Studying Low energy (e,2e) from the noble gases in the perpendicular plane. Kate L Nixon, Andrew James Murray and Christian Kaiser. Abstract: In this web page we provide detailed (e,2e) studies of electron impact ionization of noble gases including helium, neon, argon, krypton and xenon from near threshold to intermediate energies, where the outgoing electrons carry equal energy from the interaction. The experiments were conducted in the perpendicular plane, where the outgoing electrons are detected orthogonal to the incident electron beam. For electrons to emerge in this geometry they must undergo multiple scattering including scattering from the nucleus of the target, and so this provides a highly sensitive test for scattering theories. The data show the cross sections undergo complex variations as a function of incident energy, and in particular ionization of the heaviest target is at variance to all others that have been studied here. 1.0 Introduction. The (e,2e) process provides the most precise experimental data for the study of the ionization of atomic and molecular targets by electron impact. In these experiments a single electron of well defined momentum impacts with a target in the interaction region, resulting in target ionization and a scattered and ejected electron which emerge after the interaction has taken place. By determining the momenta of the scattered and ejected electrons in a time correlated coincidence measurement, the differential ionization cross section (DCS) is derived for comparison with theoretical models. Since the scattered and ejected electrons may emerge over a wide range of different angles, experiments usually define a specific scattering geometry in which to carry out the measurements. Our (e,2e) apparatus can access a wide range of geometries from a coplanar geometry, where the incident, scattered and ejected electrons all occupy the same plane, through to the perpendicular geometry, where the scattered and ejected electrons emerge in a plane orthogonal to the incident electron trajectory (figure 1). Hence it is possible to accumulate a full range of DCS measurements over all scattering angles using this spectrometer. FIG 1. The experimental geometry adopted in Manchester. The incident electron can move from the coplanar geometry (psi = 0°) to the perpendicular plane (psi = 90°), the outgoing electrons being measured in the detection plane. For the data presented here, the perpendicular plane was chosen, so only the mutual angle phi = xi1 + xi2 is relevant. In the present work, we have constrained the spectrometer to measure the DCS only in the perpendicular geometry, where psi = 90°. For both electrons to emerge in this geometry it is necessary for an interaction with the target nucleus to occur, and this provides a stringent test of current ionization models. We have taken measurements with the incident electron energy E0 ranging from near the ionization threshold of the selected target, through to energies up to 80eV above the ionization potential (IP). In this regime the probability of single ionization is a maximum, and so it is here that most ionizing collisions with electrons occur in nature. Understanding these processes is hence important in areas ranging from the production of plasmas in stellar and planetary atmospheres, through to electron impact ionization in lasers and in nuclear reactors. Low energy electron ionization has also been attributed to the production of DNA damage in biological tissue, and so it is important to understand these collisions at a fundamental level to describe the physics of these processes. We here present: results for ionization of helium from 3eV to 80eV above the IP (24.6eV), results for ionization of neon from 5eV to 50eV above the IP (21.6eV), results for ionization of argon from 2eV to 50eV above the IP (15.8eV), results for ionization of krypton from 2eV to 50eV above the IP (14.0eV), results for ionization of xenon (J=3/2, 1/2) from 2eV to 70eV above the IP (~12.1eV). For xenon, we were able to resolve the 2P1/2 and 2P3/2 ion states for incident energies up to 30eV above the IP, and so we also present examples of the measured DCS from these individual studies. In all cases results are presented for the outgoing electrons having equal energies at the analysers. The helium results are reproduced from previous studies carried out in Manchester, re-presented here for completeness. 2.0 The experimental configuration. FIG 2. The (e,2e) spectrometer used in these experiments, configured in the perpendicular plane. The apparatus can measure the DCS from the coplanar to the perpendicular plane geometry by moving the electron gun, gas jet and Faraday cup together around the interaction region (figure 2). Details can be found in other web pages on this fileserver. The electron analysers consist of a 3-element electrostatic lens that focuses electrons from the interaction region into a hemispherical energy selector. By choosing the analyser residual energy to pass electrons around the selector, a channel electron multiplier detects electrons of the specified energy for subsequent amplification and counting. In the experiments carried out here, both detectors are set to measure electrons of equal energy, given by E1 = E2 = (E0 - IP)/2. The combined coincidence energy resolution of the unselected energy electron gun and the two analysers is ~1eV. The angular range of the measured DCS is dictated by the physical size of the analysers. In the perpendicular plane, only the mutual angle phi = xi1 + xi2 is relevant, where xi1, xi2 are as shown in figure 1. For the helium data presented here, the angular range was from 50° to 310°, whereas for all other targets this range was from 70° to 290°. The data were taken over a period of several months, and the electron gun current was varied for each of the data sets so as to ensure linearity of the detection system. The data are hence normalised to unity at the peak of the DCS at each energy. The shape of the DCS for a given energy and target is then of importance, for comparison with theory. The energy of the incident electron beam was determined by observing the negative ion elastic scattering resonance in helium at ~19eV, so as to determine the offset in the gun due to contact potentials. The energies of the outgoing electrons were determined by observing inelastic scattering from the excited targets, and by considering the location of Fano resonances in the ionization continuum. 