Elastic resonant and nonresonant differential scattering of quasifree electrons from ${\mathrm{B}}^{4+}\left(1s\right)$ and ${\mathrm{B}}^{3+}\left(1s^2\right)$ ions

High-resolution doubly differential cross-section measurements and calculations for quasifree electrons elastically scattered through 180° from ground-state He- and H-like boron ions are presented. The measurements, covering the entire $1s2ln{l^{\prime }}^{}$(for ${\mathrm{B}}^{3+}$) and $2ln{l^{\prime }}^{}$(for ${\mathrm{B}}^{4+}$) Rydberg series populated by resonant excitation, were performed by zero-degree Auger projectile electron spectroscopy of $3.91\mathrm{MeV}{\mathrm{B}}^{4+}$ and $3.08–7.48\mathrm{MeV}{\mathrm{B}}^{3+}$ ions in collisions with ${\mathrm{H}}_2$ targets. The projectile energy dependence study is used to further investigate other background contributions such as direct electron capture, direct excitation, and nonresonant transfer excitation. $R$-matrix calculations, particularly sensitive at this large scattering angle to electron correlation, exchange, and interference effects, are found to be in excellent overall agreement within the electron scattering model.

Figure 1

(Color online) Schematic of the zero-degree Auger projectile electron spectroscopy setup at the JRML. The ion beam interacts with the gas target, confined by the gas-cell, transverses the spectrograph, and exits through a hole at the back. The ion beam current is recorded at the Faraday Cup for normalization purposes. The electrons emitted from the target area are focused and decelerated by the four-element lens into the hemispherical deflector analyzer (HDA) entry. After a deflection through 180°, they are detected at the two-dimensional position sensitive detector (2D-PSD).

Figure 2

(Color online) Zero-degree laboratory $\left(L\right)$ electron spectra obtained in $4.00\mathrm{MeV}{\mathrm{B}}^{3+}\left(1s^2\right)+{\mathrm{H}}_2$ collisions. The data were recorded after overlapping five different spectra segments of progressively larger tuning energies $W$. A deceleration factor of $F=4$ was used for all spectra.

Figure 3

(Color online) Blow-up of $R$-matrix calculations showing the contribution of the three individual terms of Eq. (6) for 180° elastic electron scattering from ${\mathrm{B}}^{3+}\left(1s^2\right)$. Both interference and short-range terms are seen to have substantial nonresonance contributions.

Figure 4

(Color online) Comparison between different $R$-matrix cross-section calculations for 180° electron scattering from ${\mathrm{B}}^{3+}\left(1s^2\right)$. Continuous line: full calculation of $d{\sigma }_{if}∕d\mathrm{\Omega }$ [see Eq. (6)]. Dashed-dot line: $d{\sigma }_{if}∕d\mathrm{\Omega }$ but only for the monopole contribution, i.e., just the partial wave with $l=0$ in Eq. (8) and just $\mathrm{\lambda }=0$ in Eq. (9). The monopole part is seen to contribute a substantial part (but not all) of the cross section. The extra amount is due to the anisotropic, higher-order multipole contributions. Dashed line: short-range total cross section, ${\sigma }_{if}^{\mathrm{sr}}∕4\pi$ [see Eq. (11)].

Figure 5

(Color online) Effect of finite-energy resolution on the interference structures in $4.0\mathrm{MeV}{\mathrm{B}}^{3+}\left(1s^2\right)+{\mathrm{H}}_2$ for the region around the $1s2p^2{}^{1}D$ state. Continuous line: Actual experimental resolution used in this work. Discontinuous lines: Other resolutions.

Figure 6

(Color online) Calculated SDCS for 180° electron scattering from ${\mathrm{B}}^{4+}$ using the formulation of Ref. 6 with $Z=q=4$ and the Hartree-Fock Auger rates and widths given in Table . Dashed-dot line: Rutherford term [Eq. (7)]; dotted line: resonance term; short dashed line: interference term [includes contributions from both $C_I$ and $A$ terms in Eq. (5) of Ref. 6]; and continuous line: sum of all three terms. The resonance and interference terms are seen to have negligible nonresonant contributions as opposed to the full $R$-matrix calculations based on Eq. (6).

Figure 7

(Color online) (Bottom) Comparison of the calculation shown in Fig. 6 and our $R$-matrix calculation for 180° electron scattering from ${\mathrm{B}}^{4+}$. (Top) The DDCS spectrum is compared to the two calculations which were converted to DDCS using the electron scattering model formulation and after further convolution with the spectrometer resolution. The overall discrepancy between the two theoretical results is evident (see discussion in text).

Figure 8

(Color online) Projectile rest frame absolute electron DDCS for the $21.32\mathrm{MeV}{\mathrm{F}}^{8+}+{\mathrm{H}}_2$ collisions system, near the $2p^2{}^{1}D$ resonance. Present $R$-matrix calculations, shown by the solid line, are seen to be in excellent agreement with the measurement.

Figure 9

(Color online) Resonance excitation collision strengths ${\mathrm{\Omega }}_{\mathrm{RE}}$ plotted as a function of atomic number $Z$ for the $2p^2{}^{1}D$ resonance taken in collisions of H-like ions with ${\mathrm{H}}_2$ targets. Open squares: present measurements; solid squares: values extracted from the older data of Toth et al. [58] taken with a two-stage slit spectrometer; line: Hartree-Fock calculation. Good agreement is seen for the common point of fluorine at $Z=9$.

