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Rayleigh Scattering and Raman Spectroscopy, Theory
RAYLEIGH SCATTERING AND RAMAN EFFECT, THEORY 1993
Raman Spectroscopy in Biochemistry See
Biochemical Applications of Raman Spectroscopy.
Rayleigh Scattering and Raman Effect, Theory David L Andrews, University of East Anglia, Norwich, UK Copyright © 1999 Academic Press.
Rayleigh scattering, the commonplace phenomenon which accounts for the brightness of the sky (amongst many other familiar aspects of the world we inhabit) and the Raman effect, a weaker analogue seen only at high intensities, are closely similar processes in which light is scattered by atoms or molecules. The interactions each entails differ in that the Rayleigh process is technically elastic whilst its Raman counterpart is inelastic all of the features in which the two processes significantly differ owe their origin to that fundamental difference in the energetics. Matter responsible for Rayleigh scattering neither loses nor gains energy thereby and so the scattered light has the same frequency as the radiation from which it is produced. However, atoms or molecules engaged in Raman scattering either gain or lose energy in the process, so that the frequency of the emergent light differs from that impinging on them by conservation of energy, the emergent light has either a lower or higher frequency, respectively, as a result. The two types of Raman process, known as Stokes and anti-Stokes, are illustrated schematically in the energy level or ladder diagrams of Figures 1A and 1B; the Stokes process results in a molecular transition to a state of higher energy, its anti-Stokes counterpart is a transition to a state of lower energy. Rayleigh scattering processes are represented by Figures 1C and 1D. A simple picture widely used for didactic purposes portrays Rayleigh scattering in terms of the electric field of impinging radiation generating, through its interaction with the electron cloud of the scattering molecule, an outgoing field that oscillates at the same frequency. The Raman process is considered to be the generation of an emergent field modulated by molecular vibrations. However, theory cast at that
VIBRATIONAL, ROTATIONAL & RAMAN SPECTROSCOPIES Theory
Figure 1 Energy level diagrams illustrating Raman and Rayleigh scattering, with incoming radiation on the left, scattered radiation emergent on the right. Only energy levels directly involved are depicted: (A) Stokes Raman transition, (B) anti-Stokes Raman transition; (C) and (D) Rayleigh scattering.
level is of severely limited value it fails, for example, to address the relative magnitudes of the Stokes and anti-Stokes Raman signals; and it is not well suited to processes involving electronic transitions. To fully understand those aspects we have to look further into the theory of the fundamental interactions involved in the scattering of light.
Rayleigh scattering The blue of the sky attests to more efficient Rayleigh scattering at the higher frequency, shorter wavelength end of the visible spectrum the familiar red sky before dusk and at dawn manifesting the loss of bluer light to skies in other parts of the world. In fact
1994 RAYLEIGH SCATTERING AND RAMAN EFFECT, THEORY
the efficiency of scattering has a cubic dependence on the optical frequency, the scattered intensity having a fourth power dependence because the photon energy itself then enters into the equation. Although it is a moot point for Rayleigh scattering, these dependences actually relate to the frequency of the scattered rather than the incident light, a distinction which nonetheless becomes significant in the case of Raman scattering where the same power laws apply. The mechanism for Rayleigh scattering entails the electronic polarizability D of the molecules of the sample. The polarizability itself is essentially a measure of how easily the molecular charge distribution is shifted through its interaction with electromagnetic radiation. At simplest, and in an isotropic system such as an atom, the polarizability is a quantity which represents the constant of proportionality between the electric field E of the radiation and the electric dipole moment it induces in the same direction, = E. A further guide to its nature can be gained from the polarizability volume, Dc = D/4SH0 (where H0 is the vacuum permittivity), which casts this constant in units of volume and often yields a value similar in magnitude to the molecular volume. This correctly suggests that systems such as large atoms and aromatic molecules tend to have large polarizabilities. Nonetheless in all but the highest symmetry molecules the ease of charge displacement within the molecule varies with direction, so that in general the induced dipole moment is not parallel with the applied electric field, but slanted towards the direction of least resistance. Then the polarizability is a second rank tensor and we have;
where i and j represent Cartesian coordinates for example, the dipole moment induced in the x-direction is determined by = DxxEx + DxyEy + DxzEz. It is the development of a fully quantum theoretical depiction of scattering at the molecular level which leads to the detailed structure of the electronic polarizability. Here, each Rayleigh (or Raman) scattering event is understood as involving the absorption of one photon of the incoming radiation, accompanied by the emission of one photon. It is important to recognize that the absorption and emission take place together in one concerted process; there is no measurable time delay between the two events. The energytime uncertainty relation 'E't ≥ h/2S allows for each process to take place even when there is no energy level to match the energy of the absorbed photon, as indicated by the
absence of any level at the upper end of the arrows in Figure 1. In other words the absorption does not populate a real intermediate state, since it is accompanied by emission. Thus it is, for example, that Rayleigh scattering of visible light takes place even in transparent media. Despite their widespread adoption and utility, energy diagrams such as Figure 1 are potentially misleading for any such processes involving the concerted absorption and/or emission of more than one photon for in this case they incorrectly suggest that emission takes place subsequent to absorption. All such processes are best described with the aid of time-ordered diagrams which symbolically represent such interactions as a series of photon absorptions and emissions, and which lead to a more correct theoretical representation. Both for Rayleigh and Raman scattering there are two possible sequences, depending on whether the absorption or the emission comes first. These two cases are illustrated by the time-ordered diagrams of Figures 2A and 2B, in which the vertical line represents the successive states of the molecule, and the wavy lines photons, the sequence of interactions being read upwards. Thus in both diagrams the molecular progress from an initial state m to a final state n proceeds via an intermediate state r; in (A) the transition from m to r is accompanied by absorption of a photon hQ from the incident beam, and the transition from r to n by emission of a photon hQc; in (B) this ordering of absorption and emission is reversed. In reality these processes are not separable; the diagrams simply assist development of the theory. Therefore, although the overall process is subject to energy conservation, i.e. Em + hQ = En + hQ′, energy need not be conserved in the individual absorption and emission stages. For
Figure 2 Time-ordered diagrams for light scattering, with time progressing upwards. Taken together, each applies to both Rayleigh and Raman processes; in the Rayleigh case the initial state m and the final state n of the molecule become the same, and Q′ = Q.