3.0 Experimental data. The experimental data for helium are taken from previous work carried out in Manchester re-presented here normalised to unity at the peak of the DCS to allow comparison with other noble gases. The ground state of helium is 1S0, with an electronic configuration 1s2, and so both bound electrons occupy s-shell orbitals prior to the interaction. The orbital angular momentum of each electron is hence zero, with a momentum distribution that peaks around zero a.u. This contrasts with the other noble gas targets whose levels are 1S0, but whose valence electrons occupy closed p-orbitals. For helium, neon and argon targets, the bound electrons occupy either s-orbitals or p-orbitals in their ground state. The configuration of the 10 electrons in neon is 1s22s22p6, whereas argon has 18 electrons occupying a configuration [Ne]3s23p6. By contrast, the bound electrons in both of the heavier targets also occupy d-orbitals, with a correspondingly more complex momentum distribution. Krypton has 36 electrons and has a ground state configuration [Ar]3d104s24p6, while xenon has 54 electrons with a ground state configuration [Kr]4d105s25p6. For all noble gas targets except helium, the removal of the valence p-electron during ionization leaves the ion in two possible ground states, either the 2P1/2 ion state or the 2P3/2 ion state. It was not possible to resolve these individual channels in our experiment (due to the limited energy resolution of the spectrometer) apart from for xenon, where results were taken for production of the different ion states at incident electron energies from 10eV to 30eV above the IP. 3.1 Helium. Figure 3 presents results for ionization of helium in the perpendicular plane for ten different incident energies ranging from 3eV to 80eV above the IP. The data files for these results can be downloaded as a TAB deliminated text file Helium.txt. FIG 3. Experimental results for the ionization of helium in the perpendicular plane, with incident energies ranging from 3eV to 80eV above the ionization potential for this target. At the lowest energy only a single broad peak occurs centred around 180°. This is considered to be due to the dominance of post-collisional interactions (PCI) between the outgoing electrons, which drives electrons emerging with equal energies from the interaction region asymptotically towards opposite directions. As the outgoing electron energy is lowered, the post collisional interaction time increases, resulting in an increased flux in the 180° direction. This type of threshold behaviour was first predicted by Wannier and was later considered by Peterkop, Rau and others to explain results in this regime. As the incident electron energy increases, additional peaks are seen at angles around 90° & 270°. These peaks are mirror images of each other due to symmetries in the perpendicular plane. By 20eV above the IP three distinct peaks are clearly visible, the relative strength of the central peak to outer lobes decreasing monotonically as the energy increases. At ~50eV excess energy the three lobes have equal amplitudes, whereas at higher energies the outer lobes dominate. At excess energies over 80eV above the IP the central lobe has virtually disappeared, leaving only the two outer lobes in the DCS. The physical reasons behind these structures has been described by several models, ranging from DWBA calculations through to TDCC models. The DWBA theory most easily allows the different processes involved in the interaction to be switched on and off, and this has explained the origin of these lobes. There is only one process that can produce a peak at 180¡ in the perpendicular plane without interaction with the target nucleus. This process requires the momentum of the incident electron to be exactly matched by the momentum of the bound electron, with both electrons moving in opposite directions prior to the interaction. In this case, momentum conservation requires the outgoing electrons to move in opposite directions (180°) in the perpendicular plane. This mechanism was considered as being the main reason for the peak at 180° until the work of Madison and co-workers in 2009 showed the probability of this occurring was low. They proposed a triple scattering mechanism to describe this peak, where the incident electron first scatters elastically from the nucleus so as to enter the detection plane, followed by a binary inelastic collision with a bound electron, followed by elastic scattering of one of the electrons back from the nucleus. This complex mechanism was found to emulate the cross section more closely compared to the single scattering process originally envisaged. The peaks at around 90° & 270° are found to result from elastic scattering of the incident electron from the core, followed by a single binary collision between the incident electron and a bound electron. In this case, since the particles have equal mass and energy, they emerge from the interaction region at a mutual angle of 90° & 270° as observed. It should be noted that although agreement between theory and experiment in the perpendicular plane has proven to be very good, the results in a coplanar geometry are not as satisfactory, particularly for non-equal outgoing electron energies. It is therefore clear that further work is needed to fully describe the ionization reaction, even for a simple target like helium. 3.2 Neon. New results for ionization of neon are shown in figure 4. The lowest energy studied was 5eV above the IP, with six measurements being taken up to 50eV above the IP as shown. The data files for these results can be downloaded as a TAB deliminated text file Neon.txt. FIG 4. Experimental results for the ionization of neon in the perpendicular plane, with incident energies ranging from 5eV to 50eV above the ionization potential for this target. The results at the lowest energy are similar in shape to that for helium, with a single broad peak observed centred around 180°. The width for this peak in neon is ~120°, which is broader than ~80° for the same outgoing electron energy in helium. The DCS shows no additional lobes at this energy. As the incident energy increases, the DCS for neon shows a very different character to that for helium. As the energy is raised to 10eV above the IP, the broad single peak seen at the lowest energy increases in width, and develops a flat structure at the peak. At 15eV above the IP this flat structure has evolved into two peaks with 180° now being a local minimum. As the energy increases further, the peaks separate in angle, and the minimum at 180° deepens. At the highest energy studied (50eV above the IP), the magnitude of this minimum is smaller than at the highest and lowest angles measured (40°,290°). As for all measurements in this study, the DCS must be zero for the electrons emerging at the same angle (0°, 360°), due to post collisional interactions between outgoing electrons of equal energy. The mechanisms that produce these structures in neon must be quite different to that proposed for helium, apart from at the lowest energy where PCI is expected to dominate for all targets. Al-Hagan and co-workers recently considered the low energy regime for both He and H2 from threshold to ~10eV above threshold, and concluded that for these targets PCI dominates at energies <2eV, which is also consistent with the observations for neon presented here. However, at higher energies the mechanisms proposed for helium ionization described above would be expected to produce similar structures in all targets that have a central ionic core. The results in figure 4 indicate that the interaction is clearly more complex, and may be due to the momentum distribution of the ionized p-electron and polarizability of the atom (which is 1.9 times greater for neon than for helium). 