Second-order Born approximation for the ionization of molecules by electron and positron impact

Second-order Born approximation is applied to study the ionization of molecules. The initial and final states are described by single-center wave functions. For the initial state a Gaussian wave function is used while for the ejected electron it is a distorted wave. Results of the present model are compared with recent ($e,2e$) experiments on the water molecule. Preliminary results are also presented for the ionization of the thymine molecule by electrons and positrons.

Figure 1

(a) Triple differential cross section for 250 eV electron-impact ionization of the 2$a_1$ orbital of the water molecule. The theoretical calculation is performed in the first Born approximation ($L$ = 0 partial-wave contribution). The energy of the ejected electron is $E_e$ = 10 eV and the scattered angle is 15${}^{{}^{\circ }}$. (b) Triple differential cross section for 250 eV electron-impact ionization of the 2$a_1$ orbital of the water molecule. The theoretical calculation is performed in the first Born approximation. The energy of the ejected electron is $E_e$ = 10 eV and the scattered angle is 15${}^{{}^{\circ }}$. Contribution of $L$ = 1 (solid line), of $L$ = 2 (dashed line), of $L$ = 3 (dotted line), and of $L$ = 4 (dashed-dotted line).

Figure 2

(a) Triple differential cross section for 250 eV electron-impact ionization of the 2$a_1$ orbital of the water molecule. The energy of the ejected electron is $E_e$ = 10 eV and the scattered angle is 15${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation (solid line), in the second Born approximation (dashed line), and with the eikonal approximation (dotted line). The experimental data are those of Milne-Brownlie et al. [3]. (b) Triple differential cross section for 250 eV electron-impact and positron-impact ionization of the 2$a_1$ orbital of the water molecule. The energy of the ejected electron is $E_e$ = 10 eV and the scattered angle is 15${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation (solid line) and in the second Born approximation for electron impact (dashed line) and for positron impact (dotted line).

Figure 3

(a) Summed triple differential cross section for 250 eV electron-impact ionization of the 1$b_1$ and 3$a_1$ orbitals of the water molecule. The energy of the ejected electron is respectively $E_e$ = 10 eV for the 1$b_1$ orbital and 8 eV for the 3$a_1$ orbital. The scattered angle is 15${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation (solid line) and in the second Born approximation (dashed line). The experimental data are those of Milne-Brownlie et al. [3]. (b) Summed triple differential cross section for 250 eV electron-impact ionization of the 1$b_1$ and 3$a_1$ orbitals of the water molecule. The energy of the ejected electron is respectively $E_e$ = 10 eV for the 1$b_1$ orbital and 8 eV for the 3$a_1$ orbital. The scattered angle is 15${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation (solid line) and in the second Born approximation for electron impact (dashed line) and for positron impact (dotted line).

Figure 4

(a) Triple differential cross section for 250 eV electron-impact ionization of the 1$b_1$ orbital of the water molecule. The energy of the ejected electron is $E_e$ = 10 eV and the scattered angle is 15${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation (solid line) and in the second Born approximation (dashed line). The experimental data are those of Milne-Brownlie et al. [3]. (b) Triple differential cross section for 250 eV electron-impact and positron-impact ionization of the 1$b_1$ orbital of the water molecule. The energy of the ejected electron is $E_e$ = 10 eV and the scattered angle is 15${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation (solid line) and in the second Born approximation for electron impact (dashed line) and for positron impact (dotted line).

Figure 5

(a) Triple differential cross section for 250 eV electron-impact ionization of the 1$b_2$ orbital of the water molecule. The energy of the ejected electron is $E_e$ = 10 eV and the scattered angle is 15${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation (solid line) and in the second Born approximation (dashed line). The experimental data are those of Milne-Brownlie et al. [3]. (b) Triple differential cross section for 250 eV electron-impact and positron-impact ionization of the 1$b_2$ orbital of the water. The energy of the ejected electron is $E_e$ = 10 eV and the scattered angle is 15${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation (solid line) and in the second Born approximation for electron impact (dashed line) and for positron impact (dotted line).

Figure 6

(a) Triple differential cross section for 250 eV electron-impact ionization of the first valence orbital of the thymine (ionization energy: $I_i$ = 9.14 eV). The energy of the ejected electron is $E_e$ = 20 eV and the scattered angle is 15${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation: partial wave $L$ = 1 (thin solid line), partial waves $L$ = 1 to $L$ = 2 (dashed line), partial waves $L$ = 1 to $L$ = 3 (dotted line), partial waves $L$ = 1 to $L$ = 4 (dashed-dotted line), partial waves $L$ = 1 to $L$ = 5 (dash-dot dotted line), partial waves $L$ = 1 to $L$ = 6 (short dashed line), and partial waves $L$ = 1 to $L$ = 10 (thick solid line). (b) Triple differential cross section for 250 eV electron-impact ionization of the first valence orbital of the thymine (ionization energy: $I_i$ = 9.14 eV). The energy of the ejected electron is $E_e$ = 20 eV and the scattered angle is 15${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation: partial waves $L$ = 1 to $L$ = 8 (thick solid line) and partial waves $L$ = 1 to $L$ = 10 (dashed line).

