Complex scaling calculation of phase shifts for positron collisions with positive ions (2024)

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  Figure 1

  Two sets of coordinates in ${\mathrm{e}}^{+}$-H/${\mathrm{He}}^{+}/{\mathrm{Li}}^{2+}$ scattering.

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  Figure 2

  Three sets of coordinates in ${\mathrm{e}}^{+}\text{−}{\mathrm{He}}^{+}$ scattering.

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  Problem 27 Plot the curve for \(f(x)=\frac{... [FREE SOLUTION]

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  Figure 3

  (a)Complex eigenenergies $\left\{E_k\right\}$ and $\left\{E_{0,k}\right\}$ calculated for ${\mathrm{e}}^{+}+\mathrm{Ne}$ at the complex scaling parameter $\theta =0.25$. (b)Convergence of the $S$-wave phase shift of ${\mathrm{e}}^{+}+\mathrm{Ne}$ scattering against complex scaling parameters $\theta =0.10$ (most oscillating curve), 0.13, and 0.25 (most smooth curve). The phase shifts are compared with those calculated using the Numerov method. The inset is a close-up view of the phase shift behavior in the vicinity of ${10}^{-2}\text{−}{10}^0$eV.

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  Figure 4

  Convergence of the $S$-wave phase shift ${\delta }^{\left(n\right)}\left(E_{\mathrm{col}}\right)$ [see Eq.(23) for definition] of ${\mathrm{e}}^{+}+\mathrm{Ne}$ at $E_{\mathrm{col}}=0$ and 0.1eV against the number of eigenenergies included $n$.

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  Figure 5

  (a)Complex eigenenergies $\left\{E_k\right\}$ and $\left\{E_{0,k}\right\}$ calculated for ${\mathrm{e}}^{+}+{\mathrm{He}}^{+}$ at the complex scaling parameter $\theta =0.20$. (b)Convergence of the $S$-wave phase shift ${\delta }^{\left(n\right)}\left(E_{\mathrm{col}}\right)$ of ${\mathrm{e}}^{+}+{\mathrm{He}}^{+}$ at $E_{\mathrm{col}}=0$ and 10eV against the number of eigenenergies included $n$.

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  Figure 6

  Schematic of complex eigenenergies and the related angles of depression on the complex energy plane. $(x_k,y_k)$ is selected as the closest point to $E_{0,k}$ on the $2\theta$ line.

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  Figure 7

  (a)${\mathrm{\Delta }}_{x,k},{\mathrm{\Delta }}_{y,k},{\mathrm{\Delta }}_{0x,k}$, and ${\mathrm{\Delta }}_{0y,k}$ of eigenenergies of ${\mathrm{e}}^{+}+\mathrm{Ne}$ scattering corresponding to Fig.8. (b)Reproducibility of ${\delta }_{\mathrm{diff},k}$ (at $E_{\mathrm{col}}=0$) by the first- and second-order terms calculated from ${\mathrm{\Delta }}_{x,k},{\mathrm{\Delta }}_{y,k},{\mathrm{\Delta }}_{0x,k}$, and ${\mathrm{\Delta }}_{0y,k}$ according to Eq.(30).

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  Figure 8

  $S$-wave phase shift of ${\mathrm{e}}^{+}+\mathrm{Ne}$ decomposed into ${\delta }_{\mathrm{diff},k}$ [see Eq.(24) for definition] for each pair of the complex eigenenergies. $E_{\mathrm{col}}=0.01$, 0.1, 1, and 10eV are compared with $E_{\mathrm{col}}=0$eV.

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  Figure 9

  $S$-wave phase shifts of ${\mathrm{e}}^{+}+\mathrm{Ne}$ calculated using a limited number of pairs of complex eigenenergies $n$. (a)Without calibration and (b)with calibration according to Eq.(34). The phase shifts calculated using the CSM are compared with those calculated using the Numerov method.

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  Figure 10

  Elastic scattering cross sectionsof ${\mathrm{e}}^{+}+\mathrm{X}$ (X = Ne, Ar, Kr, and Xe) calculated using the calibrated phase shifts are presented against the collision energy. The arrows of colors matching the corresponding line indicate the $\mathrm{Re}{E}_{0,n}/2$, namely, $0.5,5,50$eV from left to right, to show the expected applicable range.

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  Figure 11

  (a)Calibrated phase shifts of the ${\mathrm{e}}^{+}+{\mathrm{He}}^{+}$ scattering compared with those calculated using the Harris-Nesbet (HN) variational method[22]. For each partial wave, we use 33, 36, and 35 pairs of complex eigenenergies for the $S,P$, and $D$ waves, respectively. The arrows from right to left present the $\mathrm{Re}{E}_{0,n}/2-E_{{\mathrm{He}}^{+}\left(1\mathrm{s}\right)}$ for $P$ and $D$ waves as an indicator of the maximum applicable energy. The $\mathrm{Re}{E}_{0,n}/2-E_{{\mathrm{He}}^{+}\left(1\mathrm{s}\right)}$ for the $S$ wave is located at a much larger energy (approximately 76eV). (b)The calibrated $S$-wave phase shifts of the ${\mathrm{e}}^{+}+{\mathrm{He}}^{+}$ scattering calculated using the 33 pairs of complex eigenenergies are compared with those calculated using 27 pairs of complex eigenenergies. The black arrow presents $\mathrm{Re}{E}_{0,k=27}/2-E_{{\mathrm{He}}^{+}\left(1\mathrm{s}\right)}$, which indicates the expected approximation limit for the collision energy.

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  Figure 12

  $S$-wave phase shifts of positron scattering off H, He, ${\mathrm{He}}^{+},{\mathrm{Li}}^{+}$, and ${\mathrm{Li}}^{2+}$ atoms and ions are calculated using the CSM with calibration modification (solid lines). Points denote the previous works: $\circ$[38] and $▵$[39] for ${\mathrm{e}}^{+}$-H, $+$[40] and $\times$[38] for ${\mathrm{e}}^{+}$-He, $♦$[21] and $▽$[22] for ${\mathrm{e}}^{+}\text{−}{\mathrm{He}}^{+}$, and $⊠$[41] for ${\mathrm{e}}^{+}\text{−}{\mathrm{Li}}^{2+}$.

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