A review of electron scattering cross section measurements by use of pulsed electron beam time of flight techniques

Journal of Electron Spectroscopy and Related Phenomena 155 (2007) 1–6

A review of electron scattering cross section measurements by use of pulsed electron beam time of flight techniques R.A. Bonham ∗ Department of Biological, Chemical, and Physical Sciences, Illinois Institute of Technology, 3101 South Dearborn Street, Chicago, IL 60616, USA Available online 19 November 2006

Abstract A review of the relative sizes of cross sections for photon and electron scattering by atoms and molecules is presented. Advantages and drawbacks associated with the use of each probe is discussed. Experiments carried out by the author and his collaborators using pulsed electron beams with time of flight analysis are reviewed. © 2006 Elsevier B.V. All rights reserved. Keywords: Electron scattering; Photon scattering; Cross sections; Pulsed electron beams; Time of flight

1. Electron and photon scattering cross sections The organizers of this conference have asked me to review the work carried out in my laboratory in the Chemistry Department at Indiana University that was related to the topics of this conference. This begins with a review of the simplest theoretical treatment of electron scattering and the information that can be gleaned from such studies along with comparisons to analogous photon scattering experiments. In keeping with the themes of this conference a survey of experiments carried out by our group during the period 1975–1999 with the aim of measuring a variety of different electron impact cross sections will be reviewed. This work started because of the desire to extend our previous high energy scattering research [1] to lower energy. We decided to do this by employing a pulsed electron beam source with time of flight detection of various scattered reaction products. The development of various techniques to accomplish the goal of accurate low energy cross section measurement is reviewed. 1.1. Scattering: a comparison of simple theories The simplest nonrelativistic first Born theory for elastic electron scattering by atoms yields the scattering cross section, σ,

∗

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differential with respect to the detected solid angle d/dΩ as     dσ el dσ = [Z − F (K)]2 dΩ elastic dΩ Rutherford where el stands for electron, Z is the nuclear charge, (dσ/dΩ)Rutherford is the Rutherford cross section given in Rydberg a.u.’s (1 a.u. of energy = 13.605 eV) as (4/K4 ) with K the momentum transfer on scattering, and F(K) the X-ray scattering factor. For the case of photon scattering the equivalent result, of  is order A2 in the photons electric field A,   dσ ph = σThomson F (K)2 dΩ elastic where the Thomson cross section, σ Thomson , is 2.38 × 10−8 in Rydberg a.u.’s and ph stands for photon. The ratio, Relastic , of X-ray to electron elastic scattering in the limit as K → 0 is ph

Relastic lim K→0 =

(dσ/dΩ)elastic (dσ/dΩ)el elastic

=

2.3 × 10−7 Z2  2 r2

where r2 2 is the square of the mean square radius of the atom. In the limit as K → ∞ this same ratio approaches ph

Relastic lim K→∞ =

(dσ/dΩ)elastic (dσ/dΩ)el elastic

=

2.38 × 10−8 (dρ(r)/dr|r=0 )2 Z2 K 4

where ρ(r) is the spherically averaged electron density of the atom. Obviously, in both cases, electron scattering would appear to enjoy an enormous advantage.

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R.A. Bonham / Journal of Electron Spectroscopy and Related Phenomena 155 (2007) 1–6

For the case of total inelastic scattering both scattering types are directly proportional to the X-ray incoherent scattering factor, S(K), which cancels out in the ratio so that Rinelastic is given by ph

R

inelastic

=

(dσ/dΩ)inelastic (dσ/dΩ)el inelastic

= 5.95 × 10

−9

K

4

for which photon cross sections dominate for momentum transfers greater than 114 a.u.’s. This would correspond, for example, to carrying out an experiment with 52 keV photons at a scattering angle of 135◦ . At first glance it would appear that for scattering, as opposed to absorption, that electrons enjoy a huge advantage. Unfortunately an enormous cross section brings with it very large inter atomic or molecular multiple scattering contributions which force experiments with electrons to be carried out at target gas pressures of a few milli Torr. When one considers the higher pressures that can be used, with only small multiple scattering corrections, in photon scattering using synchrotron radiation the difference between the two probes shrinks by a factor of between 5 and 6 orders of magnitude [2,3]. Hence the advantage enjoyed by electron scattering is probably only a factor of 10–100 assuming incident projectile beams of similar intensity. 1.2. Electron exchange, spin, and photon polarization In the case of electron scattering two additional effects are worth mention. The first is the use of incident electron beams of spin aligned electrons in scattering experiments pioneered by Kessler and collaborators [4] in which the spin orientation of the scattered electron is detected. Such difficult experiments have been explained by means of relativistic partial wave analysis of scattering from the atoms making up the molecule. This theory, termed the independent atom model (IAM), treats the multiple scattering within each atom exactly assuming no distortion of the atomic electron densities due to chemical bond formation. In the Dirac or relativistic theory there are two scattering amplitudes, usually labeled f and g, where the f amplitude describes electrons scattered with retention of their original spin orientation and a g amplitude describing the scattering with electrons that have flipped their original spin orientation [5]. Molecular nuclear–nuclear interference terms are included in the scattering to lowest order in the IAM theory. A second effect, due to the indistinguishability of electrons and termed electron exchange, permits electron scattering at low energies to excite optically forbidden singlet–triplet transitions. Lassettre and co-workers pioneered this field of research [6]. In the case of photon scattering a more rigorous theory includ2  ·p ing both A2 and (A  ) terms yields the cross section for elastic scattering 