Figure 7

(a) Triple differential cross section for 250 eV electron-impact ionization of the first valence orbital of the thymine (ionization energy: $I_i$ = 9.14 eV). The energy of the ejected electron is $E_e$ = 20 eV and the scattered angle is 10${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation: partial wave $L$ = 1 (thin solid line), partial waves $L$ = 1 to $L$ = 2 (dashed line), partial waves $L$ = 1 to $L$ = 3 (dotted line), partial waves $L$ = 1 to $L$ = 4 (dash and dotted line), partial waves $L$ = 1 to $L$ = 5 (dash-dot dotted line), partial waves $L$ = 1 to $L$ = 6 (short dashed line), and partial waves $L$ = 1 to $L$ = 10 (thick solid line). (b) Triple differential cross section for 250 eV electron-impact ionization of the first valence orbital of the thymine (ionization energy: $I_i$ = 9.14 eV). The energy of the ejected electron is $E_e$ = 20 eV and the scattered angle is 10${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation: partial waves $L$ = 1 to $L$ = 8 (thick solid line) and partial waves $L$ = 1 to $L$ = 10 (dashed line).

Figure 8

(a) Triple differential cross section for 250 eV electron-impact ionization of the second valence orbital of the thymine (ionization energy: $I_i$ = 11eV). The energy of the ejected electron is $E_e$ = 20 eV and the scattered angle is 15${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation: partial wave $L$ = 1 (thin solid line), partial waves $L$ = 1 to $L$ = 2 (dashed line), partial waves $L$ = 1 to $L$ = 3 (dotted line), partial waves $L$ = 1 to $L$ = 4 (dashed- dotted line), partial waves $L$ = 1 to $L$ = 5 (dash-dot dotted line), partial waves $L$ = 1 to $L$ = 6 (short dashed line), and partial waves $L$ = 1 to $L$ = 10 (thick solid line). (b) Triple differential cross section for 250 eV electron-impact ionization of the second valence orbital of the thymine (ionization energy: $I_i$ = 11eV). The energy of the ejected electron is $E_e$ = 20 eV and the scattered angle is 15${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation: partial waves $L$ = 1 to $L$ = 8 (thick solid line) and partial waves $L$ = 1 to $L$ = 10 (dashed line).

Figure 9

(a) Triple differential cross section for 250 eV electron-impact ionization of the second valence orbital of the thymine (ionization energy: $I_i$ = 11 eV). The energy of the ejected electron is $E_e$ = 20 eV and the scattered angle is 10${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation: partial wave $L$ = 1 (thin solid line), partial waves $L$ = 1 to $L$ = 2 (dashed line), partial waves $L$ = 1 to $L$ = 3 (dotted line), partial waves $L$ = 1 to $L$ = 4 (dash and dotted line), partial waves $L$ = 1 to $L$ = 5 (dash-dot dotted line), partial waves $L$ = 1 to $L$ = 6 (short dashed line), and partial waves $L$ = 1 to $L$ = 10 (thick solid line). (b) Triple differential cross section for 250 eV electron-impact ionization of the second valence orbital of the thymine (ionization energy: $I_i$ = 11 eV). The energy of the ejected electron is $E_e$ = 20 eV and the scattered angle is 10${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation: partial waves $L$ = 1 to $L$ = 8 (thick solid line) and partial waves $L$ = 1 to $L$ = 10 (dashed line).

Figure 10

(a) Triple differential cross section for 250 eV electron-impact ionization of the third valence orbital of the thymine (ionization energy: $I_i$ = 11.4 eV). The energy of the ejected electron is $E_e$ = 20 eV and the scattered angle is 15${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation: partial wave $L$ = 0 (thin solid line), partial waves $L$ = 0 to $L$ = 1 (dashed line), partial waves $L$ = 0 to $L$ = 2 (dotted line), partial waves $L$ = 0 to $L$ = 3 (dashed-dotted line), partial waves $L$ = 0 to $L$ = 4 (dash-dot dotted line), partial waves $L$ = 0 to $L$ = 5 (short dashed line), partial waves $L$ = 0 to $L$ = 6 (short dotted line), and partial waves $L$ = 0 to $L$ = 10 (thick solid line). (b) Triple differential cross section for 250 eV electron-impact ionization of the third valence orbital of the thymine (ionization energy: $I_i$ = 11.4 eV). The energy of the ejected electron is $E_e$ = 20 eV and the scattered angle is 15${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation: partial waves $L$ = 0 to $L$ = 8 (solid line), partial waves $L$ = 0 to $L$ = 9 (dashed line), and partial waves $L$ = 0 to $L$ = 10 (dotted line).

