Single and double ionization of magnesium by electron impact: A classical study

We consider electron impact-driven single and double ionization of magnesium in the energy range of 10 to 100 eV. Our classical Hamiltonian model of these $(e,2e)$ and $(e,3e)$ processes sheds light on their total cross sections and reveals the underlying ionization mechanisms. Two pathways are at play in single ionization: delayed and direct. In contrast, only the direct process is observed in double ionization, ruling out the excitation-autoionization channel. We also provide evidence that the so-called Two-Step 2 mechanism predominates over the Two-Step 1 mechanism, in agreement with experiments.

Figure 1

Trajectory of each of the three electrons in configuration space during the double ionization of a target by electron impact. The impact electron is shown in black; the two electrons of the target, in red and blue. The position of the ionic core is shown by the cross.

Figure 2

In the upper and the lower panel, respectively, the single-ionization (SI) and the double-ionization (DI) probability of Hamiltonian (3), for 1D, 2D, and 3D models $(d=1$, 2, and 3, respectively), measured at $t_f=800\mathrm{a}.\mathrm{u}.$ The vertical line is at ${\epsilon }_0=\left|{\mathcal{E}}_g\right|$. Filled diamonds show the experimental results from Ref. [5]. Cross sections are in Mb $\left(1\mathrm{Mb}={10}^{-18}{\mathrm{cm}}^2\right)$ and ${\epsilon }_0$ is in eV.

Figure 3

Delayed single ionization. Upper panel: Trajectory of each of the three electrons in configuration space $(x,y)$. Only the trajectory for positive times (i.e., after the impact) is represented. The black trajectory corresponds to the impact electron. Red and blue trajectories correspond to the two electrons of the target. Inset: Trajectory of the electrons of the atom for $t\in [500,700]$ a.u. The instant of the collision is indicated by the pair of stars. The position of the ionic core is indicated by the cross. Lower panel: Corresponding $V_{kj}$, defined by Eq. (5), as a function of time for $(k,j)=(0,1),(0,2)$, and $(1,2)$. The impact energy is ${\epsilon }_0=40\mathrm{eV}$. Here $V_{kj}$ is in $\mathrm{eV}$ and all other quantities are in a.u.

Figure 4

Direct single ionization. Upper panel: Trajectory of each of the three electrons in configuration space $(x,y)$. Only the part of the trajectory after the impact is represented. The black trajectory corresponds to the impact electron $e_0^{-}$. Red and blue trajectories correspond to the two electrons of the target $e_1^{-}$ and $e_2^{-}$. The position of the ionic core is indicated by the cross. Lower panel: Corresponding $V_{kj}$, defined by Eq. (5), as a function of time for $(k,j)=(0,1),(0,2)$, and $(1,2)$. The impact energy is ${\epsilon }_0=40\mathrm{eV}$. Here $V_{kj}$ is in $\mathrm{eV}$ and all other quantities are in a.u.

Figure 5

Ionization scenario as a function of the initial conditions $(x_1\left(t_i\right),x_2\left(t_i\right))$ for $y_1\left(t_i\right)=y_2\left(t_i\right)=0,{\mathbf{p}}_1\left(t_i\right)=(P/\sqrt{2},-P/2)$, and ${\mathbf{p}}_2\left(t_i\right)=(\sqrt{2}P/4,\sqrt{2}P/4)$, with $P$ the solution of Eq. (6). Outside the red contour line $P$ is not defined. The impact energy is ${\epsilon }_0=40\mathrm{eV},t_i=-500\mathrm{a}.\mathrm{u}.$, and $t_f=2000\mathrm{a}.\mathrm{u}.$ For delayed single ionization, the time $\mathrm{\Delta }t$ it takes the atom to lose an electron after the impact is represented. White areas inside the contour zone (red) represent the initial conditions where the atom does not ionize (or eventually ionizes with a delay which is too long). All quantities are in a.u.

