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). 

For a reasonable set of the parameters the calculated far-infrared absorption frequency dependence presents a two-humped curve. The absorption peaks due to the librators and the rotators are situated at higher and lower frequencies with respect to each other. The absorption dependences obtained rigorously and in the above-mentioned approximations agree reasonably. An important result concerns the low-frequency (Debye) relaxation spectrum. The hat-flat model gives, unlike the protomodel, a reasonable estimation of the Debye relaxation time td. The negative result for xD obtained in the protomodel is explained as follows. The subensemble of the rotators vanishes, if u —> oo. 

Figures 22a and 22b illustrate a nondimensional absorption frequency dependence...

In the librational band we have attained now a satisfactory agreement of the theoretical and experimental absorption frequency dependences. Comparing Figs. 32a and 32b with Figs. 26a and 26c calculated in Section V for a pure ... 

It is interesting to compare the results obtained for ordinary and heavy water. To interpret the difference, we show in Fig. 33 by solid curves the total absorption attained in the R-band (i.e., near the frequency 200 cm-1). Dashed curves and dots show the components of this absorption determined, respectively, by a constant (in time) and by a time-varying parts of a dipole moment. In the case of D20, the R-absorption peak vR is stipulated mainly by nonrigidity of the H-bonded molecules, while in the case of H20 both contributions (due to vibration and reorientation) are commensurable. Therefore one may ignore, in a first approximation, the vibration processes in ordinary water as far as it concerns the wideband absorption frequency dependences (actually this assumption was accepted in Section V, as well is in many other publications (VIG), [7, 12b, 33, 34]. However, in the case of D20, where the mean free-rotation-frequency is substantially less than in the case of H20, neglecting of the vibrating mechanism due to nonrigid dipoles appears to be nonproductive. 

Figure 37. Absorption-frequency dependence, water H20 at temperature 27°C. Calculation for the HC—HO model (solid line) and for the hybrid-cosine-squared potential model (dashed-and-dotted line). Dahsed curve Experimental data [42], (b) Same as in Fig. 34c but refers to T — 300 K.

Figure 63. (a,b) Solid lines absorption frequency dependences calculated for liquid H2O (a)... 

Choosing room temperature as 20.2°C, we depict in Fig. 5a the wideband absorption frequency dependence a(v) of water H20 and in Fig. 6a we depict that of water D20. The fitted parameters of the model are presented in Table II. The total loss spectrum e"(v) is shown in Figs. 5b and 6b, respectively, for OW and HW. The solid lines in Figs. 5a,b and 6a,b mark the results of our calculations. 

Figure 5 For H20 (a) wideband absorption frequency dependence (b) total loss spectrum (c) partial absorption contributions (1-4) (d) partial loss contributions (1-4) (e) is similar to (c) but for a reduced, domain, and on a linear ordinate scale (f) is similar to (d) but on a reduced ordinate scale, (g) comprises the partial contributions to the Raman spectra in the R (v) representation (h) in the Bose-Einstein representation.

Regarding transition (A9b) —> (A9c), it appears that the mean lifetime T (water) could hardly become much shorter than 0.05ps, since the quantity (A8) now falls near its lower limit equal to h. In other words, the bandwidth Av of the absorption frequency dependence, generated by such vibrations, hardly could become much wider than the bandwidth depicted by curves 3 in Figs. 4h and 5h, respectively, for H20 and D20. 

The adopted molecular constants, along with fitted and estimated parameters, are presented in Tables IV-VI. The absorption frequency dependences are depicted in Figs. lOe, 10c, and 10a, respectively, for the lowest (7 ), room9(74), and highest (77) temperatures. The loss spectra are shown for the same temperatures in Figs. lOf, lOd, and 10b. The dash-dotted lines depict the contribution to loss spectra of transverse vibration (TV), and the dashed lines depict such spectra calculated without account of this vibration. We see that transverse vibrations play an important role in the THz region. 

Figure 46 Dimensionless absorption frequency dependences, calculated for the planar (solid lines) and spatial (dashed lines) hat models, (a) Calculation for liquid water at 27°C with the parameters / = 23°, u = 8, / = 0.8, y = 0.3. (b) Calculation for ice at -7°C with the parameters P = 23.5°, u = 8.5,/ = 0.15, y = 0.8...

Third, the velocity of atom C can be measured using the Doppler effect. If the atom C is excited with a laser, the absorption frequency depends on its velocity component relative to the direction of the laser. If LIF is used to excite the atom, the fluorescence intensity at different frequencies will contain the information about the formation of AB in different quantum states. If the atom is detected at the same time and the same position that is used to dissociate the parent molecule, Doppler broadened lines are obtained. The information about the formation of the AB product in different states is in this case contained in the edge of the Doppler broadened line. Usually it is not easy to obtain information from a noisy edge of a Doppler broadened line. 

Infrared absorption frequency depends on strength of bonds, mass of atoms and stretching or bending modes and vibrations. 

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