Classical toy models for the monopole shift and the quadrupole shift.

The penetration of s- and p(1/2)-electrons into the atomic nucleus leads to a variety of observable effects. The presence of s-electrons inside the nucleus gives rise to the isotope shift in atomic spectroscopy, and to the isomer shift in Mössbauer spectroscopy. Both well-known phenomena are manifestations of the more general monopole shift. In a recent paper (Koch et al., Phys. Rev. A, 2010, 81, 032507), we discussed the existence of the formally analogous quadrupole shift: a tensor correction to the electric quadrupole interaction due to the penetration of relativistic p(1/2)-electrons into the nucleus. The quadrupole shift is predicted to be observable by high-accuracy molecular spectroscopy on a set of 4 molecules (the quadrupole anomaly). The simple physics behind all these related phenomena is easily obscured by an elaborate mathematical formalism that is required for their derivation: a multipole expansion in combination with perturbation theory, invoking quantum physics and ideally relativity. In the present paper, we take a totally different approach. We consider three classical 'toy models' that can be solved by elementary calculus, and that nevertheless contain all essential physics of the monopole and quadrupole shifts. We hope that this intuitive (yet exact) analysis will increase the understanding about multipole shift phenomena in a broader community.