Lu-Fano graphs

Fig. 3.3. Experimental example of a Lu-Fano graph showing the avoided crossing between just two interacting series in the spectrum of Yb. The series involved are an inner-shell and a doubly-excited series this kind of excitation is discussed in chapter 7 (from M.A. Baig and J.-P. Connerade [113]).

The most obvious property of a Lu-Fano graph is its branch structure which, in the case of two series, involves only one avoided crossing. An experimental example with two series is shown in fig. 3.3. 

Fig. 3.4. Example of a Lu-Fano graph involving more than two interacting series, and consequently more than one avoided crossing. This particular graph occurs in the spectrum of Yb, and involves doubly-excited configurations, discussed in chapter 7. The number of series in the graph is equal to the number of intersections of the curves with the diagonal, i.e. 3 in this case (Kaenders and Connerade - unpublished).

An example of a two-dimensional Lu-Fano graph involving more than two interacting series is shown in fig. 3.4. A more difficult generalisation (because it cannot be represented graphically in two dimensions) is the case in which more than two series limits are present. For three limits, the... 

The idea is to incorporate some of the very important developments of MQDT made in atomic physics. An ambitious programme (not addressed here) is to develop and extend MQDT to molecular species by including full rovibronic structures for simple molecules within the framework of an extended theory by using frame transformations. Instead, we describe a far more restricted phenomenological approach, akin to MQDT for atoms, and applicable only to high n states of polyatomic species in which the rotational and most of the vibrational structure has collapsed. We can then compare the Lu-Fano graphs directly with those of corresponding atoms, and discuss both similarities and differences. 

We now plot the usual Lu-Fano graph, using, as before ... 

Fig. 3.5. Two-dimensional Lu-Fano graphs (a) for CH3I and (b) for the corresponding series in the united atom Xe. Note the structural similarity of the graphs despite the differences in magnitude of the quantum defects (from J.A.

Fig. 8.25. Influence of the combined asymmetry parameters on a typical Lu-Fano graph (a) for zero direct coupling and real values of the combined asymmetry parameter (b) for zero combined asymmetry and various values of the coupling strength. Other parameters Eooi = 71800, Eoo2 = 71890, X = —0.1, Xi — —0.2, pii = 0.4, pi-2 = 0.6 (after J.-P. Connerade [444]).

Fig. 8.27. The sequence comprising of this and the next three figures shows the influence of the q or shape parameters of the individual unperturbed series on the Lu-Fano graphs. Note the dramatic changes in magnitude and position of the avoided crossing. , = 0 Eooi = 71800, = 71890, X — —0.1, Xi — —0.2,...

Fig. 8.31. The remaining figures in this chapter show a sequence of spectra and Lu-Fano graphs in which the q parameters and coupling strengths are changed, but all other parameters are held constant. For this figure, the q = 1000 for both series, and the coupling strength = 2. A zero coupling strength, zero combined asymmetry plot is shown as a dashed curve for reference. X = 0.3, X

Such model calculations display a rich variety of effects even on a very simple model. They show that the essential structure of two-dimensional quantum defect plots is preserved, but that conclusions as to the strength of inter-series coupling cannot be reached merely by inspecting Lu-Fano graphs a simultaneous study of the spectra is also required. 

Similarly, quantum defect theory plays a very important role in modern descriptions of atomic physics, and should be included at a less rudimentary level than is found in most texts. Again, its modern developments provide an excellent illustration of many fundamental principles of scattering theory. The principles underlying the Lu-Fano graph are easily grasped, and provide excellent insight into an important aspect of the many-body problem, namely interchannel coupling. Likewise double- and inner-shell excitation are hardly discussed in textbooks, structure in the continuum receives little attention, etc, etc. 

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