POLITECNICO DI MILANO DIPARTIMENTO DI FISICA PhD course in Physics XXV cycle DEVELOPMENT AND APPLICATION OF A RAMAN MAPPING INSTRUMENT FOR THE STUDY OF CULTURAL HERITAGE Supervisor: Prof. Gianluca Valentini Tutor: Prof. Rinaldo Cubeddu PhD coordinator: Prof. Paola Taroni Candidate: BRAMBILLA ALEX 2010 - 2012
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POLITECNICO DI MILANO
DIPARTIMENTO DI FISICA
PhD course in Physics
XXV cycle
DEVELOPMENT AND APPLICATION OF A RAMAN
MAPPING INSTRUMENT FOR THE STUDY OF
CULTURAL HERITAGE
Supervisor: Prof. Gianluca Valentini
Tutor: Prof. Rinaldo Cubeddu
PhD coordinator: Prof. Paola Taroni
Candidate: BRAMBILLA ALEX
2010 - 2012
2
“In all things of nature there is something of the marvelous”
Aristotle
3
4
Summary.
Summary. 4
Abstract. 7
Introduction. 10
CHAPTER I
RAMAN SPECTROSCOPY. 13
I.1 Molecular vibrations. 13
Vibrational energy. 14
Normal modes of vibration. 15
I.2 Classical theory of Raman scattering. 19
Susceptibility and polarizability. 20
Rayleigh and Raman scattering. 21
I.3 Quantum theory of Raman scattering. 24
Time dependent perturbations: the second order transition probability. 25
A semi-classical expression for Raman effect. 28
Full quantum approach. 32
Second quantization. 33
The Kramers-Heisenberg formula. 36
General considerations. 39
I.4 The cross section. 41
I.5 Surface Enhanced Raman Scattering. 43
Bibliography. 45
CHAPTER II
THE REMOTE RAMAN SCANNER. 47
II.1. Raman spectroscopy in Cultural Heritage. 47
II.2. The aim of the project. 49
5
Raman imaging. 50
Remote Raman sensing. 51
Portable Raman instruments. 53
II.4. Remote Raman Scanner: the layout. 55
II.5. RRS: the standard components 57
Semiconductor laser. 57
Galvanometric mirrors. 59
Holographic notch filter. 60
Czerny-Turner spectrograph. 60
Charge coupled device (CCD). 61
Optical fibers. 62
II.6 RRS: the custom-made systems. 63
The optical system. 63
The software. 67
Bibliography. 71
CHAPTER III.
RRS AT WORK: TESTS AND CASE STUDIES. 77
III.1 Characterization of the performances. 77
Assessment of the depth of the field of view. 78
Assessment of the field of view. 79
Assessment of the spectral accuracy. 80
Assessment of the spatial resolution. 80
III.2 A multi-analytical approach to a model painting. 82
Reflectance spectra and reconstructed maps. 83
XRF analysis. 87
Flexible Raman mapping. 91
6
III.3 Identification of white and fluorescent pigments. 94
III.4 Analysis of samples from the “Memoriale italiano” of Auschwitz. 97
III.5 Study of a 3D plastic object 100
III.6. Mapping of a rock sample. 102
Bibliography. 106
CHAPTER IV
SERS ANALYSIS OF AMINO ACIDS WITH A PORTABLE INSTRUMENT.
108
IV.1. The aim of the work. 108
IV.2 SERS of amino acids. 109
IV.3 SERS with metal nanoparticles. 112
IV.4 The portable instrument. 115
IV.5 Tests and results. 116
Raman spectra of amino acids. 117
Analysis of liquid samples. 120
Analysis of dried solutions. 122
Dependence of L-Met spectra on pH. 124
Distinguishing two amino acids. 125
Towards a more complex sample. 128
IV.5 Perspectives. 129
Bibliography. 135
CHAPTER V
CONCLUSION. 140
Acknowledgements. 144
7
Abstract.
Raman spectroscopy is a consolidated technique for the analysis of materials: it is
based on the identification of the frequency shifts undergone by light scattered by a
molecule; this behavior is due to an interaction of the incident photons with the
vibrational quanta of the involved material. The possibility to obtain information on
a molecular scale about the chemical identity of a sample in a non-destructive way,
makes this method one of the favourite for the analysis of valuable objects, as it is in
case of Cultural Heritage. In this field of application it is particularly urgent the
necessity to monitor wide surfaces of heterogeneous objects often with an irregular
shape. Compared to a laboratory measurement, moreover, an in situ analysis implies
further requirements due to the dimensions and the portability of the employed
instrumentation, which must guarantee an adequate sensitivity to Raman signal
while keeping the necessarily limit of a non-invasive procedure.
To answer to these needs, a portable Raman spectroscopy system which works with
a non contact approach, has been conceived, designed and assembled. The device is
able to analyze a surface 5 cm wide placed at a distance of 20 cm with a flexible
mapping approach: a pair of galvanometric mirrors allows the deflection of the laser
beam, which is focused on the point of interest by a dedicated optical system. This
element was tailored to collect and guide the light scattered by the sample towards
the instruments of analysis (spectrograph and charged coupled device) which, like
the laser source, are remotely positioned and connected by optical fibers. This
solutions permits to keep the measuring probe light and small, and to mount it on a
tripod in order to ensure maximum flexibility and stability at the same time; the
instruments of analysis can be, alternatively, connected to a more conventional
Raman microscope when a deeper punctual analysis is needed. The chance of
performing remote measurements not only considerably widens the range of object
to study, but it also makes the device less affected by the vibrations which perturb
contact analysis when performed out of the laboratory environment; moreover, the
particular combination of the chosen optic elements ensures a depth of field ideal
for the analysis of three-dimensional objects.
8
The thesis treats the theoretical foundations of Raman spectroscopy to highlight its
potentialities and stress the common problematiques (in primis the low efficiency
with respect to competitive phenomena such as fluorescence); then, the phases of
the project of the Remote Raman device are reported together with the methods to
test its performances and features. One chapter is dedicated to the results of
punctual analysis and to the study of wide areas in samples of artistic or cultural
interest; whenever it is possible, these are compared to the information obtained by
other non-invasive techniques on the same objects. Last, surface-enhanced Raman
measurements on amino acids in low concentrations are reported as a preliminary
study for the analysis of organic traces in archaeological remains.
9
10
Introduction.
The application of scientific techniques to the study of Cultural heritage is wide
reaching. Not only research groups of chemists, physicists, geologists, engineers and
biologists are growingly demanded to provide diagnostic methods for several kinds
of artefacts, but museums and conservation centers now rely on scientific pools of
specialists equipped with cutting-edge analytical tools. The information furnished by
analytical methods, yielding data on the atomic and the molecular composition
through spectroscopic techniques, for example, is valuable not only for historical or
scientific interest, but also important as data can support critical evaluation, the
authentication of a masterpiece, or guide the restoration operation.
Despite the fact that the role of scientific methods in humanistic studies such as
preservation, conservation and promotion of cultural heritage is now undisputed,
the research of new methods of application continues to develop. The typical issues
related to the analysis of cultural heritage, which generally concern the need to
interact in the softest possible way with the object of interest, represent a
continuous challenge for researchers, who look for techniques capable of providing
significant information with a non-destructive or, preferably, a non-invasive
approach. These two requirements have lead to the spread of optical analytical
techniques, like reflectance spectroscopy, Infrared absorption spectroscopy and
Raman spectroscopy, for the characterization of surfaces of works of art; in most of
their applications, these methods rely on a laser as a light source, therefore they
benefit from the explosion in development and expansion of these instruments in
the last decades. The parallel improvement and the declining price of light detectors,
in addition, has led to the birth of imaging versions for optical analyses: the
possibility of studying a large set of points on a surface with a single acquisition
further promotes the application of these techniques to the study of artworks.
Within this framework, this thesis presents a novel instrument for Raman
spectroscopy, able to perform a stand-off mapping of a macroscopic surface at a safe
working distance of approximately 20 cm and thus particularly suitable for
analyzing valuable objects. Commercially available Raman Spectrometers which are
marketed for the analysis of works of art work either in contact or at very short
working distances (on the order of millimeters). The aim of the reported work has
11
been to build an instrumental apparatus ready to be operated in the laboratory as
well as on site; the need for a significant flexibility in the measurement of different
points of a representative area, led to the design and development of a dynamic
mapping device, which can selectively illuminate and stimulate the Raman
scattering of each point within its field of view, without any relative movement
between the probe head and the target object. The layout of the set-up has been
oriented towards a remote instrument, capable of being employed even in
uncomfortable environments thanks to a portable probe head and to a working
distance of 20 cm, sufficient to ensure a non-invasive approach to the object of
interest, and at the same time to reduce the sensitivity to vibrations.
