Scaled plane-wave Born cross sections for atoms and molecules

Integral cross sections for optically allowed electronic-state excitations of atoms and molecules by electron impact, by applying scaled plane-wave Born models, are reviewed. Over 40 years ago, Inokuti presented an influential review of charged-particle scattering, based on the theory pioneered by Bethe forty years earlier, which emphasized the importance of reliable cross-section data from low eV energies to high keV energies that are needed in many areas of radiation science with applications to astronomy, plasmas, and medicine. Yet, with a couple of possible exceptions, most computational methods in electron-atom scattering do not, in general, overlap each other’s validity range in the region from threshold up to 300 eV and, in particular, in the intermediate region from 30 to 300 eV. This is even more so for electron-molecule scattering. In fact this entire energy range is of great importance and, to bridge the gap between the two regions of low and high energy, scaled plane-wave Born models were developed to provide reliable, comprehensive, and absolute integral cross sections, first for ionization by Kim and Rudd and then extended to optically allowed electronic-state excitation by Kim. These and other scaling models in a broad, general application to electron scattering from atoms and molecules, their theoretical basis, and their results for cross sections along with comparison to experimental measurements are reviewed. Where possible, these data are also compared to results from other computational approaches.

Figure 1

Map showing the domains of applicability of different theoretical treatments for the process $e+A\to e^{\prime }+A^{*}$. The horizontal axis represents the dimensionless energy ${(k{a}_0)}^2$ and the vertical axis the dimensionless angular momentum $\ell (\ell +1)$, both on logarithmic scales.

Figure 2

Total cross section for ionization of H by electron impact. The abscissa is the incident electron energy $T$ in eV. Circles, experimental data by [287]; solid line, BEB cross section; short-dashed line, BEQ cross section; medium-dashed line, Gryzinski’s classical cross section ([130], [131], [132]); long-dashed line, distorted-wave Born cross section with electron-exchange correction by [351].

Figure 3

Total cross section for ionization of He by electron impact. Filled circles, experimental data by [288]; filled diamonds, data by [241]; solid line, BEB cross section (with the $t+u+1$ denominator for a neutral target); medium-dashed line, Younger’s distorted-wave Born cross section ([351]); short-dashed line, BEQ cross section.

Figure 4

Total cross section for ionization of atomic oxygen by electron impact. Shown are the BEB result for direct ionization from its ground state, ionization from excited electronic states of O within a BEB approach, and their sum to give the total BEB ionization cross section. Also shown are experimental results, at the incident electron energies indicated in the legend, from [40], [364], and [315]. See also the legend. The BEB results are taken from [189].

Figure 5

Total ionization cross sections for electron scattering from ${\mathrm{CF}}_4$. See the legend and the text for further details.

Figure 6

Total single ionization cross section of ${\mathrm{CF}}_2$ calculated using the siBED model and the experimental data. Also presented are theoretical cross sections calculated using the BEB model of [190] and the DM model ([90]). “Present expt” refers to data from [154].

Figure 7

Total ionization cross section for ${\mathrm{CCl}}_4$. Theoretical values are indicated by the solid (effective core-potential) and dashed (all-electron) curves. Experimental values are indicated by the symbols. See [286] and text for further details.

Figure 8

The configuration of a typical crossed-beam apparatus, such as was used at Sophia University, for making the DCS measurements.

Figure 9

Typical energy-loss spectrum in neon ([148]). The incident beam energy was 40 eV and the scattered electron angle was 30°.

Figure 10

Typical results in He for calibrating the analyzer transmission response at the scattered electron angle of 30° ([146]). Results using method 2 at $T=20$ (solid line), 25 (dashed line), 30 (dash-dot-dotted line), and 50 eV (dash-dotted line), and method 3 at $T=30$ (top circle, blue) and 50 eV (bottom circle, green) are shown.

Figure 11

Example GOS vs $K^2$ plot, in this case for electron-impact excitation of the $2{}^{1}P$ electronic state in He. From [145].

Figure 12

Comparison of the $1s\text{−}2p$ excitation cross sections of H. Solid curve, plane-wave Born cross section with BE scaling; dashed curve, unscaled plane-wave Born cross section; open circles, CCC cross section. From [38].

Figure 13

Electron-impact excitation of the $2{}^{1}P$ electronic state in helium. The $\mathrm{B}\mathrm{E}f$-scaled result is shown as the solid line. See also the legend for further details and references ([136]; [320]; [335]; [289]; [62]; [84]; [238]; [144]; http://www.nist.gov/pml/data/ionization).