3.3 Argon. A similar study was carried out for argon, whose outer electronic structure ([Ne]3s23p6) is similar to neon ([He]2s22p6). The calculated atomic radius of argon (71pm) is however significantly larger than that for helium (31pm) or neon (38pm) due to the n=3 shell being occupied. The static polarization of argon is eight times larger than for helium, so scattering effects due to polarization of the target are expected to be significantly larger than for neon or helium. A summary of the radii and polarization of the different targets is given in table 1. Table 1. Calculated atomic radii and static polarization of the targets under study. Target Atomic Radius (pm) [27] Static Polarization () [28] helium 31 1.38 neon 38 2.68 argon 71 11.1 krypton 88 17.7 xenon 108 27.0 In the frozen-core approximation as used in many theories, the inner electrons are considered as spectators during the interaction, and so in this approximation the structure of the DCS should be similar for neon and argon. The results presented in figure 5 however show this is not the case. The data files for these results can be downloaded as a TAB deliminated text file Argon.txt. FIG 5. Experimental results for the ionization of argon in the perpendicular plane, with incident energies ranging from 2eV to 50eV above the ionization potential for this target. For argon, nine separate energies were chosen for study, so as to reveal the complex variation in the DCS as a function of energy. The lowest energy studied was 2eV above the IP, the data at this energy having poor statistics due to experimental difficulties working with argon at this energy. However, these data show that there is an approximate 3-lobe structure to the DCS, with the result at 180° being of similar magnitude to that around 90° & 270°. It is likely that PCI is again dominating at this energy, in support of the Wannier model. Raising the electron energy to 5eV above the IP produces much better statistics, and at this energy there is now a minimum at phi = 180° in contrast to helium and neon, with clear side lobes at around 110° and 250°. This minimum flattens at 10eV above the IP and rapidly evolves into a clear peak 15eV above the IP. This central peak is visible for incident energies up to 40eV above the IP, until at the highest energy (50eV above the IP) this has once again disappeared, the side peaks having also moved outwards compared to at the lower energies. This angular shift of the side lobes is consistent with PCI playing a continuing role in the interaction, since as the energy increases the effects of PCI diminish, and the electrons move away from 180° as is observed. Once again the results for argon are not consistent with the simple models found successful for helium. The results further indicate that the use of a Ôfrozen-coreÕ approximation is unlikely to be accurate in any theory attempting to emulate these data. 3.4 Krypton. Figure 6 shows the results for krypton, which unlike helium, argon and neon also has bound electrons occupying the 3d-shell. The physical size of krypton (88pm) is slightly larger than argon due to occupation of this shell, and due to closed occupation of the 4s and 4p shells. The static polarization of krypton is ~1.5 times that of argon. The data files for these results can be downloaded as a TAB deliminated text file Krypton.txt. FIG 6. Experimental results for the ionization of krypton in the perpendicular plane, with incident energies ranging from 2eV to 50eV above the ionization potential for this target. Once again results were taken for a lowest energy 2eV above the IP, with the data having much better statistical significance compared to those for argon at the same energy. Ionization of krypton no longer shows the dominance of PCI as predicted by Wannier, since there is now a central minimum at 180°. The relative magnitude of this minimum compared to the peaks around 110° & 250° is however larger than for higher incident energies, so there is probably a contribution to the cross section at this angle due to PCI. As the energy increases, the results for krypton show a similar trend to that for argon, with the central minimum deepening until around 10eV above the IP where a central peak starts to emerge. This central peak is not as significant in krypton compared to the side lobes, and by 30eV above the IP is once again diminishing. This central peak has virtually disappeared 40eV above the IP and there is no evidence of a peak 50eV above the IP. The similarity between argon and krypton data tends to indicate that the d-shell electrons in the heavier target are not playing a significant role in the ionization process. It also appears that the variation in cross section as a function of energy follows a similar pattern for each of these targets. It might hence be expected that the cross section for the heaviest target (xenon) would follow a similar trend to that of argon and krypton. 3.5 Xenon. As the heaviest isotopically stable noble gas target, xenon has the largest calculated size (108 pm) due to full occupation of the 4d, 5s and 5p shells. The static polarizability of xenon is also highest, being ~1.6 times greater than krypton. The ionic ground states of Xe+ are sufficiently separated that it was possible to resolve ionization to individual states during the experiments, for energies up to 30eV above the IP. A coincidence energy loss spectrum is shown in figure 7 to illustrate this separation at an incident electron energy 10eV above the IP for this target. The data show that the spectrometer can reasonably resolve the ionization to each ionic ground state, and that ionization to the J=3/2 ionic ground state is ~2.3 times more likely than to the J=1/2 ground state at this energy. It was only possible to resolve these states at energies up to 30eV above the IP, and so most of the data presented below is to the dominant J=3/2 state. FIG 7. Energy loss spectrum for the ionization of xenon in the perpendicular plane, with incident energy 10eV above the ionization potential, showing the resolution of the J=3/2, 1/2 ionic ground states. The data are fitted to two Gaussians, showing that contamination from each ionic state is <10% when the spectrometer is set to each peak in the cross section. Figure 8 shows the coincidence data for the DCS obtained from ionization of xenon to the J=3/2 ion state. The results were taken from 2eV above the IP to 70eV above the IP for this target, so as to establish the variation in the cross section that is observed. The data files for these results can be downloaded as a TAB deliminated text file Xenon1.txt. FIG 8. Experimental results for the ionization of xenon resulting in the J=3/2 ionic ground state taken in the perpendicular plane, with incident energies ranging from 2eV to 70eV above the ionization potential for this target. The data are particularly surprising since they do not follow the general trend seen for all other noble gas targets at higher energies. At the lowest energy 2eV above the IP, a clear two-peak structure is observed which is similar to that observed from krypton. This indicates that PCI is once again not the dominant mechanism producing the cross section at this energy. The minimum at 180° is significantly deeper than for krypton under the same outgoing energy conditions, and unlike krypton this minimum then becomes shallower as the incident energy increases. At 7eV above the IP a peak is clearly resolved at 180°. The relative magnitude of this peak compared to the lobes at 110° and 250° then increases in magnitude as the energy increases, as with krypton and argon. However, unlike krypton and argon whose central peak reaches a maximum and then decreases, the central peak in xenon continues to dominate the cross