A thorough study of Raman scattering has been the first, indispensible step in the
design of the measurement apparatus. In Chapter 1, a description of the Raman
effect, as explained by the classic and the quantum theory, is presented, based on
some of the most common approaches followed in physics and chemistry. Particular
mention to the standard Raman event and to the enhancement to the effect provided
by the interaction with a metal substrate is made; one of the aims of the thesis,
indeed, is to determine how the adoption of suitable procedures and the
optimization of technological parameters can help the detection of the intrinsic
weak Raman scattering with a portable apparatus, whose acquisition is generally
more critical of bench-top instrument. In Chapter 2 the state of the art of Raman
mobile and stand-off devices is addressed; this study compares the proposed set-up
with available Raman mapping and imaging devices. In addition the choices
determined during the design and construction of the system are reviewed. With
Chapter 3, the performances of the completed set-up are reported: in this case, the
focus is on the capability of providing the identification of different kinds of
materials and samples of heterogeneous objects. The integration of the Raman
mapping instrument with other non-destructive analytical methods is evaluated on
a model of a painted panel, as the ultimate test of the application. In Chapter 4 tests
performed with a commercial mobile Raman instrument, chosen for a different kind
of analysis are reported: the study of organic fragments contained in ancient vessels
and pottery. For this specific purpose, research was carried out with a device with a
portable probe head but based on the traditional microscope layout. The novelty of
the chosen approach resides in the adoption of metal nanoparticles as a substrate to
12
enhance the Raman cross section of the analytes, in this case simulated by solutions
of amino acids. The exploited technique, which goes under the name of Surface-
Enhanced Raman Spectroscopy (SERS), has been applied for the first time with a
portable Raman instrument on this kind of materials: amino acids and a small
peptide, in small concentrations have been chosen for the preliminary tests to assess
the detection limits for these organic materials. The reported results are the first
step towards the determination of a routine method that could be easily applied in
situ to reveal traces of biological materials on archaeological finds.
13
CHAPTER I
RAMAN SPECTROSCOPY.
Raman spectroscopy is based on the homonym effect, postulated by Smekal in 1923
[I.1] and first observed experimentally in 1928 by C.V. Raman and Krishnan [I.2]: in
the original experiment, for which Raman was awarded the Nobel prize in physics
two years later, sunlight was focused by a telescope onto a sample which was either
a purified liquid or a dust-free vapor. A second lens was placed by the sample to
collect the scattered radiation and a system of optical filters was used to show the
existence of scattered radiation with an altered frequency from the incident light –
the basic characteristic of Raman spectroscopy [I.3]. Inelastic scattering is due to the
interaction of the electromagnetic radiation with the level structure of a molecule or
a solid: the scattering event corresponds to a virtual transition (in a sense that will
be specified later) between rotational, vibrational or electronic levels triggered by
light and mediated by quanta of electromagnetic energy called photons. Commonly,
Raman spectroscopy is focused on the second type of these processes, involving
vibrational transitions. Together with infrared absorption, this technique is used to
obtain information about the structure and properties of molecules and solids in a
non destructive way [I.4].
I.1 Molecular vibrations.
A molecule is classically described as the collection of M nuclei and N electrons [I.5]
arranged in a stable configuration: if the positions of the nuclei are considered fixed,
each molecule can be associated to a “molecular point group”, which is a symmetry
group corresponding to a fixed point, the center of mass of the molecule. In the same
way, a crystal is described as the periodic distribution of a unit cell, made out of a set
of atoms, along each point of a spatial lattice, called Bravais lattice, generated by
14
three linearly independent vectors. Each crystal is associated to a crystallographic
group according to the symmetry of the point group of its cell, and to the symmetry
of the lattice [I.6]. The symmetry properties of molecules and solids are determinant
for the vibrational transitions which are observed with Raman, and in general
vibrational spectroscopy; this is true also in a negative sense, as it is for amorphous
materials, in which regularities are mostly absent and a statistical description of the
structure is necessary.
For the present discussion, the molecule is considered as the model for the classical
and quantum systems on which the Raman effect is treated: this allows the
treatment to be relatively compact and still valid for gaseous, liquid and many solid
samples; whenever it is needed, reference to crystals and amorphous materials will
be explicitly made.
Vibrational energy.
A common approximation when describing the molecule is to consider that the
motion of the electrons can be separated by the motion of nuclei. This assumption
(called Born-Oppenheimer approximation) is realistic, considering the ratio of the
masses of the electron and a generic nucleus: the motion of the electrons is faster
than the one of nuclei at least by 3 orders of magnitude, and so the latter are almost
“at rest” with respect to these particles. Thanks to Born-Oppenheimer
approximation, the problem to determine the dynamic of the electrons is solved
while maintaining fixed the positions of the nuclei (which appear in the solution as
parameters) and the electronic contribution obtained by this calculation is used to
describe the total energy of the system [I.5]. This separation of the electronic motion
and the nuclear motions is only an approximation which may break down in certain
cases, especially for high electronic states. If there were no interaction between the
two types of motions, there would be no Raman effect of any importance. However,
the coupling is small for the lowest electronic state [I.7].
The molecule, seen as a rigid body, can translate along three arbitrary Cartesian
axes: therefore, three of its 3M degrees of freedom represent its translations in
space. If the center of mass is then fixed, the molecule has 3M – 3 remaining degrees
of freedom: rotations around the coordinate axes represent three (or two, for a
linear molecule) of these degrees of freedom. Six (five) parameters are needed, thus,
15
to describe the molecule as a whole: the remaining 3M – 6 (or 3M – 5) degrees of
freedom describe, instead, the variations among the “standard” location of the nuclei
around the center of mass, i.e. the deformations of the molecule: these are called
vibrational degrees of freedom. The global energy of the isolated molecule can be
The reported considerations refer to a molecule in absence of an external potential:
further considerations are necessary to take into account the interaction of this
system with light, as it will become clearer in the following sections.
Normal modes of vibration.
The translational motion of the molecule does not ordinarily give rise to radiation.
Classically, this happens because acceleration of charges is required for radiation.
The rotational motion causes practically observable radiation if, and only if, the
molecule has an electric (dipole) moment. The vibrational motions of the atoms
within the molecule may also be associated with radiation if these motions alter the
electric moment [I.7].
Vibrations, as we have seen, correspond to any displacement among the positions of
the atoms which do not alter the position of the center of mass nor its angular
momentum. The dynamic description of these vibrations must consider, therefore,
forces internal to the molecule: at a first approximate analysis, the intra-molecular
potential can be limited to the nearest neighbor atoms. This is equivalent to consider
the intra-molecular potential V(R) of a biatomic molecule, which is, if referred to the
equilibrium distance R0 between the two atoms, clearly asymmetric (fig.I.1); the
16
energy of the molecule tends to infinite if the atoms are too close, and instead
reaches a saturation for very long distances, when it is equal to the work necessary
to dissociate the molecule. Even though, in the range in which the molecule is stable
(below the dissociation value), it is possible to approximate the potential with a
harmonic oscillator. The expression of the intra-molecular potential can indeed be
expanded around its minimum, and it is convenient before performing this
operation, to introduce a suitable frame of reference.
Fig.I.1. The intra-molecular potential curve V(R) for a bi-atomic molecule (blue curve). Without any
loss of generality, in the graph the dissociation energy value is set equal to 0. The green curve refers
to the harmonic approximation.
