Resonances with natural and unnatural parities in positron-sodium scattering

We present an investigation of resonances in positron-sodium scattering using the complex scaling method. For the target sodium atoms, the interaction between the core and outer electron is treated using two different types of analytical model potentials. Explicitly correlated Gaussian wave functions are used to represent the correlation effects between the outer electron, the positron, and the ${\text{Na}}^{+}$ core. $S-, P-$, and $D\text{-wave}$ resonances with natural parity have been calculated for energies extending up to the positronium $n=2$ formation threshold. Resonance states for unnatural parities $P^e$ and $D^0$ have been calculated for energies extending up to the positronium $n=3$ threshold. Below both positronium thresholds we have for each symmetry identified a dipole series of resonances, with binding energies scaling in good agreement with expectations from an analytical calculation. The presented results are compared with other theoretical calculations.

Figure 1

(a) and (b) Real and (c) complex stabilization graphs of $e^{+}\text{-Na}$ states around the Na($3p$) threshold. The $3p$ threshold opens at $-0.1115$ a.u. All states which curve down as $\alpha$ is increased are continuum states, with the curvature depending on which threshold they belong to. The flat plateau at $-0.0768$ a.u. is a resonance. The $4s$ threshold opens at $-0.0716$ a.u. The flat plateau just above the $3p$ threshold is not a resonance, as is revealed by zooming in on the state (b) or going into the complex plane (c).

Figure 2

Complex scaling graph for a resonance. The crosses show the energy eigenvalues in the complex plane as a function of the complex scaling parameter $\theta$ in steps of 0.01. The stationary point, indicating the presence of a resonance, is plainly visible. Note that the real part of the energy $E_R$ is shifted up by $2.4\times {10}^{-5}$ a.u. compared to the result using the real stabilization method ($\theta =0$).