section as the energy rises. This trend continues up to the highest energy which was possible to measure (70eV above the IP), where the central peak is seen to be broad with a width ~100°. At all energies greater than 40eV above the IP, the outer peaks seen in all other targets have disappeared. These results show that the mechanism for ionization of xenon must once again be different to all other targets that have been studied. The results at higher energies are surprising, as the mechanism that produces the central peak with the electrons moving opposite each other is clearly dominating all other scattering processes. As far as we are aware this phenomenon has never been observed for any other target, and it remains to be seen why this should occur for xenon. Figure 9 finally shows the results obtained for ionization of xenon resulting in the J=1/2 Xe+ ion state. The data files for these results can be downloaded as a TAB deliminated text file Xenon2.txt. FIG 9. Experimental results for the ionization of xenon resulting in the J=3/2, 1/2 ionic ground states taken in the perpendicular plane, with incident energies ranging from 10eV to 30eV above the ionization potential for this target. The shapes of the DCS to the J=1/2 ionic ground state are very similar to those obtained for ionization to the J=3/2 ionic ground state at the same energies given in figure 8, as reproduced here. Only three energies were selected for these experiments, due to the difficulty of obtaining good statistical data. These data indicate that fine structure in the final ion state does not play a significant role in the ionization process, since the comparison between this data and that reproduced from figure 8 shows that the shapes of the cross section for each energy are very similar. This result indicates that fine structure does not appear to play a significant role in electron collision processes at these energies. 4.0 Discussion and summary. The set of data presented here for isotopically stable noble gases shows there is a large variation in the probability of ionization in the perpendicular plane. The perpendicular plane is particularly poignant for testing the most sophisticated electron impact ionization models, since multiple scattering must occur for electrons to emerge in this plane. This contrasts to the coplanar geometry most often adopted, where the dominant scattering mechanism is usually a single binary scattering between electrons, resulting in a high probability that forward scattering occurs. The perpendicular plane has proven to be a fertile ground to test recent theoretical models, which have found success for simple targets such as helium and molecular hydrogen. The results presented here will hence further test these models for heavier targets, so as to elucidate the scattering mechanisms that result in the observations. The marked contrast between results for helium and all other noble gas targets demand that these models carefully consider the intricacies of the interaction, since a simple extension of the basic physical ideas presented for helium cannot explain the experimental data. Further, the frozen core approximation often adopted in these models which treat all non-ionized electrons as spectators does not appear valid for the heavier targets when compared to neon, even thought the outer valence electrons have the same p-type character. It is hence likely that these assumptions will need to be relaxed to accurately model the data. It appears that PCI plays a role at the lowest measured energy for each target, although the significance of this process reduces as the target weight increases. Argon, krypton and xenon also all evolve a central peak as the incident electron energy increases from near threshold, which cannot be due to PCI. For argon and krypton the relative magnitude of this peak increases with respect to the side lobes until it reaches a maximum, then decreases to become insignificant at the highest energies measured. The most surprising result is for the heaviest target xenon, since the relative magnitude of the central peak continues to increase as the incident energy increases, and the side lobes reduce to insignificance. The side lobes observed previously in the perpendicular plane have been attributed to a binary collision between the incident electron elastically scattered into this plane and a bound electron. This process has always been considered as the dominant mechanism at higher incident energies, yet the new results presented here indicate these ideas need to be revisited for this heavy target. It is worth noting that the resolved data for ionization to the different Xe+ states indicate there is little variation to the measured cross section due to fine structure effects. It is hence likely that the results for neon, argon and krypton are also not significantly altered by these effects, even though the spectrometer was not capable of resolving their individual ion states. Acknowledgements. KLN thanks the Royal Society for an International Newton Research Fellowship held at the University of Manchester. CK would like to thank the University of Manchester for providing a PhD scholarship while carrying out this work. References. [1] Weigold E and McCarthy I E 1999 Electron Momentum Spectroscopy (Dordrecht: Kluwer) [2] Murray A J and Read F H 1993 Phys. Rev. A 47 3724 [3] Murray A J, Bowring N J and Read F H 1997 J. Phys. B: At. Mol. Phys. 30 387 [4] Boudaiffa B, Cloutier P, Hunting D, Huels M A and Sanche L 2000 Science 287 1658 [5] Zhang X, Whelan C T and Walters H R J 1990 J. Phys. B: At. Mol. Phys. 23 L173 [6] Rescigno T N, Baertschy M, Isaacs W A and McCurdy C W 1999 Science 286 2474 [7] Bray I, Bartschat K and Stelbovics A T 2003 Phys. Rev. A 67 060704 [8] Stelbovics A T, Bray I, Fursa D V and Bartschat K 2005 Phys. Rev. A 71 052716 [9] Colgan J and Pindzola M S 2006 Phys. Rev. A 74 012713 [10] Colgan J, Pindzola M S, Childers G and Khakoo M 2006 Phys. Rev. A 73 042710 [11] Gao J, Madison D H, Peacher J L, Murray A J and Hussey M J 2006 J. Chem. Phys. 124 194306 [12] Murray A J, Hussey M J, Gao J and Madison D H 2006 J. Phys. B: At. Mol. Phys. 39 3945 [13] Colgan J, Pindzola M S, Robicheaux F, Kaiser C, Murray A J and Madison D H 2008 Phys. Rev. Lett. 101 233201 [14] Colgan J, Al-Hagan O, Madison D H, Kaiser C, Murray A J and Pindzola M S 2009 Phys. Rev. A 79 052704 [15] Colgan J, Al-Hagan O, Madison D H, Murray A J and Pindzola M S 2009 J. Phys. B: At. Mol. Phys. 42 171001 [16] Murray A J and Read F H 1992 Phys. Rev. Lett. 69 2912 [17] Murray A J, Woolf M B J and Read F H 1992 J. Phys. B: At. Mol. Phys. 25 3021 [18] Bowring N J, Murray A J and Read F H 1997 J. Phys. B: At. Mol. Phys. 30 L671 [19] Murray A J, Bowring N J and Read F H 2000 J. Phys. B: At. Mol. Phys. 33 2859 [20] Murray A J, Turton B C H & Read F H 1992 Rev. Sci. Inst. 63 3346 [21] Wannier G H 1953 Phys. Rev. 90 817 [22] Peterkop R 1971 J. Phys. B: At. Mol. Phys. 4 513 [23] Rau A R P 1971 Phys. Rev. A 4 207 [24] Al-Hagan O, Madison D H, Murray A J, Kaiser C and Colgan J 2010 Phys. Rev. A (in press) [25] Al-Hagan O, Kaiser C, Murray A J and Madison D H 2009 Nature Phys 5 59 [26] Bray I and Colgan J Private communication (2009) [27] Clementi E, Raimondi D L and Reinhardt W P 1967 J Chem Phys 47 1300 [28] Williart A and Garcia G 2001 Phys. Scripta 64 343