RAYLEIGH SCATTERING AND RAMAN EFFECT, THEORY 1995
this reason the state r is often referred to as a virtual state, and all possible energy levels must be taken into account. Interpreting the time-ordered diagrams of Figure 2 by the rules with which they are associated, and given that the states m and n can be identified with the electronic ground state with wavefunction M0 and energy E0, the following result for the polarizability is obtained:
where the subscript on the U denotes linear polarization (synonymous with plane polarization). The value of Ul depends on the molecules responsible for the scattering and is directly expressible in terms of polarizability parameters. Specifically, if we define the polarizability mean and anisotropy J through
then for scattering by a gas or liquid we find
where the two complete terms on the left and right of the plus sign correspond directly to Figures 2A and 2B respectively. In Equation [2], Mr is the wavefunction of state r with energy Er and line width *r and i is the ith component of the electric dipole moment operator. In frequency regions close to an optical absorption band, one of the states in the summation over r will be such that ErE0 | hQ, so that the first term of Equation [2] will have a small denominator, approximating to the line width factor ih*r, and the term as a whole see Figure 2A will overwhelm all else. However, in more common off-resonant circumstances the line width factor can be neglected in each denominator. Moreover for most electronic states the Dirac brackets 〈M0, i, Mr〉 and 〈Mr, i,M0〉 are identical and can be expressed more concisely as components of the transition dipole moment Pr0. Again for conciseness, defining hQr0 = ErE0, Equation [2] then finally reduces to
Rayleigh scattering generally produces radiation with a changed polarization state; polarized incident light is to some extent depolarized by the process whilst unpolarized light is to some extent polarized. Both effects are normally characterized by a depolarization ratio defined as the intensity ratio of plane polarized components of the scattered light. For right-angled scattering of light polarized in the zdirection and incoming along the y-direction, as shown in Figure 3, the depolarization ratio Ul of light scattered in the x-direction is calculated as
with a value in the interval (0, 0.75). The lower limit corresponds to scattering with full retention of linear polarization and corresponds to J2 = 0, a case which occurs only for molecules of very high symmetry. Although introduced here for right-angled scattering, the above result is in fact independent of scattering angle. In contrast the extent of polarization introduced by the scattering of unpolarized light is an angle-dependent quantity. For right-angled scattering where the effect is largest, the corresponding depolarization ratio is given as
(where the subscript of the U stands for natural), giving
Figure 3 Scattering geometry for the usual measurement of depolarization ratios; incident radiation is z-polarized and light scattered at right-angles is resolved for its y- and z-polarization components.
1996 RAYLEIGH SCATTERING AND RAMAN EFFECT, THEORY
For scattering at other angles T (where T 0 relates to forward scattering) the result can be written as
The angle dependence of the polarization which Rayleigh scattering confers on unpolarized light is immediately evident on viewing a clear daytime sky through polarizing spectacles.
The Raman effect The Raman effect was one of the first processes whose explanation, largely through the work of Placzek in 1934, exploited and vindicated the still nascent quantum theory. As for Rayleigh scattering, each scattering event involves concerted processes of photon absorption and emission, without the need for an energy level to match the absorbed photon. The difference is that as a result of this process, the scatterer undergoes an overall transition from one energy level to another, as depicted in Figures 1A and 1B. The Raman effect is a very weak phenomenon; typically only one incident photon in ∼ 10 7 produces a Raman transition, and observation of the effect thus requires a very intense source of light. Raman scattering generally involves transitions amongst energy levels that are separated by much less than the photon energy of the incident light. The two levels, denoted by E0 and E1 in Figure 1, for example, are most often vibrational levels, whilst the energies of the absorbed and emitted photons are commonly in (or near to) the visible range hence the effect provides the facility for obtaining vibrational spectra using visible light. In general the Stokes Raman transition from level E0 to E1 results in scattering of a frequency given by
spectrum of scattered light contains a range of frequencies shifted away from the irradiation frequency. In the particular case of vibrational Raman transitions, the shifts can be identified with vibrational frequencies. Although each Stokes line and its anti-Stokes counterpart are equally separated from the Rayleigh line, they are not of equal intensity. This is because the intensity of each transition is proportional to the population of the energy level from which the transition originates; under equilibrium conditions the ratio of populations is given by the Boltzmann distribution. With the fourth-power dependence on the scattering frequency, the ratio of intensities of the Stokes line and its anti-Stokes partner in a Raman spectrum is given by
