a decay bottleneck (p ~2); Dotted line: Raman phonon population decay under conditions in which the decay bottleneck is negligible (p >>1). Dashed-dotted line: Raman phonon population decay when the secondary phonons are also created by the external perturbation like tunneling (b) Predicted Raman spectrum corresponding to the solid line in panel (a).;(c) Solid line: Modeled observed Raman spectrum obtained from convoluting the spectrum in panel (b) with a Gaussian function, Dotted line: Fit of the solid line using a Voigt profile.................................................................................................... 105

1

CHAPTER ONE ANHARMONICITY IN MATERIALS

Introduction

The simplest vibrational model in a crystal considers lattice vibrations as arising from the harmonic motion of atoms at each lattice site [1]. Vibrational interactions between the atoms give rise to waves within the solids, which have allowed wavelengths and amplitudes governed by the lattice structure in the solid. For small vibrations in one dimension about equilibrium, the potential energy U(x) of the α’th atom within the solid can be expanded as a Taylor series as follows:

, (1.1)

where the first, second, and the third term represent the zeroeth, first, and second order term respectively. Typically, to a good approximation this potential can be taken to be equal to the harmonic oscillator potential. This is the so called harmonic approximation in which the series for the potential energy is truncated at the second term. Now since each lattice site is occupied by atoms that are vibrating, these lattice sites are equivalent. Also, by symmetry, there is equal probability for motion along opposite directions. Hence the linear term in the potential expansion vanishes. The zeroeth (zero-point) term in the expansion only affects the scaling and can be duly ignored for most calculations. Thus, in the 2 3 0 1 ( )''( )... 2 U x U U x U x O x α α α α α α = + + + + ∑ ∑

2 classical picture, the potential energy of a simple harmonic oscillator contains only a term that is quadratic in displacement and can be stated as

2 2 2 1 1 2 2 U U x m x α α ω= = , (1.2)

where m is the mass of the oscillator, ω is the frequency, and x is the displacement in one direction of the oscillator. In the classical picture, the simple harmonic oscillator can be described physically with a ball and spring model, where m is the mass of the ball, x is the displacement of the ball in one direction, and k is the spring constant of the oscillator. Figure 1 shows potential energy of a harmonic oscillator plotted against position. Since the energy is proportional to the square of the displacement, it is parabolic in nature. The potential energy is symmetric about the origin at x = 0.

Figure 1.1: Potential energy versus displacement diagram for a simple harmonic oscillator.

+ x max - x max U Energy

3 While the classical “ball and spring” model can adequately explain harmonic motion between atoms, in order to get a complete description of atomic vibrational motion in a crystal lattice, one has to consider a quantum mechanical model. The energies of the quantum mechanical harmonic oscillator can be determined by the solution of Schrödinger’s equation, which is stated as

ψψ EH = (1.3)

where H is the Hamiltonian of the system, | ψ> is the wavefunction of the atom associated with the oscillator, and E is the energy of the quantum mechanical oscillator. The solution of the Schrödinger equation reveals that the energy levels of the oscillator are quantized and the energy of the n’th level is given by

ω= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ += 2 1 nE n (1.4)

The energy levels of the quantum harmonic oscillator are discrete. The harmonic potential energy can be used to approximately describe many processes in solids. In the case of vibrational motion of atoms within a solid, the harmonic potential can be used to calculate normal mode frequencies for phonons, where a phonon is a quantum of lattice vibration having energy equal to ħω. Thus with the harmonic potential, one can get a complete description of the normal vibrational modes of a crystal. However, there are drawbacks with the harmonic approximation. Certain

4 experimental observations cannot be adequately explained solely on the basis of harmonic potentials as listed below [2]- 1. Within the harmonic approximation, phonons do not interact with each other. This implies that vibrational waves, once created within the solid, would propagate infinitely and never decay. 2. A solid would not show thermal expansion. 3 Vibrational frequencies are independent of strain.

