Time-dependent calculations of transfer ionization by fast proton-helium collision in one-dimensional kinematics

We analyze a transfer ionization (TI) reaction in the fast proton-helium collision ${\mathrm{H}}^{+}+\mathrm{He}\to {\mathrm{H}}^0+{\mathrm{He}}^{2+}+ e^{-}$ by solving a time-dependent Schrödinger equation (TDSE) under the classical projectile motion approximation in one-dimensional kinematics. In addition, we construct various time-independent analogs of our model using lowest-order perturbation theory in the form of the Born series. By comparing various aspects of the TDSE and the Born series calculations, we conclude that the recent discrepancies of experimental and theoretical data may be attributed to deficiency of the Born models used by other authors. We demonstrate that the correct Born series for TI should include the momentum-space overlap between the double-ionization amplitude and the wave function of the transferred electron.

Figure 1

The TI probability density $P^{\left(1\right)}\left(k\right)$ as a function of the ejected electron momentum for the proton velocities $v_p=3$ a.u. (top), 5 a.u. (middle), and 10 a.u. (bottom). Various calculations are displayed with the following line styles: the full CPA (black solid line), the BECPA (red dashed line), and the PWFBA (green dotted line).

Figure 2

The TI probability $P$ (top) and the mean momentum of ejected electrons $\langle k\rangle$ (bottom) as functions of the projectile velocity $v_p$. Various calculations are shown as CPA (black solid line), BECPA (red dashed line), and PWFBA (green dotted line).

Figure 3

Ionization probability density $P^{\left(1\right)}\left(k\right)$ as a function of the momentum of the ejected electron for the proton velocities $v_p=3$ a.u. (top), 5 a.u. (middle), and 10 a.u. (bottom). The following calculations are shown: CPA (black solid line) and FBA (red dashed line).

Figure 4

Transfer probability as a function of the proton velocity $v_p$: CPA (black solid line), OBK (red dashed line), JSB (green dotted line), TSI (blue dotted-dashed line).

Figure 5

TI probability density $P^{\left(1\right)}\left(k\right)$ as a function of the momentum of the transferred electron for the proton velocities $v_p=3$ a.u. (top), 5 a.u. (middle), and 10 a.u. (bottom). Various calculations are shown as CPA (black solid line), TIDI (red dotted line), TIDIU (green dashed line).

Figure 6

The total TI probability $P$ (top) and the mean momentum of ejected electron $\langle k\rangle$ (bottom) as functions of $v_p$. Various calculations are shown as CPA (black solid line), TIDI (red dotted line), TIDIU (green dashed line), BECPA (blue dashed-dotted line).