Rydberg core-penetrating

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. 

Fig. 19.9 Plot of scaled total decay rates n3r of Ba 6pmn( J = l + 1 autoionizing states in atomic units vs (. For ( = 0-4 the measured rates (O) shown are the average rates from many n values. The data for the rates for > 4 are for n = 12. The solid line is a simple theoretical calculation based on the dipole scattering of a hydrogenic Rydberg electron from the 6p core electron. Note that the core penetration of the lower states reduces the actual rate from the one calculated using the dipole scattering model. The constant total decay rate for > 8 is the spontaneous decay rate of the Ba+ 6p state (from ref. 39).

Many properties in a Rydberg series scale in different ways. For example, the level spacing scales as n -1/3, which turns out to be an essential property when we come to discuss K-matrix theory in chapter 8. The same is true for core penetration, and all the properties which depend on the overlap between the core and excited electron wavefunctions (see chapters 4 and 6). The size of Rydberg states (discussed in section 2.14) scales as n 2, while transitions between adjacent levels in the Rydberg manifold, which depend on the overlap between adjacent excited states, scale as n 4. Yet more scaling rules for Rydberg series in external fields will emerge in chapter 10. 

Jakubek, Z.J., Field, R.W. Core-Penetrating Rydberg Series of BaF Single-State and Two-State Fits of New Electronic States in the 4.4 [Pg.90]

Fig. 2.1 Rydberg atoms of (a) H and (b) Na. In H the electron orbits around the point charge of the proton. In Na it orbits around the +11 nuclear charge and ten inner shell electrons. In high states Na behaves identically to H, but in low states the Na electron penetrates and polarizes the inner shell electrons of the Na+ core.

In photoabsorption spectroscopy, the distinction can also be made in a different way the probability that an electron on a Rydberg orbit is ejected depends on its penetration of the core, which scales as 1/n 3. On the other hand, the Auger electron has a constant probability Tauger of being ejected. Thus, if we observe a Rydberg series of autoionising resonances in photoabsorption, then the total width rn of the nth member is given by... 

Core-non-penetrating Rydberg states of molecules that contain heavy atoms and have a 1 + ion-core ground state, such as HgF or BaF, are likely to exhibit a level pattern closer to case (b+, e) than (b+, d). The 2F7/2 — 2F5/2 spin-orbit splitting in Ba+ is 225 cm-1 for 4/ and 16 cm-1 for 11/, which is in reasonable agreement with the n 3-scaling prediction of 225(4/ll)3 = 11 cm-1. 

The difference between the spin-orbit constants of the main constituent atomic orbital, A(atom, nl), in the Rydberg molecular orbital and that of the molecular Rydberg state, Ar, is due to penetration of the Rydberg MO into the molecular core, i.e., to the contribution of the (n — l) atomic orbital responsible for the orthogonality between the Rydberg MO and the molecular core orbitals. 

The quantum defect is about 1.0 for user orbitals of molecules composed of atoms between Li and N, since these orbitals penetrate into the molecular core (cf. Mulliken, 1964) and consequently are strongly modified by the presence of inner electrons of the same sa symmetry. The quantum defect is of the order 0.7 for npa and npn orbitals, which are less modified by the core, and nearly zero for ndcr, ndir, and ndS orbitals, which have no precursors in the core. For a given molecule this quantum defect is roughly independent of the ion state to which the Rydberg series converges (see, for example, for O2, Table 1 of Wu, 1987). 

The Rydberg-Ritz formula can be established empirically not only for the terms of the outer orbits, but also for orbits which penetrate the core and which we shall call penetrating orbits. It may in fact be derived theoretically for very general cases. 

In 26 we have ascribed the large values of the Rydberg corrections to the fact that the electron penetrates deeply into the atomic core, and is thus subjected to an increased nuclear influence. 

The Rydberg correction for penetrating orbits is not very different2 from the radial action integral of the largest orbit, completely contained in the core, divided by h. 

Our model, in which the electron under consideration moves in a central field, gives rosettes for the electron paths, and these are determined by two quantum numbers n and k. Orbits with different values of n must in fact occur in the interior of the atom. The behaviour of the Rydberg corrections shows that, for almost all elements, the p-orbiljp penetrate in order that this may be possible the core must at least contain orbits with n=2. Of the orbits in the core those with n— k=l) are nearest to the nucleus, then... 

In the case of the very light elements, the p-orbits are still external the smallness of the Rydberg corrections, and small core radii, suggest that this may also be the case for Cu, Ag, and Au, but the magnitude of the doublet separations and the variation of the value of the correction for different atoms of the same atomic structure (Cu, Zn+, Ga++, Ge+++, etc.1) seem to show conclusively that the p-orbits penetrate., The apparently small Rydberg corrections for... 

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