Differential optical cross-section

In view of Equation 37, r is referred to as a full optical cross-section (in the atmosphere optics literature), and Rayleigh s ratio differential optical cross-section... 

K) (a) optical micrograph of the related unfunctiO-nalized polymer blend (b). The scattering vector, qx (41T/X) sin (0/2) where 0 is the observation angle. dS/dn is the differential scattering cross-section per atom with respect to the solid angle, as normalized to a unit volume. 

The differential scattering cross sections given in Equations (5.21) and (5.31) allow us to deduce the optical wavelength dependence. Note that, basically, for nematics... 

It should be noted that low-loss spectra are basically connected to optical properties of materials. This is because for small scattering angles the energy-differential cross-section dfj/dF, in other words the intensity of the EEL spectrum measured, is directly proportional to Im -l/ (E,q) [2.171]. Here e = ei + iez is the complex dielectric function, E the energy loss, and q the momentum vector. Owing to the comparison to optics (jqj = 0) the above quoted proportionality is fulfilled if the spectrum has been recorded with a reasonably small collection aperture. When Im -l/ is gathered its real part can be determined, by the Kramers-Kronig transformation, and subsequently such optical quantities as refraction index, absorption coefficient, and reflectivity. 

Owing to the symmetry property of an optical dipole transition, the data analysis for a photodissociation study is greatly simplified. The center-of-mass differential cross-section for a single-photon, dissociative process can be expressed as38,39... 

The relative success of the binary encounter and Bethe theories, and the relatively well established systematic trends observed in the measured differential cross sections for ionization by fast protons, has stimulated the development of models that can extend the range of data for use in various applications. It is clear that the low-energy portion of the secondary electron spectra are related to the optical oscillator strength and that the ejection of fast electrons can be predicted reasonable well by the binary encounter theory. The question is how to merge these two concepts to predict the full spectrum. 

Figures 4.35, 4.36, and 4.37 show the absorption spectra of the free radicals CIO, BrO, and IO, respectively (Wahner et al., 1988 DeMore et al., 1997 Laszlo et al., 1995). All have beautifully banded structures at longer wavelengths and large absorption cross sections, which allows one to measure these species in laboratory and atmospheric systems using differential optical absorption spectrometery (DOAS) (see Chapter 11.A.Id). However, as in the case of HCHO, adequate resolution is an important factor in obtaining accurate cross sections.

In this case, a is the differential optical absorption cross section for the absorption band. In practice, of course, there are many different absorbers, t, present at different concentrations and absorbing at different wavelengths over the path length L. 

The classical theory makes especially clear the inherent ambiguity of data analysis with the optical model, and this ambiguity carries over into the quantum model. If we wish to use experimental differential cross sections to gain information about V0(r) and P(b) or T(r), we must assume a reasonable parametric form for V0(r) that determines the shape of the cross section in the absence of reaction. The value P(b) is then determined [or T(r) chosen] by what is essentially an extrapolation of this parametric form. In the classical picture a V0(r) with a less steep repulsive wall yields a lower reaction probability from the same experimental cross-section data. The pair of functions V0 r), P b) or VQ(r), T(r) is thus underdetermined. The ambiguity may be relieved somewhat (to what extent is not yet known) by fitting several sets of data at different collision energies and, especially, by fitting other types of data such as total elastic and/or reactive cross sections simultaneously. 

Wakiya, K. (1978). Differential and integral cross sections for the electron impact excitation of O2 I. Optically allowed transitions from the ground state. J. Phys. B At. Mol. Phys. 11 3913-3930. 

The measurement of absolute multidimensional cross sections, such as (e,2e), is usually difficult to achieve with high accuracy due to the various experimental difficulties associated with low pressure gas targets and electron optics. Relative cross sections to different ion states can be obtained with much greater accuracy than the absolute values. The relationship between the (e,2e) differential cross section and the experimentally observable parameters is given by (see section 2.3.3)... 

Fig. 8.2. Differential cross section for the elastic scattering of electrons on hydrogen. Circles, Williams (1975) solid curve, coupled-channels-optical calculation long-dashed curve, one channel with discrete polarisation potential only short-dashed curve, one channel without polarisation potential. Adapted from Bray et al. (1991h).

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