Studying Low energy (e,2e) from the noble gases

in the perpendicular plane.

Kate L Nixon, Andrew James Murray and Christian Kaiser.

Abstract:

In this web page we provide detailed (e,2e) studies of electron impact ionization of noble gases including helium, neon, argon, krypton and xenon from near threshold to intermediate energies, where the outgoing electrons carry equal energy from the interaction.

The experiments were conducted in the perpendicular plane, where the outgoing electrons are detected orthogonal to the incident electron beam. For electrons to emerge in this geometry they must undergo multiple scattering including scattering from the nucleus of the target, and so this provides a highly sensitive test for scattering theories.

The data show the cross sections undergo complex variations as a function of incident energy, and in particular ionization of the heaviest target is at variance to all others that have been studied here.

1.0 Introduction.

The (e,2e) process provides the most precise experimental data for the study of the ionization of atomic and molecular targets by electron impact. In these experiments a single electron of well defined momentum impacts with a target in the interaction region, resulting in target ionization and a scattered and ejected electron which emerge after the interaction has taken place.

By determining the momenta of the scattered and ejected electrons in a time correlated coincidence measurement, the differential ionization cross section (DCS) is derived for comparison with theoretical models.

Since the scattered and ejected electrons may emerge over a wide range of different angles, experiments usually define a specific scattering geometry in which to carry out the measurements. Our (e,2e) apparatus can access a wide range of geometries from a coplanar geometry, where the incident, scattered and ejected electrons all occupy the same plane, through to the perpendicular geometry, where the scattered and ejected electrons emerge in a plane orthogonal to the incident electron trajectory (figure 1). Hence it is possible to accumulate a full range of DCS measurements over all scattering angles using this spectrometer.

FIG 1. The experimental geometry adopted in Manchester. The incident electron can move from the coplanar geometry (psi = 0°) to the perpendicular plane (psi = 90°), the outgoing electrons being measured in the detection plane. For the data presented here, the perpendicular plane was chosen, so only the mutual angle phi = xi1 + xi2 is relevant.

In the present work, we have constrained the spectrometer to measure the DCS only in the perpendicular geometry, where psi = 90°.

For both electrons to emerge in this geometry it is necessary for an interaction with the target nucleus to occur, and this provides a stringent test of current ionization models.

We have taken measurements with the incident electron energy E0 ranging from near the ionization threshold of the selected target, through to energies up to 80eV above the ionization potential (IP). In this regime the probability of single ionization is a maximum, and so it is here that most ionizing collisions with electrons occur in nature.

Understanding these processes is hence important in areas ranging from the production of plasmas in stellar and planetary atmospheres, through to electron impact ionization in lasers and in nuclear reactors. Low energy electron ionization has also been attributed to the production of DNA damage in biological tissue, and so it is important to understand these collisions at a fundamental level to describe the physics of these processes.

We here present:

•results for ionization of helium from 3eV to 80eV above the IP (24.6eV),
•results for ionization of neon from 5eV to 50eV above the IP (21.6eV),
•results for ionization of argon from 2eV to 50eV above the IP (15.8eV),
•results for ionization of krypton from 2eV to 50eV above the IP (14.0eV),
•results for ionization of xenon (J=3/2, 1/2) from 2eV to 70eV above the IP (~12.1eV).

For xenon, we were able to resolve the 2P1/2 and 2P3/2 ion states for incident energies up to 30eV above the IP, and so we also present examples of the measured DCS from these individual studies.

In all cases results are presented for the outgoing electrons having equal energies at the analysers. The helium results are reproduced from previous studies carried out in Manchester, re-presented here for completeness.

2.0 The experimental configuration.

FIG 2. The (e,2e) spectrometer used in these experiments, configured in the perpendicular plane.

The apparatus can measure the DCS from the coplanar to the perpendicular plane geometry by moving the electron gun, gas jet and Faraday cup together around the interaction region (figure 2). Details can be found in other web pages on this fileserver.

The electron analysers consist of a 3-element electrostatic lens that focuses electrons from the interaction region into a hemispherical energy selector. By choosing the analyser residual energy to pass electrons around the selector, a channel electron multiplier detects electrons of the specified energy for subsequent amplification and counting. In the experiments carried out here, both detectors are set to measure electrons of equal energy, given by E1 = E2 = (E0 - IP)/2. The combined coincidence energy resolution of the unselected energy electron gun and the two analysers is ~1eV.

