Projectile and target scaling of the total ionization cross sections of atoms and molecules

A scaling for the total ionization cross sections of atomic and molecular targets by neutral to highly charged projectiles with energies ranging from a few keV/u to many MeV/u is presented, with few free parameters, all velocity independent, and whose values do not vary appreciably for the projectiles and atomic and molecular targets considered here. Another important feature of the proposed scaling is that it has been built considering the characteristics of the collision systems, taking into account the various physical processes that lead to target ionization and the velocity regions where each of them dominates, and performing a smooth transition between them. This scaling describes quite well the existent experimental data covering a broad range of collision systems as diverse as H${}^{+}$ on H, Xe${}^{30+}$ on He, or O${}^{6+}$ on uracil.

Figure 1

Scaled total (single $+$ double) ionization cross sections of He by bare projectiles. (a) Born scaling: $\sigma /q^2$ in Mb versus $E/M$ in keV/u; (b) Olson scaling: $\sigma /q$ in Mb versus $(E/M)/q$ in keV/u; (c) Scaling from Ref. [13]: $\sigma /q^{4/3}$ in Mb versus $(E/M)/q^{2/3}$ in keV/u. Experiment: black squares, H${}^{+}$ from Refs. [3, 4]; red circles, He${}^{2+}$ from Ref. [3]; blue up triangles, Li${}^{3+}$ from Refs. [3, 5, 6]; magenta diamonds, B${}^{5+}$; green down triangles, C${}^{6+}$; and brown stars, O${}^{8+}$, all from Ref. [6]; black asterisks, Ar${}^{16+}$ from Ref. [7]; and red crosses, I${}^{5+\mathrm{to}27+}$ from Ref. [8].

Figure 2

Scaled total ionization cross sections [given by Eq. (3)] of atomic and molecular targets by protons in Mb as a function of the scaled energy $(E/M)/I_{nl}$ in keV/u. Experiment: black open squares, H from Ref. [28]; red open circles, He; green open up triangles, Ne; blue open down triangles, Ar; and magenta open diamonds, Kr, all from Refs. [22, 24, 29]; black crosses, Xe from Refs. [24] and [30, 31] as cited by Rudd [4]; red asterisks, H${}_2$, from Refs. [29, 32]; green solid up triangles, N${}_2$, from Ref. [29]; black solid squares, H${}_2$O, from Ref. [33]; red solid circles, CH${}_4$, from Refs. [29, 34]; and blue solid stars, uracil, from Iriki et al. as cited by Galassi et al. [35]. The solid line is the scaling function given by Eq. (4) and the dotted lines represent a deviation of $\pm 30\%$ of the solid line (see text).

Figure 3

The same as in Fig. 2 for antiprotons. Experiment: red circles, He from Refs. [39, 40, 41]; blue up triangles, Ne from Ref. [42]; black squares, Ar from Refs. [41, 42]; green down triangles, Kr, and magenta stars, Xe, both from Ref. [42]; and brown diamonds, H${}_2$ from Ref. [40]. The solid line is the scaling function given by Eq. (4) and the dotted lines represent a deviation of $\pm 40\%$ of the solid line (see text).

Figure 4

Scaled total ionization cross sections of Ar, in Mb, for various projectiles. (a) Born scaling: $\sigma /q^2$ in Mb versus $E/M$ in keV/u; (b) scaling from Ref. [13]: $\sigma /q^{4/3}$ in (Mb) versus $(E/M)/q^{2/3}$ in keV/u. Experiment: green squares, H${}^{+}$ from Ref. [4]; blue up triangles, H${}^{+}$ from Ref. [24]; red circles, He${}^{+}$ from Ref. [43]; orange down triangles, He${}^{2+}$ from Ref. [44]; magenta stars, C${}^{3+}$ from Ref. [45]; black cross, C${}^{3+}$ from Ref. [46]; and violet diamonds, Ar${}^{q+}$, with $q$ ranging from 6 to 14, from Ref. [47]. The solid line is the scaling function given by Eqs. (3) and (4).

Figure 5

Scaled ionization cross sections (${\sigma }_{\mathrm{ion}}$/q) of He by several projectiles as a function of the scaled velocity ($v/q^{1/4}$). Experiment: orange squares, H${}^0$ [53]; orange up triangles, C${}^0$; orange down triangles, N${}^0$; orange diamonds, O${}^0$, all from [48]; orange circles, He${}^0$, and orange stars, Ne${}^0$, both from [54]; purple squares, He${}^{+}$ [55]; purple circles, C${}^{+}$; purple triangles, N${}^{+}$; and purple diamonds, O${}^{+}$, all from [48]; green squares, He${}^{2+}$ [3]; green circles, C${}^{2+}$; green triangles, N${}^{2+}$, both from [48]; magenta squares, Li${}^{3+}$ [3]; magenta triangles, N${}^{3+}$ [48]; blue triangles, I${}^{5+\mathrm{to}16+}$ [8]; red circles, Ar${}^{16+}$ and black squares, Xe${}^{30+}$, both from [7]. The solid line is the scaling function given by Eq. (12).

Figure 6

$g(v,q)$, given by Eq. (16) as a function of the scaled velocity ($v/q^{1/4}$). Experiment: red squares, Ar${}^{16+}$, and blue circles, Xe${}^{30+}$, both from [7]. The solid line is the linear fit given by Eq. (17) (see text).