The above features can be explained in part by an anharmonic theory which takes account of the terms in the potential energy which are higher than the quadratic terms. In that case, the Hamiltonian of the oscillator can be split into the harmonic (quadratic) and anharmonic (cubic and higher terms) parts, and a quasi-harmonic treatment could then be applied to the oscillator. Within this framework, phonons within a solid are considered as traveling waves that can interact with each other. The anharmonic component of the oscillator potential causes an interchange of energy between these traveling waves, thus resulting in the decay of phonons, and hence finite phonon lifetimes. The inclusion of a third order term causes the energy of the anharmonic oscillator to deviate from the parabolic shape of the harmonic oscillator and looks like the oscillator energy shown in Fig. 1.2 below.

5

Figure 1.2: Potential energy versus position diagram for a harmonic and anharmonic oscillator.

Experimental observations of anharmonicity

Anharmonic effects in materials been observed by a variety of methods. Shiren [3] described an experiment in which a beam of longitudinal phonons of frequency 9.2 GHz interact in a magnesium oxide crystal with a parallel beam of longitudinal phonons at 9.18 GHz. The interaction of these two beams produced a third beam of longitudinal phonons at 9.2 + 9.18 = 18.38 GHz[3]. Also, anharmonic interactions between phonons lead to phonon decay, and hence finite lifetimes. Phonon lifetimes have been observed in several materials through various spectroscopic methods such as inelastic x-ray scattering [4], Raman scattering [5] and neutron scattering [6]. The linewidth of a Raman mode is inversely proportional to its lifetime, and anharmonic linewidths (lifetimes) increase (decrease) with increase in x Energy

6 temperatures. For example, Šćepanović et al. [7] observed changes in the linewidth of the 143 cm -1 mode in nanocrystalline TiO 2 powder. Such changes are normally attributed to phonon confinement effects, but they show that the anharmonic contribution to the change in linewidth (phonon lifetime) with temperature is greater than that of confinement. Fig. 1.3 shows their calculated contributions due to anharmonicity and confinement, and while the confinement contribution does not change significantly with temperature, the anharmonic broadening of the 143 cm -1 mode with increasing temperature is quite large. More details about phonon lifetimes and anharmonic broadening of Raman linewidths due to increasing temperature in carbon nanotubes are addressed in chapter 4.

Figure 1.3: The calculated contribution of confinement (open circles) and anharmonic (solid circles) effects to the linewidth of the E g mode (143 cm -1 ) in TiO 2 nanoparticles at different temperatures [7].

7 As mentioned above, the harmonic approximation does not explain thermal lattice expansion or contraction. Similarly, changes in a lattice due to inherent strains induced during growth cannot be explained within the harmonic potential picture, and an anharmonic potential must be taken into account to fully explain the experimentally observed data. In such cases, Raman spectroscopy is a powerful tool since strains cause Raman peaks to move and broaden. The Raman spectra of samples under strain can be then modeled using anharmonic potentials. Peercy et al. [8] studied the uniaxial stress dependence in the Raman-active phonons in TiO 2 using experiment as well as anharmonic theory. Fig. 1.4 shows the room temperature frequency shifts of 3 different Raman modes in TiO 2 as a function of uniaxial pressure. A uniaxial stress along the a axis of TiO 2 causes the A 1g and E g mode to upshift due to compressive strain, while the B 1g mode downshifts due to tensile strain.

Figure 1.4: Experimental (dots) and calculated (lines) data showing the behavior of the Raman modes of TiO 2 with increasing uniaxial stress along the a axis [8].

8 In this dissertation, we analyze the effect of compressive strains in gallium oxide nanowires via micro-Raman scattering. Gao et al. [9] synthesized β-Ga 2 O 3

nanowires having a [ 40 1 ] growth direction and

the Raman mode frequencies of the [ 40 1 ] nanowires are red-shifted (shifted lower in energy) relative to corresponding frequencies in bulk β-Ga 2 O 3 by 4-23 cm -1 (Fig. 1.5). On the other hand, Choi et al. [10] grew β-Ga 2 O 3 nanowires having a [001] direction and they saw no changes between the Raman spectra of their nanowires compared to that of bulk β-Ga 2 O 3 (Fig. 1.6).

