Low-energy electron-impact ionization of helium

E. Schow, K. Hazlett, J. G. Childers, C. Medina,* G. Vitug, I. Bray, D. V. Fursa, and M. A. Khakoo Department of Physics, California State University, Fullerton, California 92834, USA Centre for Atomic, Molecular and Surface Physics, School of Mathematical and Physical Sciences, Murdoch University, Perth 6150, Australia Received 7 August 2005; revised manuscript received 3 October 2005; published 27 December 2005


I. INTRODUCTION
Helium has been a popular target for electron scattering studies, including pioneering studies of fundamental electron scattering processes, e.g., elastic ͓1͔ and inelastic ͓2͔ scattering, resonant scattering ͓3͔, electron-photon coincidence ͓4͔, and ionization ͓5͔.Helium is frequently used as a calibration standard in experimental electron collision studies, since it is an inert gas that is easily handled in vacuum systems.Accurate elastic and inelastic differential cross sections ͑DCSs͒ for electron scattering from helium exist due to the considerable effort on the part of both theory and experiment.In the past decade, the convergent close-coupling ͑CCC͒ method has been applied with great success to the calculation of electron-impact excitation of atomic helium ͑see Fursa and Bray ͓6͔,Bray et al. ͓7͔,Fursa and Bray ͓8͔,and Röder et al. ͓9͔͒.The CCC method has also been applied to the calculation of electron-impact ionization of helium ͑Röder et al. ͓10͔ and Bray and Fursa ͓11͔͒.Generally, the agreement with experiments here is excellent considering the difficulty in both the theory and experiment.However, some discrepancies remain between experimentally determined doubly differential cross sections ͑DDCS͒ and the results of the CCC theory at the lower energies.
Very recently, two additional models, namely the timedependent close-coupling theory ͑TDCC͒ of Pindzola et al. ͓12͔ and the exterior complex scaling model ͑ECS͒ of Horner et al. ͓13͔, have been applied to electron-helium scattering.However, as yet we are unaware of the ECS and TDCC approaches being applied to calculate electron-helium differential ionization cross sections.We note here that the CCC and ECS methods have been successful in modeling the elec-tron impact of atomic hydrogen, and excellent agreement between theory and experiment over a wide range of conditions has been observed ͑see, e.g., Childers et al. ͓14͔͒.
Triply differential ionization cross-section measurements ͑TDCS͒ have been carried out by the Kaiserslautern group of Ehrhardt and co-workers ͓15-19͔ for the ionization of helium at a range of incident energies ͑E 0 ͒.Jones et al. ͓20͔, using the distorted wave Born approximation, showed some agreement with the equal energy-sharing TDCSs even for near-threshold ionization.The distorted partial wave method of Pan and Starace ͓21͔ yields TDCSs only for the geometry of back-to-back outgoing electrons.This approach gives excellent agreement with such experimental TDCS taken at E 0 values of 32.6, 44.6, 64.6, and 104.6 eV ͓19͔.Most recently, the CCC method has been applied extensively to e-He ionization and was found to give excellent quantitative agreement with experiment for all incident energies and geometries considered ͑Stelbovics et al. ͓22͔͒.
There have been several measurements of doubly differential ionization cross sections ͑DDCSs͒ for the electronimpact ionization of helium.Pichou et al. measured nearthreshold DDCSs for helium at incident energies of 25.4, 26.0, 27.0, 28.2, and 30.6 eV for scattering angle ͑͒ values of 10°-110°.The DDCSs of Pichou et al. were normalized to the elastic electron scattering differential cross sections of helium from Andrick and Bitsch ͓25͔ using the helium ionization continuum symmetry properties ͓i.e., flatness of the continuum energy-loss spectrum for residual energies ͑E R ͒ between 0.2 and 0.7 eV for an incident energy of 25.38 eV͔ to correct for their spectrometer transmission, extending the measurements piecewise to higher E R values using higher E 0 settings.Background determinations in the Pichou et al. experiments were accomplished by an ad hoc method of constructing a linear background joining the baseline between the n = 3 and 4 energy-loss features and extrapolating it to the end of the ionization continuum.This method has been shown in the present set of experiments to be unreliable; it yields systematically higher DDCSs by including unresolved secondary electron backgrounds.The extensive DDCSs of Röder et al. ͓24͔ were taken at E 0 values of 28.6, 32.6, 40, 50, 64.6, 100, and 200 eV and scattering angles of 20°-145°.Their data were relatively normalized ͑via measurement of the total helium ion current in the experiment͒ to the total ionization cross sections of Shah et al. ͓26͔ and then absolutely normalized to the E 0 = 100 eV DDCSs of the CCC theory ͓27͔.However, there remain significant disagreements between the most recent CCC results ͓28͔, Röder et al. ͓24͔,and Pichou et al. ͓23͔. For example,at 32.6 eV,Röder et al. observe good qualitative agreement between their DDCS angular distributions and those of CCC, but there are consistent quantitative disagreements between their results and CCC.The near-threshold energy region is important in that the outgoing electrons experience large correlations in this energy regime as they slowly recede from the core He + ion.Also, low-energy electron collisions are important in plasma environments such as planetary atmosphere physics and man-made industrial plasmas including microelectronics and lighting.
Additionally, for reference, DDCSs taken at E 0 Ͼ 40 eV ͑outside the range of this work͒ have been measured in the past by Opal et al. ͑100-2000 eV ͓29͔͒, Oda ͑500 eV ͓30͔͒, Rudd and Dubois ͑100 and 200 eV ͓31͔͒, Shyn and Sharp ͑50,100,200,and 300 eV ͓32͔͒,200,300,400,500,and 600 eV ͓33͔͒,and Rösel et al. ͑100,200,300,400,500, and 600 eV ͓34͔͒.Based on the need for reliable quantitative low-energy electron-impact DDCSs for helium, we have measured new DDCSs for helium at E 0 values of 26.3, 28.3, 30.3, 32.5, 34.3, 36.5, and 40.7 eV.Our measurements are compared to the corresponding DDCSs of the CCC theory and available experiments.