where g0 is the degeneracy of the ground state and g1 that of the upper level and hence the anti-Stokes line is almost invariably weaker in intensity. The dependence of this ratio on the absolute temperature T can, for example, be employed as a means of flame thermometry. However, since the Stokes and antiStokes lines give precisely the same information on molecular frequencies, it is usually only the stronger (Stokes) part of the spectrum that is recorded. A development of detailed theory, again based on the time-ordered diagrams of Figure 2, establishes the dependence of Raman scattering on a transition tensor which takes on the same role as the polarizability in Rayleigh scattering. Once the usual Born Oppenheimer separation of electronic and vibrational wavefunctions has been effected, then for the vibrational Raman transition v → v′ involving a normal mode of vibration O, this tensor takes the form
where 'E = (E1 E0), and the corresponding antiStokes transition from E1 to E0 produces a frequency
Thus each allowed Raman coupling generally produces two frequencies in the spectrum of scattered light, shifted to the negative and positive sides of the dominant Rayleigh line by the same amount, 'Q = 'E/h. For this reason Raman spectroscopy is concerned with measurements of frequency shifts, rather than absolute frequencies. In most cases a number of Raman transitions can take place, involving various molecular energy levels, and the
where, for example, the vibrational wavefunction F denotes a state with a quantum number v″ in the vibrational mode P, within the set of levels associated with electronic state r. Here also E and * relate to the total (electronic plus vibrational) energy and the damping, respectively, of that state. Away
RAYLEIGH SCATTERING AND RAMAN EFFECT, THEORY 1997
from resonance, in other words when using frequencies Q well removed from any optical absorption bands, then the vibrational energy contributions in each denominator term of Equation [13] can safely be neglected. Then, using the completeness relation of quantum mechanics, the ~F ² ¢F ~sum can be effected to give
using Equation [2]. The result thus involves the dependence of the electronic polarizability on the nuclear coordinate QO relating to the excited vibration. Although all molecules have a finite polarizability, that is not the case for D but no Raman signal will emerge when the latter is zero. Here a powerful symmetry rule emerges: any Raman-active vibration must transform under an irreducible representation spanned by components of the polarizability tensor (transforming as the quadratic variables x2, xy, etc., or one of their combinations). Some of the broad implications of this are highlighted below. To obtain the major selection rules for Raman scattering we can first expand Equation [14] in a Taylor series about the equilibrium configuration Qe;
during the vibration, as the molecule passes through its equilibrium configuration. This is the key selection rule for the Raman effect, illustrated for CO2 in Figure 4. It is immediately apparent that Raman transitions are governed by different selection rules from absorption or fluorescence. The case of CO2 illustrates a general principle applicable to all centrosymmetric molecules, which is that only gerade vibrations (those which are even with respect to inversion symmetry) appear in the Raman spectrum, whilst only ungerade vibrations (odd with respect to inversion) show up in infrared absorption. This illustrates the so-called mutual exclusion rule for centrosymmetric molecules, which states that vibrations active in the infrared spectrum are inactive in the Raman, and vice versa. Even for complex polyatomic molecules lacking much symmetry, the intensities of lines resulting from the same vibrational transition may be very different in the two types of spectrum, so that in general there is a useful complementarity between the two methods. Generally it is the vibrations of the most polarizable groups which are strongest in the Raman spectrum, those of the most polar groups being strongest in the infrared, as nicely illustrated in the spectra of the drug acetaminophen (UK paracetamol; p-hydroxyacetanilide) shown in Figure 5. Further information on the symmetry properties of Raman-active molecular vibrations can be obtained by measurement of the depolarization ratios of the lines in the Raman spectrum see Figure 3 and Equation [4]. Interpretation of the results here invokes Equation [17]:
and hence we have
The first term on the right is non-zero only when the initial and final states are identical which relates back to Rayleigh scattering. It is the second term which is significant for the Raman process and its detailed form establishes two rules governing Raman-allowed transitions, since both of its factors must then be non-zero. For the Dirac bracket to be non-zero dictates vc v ± 1, as in infrared absorption spectroscopy. For the polarizability derivative to be non-zero, the polarizability must change
Figure 4 Variation of polarizability in the course of three normal modes of vibration of carbon dioxide: (A) symmetric stretch, (B) bending mode and (C) antisymmetric stretch. The slope wD/wQO on crossing the vertical axis is non-zero only for the symmetric stretch, and hence only this vibration gives a Raman signal.
1998 RAYLEIGH SCATTERING AND RAMAN EFFECT, THEORY
Figure 5 Fourier-transform spectra of paracetamol: (A) Raman, and (B) infrared. Stretch vibrations of the non-polar C–H groups, close to 3000 cm–1, show up well in the Raman spectrum. In the infrared spectrum this whole region is dominated by stretching vibrations of the highly polar O–H and N–H groups, much broadened through association with hydrogen bonding. Reproduced with permission of Nicolet Instruments.
The prime on the mean and anisotropy parameters, and Jc, respectively, denote values obtained from the polarizability derivative defined in the sense of Equation [5], but in components of the tensor wDij / wQO rather than Dij itself. In the case of gases and liquids, Ul is lower than for vibrations that are totally symmetric (vibrations transforming under the totally symmetric representation of the molecular point group), but exactly for for other vibrations that lower the molecular symmetry, since is then zero.