Figure 1.5: Raman spectra of β-Ga 2 O 3 nanorods (top trace) synthesized in a RF- induction furnace and β-Ga 2 O 3 powder (bottom trace) [9].

9

Figure 1.6: Raman spectra of β-Ga 2 O 3 nanowires produced by arc- discharge and β-Ga 2 O 3 powder [10].

Chapter 2 outlines the quantum mechanical description of the Raman effect and the experimental setup used in our lab to probe the vibrational properties nanomaterials described in this dissertation. Chapter 3 introduces the experimentally observed blueshift in the Raman mode frequencies of our strained gallium oxide nanowires. Both theory and experiment are used to evaluate the Raman peak upshifts and the theoretical framework is described which provides insight into the origin for this blueshift.

10

Figure 1.7: (a) Schematic diagram showing the set-up for performing spectroscopy on suspended SWNTs. A voltage is applied to the substrate with respect to the tip, and the current flowing from the substrate through the SWNT to the tip is measured. (b), STM image of a nanotube crossing a trench. Scale bar, 25 nm. The apparent width of the 2-nm-diameter tube is enlarged by tip convolution. (c), High-resolution image of the suspended portion of the SWNT showing atomic resolution. Scale bar, 2 nm [11].

In chapter 4, the analysis of anharmonic phonon lifetimes in suspended single-walled carbon nanotubes is presented. Recent scanning tunneling microscopy (STM) experiments show that electrons tunneling into a metallic single-walled carbon nanotube (SWNT) lead to a non-equilibrium phonon population for the radial breathing mode (RBM). LeRoy et al. [11] injected electrons into a suspended single-walled carbon nanotube through an STM tip (at 5K) that was brought into close contact with the nanotube (Fig. 1.7). They found that the injection of a large number of electrons into the nanotube causes a build- up of non-equilibrium phonons. By measuring the differential conductance of the injected electrons with varying sample voltage, they found that small peaks

11 appear at ~25 meV, which are caused due to phonon-assisted tunneling. Interestingly, the energy of 25 meV is similar to the energy of the radial breathing mode phonon. Thus they were able to excite the RBM by injecting electrons into a SWNT. Furthermore, they analyzed various semiconducting and metallic nanotubes, and found an inverse relationship between the energy of the phonon- assisted tunneling peak and the nanotube diameter. By measuring the rate of electrons tunneling into the nanotube, they were able to estimate the anharmonic lifetime of the RBM to be τ ≈ 10 ns. This corresponds to a Raman linewidth of 5×10 -4 cm -1 . To the best of our knowledge, the smallest Raman linewidth that has been experimentally measured (at a low temperature ~20 K) for the RBM of the inner tube in a double-walled carbon nanotube is 0.4 cm -1 [12]. In this dissertation, a study on the linewidth measurements in suspended SWNTs is presented in chapter 4. In our study the Raman linewidths are lower than previously measured in isolated SWNTs, and a model is discussed to explain the discrepancy in the linewidth values measured through optical and tunneling experiments. Finally, at the end of this dissertation, a glossary of abbreviated terms used throughout the manuscript has been compiled in the appendix.

CHAPTER TWO RAMAN SPECTROSCOPY

Introduction

Raman spectroscopy is based upon the Raman effect, which may be described as the inelastic scattering of light from a gas, liquid or solid. Typically, the sample is excited with a monochromatic light source (for, e.g., a laser) and the Raman spectrum is detected as the scattered light intensity found at energies below and above the excitation energy. Discovered by the Indian physicist C. V. Raman in 1928 [13], it has also been called the Smekal-Raman effect [14], the former investigator having made some earlier theoretical predictions about it. Raman spectroscopy is based on the interaction of light with molecules in some medium. When light hits a molecule it scatters off the molecule as Rayleigh scattering or as Raman scattering (Fig. 2.1). The radiation excites the molecule, distorting the shape of the molecule's electron cloud. When the electron cloud returns to its original shape, the energy in the molecule may have increased or decreased slightly, changing the energy of the scattered Raman radiation. Viewed in terms of energy levels, the electrons reside in the ground vibrational and electronic states before excitation. The monochromatic laser source excites the electrons to a virtual state, equal to the energy of the laser. When the electrons relax back to the ground electronic state, most go back to the ground vibrational state, giving back the same energy. This is Rayleigh (elastic) scattering. The small portion that relaxes back to an upper or lower vibrational state is the

