Resonances in positron scattering on a supercritical nucleus and spontaneous production of $$e^{+}e^{-}$$ e + e - pairs

Abstract

We re-examine the physics of supercritical nuclei, specially focusing on the scattering phase δ and its dependence on the energy ε of the diving electronic level, for which we give both exact and approximate formulas. The Coulomb potential Z α/r is rounded to the constant Z α/R for r < R. We confirm the resonant behavior of δ that we investigate in detail. In addition to solving the Dirac equation for an electron, we solve it for a positron, in the field of the same nucleus. This clarifies the interpretation of the resonances. Our results are compared with claims made in previous works.

On the resonances near the continua boundaries of the Dirac equation with a short-range interaction

“…The fact that the partial phase δ κ (k) is real ensures unitarity of the partial matrix S κ (k) of elastic scattering, hence absence of any inelastic processes including spontaneous production of e + e − -pairs. Note that all the formal results of [15] have been confirmed in [19] by independent calculations.…”

“…However, the authors of [19] did not share the conclusion of [15], still retaining a belief that resonances in scattering of positrons by a supercritical nucleus are namely the indicators of spontaneous production of e + e − -pairs and leaning towards the description of the phenomena with Z > Z cr stated in the review [9]. Here we continue the discussion by downgrading to the simplest model of a spherical rectangular well, for the first time considered from this perspective in [1].…”

Critical nucleus charge in a superstrong magnetic field

“…In relatively recent paper [23] the scattering of positrons on a supercritical nuclei was studied and it was claimed that there is no spontaneous e + e − pair production from a supercritical nucleus. This made us to reexamine the physics of supercritical nuclei in paper [4] and in this section we briefly discuss the results obtained there.…”

“…In the paper [4] results with an exact dependence on the parameter m × R have been obtained on the both sheets of the complex energy plane in the form convenient for numerical evaluation. However one can hardly hope to study this phenomenon experimentally: even if a supercritical nucleus can be produced in heavy ions collisions, its life time will be so short that it makes impossible to scatter a positron on it, not to mention the still bigger challenge of making a target with supercritical nuclei.…”

“…However, it can be done if the finite size of a nucleus is taken into account (see, for example, [1][2][3]; more references can be found in [4]). At Z = Z cr ∼ 170 ground energy level reaches lower continuum (ε = −m, where m is the electron mass) and spontaneous electron-positron pair production occurs.…”