Selection Rules for Raman Scattering

Inspection of Fig. 12.2 shows that some of the bands in an IR absorption spectrum also feature in the off-resonance Raman emission spectrum. The relative intensities of the bands are different, however, and there generally are bands that appear in 

We saw in Chap. 6 that there are two main selection rules for direct excitation of a harmonic oscillator from level n to level m first, w = n 1, and second, the vibration must change the permanent dipole moment of the molecule. Arguments parallel to those we used to find the selection mles for IR absorption can be used to predict qualitatively whether or not a particular vibrational mode will contribute to off-resonance Raman scattering. The difference is that for Raman scattering we relate the scattering matrix element ai,a to the molecular polarizability (a) rather than the permanent dipole moment. If the polarizability is expanded in a Taylor s series as a function of the normal coordinate (x) for the mode, the matrix element for Raman scattering becomes 

Resonance Raman scattering emphasizes the vibrational modes that are coupled most strongly to the resonant electronic transition i.e., modes with the largest displacement in the excited state). This tends to makes resonance Raman spectra much more selective than off-resonance Raman spectra, as we saw in Fig. 12.2. At the same time, the coupling to an electronic transition also relaxes the requirement 

12 Raman Scattering and Other Multi-photon Processes 

This usually is larger than the corresponding integral for scattering to b,( ) = 2 because, in the wave-packet picture, Xi(t) builds up overlap withxi,i(g) more quickly than it does f/iSn Xix(g) (Fig- 12.6). Dephasing of the wavepacket frustrates the rise of Cf2,ife)P (0)- In addition, the build up of ixi,ng) At)) must occur while the wavepackets for all the other modes retain good overlap with the initial, 

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Numerous SERS studies of adsorbed molecules have appeared in the literature. Obviously, it is a useful method for the identification of species at the interface, and its inherent surface sensitivity is an attractive feature. In this context it should be noted that the adsorption of a molecule can change the selection rules for Raman scattering, and modes that are Raman inactive in the isolated molecule may show up in SERS. 

To obtain the major selection rules for Raman scattering we can first expand Equation [14] in a Taylor series about the equilibrium configuration Q ... 

For a harmonic oscillator, the selection rule for the Raman effect is the following. Let us assume that the harmonic oscillator is originally in the state a with quantum number n. Then the matrix element (ci R y) will be different from zero only if the state j has the quantum number nil. Similarly, if state h has the quantum number nij 0 R ) will be different from zero only if state y has the quantum number mil. Both matrix elements will be simultaneously different from zero only if m = n or m = n 2, so that we may conclude that the selection rule for Raman scattering by a harmonic oscillator is An = 0, 2. The first possibility corresponds to scattering of light of the incident frequency v the second corresponds to scattering of light of frequency v db 2vg, where is the fundamental frequency of the harmonic oscillator. 

According to 8 72, the frequency v + Vah will appear in the Raman effect if any matrix element of the type (0 xiX2 b)y where Xi, X2 equal Xy 2/, or Zy is different from zero. We shall use this formulation of the selection rules for Raman scattering. 

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. 

S,R,Q,P,0 branches of the rotational Raman spectrum. The treatment outlined here is a little more general than usual since we have retained the antisymmetric operator contribution (2,.39b), which generates the less familiar selection rules for antisymmetric scattering, namely at = I with AX = O forbidden if K = 0 [18]. 

Because Raman scattering is also a two-photon process the selection rules for two-photon absorption are the same as for vibrational Raman transitions. For example, for a two-photon electronic transition to be allowed between a lower state j/" and an upper state... 

The selection rule for rotational Raman transitions are AJ = 2. This result relates to the involvement of two photons, each with angular momentum h, in the scattering process. Also allowed is A J = 0, but since such a transition implies zero change in energy it represents Raleigh scattering only. 

The selection rules for fi hyper-Raman scattering were derived by Cyvin, Rauch, and Decius, 02) and those for y hyper-Raman scattering, which has not yet been detected experimentally, by Christie and Lockwood 103>. From their tables one can see that silent modes become 3-active for such important point groups as C6, D6, C3v, C6v, C. D, 0 and Oh. Examples of additional 7 activity can be found in the point groups C4v, C. D. D, and Oh. Long and Stanton 104) have derived a quantum-mechanical theory of the hyper-Raman effect which indicates several possibilities for resonance enhancement of hyper-Raman intensities. Iha and Woo 105) extended the theory of nonlinear... 

The selection rules for molecular vibrations involved in hyper Raman scattering are summarized by... 

The selection rules for CARS are precisely the same as for spontaneous Raman scattering but CARS has the advantage of vastly increased intensity. 

The experimental techniques most commonly used to measure the phonon distributions are IR absorption, Raman scattering and neutron scattering. The IR and Raman spectra of crystalline silicon reflect the selection rules for optical transitions and are very different from the phonon density of states. The momentum selection rules are relaxed in the amorphous material so that all the phonons contribute to the spectrum. 

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