Although the frequency of radiation used for the study of Raman scattering is generally well removed from any absorption band of the sample, to forestall problems associated with absorption and subsequent fluorescence, special features become apparent on irradiation at a frequency close to a broad and intense optical absorption band. Quite simply, the closer the approach, the greater is the intensity of the Raman spectrum. Spectra obtained under such conditions are known as resonance Raman spectra. In the case
RAYLEIGH SCATTERING AND RAMAN EFFECT, THEORY 1999
of large polyatomic molecules where any electronic absorption band may be due to localized absorption in a particular chromophore, the vibrational Raman lines which experience the greatest amplification are those of the appropriate symmetry involving vibrations of nuclei close to the groups responsible for the resonance. Equation [13] correctly represents the Raman tensor even under resonance or pre-resonance conditions and the resonance enhancement is clearly attributable to the fact that if there is an excited state for which E E is close to hQ, the first term of that equation has a denominator of greatly diminshed magnitude. However, the subsequent development of theory leading to Equation [16] is no longer valid under such conditions; for example, the vc v ± 1 selection rule breaks down and overtones commonly appear. Other vibrations (those which transform like the rotations Rx, Ry and Rz) can also become active through changed selection rules, associated with the fact that the Raman tensor is no longer real and index-symmetric, but complex and non-symmetric. As a result the equations for the Raman depolarization ratio also require modification to the following form
differential scattering in a region of the spectrum associated with a particular group frequency can be interpreted in terms of the chiral environment of the corresponding functional group. In contrast to the theory developed here, the scattering entails not only electric dipole but also the much weaker magnetic dipole and electric quadrupole interactions. At the high intensities now available from laser sources, numerous other variants of the Raman effect can be observed, many associated with optically nonlinear behaviour. For analytical purposes the most important of these come under the heading of coherent Raman spectroscopy, of which the process known as CARS (coherent anti-Stokes Raman spectroscopy) is the most common. Here, two beams are directed into the sample: one has a fixed frequency playing the role of Q and the other, a frequency Qc tunable across the Stokes range. As Qc tunes into each Stokes frequency QS a four-photon process occurs, essentially combining all the elements of Figures 1A and 1B, and generating coherent emission at the corresponding anti-Stokes frequency QAS. The laser-like nature of this output facilitates its collection for spectroscopic analysis, and permits the analysis of microscopic samples.
List of symbols where
is a measure of the degree of antisymmetry in the Raman tensor. One consequence of including this factor in Equation [18] is the possibility of depolarization ratios exceeding the normal upper bound of in some cases indeed yielding an infinite result (complete depolarization). Generally, the use of circularly polarized light in studies of Rayleigh or Raman scattering offers no further information beyond that provided by plane polarizations. However, optically active (chiral) compounds in the liquid or solution state respond differentially to circularly polarized light, according to its handedness, making it possible to obtain a spectrum showing a marginal difference in the Raman intensity IRIL as a function of scattering frequency. The extent of this differential for each molecular vibration is directly related to the detailed stereochemical structure responsible for the manifestation of chirality. In particular, the extent of
E = electric field; Em = energy of level m; g0 = degeneracy of ground state; g1 = degeneracy of upper level; h = Plancks constant; I = intensity; k = Boltzmanns constant; Q = Nuclear coordinate; t = time; T = absolute temperature; = electronic mean polarizability polarizability; J = polarizability anisotropy; *r = damping of level r; H0 = vacuum permittivity; T = scattering angle; O = normal mode of vibration; Pind = induced electric dipole moment; Pr0 = transition dipole moment; Q = frequency of incident radiation; Qc = frequency of emitted radiation; QS = Stokes frequency; QAS = antiU = depolarization ratio; Stokes frequency; I0 = wavefunction with energy E0; F = vibrational wavefunction. See also: Biochemical Applications of Raman Spectroscopy; Nonlinear Optical Properties; Raman Optical Activity, Applications; Raman Optical Activity, Spectrometers; Raman Optical Activity, Theory; Raman Spectrometers.
Further reading Andrews DL (1997) Lasers in Chemistry , 3rd edn, pp 128149. Berlin: Springer-Verlag.
2000 RELAXOMETERS
Barron LD (1982) Molecular Light Scattering and Optical Activity. Cambridge: Cambridge University Press. Craig DP and Thirunamachandran T (1984) Molecular Quantum Electrodynamics. London: Academic Press. Long DA (1977) Raman Spectroscopy. New York: McGraw-Hill. Placzek G (1934) Rayleigh-Streuung und Raman-Effekt. In: Marx E (ed) Handbuch der Radiologie, Vol. 6, Part 2, pp 205374. Leipzig: Akademische Verlag.
Raman CV and Krishnan KS (1928) A new type of secondary radiation. Nature 121: 501. Sheppard N (1990) Chemical applications of molecular spectroscopy A developing perspective. In: Andrews DL (ed) Perspectives in Modern Chemical Spectroscopy, pp 141. Berlin: Springer-Verlag.
Regulatory Authority Requirements See
Calibration and Reference Systems (Regulatory Authorities).
Relaxometers Ralf-Oliver Seitter and Rainer Kimmich, Universität Ulm, Germany Copyright © 1999 Academic Press
Purpose and classification of NMR relaxometers Nuclear magnetic relaxation, that is, thermal equilibration of the spin systems with respect to longitudinal or transverse magnetization components, multiple-quantum spin coherences and longitudinal dipolar, quadrupolar or scalar order, comprises a vast variety of different experimental protocols and phenomena. In a typical relaxation experiment, one first produces a nonequilibrium population of the spin states, often combined with spin coherences. It is then a matter of the fluctuations of the spin couplings to induce spin transitions towards thermal equilibrium. Equilibrium means (i) populations following Boltzmann's distribution, and (ii) completely vanishing spin coherences. Consequently there are three elements inherent in a typical relaxation experiment: Preparation of a nonequilibrium state of the spin systems; the variable evolution interval allowing for the induction of spin transitions; and the detection of the populations, longitudinal order, or coherences after the evolution interval. The time constants with which the observable approaches
MAGNETIC RESONANCE Methods & Instrumentation
equilibrium during the evolution interval are the relaxation times, such as the spinlattice relaxation time T1, the transverse relaxation time T2, the rotating-frame relaxation time T1U , the dipolar-order relaxation time Td, and so on. In the following we will focus on T1 in particular. The spinlattice relaxation rate of dipolar coupled homonuclear two-spin I systems, for instance, is given by
where P0 is the magnetic field constant, J is the gyromagnetic ratio, and J(i)(Z) is the intensity function of the Larmor frequency, Z = JB0, depending on the flux density of the external magnetic field, B0. The intensity function is given as the Fourier transform of the dipolar autocorrelation function Gi (W),
Raman Spectroscopy in Biochemistry See
Biochemical Applications of Raman Spectroscopy.
