Magnetic field-free measurements of the total cross section for positrons scattering from helium and krypton

An electrostatic beam has been used to perform scattering measurements with an angular-discrimination of ≲ 2 ° ?> . The total cross sections of positrons scattering from helium and krypton have been determined in the energy range (10–300) eV. This work was initially stimulated by the investigations of Nagumo et al (2011 J. Phys. Soc. Japan 80 064301), the first positron field-free measurements performed with a similarly high resolution, which found significant discrepancies at low energies with most other experiments and theories. The present results show good agreement with theories and several other measurements, even those characterized by a much poorer angular discrimination, implying a small contribution from particles elastically scattered at forward angles, as theoretically predicted for He but not for Kr.

The opposite signs of the static and polarization interactions for positrons tend to reduce their scattering probability at low energies in comparison to electrons (e.g. [15]), despite the presence of extra channels (annihilation and Ps formation) available to positrons in encounters with matter. Annihilation is considered generally negligible, except in the limit of zero velocity relative to that of an electron (e.g. [16]) or by attachment (e.g. via vibrational Feshbach resonances [2,17]). Ps formation, however, may account for up to ∼50% of the total scattering probability and be a significant channel even when formation occurs in an excited-state [18] or accompanied by excitation of the residual ion [19]. Polarization often enhances positron-impact direct ionization (e.g. [20]), so that the total ionization cross-section by e + may exceed that by electrons [3].
The present work was initially stimulated by the total cross section results for positrons colliding with helium [21] and neon [22] determined for the first time in the absence of a magnetic field, employed for beam transport in all previous experiments. Significant discrepancies were found between (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) eV with earlier results and, given that in transmission experiments, a major source of systematic error arises from ascribing particles elastically scattered to small forward angles (FSP) to the unscattered beam, it was noted that the high (energy-independent) angular resolution associated with electrostatic systems should help to minimize this effect.
In the current experiment, the total cross sections for positron scattering from helium and krypton have also been performed in a field-free region using an electrostatic beam transport. Measurements have been obtained in a range of energies between (10-300) eV and are here compared with previous experimental and theoretical results.

Experimental apparatus
The equipment used for this experiment is illustrated in figure 1 and has been described in detail elsewhere [23]. Briefly, positrons from a 22 Na source are moderated by a stack of three annealed tungsten meshes (20 m m wire and 70% transmission) [24]. A set of primary lenses transports the beam at 2 keV from the moderator and focus it to a small beam spot of radius ∼1 mm at the re-moderator. This is an annealed W 100 ( ) foil (thickness 50  nm) with a remoderation efficiency of 0.1 [23]. The re-moderator is floated at a potential (V Rm ) to accelerate the positrons to the required beam energy (E eV Rm | | f = + + ), where 2.7 0.1 f = - eV is the positron work function for the current re-moderator [23] and e is the positron electric charge. The positrons are then transported around a 90 • corner through a cylindrical mirror analyser before reaching the interaction region.
An aluminium cylindrical cell is situated after the exit lenses. It has a length of L=53 mm and inner radius 25.4 mm. Two aperture radii were used (R 0.5 mm a = and 1 mm for He and Kr, respectively) in order to retain a high angular discrimination and a sharp gas density profile [25]. In front of the detector, two grids are mounted that enable retarding potential analysis (RPA) and are also used, during the total cross section measurements, to reflect inelastically forward scattered particles. The beam has an angular divergence of 1°and longitudinal energy spread of 1% [23]. The angular acceptance is set by geometrical constraint 3 A comparison between some of the characteristics of the current and previous experimental set-ups is made in table 1.
The beam is also equipped with a time of flight (TOF) device which is started by secondary electrons released at the re-moderator (detected by an off-axis channel electron multiplier) and ended by the PSD signal. A timing efficiency of approximately 10% has been obtained together with a resolution of 6  ns [23]. The PSD signal may be set in coincidence with that from the TOF (centered on the arrival time of the incident beam) to obtain a timed PSD distribution. This method reduces the random background to essentially zero.

Experimental method
The total cross section, T s , is determined using the Beer-Lambert law:

k T Pl
where I 0 and I are the incident and transmitted (unscattered) beam intensities respectively, P and T are the gas pressure and temperature respectively, l is the length of the positron path through the gas and k B the Boltzmann constant.
The pressure in the gas cell was measured with a baratron (MKS 627D capacitance manometer), temperature stabilized to 45°C. A thermal transpiration correction ( 3%  ) has been applied using the method described in [43], ambient temperature being on average 17 1 ( ) °C. A range of gas pressures (0.1-1 Pa for Kr and 4-10 Pa for He) was used to verify pressure independence of the final cross section values, as expected from equation (1) and as seen in figure 2.
In order to ensure similar positron-scattering probabilities outside the cell (estimated to be 1.5% < ), the I 0 measurements were conducted under the same vacuum conditions as the I measurements by leaking gas into the system through a bypass gas line. The system pressure was monitored throughout the runs using an ion gauge above the remoderator chamber. Consecutive I and I 0 measurements were made, and the average of I 0 on either side of I (and vice versa) was used to calculate the total cross-section. Measurements of the background were frequently repeated throughout the runs by biasing off the beam with the repelling grid in front of the detector. To minimize possible contaminants introduced into the cell with the gases under investigation (nominal purity: 99.9995% for He and 99.9999% for Kr), the gas lines were periodically baked at temperature of 60 C   and repeatedly flushed. A quadrupole mass spectrometer was used to monitor the effective gas purity in the system which was found to be 99.8% > throughout the measurements.