Figure 7

Ratio $g/g_{\mathrm{lin}}$ as a function of the $R=r_{\mathrm{rel}}/r_{\mathrm{adiab}}$. Experiment: orange stars, $q=0$ [48, 53, 54]; brown down triangles, $q=1$ [48, 55]; green asterisks, $q=2$ [3, 48]; magenta diamonds, $q=3$ [3, 48]; blue triangles, $q=5-16$ [8]; red circles, $q=16$; and black squares, $q=30$, both from [7].

Figure 8

Scaled experimental ionization cross sections (${\sigma }_{\mathrm{ion}}^{\exp }/q_{\mathrm{eff}}^2$) of He by several projectiles as a function of $E/M$. Experiment: black up triangles, H${}^0$ [48]; brown up triangles, H${}^0$ [53]; blue up triangles, H${}^{+}$ [3]; black squares, He${}^0$ [48]; orange squares, He${}^0$ [54]; red squares, He${}^{+}$ [55]; blue squares, He${}^{2+}$ [3]; blue circles, Li${}^{3+}$ [3]; purple squares, I${}^{5+\mathrm{to}16+}$ [8]; and green inverted triangles, Ar${}^{16+}$ and green diamonds, Xe${}^{30+}$, both from [7]. The solid line is the scaled ionization cross sections for singly charged projectiles ${\sigma }_{\mathrm{p}}$ given by Eq. (3).

Figure 9

Scaled experimental ionization cross sections including the binding correction ${\sigma }_{\mathrm{ion}}^{\exp }{\overline{\epsilon }}^2/q_{\mathrm{eff}}^2$ of He by several projectiles as a function of $E_{\mathrm{corr}}=E/\left(M\overline{\epsilon }\right)$ (see text). The experimental data are the same as in Fig. 8. The solid line is the scaled ionization cross sections for singly charged projectiles ${\sigma }_{\mathrm{p}}$ given by Eq. (3) and the dotted lines represent a deviation of $\pm 40\%$ of the solid line (see text).

Figure 10

The same as in Fig. 8 for the H target. Experiment: black squares, H${}^{+}$ and green circles, He${}^{2+}$, both from [28]; red squares, Li${}^{+}$; red circles, Li${}^{2+}$; and red triangles, Li${}^{3+}$, all from [69]; blue squares, C${}^{2+}$; blue circles, C${}^{3+}$; blue triangles, C${}^{4+}$; magenta squares, O${}^{2+}$; magenta circles, O${}^{3+}$; and magenta triangles, C${}^{4+}$, all from [70]; orange squares, Ar${}^{3+}$; orange circles, Ar${}^{4+}$; orange triangles, Ar${}^{5+}$; and orange diamonds, Ar${}^{6+}$, all from [71].

Figure 11

The same as in Fig. 8 for the Ne target. Experiment: black squares, H${}^{+}$ [4]; red squares, H${}^{+}$ [72]; blue up triangles, He${}^{+}$ [43]; blue circles, He${}^{2+}$ [19]; orange squares, Li${}^{3+}$ [73]; green inverted triangles, B${}^{2+}$ [74]; magenta diamonds, C${}^{3+}$ [75]; brown circles, Ne${}^{3+}$ [76]; and purple stars, C${}^{6+}$ [11].

Figure 12

The same as in Fig. 8 for the Ar target. Experiment: black squares, H${}^{+}$ [4]; red squares, H${}^{+}$ [72]; blue up triangles, He${}^{+}$ [43]; blue circles, He${}^{2+}$ [19]; orange up triangles, C${}^{3+}$ [45]; magenta diamonds, C${}^{3+}$ [46]; green inverted triangles, Ar${}^{4+\mathrm{to}14+}$ [47]; orange stars, C${}^{6+}$; purple stars, Nb${}^{23+,31+,34+}$; and gray stars, Pb${}^{54+}$, all from Ref. [11].

Figure 13

The same as in Fig. 8 for the water target. Experiment: red squares, H${}^0$ and black squares, H${}^{+}$ [33]; blue up triangles, He${}^{+}$ [77]; magenta circles, Li${}^{3+}$ [78]; green inverted triangles, C${}^{3+}$ [79]; purple stars, C${}^{6+}$ [80]; and orange diamonds, Xe${}^{44+}$ [81].

Figure 14

The same as in Fig. 8 for the uracil target. Experiment: red squares, H${}^{+}$ from Iriki et al. as cited by Galassi et al. [35]; blue circles, C${}^{4+}$ [82]; and green triangles, O${}^{6+}$ [82].

Figure 15

Total positive ion production for collisions of C${}^{4+}$ projectiles and the uracil target. Experiment: blue squares, [82]. Theory: red dashed line, ionization cross sections using the present scaling, Eq. (32); blue dash-dotted line, capture cross sections using the Bohr-Lindhard model [57]; and full line, sum of the ionization and capture cross sections.

Figure 16

Projectile effective charge as a function of $E/M$ in keV/u for collisions of C${}^{3+}$ projectiles on Ne. The black line is the value of $q_{\mathrm{eff}}$ obtained using Eq. (23). Experiment: blue squares, $q_{\mathrm{eff}}$ obtained from Eq. (34) for the data of Ref. [75].