Spectral function absorption frequency dependence

Using the so-called planar libration-regular precession (PL-RP) approximation, it is possible to reduce the double integral for the spectral function to a simple integral. The interval of integration is divided in the latter by two intervals, and in each one the integrands are substantially simplified. This simplification is shown to hold, if a qualitative absorption frequency dependence should be obtained. Useful simple formulas are derived for a few statistical parameters of the model expressed in terms of the cone angle (5 and of the lifetime x. A small (3 approximation is also considered, which presents a basis for the hybrid model. The latter is employed in Sections IV and VIII, as well as in other publications (VIG). 

At high temperatures, vibrational states must also be included in the partition sum above. The nuclear weights are gj for hydrogen we have, for example, gj = 1 for even j, and gj = 3 for odd j. However, we mention that in low-temperature laboratory measurements as well as in astrophysical applications, para-H2 and ortho-H2 abundances may actually differ from the proportions characteristic of thermal equilibrium (Eq. 6.53). In such a case, at any fixed temperature T, one may account for non-equilibrium proportions by assuming gj values so that the ratio go/gi reflects the actual para to ortho abundance ratio. Positive frequencies correspond to absorption, but the spectral function g(co T) is also defined for negative frequencies which correspond to emission. We note that the product V g a> T) actually does not depend on V because of the reciprocal F-dependence of Pt, Eq. 6.52. 

The role of specific interactions in the plasticization of PVC has been proposed from work on specific interactions of esters in solvents (eg, hydrogenated chlorocarbons) (13), work on blends of polyesters with PVC (14—19), and work on plasticized PVC itself (20—23). Modes of interaction between the carbonyl functionality of the plasticizer ester or polyester were proposed, mostly on the basis of results from Fourier transform infrared spectroscopy (ftir). Shifts in the absorption frequency of the carbonyl group of the plasticizer ester to lower wave number, indicative of a reduction in polarity (ie, some interaction between this functionality and the polymer) have been reported (20—22). Work performed with dibutyl phthalate (22) suggests an optimum concentration at which such interactions are maximized. Spectral shifts are in the range 3—8 cm-1. Similar shifts have also been reported in blends of PVC with polyesters (14—20), again showing a concentration dependence of the shift to lower wave number of the ester carbonyl absorption frequency. 

It follows from Eq. (32) that the spectral function L(z) actually determines the absorption coefficient. At high frequencies,13 such thatx y, this coefficient is proportional to xlm[x (x)]. In other limit, at low frequencies, one may neglect the frequency dependence L(z) by setting L(z) = L(iy). In this approximation, Eq. (32) yields the Debye-relaxation formula (VIG, p. 194) ... 

Coherent light sources are characterized by a spectral intensity distribution E(oj) and a frequency-dependent phase 0(w). According to first-order perturbation theory, linear absorption probabilities are given by the overlap between the spectrum of the light source E(w) and the optical transition, and are independent of the phase function In non-linear processes (2nd... 

The analysis of the dynamics and dielectric relaxation is made by means of the collective dipole time-correlation function (t) = (M(/).M(0)> /( M(0) 2), from which one can obtain the far-infrared spectrum by a Fourier-Laplace transformation and the main dielectric relaxation time by fitting < >(/) by exponential or multi-exponentials in the long-time rotational-diffusion regime. Results for (t) and the corresponding frequency-dependent absorption coefficient, A" = ilf < >(/) cos (cot)dt are shown in Figure 16-6 for several simulated states. The main spectra capture essentially the microwave region whereas the insert shows the far-infrared spectral region. 

Eq. (3.2-16) shows that the frequency dependence of 6 is manifested through the dependence of the refractive index n on the radiation frequency. Since the spectral range in which optical materials are used is far enough from their absorption bands the dependence of n and hence of 6 on the frequency is a slightly and smoothly changing function in this spectral interval. Therefore, totally reflecting retarders are almost achromatic optical devices. 

In Figure 9 we depict the frequency dependences of the partial absorption coefficients aq(v) and a (v) pertinent to two harmonic-vibration modes. These frequency dependences are calculated from formulas (A6), (21) [24], (25), (28), and (29). When the above-mentioned coupling is accounted for (solid lines in Fig. 9), the spectral functions are taken from Eq. (Al). On the other hand, when the coupling is neglected (open circles in Fig. 9), then Lq and L are found from Eq. (19). We see from Fig. 9a that for both cases the calculated partial absorption a (v) practically coincide. The same assertion is valid also for the partial absorption ocq(v) depicted in Fig. 8b. Hence, there is no practical need to account for the coupling between the harmonic reorientation and vibration of HB molecules for calculation of spectra in liquid water. However, the effect of such coupling becomes noticeable (being, however, a rather small) in the case of ice, where the absorption lines are much narrower. 

The spectral profile, J(term depends on the product of real and imaginary parts of A, 91 A and 3 X, Eq. 5.92, which will be positive at low frequencies if both, the dipole function and the interatomic potential are short range [236]. In other words, the intercollisional absorption profile is negative under such conditions an absorption dip is obtained, in agreement with the observations. 

The u.v.-visible spectra, in the linear absorption regime, would also yield a linear dependence of the absorbance as a function of the number of layers, provided the structural correlation from layer to layer is maintained. The results of the quartz-crystal microbalance and the u.v.-visible spectral studies for L-B films of a macrocycle, tetrakis-butyl-phenoxy-phthalocyanine show a linear relation in each case,indicating a uniform transfer from layer to layer with the same molecular structure. In contrast, the results on the L-B films of poly-(3 dodecyl-thiophene) reveal a uniform mass transfer (linear relation between the frequency change and the number of layers), but the structural correlation from layer-to-layer changes around the tenth layer, as evidenced by the change of the slope of the absorbance curve, as a function of the number deposited layers. 

A detailed consideration [11-17] shows that the time evolution of the laser intensity in a specific mode q a)) with frequency (d after the start of the pump pulse depends on the gain profile of the laser medium, the absorption a o)) of the intracavity sample, and the mean mode lifetime fm- If the broad gain profile with the spectral width Acpg and the center frequency coq can be approximated by the parabolic function... 

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