Quantum defect theory QDT

In a Rydberg state of any atom but H, when the distance, r, of the Rydberg electron from the ion core is greater than a core radius, rc, the potential is a coulomb potential, but for r rc the potential is usually deeper than a coulomb potential. The effect of the deeper potential is that if an incoming coulomb wave scatters from the ion core, the reflected wave has a phase shift of nfi compared to what it would have if it scattered from a proton. In other words the standing wavefunction for all r rc is given by 

Although the properties of the/and g functions are outlined in chapter 2, it is worth summarizing their properties here.8 The / and g coulomb functions are termed regular and irregular since asr— 0,/ rt+ and g oc r (. Due to the r = 0 behavior of the g function, in H only the/wave exists. As r — for Wt 0 the/ and g waves are sine and cosine functions, and if Wt 0, jiv simply specifies the phase of the wavefunction relative to the hydrogenic/wave. If Wt 0 the/and g waves both have exponentially increasing and decreasing parts, and, as we have seen in Chapter 2, only if 

Although as r oo the wavefunctions of different collision channels are very different, at small r they are similar, and it is possible to find normal modes of the scattering from the ionic core. Over an energy range AW we can find a radius R, R rc, such that for r R the wavefunctions of Eq. (20.4) are energy 

The wavefunctions of the normal scattering modes are the standing waves produced by a linear combination of incoming coulomb wavefunctions which is reflected from the ionic core with only a phase shift. The composition of the linear combination is not altered by scattering from the ionic core. These normal modes are usually called the a channels, and have wavefunctions in the region rc r R given by 

The normal modes with the wavefunctions defined by Eq. (20.6) are in essence the eigenfunctions which match the boundary condition at rc, but they do not match the r — °° boundary condition if there are any closed channels. In contrast the fl, wavefunctions match the r — °° boundary condition but not the r = rc boundary condition. In general for r rc the wavefunction can be written as a linear combination of either the or wavefunctions. Explicitly 

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Quantum defect theory (QDT) was developed by Seaton [111] and his collaborators, from ideas which can be traced to the origins of quantum mechanics, through the work of Hartree and others. They relate to early attempts to extend the Bohr theory to many-electron systems (see e.g. [114]). 

Fig. 5.14. Quantum defect plots for the centrifugally distorted nf series in Ba+ (a) shows the ordinary QDT plot, while (b) shows the plot obtained from the same experimental data using the generalised theory in the text. Note that the lowest point in the nf channel lies off the graph this is normal, since it has no node except at the origin and the corresponding wavefunction lies mostly in the non-Coulombic part of the potential (c) shows how the energy of the bound state can also be obtained by fitting a Morse potential to the Hartree-Fock potential of the inner well (after J.-P. Connerade [217]).

A linear combination of the hydrogenic functions mentioned above can be used to describe the asymptotic part of any atomic state where the Coulomb potential dominates beyond a chosen (rather arbitrarily) radius, rg. Then, by matching this solution with the inner region of the wavefunction, one can obtain, depending on the value of energy, the phase shifts or the quantum defects and from there produce information on spectra. This information is usually produced via the use of QDT parameters which are obtained empirically. In cases where the aim is for the standard QDT to be implemented from first principles, R-matrix-type theories have been employed, e.g.. Ref. [130]. 

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