Raman scattering matrix element

The transition polarizibility a s (Raman scattering matrix element) can be expressed in terms of ... 

For GaAs values for the EP scattering matrix element from r-X and r-L have been deduced by various experiments such as transferred electron effects and time-dependent Raman scattering. In general, these workers find that 2... 

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... 

The importance of the hyper Raman effect as a spectroscopic tool results from its symmetry selection rules. These are determined by products of three dipole moment matrix elements relating the four levels indicated in Fig. 3.6-1. It turns out that all infrared active modes of the scattering system are also hyper-Raman active. In addition, the hyper Raman effect allows the observation of silent modes, which are accessible neither by infrared nor by linear Raman spectroscopy. Hyper Raman spectra have been observed for the gaseous, liquid and solid state. A full description of theory and practice of hyper-Raman spectroscopy is given by Long (1977, 1982). 

From the above discussion when dpjdq, or more rigorously, VqPy is zero or has a slope discontinuity, there are likely to be slope discontinuities in the combined density of states, as revealed by infrared and Raman spectra of two-phonon processes. Points in the Brillouin zone where each of the components of VqP = 0 are known as critical points. The intensity of infrared absorption or Raman scattering depends upon quantum mechanical matrix elements which are, in general, not simple to evaluate. However, by using symmetry considerations and group theoretical methods, the various modes can be assigned as infrared or Raman active. 

The second factor in (2.66) describes quite generally the transition probability for all possible two-photon transitions such as Raman scattering or two-photon absorption and emission. Figure 2.30 illustrates schematically three different two-photon processes. The important point is that the same selection rules are valid for all these two-photon processes. Equation (2.66) reveals that both matrix elements D,- and Dkf must be nonzero to give a nonvanishing transition probability A,/. This means that two-photon transitions can only be observed between two states i) and I/) that are both connected to intermediate levels fe) by allowed single-photon optical transitions. Because the selection rule for single-photon transitions demands that the levels i) and A ) or A ) and /) have opposite parity, the two levels i) and I/) connected by a two-photon transition must have the same parity. In atomic two-photon spectroscopy s s or s d transitions are allowed, and in diatomic homonuclear molecules Eg Eg transitions are allowed. 

The intensity of the Raman lines is proportional to the product of the Raman scattering cross section aR, which depends according to (3.12) on the matrix elements atj) of the polarizability tensor and the density Ni of molecules in the initial state. If the cross sections aR have been determined elsewhere, the intensity of the Raman lines can be used for measurements of the population densities N(v, J). Assuming a Boltzmann distribution (3.11a), the temperature T of the sample can be derived from measured values of N(v, J). This is frequently used for the determination of unknown temperature profiles in flames [325] or of unknown density profiles in liquid or gaseous flows [326] at a known temperature (Sect. 3.5). 

Figure 9 (Henneker et al., 1978a) shows an example of Raman scattering by a non totally symmetric mode involved in (linear) Herzberg-Teller coupling of two excited states, d> and <1> . It is an elaboration of Fig. 7 in that the result of resonance with both excited states is shown. For convenience, we have assumed that both transitions -> and <5, -> are allowed and have the same transition moment, but different polarization. This assumption makes the Rayleigh profile symmetric relative to the two states, but has no effect on the structure of the fundamental REP since its matrix elements are proportional to the product of the two transition moments. The difference in intensity of the overtones in the and <1> band region reflects the frequency differences cOg — co and cOg — co . Figure 9 is based...

The selection rules for the Raman scattering process are imposed by the susceptibility tensor in eqs. (1) and (6). Consider the matrix element f Xab i), where i) and /) are the initial and final states of the scattering medium, which transform like the irreducible representations of the material s point group, T, and Tf, respectively, and where the susceptibility tensor x associated with an excitation is decomposed into irreducible representations T of the point group of the crystal,... 

Although evaluating the matrix element for resonance Raman scattering by Eq. (12.13) is staightforward for a molecule with only one or two vibrational modes, it rapidly becomes intractable for larger molecules, and a wavepacket treatment similar to the one described for absorption in Chap. 11 becomes increasingly useful. To recast Eq. (12.13) in a time-dependent form, we first note that the factor... 

The matrix element for resonance Raman scattering thus is proportional to a half-Fourier transform of flie overlap of the final vibrational wavefunction (Xh(g)) with the time-dependent wavepacket X(t) created by exciting the molecule with white light in the ground state. See [6] and [8] for more complete proofs of this relationship, and [10] for a review of some of its extensions and applications. The resonance Raman excitation spectrum is proportional to as explained above. 

The matrix element for two-photon absorption is essentially the same as that for off-resonance Raman scattering (Eq. 12.11). Assuming that the two photons have... 

The Raman scattering cross section is also constrained by the selection rules imposed by the susceptibility tensor in Eq. (6). For example, the transition susceptibility matrix element (f Xabl ) nonzero only if the decomposition of the direct product Tf <8> <8> Fi contains the totally symmetric repre-... 

H) = component of with Cartesian index fia y, Rab component along fi of the transition dipole matrix element for transitions from a to b Tj = population relaxation characteristic time T2 = phase relaxation characteristic time V t) = interaction operator (a,((o))g= Raman scattering tensor for transitions from state g to state / ... 

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. 

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