To start with, a standard Cartesian system of coordinates xi is chosen. A molecule
constituted by M atoms is described by a set of 3M coordinates, so i runs from 1 to
3M; at the equilibrium configuration, these respectively assume the values ci. The
displacement along each coordinate, therefore, will be calculated as the difference
Δxi = xi – ci; clearly, the time derivatives (indicated by a dot over the letter) of the
initial Cartesian coordinates are exactly equal to the corresponding time derivative
17
of the displacement coordinates. A set of the so-called mass-weighted coordinates qi
is constructed according to the relationship
q m
where, obviously, the mis are equal in groups of three in a row because they refer to
the same atom. This choice permits to simplify the kinetic energy expression
q
The series expansion of the potential energy is, accordingly, carried out in powers of
qi:
q
q
q q
q q
,
A comfortable choice of V0 is to set it to zero, without any loss of generality. The
second term in the right-hand side of the equation is automatically zero for the
definition of the equilibrium position (the derivative is evaluated for qi = 0). If the
expansion is arrested at the second order, therefore, only the last term is surviving,
and it can be synthetically written as
q q
q q
,
f q q
where the second order derivatives of V, evaluated in the equilibrium position, can
be considered as the elements fij of a 3M × 3M dynamic matrix F. The equations of
motion of the molecule are described by the solutions of the 3M Lagrange equations
d
dt
q
q
which, with the chosen approximation, can be written as a set of 3M simultaneous
second-order linear differential equations, each one closely resembling the well-
known equations of the harmonic oscillator:
q f q
(1.4)
(1.5)
(1.6)
(1.7)
(1.8)
(1.9)
18
If a tentative solution of the form
q cos t
where Ai, and are properly chosen constants, is substituted, the set of differential
equations is transformed in a set of 3M simultaneous algebraic equations in Ai
f
Where the ij symbol stands for the Kronecker delta. The solutions of these
equations are not trivial only for a special set of s (which have the physical
dimensions of pulsations): clearly, this is but a secular equation for the dynamic
matrix F. For each particular value of , namely k, a certain set of amplitude Ai,
namely Aik, is obtained; however, it can be shown that only the ratios between the
different Ais is determined per each k. Therefore, the choice of convenience, if there
is no strict constraint given by the initial conditions on qi, is to set the values for
which the modulus of the vector [A1k, A2k, 3Mk] is unitary. These normalized
vectors are indicated as aik, and they can be grouped in a matrix called A.
It can be demonstrated, as well, that six out of the 3M eigenvalues of the matrix F are
zero: the null roots of the secular equation correspond to null pulsations, which
means that six degrees of freedom, as previously reported, are not ascribable as
vibrations. This means that, through a proper choice of the coordinates, this can be
reduced to a 3M – 6 square matrix without any loss of information. It can be easily
shown that this transformation is performed by multiplying the qi vector by the
matrix A of the normalized eigenvectors. Since there are only 3M – 6 eigenvectors,
each one made of 3M elements, the transformation produces naturally a new system
of 3M – 6 coordinates Qk, called vibrational coordinates. On the basis of the
vibrational coordinates, the dynamic matrix is diagonal, i.e. the harmonic oscillators
are decoupled.
This rather laborious analysis of the intra-molecular potential has the advantage to
introduce all the necessary elements to thoroughly describe molecular vibrations.
From the initial truncation of the series expansion, called Mechanic Harmonic
approximation, it was possible to derive the dynamic matrix F. The eigenvalues k of
this matrix correspond to the pulsations of the oscillations of the atoms around their
(1.10)
(1.11)
19
equilibrium positions, their amplitude along the coordinate axis (except for a
multiplicative factor) given by the eigenvectors Aik. The equations (1.10) therefore,
describe motions in which all the atoms vibrate with a single frequency ωk = k½ in
phase: these are called vibrational modes of the molecule, and are described by the
vibrational coordinates Qk as independent harmonic oscillators.
Qk = Qk0 cos (ωkt + k) (1.12)
For the properties here described, the modes are orthogonal and normal: this
property allows the decomposition of any collective displacement of the nuclei of the
molecule in a series built over a basis of independent oscillations. In addition, it is
worth to anticipate that the decomposition along the normal modes of vibration is
necessary for the quantum description of Raman effect, because only within this
frame of reference the vibrational eigenfunction Ψvibrational can be written as the
product of independent quantum oscillators.
It is possible to calculate the force constants of each harmonic oscillator starting
from ab initio Hartree-Fock or density functional theory (DFT) methods: the
symmetry of the potential function will allow the reduction in size of the dynamic
matrix to a group of smaller matrices, one for each irreducible representation of the
molecular point group. Each normal mode of a molecule forms a basis for an
irreducible representation of the point group of the molecule itself [I.8]. Group
theory can be used to recover the normal mode coordinates starting from the
geometry of the molecule: the use of the so-called character table permits not only to
list the vibrational modes, but it also assigns them to the respective class of the
symmetry operations that build up the point group [I.8]. At the same time, the study
of the molecule symmetry allows the identification of degenerate vibration
frequencies and it is crucial for the determination of a mode’s activity in the Infrared
or Raman spectrum.
I.2 Classical theory of Raman scattering.
Raman effect is classically explained in terms of the dependence of the polarizability
of the molecule on the normal modes of vibration of the molecule. The scattering of
an electromagnetic wave, indeed, can be expressed in terms of the polarization
20
induced in the material by the incoming electric field (the role of the magnetic field
can be neglected for the present treatment). It is known, in fact, that an oscillating
dipole absorbs end emits energy in the form of electromagnetic waves. Polarization,
indicated by the letter P, is a vector which describes the average (and hence,
macroscopic) contribution of the electric dipoles p which can be associated to the
molecules constituting the material. There are two possible models to describe this
quantity: one introduces the concept of susceptibility and is more suitable for
describing the behavior of a solid, being it an isolator or a metal; whereas for
molecules it is convenient to introduce the polarizability. Both these descriptions
consider only the linear dependence in the electric field E, but a more thorough
theory of scattering must include higher order terms1 as well (which are necessarily
taken into account for an explanation of phenomena such as hyper Raman or
stimulated Raman scattering, not considered here).
Susceptibility and polarizability.
The first approach follows a global description of the scattering material.
Polarization, then, can be written as the difference between the dielectric
displacement vector, D and the field in the vacuum multiplied by the permittivity of
the vacuum ε0. Polarization can be expressed, as well, making explicit the
proportionality to the electric field: the constant which relates the vector P and E is,
apart from a factor of ε0, known as susceptibility and written as χ
P = D - ε0E ε0 εR -1) E ε0 χ E (1.13)
χ is dependent only on the properties of the materials and on the frequency of the
electric field; in the most general case, where P is not necessarily parallel to E, it is a
tensor.
If the description of the electric induction is measured on the single molecule, it is
convenient to express the polarization in terms of the induced dipole momentum p;
1 The total time-dependent induced electric dipole moment vector of a molecule may be written as the sum of a series of time-dependent vectors as follows: p = p
(1) + p
(2) + p
(3) + …
where p(1)
= αE, p(2)
= ½ βEE, p(3)
= ⅙ γEEE …[I.12] and the order of magnitude of the addends
p(n) decreases dramatically with the order n.
21
this is related to the electric field through the polarizability α, which is again a tensor
(p can be non parallel to E).
p = α E (1.14)
where α is measured in C m2/V unity and its components are indicated with the
notation αρσ, the indices ρ and σ assume the values x, y, z. Except in very special
circumstances, the Raman polarizability tensor is real and symmetric, i.e. αρσ = ασρ;
it is always possible, moreover, to make it diagonal by a proper choice of the
coordinate frame of reference.
If corrections for the so-called local field are negligible, it is possible to correlate the
two expressions 1.1 and 1.2 through the definition of polarization P as the average
dipole momentum multiplied by the number of molecules per unit of volume.
P = <p> N/V (1.15)
Only in particularly simplified cases, however (for example, a gas in standard
conditions of temperature and pressure [I.9]), it is possible to trace an easy
relationship between susceptibility and the polarizability tensor,
χ = αN/ ε0V (1.16)
where α is the scalar corresponding to the average of the elements αii of the
polarizability matrix in its diagonal form [I.10]. More generally, it can be said that
the electric susceptibility is the sum of the polarizabilities of the single molecules
contained in the unit volume, divided per the dielectric constant and the volume
itself.
Rayleigh and Raman scattering.
For the present description it is sufficient to refer to the induced dipole moment of
the molecule, and therefore, to the polarizability. This quantity is dependent on the
distribution of the electrons around the ions of the molecule; the perturbation
introduced by the incident electric field produces a displacement of the charge
distributions with respect to their position at the equilibrium: this phenomenon can
be expressed by the concept of an induced dipole. Polarizability depends also on the
characteristics of the exciting field: the response of the charge distribution is not
instantaneous [I.11], so the effect of applying a constant perturbation is not the same
22
of an electric field which varies with a pulsation ω. In synthesis, from a
phenomenological viewpoint, it is possible to state that polarizability is a function of
the angular frequency of the electric field
p(ω) = α(ω)E(ω) (1.17)
for the simplest description of the elastic (or Rayleigh) scattering, that is of the
scattering of light at the same frequency of the exciting electromagnetic wave. The
actual form of this tensor can be derived by a series of descriptions, one of which is
represented by the Lorentz model of the atom as a harmonic oscillator with a
natural pulsation ω0.