A. Experiment
The present measurements were carried out on two instruments, both of which have been discussed in previous publications.The first was equipped with a double hemisphere electron monochromator and double hemisphere electron analyzer that normally operates with an energy resolution of 40 meV or below, but was operated at a degraded resolution of 100 meV with a current of 100-120 nA ͑by opening the central slit between the hemispheres from 1 mm width to 2.5 mm and raising the sphere pass voltages from typically 2 to 5 eV͒.This instrument has been described in our recent experiments with argon and molecular nitrogen ͑see Khakoo et al. ͓35͔ and Khakoo et al. ͓36͔͒ and was used for initial measurements at E 0 = 26 and 28 eV.The second spectrometer was a single hemisphere device in both the monochromator and the analyzer, but the system was equipped with a microwave discharge atomic hydrogen source.This apparatus has been described in Childers et al. ͓37,38͔.This spectrometer operated with a current of 50-100 nA with a resolution of Ϸ140 meV.In both systems, the electron beam crossed an effusive source of gas emitted by a capillary needle source of diameter ranging from 0.5 to 0.8 mm.Surfaces surrounding the collision region ͑including the needle͒ were heavily sooted with an acetylene flame to suppress the production of secondary electrons by the primary electron beam colliding with these surfaces.In both spectrometers, the depth of field of the detector was restricted to a small region around the collision center ͑5 -6 mm region͒ by using an additional pupil in the input electron lens stack of the detector.Both spectrometers were baked to greater than 120 °C to maintain stability of the electron beam and the detector.Both instruments were housed in double mu-metal shields to reduce the ambient magnetic field at the collision region to below 5 mG.
Both systems were equipped with a moveable gas target source ͓39͔.Using the moveable source, it is possible to obtain background-free electron-energy-loss spectra for both discrete and continuum features.To ensure that the movable source did not affect the electron beam, the tip of the needle was kept at least 4 mm away from the center of the collision region, and variations of the transmitted electron beam were monitored, using an electron collecting plate arranged downstream of the collision region, as the needle was moved into and out of alignment with the incident electron beam.No variations were observed when the electron beam was placed ͑by spectrometer deflection plates͒ at the collision region center.At small scattering angles, we observed an additional source of secondary electrons possibly due to the electron beam slightly changing size ͑due to space-charge neutraliza-tion͒ at the collision region when the gas beam was moved into and out of alignment.This effect produced an additional source of secondary electrons from the analyzer shielding plates ͑for E R below 1 eV͒ that could not be systematically removed by subtracting the electron-energy-loss spectra with the gas in and out of alignment with the incident electron beam.This added an additional 15% uncertainty to the signal for these low E R electrons.
We calibrated the transmission of our analyzer as follows.First, the detector transmission was made as uniform as possible for electrons with different E R values by tuning the analyzer and monitoring the flatness of the helium continuum at E 0 Ϸ 30 eV.The fact that the helium continuum energy-loss spectrum is flat was established by Pichou et al. ͓23͔, and was used successfully in previous work by our group in atomic hydrogen ͓14͔.At larger E R , this transmission was found to be stable during the course of measurements as long as the analyzer was not retuned.However, closer to zero E R this uniform transmission could not be achieved.Accounting for the analyzer transmission at small E R values was found to be the most difficult problem in this experiment, resulting in increased error bars at these small E R values.We recalibrated the overall transmission as follows: Atomic hydrogen spectra were obtained ͑using our microwave source͒ in the manner described in Childers et al. ͓38͔ and the background-free atomic hydrogen continuum energy-loss spectrum was normalized to the exterior complex scaling ͑ECS͒ theory ͓40,41͔, which was previously established as accurate by the experimental work of Childers et al. ͓38͔.By way of obtaining background-free atomic hydrogen spectra, we also obtained discharge-off background-free molecular hydrogen spectra in the same way as ͓38͔.These H 2 spectra were calibrated for transmission using the transmission obtained from the atomic hydrogen spectra at the same E 0 and from the ECS theory.This transmission-corrected H 2 spectrum was used to calibrate the helium energy-loss spectrum by a relative flow method ͓42͔, where both He and H 2 were passed through the capillary with approximately the same mean free path ͑i.e., at drive pressures inversely proportional to the square of their molecular diameters, 2.18ϫ 10 −8 cm for helium and 2.74ϫ 10 −8 cm for H 2 ; see Rugamas et al. ͓43͔͒, so that