Relativistic Effects on Photoabsorption Cross Sections of Highly Charged Ions

Abstract

The study of highly charged ions offers a unique platform for probing the breakdown of non-relativistic theory under the influence of extreme electromagnetic environments. Here, we investigate the photoabsorption of highly charged ions within the dipole approximation using both the time-dependent Schrödinger equation (TDSE) and the time-dependent Dirac equation (TDDE), modelling the external field as an instantaneous broadband excitation. Nonrelativistic scaling relations with respect to the nuclear charge are utilized as a diagnostic tool to systematically identify and quantify relativistic contributions. Within the purely nonrelativistic TDSE framework, these scaling relations hold exactly, allowing the absorption spectra of arbitrary highly charged ions to be inferred directly from a neutral hydrogenic reference. However, as the nuclear charge increases, relativistic effects become dominant through a sizeable blue shift in the absorption cross section, due to the relativistic enhancement of the binding energy. We further evaluate semi-relativistic TDSE approximations by direct comparison with full TDDE simulations, assessing their predictive power and establishing the regimes where a full Dirac treatment is indispensable for quantitative accuracy.

Figures (5)

• Figure 1: (a) Schematic of $n=1 \to 2$ transitions for nonrelativistic (blue) and relativistic (red) systems. The relativistic contraction of levels increases the transition energy ($\Delta E_{\rm r} > \Delta E_{\rm nr}$). (b) Scaling of binding energies relative to $Z=1$ as a function of nuclear charge $Z$. The relativistic factor $S^{-1}(Z)$ (red) grows faster than the nonrelativistic $Z^{2}$ scaling (blue). Inset: The ratio $S^{-1}(Z)/Z^{2}$ highlights the relativistic enhancement as $Z$ increases.
• Figure 2: Evolution of the dipole response from the non-relativistic to the relativistic regime. (a) At $Z = 1$ ($F'_0 = 0.01$ a.u.), TDSE and TDDE results coincide. (b) For $Z = 10$ ($F_0 = 0.1$ a.u.), the dynamics accelerate according to the $Z^2$ scaling, with the semi-relativistic TDSE($Z' = 10.007$) tracking the full TDDE result. (c) At $Z = 25$ ($F_0 = 0.25$ a.u.), relativistic influences emerge as a cumulative phase deviation over extended time scales. (d) For $Z = 50$ ($F_0 = 0.5$ a.u.), the relativistic signature becomes dominant, manifesting as a significant temporal compression and frequency up-conversion that the TDSE($Z' = 50.885$) model only partially recovers.
• Figure 3: Temporal evolution of the scaled dipole moment $d(t) \cdot Z$ plotted against scaled time for increasing nuclear charges computed from (a) TDSE, (b) TDDE, and (c) TDSE($Z'$). All simulations are performed for a reference field amplitude $F'_0 = 0.01$ a.u.
• Figure 4: Scaled photo-absorption cross-section as a function of scaled energy for hydrogen-like ions with different nuclear charges (a) $Z =10$, (b) $Z=25$, and (c) $Z=50$ calculated using nonrelativistic and relativistic formulations. The amplitude parameter is fixed at $F'_0 = 0.01$ a.u. for the hydrogen atom ($Z=1$).
• Figure 5: Scaled photoabsorption cross-section as a function of scaled energy for hydrogen-like ions with various nuclear charges $Z$. The adopted semi-relativistic framework incorporates kinematic corrections that effectively realign the absorption peaks for moderate $Z$. The inset highlights the residual blue shift that reappears for $Z = 50$, marking the transition into the fully relativistic regime where simple frequency mapping is no longer sufficient. The reference amplitude parameter is fixed at $F'_0 = 0.01$ a.u. for the $Z = 1$ case.