Rayleigh Scattering and Raman Effect, Theory David L Andrews, University of East Anglia, Norwich, UK Copyright © 1999 Academic Press.
Rayleigh scattering, the commonplace phenomenon which accounts for the brightness of the sky (amongst many other familiar aspects of the world we inhabit) and the Raman effect, a weaker analogue seen only at high intensities, are closely similar processes in which light is scattered by atoms or molecules. The interactions each entails differ in that the Rayleigh process is technically elastic whilst its Raman counterpart is inelastic all of the features in which the two processes significantly differ owe their origin to that fundamental difference in the energetics. Matter responsible for Rayleigh scattering neither loses nor gains energy thereby and so the scattered light has the same frequency as the radiation from which it is produced. However, atoms or molecules engaged in Raman scattering either gain or lose energy in the process, so that the frequency of the emergent light differs from that impinging on them by conservation of energy, the emergent light has either a lower or higher frequency, respectively, as a result. The two types of Raman process, known as Stokes and anti-Stokes, are illustrated schematically in the energy level or ladder diagrams of Figures 1A and 1B; the Stokes process results in a molecular transition to a state of higher energy, its anti-Stokes counterpart is a transition to a state of lower energy. Rayleigh scattering processes are represented by Figures 1C and 1D. A simple picture widely used for didactic purposes portrays Rayleigh scattering in terms of the electric field of impinging radiation generating, through its interaction with the electron cloud of the scattering molecule, an outgoing field that oscillates at the same frequency. The Raman process is considered to be the generation of an emergent field modulated by molecular vibrations. However, theory cast at that
VIBRATIONAL, ROTATIONAL & RAMAN SPECTROSCOPIES Theory
Figure 1 Energy level diagrams illustrating Raman and Rayleigh scattering, with incoming radiation on the left, scattered radiation emergent on the right. Only energy levels directly involved are depicted: (A) Stokes Raman transition, (B) anti-Stokes Raman transition; (C) and (D) Rayleigh scattering.
level is of severely limited value it fails, for example, to address the relative magnitudes of the Stokes and anti-Stokes Raman signals; and it is not well suited to processes involving electronic transitions. To fully understand those aspects we have to look further into the theory of the fundamental interactions involved in the scattering of light.
Rayleigh scattering The blue of the sky attests to more efficient Rayleigh scattering at the higher frequency, shorter wavelength end of the visible spectrum the familiar red sky before dusk and at dawn manifesting the loss of bluer light to skies in other parts of the world. In fact
1994 RAYLEIGH SCATTERING AND RAMAN EFFECT, THEORY
the efficiency of scattering has a cubic dependence on the optical frequency, the scattered intensity having a fourth power dependence because the photon energy itself then enters into the equation. Although it is a moot point for Rayleigh scattering, these dependences actually relate to the frequency of the scattered rather than the incident light, a distinction which nonetheless becomes significant in the case of Raman scattering where the same power laws apply. The mechanism for Rayleigh scattering entails the electronic polarizability D of the molecules of the sample. The polarizability itself is essentially a measure of how easily the molecular charge distribution is shifted through its interaction with electromagnetic radiation. At simplest, and in an isotropic system such as an atom, the polarizability is a quantity which represents the constant of proportionality between the electric field E of the radiation and the electric dipole moment it induces in the same direction, = E. A further guide to its nature can be gained from the polarizability volume, Dc = D/4SH0 (where H0 is the vacuum permittivity), which casts this constant in units of volume and often yields a value similar in magnitude to the molecular volume. This correctly suggests that systems such as large atoms and aromatic molecules tend to have large polarizabilities. Nonetheless in all but the highest symmetry molecules the ease of charge displacement within the molecule varies with direction, so that in general the induced dipole moment is not parallel with the applied electric field, but slanted towards the direction of least resistance. Then the polarizability is a second rank tensor and we have;
where i and j represent Cartesian coordinates for example, the dipole moment induced in the x-direction is determined by = DxxEx + DxyEy + DxzEz. It is the development of a fully quantum theoretical depiction of scattering at the molecular level which leads to the detailed structure of the electronic polarizability. Here, each Rayleigh (or Raman) scattering event is understood as involving the absorption of one photon of the incoming radiation, accompanied by the emission of one photon. It is important to recognize that the absorption and emission take place together in one concerted process; there is no measurable time delay between the two events. The energytime uncertainty relation 'E't ≥ h/2S allows for each process to take place even when there is no energy level to match the energy of the absorbed photon, as indicated by the
absence of any level at the upper end of the arrows in Figure 1. In other words the absorption does not populate a real intermediate state, since it is accompanied by emission. Thus it is, for example, that Rayleigh scattering of visible light takes place even in transparent media. Despite their widespread adoption and utility, energy diagrams such as Figure 1 are potentially misleading for any such processes involving the concerted absorption and/or emission of more than one photon for in this case they incorrectly suggest that emission takes place subsequent to absorption. All such processes are best described with the aid of time-ordered diagrams which symbolically represent such interactions as a series of photon absorptions and emissions, and which lead to a more correct theoretical representation. Both for Rayleigh and Raman scattering there are two possible sequences, depending on whether the absorption or the emission comes first. These two cases are illustrated by the time-ordered diagrams of Figures 2A and 2B, in which the vertical line represents the successive states of the molecule, and the wavy lines photons, the sequence of interactions being read upwards. Thus in both diagrams the molecular progress from an initial state m to a final state n proceeds via an intermediate state r; in (A) the transition from m to r is accompanied by absorption of a photon hQ from the incident beam, and the transition from r to n by emission of a photon hQc; in (B) this ordering of absorption and emission is reversed. In reality these processes are not separable; the diagrams simply assist development of the theory. Therefore, although the overall process is subject to energy conservation, i.e. Em + hQ = En + hQ′, energy need not be conserved in the individual absorption and emission stages. For
Figure 2 Time-ordered diagrams for light scattering, with time progressing upwards. Taken together, each applies to both Rayleigh and Raman processes; in the Rayleigh case the initial state m and the final state n of the molecule become the same, and Q′ = Q.