the collision region geometry did not affect the transmission.These calibrations were undertaken at E 0 = 30 eV for = 60°and 90°and for E 0 = 40 eV for = 50°and 90°.However, the best reproducibility for this calibration was obtained for the full ionization continuum at E 0 = 30 eV for = 90°, which was adopted as our most accurate determination of the transmission calibration ͑to approximately 8% on a relative scale͒, and for E R Ͼ 10 eV at E 0 = 40 eV, which is more than half of the continuum at this incident energy.Using the above calibrations, we were able to accurately correct for the transmission of our detector.Finally, the spectrometer transmission was extrapolated into the He ͑n =2͒ energy-loss region using a polynomial of order ഛ2, to enable us to normalize the energy-loss spectrum to the electron impact DCS for excitation of the He ͑n =2͒ energy-loss feature in our electronenergy-loss spectra.
In general, many electron-energy-loss spectra of helium were taken ͑usually at least three spectra per initial value of E 0 and , and in some cases up to 10, especially for the = 90°spectra, which were used to check the instrumental transmission͒ spanning the energy-loss range of 19 eV to E 0 + 1 eV and covering the n = 2 features.Sample helium energy-loss spectra showing the moveable source background removal method are shown in Figs.1͑a͒ and 1͑b͒.The = 90°spectra taken at E 0 = 30 eV served to determine the transmission characteristics of the analyzer using the H-source calibrations discussed above and these calibrations were used to recalibrate spectra taken at other angles during the same electron beam and gas beam conditions.To determine the continuum DDCS from a given energy-loss spectrum, we normalized the transmission-corrected spectrum ͓extrapolated into the He ͑n =2͒ energy-loss region via a polynomial of order ഛ2͔ to the experimental He ͑n =2͒ DCSs ͑summed DCSs for the electron-impact excitation of the 1 1 S → 2 3 S, 2 1 S, 2 3 P, and 2 1 P transitions͒ measured by Hall et ͓48͔, which were in excellent agreement in most cases to within 15% and with a conservative average standard deviation of 12%.For incident energies not covered in the previous He ͑n =2͒ DCS measurements, the DCS data were interpolated.The interpolated values agreed well with the CCC He ͑n =2͒ DCSs at lower values, but deviated from the CCC by as much as 16% for close to 130°at E 0 values near 30 eV.These helium ͑n =2͒ DCSs are given in Table I.Our DDCS were obtained from the transmission-corrected energy-loss spectra by the formula where N͑E, continuum͒ is the height of the continuum ͑number of electron scattering events͒ at the position E ͑energy-loss͒ in the continuum, ⌬E ͑typically set to Ϸ0.0217 and 0.0407 eV for the double hemispherical and single hemispherical setups, respectively͒ is the energy step width per channel in the energy-loss spectrum, N͑He, n =2͒ is the total number of electron scattering events under the He ͑n =2͒ energy-loss features, and d / d⍀ is the He ͑n =2͒ DCS from Table I.The value of N ͑E,continuum͒ was determined by fitting the continuum to a polynomial series in E of order ഛ2.The calibration of the spectrum was dependent upon an accurate determination of the value of ⌬E.This value was determined from the energy-loss spectra in which the start energy-loss and end energy-loss values were recorded.The location of the helium 2 3 S feature in the spectrum at 19.814 eV energy-loss served to calibrate E for the whole spectrum.The linearity of the energy-loss scale ramp voltage supply was determined in a separate experiment and was found to be Ϸ0.2% of the energy-loss covered in the full energy-loss spectrum.The incident energy of the electron beam was determined from the spectrum by using the cutoff energy-loss value ͑see Fig. 1͒ of the continuum.This method served to determine E 0 to an accuracy of ±0.12 eV.The quoted E 0 values in our DDCSs are an average of these measurements.
Unfortunately, using the polynomial series in E of order ഛ2 to extrapolate the instrumental transmission determined for the He ionization continuum into the He ͑n =2͒ energyloss region, a region inaccessible to our calibration techniques, resulted in a systematic error.To compensate for this error, the ͑relative͒ DDCSs were extrapolated to = 0°and 180°and integrated over to obtain singly differential cross sections ͑SCDSs͒ or d / dE͑E 0 , E͒.These SDCSs were then multiplied by factor 1/2 ͑to account for the scattered electron only͒ and integrated over E to obtain the total ionization cross section.This enabled us to normalize our results to the well-established and accurate ͑±3% uncertainties͒ total ionization cross sections measured by Shah et al. ͓26͔.In the determination of our SDCSs, we extrapolated our DDCSs to small and large scattering angles using the form of the CCC.The error in this extrapolation was estimated by repeating the integration, but instead employing flat DDCS extrapolations to = 0°and 180°from the end points of the angular distribution of the DDCSs.The error estimate was determined from the difference of the two extrapolations and added in quadrature with the mean error of our DDCSs.

