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The essential principles of infrared absorption and Raman scattering
viii of laser radiation make it easy to measure the polarization properties of the Raman lines and their absolute intensity (the scattering cross-section), while the highly monochromatic nature of the radiation simplifies the study of line shapes and fine structure. The number of publications on the application of the Raman scattering method has grown rapidly and now the ratio of IR to Raman investigations is close to one. These two methods complement each other in studies of the structure and physico-chemical properties of crystals and molecular systems. Raman spectroscopy has been limited in its applications by one major p o i n t fluorescence. As a phenomenon, fluorescence is approximately 106 - 108 times stronger than Raman scattering. Often, when one tries to excite a Raman spectrum, the fluorescence is the only phenomenon observed. Trace impurities, coatings on polymers, additives, etc., may fluoresce so strongly that it is impossible to observe the Raman spectrum of a major component. The use of UV or near-IR excitation has proved to be effective in reducing this problem. Its main reduction is related to the widespread application of FT-Raman spectroscopy.
The essential principles of infrared absorption and Raman scattering The simplest way of describing the mechanism of Raman spectroscopy is via an energy level diagram. An incident photon of energy hv0 interacts with a molecule having vibrational energy levels v~, v2, etc. Most of the incident radiation is unchanged in energy. It is transmitted, refracted, reflected, or even scattered, but at the same energy (frequency). A small portion of the energy, however, is lost to the vibrational energy levels and appears as h(v0-v0, h(v0-v2) ,
etc. This is the Raman-scattered radiation. If Vl,V2, etc., are relatively close to the
ground state, at ordinary temperatures these levels will have a significant population determined by the Boltzmann distribution. In this case, molecules in the vibrationally excited states can interact with the incident radiation and return to the ground state. This will result in energies of (v0+vl), (v0+v2), etc., being observed. The shifts to lower and higher energy are known as Stokes and anti-Stokes Raman scattering, respectively: the first type is used most frequently. In all spectroscopy there is a mechanism by which the incident radiation interacts with the molecular energy levels. For infrared (IR) absorption spectroscopy, which is associated with molecular vibrational energy levels, it is the change in dipole moment during the vibration. For Raman spectroscopy, the mechanism has its origins in the general phenomenon of light scattering, in which the electromagnetic radiation interacts with a pulsating, deformable (polarizable) electron
ix
cloud. In the specific case of vibrational Raman scattering, this interaction is modulated by the molecular vibrations. Suppose that the incident electric field associated with the light, which is the wave phenomenon, is represented by E = Eo cos 2~rvt, where E is the time-dependent intensity, Eo the maximum amplitude, and v is the frequency. This field induces a dipole ~t, such that Ia = a E = aEo cos 2rcvt,
where the proportionality constant et is known as the polarizability. The classical theory gives the average rate of total radiation as I = (16 "JT,4/3C 3) V4 ~0 2,
where l.t0 is the amplitude of ~t. For this case the scattered radiation has the same frequency as the incident. The expression for ~t can be rewritten in terms of Cartesian components; in its most general form: btx= ~xxEx+fl;xyEy+0~xzEz ).ty=~yxEx+%yEy+%zEz ).tz=~zxEx+~zyEy+~zzEz
Iil Ixxxy IExl
and this can be rewritten as a matrix equation ~t=~E, that is
=
[~yx ~yy
~yz
[~zx
[~zz
~zy
X
Ey
Ez
For almost every case, ~ is a symmetric matrix
([~xy=l~yx, etc.). Now suppose that the
scattering body is not just a polarizable sphere but has vibrational modes of its own - normal modes, Q, described by Qk = Qk~
2nVkt.
These oscillations can affect the polarizability, and this effect can be written as t~ = t~o+(Ot~/OQk~ Qk + higher-order terms. Multiplying by E gives
~E=ltt=~oE+(0~/0Qk~
E.
l-he expression for ~t now becomes ~t = aoEocos 2~vt + EoQk~
cos 27rvt cos 2~Vkt.
Using a trigonometric identity for the product of two cosines, this can be rewritten as ~t = ~oEocoS 2~vt + 0.5EoQk~
27r(v + v0t + cos 2~(v - Vk)t].