More generally, the induced dipole possesses a spectral distribution which is not
simply a Dirac delta centered on the angular frequency of the exciting electric field.
This fact implies that the polarizability not only is a function of this variable, but it is
also relevantly influenced by the natural pulsations ωk of the vibrations of the
molecule. The induced dipole emitting at a generic angular frequency ωS, where S
stands for Scattering, can be described by the equation
p(ωS) = α(ω, ωk)E(ω) (1.18)
given vibration will hence induce a “modulation” of the linear optical polarizability
with a frequency equal to that of the internal vibration. Each of this motions can be
decomposed, as we have seen, in a basis of orthonormal vibrational coordinates Qk.
It is possible to expand in a Taylor series every matrix element of α according to the
normal coordinates of vibration Qk of the molecule and neglecting, for clarity of the
notation, the dependence on the pulsation of the exciting field. The expansion of the
polarizability component αij, where the indices i and j can vary among the chosen
Where the quantity α/ Qk)0 is called derived polarizability tensor or Raman
tensor of the normal mode k. It is apparent how, besides the presence of the so-
called Rayleigh component at ω, the emergence of two symmetric terms ω – ωk
(Stokes) and ω + ωk (anti-Stokes) per each vibrational mode is justified simply by
trigonometry [I.12]. The classical expression for Raman effect, therefore, is related
to the polarizability components which have a non-zero derivative with respect to a
molecular vibration at the equilibrium position of the molecule. The main
consequence of this fact is that not all vibrational modes can be investigated by the
detection of Raman scattering, but vibrations which correspond to a zero value for
the Raman tensor are said to be Raman-inactive.
The classical interpretation of Rayleigh and Raman scattering can, in conclusion, be
summarized as follows. If the molecule is at rest, the induced moment, and therefore
the scattered light, has the same frequency as the incident light. If, however, the
molecule is rotating or vibrating, this is not necessarily the case, because the
amplitude of the induced electric moment may depend on the orientation of the
molecule and the relative positions of its atoms. Since the configuration changes
periodically because of rotation and vibration, the scattered radiation is
''modulated" by the rotational and vibrational frequencies so that it consists of light
of frequencies equal to the sum and to the difference of the incident frequency and
the frequencies of the molecular motions, in addition to the incident frequency [I.7].
Another difference is that Rayleigh scattering happens always in phase with the
incident radiation, whereas Raman scattering bears an arbitrary phase relation due
to the phase k of the involved molecular vibrations [I.12].
(1.20)
24
Fig. I.2. Example of dependence of the polarizability from the vibrational modes and correspondent
Raman activity for a linear molecule of the type A-B-A (for example, CO2)[I.12]
I.3 Quantum theory of Raman scattering.
It has been shown that a classical theory of Raman scattering is possible, even if it
adopts a phenomenological viewpoint. Nonetheless, some of the conclusions which
can be drawn, for example, from equation 1.20 can be misleading. For instance,
correlating the Raman intensity (proportional to the square of the scattered field) to
the Raman tensor would predict equal intensities for the inelastic and superelastic
component of the spectrum, since the magnitude of the derivative of the
polarizability is the same for both the branches: prediction that is strongly denied by
the actual spectroscopic measurements, in which Stokes scattering is most easily
detected. Moreover, dependence of Raman on the vibrations of the molecule would
infer that no Raman spectrum can be obtained by molecules on the vibrational
ground state, contrarily to what is observed. This deficiency of the classical
treatment, in particular, is to be related to the absence of the zero-point energy in a
harmonic oscillator problem, which leads to the impossibility to describe
phenomena like spontaneous emission of light. A quantum treatment of the
25
scattering process is, therefore, necessary, even for a treatment that does not
include nonlinear phenomena such as hyper Raman scattering.
From a quantum viewpoint, Raman scattering may be treated as the inelastic
collision of an incident photon ħωo with a molecule on the initial energy level Enm
[I. 3]. Following the collision, a photon ħωs with different energy is detected and the
molecule is found on another energy level En. The energy difference ΔE = En-Em =
ħωnm, when positive, may appear as vibrational, rotational of electronic energy of
the molecule: this variation is called Stokes shift. If the molecule is initially in a
excited state, the difference can be negative and the superelastic photon scattering is
indicated as an anti-Stokes event. If the character of a vibration to be or not Raman
active can in principle be derived by classical or semi-classical description, the
intensity of a Raman band is not easily predicted by these treatments. For this
reason, a more correct formula for the Raman cross section of a vibration must be
the aim of the full quantum theory of light scattering.
Time dependent perturbations: the second order transition
probability.
In a quantum system described by the time independent Hamiltonian ℋ0, every
eigenstate is described by the symbol |u⟩ and corresponds to an eigenfucntion Ψu
and to an eigenvalue Eu: this eigenvalue governs the exponential time dependence of
the eigenfunction
Ψ ⟩ ⟩ e
/
Since the probability that the quantum system is in a state n is equal to the square
modulus of |Ψn⟩, it does not change with time: therefore, the description of any
transitions between two states must take into account the time dependence.
In Schrödinger’s representation the operators are constant whereas the wave
functions evolve with time. If a perturbation ℋi = ℋi(t) is introduced, so that ℋi(t)
Ŵ t > , the states u⟩ (from now on |u⟩), are still an orthonormal base for the
vector space. This means that a generic state |Ψ(t)⟩ can be expressed as
Ψ t ⟩ au t e i ut/ u⟩
u
(1.21)
(1.22)
26
provided that the complex coefficients au are functions of the time. As for the time-
independent case, their square modulus is proportional to the probability to find the
system in a state |n⟩ at the time t. If the initial state of the system is the state |m⟩, the
summation at time t = 0 is degenerate. Thanks to this property, an expression for an
can be obtained by substitution of 1.1 into the Schrödinger equation; this, after a
simple multiplication by ⟨n| eiEnt/ of both sides leads to
an
t
i au t e
– i u – n t/
u
n W u
Exact solution for this set of equations is possible only for limited cases (for
example, a two-level system). In general, it is necessary to proceed by successive
approximations by developing each coefficient au as a sum of different order terms.
au = au(0) + au(1) + au(2)
this approach is known as perturbative method. The validity of this method is
confirmed, in the case here discussed, as long as the applied electric field is a small
perturbation from the typical electric fields felt by electrons and atoms in molecules
(which is of the order of 1010 V/m)[ I.11]. A Raman scattering description must
consider at least to stop this expansion to the second order, because from a particle
viewpoint this phenomenon involves two photons. Since the perturbation is small,
the first order solution for a generic final state f is obtained by neglecting all terms
with u ≠ m in the equation . 3 they are strictly null at t , thus reducing the
summation to a single element
an
t
i e – i m – n t/ n W m
and integrating with time τ (with the consideration that the perturbing Hamiltonian
matrix has zero diagonal elements, i.e ⟨u Ŵ u⟩ = 0).
