Cross autoionization

It is well known that the value of the p parameter, more than the cross-section a, often shows a strong response to resonant structure embedded in the continuum. Given the sensitivity exhibited by the parameter in the foregoing there must be an a priori expectation that it would also show a strong response to resonant behavior. Computational methods do not yet exist to deal with autoionization phenomena in the systems of interest here, but one electron shape resonances can, in principle, be examined. 

If we suppose that these Rydberg states have non-negligible lifetimes against autoionization, as could occur via dispersion of the potential energy of the molecule into various vibrational modes, then these states would be very sensitive to electric fields in the measuring apparatus. It is suggested here that this is the reason for the large discrepancies between cross sections measured in the various beam experiments. 

The Bell equation gives the correct behavior for the ionization cross section at both high- and low-impact energies. In cases where autoionization is important it is not always possible to reproduce the cross section from the single equation above, but if it is used in two separate fits, one from the ionization threshold to the autoionization threshold, and the second above the autoionization threshold, a good fit to the cross section may be obtained over the entire range. 

According to the theoretical investigations by Nakamura [126-128] and Miller [129], the collisional energy dependence of these processes can be calculated if the interaction potential V(R) for the system He(2 S)-M and the autoionization rate r K)jh from the intermediate quasi-molecule [He(2 S)M] to the resulting quasi-molecular ion [HeM] are known. For example, by the classical formula of Miller [129], we have theoretical cross sections for Penning ionization as ... 

The H2 molecule is a system for which quite recently it has been possible to measure in unprecedented detail state-selected vibrationally and rotation-ally resolved photoionization cross sections in the presence of autoionization [27-29]. The technique employed has been resonantly enhanced multiphoton ionization. The theoretical approach sketched above has been used to calculate these experiments from first principles [30], and it has thus been possible to give a purely theoretical account of a process involving a chemical transformation in a situation where a considerable number of bound levels is embedded in an ensemble of continua that are also coupled to one another. The agreement between experiment and theory is quite good, with regard to both the relative magnitudes of the partial cross sections and the spectral profiles, which are quite different depending on the final vibrational rotational state of the ion. 

Figure 29. Ratio of cross section for excitation transfer followed by atomic autoionization (AAI), to total ionization cross section, as function of collision energy for systems He -Ar,Kr,Xe.77...

Figure 37. Variation with collision energy of cross section, leading by autoionization to main electron peak at 8.25 eV. Solid line represents result of Landau-Zener calculation (see text).

As mentioned in Section II.A, the Pgl process is ideal for the application of the optical model. This is clear in the classical and semiclassical Pgl theory,24,25 for which opacity and cross-section formulas are completely equivalent to those given earlier in this chapter. The quantal optical model is also rigorously related to the elastic component of the quantal Pgl theory. Miller49 has shown that T(r), identified in Pgl as the autoionization width of the excited electronic state, may be accurately obtained by a standard Born-Oppenheimer electronic structure calculation as... 

The most important implication of not being a good quantum number is that blue and red states are coupled by their slight overlap at the core. In the region below the classical ionization limit blue and red states of adjacent n do not cross as they do in H, but exhibit avoided crossings as a result of their being coupled. Above the classical ionization limit blue states, which would be perfectly stable in H, are coupled to degenerate red states, which are unbound, and ionization occurs rapidly compared to radiative decay. It is really an autoionization process in which the blue state is coupled to the red continuum state at the ionic core. 

For the optical transition from the bound 6snt state to the autoionizing 6pn state the optical cross section is given by the Lorentzian form... 

If we again consider Fig. 21.4, we can see that the cross section vanishes at v2 = 0.32 and that the profile does not match the spectral density, A2, of the autoionizing state. The Beutler-Fano profile of Fig. 21.4 is periodic in v2 with period 1, so the spectrum from the ground state consists of a series of Beutler-Fano profiles. At higher values of v2 the profiles become compressed in energy since dW/dv2 = l/vf. Fig. 19.2 shows two regular series of Beutler-Fano profiles between the Ba+ 6p1/2 and 6p3/2 limits. In this case the absorption never vanishes because there is more than one continuum. 

The wavefunction T of the autoionizing state is given by Eq. (21.1), and to calculate the photoionization cross section we need the dipole matrix element ( bl l ) = (Tb /i A202)- We can write (Tb and 02) as product wave functions ... 

All the energy dependence in the cross section arises from the factors A2 and (vb B v2 ) 2. These are, respectively, the spectral density of the channel 2 autoionizing states, and the overlap integral between the bound and continuum ni states with effective quantum numbers vb and v2. We have already seen that A is simply given by -dvl/dv2, the derivative of the quantum defect surface, and repeats modulo 1 in v2. The overlap integral is given is closed form by5... 

Fig. 21.6 Calculated cross section o from a bound state of effective quantum number vb = 12.35 to an autoionizing Rydberg series with width T = O.llr-3 and quantum defect d2 = 0.15 so that the autoionizing states are found at v2 = 0.85(mod 1) (from ref. 6).

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