The three terms of this equation represent the three major phenomena observed in a simple Raman spectroscopy experiment: the first term is elastic scattering (without frequency change), known as Rayleigh scattering, the second term, of frequency (V+Vk), is anti-Stokes Raman scattering, and the third one is the Stokes Raman scattering. The classical description gives only a very limited insight into the relative intensities of each of these phenomena. One does expect that 0a/0Qk will be much smaller than a0, so that the Raman scattering should be less intense than the Rayleigh scattering. This is in fact the case. Moreover, the classical prediction indicates a simple, linear dependence of Raman scattering on incident beam intensity and sample concentration, again consistent with experiment except for certain special cases. The relative intensities of the Stokes - and anti-Stokes-scattering are only predicted to differ by the ratio of [(V-Vk)/(V+Vk)]4, which is not in accord with observation. The Boltzmann distribution will be the major factor in determining the relative intensities of these two phenomena. The population of any excited level is always less than that of the ground state, making the Stokes Raman scattering always more intense than the anti-Stokes. A full quantum mechanical treatment of the Raman effect is usually done using time-dependent perturbation theory (see Long [6]) and only certain key results will be given here. From the classical approach it can be appreciated that the geometry of the sample and that of experiment (incident and observing directions) will affect the observations. For analytical purposes, the most important samples are liquids and randomly oriented solids. The
commonly used experimental geometry has the
observation at right angles to the excitation, although there are occasionally good reasons for observing the scattering in other directions, particularly at 180 ~ to the direction of excitation. A special case of interest, that of oriented polymers, is discussed in ref. [7]. Placzek [8] originally derived the expressions for Raman scattering with different geometries, including the conventional 90 ~ scattering, and put these into a convenient form. In these expressions, the polarizability (a) is divided into two parts:
[~=~s+~a, where t~S is the symmetric or isotropic part and t~a is the asymmetric or anisotropic part. These are defined as 30r = ~xx+ ~yy"b ~zz
2(tta) 2= [(ttxx- [~yy)27t- (~yy- 0~zz)2-[- ([~zz- ~xx) 27t- 6(t~xy2+ 0~yx2 d" [~zx2)]. It is possible to make a transformation from Cartesian coordinates to principal axes so that these expressions take the simpler forms: t~'= 1/3(t~ + t~2+ t~3) and
xi
2(=a)2 = (=,_ %)2 + (~2- %)2 + (%_ =,)2. For molecular vibrations, it is not the polarizabilities themselves that we are dealing with, but rather the elements of the matrix of polarizability derivatives, (O=/OQ), usually designated as ='. Placzek's result for Raman scattering at right angles, in terms of these components of the polarizability derivatives connecting a molecule initially in state m and finally in state n, is I = constant[(v +
Vmn)4/Vmn]X[NIo/1
-
exp(-hVmJkt)]X[45(='s) 2+ 13(l~'a)2]
where N is the number of molecules in state m, and I0 is the incident intensity. The constants 45 and 13 arise from the orientational averaging process (see [6] for details) and are a consequence of the experimental geometry. This yields the ratio of Stokes - to anti-Stokes - intensity Is/Ias = [(Vo- Vmn)4/(VO -[- Vmn )4 IX exp(hvmjkT), which is verified experimentally at thermal equilibrium. These expressions assume that v0 is far from any electronic energy levels of the molecule. What was done here so far [9] only gives us the terms in the expression for Raman intensity. It does not say whether the key terms in this expression, the t~'s, are non-zero for a particular vibrational mode. In fact, this is very difficult to predict. But group-theory allows us to predict whether these terms can be non-zero, using information about the symmetry of a molecule or crystal. In each case, group-theory is used to predict whether a transition moment integral can be non-zero. These integrals contain the product of three terms: the wave functions for the ground- and excited-states, and the operator (in this case, the components of the polarizability derivatives) that connects these two states. For a transition to be observed, the product of these three terms must be totally symmetric; that is, it must leave the original molecular symmetry unchanged. One finds that, in molecules of high symmetry, both IR and Raman spectroscopy are needed to observe the vibrational modes. Even with both techniques, there may still be some vibrations that are totally forbidden. The best known selection rule for IR and Raman spectroscopy is known as the "Rule of Mutual Exclusion", which states that if a molecule has a centre of symmetry, vibrations cannot be active in both IR and Raman spectroscopy. This rule has often been applied in molecular structure investigations to determine whether a centre of symmetry is present. In general, vibrations that do not distort the molecular symmetry, "symmetric vibrations", are intense in the Raman spectrum while those that maximize the distortion are most intense in the IR spectrum. If the atoms involved in these vibrations are highly polarizable (e.g., sulfur or iodine) then the Raman intensity is high. Some examples of
