Unified calculation of generalized oscillator strength of argon ranging from bound to continuum states

The electron and photon scattering data of an atom are crucial for many scientific fields, including plasma physics, astrophysics, and so on. For high enough but nonrelativistic incident energies, the first Born approximation is applicable for calculating these data, in which the key physics quantity is the generalized oscillator strength (GOS). In high-energy electron impact excitation processes, atoms will be excited into various excited states including strongly perturbed Rydberg and adjacent continuum states. How to calculate these quantities of a nontrivial many-electron atom rapidly and accurately is still a great challenge. Based on our eigenchannel $R$-matrix method $R$-eigen, we further extend it to calculate the GOS of a whole channel in an atom, which includes all Rydberg and adjacent continuum states. The $J^{\pi }=1^{-}$ states of argon are chosen as an illustrating example. The calculation results are in good agreement with the available benchmark absolute experimental measurements. The calculated eigenchannel GOS matrix elements are smooth functions of the excitation energy and momentum transfer. From such smooth eigenchannel GOS matrix elements, we can obtain the GOS of any specific excited state through multichannel quantum defect theory, e.g., infinite Rydberg (including a strongly perturbed one), autoionization, and continuum states.

Figure 1

Eigen-quantum defects ${\mu }_{\alpha }$, Euler angles ${\theta }_k$ for $U_{i\alpha }$ matrix in $J^{\pi }=1^{-}$ symmetry of Ar. The vertical dotted blue line indicates the connection position of two channels and five channels. The vertical dashed red line is the position of the ionization threshold of ${}^2{P}_{3/2}$.

Figure 2

Upper: Quantum defect $-{\nu }_{3/2}$(mod 1) vs ${\nu }_{1/2}$ plot of Ar. Various open points are experimental level positions. The solid periodic curves for Eq. (4) and dashed lines for Eq. (5) are shown. The vertical dashed red line indicates the $I_{3/2}$ threshold and the vertical dotted blue line denotes the smooth connection between the two-physical channel calculation and the five-physical channel calculation. Lower: The corresponding oscillator strength densities, where the symbols are consistent with the one in the upper panel.

Figure 3

The eigenchannel GOS matrix elements of $J^{\pi }=1^{-}$ symmetry of Ar.

Figure 4

The generalized oscillator strength densities of Ar from initial state $3p^6{J}^{\pi }=0^{+}$ to final channel $J^{\pi }=1^{-}$.

Figure 5

The comparison of the generalized oscillator strength of Rydberg states $3p^{5}4s[3/2{]}_1,3p^{5}4s^{\prime }[1/2{]}_1,3p^{5}3\mathrm{d}[3/2{]}_1$, and $3p^{5}3{\mathrm{d}}^{\prime }[3/2{]}_1$ with the experimental measurement.