an
i e – i m – t/
τ
n W m dt
For higher orders, the procedure is iterative: the second order solution is given by
substituting into 1.2.2 the first order expression of the an. The resulting expression is
(1.24)
(1.25)
(1.26)
(1.23)
27
a
i a
t
e – – /
n W u dt
in which the expression of the first order solution can be made explicit. This
equation is valid for each choice of interacting Hamiltonian, provided that the
perturbation is “turned on” at time t . The electromagnetic field incident on the
molecule can be represented as a periodic potential e pressed by the function Ŵ t
Ŵ+ e–iωt Ŵ- eiωt, with Ŵ+ = Ŵ-† to make sure it is Hermitian and, hence, physically
meaningful. Spatial dependence of the electromagnetic field is thought as negligible
along the more significant dimension of the considered system, as it is for a molecule
exposed to NIR or visible light. If the energy difference between the state |n⟩ and |m⟩
is written as ħωmn, the expression for the first order coefficient is
a
n W m
e –
ω – ω n W m
e
ω ω
which is able to describe optical transitions where a single photon is involved
(absorption). The second order coefficient, substituting 1.28 into 1.27 and exploiting
the exponential dependence of the perturbing Hamiltonian with time, can be
synthetically written as
a
i
n W u u W m
ω ω
e
e dt
This expression can be separated in the sum of six summations: according to the
presence of one or the opposite sign, it is possible to distinguish different domains
for the energy values of the involved eigenstates. The sign of Ŵ is plus if n > Em,
minus vice versa, whereas the sign between the angular frequencies is chosen
according to the position of the intermediate energy level |u⟩: in particular, it is +
when Eu > Em. The transition probability Pmn is evaluated as the square modulus of
the coefficient an, which according to eq. 1.24 is given by the sum of the two
components an(1) and an(2),
Pmn ≈ an(1) + an(2) |2
This is the complete probability that a transition between the levels m and n,
mediated by an electromagnetic field represented by the perturbing Hamiltonian Ŵ,
occurs: all phenomena which involve one or two photons interacting with the
system (atom or molecule) are therefore included. Despite its generality, it is
(1.29)
(1.30)
(1.27)
(1.28)
28
already possible to highlight some peculiarities in the formula above reported. First
of all, the presence of differences at the denominators hints at the possibility of
indefinitely high value (in practice limited by the uncertainty at which the energy of
the quantum states can be measured) for the probability transitions: in other words,
the expression takes into account the resonant behavior which characterizes the
interaction of an electric field whose frequency is correspondent to a “jump”
between two states in the quantum system considered. Additionally, the transition
probability is shown to be dependent on the off-diagonal matrix elements of the
perturbing Hamiltonian Ŵ, which can be written as an integral quantity:
W u W v W d
This matrix expression reveals how the introduced perturbation determines which
transition between the unperturbed set of quantum states is or not allowed. A
theory of Raman scattering must start from a meaningful expression of the
interaction of the exciting light with the molecule or solid and then consider all the
terms inside 1.30 corresponding to inelastic (or superelastic) scattering, i.e. where
the angular frequency of the incident electric field is shifted.
A semi-classical expression for Raman effect.
This description treats the interaction of a quantum system (the molecule), whose
Hamiltonian H0 can be expressed as in equation 1.3, with a classical electromagnetic
field: the energy contribution of this interaction is represented by the symbol ℋi.
The interaction Hamiltonian for a multiparticle system exposed to an
electromagnetic field does not show an easy to handle expression, so it is often
necessary to approximate it with the so-called Electric-Dipole (ED) Hamiltonian:
ℋi = ℋED = – E p
which is by far the most important contribution to the perturbing Hamiltonian; the
other terms (magnetic dipole, electric quadrupole etc.) arising from the interaction
are negligible as far as ℋED is not zero, so in most cases the approximation is
(1.32)
(1.31)
29
excellent2. The electric field E is supposed to be a plane monochromatic wave of
angular frequency ω, whereas the operator p is defined as the sum of all the
coordinate vectors of the nuclei and electrons,
where q is equal to ±e if a proton or an electron charge is considered. For Raman
scattering of molecule this definition works also in a simpler depiction, with rj being
the atoms coordinates and q the effective charges they carry on. Sometimes,
moreover, the total dipole moment of the molecule can be considered as the sum of
biatomic dipole contributions [I.12] constituted by the couple of nearest neighbor
atoms. The vector components of p, px, py, pz are similarly defined with respect to the
projection of rj and r along the Cartesian coordinates.
The following treatment is a synthesis of the approach followed by Placzek [I.14]: his
description of the Raman scattering does not explicitly proceed through the
transition probability expression, but an analogous method is used to determine the
form of the electric polarizability. A transition dipole, equivalent to the matrix
element of the dipole operator among the time dependent wave functions Ψ’n, Ψ’n of
the perturbed system, is introduced
(p)mn = ⟨Ψ’n| p |Ψ’m⟩
This transition dipole can be considered as the quantum equivalent of the
oscillating electric dipole of the classic description. Thanks to perturbation theory,
an expression of the transition dipole in terms of the non-perturbed eigenstates |u⟩
can be recovered. By arresting the expansion of the perturbed functions to the first-
order, and after grouping all the terms with the same frequency dependence, the
expression of the real component of the component ρ of the vector p, with the choice
of the symbol ωmn = (En – Em)/ , results3:
2 For linear processes in E such as standard Raman scattering it is sufficient to consider the electric dipole term: for a standard molecular transition, the electric quadrupole and magnetic dipole terms, which follow in the expansion of ℋi, are of the order of the so-called fine structure constant α ≈ 1/137 [2]. 3 A more rigorous expression takes into account the decay time of each intermediate state |u⟩, which, differently from the initial and final ones, does not need to be stationary. Considering these decays as constants removes the divergence caused by the difference at the denominators.
(1.33)
(1.34)
30
p
n p u u p m
ω ω e
n p u u p m
ω ω
e
∑
e
e c. c.
where c.c. stands for complex conjugate; the expression, rather cumbersome,
manages to express the transition dipoles in terms of the matrix element of the
dipole operator, evaluated on the unperturbed states; the terms can be grossly
divided in two, the ones oscillating at the frequency sum and the ones at the
difference. The latter take into account the three kinds of scattering, provided that
the quantity ω – ωnm is positive: Rayleigh is obtained for |m⟩ = |n⟩; the terms with
the frequency (ω + ωnm) at the exponential, instead, do not correspond to any
scattering process. Recalling the classical expression which relates the induced
dipole to the electric field, it is possible to derive an equation defining the so-called
general transition polarizability (α)fi, whose components are of the form
α
n p u u p m
ω ω
n p u u p m
ω ω
Fig. I.3. Pictorial representation of the vibrational transitions corresponding to the three kinds of
scattering. For the anti-stokes transition the order of the n and m states is reversed with respect to
the notation adopted here.
(1.36)
(1.35)
31
In principle, the calculus of polarizability, and hence of the Raman scattering cross
section, is made possible only through an accurate estimation of every energy level,
wave function and lifetime of the molecule. Fortunately, a series of general
assumptions can be made to apply this formula to concrete cases of studies. If the
energy of the incident photon, for example, is not sufficient to excite the molecule in
its first electronic level, and the ground electronic state is not degenerate, the
eigenstates involved in the matrix elements can be identified with the set of the
vibrational eigenfunctions (vi). The transition polarizability, therefore, can be
written in the compact form
α v α v
Other simplifications stand in less favorable cases, introducing the coupling between
electronic and nuclear motion can be as a second perturbation. In any case, it is
possible in the end to come to a more manageable formula of this sort, for the
scattered intensity summed over all the space directions:
This form, much like the Raman tensor of the classic description, allows a rapid
determination of the so-called selection rules which determine if a vibrational
transition can be triggered by Raman scattering. In the mechanical and electric
approximation, the transitions between pure vibrational states are possible if, and
only if, the variation of the quantum number is ±1. A more general formulation of
this constraint regards the symmetry representations of the elements involved in
the equation of the transition polarizability. These components transform as the
binary products x2, y2, z2, xy, xz and yz. Group theory characters tables, therefore,
show which irreducible representation correspond to Raman or IR active mode
according to the presence of these functions.
Γ1⊂Γ(vm)Γ(αρσ)Γ(vn)
Where Γ1 is the totally symmetric representation [I.8, I.12]. This general rule makes
no statement as to the intensity with which a permitted transition will appear in the
(1.37)
(1.38)
(1.39)
32
Raman spectrum; in a mono-dimensional case it can be described as an argument
based on the parity of the functions which concur to the product inside the integral
(1.37).
Full quantum approach.
The full quantum approach becomes necessary whenever a resonance phenomenon
is affecting the Raman scattering [I.15]: it is through this formulation, indeed, that
the role of the coupling between the photons and the electronic states of the
material can be evaluated.
In a multi-level atom or molecule, several processes can occur in presence of an
incoming electromagnetic radiation (fig. I.4). In terms of the quantum theory of light,
Raman scattering consists in the destruction of an incoming photon (of frequency ν,
pulsation ω = 2πν and wave vector k, k= w n(ω)/c) in correspondence with the
creation of a photon of a difference pulsation ωs and wave vector ks: the process
leaves the quantum system on a different energy level and can be considered
instantaneous [I.16]. It is, as well, a coherent event [I.4], in the sense that the two
phases must satisfy conservation of energy and momentum: the differences ω – ωs, k
– ks, are therefore associated with a transfer of energy and momentum to the
molecule. This transfer results, in the second case, in the excitation of one or more
vibrational modes; if the scattering material is a crystal, the Raman scattering
corresponds to the excitation of a so-called phonon, i.e. a quantized oscillation
distributed along the whole lattice. The key difference between Raman scattering
and fluorescence is that in Raman scattering the incident photon is not fully
absorbed and instead perturbs the molecule exciting or de-exciting vibrational or
rotational energy states. Contrastingly, in fluorescence the photon is completely
absorbed causing the molecule to jump to a higher electronic state, and then the
emitted photon is due to the molecule’s decay back to a lower energy state [I. 7].