B. Theory
The details of the application of the CCC theory to calculating differential e-He ionization cross sections have been recently discussed by Bray et al. ͓28͔ and Stelbovics et al. ͓22͔.Briefly, the total electron-He wave function is expanded using a set of square-integrable target states with the resulting coupled integral equations solved in momentum space.The target states may be obtained using Laguerre-or boxbased one-electron orbitals.These orbitals are used to define a configuration-interaction ͑CI͒ expansion.For dominant one-electron transitions, we find that the CI expansions need not include two-electron excitations, and so we may restrict one of the He electrons to remain in the 1s orbital of the He + ion while maintaining the required symmetries.Ionization amplitudes in the CCC theory are obtained directly from excitation amplitudes of the positive-energy target states ͓11͔.The numerical parameters used in the present calculations are much the same as used earlier ͓28͔, except they were performed at the kinematics of the present experiment.

B. Singly differential cross-sections
Figures 7͑a͒-7͑f͒ show our normalized SDCSs compared to the CCC calculations.As mentioned before, these experimental SDCSs were normalized to the total cross sections of Shah et al. ͓26͔.In all cases, excellent agreement with CCC is observed, albeit within the large error bars of our SDCSs.We note that for all E 0 values, our SDCSs do not show the curved profiles displayed by CCC, but instead preferentially show a flat profile.We also note the increased uncertainties at small E R due to uncertainties in the DDCS extrapolations during the integration of these data to yield SDCSs ͑see Sec.II for discussion͒.