Figure 6

Energy of the atom $E$ at $t_f=2000\mathrm{a}.\mathrm{u}.$ as a function of the initial conditions $(x_1\left(t_i\right),x_2\left(t_i\right))$ for $y_1\left(t_i\right)=y_2\left(t_i\right)=0,{\mathbf{p}}_1\left(t_i\right)=(P/\sqrt{2},-P/2)$, and ${\mathbf{p}}_2\left(t_i\right)=(\sqrt{2}P/4,\sqrt{2}P/4)$, with $P$ the solution of Eq. (6). Outside the red contour line $P$ is not defined. The impact energy is ${\epsilon }_0=40\mathrm{eV}$, and $t_i=-500\mathrm{a}.\mathrm{u}.$ Here $E$ is in $\mathrm{eV}$ and all other quantities are in a.u.

Figure 7

Upper panel: Single-ionization (SI) probability as a function of the impact energy ${\epsilon }_0$ by delayed and direct mechanisms. Lower panel: Double-ionization (DI) probability as a function of the impact energy ${\epsilon }_0$ by TS1 and TS2 mechanisms. The vertical line is at ${\epsilon }_0=\left|{\mathcal{E}}_g\right|$. The integration time is $t_f=800\mathrm{a}.\mathrm{u}.$ Here ${\epsilon }_0$ is in $\mathrm{eV}$.

Figure 8

Probability density function of $E$ in logarithmic scale as a function of the impact energy ${\epsilon }_0$, where $E$ is the energy of the atom when at least one electron has reached the detector. The white line represents $\langle E\rangle$, the average of $E$ for a fixed ${\epsilon }_0$, with bars indicating one standard deviation. Inset: Probability density function as a function of the final energy of the atom $E$ for fixed impact energy ${\epsilon }_0$. Here $E$ and ${\epsilon }_0$ are in $\mathrm{eV}$.

Figure 9

Upper panel: Sample trajectory undergoing TS1 in configuration space. Only the part of the trajectory after the impact is represented. Lower panel: Corresponding $V_{kj}\left(t\right)$ as a function of time for $(k,j)=(0,1),(0,2)$, and $(1,2)$. The position of the ionic core is identified by the cross. The impact energy is ${\epsilon }_0=40\mathrm{eV}$. Here $V_{kj}$ is in $\mathrm{eV}$ and all other quantities are in a.u.

Figure 10

Upper panel: Sample trajectory undergoing TS2 in configuration space. Only the part of the trajectory after the impact is represented. Lower panel: Corresponding $V_{kj}\left(t\right)$ as a function of time for $(k,j)=(0,1),(0,2)$, and $(1,2)$. The position of the ionic core is identified by the cross. The impact energy is ${\epsilon }_0=40\mathrm{eV}$. Here $V_{kj}$ is in $\mathrm{eV}$ and all other quantities are in a.u.

Figure 11

Upper panels: Total single-ionization (SI) probability as a function of the impact energy ${\epsilon }_0$ and its decomposition in delayed and direct mechanisms. Lower panels: Total double-ionization (DI) probability as a function of the impact energy ${\epsilon }_0$ and its decomposition in TS1 and TS2 mechanisms. Single and double ionization both computed with $b=0.3$ a.u. Left panels: The measurement is performed at $t_f=800\mathrm{a}.\mathrm{u}.$ after the impact. Right panels: The measurement is performed at $t_f=3000\mathrm{a}.\mathrm{u}.$ after the impact. The vertical line is at ${\epsilon }_0=\left|{\mathcal{E}}_g\right|$. Here ${\epsilon }_0$ is in $\mathrm{eV}$.

Figure 12

Upper panel: Total single-ionization (SI) probability as a function of the impact energy ${\epsilon }_0$ and its decomposition in delayed and direct mechanisms. Lower panel: Total double-ionization (DI) probability as a function of the impact energy ${\epsilon }_0$ and its decomposition in TS1 and TS2 mechanisms. Single and double ionization both computed with $b=1\mathrm{a}.\mathrm{u}.$ and $t_f=800\mathrm{a}.\mathrm{u}.$ For each trajectory, the impact parameter $y_0\left(t_i\right)$ is chosen randomly in the range $[0,3]\mathrm{a}.\mathrm{u}.$ Here ${\epsilon }_0$ is in $\mathrm{eV}$, and all other quantities are in a.u.