Three terms contribute to the transition rate associated with this event (creation
followed by destruction, destruction followed by creation, creation and destruction
simultaneously occurring) according to the quantum mechanical description: in this
formalism there is no difference between the order in which the two events occur,
provided that the energy conservation is satisfied by their combination [I.16]. Each
of these terms, indeed, is multiplied by a Dirac delta along the ωs function centered
33
at ω – ωmn, where ωmn is the pulsation of the final state. However, as already
emphasized, the energy ω does not necessarily correspond to any electronic
transition energy and the photon of energy ωmn is not absorbed in the strict
spectroscopic sense, as there is no conservation of energy in this stage of the
process. The role of the incident radiation is rather to perturb the molecule and
open the possibility of spectroscopic transitions other than direct absorption. If ω
approaches an electronic transition energy, enhancement of the scattered intensity
is observed [I.12]. It is important to note that Raman scattering is not the absorption
of the photon, followed by the emission of a photon of less energy, that instead
describes fluorescence.
Fig.I.4. A comparison between the most important different transitions involving the interaction of a molecule
with photons in the visible-near infrared range, on a schematic energy diagram [I.23].
Second quantization.
A step further in the quantum theory of scattering needs the introduction of proper
quantum operator for electric field. Let us consider for the moment only one mode
of oscillation. Electromagnetic radiation with a single frequency ωk and a wave
vector k is fully described by the potential vector Ak(t) = Ak e-i(ωt-kr) + Ak* ei(ωt-kr). It is
34
associated with an energy term which can be suitably expressed as the energy of a
classical harmonic oscillator if quantities
are introduced. The expression for the energy is, therefore,
2ε0AkAk*V = ½ (pk2 + qk2ω2). (1.41)
The quantum solution for this kind of problems requires the replacement of the
classical quantities with suitable operators qk q , pk
p , which can be considered as
a coordinate and its momentum, since the commutation relationship gives [p k,q k] =
– i . The Schrödinger equation for this kind of potential gives as a solution a set of
discrete eigenvalues for the radiation energy En,k = ωk (nk + ½), where nk is an
integer. For this reason, it is possible to consider each contribute of ωk as a
quantum of energy, called photon; in this way, to each eigenvalue for energy a wave
function called “number state” or Fock state and corresponding to the presence of n
photons can be found. This number states are eigenfunctions not only of the
Hamiltonian, but of the number operator n k = âk †âk, given by the product of the so
called operators “creation” and “destruction” whose definition is respectively
These two operators are not observables, but they correspond to the addition and
subtraction of one quantum of energy to the eigenstates of the Hamiltonian,
provided that a minimum amount of energy ( ωk/2, zero-point energy) is
conserved, transforming them from |nk⟩ to |nk+1⟩ and |nk-1⟩ respectively.
The set of all the possible modes of the electromagnetic field in a cavity (or in free
space) can be regarded as the cumulative wave function of all the number states.
|{nk}⟩ = |nk1⟩|nk2⟩
(1.40)
(1.40b)
(1.42)
(1.42b)
(1.43)
35
There is always an infinite number of oscillators, so the |{nk}⟩ form a complete set of
state for the electromagnetic field in the cavity.
Starting from these functions it is possible to recover the expression for the
Potential vector with straightforward substitutions, and hence the quantum
expression for electric and magnetic field. Equations 1.44 and 1.44b give the
expression for the k mode of both fields.
These expressions are used when the perturbation is added to the Hamiltonian of
the molecule. If the interaction between the incoming electromagnetic wave and the
molecule is limited to ℋED, then the dipole momentum gives rise to the transition
dipole pfi,
To make explicit the transition nature of this quantity, the dipole is associated to the
transition operator |i⟩⟨j|, which, when applied to |j⟩ gives |i⟩ and gives zero with all
the other eigenstates. Thanks to the closure theorem, the expression of the dipole
operator p can be rewritten in a more significant form.
The explicit form of the quantum operator p in ℋED is, then
⟩ ⟨ ⟩ ⟨ ⟩⟨
,
and therefore the perturbing Hamiltonian can be written as
ℋ
,
⟩
⟨
This e pression is responsible of the transitions between the states j and I, with I ≠ j
since the terms with I = j are null because of integral of odd functions. From the
(1.44)
(1.44b)
(1.45)
(1.46)
(1.47)
36
matrix elements it is possible to calculate absorption and emission rates for a
generic atom and photon distributions.
Before moving to the application of this formula to the study of the interaction of
particles with electromagnetic field, it is convenient to remind that the harmonic
oscillator problem is actually a common approximation for many physical problems.
The same vibrational energy contribution within the global molecular Hamiltonian,
as it was shown, can be expanded up to the second order and hence, each vibrational
degree of freedom Qk possesses an energy
Ev,k = (vk + ½) ωk, vk , , ,
where k, running from 1 to 3M-6, is the index of the vibrational modes and not a
wave vector. Vibrations, therefore, are ruled by the same quantum operators
introduced for photons, applied to the normal modes Qk: q k, q k†for the destruction
and the creation of a vibrational quantum respectively. Equivalently to quanta of
electromagnetic energy, quanta of vibrations can be treated as particles, which
statistically behave as bosons.
The Kramers-Heisenberg formula.
In principle, the forces between the atoms can be calculated a priori from the
electronic wave equation, but in practice this is not mathematically feasible (except
for H2), and it is necessary to postulate the forces in such a manner as to obtain
agreement with experiments. Therefore, although it is theoretically possible to start
with a model consisting of electrons and nuclei interacting by a Coulomb potential
and obeying the laws of quantum mechanics, in practice it is necessary to assume
the nature of the equilibrium configuration and of the forces between the atoms, so
that it seems more desirable to start with the model in which the atoms are the units
[20]. This system is then exposed to the perturbation of the electromagnetic field, in
the form of the quantum operators obtained by the second quantization process.
The following treatment allows the calculation of the cross-section for a multi-level
molecule by time-dependent perturbation theory. The hypotheses are the
assumption that the incident beam is weak enough to be considered a perturbing
element of the Hamiltonian, and that the broadening of the energy level is negligible.
(1.48)
37
Fig. I.5. Feynman diagrams of the two channels involved in the quantum description of Raman
scattering: the arrows describe the photons created or annihilated during the event, whereas on the
horizontal line the sequence of the molecule states involved in the transition are reported.
The transition rate (1/τ) for a process in which an incident photon of energy ω is
destroyed and another one of energy ωs is produced is calculated according to the
perturbation theory, arresting the expansion of the probability to the second order:
this quantity corresponds to the transition probability between two states m and n,
divided by time. The global transition rate for the scattering phenomenon is
summed over all the possible final states: moreover, in the first passages the
contribution of all the possibly involved phenomena are considered.
ℋ
ℋ ℋ
The expression contains both the first and the second order terms of the transition
probability and it is shown to be dependent only on the perturbation Hamiltonian
ℋi = ℋED. The first term inside the modulus reproduces the Fermi golden rule
expression and provides the rate of first-order transition form m to n; the second
(1.49)
38
term provides the indirect transitions from state m to state n via a range of virtual
intermediate state u [I.16].
A general scattering event does not involve a single mode of the radiation field, but
the creation and the destruction operator affect different modes of the
electromagnetic field: thus, it is necessary to introduce â†, â, â†s,âs. A quantum state
describing the interacting system is indicated by the bra |l, ls, i> where the first two
letters are the number of photons of the incident and the scattering fields
respectively, while the third indicates the molecule energy level. Adapting the
equation 1.49 for the transition rate to new notation, the expression for an initial
state of l incident photons and the molecule in the ground state (m = 0) results4 in
, , ℋ , , ℋ , ,
This kind of equation belongs to the class of integral equations that give the
solutions for quantum mechanical problems of scattering theory [I.18]. The second-
order transition rate can be simplified and a more explicit expression shows the two
concurring paths for the interaction of the two photons:
τ
πe ωωs3l
s nu um
ωu ω
nu s um
ωn ωu
ω ωs ωnm
ksn
where , s are the unit polarization vectors of the incident and scattered photons
and Dnm is the matrix element m|D|m . This expression correctly predict the
dependence on the third power of the scattered pulsation, which, multiplied by the
angular frequency of the incident wave, recover the fourth power typical of the
4 A more rigorous form can be found in the 2nd edition of Loudon [I.16]: the presence of the nonlinear Hamiltonian gives rise to an additional term inside the square modulus. The first term at the right hand side of the equation corresponds to a non linear event which destroys a photon k and creates ks: the Hamiltonian in this case is proportional to A2[5]. The second term (proportional to Ap) is implicitly made out of two contributes which differ in the type of the intermediate state l: one produces the operators in the order âs
†â, the other â, âs†. Separately the two contributes do not
conserve the energy, but their combination does, so can it be accepted; in the time between the two events the molecule belongs to a state l which is different from both the initial and the final state. Due to the relationship B2 = E2/c2, the nonlinear term can be neglected in most cases (when the denominator in the second term is different from zero).
