Estimating the radiative part of QED effects in superheavy nuclear quasimolecules

A method for calculating the electronic levels in the compact superheavy nuclear quasimolecules, based on solving the two-center Dirac equation using the multipole expansion of two-center potential, is developed. For the internuclear distances up to $d\sim 100$ fm, such technique reveals a quite fast convergence and allows for computing the electronic levels in such systems with accuracy $\sim {10}^{-6}$. The critical distances $R_{cr}$ between the nuclei for $1{\sigma }_g$ and $1{\sigma }_u$ electronic levels in the region $Z\simeq 87-100$ are calculated. By means of the same technique, the shifts of electronic levels due to the effective interaction $\mathrm{\Delta }{U}_{AMM}$ of the electron's magnetic anomaly with the Coulomb field of the closely spaced heavy nuclei are evaluated as a function of the internuclear distance and the charge of the nuclei, nonperturbatively both in $Z\alpha$ and (partially) in $\alpha /\pi$. It is shown that the levels' shifts near the lower continuum decrease with the enlarging size of the system of Coulomb sources, both in the absolute units and in units of $Z^4{\alpha }^5/\pi n^3$. The last result is generalized to the whole self-energy contribution to the level shifts and so to the possible behavior of the radiative part of QED effects with virtual photon exchange near the lower continuum in the overcritical region.

Figure 1

The multipoles (a),(b) $U_n$ and (c),(d) $V_n$ of the $U-U$ system (the nuclear radius is $R\simeq 7.54$ fm) for the distances between the centers of the nuclei $d=30$ fm in the interval $(a-R,a+R)$. The multipoles of (a),(c) order $n=0,2,4,6$ (the multipole moment $V_0$ is scaled by a factor 1/50) and (b),(d) order $n=8,10,12,14$ (in units of inverse Compton wavelength for the electron $1/{}^{-}{\lambda }_e=mc/\hslash$).

Figure 2

The electronic levels $1{\sigma }_g$ and $1{\sigma }_u$ in dependence on the internuclear distance $d=2a$ in two-nuclei quasimolecules ${\mathrm{U}}_2^{183+}$ (blue) and ${\mathrm{Cm}}_2^{191+}$ (red) in the pure Coulomb problem for the pointlike (dashed) and extended (solid) nuclei.

Figure 3

The function $F_{nj}^{AMM}$ and the rate of growth $n\left(Z\right)$ for the shift due to $\mathrm{\Delta }{U}_{AMM}$ of the (a),(c) $1{\sigma }_g$ and (b),(d) $1{\sigma }_u$ levels in a symmetrical two-nuclei quasimolecule as a function of the total charge $Z$ for the fixed internuclear distances $d=15.5,20,30,40$ fm. For comparison, the same functions for the $1s_{1/2}$ and $2p_{1/2}$ levels in an H-like atom from Ref. [9] are shown.