RAYLEIGH SCATTERING AND RAMAN EFFECT, THEORY 1995
this reason the state r is often referred to as a virtual state, and all possible energy levels must be taken into account. Interpreting the time-ordered diagrams of Figure 2 by the rules with which they are associated, and given that the states m and n can be identified with the electronic ground state with wavefunction M0 and energy E0, the following result for the polarizability is obtained:
where the subscript on the U denotes linear polarization (synonymous with plane polarization). The value of Ul depends on the molecules responsible for the scattering and is directly expressible in terms of polarizability parameters. Specifically, if we define the polarizability mean and anisotropy J through
then for scattering by a gas or liquid we find
where the two complete terms on the left and right of the plus sign correspond directly to Figures 2A and 2B respectively. In Equation [2], Mr is the wavefunction of state r with energy Er and line width *r and i is the ith component of the electric dipole moment operator. In frequency regions close to an optical absorption band, one of the states in the summation over r will be such that ErE0 | hQ, so that the first term of Equation [2] will have a small denominator, approximating to the line width factor ih*r, and the term as a whole see Figure 2A will overwhelm all else. However, in more common off-resonant circumstances the line width factor can be neglected in each denominator. Moreover for most electronic states the Dirac brackets 〈M0, i, Mr〉 and 〈Mr, i,M0〉 are identical and can be expressed more concisely as components of the transition dipole moment Pr0. Again for conciseness, defining hQr0 = ErE0, Equation [2] then finally reduces to
Rayleigh scattering generally produces radiation with a changed polarization state; polarized incident light is to some extent depolarized by the process whilst unpolarized light is to some extent polarized. Both effects are normally characterized by a depolarization ratio defined as the intensity ratio of plane polarized components of the scattered light. For right-angled scattering of light polarized in the zdirection and incoming along the y-direction, as shown in Figure 3, the depolarization ratio Ul of light scattered in the x-direction is calculated as
with a value in the interval (0, 0.75). The lower limit corresponds to scattering with full retention of linear polarization and corresponds to J2 = 0, a case which occurs only for molecules of very high symmetry. Although introduced here for right-angled scattering, the above result is in fact independent of scattering angle. In contrast the extent of polarization introduced by the scattering of unpolarized light is an angle-dependent quantity. For right-angled scattering where the effect is largest, the corresponding depolarization ratio is given as
(where the subscript of the U stands for natural), giving
Figure 3 Scattering geometry for the usual measurement of depolarization ratios; incident radiation is z-polarized and light scattered at right-angles is resolved for its y- and z-polarization components.
1996 RAYLEIGH SCATTERING AND RAMAN EFFECT, THEORY
For scattering at other angles T (where T 0 relates to forward scattering) the result can be written as
The angle dependence of the polarization which Rayleigh scattering confers on unpolarized light is immediately evident on viewing a clear daytime sky through polarizing spectacles.
The Raman effect The Raman effect was one of the first processes whose explanation, largely through the work of Placzek in 1934, exploited and vindicated the still nascent quantum theory. As for Rayleigh scattering, each scattering event involves concerted processes of photon absorption and emission, without the need for an energy level to match the absorbed photon. The difference is that as a result of this process, the scatterer undergoes an overall transition from one energy level to another, as depicted in Figures 1A and 1B. The Raman effect is a very weak phenomenon; typically only one incident photon in ∼ 10 7 produces a Raman transition, and observation of the effect thus requires a very intense source of light. Raman scattering generally involves transitions amongst energy levels that are separated by much less than the photon energy of the incident light. The two levels, denoted by E0 and E1 in Figure 1, for example, are most often vibrational levels, whilst the energies of the absorbed and emitted photons are commonly in (or near to) the visible range hence the effect provides the facility for obtaining vibrational spectra using visible light. In general the Stokes Raman transition from level E0 to E1 results in scattering of a frequency given by
spectrum of scattered light contains a range of frequencies shifted away from the irradiation frequency. In the particular case of vibrational Raman transitions, the shifts can be identified with vibrational frequencies. Although each Stokes line and its anti-Stokes counterpart are equally separated from the Rayleigh line, they are not of equal intensity. This is because the intensity of each transition is proportional to the population of the energy level from which the transition originates; under equilibrium conditions the ratio of populations is given by the Boltzmann distribution. With the fourth-power dependence on the scattering frequency, the ratio of intensities of the Stokes line and its anti-Stokes partner in a Raman spectrum is given by