(1.50)
(1.51)
39
scattering events.The transition rate can be converted to a cross-section by dividing
it for the flux density c/V
This expression, which was obtained first by Kramers and Heisenberg, is suitable for
describing both Rayleigh (m = n, and Thomson, for high frequency) and Raman
scattering. The summation is limited to ωf < ω so that the delta function is not
needed but the conservation of energy requirement is automatically fulfilled.
Complete formula takes into account the two independent photon polarizations
[I.16]. Both this equation and the semi-classical formula obtained by Placzek [I.14]
provide a general framework, and can be simplified according to the range of
energies involved by the scattering, or to the kind of molecular functions involved
(purely vibrational, vibronic, etc.).
General considerations.
- Raman scattering is much less effective than the elastic scattering, which normally
regards a small fraction (10-3, Raman scattering Theory, David W. Hahn) of the
incoming photons. Thus, high intensity light source are required for observing it.
Another strict constraint is, naturally, the monochromatic character of the exciting
electric field. For best resolution performances, a narrow emission-line laser is the
choice of preference.
- Raman spectra provide the same kind of information of IR absorption spectra, but
they obey to different selection rules: therefore, in many cases the two techniques
work in a complementary way to characterize the structural and chemical
properties of a material.
- Stokes branch is more intense than anti-Stokes: the main reason is
thermodynamical, and a proper evaluation of the ratio between symmetric bands
can be obtained with the quantum mechanic treatment.
- Vibrational spectrum are generally reported in terms of the relative shift,
expressed in wavenumbers or inverse wavelengths, between the energy of the
detected photon and the one of the exciting photon. The most common unit for the
inverse wavelength coordinate is the cm-1. This allows an easy comparison not only
(1.52)
40
among spectra of the same method obtained with different excitation wavelengths,
but also between Infrared and Raman spectra of the same material, despite
corresponding to different techniques, can be put side by side. On the other hand,
one must not forget that the spectral extension of the spectra is, in the absolute
frequencies scale, much different, with evident consequences on the potential
resonances effect of the chosen source of radiation and on the sensitivity of the
detector. If, for example, the N2 vibrational mode is considered, corresponding to
2331 cm-1, this will be found at 387 nm if the exciting wavelength is 355 nm; if,
instead, the incident wavelength is 632 nm, the mode will be shifted at 741nm [I.19].
- Even if only a full reconstruction of the vibrational levels of the molecule can
provide the correct description of every spectral feature, in many practical uses
Raman spectroscopy can be set aside by an extensive approach. In many molecules,
indeed, atoms can be grouped in sub-units which often possess their own vibrational
modes that are not strongly affected by the global geometry of the molecule. On the
basis of this consideration, it is possible to report in tables, now available for many
class of materials, the typical energy interval in which the vibrational mode
involving the sub-group can be found. An example is shown in figure I.6.[ I.3]
- Since Raman spectroscopy traditional probes use light sources in the visible or
near infrared wavelength range, Raman scattering is not the only form of radiation-
molecule interaction. On the contrary, it is much likely that the material shows
absorption for the chosen wavelength, and therefore luminescence (excitation of
electronic upper energy levels, followed by a radiative decay) is often a competitive
phenomenon for Raman. Different approaches have been proposed and adopted to
overcome, or at least to limit this problem: the last paragraph of this chapter
describes Surface-Enhanced Raman Spectroscopy as an increasingly convenient
option in this direction.
- The treatment reported here below does not consider the angular dependence of a
Raman scattering event. While useful consideration could be drawn from this
description, introducing the formalism needed for this part of the problem will
exceed the scope of this thesis. We refer to Griffiths [I.20] and to Sakurai [I.21] for a
complete theory of scattering.
41
Fig. I.6. Energy distribution of the most common vibrational modes in the spectral region 0-700 cm-1
[I.3].
I.4 The cross section.
Experimentally, the quantity that can be revealed by a detector of electromagnetic
radiation is the intensity; it is convenient, therefore, to quantify the probability that
a Raman event takes place not in terms of the scattered field, but of its square
modulus. Indeed, the classical expression [I.10] for the intensity associated to a
radiating dipole, along a direction which creates with the dipole p an angle ϑ is
IS = ωS4 |p|2 sin2ϑ /(32 π2 ε0 c3)
Where the temporal variation of p is averaged over an oscillation cycle. This quantity
can be also written in terms of the scattering direction, represented by the unit
vector eS
(1.53)
42
IS = ωS4 |eSp|2 /(32 π2 ε0 c3)
There is a way to correlate the intensity of the scattered field with the one incident
on the area of interest: cross section (σ) gives at a glance an indication on the
efficiency of the considered scattering process. Loudon defines σ as the rate at which
energy is removed from the incident beam by scattering, divided by the rate at
which the energy in the incident photon beam crosses a unit area perpendicular to
its propagation direction [I.16].
σ ωs
ωI Isr
d
In case of Raman scattering, the value of σ is not only related to the exciting laser
wavelength, but to a particular vibrational mode of the selected molecule of angular
frequency ω – ωs. This quantity can be expressed in terms of the transition rate τ of
a scattering event, provided that the volume V which participates to the scattering is
well defined as the product of the cross section times the length crossed in that time
by the light in the material, nk c τ.
A differential cross section, dσ/d can be introduced when the cross section angular
distribution is of interest: this is often the case for an electric dipole, where the
scattered radiation is not isotropic. A possible definition for the differential
scattering cross section is given by Cardona, as the ratio between the energy emitted
per unit time by an electric dipole M and the energy incident on that dipole divided
per unit area and unit time, [I.22]
dW = d |es α el|2 EL2ω4/(4π)2 ε0c3
by the energy incident on that dipole per unit area and unit time, W = ε0c EL2, and
integrating over the solid angle ; this definition can be proven to be equivalent to
the. The unitary vectors el and es respectively refer to the polarization of incoming
and scattered of (laser) light. The expansion of the polarizability carried out in 1.5
easily leads to an expression for the Raman component of the scattering cross
section. In equation I.56 the differential cross section for a Stokes-shifted line at
angular frequency ωs = ω0 – ωk is reported. [I.5, I.11]
dσs/d = |es α/ Qn el|2<QkQk*> ωs4/(4πε0)2c4
where < > represents the thermodynamical average over the ground state of the
molecule [I.22], needed to take into account the different population of the
(1.54)
(1.54)
(1.55)
(1.56)
43
vibrational levels in an ensemble of many molecules. This formula closely resembles
the intensity formula (1.37) obtained by the semi classical theory, and reveals the
explicit dependence of the polarizability tensor on the normal coordinates Qk.
I.5 Surface Enhanced Raman Scattering.
Though Raman spectroscopy gives detailed chemical and conformational
information, the small scattering cross-section has limited widespread use.
Experiments performed in the mid 1970s demonstrated that molecules adsorbed to
a roughened metal surface generated anomalously large Raman intensities [21]. This
phenomenon is known as Surface-Enhanced Raman Scattering (SERS), where the
last S can also be used to refer to Spectroscopy, where the attention is given to the
technique more than to the effect exploited. Enhancements varying from 103 to
1011[I.23] have been reported for molecules adsorbed to nano-structured silver,
gold and copper, in order of width of use, but other metals have found application in
minor cases. While there is still occasion to debate about a commonly accepted
theory of the phenomenon, general agreement is given to the attribution of the main
reason for this boost in the Raman cross section to a so-called Electromagnetic
Mechanism (EM). Occasionally, this factor can be helped by a Chemical Mechanism
(CHEM), which nonetheless is of lesser importance.