xii
vibrational modes that are of importance in the Raman scattering of polymers, and their frequency ranges, are shown in Table 1. There are four main generalizations of the common observations about Raman spectral intensities [9]: 1. Stretching vibrations associated with chemical bonds should be more intense than deformation vibrations. 2. Multiple chemical bonds should give rise to intense stretching modes. For example, a Raman band corresponding to a C=C (or C=C) vibration should be more intense than that from a C-C vibration. 3. Bonds involving atoms of large atomic mass are expected to give rise to stretching vibrations of high Raman intensity. The S-S linkages in proteins are good examples of this [ 10]. 4. Those Raman features arising from normal co-ordinates involving two in-phase bondstretching motions are more intense than those involving a 180 ~ phase difference. Similarly, for cyclic compounds, the in-phase "breathing" mode is usually the most intense.
Important advantages of Raman spectroscopy 1. The "transparency" of water and glass: the very low Raman scattering of water (which is important for living systems) and of glass make it suitable for dilute aqueous solutions of substances as well as for hygroscopic materials, and permits the use of standard glass cuvettes and capillaries.
2. Non-destructivity, and the absence of need of very sophisticated sample preparation. Raman spectroscopy is equally suitable for the analysis of gases, liquids, fibres, single crystals, surface features, etc. Intact measurements permit one to investigate the native molecular structure in biopolymers, living and other systems. It permits studies of eye lenses, the end processes of muscle contraction, components of living cells, and of ancient manuscripts and art objects, etc. The crystallinity of polymeric materials and orientation effects in fibres, monitored by FT-Raman spectra, could be very useful in technological control and in forensic science.
3. Symmetrical bonds such as C-C, C=C, C=C, N=N, 0-0, S-S, manifest themselves by giving the most intensive bands in Raman spectra, and especially structures with the latter heavy atoms, while they are inactive in the infrared. Among spectral methods Raman, is exceptional in showing the structure of natural S-S cross-linkages in biomolecules, artificial ones in vulcanized
The essential principles of infrared absorption and Raman scattering The simplest way of describing the mechanism of Raman spectroscopy is via an energy level diagram. An incident photon of energy hv0 interacts with a molecule having vibrational energy levels v~, v2, etc. Most of the incident radiation is unchanged in energy. It is transmitted, refracted, reflected, or even scattered, but at the same energy (frequency). A small portion of the energy, however, is lost to the vibrational energy levels and appears as h(v0-v0, h(v0-v2) ,
etc. This is the Raman-scattered radiation. If Vl,V2, etc., are relatively close to the
ground state, at ordinary temperatures these levels will have a significant population determined by the Boltzmann distribution. In this case, molecules in the vibrationally excited states can interact with the incident radiation and return to the ground state. This will result in energies of (v0+vl), (v0+v2), etc., being observed. The shifts to lower and higher energy are known as Stokes and anti-Stokes Raman scattering, respectively: the first type is used most frequently. In all spectroscopy there is a mechanism by which the incident radiation interacts with the molecular energy levels. For infrared (IR) absorption spectroscopy, which is associated with molecular vibrational energy levels, it is the change in dipole moment during the vibration. For Raman spectroscopy, the mechanism has its origins in the general phenomenon of light scattering, in which the electromagnetic radiation interacts with a pulsating, deformable (polarizable) electron
ix
cloud. In the specific case of vibrational Raman scattering, this interaction is modulated by the molecular vibrations. Suppose that the incident electric field associated with the light, which is the wave phenomenon, is represented by E = Eo cos 2~rvt, where E is the time-dependent intensity, Eo the maximum amplitude, and v is the frequency. This field induces a dipole ~t, such that Ia = a E = aEo cos 2rcvt,
where the proportionality constant et is known as the polarizability. The classical theory gives the average rate of total radiation as I = (16 "JT,4/3C 3) V4 ~0 2,
where l.t0 is the amplitude of ~t. For this case the scattered radiation has the same frequency as the incident. The expression for ~t can be rewritten in terms of Cartesian components; in its most general form: btx= ~xxEx+fl;xyEy+0~xzEz ).ty=~yxEx+%yEy+%zEz ).tz=~zxEx+~zyEy+~zzEz
Iil Ixxxy IExl
and this can be rewritten as a matrix equation ~t=~E, that is
=
[~yx ~yy
~yz
[~zx
[~zz
~zy
X
Ey
Ez
For almost every case, ~ is a symmetric matrix
([~xy=l~yx, etc.). Now suppose that the
scattering body is not just a polarizable sphere but has vibrational modes of its own - normal modes, Q, described by Qk = Qk~
2nVkt.