C. Shape of DDCSs at = 90°T
he threshold ionization process has been treated theoretically in several different approaches.In the well-known work of Wannier ͓49͔, which was later extended by Vinkalns and Gailitis ͓50͔, it is assumed that in the inner reaction zone the trajectories of the outgoing electrons fill the available phase space with a smooth and nonsingular probability; when outside this reaction zone, the electron trajectories are classical.One of the predictions of this theory including several quantum-mechanical models ͑see, e.g., Rau ͓51͔͒ is that the kinetic energy distribution of the electrons is uniform.Detailed studies of the dynamics of the Wannier nearthreshold behavior were performed by Cvejanović and Read ͓52͔.They showed that at the E 0 value of 25.4 eV ͑0.8 eV above threshold͒, for the ionized electrons emitted coincidentally in directions of 90°and 270°from the incident electron direction, the energy partitioning between the two outgoing electrons was uniformly distributed within E R .This was further investigated by Pichou et al. ͓23͔ using ad hoc background-corrected continua electron-energy-loss spectra with energies up to 6 eV above threshold.Their results showed a reasonably uniform distribution of ejected electron energy partitioning at = 90°for E 0 up to 30.6 eV.This important observation by Pichou et al. has often been used to calibrate the transmission of electron spectrometers for scattered electrons ͑see Nickel et al. ͓42͔͒.In Fig. 8, we show our DDCSs at 90°for all our E 0 values in this experiment and compare these to the results of the DDCSs of Pichou et al., Röder et al. and CCC.We note here that the smoothness of our results is due to the averaging of the polynomial fits to our DDCS data, as discussed earlier.
We note several trends in the DDCSs.First, we are only able to reliably go to an E R minimum of 0.5 eV at low E 0 values.Secondly, the DDCSs remain essentially flat up until E 0 = 32.5 eV within a deviation of 5%, and then progressively slope downward as a function of E R at higher E 0 values.For example, at E 0 = 40.7 eV ͓Fig.8͑f͔͒, the DDCSs drop by about 15% as E R goes from 1.0 to 15.0 eV.This drop is almost linear.Agreement with the results of Pichou et al. and Röder et al. is mixed at low E 0 values below 30 eV, but improves as E 0 increases above this.In general, the shapes stay in very good agreement within the 15% error bars at low E 0 , but improve to within 8% at 40 eV.The DDCS taken at E 0 = 40.7 eV significantly extends the use of the helium continuum and discrete energy-loss spectrum for spectrometer transmission calibrations to an E R range of 1.0 eV up to approximately 20.0 eV ͑1 1 S → 2 3 S line͒, which is almost double the range at E 0 = 30.6eV provided by Pichou et al.We note also that at E 0 = 30.3eV, the results of CCC and Pichou et al. ͓23͔ show a significant increase ͑Ϸ50% ͒ in the DDCS for E R Ͻ 1; this is also suggested by our results, albeit with larger error bars.The Röder et al. data show this increase in all their 90°data.This feature needs to be investigated in detail in future studies.

IV. CONCLUSIONS
We have systematically determined quantitative DDCSs from background-corrected electron-He energy-loss spectra at energies from 1.7 eV above threshold to 16.1 eV above threshold.This was made possible by our moveable source method.Our results show very good quantitative agreement with the present CCC calculations and are expected to be an improvement over earlier measurements.In most cases, qualitative agreement with the DDCSs of Röder et al. ͓24͔ is very good, as is the agreement with the DDCSs of Pichou et al. ͓23͔, though not always quantitatively.At the highest E 0 and larger E R we expect our DDCSs to suffer the least from systematic errors and to be a good place to test theoretical models.At small E R , difficulties in quantitatively reproducing the He continuum spectrum, especially at small , due to larger secondary electron backgrounds lead to increased uncertainties in the results and prevent the present experiment from providing very precise data to test theory in this important range.This problem will be investigated in depth in future experiments.

FIG. 7 .
FIG. 7. Present normalized helium SDCSs.The present SDCSs were normalized to the total ionization cross sections of Shah et al. ͓26͔.See text for details.͑a͒ Legend: b, present work at E 0 = 26.3eV; -, CCC at E 0 = 26.0eV.͑b͒ Same as ͑a͒ but for the present work at E 0 = 28.3eV and CCC at E 0 = 28.0 eV.͑c͒ Same as ͑a͒ but for the present work at E 0 = 30.3eV and CCC at E 0 = 30.0eV.͑d͒ Same as ͑a͒ but for the present work at E 0 = 32.5 eV and CCC at E 0 = 32.0 eV.͑e͒ Same as ͑a͒ but for the present work at 36.5 eV and CCC at E 0 = 36.0eV.͑f͒ Same as ͑a͒ but for the present work at E 0 = 40.7 eV and CCC at E 0 = 40.0eV.

TABLE I .
DCSs for the summed excitation of the n = 2 levels of helium used in this work.Uncertainties for DCSs at E 0 of 22, 24, 30, and 40 eV are ±12% ͑one standard deviation͒.For other E 0 values ͑26, 28, 32, 34, and 36 eV͒, which were interpolated from the above E 0 values using the CCC as a guide, the uncertainties are ±15%.Units are in 10 −19 cm 2 sr −1 .See text for discussion.