Cooper minimum

Cooper minima axe also responsible for non Franck-Condon vibrational intensity distributions and intensity anomalies observed in both PES and ZEKE spectra. As for atoms, when the photoionization transition corresponds to excitation from a Rydberg orbital having at least one radial node in its wavefunction, the 

For molecules, the problem is more complex than for atoms because the continuum must usually be expressed as a sum of many partial waves. It is highly unlikely that the Cooper minima for all of the important /-partial wave components will occur near the same value of e. An example of a Cooper minimum in a molecular photoionization cross section is observed for HI, corresponding to ionization from the 5p7t orbital (Carlson et al., 1984). Other examples of Cooper 

In Section 8.11, the influences of shape resonances and Cooper minima on photoelectron angular distributions will be discussed. 

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The values of r0 and a are presented in Table 4.1. As shown by Table 4.1, the values of a are all near 3, as expected from Eq. (4.11). The biggest discrepancies occur in the Li and K np series which have their Cooper minima nearest the ionization limit. It is also apparent that the np lifetimes are the longest, as expected from Figs. 4.1 and 4.2. In using the values of Table 4.1 it is worth remembering that for high n states the black body decay rate is (independent, so the lifetimes of the longest lived states are the most affected by black body radiation.32... 

Table 1 RRPA calculated [29] photoelectron energy positions of the Cooper minima (au) for the free and encaged Ca, Sr and Ba...

In addition, as seen in the same Figure 27, (i) the positions of deep (Cooper) minima in /fes of free Hg and pfsAb of Hg are noticeably different with regard to each other and (ii) f(n has a noticeably deeper and... 

For orbitals with a radial node, there is a minimum in the cross section, known as a Cooper minimum. At some photon energy, the contributions from the inner and outer parts of the orbital will tend to cancel one another (Figure 7(d)). Figure 7(e) illustrates the occurrence of such a minimum in the cross section of the 3p electrons of the elements Si to Cl. Figure 7(f) shows the cross section of the ti band of TiCfr, which shows a Cooper minimum at 40 eV " by symmetry, this is pure Cl 3p in character (see also Section 4.2.4). If an orbital has more than one radial node, there may be a corresponding number of Cooper minima in the cross section. The minima may occur just above threshold, as is the case for the 3s orbital of Na, or several hundred eV, as is found for 4d and 5f electrons of transition elements. Rules for Cooper minima, which have been arrived at by experimental and theoretical investigations, are as follows ... 

Fig. 4.4. Seaton-Cooper minima for a selection of alkali spectra (schematic), showing their evolution as a function of the atomic species. Theoretical curves are shown on the left and corresponding plots of the experimental photoionisation cross sections on the right. The figure demonstrates the large difference between the alkalis (after M. Nawaz et al [166]).

The possibility of such minima in the observed cross section, and the explanation of how they arise were first discussed by Rudkjpbing [138], in a pioneering paper which is unfortunately hard to find and has therefore been rarefy quoted. Accurate calculations of cross sections near such minima in the alkalis were first reported by Seaton [139]. Similar calculations were also performed by Cooper [140]. In line with current usage, we shall refer to such features as Seaton-Cooper minima. They are not to be confused with another type of minimum of the photoionisation cross section (the Combet-Farnoux minimum), which occurs in the presence of a centrifugal barrier (see section 5.4). 

For a more recent example of how Seaton-Cooper minima arise in laser spectroscopy and a more up-to-date comparison between theory and experiment, see [141]. 

The Seaton-Cooper minima of alkali spectra are the best known. Because of the continuity between discrete and continuous spectra noted in the previous subsection, if a Seaton-Cooper minimum drops below the threshold, it will turn into a minimum in the course of intensities of the corresponding Rydberg series, in such a manner that continuity of the df/dE plot is preserved (see above). In the discrete part of the spectrum, one may find a minimum rather than a zero, because it is not necessary that a transition should exist at precisely the energy where cancellation occurs, i.e. the Seaton-Cooper minimum may very well fall between two members of a Rydberg series. However, the anomaly in the course of intensities in the series will be apparent. 

The delayed onset effect was discovered experimentally by Ederer [180] in 1964 in the photoionisation spectrum of Xe gas. A more recent compilation of results for the same spectrum is presented in fig. 5.1. If one considers the ionisation continuum of H, then just above the ionisation threshold, there is a maximum in the cross section, which is followed by a monotonic decline with increasing energy. Such situations are illustrated in fig. 4.1 until the observations of Ederer, it had been believed that this would be the most general behaviour of continuum cross sections and that, apart from the Seaton-Cooper minima discussed in section 4.4, or the local influence of interchannel perturbations, revealed in spectra such as that of fig. 4.3, unperturbed continuum cross sections would usually follow a monotonic course of declining intensity comparable to that of H. 

Continuum effects Seaton-Cooper minima in solids 407... 

Av = 0 propensity but unusually strong transitions with AN = even, particularly AN = 0 [9, 17]. Combining the experimental results with quantum-chemical calculations of ground-state, resonant-state, and photoelectron orbitals, a number of Cooper minima were predicted to occur in various photoelectron channels of the f, g, and h states calculations of the photoelectron angular momentum distributions and the angular momentum compositions of photoelectron matrix elements also provided insight into the origin of these Cooper minima [17]. 

Quantum-chemical calculations for (3+1) REMPI via the B 11 ( NH X 3pa) Rydberg state predicted Cooper minima resulting in non-Franck-Condon vibrational distributions [19] and unusual rotational branching ratios [20] for various vibrational transitions. 

In the soft X-ray region, simple hydrogenic models in a single-electron process break down, as shown by comparing experimental observation of X-ray absorption spectra to theoretical predictions. Outer subshells make relatively important contributions to cross sections giving rise to Cooper minima [2 to 5]. 

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