Frequency dependent absorption coefficients

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. 

Figure 16-6. Left collective dipole time correlation for SCW states. Right corresponding frequency-dependent absorption coefficients [26]...

On crossing the X-ray K absorption edge from lower to higher frequencies the absorption coefficient p will increase by the factor 63.868/Z . Since the spatial distribution of the core electrons is confined to a very small volume near the nucleus, f" (and f) shows a relatively weak dependence on the scattering angle. Few experiments have been done to confirm theoretical predictions, e.g. on Barium... 

The theoretical models for interpretation of infrared intensities presented in the subsequent chapters have been largely applied in analyzing gas-phase experimental data. Gas-phase intensities provide an unique opportunity to study in a unifonn approach the interrelations between molecular structure and intensity parameters. This is due to the fact that, in contrast to vibrational frequencies, the absorption coefficients depend strongly on the phase state and on solvent effects. Intensities of different modes of the same molecule are not influenced in a systematic way by the solvent. The variations of absorption coefficients may reach tens and hundreds percent. Accurately determined gas-phase intensities are, therefore, of fundamental importance as a source of experimoital information on intramolecular properties. 

The sound absorption of materials is frequency dependent most materials absorb more or less sound at some frequencies than at others. Sound absorption is usually measured in laboratories in 18 one-third octave frequency bands with center frequencies ranging from 100 to 5000 H2, but it is common practice to pubflsh only the data for the six octave band center frequencies from 125 to 4000 H2. SuppHers of acoustical products frequently report the noise reduction coefficient (NRC) for their materials. The NRC is the arithmetic mean of the absorption coefficients in the 250, 500, 1000, and 2000 H2 bands, rounded to the nearest multiple of 0.05. 

Sometimes the atoms (or molecules) in molecular beams are put into selected electronic, vibrational and rotational states. The initial state selection can be made with lasers. A laser beam of appropriate frequency is shined onto a molecular beam and the molecule goes onto an appropriate excited state. The efficiency of selection depends upon the absorption coefficient. We can attain sufficient absorption to get highly vibrationally excited molecule with the laser. A spin forbidden transition can also be achieved by using a laser. 

It is known that measnring the absorption coefficient (and thns the extinction coefficient) over the whole freqnency range, 0 < real part of N(co) - that is, the normal refractive index ( >) - can be obtained by nsing the Kramers-Kronig relationships (Fox, 2001). This is an important fact, because it allows us to obtain the frequency dependence of the real and imaginary dielectric constants from an optical absorption experiment. 

The reason for the different dependencies of absorption on the concentration can be seen starting with the Beer-Lambert Law. If L° is the radiance of frequency v incident on an absorber with absorption coefficient a present at concentration N and the effective path length is /, the transmitted radiance Lv is given by... 

The amount of light absorbed is a function of the so-called absorption coefficient (A ) and of the optical pathlength in the atomiser cell (ft) k depends on the frequency of the selected analytical line and on the concentration of the analyte absorbing atoms. The general absorbance law (Lambert Beer Bouguer law) relates transmittance (and so measured intensities I and If) to k and b through the following equation ... 

For most treatments, the spectral density, J(a>), Eq. 2.86, also referred to as the spectral profile or line shape, is considered, since it is more directly related to physical quantities than the absorption coefficient a. The latter contains frequency-dependent factors that account for stimulated emission. For absorption, the transition frequencies ojp are positive. The spectral density may also be defined for negative frequencies which correspond to emission. 

Intercollisional interference. We note that at the lowest frequencies the simple proportionality between absorption coefficient and product of gas densities breaks down. Under such conditions, certain many-body interactions affect the observations and modify the shape or intensities of the binary spectra, often quite strikingly. An example is shown in Fig. 3.3, a measurement of the absorption in a neon-xenon mixture in the microwave region, at the fixed frequency of 4.4 cm-1. Because of the frequency-dependent factor of g(v) that falls off to zero frequency as v2, absorption is extremely small at such frequencies, Eq. 3.2. As a consequence, it has generally been necessary to use sensitive resonator techniques for a measurement of the absorption at microwave frequencies... 

We have introduced the effective complex susceptibility x ( ) = X,( )+ X ) stipulated by reorienting dipoles. This scalar quantity plays a fundamental role in subsequent description, since it connects the properties and parameters of our molecular models with the frequency dependences of the complex permittivity s (v) and the absorption coefficient ot (v) calculated for these models. 

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) ... 

At a y the frequency dependence of the normalized absorption coefficient can be represented as... 

Figure 25. Frequency dependence of the absorption coefficient (a) and dielectric loss (b). Liquid fluoromethane CH F at 133 K calculated for that hat-curved model (solid lines). Dashed curve in Fig. (a) refers to the experimental [43] data, vertical line in Fig. (b) marks the experimental position of the maximum dielectric loss. The parameters of the hat-curved model are presented in Table VIII.

Figure 28. Experimental frequency dependences of dielectric parameters recorded for liquid water (a) Real (curve 1) and imaginary (curve 2) parts of the complex permittivity at 27°C. The data are from Refs. 42 (solid lines) and 17 (circles), (b) Absorption coefficient. Solid line and crosses 1 refer to 1°C filled circles 2 refer to 27°C dashed line and squares 3 refer to 50°C. For lines the data from Ref. 17 were employed, for circles the data are from Ref. 42, for crosses and squares the data are from Ref. 53.

Figs. 32a-c illustrate the absorption spectra, calculated, respectively, for water H20 at 27°C, water H20 at 22.2°C, and water D20 at 22.2°C dotted lines show the contribution to the absorption coefficient due to vibrations of nonrigid dipoles. The latter contribution is found from the expression which follows from Eqs. (242) and (255). The experimental data [42, 51] are shown by squares. The dash-and-dotted line in Fig. 32b represents the result of calculations from the empirical formula by Liebe et al. [17] (given also in Section IV.G.2) for the complex permittivity of H20 at 27°C comprising double Debye-double Lorentz frequency dependences. 

Figure 34. (a, b) Wideband frequency dependence of the absorption coefficient of ordinary (a)... 

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