The angular range of the measured DCS is dictated by the physical size of the analysers. In the perpendicular plane, only the mutual angle phi = xi1 + xi2 is relevant, where xi1, xi2 are as shown in figure 1. For the helium data presented here, the angular range was from 50° to 310°, whereas for all other targets this range was from 70° to 290°.

The data were taken over a period of several months, and the electron gun current was varied for each of the data sets so as to ensure linearity of the detection system. The data are hence normalised to unity at the peak of the DCS at each energy. The shape of the DCS for a given energy and target is then of importance, for comparison with theory.

The energy of the incident electron beam was determined by observing the negative ion elastic scattering resonance in helium at ~19eV, so as to determine the offset in the gun due to contact potentials.

The energies of the outgoing electrons were determined by observing inelastic scattering from the excited targets, and by considering the location of Fano resonances in the ionization continuum.

3.0 Experimental data.

The experimental data for helium are taken from previous work carried out in Manchester re-presented here normalised to unity at the peak of the DCS to allow comparison with other noble gases.

The ground state of helium is 1S0, with an electronic configuration 1s2, and so both bound electrons occupy s-shell orbitals prior to the interaction. The orbital angular momentum of each electron is hence zero, with a momentum distribution that peaks around zero a.u. This contrasts with the other noble gas targets whose levels are 1S0, but whose valence electrons occupy closed p-orbitals.

For helium, neon and argon targets, the bound electrons occupy either s-orbitals or p-orbitals in their ground state. The configuration of the 10 electrons in neon is 1s22s22p6, whereas argon has 18 electrons occupying a configuration [Ne]3s23p6. By contrast, the bound electrons in both of the heavier targets also occupy d-orbitals, with a correspondingly more complex momentum distribution. Krypton has 36 electrons and has a ground state configuration [Ar]3d104s24p6, while xenon has 54 electrons with a ground state configuration [Kr]4d105s25p6.

For all noble gas targets except helium, the removal of the valence p-electron during ionization leaves the ion in two possible ground states, either the 2P1/2 ion state or the 2P3/2 ion state.

It was not possible to resolve these individual channels in our experiment (due to the limited energy resolution of the spectrometer) apart from for xenon, where results were taken for production of the different ion states at incident electron energies from 10eV to 30eV above the IP.

3.1 Helium.

Figure 3 presents results for ionization of helium in the perpendicular plane for ten different incident energies ranging from 3eV to 80eV above the IP. The data files for these results can be downloaded as a TAB deliminated text file Helium.txt.

FIG 3. Experimental results for the ionization of helium in the perpendicular plane, with incident energies ranging from 3eV to 80eV above the ionization potential for this target.

At the lowest energy only a single broad peak occurs centred around 180°. This is considered to be due to the dominance of post-collisional interactions (PCI) between the outgoing electrons, which drives electrons emerging with equal energies from the interaction region asymptotically towards opposite directions. As the outgoing electron energy is lowered, the post collisional interaction time increases, resulting in an increased flux in the 180° direction. This type of threshold behaviour was first predicted by Wannier and was later considered by Peterkop, Rau and others to explain results in this regime.

As the incident electron energy increases, additional peaks are seen at angles around 90° & 270°. These peaks are mirror images of each other due to symmetries in the perpendicular plane.

By 20eV above the IP three distinct peaks are clearly visible, the relative strength of the central peak to outer lobes decreasing monotonically as the energy increases. At ~50eV excess energy the three lobes have equal amplitudes, whereas at higher energies the outer lobes dominate. At excess energies over 80eV above the IP the central lobe has virtually disappeared, leaving only the two outer lobes in the DCS.

The physical reasons behind these structures has been described by several models, ranging from DWBA calculations through to TDCC models. The DWBA theory most easily allows the different processes involved in the interaction to be switched on and off, and this has explained the origin of these lobes.

There is only one process that can produce a peak at 180¡ in the perpendicular plane without interaction with the target nucleus. This process requires the momentum of the incident electron to be exactly matched by the momentum of the bound electron, with both electrons moving in opposite directions prior to the interaction. In this case, momentum conservation requires the outgoing electrons to move in opposite directions (180°) in the perpendicular plane. This mechanism was considered as being the main reason for the peak at 180° until the work of Madison and co-workers in 2009 showed the probability of this occurring was low. They proposed a triple scattering mechanism to describe this peak, where the incident electron first scatters elastically from the nucleus so as to enter the detection plane, followed by a binary inelastic collision with a bound electron, followed by elastic scattering of one of the electrons back from the nucleus. This complex mechanism was found to emulate the cross section more closely compared to the single scattering process originally envisaged.

The peaks at around 90° & 270° are found to result from elastic scattering of the incident electron from the core, followed by a single binary collision between the incident electron and a bound electron. In this case, since the particles have equal mass and energy, they emerge from the interaction region at a mutual angle of 90° & 270° as observed.

It should be noted that although agreement between theory and experiment in the perpendicular plane has proven to be very good, the results in a coplanar geometry are not as satisfactory, particularly for non-equal outgoing electron energies. It is therefore clear that further work is needed to fully describe the ionization reaction, even for a simple target like helium.

3.2 Neon.

New results for ionization of neon are shown in figure 4. The lowest energy studied was 5eV above the IP, with six measurements being taken up to 50eV above the IP as shown. The data files for these results can be downloaded as a TAB deliminated text file Neon.txt.

FIG 4. Experimental results for the ionization of neon in the perpendicular plane, with incident energies ranging from 5eV to 50eV above the ionization potential for this target.

The results at the lowest energy are similar in shape to that for helium, with a single broad peak observed centred around 180°. The width for this peak in neon is ~120°, which is broader than ~80° for the same outgoing electron energy in helium. The DCS shows no additional lobes at this energy.

