Raman scattering correlation function

The autocorrelation function G(t) corresponds to the correlation of a time-shifted replica of itself at various time-shifts (t) (Equation (7)).58,65 This autocorrelation defines the probability of the detection of a photon from the same molecule at time zero and at time x. Loss of this correlation indicates that this one molecule is not available for excitation, either because it diffused out of the detection volume or it is in a dark state different from its ground state. Two photons originating from uncorrelated background emission, such as Raman scattering, or emission from two different molecules do not have a time correlation and for this reason appear as a time-independent constant offset for G(r).58... 

Time-dependent correlation functions are now widely used to provide concise statements of the miscroscopic meaning of a variety of experimental results. These connections between microscopically defined time-dependent correlation functions and macroscopic experiments are usually expressed through spectral densities, which are the Fourier transforms of correlation functions. For example, transport coefficients1 of electrical conductivity, diffusion, viscosity, and heat conductivity can be written as spectral densities of appropriate correlation functions. Likewise, spectral line shapes in absorption, Raman light scattering, neutron scattering, and nuclear jmagnetic resonance are related to appropriate microscopic spectral densities.2... 

Liquids are difficult to model because, on the one hand, many-body interactions are complicated on the other hand, liquids lack the symmetry of crystals which makes many-body systems tractable [364, 376, 94]. No rigorous solutions currently exist for the many-body problem of the liquid state. Yet the molecular properties of liquids are important for example, most chemistry involves solutions of one kind or another. Significant advances have recently been made through the use of spectroscopy (i.e., infrared, Raman, neutron scattering, nuclear magnetic resonance, dielectric relaxation, etc.) and associated time correlation functions of molecular properties. 

The shape of the vibration-rotation bands in infrared absorption and Raman scattering experiments on diatomic molecules dissolved in a host fluid have been used to determine2,15 the autocorrelation functions unit vector pointing along the molecular axis and P2(x) is the Legendre polynomial of index 2. These correlation functions measure the rate of rotational reorientation of the molecule in the host fluid. The observed temperature- and density-dependence of these functions yields a great deal of information about reorientation in solids, liquids, and gases. These correlation functions have been successfully evaluated on the basis of molecular models.15... 

Abstract Electronic correlation effects in La2.ISrINi04 lead to spontaneous phase separation into microscopic spin/charge stripes with commensurate or incommensurate order. Raman scattering experiments on such single crystalline materials show a rich phenomenology of phonon and magnon anomalies due to the new, self-organized periodicities. These effects are observable as function of temperature. 

Raman scattering depends on the time correlation function of the many-body polarizability of the liquid, collective dipole moment. In the case of Raman scattering, an external electric field (from a laser) generates an induced collective dipole in the liquid ... 

Several complementary techniques exist for the experimental study of dephasing, and here we will underline only the differences between them. The simplest experimental access to dephasing is by spontaneous Raman scattering. A laser of frequency scattered light at to, around cuj = frequency resolved. The isotropic Raman scattering cross section is directly related to the vibrational correlation function... 

Coefficients (pj 1 - 2 Involved in Raman Isotropic and Depolarized Light Scattering Correlation Functions Fu.(t) = Y.N,h,h wli 1 12 SN(t)Rj1(t)Rh(t)F+ for Successive Multipolar Induction Operators... 

The nonresonant term may be obtained from the resonant term by the replacement cj, - tus, and, henceforth, will be neglected. Equation (2.9) states that the scattering amplitude is the half-Fourier transform of the overlap of the time-evolving wavepacket with the final state of interest (multiplied by the transition moment). Equation (2.9) bears a close resemblance to Eq. (2.3) for the absorption cross section, but there are three differences to note (1) the cross-correlation function of the moving wavepacket with the final vibrational state of interest is required, rather than the autocorrelation function (2) an integral over the range [0, oo], not [-00,00], is required for the Raman amplitude (3) The cross-section / " (to) is proportional to the absolute value squared of a ... 

This time-dependent method allows one to nicely connect the theoretical and experimental observations. As mentioned earlier, the correlation function and its generalizations yield the spectra for a large number of other photospectroscopy processes, such as Raman processes [M], as well as molecular scattering [73, 74]. 

The final application we discuss is one where the maximum entropy formalism is used not only to fit the spectrum but also to extract new results. Specifically we discuss the determination of the time cross-correlation function, Cf, t) (Eq. (43)), which is the Fourier transform of the Raman scattering amplitude a/((Tu) (Eq. (44)) when what is measured is the Raman scattering cross section afi(m) a/((Tii) 2. The problem is that the experiment does not appear to determine the phase of the amplitude. The application proceeds in two stages (i) Representing the Raman spectrum as one of maximal entropy, using as constraints the Fourier transform of the observed spectrum. At the end of this stage one has a parametrization of a/,( nr) 2 whose accuracy can be determined by how well it fits the observed frequency dependence, (ii) The fact that the Raman spectrum can be written as a square modulus as in Eq. (97) implies that it can be uniquely factorized into a minimum phase function... 

There is an indirect way to detect intermittent local collective motions. In the case of depolarized Raman scattering, the depolarization ratio is sensitive to low-frequency fluctuations in water. Depolarization is the scattering of the polarization of the electric field of light in a direction perpendicular to the original direction of polarization. Each fluctuating state has a distinct depolarization ratio. The intermittent character of the dynamics is known to appear as a so-called 1,/ frequency (f) dependence in a power spectrum. The power spectmm is obtained by Fourier transforming a time correlation function. 

Raman spectroscopy has been used to probe interactions occurring in PAni nanotube [23-24] composites, the orientation of nanotube bundles within a matrix [25, 26], and the efficiency of load transfer from the host matrix to SWCNTs [27,28]. Unlike X-ray diffraction (XRD) methods [12], Raman spectroscopy can detect very low concentrations of SWCNTs in a polymer matrix [29,30]. The degree of orientation of aligned nanotubes can be estimated by polarized Raman spectroscopy due to the presence of a strong resonance Raman scattering effect [31,32]. Polarized Raman spectroscopy in combination with a mathematical model [33] has been employed to characterize the orientational order of nanotubes in polymers [34]. Using this model, the polarized Raman intensity of nanotubes is correlated with the orientation order parameters of SWCNTs in a utuaxially oriented system. An orientation distribution function can then be obtained. 

Conductivity can be deduced from vibrational spectra in IR spectroscopy, the absorption coefficient a(co) is related to tr(co) a(o)) = 4no(o))/nc, n being the refractive index and c the velocity of light. In Raman spectroscopy, the scattered intensity /(m) is related to conductivity by a(o ) oc o)I (o)/n(a)) + 1, n(co) being the Bose-Einstein population factor . Finally, the inelastic incoherent neutron scattering function P(o)) is proportional to the Fourier transform of the current correlation function of the mobile ions. P co) is homogeneous with a) /(cu) formalism. However, since P(co) reflects mainly single particle motions, its comparison with ff(co) could provide a method for the evaluation of correlation effects. (For further discussion, see also Chapter 9 and p. 333.)... 

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