Data analysis
In order to discriminate against the possible detection of FSP, the variation of the computed T s was examined versus the radius (r) of the beam spot on the PSD. The beam center was determined in two ways: the weighted center of the intensity distribution on the PSD and the center of the timed PSD distribution. Calculation of the total cross section using each center gave results in agreement to within 1% in all cases except at 15 eV in Kr where 5% applies.
The cut-off radius R c ( ) of the beam intensity distribution has been investigated using the cumulative radial profiles for I and I 0 after background subtraction (the 'cumulative' profiles are obtained by considering concentric circular regions and summing up the count rates as the radius of the region is increased). An example for a specific run is shown in figure 3, including the variation of the cross section with r. At low radii, large fluctuations in the total cross section can be seen which decrease with increasing r, as this approaches the beam edge. The origin of the fluctuations is partly statistical and partly systematic, the latter related to possible errors in the determination of the centers in each I and I 0 measurement. In this work, the cross section value with the smallest statistical error was chosen, the corresponding R c ( ) value agreeing with that of the measured I 0 profile, as illustrated in figure 3.
Using the theoretical differential elastic cross sections of [26,27] for He and [28] for Kr, it is possible to estimate the  [26,27], while for Kr the theoretical values in [28] have been used.  potential systematic errors due to elastic FSP on the total cross section measurements, theoretical predictions for He being in very good agreement with each other [26,28,44]. Table 1 shows a comparison among various experiments. Only the ANU group [41] has explicitly applied this type of correction to their measurements.

Results and discussion
The positron total cross section results for helium and krypton in the energy range (10-300) eV are shown, respectively, in figures 4 and 5 where they are compared with other available experiments and theories.
For He, at 30 eV and above, the present results are systematically larger than those of Griffith et al [33], Canter et al [32] and Kauppila et al [37]; they agree within errors with those of Nagumo et al [21] and Caradonna et al [45] except in the region 57-60 eV, where they are higher and in better accord with the measurements of Machacek et al [46] who searched for-but could not confirm-the resonances predicted to arise from the binding of a positron to doubly excited He [47].
In the range 10-20 eV, the present data disagree with those of Nagumo et al [21] and of Jaduszliwer et al [29] also  performed with a high angular discrimination and, to a smaller extent, of Karwasz et al [40]. They are in accord with those of the other experiments shown [31,36,38,42,48,49], despite the associated angular discrimination being lower in most cases. This finding supports the theoretical predictions [26,44,50] of a small contribution from elastically FSP in this energy range.
Included in figure 4 are the results of various theoretical calculations [44,[50][51][52]. Below the Ps formation threshold (17.79 eV), a good agreement is found between the current results and those of the close coupling calculation (CC) [51], the convergent close coupling calculation (CCC) [50] and Kohn variational method [44], the same level of agreement extending over the whole range investigated theoretically by the latter [44] and up to 50 eV for CCC [50]. At the highest energies, a systematic deviation ( 10% ) is observed from both CCC [50] and CC [51], the latter also being higher than experiments between (20-40) eV. The complex optical potential (COP) calculation of Baluja and Jain [52] is close to the present measurements both in shape and magnitude, except at its lowest (20 eV) and highest (300 eV) energy investigated.
For Kr above 30 eV, the present results are in good agreement with those of Makochekanwa et al [41], while they are systematically higher than those of Canter et al [32], Dababneh et al [34,35] and Zecca et al [39]. Below 15 eV, the data of Canter et al [32], Zecca et al [39] and Makochekanwa et al [41] are generally higher than the present results 4 while a good agreement is found with the results of Dababneh et al [34] and Jay and Coleman [42]; given their poorer angular discrimination, this might imply that the predicted corrections in table 1 are too large.
Theoretical calculations for Kr are also shown in figure 5. As in the case of He, a good agreement is found between the present results and the predictions of the COP approach of Baluja and Jain [52]. The accord with the CCC calculation [28,41] and with the complex scattering potential-ionization theory [53] is weaker.

Conclusion and outlook
We have presented new data for the total cross section of positrons scattering from helium and krypton in the range (10-300) eV obtained with a high (energy-independent) angular discrimination. For both targets, the present results do not show a systematic deviation from previous determinations obtained using magnetic transport and often poorer angular discrimination. We must thus conclude that residual discrepancies among experiments are probably due to other sources of systematic errors. Additionally, in the case of helium, in view of the significant increase in the total cross section between 50 and 57 eV (close to the energies where resonances are predicted to arise due to positron complexes [47]), further investigations may be worthwhile.
Finally, we hope to use the high angular resolution of the present system to investigate polar molecules such as water for which forward elastic scattering currently introduces severe experimental uncertainties [54] and, following a planned upgrade of the positron source, extend measurements to lower energies and to differential cross sections.