Theory of Raman Scattering

Because Raman scattering involves vibrational and rotational modes within a sample, its explanation must necessarily involve a quantum mechanical treatment [21]. This is certainly true when the incident light corresponds to an intrinsic region of absorption in the sample, but it is also required for a quantitative analysis of the simpler Stokes and anti-Stokes Raman scattering, which is the subject of the discussion in this chapter. A detailed quantum mechanical understanding of Raman scattering, however, is not necessary for the applications that are of interest in this book, and for that reason, only a brief account is offered here. 

When light having a frequency that is removed from any absorption band impinges on a sample, the exiting radiation will be the result of several possible processes. Since, in this discussion, the material is transparent, the majority of the light will be simply trans- 

Energy level diagram for Rayleigh (left) and Raman (right) scattering. 

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Lee S-Y and Heller E J 1979 Time-dependent theory of Raman scattering J. Chem. Rhys. 71 4777... 

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). 

L.A. Nafle, Theory of Raman scattering, in Handbook of Raman Spectroscopy From the Research Laboratory to the Process Line, I.R. Lewis and H.G.M. Edwards (Eds), Practical Spectroscopy Series 28, Marcel Dekker, New York, 2001. 

Long, D. A., The Raman Effect A Unified Treatment of the Theory of Raman Scattering by Molecules, J. Wiley Sons, New York, 2002. 

Heather, R.W. and Metiu, H. (1989). Time-dependent theory of Raman scattering for systems with several excited electronic states Application to a H3" model system, J. Chem. Phys. 90, 6903-6915. 

As previously mentioned, the evanescent wave could interact with the optically rare medium not only by being absorbed but also by being scattered either elastically (Rayleigh Scattering) or inelastically (Raman Scattering). Because it is not within the scope of this paper to review the complete history and theory of Raman scattering, further information is indicated in Ref. 

Barker AS, Loudon R. Response functions in the theory of Raman scattering by vibrational and polariton modes in dielectric crystals. Rev Modern Phys 1972 44 18. 

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). 

This chapter covers the electromagnetic radiation scattering behavior of semiconductor nanowires as it impacts their Raman scattering properties, as well as how the basic theory of Raman scattering in semiconductor crystals may be applied to the corresponding nanowire forms, with specific examples from the recent literature. The discussion then turns to more specific ways in which Raman spectroscopy can be used to precisely determine crystal structure, crystal size (or quality), composition, impurity concentration, strain, and temperature with submicron resolution, as demonstrated in recent literature examples. 

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... 

Cowley RA (1964) The theory of Raman scattering from crystals. Proc Phys Soc 84 281-296 Ghose S, Ito Y, Hatch DM (1991) Paraelectric-antrferroelectric phase transition in titanite, CaTiSiOs. I. A high temperature X-ray diffraction study of the order parameter and transition mechansim. Phys Chem Miner 17 591-603... 

The original Placzek theory of Raman scattering [30] was in terms of the linear, or first order microscopic polarizability, a (a second rank tensor), not the third order h3q)erpolarizability, y (a fourth rank tensor). The Dirac and Kramers-Heisenberg quantum theory for linear dispersion did account for Raman scattering. It turns out that this link of properties at third order to those at first order works well for the electronically nonresonant Raman processes, but it cannot hold rigorously for the fully (triply) resonant Raman spectroscopies. However, provided one discards the important line shaping phenomenon called pure dephasing , one can show how the third order susceptibility does reduce to the treatment based on the (linear) polarizability tensor [6, 27]. 

The theory of Raman scattering is virtually identical to that for two-photon absorption the principal difference is that E -D for the incident wave need not be identical to Ei D for the scattered wave. Thus ( 7 2) is not necessarily zero and terms involving can appear in the effective Hamiltonian that drives the transition. Richman et al. (1963) observed the Raman spectrum of LaClj and... 

In this section we review the basic theory of Raman scattering and list or derive the formulas necessary for the calculation of scattering cross sections... 

Raman scattering selection rules 516 5.2. Theory of Raman scattering in ... 

Raman scattering is a fundamental form of molecular spectroscopy [1,2]. Together with infrared (IR) absorption, Raman scattering is used to obtain information about the structure and properties of molecules from their vibrational transitions [3-5], The theory of Raman scattering is more complex that the theory of IR absorption, but there are a number of close parallels between the two theories [6-10],... 

In this chapter, we will provide a very basic, unified treatment of the theory of Raman scattering and ROA. More extensive descriptions of theory of Raman and ROA scattering can be found elsewhere [6-10,12,14,15]. 

The theory of Raman scattering simplifies dramatically in the far-from-resonance (FFR) limit, where the exciting laser radiation is far from the lowest allowed excited electronic state of the molecule [15]. In this limit, the interaction of the light with the molecule is approximately the same for both the incident and the scattered radiation. There, the Raman tensor becomes symmetric and this symmetry reduces the number of Raman invariants from three to two, the isotropic and (symmetric) anisotropic invariants. The antisymmetric anisotropic invariant vanishes, by its antisymmetric definition in Eqs. (7) and (9), and we have... 

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