As the incident energy increases, the DCS for neon shows a very different character to that for helium. As the energy is raised to 10eV above the IP, the broad single peak seen at the lowest energy increases in width, and develops a flat structure at the peak. At 15eV above the IP this flat structure has evolved into two peaks with 180° now being a local minimum. As the energy increases further, the peaks separate in angle, and the minimum at 180° deepens. At the highest energy studied (50eV above the IP), the magnitude of this minimum is smaller than at the highest and lowest angles measured (40°,290°). As for all measurements in this study, the DCS must be zero for the electrons emerging at the same angle (0°, 360°), due to post collisional interactions between outgoing electrons of equal energy.

The mechanisms that produce these structures in neon must be quite different to that proposed for helium, apart from at the lowest energy where PCI is expected to dominate for all targets.

Al-Hagan and co-workers recently considered the low energy regime for both He and H2 from threshold to ~10eV above threshold, and concluded that for these targets PCI dominates at energies <2eV, which is also consistent with the observations for neon presented here. However, at higher energies the mechanisms proposed for helium ionization described above would be expected to produce similar structures in all targets that have a central ionic core.

The results in figure 4 indicate that the interaction is clearly more complex, and may be due to the momentum distribution of the ionized p-electron and polarizability of the atom (which is 1.9 times greater for neon than for helium).

3.3 Argon.

A similar study was carried out for argon, whose outer electronic structure ([Ne]3s23p6) is similar to neon ([He]2s22p6). The calculated atomic radius of argon (71pm) is however significantly larger than that for helium (31pm) or neon (38pm) due to the n=3 shell being occupied. The static polarization of argon is eight times larger than for helium, so scattering effects due to polarization of the target are expected to be significantly larger than for neon or helium. A summary of the radii and polarization of the different targets is given in table 1.

Table 1. Calculated atomic radii and static polarization of the targets under study.

Target Atomic Radius (pm) [27] Static Polarization (

) [28]

helium 31 1.38

neon 38 2.68

argon 71 11.1

krypton 88 17.7

xenon 108 27.0

In the frozen-core approximation as used in many theories, the inner electrons are considered as spectators during the interaction, and so in this approximation the structure of the DCS should be similar for neon and argon. The results presented in figure 5 however show this is not the case. The data files for these results can be downloaded as a TAB deliminated text file Argon.txt.

FIG 5. Experimental results for the ionization of argon in the perpendicular plane, with incident energies ranging from 2eV to 50eV above the ionization potential for this target.

For argon, nine separate energies were chosen for study, so as to reveal the complex variation in the DCS as a function of energy. The lowest energy studied was 2eV above the IP, the data at this energy having poor statistics due to experimental difficulties working with argon at this energy. However, these data show that there is an approximate 3-lobe structure to the DCS, with the result at 180° being of similar magnitude to that around 90° & 270°. It is likely that PCI is again dominating at this energy, in support of the Wannier model.

Raising the electron energy to 5eV above the IP produces much better statistics, and at this energy there is now a minimum at phi = 180° in contrast to helium and neon, with clear side lobes at around 110° and 250°. This minimum flattens at 10eV above the IP and rapidly evolves into a clear peak 15eV above the IP. This central peak is visible for incident energies up to 40eV above the IP, until at the highest energy (50eV above the IP) this has once again disappeared, the side peaks having also moved outwards compared to at the lower energies. This angular shift of the side lobes is consistent with PCI playing a continuing role in the interaction, since as the energy increases the effects of PCI diminish, and the electrons move away from 180° as is observed.

Once again the results for argon are not consistent with the simple models found successful for helium. The results further indicate that the use of a Ôfrozen-coreÕ approximation is unlikely to be accurate in any theory attempting to emulate these data.

3.4 Krypton.

Figure 6 shows the results for krypton, which unlike helium, argon and neon also has bound electrons occupying the 3d-shell. The physical size of krypton (88pm) is slightly larger than argon due to occupation of this shell, and due to closed occupation of the 4s and 4p shells. The static polarization of krypton is ~1.5 times that of argon. The data files for these results can be downloaded as a TAB deliminated text file Krypton.txt.

FIG 6. Experimental results for the ionization of krypton in the perpendicular plane, with incident energies ranging from 2eV to 50eV above the ionization potential for this target.

Once again results were taken for a lowest energy 2eV above the IP, with the data having much better statistical significance compared to those for argon at the same energy. Ionization of krypton no longer shows the dominance of PCI as predicted by Wannier, since there is now a central minimum at 180°. The relative magnitude of this minimum compared to the peaks around 110° & 250° is however larger than for higher incident energies, so there is probably a contribution to the cross section at this angle due to PCI.

As the energy increases, the results for krypton show a similar trend to that for argon, with the central minimum deepening until around 10eV above the IP where a central peak starts to emerge. This central peak is not as significant in krypton compared to the side lobes, and by 30eV above the IP is once again diminishing. This central peak has virtually disappeared 40eV above the IP and there is no evidence of a peak 50eV above the IP.

The similarity between argon and krypton data tends to indicate that the d-shell electrons in the heavier target are not playing a significant role in the ionization process. It also appears that the variation in cross section as a function of energy follows a similar pattern for each of these targets. It might hence be expected that the cross section for the heaviest target (xenon) would follow a similar trend to that of argon and krypton.

3.5 Xenon.

As the heaviest isotopically stable noble gas target, xenon has the largest calculated size (108 pm) due to full occupation of the 4d, 5s and 5p shells. The static polarizability of xenon is also highest, being ~1.6 times greater than krypton.

