Raman scattering classical theory

Both quantum mechanical and classical theories of Raman scattering have been developed. The quantum mechanical treatment of Kramers and Heisenberg 5) preceded the classical theory of Cabannes and Rochard 6). 

The classical theory of scattering provides us with a relatively simple selection rule for Raman activity which can be compared with that for infrared activity. 

Raman effect (continued) spectral activity, 339-341 terminology of, 295 vibrational wavefunctione, 339-341 Raman lines, 296 weak, 327-330 Raman scattering, 296 classical theory, 297-299 quantum mechanical theory, 296, 297 Raman shift, 296... 

According to classical theory, in Eq. (16.16), the first term represents an oscillating dipole that radiates light of frequency v0, that is, Rayleigh scattering. The second term is associated with the Raman scattering of frequency v0 - vm (Stokes process) and v0 + vm (anti-Stokes process). If (daldq)0 is zero, the vibration is not Raman active (Ferraro and Nakamoto, 1994). 

Similarly to IR, classical theories have also been proposed in the literature for Raman intensities in solution [29,32-38], The starting point is again the definition of the local field Eso1 acting on the molecule. In all cases the local field factor is defined as / = S-i/S-c, with 5 sc being the scattering intensity. 

According to classical theory, Raman scattering can be explained as follows The electric field strength (E). of the electromagnetic wave (laser heard) fluctuates with time (t) as shown by Eq. (1-1) ... 

Finally, we like to mention that equivalent to the conventional energy frame KHD formulation, the time-dependent theory of Raman scattering is free from any approximations except the usual second order perturbation method used to derive the KHD expression. When applied to resonance and near resonance Raman scattering, the time-dependent formulation has shown advantages over the static KHD formulation. Apparently, the time-dependent formulation lends itselfs to an interpretation where localized wave packets follow classical-like paths. As an example of the numerical calculation of continuum resonance Raman spectra we show in Fig. 6.1-7 the simulation of the A, = 4 transitions (third overtone) of D excited with Aq = 488.0 nm. Both, the KHD (Eqs. 6.1-2 and 6.1-18) as well as the time-dependent approach (Eqs. 6.1-2 and 6.1-19) very nicely simulate the experimental spectrum which consists mainly of Q- and S-branch transitions (Ganz and Kiefer, 1993b). 

Both classical and quantum mechanical treatments of Raman scattering are based on Eq. (2.1), and such treatments are very valuable in understanding the effect and interpreting spectra (1-5). One of the more analytically important results of Raman theory is the Raman scattering cross section, aj, which will be discussed at some length below. Before considering the factors that affect CT, it is useful to review several aspects of Raman theory. 

The classical theory of Raman scattering offered a mechanistic way of understanding the Raman effect but was unable to provide information on the intensity of Raman... 

The purpose of this section is to provide some background to the theories by introducing theoretical expressions for Raman scattering intensities and outlining some of the more important features of the various theories classical electromagnetic enhancement (EM enhancement) and nonclassical contributions (chemical effects). 

In these lectures I have presented a survey of modern Raman spectroscopy, emphasising the applications to gases. Since it is necessary to have some idea of the theory in order to fully appreciate the various applications, I have outlined the semi-classical theory of both conventional and nonlinear Raman processes. Quite a lot of space is devoted to the basis of conventional Raman scattering, not only because it provides an essential foundation for the semi-classical theory of nonlinear processes, but also to emphasise that, using modern electro-optical technology, conventional Raman is still very much the preferred technique for many applications, and that the new nonlinear techniques should only be used when conventional Raman fails or when very high resolution is essential. 

According to the classical theory of radiation [1], an oscillating dipole moment p will emit radiation with an electric field proportional to its second time derivative p. Equations 11.1 and 11.5 then imply that radiation will be scattered at the frequencies coq, coq — co, and coq + co, corresponding to Rayleigh, Stokes Raman, and anti-Stokes Raman scattering, respectively. The scattered electric fields are proportional to Eq, so that the Rayleigh and Raman intensities are linear in the incident laser intensity. Expressions similar to Eq. 11.5 are... 

Kramers and Heisenberg [2], who predicted the phenomenon of Raman scattering several years before Raman discovered it experimentally, advanced a semiclas-sical theory in which they treated the scattering molecule quantum mechanically and the radiation field classically. Dirac [3] soon extended the theory to include quantization of the radiatiOTi field, and Placzec, Albrecht and others explored the selection rules for molecules with various symmetries [4, 5]. A theory of the resonance Raman effect based on vibratiOTial wavepackets was developed by Heller, Mathies, Meyers and their colleagues [6-11]. Mukamel [1, 12] presented a comprehensive theory that considered the nonlinear response functions for pathways in LiouvUle space. Having briefly described the pertinent pathways in Liouville space above, we will first develop the Kramers-Heisenberg-Dirac theory by a second-order perturbation approach, and then turn to the wavepacket picture. 

Skoog et al. [2] emphasized the instmmentation advances of the 1980s and early 1990s. Both Fourier-transform Raman (FT-Raman) spectrometers and single-stage spectrographs are discussed. There is some discussion of optical fiber probes, but none of the Raman microprobe. The authors sketch the theory of Raman scattering and present a classical (polarizability derivative) treatment of selection rules and intensity. Resonance enhancement and surface enhancement are treated briefly. In a textbook noted for its emphasis on instrumentation, there is little discussion of current applications. 

Fig. 1.32. Classical theory for the origin of the Raman effect. In (a) we see a diatomic molecule where represents protons and represents the center of gravity of the electrons. The electrons are displaced by the external field of the photon and an induced dipole moment is generated which changes when the bond length changes during the molecular vibration. The induced dipole moment is plotted in (b) as an amplitude modulated wave with steady amplitude components shown in (c) from which scattered radiation is generated.

---