Absorption cross section partial

We will be interested not only in the total absorption cross section [Eqs. (4.1) and (4.2)], which gives us a measure of the total probability that the molecule will absorb light and dissociate, but also in the probability that different product quantum states will be formed. This probability is given by a partial cross section cj/( ). From Eq. (4.1) we see that this partial integral cross section may be written as... 

In this section we derive an approximate expression for the absorption cross section of a large weakly absorbing sphere. We assume that the incident plane wave can be subdivided into a large number of rays the behavior of which at interfaces is governed by the Fresnel equations and Snell s law (Section 2.7). A representative ray incident on the sphere at an angle 0, is shown in Fig. 7.1. At point 1 on the surface of the sphere the incident ray is divided into externally reflected and internally transmitted rays these lie in the plane of incidence, which is determined by the normal to the sphere and the direction of the incident ray. If the polar coordinates of point 1 are (a, 0f, ), the plane of incidence is the plane = constant. At point 2 the transmitted ray encounters another boundary and therefore is partially reflected and partially transmitted. In a like manner we can follow the path of the rays within the sphere, a path that does not deviate outside the plane of incidence. At each point where a ray encounters a boundary it is partially reflected internally and partially transmitted into the surrounding medium. On physical grounds we know that the absorption cross section cannot depend on the polarization of the incident... 

In the frame of the theoretical formulation, in which the Penning process is described by the local quantities V+ R), T(/ ), and V+(R), the total cross section can be calculated as either (1) total absorption cross section atotaI from the complex phase shift for scattering by the complex potential V(R)= V (R)- r(R) or (2) as the sum of the partial cross sections a(Pgl), a(AI), and a(QI), into whose calculation also V+(R) enters in the form of matrix elements involving nuclear wave functions in this potential. 

Figure 4.19 The partial-wave singlet (full curves) and triplet (broken curves) absorption cross sections in e+ + H(1s) collisions, plotted versus the incident positron energy measured from the threshold energy for positronium formation. Results of hyperspherical closecoupling calculations including the absorption potential —iVabs in the Hamiltonian. Note that the thresholds Etu for the full and broken curves are different by 0.841 meV, the hyperfme splitting. Figure from Ref. [16].

Based on this relation, the relative partial cross sections absorption cross section relative values can then be placed on an absolute scale, because in the energy range of interest the total absorption cross section is identical to the total... 

Some partial photoionization cross sections, derived in this way for neon, are shown in Fig. 2.11 as a function of photon energy. The uppermost curve is the total absorption cross section. At the onset of the ionization thresholds for the ejection of Is, 2s and 2p electrons this quantity shows the corresponding absorption edges (see the discussion related to equ. (2.11)). The partition of the total cross section into partial contributions cr(i) clearly demonstrates that the dominant features are due to main photoionization processes described by the partial cross sections satellite transitions from multiple photoionization processes are also present. If these are related to a K-shell ionization process, they are called in Fig. 2.11 multiple KL where the symbol KLX indicates that one electron from the K-shell and X electrons from the L-shell have been released by the photon interaction. Similarly, multiple I/ stands for processes where X electrons from the L-shell are ejected. Furthermore, these two groups of multiple processes are classified with respect to ionization accompanied by excitation, (e, n), or double ionization, ( ,e). If one compares in Fig. 2.11 the magnitude of the partial cross sections for 2p, 2s and Is photoionization at 1253.6 eV photon energy (Mg Ka radiation) and takes into account the different... 

In the following we will call the a u,n,j) partial photodissociation cross sections.t They are the cross sections for absorbing a photon with frequency u and producing the diatomic fragment in a particular vibrational-rotational state (n,j). Partial dissociation cross sections for several photolysis frequencies constitute the main body of experimental data and the comparison with theoretical results is based mainly on them. Summation over all product channels (n,j) yields the total photodissociation cross section or absorption cross section ... 

Let us assume that the upper state is degenerate with substates F, all corresponding to the same total energy Ef. The photon excites each of these states simultaneously because the resonance condition ujfi ui holds for all of them. The absorption cross section is consequently composed of several partial absorption cross sections cr(u), [3) each being defined as in (2.27) with Ff) replaced by F ). We will come back to this in Section 2.5 when discussing photodissociation. 

After having defined the partial dissociation wavefunctions l>(R,r E,n) as basis in the continuum, the derivation of the absorption rates and absorption cross sections proceeds in the same way as outlined for bound-bound transitions in Sections 2.1 and 2.2. In analogy to (2.9), the total time-dependent molecular wavefunction T(t) including electronic (q) and nuclear [Q = (R, r)] degrees of freedom is expanded within the Born-Oppenheimer approximation as... 

In the time-independent approach one has to calculate all partial cross sections before the total cross section can be evaluated. The partial photodissociation cross sections contain all the desired information and the total cross section can be considered as a less interesting by-product. In the time-dependent approach, on the other hand, one usually first calculates the absorption spectrum by means of the Fourier transformation of the autocorrelation function. The final state distributions for any energy are, in principle, contained in the wavepacket and can be extracted if desired. The time-independent theory favors the state-resolved partial cross sections whereas the time-dependent theory emphasizes the spectrum, i.e., the total absorption cross section. If the spectrum is the main observable, the time-dependent technique is certainly the method of choice. 

Fig. 6.5. Comparison of the measured and the calculated absorption spectra for CH3SNO(5i). The theoretical absorption cross section is decomposed into the various partial cross sections a(E, n) (dashed curves) for producing NO in final vibrational states n — 0 and 1. The experimental and the theoretical curves are vertically shifted for clarity. Adapted from Schinke et al. (1989).

As estimated from the widths of the partial absorption cross sections, the lifetime varies from 15 fe to 70 fs which corresponds, on the average, to merely one to four vibrational periods of NO within the T complex. The photodissociation of ClNO(Ti) is thus a hybrid of direct and indirect dissociation. 

The ProDos differs from the usual partial density of states (PDOS) calculated in the previous works in the definition (6). The summation in the Eq. (7) is over the unoccupied final states, thus N iE) will be proportional to the absorption cross section approximately (5). However, ProDos and PDOS are nearly the same, as far as the localized bound states are concerned. We have found that the ProDos of our clusters are very similar to PDOS of them. In this report, we concentrate our discussions on ProDOS of the central atom Ti to compare them with the XAS results of Ti40, (7). 

Hartree-Fock orbitals of NO constructed In a minimal cartesian gau-sslan basis set. These results provide satisfactory representations of the general spatial characteristics of the Indicated orbitals, although the associated orbitals and energies, of course, do not correspond to Hartree-Fock limits. It should be noted that the unoccupied 6a(a") orbital appears at a positive energy, and on this basis alone can be expected to contribute to partial photolonlzatlon cross sections of ka final-state S3nnmetry. Since the 2ir(7r ) orbital Is singly occupied In this case. It Is bound and can be expected to contribute to the absorption cross section as a final state. 

A focused laser beam (0 = 0.4 mm) with EmW input power at X = 623 nm crosses a molecular beam (0=1 mm) perpendicularly. The absorbing molecules with a partial flux density Ni iJ= 10 /(scm ) have the absorption cross section a = 10 cm. ... 

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