The polarizability of a confined atomic system: An application of Dalgarno-Lewis method

TL;DR

This paper applies the Dalgarno-Lewis method to calculate the polarizability of a model atom near a substrate, revealing its extreme sensitivity to distance, which can be used for nanometric measurements.

Contribution

It provides an exact expression for static and dynamic polarizabilities of a confined atomic system using Dalgarno-Lewis theory, bypassing traditional sum-over-states methods.

Findings

Polarizability is highly sensitive to atom-substrate distance.

Dynamic polarizability line shape varies with distance.

Method offers potential for nanometric distance measurement.

Abstract

In this paper we give an application of Dalgarno-Lewis method, the latter not usually taught in quantum mechanics courses. This is very unfortunate since this method allows to bypass the sum over states appearing in the usual perturbation theory. In this context, and as an example, we study the effect of an external field, both static and frequency dependent, on a model-atom at fixed distance from a substrate. This can happen, for instance, when some organic molecule binds from one side to the substrate and from the other side to an atom or any other polarizable system. We model the polarizable atom by a short range potential, a Dirac$-\delta$ and find that the existence of a bound state depends on the ratio of the effective "nuclear charge" to the distance of the atom to the substrate. Using an asymptotic analysis, previously developed in the context of a single $\delta-$function potential in an infinite medium, we determine the ionization rate and the Stark shift of our system. Using Dalgarno-Lewis theory we find an exact expression for the static and dynamic polarizabilities of our system valid to all distances. We show that the polarizability is extremely sensitive to the distance to the substrate creating the possibility of using this quantity as a nanometric ruler. Furthermore, the line shape of the dynamic polarizability is also extremely sensitive to the distance to the substrate, thus providing another route to measure nanometric distances. The ditactic value of the $\delta-$function potential is well accepted in teaching activities due to its simplicity, while keeping the essential ingredients of a given problem.