These oscillations can affect the polarizability, and this effect can be written as t~ = t~o+(Ot~/OQk~ Qk + higher-order terms. Multiplying by E gives
~E=ltt=~oE+(0~/0Qk~
E.
l-he expression for ~t now becomes ~t = aoEocos 2~vt + EoQk~
cos 27rvt cos 2~Vkt.
Using a trigonometric identity for the product of two cosines, this can be rewritten as ~t = ~oEocoS 2~vt + 0.5EoQk~
27r(v + v0t + cos 2~(v - Vk)t].
The three terms of this equation represent the three major phenomena observed in a simple Raman spectroscopy experiment: the first term is elastic scattering (without frequency change), known as Rayleigh scattering, the second term, of frequency (V+Vk), is anti-Stokes Raman scattering, and the third one is the Stokes Raman scattering. The classical description gives only a very limited insight into the relative intensities of each of these phenomena. One does expect that 0a/0Qk will be much smaller than a0, so that the Raman scattering should be less intense than the Rayleigh scattering. This is in fact the case. Moreover, the classical prediction indicates a simple, linear dependence of Raman scattering on incident beam intensity and sample concentration, again consistent with experiment except for certain special cases. The relative intensities of the Stokes - and anti-Stokes-scattering are only predicted to differ by the ratio of [(V-Vk)/(V+Vk)]4, which is not in accord with observation. The Boltzmann distribution will be the major factor in determining the relative intensities of these two phenomena. The population of any excited level is always less than that of the ground state, making the Stokes Raman scattering always more intense than the anti-Stokes. A full quantum mechanical treatment of the Raman effect is usually done using time-dependent perturbation theory (see Long [6]) and only certain key results will be given here. From the classical approach it can be appreciated that the geometry of the sample and that of experiment (incident and observing directions) will affect the observations. For analytical purposes, the most important samples are liquids and randomly oriented solids. The
commonly used experimental geometry has the
observation at right angles to the excitation, although there are occasionally good reasons for observing the scattering in other directions, particularly at 180 ~ to the direction of excitation. A special case of interest, that of oriented polymers, is discussed in ref. [7]. Placzek [8] originally derived the expressions for Raman scattering with different geometries, including the conventional 90 ~ scattering, and put these into a convenient form. In these expressions, the polarizability (a) is divided into two parts:
[~=~s+~a, where t~S is the symmetric or isotropic part and t~a is the asymmetric or anisotropic part. These are defined as 30r = ~xx+ ~yy"b ~zz
2(tta) 2= [(ttxx- [~yy)27t- (~yy- 0~zz)2-[- ([~zz- ~xx) 27t- 6(t~xy2+ 0~yx2 d" [~zx2)]. It is possible to make a transformation from Cartesian coordinates to principal axes so that these expressions take the simpler forms: t~'= 1/3(t~ + t~2+ t~3) and
xi
2(=a)2 = (=,_ %)2 + (~2- %)2 + (%_ =,)2. For molecular vibrations, it is not the polarizabilities themselves that we are dealing with, but rather the elements of the matrix of polarizability derivatives, (O=/OQ), usually designated as ='. Placzek's result for Raman scattering at right angles, in terms of these components of the polarizability derivatives connecting a molecule initially in state m and finally in state n, is I = constant[(v +
Vmn)4/Vmn]X[NIo/1
-
exp(-hVmJkt)]X[45(='s) 2+ 13(l~'a)2]
