Quantum Defects-Core Polarization

Early we noted that if we could measure several A/ intervals we could parametrize all the higher A/ intervals in terms of the polarizability of the ionic core. As an example of the application of the static polarization model let us consider the Cs A/ intervals. It is clear that Eq. (4) may be recast in terms of the intervals between adjacent / states. Let us label the difference in the polarization enei ies of the / and / states, Wp i/ - Wpou- as A,/-. It is convenient to adopt the convention of Edlen and write An- in cm and define for / the terms Pi = where R is the Rydberg constant and 

It is interesting to compare the results of the core polarization measurements from Rydberg atom A/ intervals with calculated values. As we mentioned earlier the polarizabilities from the measurements are expected to differ from the calculated values aj and a, because of the nonadiabatic effects. In Table 2 we present the measured and calculated  

Ficiiiell. Plot of the quantum defects /t of the alkali atoms vs. l. LiC), Na(A), K(D), Rb (O), and Cs (B). Note the r dependence for high / where the dipole polarizability dominates. 

Of particular interest is the fact that for the nonpenetrating high / states the quantum defect varies as the leading term in the core polarization energy shift. In fact it is the departure from the 1 behavior as I is decreased that indicates the onset of core penetration. This is particularly apparent for the Cs / states and Na d states, although no evidence for this is seen in the Li quantum defects. 

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In the first two chapters we have seen that the Na atom, for example, differs from the H atom because the valence electron orbits about a finite sized Na+ core, not the point charge of the proton. As a result of the finite size of the Na+ core the Rydberg electron can both penetrate and polarize it. The most obvious manifestation of these two phenomena occurs in the lowest states, which are substantially depressed in energy below the hydrogenic levels by core penetration. Core penetration is a short range phenomenon which is well described by quantum defect theory, as outlined in Chapter 2. 

Since the accuracy of the asymptotic expansion rapidly gets even better with increasing L, there is clearly no need to perform numerical solutions to the Schrodinger equation for L > 7. The entire singly excited spectrum of helium is covered by a combination of high precision variational solutions for small n and L, quantum defect extrapolations for high n, and asymptotic expansions based on the core polarization model for high L. The complete asymptotic expansion for helium up to (r-10) is [36,29]... 

In alkaline earth atoms, on the other hand, the static core polarization model clearly does not reproduce the energies of the atomic levels, because of the magnitude of the nonadiabatic effects/ Several theories have been formulated for the nonadiabatic effects. For example, the validity of the approach of Eissa and Opik has recently been verified by Vaidyanathan and Shorer by comparing calculated and measured quantum defects ofCa. 

Since the quantum defects of low angular momentum states are well known from optical spectroscopy and the high / quantum defects can be easily derived from the core polarization model, it is useful to present all the results together. In Figure 11 we show a plot of the alkali quantum defects vs. / on a logarithmic scale modified to include the s and p states. 

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