14 Raman scattered light and is about 5 - 7 orders of magnitude less intense than Rayleigh scattered light [15].

Figure 2.1 – Rayleigh and Raman scattered light off a sample excited with the incident monochromatic light.

The Raman scattered light that is adjusted up in wavelength is called the Stokes Raman scattering and that light which is adjusted down in wavelength is called the anti-Stokes Raman scattering (Fig. 2.2). Stokes Raman scattering occurs when some energy is absorbed from the photon of incident light into the molecule’s rotational and vibrational energy and consequently a new photon of light with less energy is released. Anti-Stokes scattering occurs when the new photon formed gains energy compared to the incident photon via the absorption of energy from a previously excited molecule. Since the probability for an electron to be in an excited state before the scattering process is not high, the cross-section

15 for anti-Stokes scattering is much less than that of Stokes scattering and, consequently, the intensity of anti-Stokes scattered light is also much lower.

Figure 2.2 - Energy diagram comparing Rayleigh, Stokes and anti-Stokes Raman scattering light. ∆E and ∆E’ denote the incident and scattered photon energies respectively.

A theoretical treatment of Raman scattering is essential for the understanding, interpretation and appreciation of the experimental Raman spectra from nanotube samples which will be presented later. In this section a general theory of Raman scattering is presented without any reference to the specific properties of the sample. By considering the general theory we gain insight into what information the Raman spectrum carries about the sample. The relevant properties of gallium oxide and carbon nanotubes are presented in chapters 3 and 4 respectively. Energy

Rayleigh Scattering (elastic) Stokes Scattering Anti-Stokes Scattering hν 0 h ν

ν

ν

ν

−

ν ν 0 +hν E 0

E 0 +hν m Virtual State IR Absorption Raman (inelastic)

16 Raman scattering can be described theoretically using classical as well as quantum mechanical physics. In the classical description, when light strikes a molecule, it causes an induced electric dipole in the molecule. The electric dipole moment is related to the electric field of the incoming radiation through the polarizability of the molecule. The polarizability is a tensor which in general is a function of the interatomic distances, and will therefore change if the molecule is vibrating. If we expand the polarizability as a Taylor series, we can see that the light scattering causes the vibrating molecule to emit light having frequencies above and below the frequency of the incoming light. The polarizability, α ij can be expressed as

2 0 , 0 0 1 ( )... 2 ij ij ij ij k k l k k l k k l q q q q q q α α α α ⎛ ⎞ ∂ ∂ ⎛ ⎞ = + + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ∑ ∑ , (2.1)

where the position-dependent polarizability α ij has been expanded as a function of the generalized coordinates q k and q l . The first two terms in the above equation can be grouped together and written as

' 0k k k qα α α= + (2.2)

Assuming that the molecules are undergoing simple harmonic motion, the generalized position coordinate of the k’th molecule can be written as ' 0k k k qα α α= +

17

0 cos( ) k k k k q q t ω δ= + (2.3)

Combining eqs. (2.3) and (2.2), we get

' 0 0 cos( ) k k k k k q t α α α ω δ= + + (2.4)

The polarizability is related to the molecules electric dipole moment and the electric field intensity of the incident radiation as

(1) k P Eα= ⋅ , with 0 0 cosE E t ω = , (2.5)

where the ω 0 is the frequency of the incident radiation. Thus the dipole moment can be written as

(1) 0 0 cos k P E t α ω= ⋅ (2.6)

From eqs. (2.6) and (2.4), we get

(1)' 0 0 0 0 0 0 cos cos cos( ) k k k k P E t E q t t α ω α ω ω δ= ⋅ + ⋅ + (2.7)