Figure 10

(Color online) (Top) Data: Projectile frame DDCS electron spectra for $3.91\mathrm{MeV}{\mathrm{B}}^{4+}+{\mathrm{H}}_2$ collisions. Solid line: ESM-$R$-matrix calculations [Eq. (3) and bottom] are seen to be in excellent agreement with the measurement. (Bottom) SDCS $R$-matrix calculations for 180° elastic electron scattering from ${\mathrm{B}}^{4+}\left(1s\right)$ ions, plotted as a functions of the incident electron energy. The ${\mathrm{B}}^{3+}\left(2ln{l}^{\prime }\right)$ resonances are seen superimposed on top of the broad nonresonant continuum.

Figure 11

(Color online) Same as in Fig. 10, but for $4.00\mathrm{MeV}{\mathrm{B}}^{3+}+{\mathrm{H}}_2$ collisions. Now the ${\mathrm{B}}^{2+}\left(1s2ln{l}^{\prime }\right)$ resonances produced by resonant electron scattering are seen to be superimposed on top of the broad nonresonant continuum (bottom). The presence of the $1s2s2p{}^{4}P$ and $2s2p{}^{3,1}P$ lines in the data is due to capture and excitation processes with the small contaminant $(\lesssim 3\%)$ of $1s2s{}^{3}S$ metastable ions not included in the $R$-matrix calculations.

Figure 12

(Color online) Energy dependence study of the $KLL$ resonances produced in collisions of ${\mathrm{B}}^{3+}$ ions with ${\mathrm{H}}_2$ gas targets. ESM–$R$-matrix calculations (solid lines) for $e^{-}+{\mathrm{B}}^{3+}\left(1s^2\right)$ scattering are seen to reproduce very well the $1s2p^2{}^{2}D$ resonance. The nonresonant contributions have been subtracted. The discrepancies observed around the ${}^{2}P$ resonances arise from $2p$ electron capture to the small amount of $1s2s{}^{3}S$ component of the beam (see text).

Figure 13

(Color online) Same as in Fig. 12 but for the $KLn(n>2)$ spectral region. The agreement between the ESM–$R$-matrix calculations and the measurements is extremely good except at the lowest collision energy of $3.08\sim \mathrm{MeV}$. Unaccounted peaks are due to inelastic contributions primarily formed by $1s\to 2p$ excitation of the $1s2s{}^{3}S$ metastable component of the beam, as, for example, those present at electron energies of $187.5\left(2s2p{}^{3}P\right)$ and $193.5\left(2s2p{}^{1}P\right)\mathrm{eV}$.

Figure 14

(Color online) Experimental data (open circles) and ESM–$R$-matrix calculations (solid lines) for $4\mathrm{MeV}$ ${\mathrm{B}}^{3+}+{\mathrm{H}}_2$ collisions in the $KLL$ spectral region. [Top] Metastable fraction $f_{{}^{3}S}=25\%$. [Bottom] $f_{{}^{3}S}<3\%$. The enhancement of the ${}^{4}P$ and ${}^{2}P$ lines in the case of $f_{{}^{3}S}=25\%$ is due to the capture of a $2p$ electron to the $1s2s{}^{3}S$ ion core. For both spectra the electron analyzer was used with the same identical parameters. In the top spectrum, however, the lines are seen to be slightly broadened due to energy straggling of the boron ions in the carbon post-stripper foil utilized to produce a high metastable content. No post-stripping was used in the bottom spectrum. A spectrometer resolution of $0.59\mathrm{eV}$ and $0.35\mathrm{eV}$ was used, for the top and bottom spectrum, respectively.

Figure 15

(Color online) Scaling of Coulomb DDCS: The IA-BEe calculations for bare boron $\left({\mathrm{B}}^{5+}\right)$ computed from the Coulomb formula [Eq. (7)] (short dashed line) are multiplied by the enhancement factor $\mathcal{R}=1.50$ (dashed line) to match the $R$-matrix results (solid line) over the nonresonant part of the DDCS spectrum.

Figure 16

(Color online) Boron energy level diagram: RES from ${\mathrm{B}}^{4+}\left(1s\right)$ leads to ${\mathrm{B}}^{3+}\left(2l2l^{\prime }\right)$ doubly excited states and from ${\mathrm{B}}^{3+}\left(1s^2\right)$ leads to ${\mathrm{B}}^{2+}\left(1s2ln{l}^{\prime }\right)$ doubly excited states. The quasifree electron “beam” spectral profiles in $3.91\mathrm{MeV}{\mathrm{B}}^{3+}$ and $4.00\mathrm{MeV}{\mathrm{B}}^{4+}$ (shaded areas) have a mean electron “beam” energy [see Eq. (2) with $p_z=0$] of $⟨\epsilon ⟩=179.5$ and $184\mathrm{eV}$, respectively, above each of the initial states with a broad FWHM of more than $100\mathrm{eV}$, seen to overlap many resonances. The corresponding Auger electron energies for transitions to the ground state are given on the right-hand scale.