Figure 14

GOS vs $K^2$ for excitation of the $3s{[3/2]}_1$ and $3s^{\prime }{[1/2]}_1$ states in neon. Data from [146] and see also the legend. Fits to the GOS data using the formalism of [330] (—) and the result of a relativistic distorted-wave calculation from [328] are also plotted.

Figure 15

ICSs from [148], both experimental and $\mathrm{B}\mathrm{E}f$-scaling calculations, compared to previous results. See the legend and text for further details.

Figure 16

Comparison of the $2s\text{−}2p$ excitation cross sections of Li. Dashed curve: unscaled plane-wave Born ${\sigma }_{\mathrm{P}\mathrm{W}\mathrm{B}}$; solid curve: BE- and $f$-scaled ${\sigma }_{\mathrm{P}\mathrm{W}\mathrm{B}}$; solid dots: experimental data for optical emission from [211]; and circles: CCC theory of [285].

Figure 17

Comparison of the $2s^2\text{−}2s2p{}^{1}P$ excitation cross sections of Be. Circles: CCC cross section ([116]); short-dashed curve: unscaled plane-wave Born cross section from uncorrelated wave functions; medium-dashed curve: unscaled plane-wave Born cross section from multiconfiguration wave functions; long-dashed curve: BE-scaled plane-wave Born cross section from multiconfiguration wave functions; solid curve: BE- and $f$-scaled plane-wave Born cross sections from uncorrelated wave functions.

Figure 18

Comparison of the $6s^2\text{−}6s6p{}^{1}P$ excitation cross sections of Hg. Short-dashed curve: unscaled plane-wave ${\sigma }_{\mathrm{P}\mathrm{W}\mathrm{B}}$; medium-dashed curve: BE-scaled ${\sigma }_{\mathrm{P}\mathrm{W}\mathrm{B}}$; solid curve: BE- and $f$-scaled ${\sigma }_{\mathrm{P}\mathrm{W}\mathrm{B}}$; squares: relativistic distorted-wave Born theory of [296]; upright triangles: experimental data for direct excitation from [260]; and inverted triangles: data for direct excitation from [258].

Figure 19

Electron impact on ${\mathrm{H}}_2$: (a) Typical electron energy-loss spectrum and its associated spectral deconvolution ([174]) for 40 eV, at 10°. Also plotted are the relevant vibrational sublevels of the respective $B{{{}^1\mathrm{\Sigma }}_u}^{+}$ and $C{{}^1\mathrm{\Pi }}_u$ electronic states. (b) GOS as a function of $K^2$ for the $B{{{}^1\mathrm{\Sigma }}_u}^{+}$ electronic state for two different impact kinetic energies, illustrating the procedure for determining the OOS in the limit of small $K$ and the form of the GOS from which the integrated cross section can be determined. (c) ICSs for the $B{{{}^1\mathrm{\Sigma }}_u}^{+}$ electronic state. (d) ICSs for the $C^1{\mathrm{\Pi }}_u$ electronic state. (c) and (d) See the legend and the text for details. (e) Recommended cross-section data from Itikawa and colleagues ([347]), updated to include recent ICS ([174]) and with corresponding $\mathrm{B}\mathrm{E}f$-scaling results ([12]).

Figure 20

Electron impact on CO: (a) Typical electron energy-loss ($\mathrm{\Delta }E$) spectrum, $\mathrm{\Delta }E\sim 7.5–15.5\mathrm{eV}$, for electronic-state excitation. The incident energy is 100 eV and the scattered electron angle is 4.3°. (b) GOS vs $K^2$ for the $A{}^1\mathrm{\Pi }({\nu }^{\prime }=2)$ electronic state. A Vriens-type fit ([330]) to the measured data at 100 and 200 eV is shown. From [175]. (c) ICSs for the vibrational sublevels of the $A^1\mathrm{\Pi }$ electronic state. (d) ICSs for the $C{{}^1\mathrm{\Sigma }}^{+}+c{}^3\mathrm{\Pi }$ electronic states. (e) ICSs for the $E{}^1\mathrm{\Pi }$ electronic state. (c)–(e) See the legend and the text for details. (f) Recommended cross-section data set from [356], updated to include the recent ICS ([173]; [176]) and with corresponding $\mathrm{B}\mathrm{E}f$-scaling results ([12]).