Electromagnetic Mechanism is mostly related to the excitation of a localized surface
Plasmon resonance by the laser source. Briefly, the nanometric features of the metal
substrate on which the molecule is adsorbed are characterized by collective
oscillations of charges (typically conduction electrons), which can be selectively
e cited by resonant laser wavelengths. These “plasma waves” are not only localized
inside the roughness features of the conductor, but naturally they are confined to its
surface: hence comes the name of Surface Plasmon Polaritons. This allows the
Raman cross section of the modes of the adsorbed material, which are parallel to the
excited plasmons, to benefit of the huge electric field which is localized along the
metal surface. CHEM enhancement, instead, is not only site-specific but it also
depends on the analyte: it is present when a charge-transfer state is created
between the metal and the adsorbed molecule. If the laser wavelength is not only in
resonance conditions with the metal nano-features, but with an electronic transition
44
of the molecule under analysis, the effect experiences a further enhancement and it
is called SERRS, where the first R of the acronym stands for Resonance.
The character of resonance technique is, however, implicit in the description of the
Surface-Enhanced Raman effect: for this reason, even if the enormous gain given to
the signal to noise ratio of the vibrational spectrum makes this technique preferable
for samples in very low concentrations, or for cases in which fluorescence would be
an unavoidable obstacle, some caveats must be considered whenever a SER analysis
is planned. First of all, the enhancement provided by the interaction with the
plasmon is naturally very selective: not all the vibrational modes are affected in the
same way, but, as previously reported, geometry plays a fundamental role in
determining the effect, due to the relative orientations of the vibrational modes with
the excited plasmons, the polarization of the incoming field, the distance between
the roughened surface and the different parts of the molecule, etc. Without going too
much into details, these variables result in a distortion of the SER spectra with
respect to the normal Raman ones, which can be traced in terms of a different
intensity ratioes between the peaks or, if the local environment of the molecule has
significantly changed, in broadening or shifts of the vibrational features. It must be
noted, moreover, that there does not exist a perfect SER substrate, which can be
used for every sample, but different materials and shapes produce variable effects
according to the wavelength of the chosen laser and, of course, to the analyte.
Nonetheless, the study and the exploitation of SERS are growing: in the last ten years
the number of published works dedicated to or containing the SER technique more
than doubled, and the trend does not seem to decrease [I.24]. The development and
consequently the availability of cutting-edge methods permits the researcher to
adopt an increasing number of possible geometries, shapes of the structures,
materials and, in general, probing methods. If the stability still makes the traditional
silver or gold nanoparticles a reliable starting point for a SER measurement, the
current tendency is to design the substrate parameters for a particular analyte,
making SERS a truly tailored detection and identification method.
45
Bibliography.
[I.1] A. Smekal, Naturwiss. 16, 873 (1923);
[I.2] C.V. Raman and K.S. Krishnan, Nature, 121, 501 (1928);
[I.3] E. Smith, G. Dent, Modern Raman spectroscopy: a practical approach, John Wiley
& Sons(2005);
[I.4] I.R. Lewis, H.W. Edwards. Handbook of Raman Spectroscopy. From the Research
Laboratory for the Process Line, CRC Press (2001);
[I.5] R. Aroca, Surface Enhanced Vibrational Spectroscopy, John Wiley & Sons
(2007);
[I.6] N. Ashcroft, M. David , Solid State Physics, Cengage Learning Emea (2000);
[I.7] E.B. Wilson, J.C. Decius, P.C. Cross, Molecular Vibrations: The Theory of Infrared
and Raman Vibrational Spectra, Dover Publications (1955);
[I.8] D.C. Harris, M.C. Bertolucci, Symmetry and spectroscopy: An Introduction To
Vibrational and Electronic Scattering, Dover Publications (1989);
[I.9] G.Moruzzi, Dispense di struttura della materia. University of Pisa
[I.10] P. Mazzoldi, M. Nigro, C. Voci, Fisica – Volume 2: Elettromagnetismo – Onde,
Edises (2000);
[I.11] E. Le Ru, P. Etchegoin, Principles of Surface Enhanced Raman Spectroscopy
and related plasmonic effect, Elsevier Science (2008);
[I.12] D. A. Long, The Raman effect. A unified treatment of the theory of Raman
scattering by molecules, John Wiley & Sons (2002);
[I.13] W. Demtröder, Laser Spectroscopy - Basic concepts and instrumentation,
Springer (1996);
46
[I.14] Placzek, G. Rayleigh-Streuung und Raman-Effekt, in Handbuch der Radiologie,
E. Marx (ed.), 6, 205–374. Academische Verlag: Leipzig (1934);
[I.15]W. H. Weber, R. Merlin, Raman Spectroscopy in Materials Science, Springer
(2000);
[I.16]R. Loudon, The quantum theory of light. Second edition, Clarendon Press,
Oxford,(1983);
[I.17]H.A. Szymanski, Raman Spectroscopy: Theory and Practice, Plenum Press,
(1967);
[I.18] E. Merzbacher, Quantum mechanics, 3rd edition, John Wiley & Sons (1998);
[I.19] David W. Hahn, Raman Scattering Theory, University of Florida (2007);
[I.20] D. Griffiths, Introduction to Quantum Mechanics, Prentice Hall, (1995);
[I.21] J. J. Sakurai, Modern Quantum Mechanics, Yoshioka Shoten, Kyoto, (1989);
[I.22] M. Cardona, G. Guntherodt, Light Scattering in Solids, Vol. 2, Basic concepts
and instrumentation, 3rd edition, Springer (1989);
[I.23] R. P. Van Duyne, C. Haynes. Raman Spectroscopy, Northwestern University
(review available online);
[I.24] B. Sharma, R. R Frontiera, A.I. Henry, E. Ringe, R. P. Van Duyne, SERS:
Materials, Applications, and the Future Surface Enhanced Raman Spectroscopy,
Materials Today, 15 (2012), 16–25.
47
CHAPTER II
THE REMOTE RAMAN SCANNER.
The determination of the physical processes resulting in a Raman scattering event,
carried out in the first chapter, enables the researchers to consider all the necessary
parameters in the preparation of an experiment. It is even more crucial, when
designing a novel instrument which relies on this effect, to determine the physical
quantities it involves, and hence the most suitable techniques for the detection of
this precious but weak signal, the critical points and the advantages of the possible
approaches and the recent discoveries and innovative methods concerning Raman
spectroscopy. This chapter focuses on the development of the Remote Raman
Scanner instrument, starting from its objectives to the phases of the project to the
construction. An introductory section, however, is dedicated to the application of
Raman spectroscopy to the field of Cultural heritage, in order to appreciate in which
direction the proposed device should be situated.
II.1. Raman spectroscopy in Cultural Heritage.
Identification of materials in and on objects of cultural and artistic interest has long
been recognized as a task necessary for uncovering the history of such objects and
finding out possibly how, where and when they were made, what was their use and
even how they have interacted with their environment during their lifetime.
Furthermore, knowing the materials not only opens a window to the past for
archaeologists and historians but also enables conservators to pave a safe way to the
future by means of proper preservation [II.1]. The main challenge in this type of
work is the vast diversity and variety of materials encountered, which include:
pigments, mineral or organic, natural or synthetic; metals or metal alloys and their
corrosion products; stone and glass; bio-organic materials such as wood, leather,
48
paper, parchment, but also oil, protein or sugar-based binding media and glues and
synthetic or natural varnish coatings and associated by-products arising during their
aging [II.2]. In this context, Raman spectroscopy proves to be an effective option,
providing a deep insight on the molecular [II.3] and structural [II.4] properties of a
material without altering or damaging it. As an optical measurement, it does not
need any contact between the probe and the sample and the stimulation of scattered
light does not imply high light intensities, so the power of the exciting
electromagnetic field can be kept well below the threshold of polymerization,
ablation or burning, with respect to the possible risks concerning different
materials. The detection of highly specific fingerprints such as the frequencies of the
vibrational modes enables the researchers to distinguish among similar materials
and even among different crystal phases of the same compound, such as in the case
of minerals; moreover, the vibrational frequencies depend on the strain undergone
by a solid sample, so they can be used as a stress indicator (showing deformations
before they become irreversible); furthermore, the influence of thermal energy on
the ratio between the Stokes and anti-Stokes spectral branches, can be measured to
monitor the temperature evolution of a sample, for example due to exposure to the
laser beam [II.5].
A huge number of publications attests in the last 20 years the widespread
possibilities of employing Raman spectroscopy on objects of art such as paintings