Figure 11

(a) Triple differential cross section for 250 eV electron-impact ionization of the third valence orbital of the thymine (ionization energy: $I_i$ = 11.4 eV). The energy of the ejected electron is $E_e$ = 20 eV and the scattered angle is 10${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation: partial wave $L$ = 0 (thin solid line), partial waves $L$ = 0 to $L$ = 1 (dashed line), partial waves $L$ = 0 to $L$ = 2 (dotted line), partial waves $L$ = 0 to $L$ = 3 (dashed-dotted line), partial waves $L$ = 0 to $L$ = 4 (dash-dot dotted line), partial waves $L$ = 0 to $L$ = 5 (short dashed line), partial waves $L$ = 0 to $L$ = 6 (short dotted line), and partial waves $L$ = 0 to $L$ = 10 (thick solid line). (b) Triple differential cross section for 250 eV electron-impact ionization of the third valence orbital of the thymine (ionization energy: $I_i$ = 11.4 eV). The energy of the ejected electron is $E_e$ = 20 eV and the scattered angle is 10${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation: partial waves $L$ = 0 to $L$ = 8 (solid line), partial waves $L$ = 0 to $L$ = 9 (dashed line), and partial waves $L$ = 0 to $L$ = 10 (dotted line).

Figure 12

(a) Triple differential cross section for 250 eV electron-impact ionization of the fourth valence orbital of the thymine (ionization energy: $I_i$ = 12.16 eV). The energy of the ejected electron is $E_e$ = 20 eV and the scattered angle is 15${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation: partial wave $L$ = 0 (thin solid line), partial waves $L$ = 0 to $L$ = 1 (dashed line), partial waves $L$ = 0 to $L$ = 2 (dotted line), partial waves $L$ = 0 to $L$ = 3 (dashed-dotted line), partial waves $L$ = 0 to $L$ = 4 (dash-dot dotted line), partial waves $L$ = 0 to $L$ = 5 (short dashed line), partial waves $L$ = 0 to $L$ = 6 (short dotted line), and partial waves $L$ = 0 to $L$ = 10 (thick solid line). (b) Triple differential cross section for 250 eV electron-impact ionization of the fourth valence orbital of the thymine (ionization energy: $I_i$ = 12.16 eV). The energy of the ejected electron is $E_e$ = 20 eV and the scattered angle is 15${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation: partial waves $L$ = 0 to $L$ = 10 (solid line), partial waves $L$ = 0 to $L$ = 12 (dashed line), and partial waves $L$ = 0 to $L$ = 30 (dotted line).

Figure 13

(a) Triple differential cross section for 250 eV electron-impact ionization of the fourth valence orbital of the thymine (ionization energy: $I_i$ = 12.16 eV). The energy of the ejected electron is $E_e$ = 20 eV and the scattered angle is 10${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation: partial wave $L$ = 0 (thin solid line), partial waves $L$ = 0 to $L$ = 1 (dashed line), partial waves $L$ = 0 to $L$ = 2 (dotted line), partial waves $L$ = 0 to $L$ = 3 (dashed-dotted line), partial waves $L$ = 0 to $L$ = 4 (dash-dot dotted line), partial waves $L$ = 0 to $L$ = 5 (short dashed line), partial waves $L$ = 0 to $L$ = 6 (short dotted line), and partial waves $L$ = 0 to $L$ = 10 (thick solid line). (b) Triple differential cross section for 250 eV electron-impact ionization of the fourth valence orbital of the thymine (ionization energy: $I_i$ = 12.16 eV). The energy of the ejected electron is $E_e$ = 20 eV and the scattered angle is 10${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation: partial waves $L$ = 0 to $L$ = 8 (solid line), partial waves $L$ = 0 to $L$ = 9 (dashed line), and partial waves $L$ = 0 to $L$ = 10 (dotted line).

Figure 14

Triple differential cross section for 250 eV electron-impact and positron-impact ionization of the first valence orbital of the thymine (ionization energy: $I_i$ = 9.14 eV). The energy of the ejected electron is $E_e$ = 20 eV and the scattered angle is 10${}^{{}^{\circ }}$. The theoretical calculations are performed in the first Born approximation (solid line) and in the second Born approximation for electron impact (dashed line) and for positron impact (dotted line).