A new angle on resonances in confined atoms and ions

Confined atoms are rich in new effects of which some remain to be discovered. They offer new insights into atomic physics. Reference [1] is a review paper summarising work by many authors (other than the present one) on atomic compression and confinement since the origin of the subject. The present author has a long-standing interest in orbital collapse effects in atomic physics, which has developed from synchrotron radiation and laser spectroscopy of atoms and extended into the new subject of atomic confinement. Experimentally, the observations are very challenging, but some quite simple theory can already be advanced. Thus, in a paper with Dolmatov and Manson [2], we have distinguished between three types of resonance which can occur, viz. (a) those due to the atom, modified by confinement, (b) resonances due to the confining potential, modified by the presence of the confined atom and (c) resonances of molecular origin. In the present paper, I would like to suggest yet a fourth situation. Whereas (a), (b) and (c) is a general distinction for confined atoms, because spherical confinement necessarily produces confinement resonances, the new situation I will describe in the present note is of a unique kind, because it is a rare one and can only occur for atoms or ions in specific positions of the Periodic table.

The excitations I consider here occur in an unusual atomic situation, for which it becomes necessary to examine the full excitation channel, viz. a complete Rydberg series, rather than simply a single atomic state.

In a joint paper with Dolmatov [3], we have discussed similarities between 'external' and 'internal' confinement of the atom. What emerges theoretically is that, for inner reaches of the effective atomic potential, the effects of confinement within a perturbing sphere and the effects of a small (non-integral) change in nuclear charge are very similar and act similarly on the inner loops of wavefunctions. There is of course an important difference between them: changing the nuclear charge of a free atom by non-integral amounts is not possible experimentally, whereas confining an atom in a spherical cavity (for example, in an endohedral fullerene), is in principle fully open to experiment. Nonetheless, as shown in [3] 'internal' perturbation by an incremental increase of the nuclear charge is a useful and simple way of representing the effects of spherical confinement, as illustrated by detailed calculations for Ca⁺.

From the experimental point of view, dealing now with free atoms and ions, there are several ways of detecting 'orbital collapse' effects arising from the radial (rather than any angular) parts of the Schrödinger equation. First, one may consider simply horizontal sequences in the Periodic table and look for sudden changes, like the filling of the 4f subshell. We find that collapse occurs at Ce for neutral atoms. Second, we may study inner shell excitation of the same atoms, which reinforces the attractivity of the inner well of the double-well effective f-potential. Experiment reveals that a 4f giant resonance occurs at Ba with 4d excitation. We may also set up isoelectronic sequences such as Cs, Ba⁺, La⁺⁺, etc in which case we find that 5f is 'collapsed' in the inner well for Ba⁺. Thus, the effect of increasing the nuclear charge by one and the effect of making a 'hole' in the 4d subshell are found to be somewhat different. We can also consider data for extended homologous sequences [4], i.e. look at similar external configurations and study columns in the Periodic table. This turns out to be more useful for the systematic study of d-orbital collapse, by using experimental energies of the ns states as internal 'markers' of the atomic potential in absence of the centrifugal barrier.

In addition to all these methods of study, we now have the new approach of atomic confinement [1], which can produce many intermediate situations not actually achievable in experiment by the rather blunt tools of direct ionisation and inner shell excitation.

I now turn to an experimental effect originally noticed by Saunders et al a very long time ago [5] based on experiments by Rasmussen [6]. What they realized was that there exists, in the nf Rydberg series of Ba⁺, a very unusual perturbation, despite the absence of any plausible perturber in the vicinity. The curious effect they found, subsequently confirmed in more detail experimentally [7], is that the quantum defects in this series are not at all constant, in contrast with the usual behaviour for unperturbed Rydberg series, but undergo wide fluctuations, and that the spin–orbit splittings are also quite anomalous, facts for which Saunders et al [4] could suggest no explanation, beyond drawing attention to the anomaly.

The origin of these effect was discussed in a joint paper with Mansfield [8] who performed Hartree–Fock single channel calculations and showed that Ba is a unique case in the Periodic table for which the ion possesses a double-well potential so finely balanced that the inner loop of the 5f function becomes resonant in the inner well, whereas 4f and nf functions are more external and have lower amplitudes in the inner well. This was followed up by developing a single channel quantum defect theory for double-well potentials [9] in which it was shown that the n-dependent resonance of the inner loop indeed accounts for the unusual curvature of the quantum defect plot, in terms of a virtual state of the inner well.

In a more detailed theoretical investigation [10], both by the g-Hartree method (a spherical distortion of the Hartree–Fock potential based on principles of Field theory) and by the introduction of an 'internal' perturbation, it was shown that essentially complete agreement between theory and experiment for this unique situation in the free ion is achieved by a slight increase in the effective nuclear charge. We thus have a very interesting case, for which the increase in nuclear charge is no longer simply an artifice of theory, but actually brings calculations in line with experimental observations for the free ion system.

It follows that, if Ba⁺ is confined within an repulsive or attractive shell, we have a system for which the inner well of the double-well potential can now be controlled, so that the resonance of the inner loop of the wavefunction in the Rydberg sequence may be made to travel up or down at will in the excited nf manifold as a function of external confinement. Thus, in effect, confinement can profoundly alter or even destroy Rydberg recapitulation for the sequence.

This is not in any way a ubiquitous effect in atoms. Such behaviour is unique for specific atoms or ions poised on the verge of orbital collapse in the Periodic table, Ba⁺ being the most prominent example. The interesting point is that such Rydberg-resonant behaviour is of atomic origin, but occurs for different states in the free system and under confinement. Basically, it is of type (a), but instead of being a 'stable' atomic situation, concerning just one resonance (e.g. 4f) slightly modified by confinement, the spectrum is now completely transformed by the radial perturbation, leading to different resonances under different degrees of confinement and varying degrees of phase shift. Experimentation of this kind would also be of interest to solid state physicists wishing to explore the connection between intermediate valence in condensed matter and atomic orbital collapse [11].

The cases considered above raise the more general question of Rydberg series in confined systems for which the inner loops of wavefunctions lie inside the confining shell and the outer loops lie outside, leading to the need for a new type of quantum defect theory [9] in which, even for a single channel, the quantum defect will exhibit unusual, sensitive and interesting variations due to perturbation by the confining shell.

The present paper was prepared as a comment, to mark Steve Manson's 80th birthday and remember many happy days spent with him at the Starodubtsev Institute in Tashkent (Uzbekistan) and at Georgia State University, Atlanta, (USA).

All data that support the findings of this study are included within the article (and any supplementary files).