where g0 is the degeneracy of the ground state and g1 that of the upper level and hence the anti-Stokes line is almost invariably weaker in intensity. The dependence of this ratio on the absolute temperature T can, for example, be employed as a means of flame thermometry. However, since the Stokes and antiStokes lines give precisely the same information on molecular frequencies, it is usually only the stronger (Stokes) part of the spectrum that is recorded. A development of detailed theory, again based on the time-ordered diagrams of Figure 2, establishes the dependence of Raman scattering on a transition tensor which takes on the same role as the polarizability in Rayleigh scattering. Once the usual Born Oppenheimer separation of electronic and vibrational wavefunctions has been effected, then for the vibrational Raman transition v → v′ involving a normal mode of vibration O, this tensor takes the form
where 'E = (E1 E0), and the corresponding antiStokes transition from E1 to E0 produces a frequency
Thus each allowed Raman coupling generally produces two frequencies in the spectrum of scattered light, shifted to the negative and positive sides of the dominant Rayleigh line by the same amount, 'Q = 'E/h. For this reason Raman spectroscopy is concerned with measurements of frequency shifts, rather than absolute frequencies. In most cases a number of Raman transitions can take place, involving various molecular energy levels, and the
where, for example, the vibrational wavefunction F denotes a state with a quantum number v″ in the vibrational mode P, within the set of levels associated with electronic state r. Here also E and * relate to the total (electronic plus vibrational) energy and the damping, respectively, of that state. Away
RAYLEIGH SCATTERING AND RAMAN EFFECT, THEORY 1997
from resonance, in other words when using frequencies Q well removed from any optical absorption bands, then the vibrational energy contributions in each denominator term of Equation [13] can safely be neglected. Then, using the completeness relation of quantum mechanics, the ~F ² ¢F ~sum can be effected to give
using Equation [2]. The result thus involves the dependence of the electronic polarizability on the nuclear coordinate QO relating to the excited vibration. Although all molecules have a finite polarizability, that is not the case for D but no Raman signal will emerge when the latter is zero. Here a powerful symmetry rule emerges: any Raman-active vibration must transform under an irreducible representation spanned by components of the polarizability tensor (transforming as the quadratic variables x2, xy, etc., or one of their combinations). Some of the broad implications of this are highlighted below. To obtain the major selection rules for Raman scattering we can first expand Equation [14] in a Taylor series about the equilibrium configuration Qe;
during the vibration, as the molecule passes through its equilibrium configuration. This is the key selection rule for the Raman effect, illustrated for CO2 in Figure 4. It is immediately apparent that Raman transitions are governed by different selection rules from absorption or fluorescence. The case of CO2 illustrates a general principle applicable to all centrosymmetric molecules, which is that only gerade vibrations (those which are even with respect to inversion symmetry) appear in the Raman spectrum, whilst only ungerade vibrations (odd with respect to inversion) show up in infrared absorption. This illustrates the so-called mutual exclusion rule for centrosymmetric molecules, which states that vibrations active in the infrared spectrum are inactive in the Raman, and vice versa. Even for complex polyatomic molecules lacking much symmetry, the intensities of lines resulting from the same vibrational transition may be very different in the two types of spectrum, so that in general there is a useful complementarity between the two methods. Generally it is the vibrations of the most polarizable groups which are strongest in the Raman spectrum, those of the most polar groups being strongest in the infrared, as nicely illustrated in the spectra of the drug acetaminophen (UK paracetamol; p-hydroxyacetanilide) shown in Figure 5. Further information on the symmetry properties of Raman-active molecular vibrations can be obtained by measurement of the depolarization ratios of the lines in the Raman spectrum see Figure 3 and Equation [4]. Interpretation of the results here invokes Equation [17]:
and hence we have
The first term on the right is non-zero only when the initial and final states are identical which relates back to Rayleigh scattering. It is the second term which is significant for the Raman process and its detailed form establishes two rules governing Raman-allowed transitions, since both of its factors must then be non-zero. For the Dirac bracket to be non-zero dictates vc v ± 1, as in infrared absorption spectroscopy. For the polarizability derivative to be non-zero, the polarizability must change
Figure 4 Variation of polarizability in the course of three normal modes of vibration of carbon dioxide: (A) symmetric stretch, (B) bending mode and (C) antisymmetric stretch. The slope wD/wQO on crossing the vertical axis is non-zero only for the symmetric stretch, and hence only this vibration gives a Raman signal.
1998 RAYLEIGH SCATTERING AND RAMAN EFFECT, THEORY
Figure 5 Fourier-transform spectra of paracetamol: (A) Raman, and (B) infrared. Stretch vibrations of the non-polar C–H groups, close to 3000 cm–1, show up well in the Raman spectrum. In the infrared spectrum this whole region is dominated by stretching vibrations of the highly polar O–H and N–H groups, much broadened through association with hydrogen bonding. Reproduced with permission of Nicolet Instruments.
The prime on the mean and anisotropy parameters, and Jc, respectively, denote values obtained from the polarizability derivative defined in the sense of Equation [5], but in components of the tensor wDij / wQO rather than Dij itself. In the case of gases and liquids, Ul is lower than for vibrations that are totally symmetric (vibrations transforming under the totally symmetric representation of the molecular point group), but exactly for for other vibrations that lower the molecular symmetry, since is then zero.