where N is the number of molecules in state m, and I0 is the incident intensity. The constants 45 and 13 arise from the orientational averaging process (see [6] for details) and are a consequence of the experimental geometry. This yields the ratio of Stokes - to anti-Stokes - intensity Is/Ias = [(Vo- Vmn)4/(VO -[- Vmn )4 IX exp(hvmjkT), which is verified experimentally at thermal equilibrium. These expressions assume that v0 is far from any electronic energy levels of the molecule. What was done here so far [9] only gives us the terms in the expression for Raman intensity. It does not say whether the key terms in this expression, the t~'s, are non-zero for a particular vibrational mode. In fact, this is very difficult to predict. But group-theory allows us to predict whether these terms can be non-zero, using information about the symmetry of a molecule or crystal. In each case, group-theory is used to predict whether a transition moment integral can be non-zero. These integrals contain the product of three terms: the wave functions for the ground- and excited-states, and the operator (in this case, the components of the polarizability derivatives) that connects these two states. For a transition to be observed, the product of these three terms must be totally symmetric; that is, it must leave the original molecular symmetry unchanged. One finds that, in molecules of high symmetry, both IR and Raman spectroscopy are needed to observe the vibrational modes. Even with both techniques, there may still be some vibrations that are totally forbidden. The best known selection rule for IR and Raman spectroscopy is known as the "Rule of Mutual Exclusion", which states that if a molecule has a centre of symmetry, vibrations cannot be active in both IR and Raman spectroscopy. This rule has often been applied in molecular structure investigations to determine whether a centre of symmetry is present. In general, vibrations that do not distort the molecular symmetry, "symmetric vibrations", are intense in the Raman spectrum while those that maximize the distortion are most intense in the IR spectrum. If the atoms involved in these vibrations are highly polarizable (e.g., sulfur or iodine) then the Raman intensity is high. Some examples of
xii
vibrational modes that are of importance in the Raman scattering of polymers, and their frequency ranges, are shown in Table 1. There are four main generalizations of the common observations about Raman spectral intensities [9]: 1. Stretching vibrations associated with chemical bonds should be more intense than deformation vibrations. 2. Multiple chemical bonds should give rise to intense stretching modes. For example, a Raman band corresponding to a C=C (or C=C) vibration should be more intense than that from a C-C vibration. 3. Bonds involving atoms of large atomic mass are expected to give rise to stretching vibrations of high Raman intensity. The S-S linkages in proteins are good examples of this [ 10]. 4. Those Raman features arising from normal co-ordinates involving two in-phase bondstretching motions are more intense than those involving a 180 ~ phase difference. Similarly, for cyclic compounds, the in-phase "breathing" mode is usually the most intense.
Important advantages of Raman spectroscopy 1. The "transparency" of water and glass: the very low Raman scattering of water (which is important for living systems) and of glass make it suitable for dilute aqueous solutions of substances as well as for hygroscopic materials, and permits the use of standard glass cuvettes and capillaries.
2. Non-destructivity, and the absence of need of very sophisticated sample preparation. Raman spectroscopy is equally suitable for the analysis of gases, liquids, fibres, single crystals, surface features, etc. Intact measurements permit one to investigate the native molecular structure in biopolymers, living and other systems. It permits studies of eye lenses, the end processes of muscle contraction, components of living cells, and of ancient manuscripts and art objects, etc. The crystallinity of polymeric materials and orientation effects in fibres, monitored by FT-Raman spectra, could be very useful in technological control and in forensic science.
3. Symmetrical bonds such as C-C, C=C, C=C, N=N, 0-0, S-S, manifest themselves by giving the most intensive bands in Raman spectra, and especially structures with the latter heavy atoms, while they are inactive in the infrared. Among spectral methods Raman, is exceptional in showing the structure of natural S-S cross-linkages in biomolecules, artificial ones in vulcanized