Photodissociation cross section integral

B. The Time-Dependent Formulation Total Integral Photodissociation Cross Section... 

In particular, Shapiro and others calculated state-to-state photodissociation cross sections from vibrationally excited states of HCN and DCN [58], N2O [59], and O3 [60]. Eor instance, the detailed product-vibrational state distributions and absorption spectra of HCN(DCN) were compared [58]. These results were obtained employing a half-collision approximation, where the photodissociation could be depicted as consisting of two steps, that is, absorption of the photon and the dissociation, as well as an exact numerical integration of the coupled equations. In particular, it was predicted that large isotope effects can be obtained in certain regions of the spectrum by photodissociation of vibrationally excited molecules. 

The main virtue of the adiabatic approximation is its interpretative power. The separation of the various degrees of freedom leads to a factorization of the photodissociation cross sections into a one-dimensional bound-bound and a one-dimensional bound-free overlap integral. Let us assume, for simplicity, a separable ground-state wavefunction of the form... 

Therefore, we can use this somewhat artificial wavepacket, when it has been determined by direct integration of the time-dependent Schrodinger equation, to extract the transition amplitudes and hence the photodissociation cross sections for all energies. 

The calculation of photodissociation cross sections requires the overlap of the continuum wavefunctions with the bound-state wavefunction multiplied by the transition dipole function. Employing for both wave-functions the expansion in terms of the Qj p and utilizing (11.13) leads to radial integrals of the form... 

The mathematical tools needed to calculate photodissociation cross sections were first presented by Shapiro, and an exact treatment of rotational predissociation in the Ar-N2 system was subsequently performed by Beswick and Shapiro. These calculations clearly show how the theory can account for both symmetric and asymmetric resonance line shapes as well as overlapping nonisolated resonance lines. The theory has also been applied to the Ar-H2 and Ar-HD system for which benchmark calculations were performed. Several other methods of computing the exact bound-continuum integrals needed in the evaluation of the expression for the photodissociation cross section have also been presented (see refs 7 and 13). 

In deriving this relation we used (4.4) and (4.5). S(t) on the left-hand side of (4.6) is the autocorrelation function. It is the central quantity in the time-dependent formulation of spectroscopy and photodissociation. Multiplying (4.6) by etEft/nj integrating over t, and using Equations (2.56), (2.57), and (2.67) yields the final expression for the total absorption cross section,... 

The workhorse for the calculation of cross sections in full collisions is the so-called Monte Carlo technique (Schreider 1966 Porter and Raff 1976 Pattengill 1979). The application to photodissociation proceeds in an identical fashion. Within the Monte Carlo method an integral over a function f(x) is approximated by the average of the function over N values Xk randomly selected from a uniform distribution,... 

Fig. 7.21. One-dimensional illustration of the transition from direct, (a) and (b), to indirect, (d) and (e), photodissociation. While the spectra in (a) and (b) reflect essentially the initial wavefunction in the electronic ground state, the spectra in (d) and (e) mirror the resonance states of the upper electronic state. The spectrum in (c) illustrates an intermediate case. According to (2.72), the integrated cross sections are the same in each case. Reproduced from Schinke et al. (1989).

In the troposphere, the temperature decreases with altitude increase up to the tropopause (approximately 10 km altitude). In this zone the temperature is close to 220 K and precisely, under these conditions, the peroxynitrate UV absorption cross sections are the least. This means that its photodissociation rate will be lower than at the surface. The precise value integrated for the whole absorption region will depend, of course, on the solar flux, that is richer in short wavelength radiation than the surface flux. 

A simple measurement of the total photoionization cross section (isotropic sample, cross section measured at a specified photon energy, integrated over all photoejection angles, without specification of the internal state of the photoion, without determination of ms of the ejected electron) contains no information about the distribution of Z, m -values of the ejected electron. However, measurable properties of the photoionization event can provide information about the mechanism of the photoionization process. The frequently measured quantities included f3, Aq, and A. The ft quantity describes the angular distribution of the photoelectrons and is defined analogously to the (3 for photodissociation (see Section 7.2.4), Ajp is the alignment ( Mm distribution) of the photoion. A (not to be confused with the spin-orbit constant) or alternatively P, is the spin-polarization of the ejected electron, which is relevant when the photoion has nonzero spin. 

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