Absorption-peak frequency estimation

Figure 6. Frequency dependence of dimensionless absorption. Dimensionless collision frequency y = 0.3. (a) Calculation from rigorous formulas (70a) and (70b) (solid lines) and from the PL-RP approximation, Eqs. (78-80b) (dashed lines). Curves 1 refer to P = ji/8 and curves 2 to (3 = ti/4. Vertical lines mark the values of the absorption-peak frequencies estimated by Eqs. (85) and (86). (b) Comparison of the total absorption (solid line) with contribution of the precessional component (dashed line). Calculation for the PL-RP approximation, P = ji/8.

Mean Number of Reflections. Estimation of Absorption-Peak Frequency... 

We also give here the formulas for the proportion r of the rotators, for estimation of the absorption-peak frequency x mentioned in Section IV.D.3, and for the mean number m of the reflections performed during the lifetime. 

Then the absorption-peak frequency vUb (in cm ) could be estimated as a frequency of a periodic motion averaged over ensemble of the dipoles ... 

In Figs. 66 and 68 the calculated absorption and loss spectra are depicted for ordinary water at the temperatures 22.2°C and 27°C and for heavy water at 27°C. The solid curves refer to the composite model, and the dashed curves refer to the experimental spectra [42, 51]. For comparison of our theory with experiment at low frequencies, in the case of H20 we use the empirical formula [17] comprising double Debye-double Lorentz frequency dependences. In the case of D20 we use empirical relationship [54] aided by approximate formulae given in Appendix 3 of Section V. The employed molecular constants were presented in previous sections, and the fitted/estimated parameters are given in Table XXIV. The parameters of the composite model are chosen so that the calculated absorption-peak frequencies ilb and vR come close to the... 

The separation of the two stages is easier to discern when the rates of the two processes are so different, but it can also be seen in the ultrasonic spectra of metal-sulfate systems (Sec. 3.4.4). Ultrasonic absorption peaks can be attributed to formation of outer-sphere complexes (at higher frequency, shorter t) and collapse of outer-sphere to inner-sphere complexes (at lower frequency). In addition to uv spectral and ultrasonic detection, polarimetry and nmr methods have also been used to monitor and measure the strength of the interaction. There are difficulties in assessing the value of ATq, the outer-sphere formation constant. The assemblage that registers as an ion pair by conductivity measurements may show a blank spectroscopically. The value of Aq at T" K may be estimated using theoretically deduced expres-... 

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. 

We remind the Reader that we use the hat-flat model in order to describe a two-humped absorption spectmm characteristic for water. The larger peak frequency (x) we refer to the librators and the smaller one (x) to the rotators of both types. We estimate x by the same way as in Eq. (85) ... 

As for the rotational-peak frequency x, we estimate it in the same fashion as was used for derivation of Eq. (86). Namely, we find x as the frequency, at which the rotational-absorption component A (x) exhibits maximum in the limit y —> 0. It follows from Eq. (107) that in this limit... 

Using this result, we may simplify calculation of the spectral function Liz) by neglecting the precessional contribution to L. We shall estimate also in this approximation the peak frequencies X ib and xrot of the absorption bands determined by the librational and the rotational subensembles. 

The estimated positions of maximum absorption (see vertical lines in Figs. 22c and 22d are close to the peak frequencies found from Eqs. (171)-(173) and (179). The peak frequency (190a) is inversely proportional to p it increases with the decrease in the form factor/or with the increase of the well depth u, see, respectively, Fig. 23a and 23b. The proportions of the librators, precessors, and rotators are calculated from Eq. (184) and presented in Fig. 23c and 23d by solid, dash-dotted, and dashed curves, respectively. Figure 23c shows relevant functions of the reduced well depth u and Fig. 23d—of the form factor/. 

Figure 23. (a, b) Dependence of the estimated peak frequency of the librational absorption... 

A. Previous models of water (see 1-6 in Section V.A.l) and also the hat-curved model itself cannot describe properly the R-band arising in water and therefore cannot explain a small isotope shift of the center frequency vR. Indeed, in these models the R-band arises due to free rotors. Since the moment of inertia I of D20 molecule is about twice that of H20, the estimated center of the R-band for D20 would be placed at y/2 lower frequency than for H20. This result would contradict the recorded experimental data, since vR(D20) vR(H20) 200 cm-1. The first attempt to overcome this difficulty was made in GT, p. 549, where the cosine-squared (CS) potential model was formally (i.e., irrespective of a physical origin of such potential) applied for description of dielectric response of rotators moving above the CS well (in this work the librators were assumed to move in the rectangular well). The nonuniform CS potential yields a rather narrow absorption band this property agrees with the experimental data [17, 42, 54]. The absorption-peak position Vcs depends on the field parameter p of the model given by... 

At this point it is necessary to relate the relaxation time for the chemical process to the spin-spin relaxation process which is always present when the system absorbs energy from the radio frequency wave. Both processes lead to broadening of the NMR absorption peak. If T, is the total relaxation time for process i, it may be estimated from T2 and x,- using the relationship... 

The dielectric response shown in Fig. 43a resembles a highly damped resonance curve with the peak frequency v m located in the THz region. The s// L(v) spectra in Fig. 43b are compared for y =2.8 (curve 1) andy L = 2 (curve 2), for which the lifetimes t are respectively 0.089 and 0.14 ps. For greater y L (i.e., for shorter lifetime t l) the absorption curves become shallower. The frequency vj m of the loss-peak intensity, marked in Fig. 43b by the vertical lines, agrees with the estimate (193) only approximately. It is worthwhile to emphasize the following ... 

Figure 10 shows the absorption spectrum of the gold particles in solution (dashed curve) and the fluorescence spectrum of a solution of lissamine fluorophores both before and after they are attached to the gold particles. As expected, the gold particles in solution show an absorption peak centered at the dipolar plasmon frequency. It is seen that the fluorescence of the lissamine is strongly quenched by presence of the gold nanoparticles. It was estimated that the quenching efficiency exceeded 99%, when the residual fluorescence of unbound fluorophores was taken into account. 

The PL-RP approximation allows us to estimate this quantity in another fashion. It is seen in Fig. 6b that near the peak-absorption frequency the contribution of the precessional component Lpp is negligible, so that L = Lpl at A = xm. We shall estimate xm by considering the limit y > 0. The integrand in Eq. (79) comprises the Lorentz-like terms. In this limit their imaginary... 

Figure 13. Dimensionless absorption versus normalized frequency calculated rigorously (solid lines), from the PL-RP approximation (dashed lines), and for the hybrid model (dashed-and-dotted lines). The cone angle P = tt/8 and the reduced collision frequency y = 0.2. The reduced well depth u = 3.5 (a) and 5.5 (b). Left and righ vertical lines mark the frequency peaks estimated, respectively, in the rotational and librational ranges.

If we (1) take a reduced mass of two oscillating molecules equal to (1/2) ntgM— that is, equal to half of the mass of one water molecule (mH is mass of a proton and M is the molecular mass)—and (2) set il = 2ncvR, where vR is a peak-absorption frequency of the R-band (v,r 190 cm ), then we find for water the following estimation ... 

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