Figure 21

Electron impact on ${\mathrm{O}}_2$: (a) Typical electron energy-loss spectrum, $\mathrm{\Delta }E\sim 4.9–14.4\mathrm{eV}$, for electronic-state excitation. The incident electron energy was 100 eV and the scattered electron angle was 5.1°. From [301]. (b) Potential energy (eV) as a function of internuclear distance (Å) for the electronic states of ${\mathrm{O}}_2$ and ${{\mathrm{O}}_2}^{-}$. From [61]. (c) The effect of the Rydberg-valence interactions leading to avoided level crossings. From [214]. (d) ICSs for excitation of the Schumann-Runge continuum. (e) ICSs for excitation of the longest band. (f) ICSs for excitation of the second band. (d)–(f) See the legend and the text for details. (g) Recommended cross-section data set from [164], updated to include the recent ICSs from [301] and their corresponding $\mathrm{B}\mathrm{E}f$-scaling results.

Figure 21a

Continued.

Figure 22

Electron impact on ${\mathrm{CO}}_2$. (a) Typical electron energy-loss spectrum, $\mathrm{\Delta }E\sim 6.0–14\mathrm{eV}$, for electronic-state excitation. The incident energy is 100 eV and the scattered electron angle is 15°. (b) GOS vs $K^2$ for the ${{{}^1\mathrm{\Sigma }}_u}^{+}$ electronic state. A Vriens-type fit ([330]) to the available experimental data ([193]; [177]) is also shown. (c) ICSs for excitation of the ${{{}^1\mathrm{\Sigma }}_u}^{+}$ electronic state. (d) ICSs for the excitation of the ${{}^1\mathrm{\Pi }}_u$ electronic state. (b)–(d) See the legends and the text for details. (e) Recommended cross-section data from [161], updated to include recent ICSs from [177] and corresponding $\mathrm{B}\mathrm{E}f$-scaling results.

Figure 23

Electron impact on ${\mathrm{N}}_2\mathrm{O}$. (a) Typical electron energy-loss spectrum $\mathrm{\Delta }E\sim 4–13\mathrm{eV}$ for electronic-state excitation. The incident electron energy is 100 eV and the scattered electron angle is 4.3°. (b) GOS vs $K^2$ for the $D{{}^1\mathrm{\Sigma }}^{+}$ electronic state. A Vriens-type fit ([330]) to the available experimental data ([178]) is also shown. (c) ICSs for excitation of the $C{}^1\mathrm{\Pi }$ electronic state. (d) Similar to (c) but for the $D{{}^1\mathrm{\Sigma }}^{+}$ electronic state. (b)–(d) See the legends and the text for details. (e) Recommended cross-section data set from Zecca and colleagues ([171]), updated to include recent ICSs ([178]) and corresponding $\mathrm{BE}f$-scaling results.

Figure 24

Electron impact on ${\mathrm{H}}_2\mathrm{O}$. (a) Typical electron energy-loss spectrum $\mathrm{\Delta }E\sim 6–11\mathrm{eV}$ for electronic-state excitation. The incident electron energy is 200 eV and the scattered electron angle is 3°. (b) GOS vs $K^2$ for the $A{{}^{1}B}_1$ electronic state. A Vriens-type fit ([330]) to the 100 and 200 eV data ([319]) is also shown. (c) ICSs for excitation of the $A{{}^{1}B}_1$ electronic state. (b), (c) See the legends and the text for further details.

Figure 25

Electron impact on ${\mathrm{BF}}_3$. (a) GOS vs $K^2$ for excitation of the $1e^{\prime \prime }(\pi )\to 2a_2^{\prime \prime }({\pi }^{*})$ transition from [98]. See the legend and the text for further details. (b) Integral cross sections (${10}^{-16}{\mathrm{cm}}^2$) for excitation of the ${\pi }^{*}$ transition. Measured data (•) and the unscaled Born (dashed curve) and $\mathrm{B}\mathrm{E}f$-scaling results (solid curve) are all from [98].

Figure 26

Electron impact on ${\mathrm{C}}_6{\mathrm{H}}_6$. (a) Typical electron energy-loss spectrum at an impact energy of 100 eV and scattering angle of 5°. The energy-loss range is $\mathrm{\Delta }E\sim 4.5–14\mathrm{eV}$. From [172]. (b) GOS vs $K^2$ for the ${{}^{1}E}_{1u}$ electronic state. A Vriens-type fit ([330]) to the available experimental data ([172]) is also shown. (c) ICSs for excitation of the ${{}^{1}B}_{1u}+{{}^{3}E}_{2g}$ electronic states. See the legend and the text for details. (d) Similar to (c) but for the ${{}^{1}E}_{1u}$ electronic state.