Although the frequency of radiation used for the study of Raman scattering is generally well removed from any absorption band of the sample, to forestall problems associated with absorption and subsequent fluorescence, special features become apparent on irradiation at a frequency close to a broad and intense optical absorption band. Quite simply, the closer the approach, the greater is the intensity of the Raman spectrum. Spectra obtained under such conditions are known as resonance Raman spectra. In the case
RAYLEIGH SCATTERING AND RAMAN EFFECT, THEORY 1999
of large polyatomic molecules where any electronic absorption band may be due to localized absorption in a particular chromophore, the vibrational Raman lines which experience the greatest amplification are those of the appropriate symmetry involving vibrations of nuclei close to the groups responsible for the resonance. Equation [13] correctly represents the Raman tensor even under resonance or pre-resonance conditions and the resonance enhancement is clearly attributable to the fact that if there is an excited state for which E E is close to hQ, the first term of that equation has a denominator of greatly diminshed magnitude. However, the subsequent development of theory leading to Equation [16] is no longer valid under such conditions; for example, the vc v ± 1 selection rule breaks down and overtones commonly appear. Other vibrations (those which transform like the rotations Rx, Ry and Rz) can also become active through changed selection rules, associated with the fact that the Raman tensor is no longer real and index-symmetric, but complex and non-symmetric. As a result the equations for the Raman depolarization ratio also require modification to the following form
differential scattering in a region of the spectrum associated with a particular group frequency can be interpreted in terms of the chiral environment of the corresponding functional group. In contrast to the theory developed here, the scattering entails not only electric dipole but also the much weaker magnetic dipole and electric quadrupole interactions. At the high intensities now available from laser sources, numerous other variants of the Raman effect can be observed, many associated with optically nonlinear behaviour. For analytical purposes the most important of these come under the heading of coherent Raman spectroscopy, of which the process known as CARS (coherent anti-Stokes Raman spectroscopy) is the most common. Here, two beams are directed into the sample: one has a fixed frequency playing the role of Q and the other, a frequency Qc tunable across the Stokes range. As Qc tunes into each Stokes frequency QS a four-photon process occurs, essentially combining all the elements of Figures 1A and 1B, and generating coherent emission at the corresponding anti-Stokes frequency QAS. The laser-like nature of this output facilitates its collection for spectroscopic analysis, and permits the analysis of microscopic samples.
List of symbols where
is a measure of the degree of antisymmetry in the Raman tensor. One consequence of including this factor in Equation [18] is the possibility of depolarization ratios exceeding the normal upper bound of in some cases indeed yielding an infinite result (complete depolarization). Generally, the use of circularly polarized light in studies of Rayleigh or Raman scattering offers no further information beyond that provided by plane polarizations. However, optically active (chiral) compounds in the liquid or solution state respond differentially to circularly polarized light, according to its handedness, making it possible to obtain a spectrum showing a marginal difference in the Raman intensity IRIL as a function of scattering frequency. The extent of this differential for each molecular vibration is directly related to the detailed stereochemical structure responsible for the manifestation of chirality. In particular, the extent of
E = electric field; Em = energy of level m; g0 = degeneracy of ground state; g1 = degeneracy of upper level; h = Plancks constant; I = intensity; k = Boltzmanns constant; Q = Nuclear coordinate; t = time; T = absolute temperature; = electronic mean polarizability polarizability; J = polarizability anisotropy; *r = damping of level r; H0 = vacuum permittivity; T = scattering angle; O = normal mode of vibration; Pind = induced electric dipole moment; Pr0 = transition dipole moment; Q = frequency of incident radiation; Qc = frequency of emitted radiation; QS = Stokes frequency; QAS = antiU = depolarization ratio; Stokes frequency; I0 = wavefunction with energy E0; F = vibrational wavefunction. See also: Biochemical Applications of Raman Spectroscopy; Nonlinear Optical Properties; Raman Optical Activity, Applications; Raman Optical Activity, Spectrometers; Raman Optical Activity, Theory; Raman Spectrometers.
Further reading Andrews DL (1997) Lasers in Chemistry , 3rd edn, pp 128149. Berlin: Springer-Verlag.
2000 RELAXOMETERS
Barron LD (1982) Molecular Light Scattering and Optical Activity. Cambridge: Cambridge University Press. Craig DP and Thirunamachandran T (1984) Molecular Quantum Electrodynamics. London: Academic Press. Long DA (1977) Raman Spectroscopy. New York: McGraw-Hill. Placzek G (1934) Rayleigh-Streuung und Raman-Effekt. In: Marx E (ed) Handbuch der Radiologie, Vol. 6, Part 2, pp 205374. Leipzig: Akademische Verlag.
Raman CV and Krishnan KS (1928) A new type of secondary radiation. Nature 121: 501. Sheppard N (1990) Chemical applications of molecular spectroscopy A developing perspective. In: Andrews DL (ed) Perspectives in Modern Chemical Spectroscopy, pp 141. Berlin: Springer-Verlag.
Regulatory Authority Requirements See
Calibration and Reference Systems (Regulatory Authorities).
Relaxometers Ralf-Oliver Seitter and Rainer Kimmich, Universität Ulm, Germany Copyright © 1999 Academic Press
Purpose and classification of NMR relaxometers Nuclear magnetic relaxation, that is, thermal equilibration of the spin systems with respect to longitudinal or transverse magnetization components, multiple-quantum spin coherences and longitudinal dipolar, quadrupolar or scalar order, comprises a vast variety of different experimental protocols and phenomena. In a typical relaxation experiment, one first produces a nonequilibrium population of the spin states, often combined with spin coherences. It is then a matter of the fluctuations of the spin couplings to induce spin transitions towards thermal equilibrium. Equilibrium means (i) populations following Boltzmann's distribution, and (ii) completely vanishing spin coherences. Consequently there are three elements inherent in a typical relaxation experiment: Preparation of a nonequilibrium state of the spin systems; the variable evolution interval allowing for the induction of spin transitions; and the detection of the populations, longitudinal order, or coherences after the evolution interval. The time constants with which the observable approaches
MAGNETIC RESONANCE Methods & Instrumentation
equilibrium during the evolution interval are the relaxation times, such as the spinlattice relaxation time T1, the transverse relaxation time T2, the rotating-frame relaxation time T1U , the dipolar-order relaxation time Td, and so on. In the following we will focus on T1 in particular. The spinlattice relaxation rate of dipolar coupled homonuclear two-spin I systems, for instance, is given by
where P0 is the magnetic field constant, J is the gyromagnetic ratio, and J(i)(Z) is the intensity function of the Larmor frequency, Z = JB0, depending on the flux density of the external magnetic field, B0. The intensity function is given as the Fourier transform of the dipolar autocorrelation function Gi (W),