The ionic ground states of Xe+ are sufficiently separated that it was possible to resolve ionization to individual states during the experiments, for energies up to 30eV above the IP. A coincidence energy loss spectrum is shown in figure 7 to illustrate this separation at an incident electron energy 10eV above the IP for this target. The data show that the spectrometer can reasonably resolve the ionization to each ionic ground state, and that ionization to the J=3/2 ionic ground state is ~2.3 times more likely than to the J=1/2 ground state at this energy. It was only possible to resolve these states at energies up to 30eV above the IP, and so most of the data presented below is to the dominant J=3/2 state.

FIG 7. Energy loss spectrum for the ionization of xenon in the perpendicular plane, with incident energy 10eV above the ionization potential, showing the resolution of the J=3/2, 1/2 ionic ground states. The data are fitted to two Gaussians, showing that contamination from each ionic state is <10% when the spectrometer is set to each peak in the cross section.

Figure 8 shows the coincidence data for the DCS obtained from ionization of xenon to the J=3/2 ion state. The results were taken from 2eV above the IP to 70eV above the IP for this target, so as to establish the variation in the cross section that is observed. The data files for these results can be downloaded as a TAB deliminated text file Xenon1.txt.

FIG 8. Experimental results for the ionization of xenon resulting in the J=3/2 ionic ground state taken in the perpendicular plane, with incident energies ranging from 2eV to 70eV above the ionization potential for this target.

The data are particularly surprising since they do not follow the general trend seen for all other noble gas targets at higher energies. At the lowest energy 2eV above the IP, a clear two-peak structure is observed which is similar to that observed from krypton. This indicates that PCI is once again not the dominant mechanism producing the cross section at this energy.

The minimum at 180° is significantly deeper than for krypton under the same outgoing energy conditions, and unlike krypton this minimum then becomes shallower as the incident energy increases. At 7eV above the IP a peak is clearly resolved at 180°. The relative magnitude of this peak compared to the lobes at 110° and 250° then increases in magnitude as the energy increases, as with krypton and argon. However, unlike krypton and argon whose central peak reaches a maximum and then decreases, the central peak in xenon continues to dominate the cross section as the energy rises. This trend continues up to the highest energy which was possible to measure (70eV above the IP), where the central peak is seen to be broad with a width ~100°. At all energies greater than 40eV above the IP, the outer peaks seen in all other targets have disappeared.

These results show that the mechanism for ionization of xenon must once again be different to all other targets that have been studied. The results at higher energies are surprising, as the mechanism that produces the central peak with the electrons moving opposite each other is clearly dominating all other scattering processes.

As far as we are aware this phenomenon has never been observed for any other target, and it remains to be seen why this should occur for xenon.

Figure 9 finally shows the results obtained for ionization of xenon resulting in the J=1/2 Xe+ ion state. The data files for these results can be downloaded as a TAB deliminated text file Xenon2.txt.

FIG 9. Experimental results for the ionization of xenon resulting in the J=3/2, 1/2 ionic ground states taken in the perpendicular plane, with incident energies ranging from 10eV to 30eV above the ionization potential for this target. The shapes of the DCS to the J=1/2 ionic ground state are very similar to those obtained for ionization to the J=3/2 ionic ground state at the same energies given in figure 8, as reproduced here.

Only three energies were selected for these experiments, due to the difficulty of obtaining good statistical data. These data indicate that fine structure in the final ion state does not play a significant role in the ionization process, since the comparison between this data and that reproduced from figure 8 shows that the shapes of the cross section for each energy are very similar. This result indicates that fine structure does not appear to play a significant role in electron collision processes at these energies.

4.0 Discussion and summary.

The set of data presented here for isotopically stable noble gases shows there is a large variation in the probability of ionization in the perpendicular plane.

The perpendicular plane is particularly poignant for testing the most sophisticated electron impact ionization models, since multiple scattering must occur for electrons to emerge in this plane. This contrasts to the coplanar geometry most often adopted, where the dominant scattering mechanism is usually a single binary scattering between electrons, resulting in a high probability that forward scattering occurs.

The perpendicular plane has proven to be a fertile ground to test recent theoretical models, which have found success for simple targets such as helium and molecular hydrogen. The results presented here will hence further test these models for heavier targets, so as to elucidate the scattering mechanisms that result in the observations. The marked contrast between results for helium and all other noble gas targets demand that these models carefully consider the intricacies of the interaction, since a simple extension of the basic physical ideas presented for helium cannot explain the experimental data. Further, the frozen core approximation often adopted in these models which treat all non-ionized electrons as spectators does not appear valid for the heavier targets when compared to neon, even thought the outer valence electrons have the same p-type character. It is hence likely that these assumptions will need to be relaxed to accurately model the data.

It appears that PCI plays a role at the lowest measured energy for each target, although the significance of this process reduces as the target weight increases. Argon, krypton and xenon also all evolve a central peak as the incident electron energy increases from near threshold, which cannot be due to PCI. For argon and krypton the relative magnitude of this peak increases with respect to the side lobes until it reaches a maximum, then decreases to become insignificant at the highest energies measured.

The most surprising result is for the heaviest target xenon, since the relative magnitude of the central peak continues to increase as the incident energy increases, and the side lobes reduce to insignificance. The side lobes observed previously in the perpendicular plane have been attributed to a binary collision between the incident electron elastically scattered into this plane and a bound electron. This process has always been considered as the dominant mechanism at higher incident energies, yet the new results presented here indicate these ideas need to be revisited for this heavy target.

It is worth noting that the resolved data for ionization to the different Xe+ states indicate there is little variation to the measured cross section due to fine structure effects. It is hence likely that the results for neon, argon and krypton are also not significantly altered by these effects, even though the spectrometer was not capable of resolving their individual ion states.

Acknowledgements.

KLN thanks the Royal Society for an International Newton Research Fellowship held at the University of Manchester. CK would like to thank the University of Manchester for providing a PhD scholarship while carrying out this work.

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