dσ dΩ

ph = σThomson |(ˆni · nˆ s )2 (F (K) + f1 ) elastic

+(ˆni · ks )(ˆns · ki )f2 |2

Table 1 Values of the complex anomalous dispersion amplitudes at two different photon energies Incident photon energy

f1

f2

849 eV 4.39 keV

0.027−i0.0020 0.0011−i0.00019

0.0038−i0.00062 0.00015−i0.000025

where nˆ i , nˆ s are unit vectors in the directions of the polarization of the incident and scattered photon and ki , ks are the momentum vectors of the incident and scattered photon. The complex ampli ·p tude, f1 , is a second order correction from the A  term of the photons electric field and is usually termed the anomalous dispersion term while the complex f2 term is also of the same order in the electric field but with a very different dependence on the photon polarizations. Gavrila and Costescu have evaluated both f1 and f2 exactly, with all terms in the retardation expansion of the photon plane wave included, for atomic hydrogen [7]. It was found that f1 exhibits resonances at the same energies as found for excitations by photo-absorption. In addition the resonances found in the amplitude f2 are missing the transition between n = 1 and n = 2. A more recent study [8] has shown that f2 obeys quadrupole transition like selection rules in the small angle limit and suggested that it might be experimentally possible to measure f2 directly by carrying out experiments with nˆ i ⊥ nˆ s . In addition this work suggested the possibility of extending the exact H atom case to many electron systems. The case of atomic He has been treated by expanding the two electron Green’s function as a sum of hydrogenic one electron Green’s functions and neglecting terms depending on the electron–electron repulsion term in the Hamiltonian [9]. A comparison of the values for f1 and f2 for zero angle scattering using this approximate expansion of the Green’s function is given in Table 1 using an open shell (OS) wave function to describe the He ground state. The OS description of the He ground state yields about 30% of the correlation energy for the atom. Although f2 is about 7 times smaller than f1 it should be noted that the coefficient of f2 can approach ki2 in magnitude. The angular dependence of these terms have been investigated from 0◦ to 180◦ . In the case of 849 eV photons both the scattered amplitudes are nearly isotropic with a fall off of about 2–4% over the entire angular range. For 4.39 keV photons both amplitudes fall off uniformly by 35–45% by 180◦ from their values at 0 angle. The focus here on anomalous dispersion is partly because polarization studies with photons would seem to be easier to carry out than their electron counterparts and partly because such studies in resonant spectral regions using a polarized source appears to be an under-utilized spectroscopic tool. 1.3. Is elastic scattering really elastic? Consider an electron scattered elastically by an atom of mass M. If energy and momentum are conserved in the scattering then in Rydberg a.u.’s energy conservation can be expressed as ki2 +

p2 p 2 = ks2 + M M

R.A. Bonham / Journal of Electron Spectroscopy and Related Phenomena 155 (2007) 1–6

where p  and p are the initial and final momenta of the atom and ki , ks are the initial and final momenta of the scattered electron. Conservation of momentum then takes the form ki + p  = ks + p

which can be used to eliminate the dependence on the final momentum of the atom. Introducing the energy transferred to the atom by the incident electron, called the energy loss E, where E = ki2 − ks2 , and solving for the energy loss yields    ·p 2 K  K2 E= + . M M For a static gas sample the average over the initial atom momenta should vanish or, in the case of sub sonic expanding jets, should be small in comparison to the first term. For atomic He scattering 17 keV electrons by 90◦ , a momentum transfer of 50 a.u., this formula predicts an energy loss of 4.6 eV. The case of elastic scattering from molecules has been investigated theoretically with the prediction that the elastic peak for most molecules would split into a doublet for sufficiently high momentum transfers with the lowest energy loss peak due to translational and pure rotational excitation of the molecule as a whole while the second higher energy loss peak would be due to ro-vibrational excitation of the molecule [10]. This scattering was dubbed translational and vibrational Compton scattering. At the time this paper was published no experiment appeared to be capable of observing scattering requiring such high energy resolution for such large momentum transfer. This situation has recently changed with the observations of split elastic peaks in both gas [11] and solid targets [12]. In fact the reporting of vibrational Compton scattering or splitting of the elastic peak in electron scattering by a molecule in the gas phase was reported for the first time in this conference by Hitchcock and co-workers. These authors interpret their data as translational Compton scattering from each individual atom. The prediction of this theory is that there would be as many elastic Compton peaks as there are different atoms in the molecule. The number of peaks predicted by the theory of Ref. [10] would always be two, one for translational excitation of the molecule as a whole and a second for ro-vibrational excitation.

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and the Beer-Lambert law it seemed it should be possible to carry out measurement of the total absorption cross section of the target gas simultaneously over an extended range of impact energies. This idea lead to a number of successful measurements described in the next section and encouraged us to explore the possibility of making other types of cross section measurements which are discussed in following sections. The impetus for starting this work came primarily from theorists needing accurate cross sections to model radiation damage. As time went by we were made aware of the need for cross section data by members of the plasma modeling community. It was the latter that moved us to attempt experiments on the break up of molecules as the result of electron impact. 2.1. The measurement of total electron impact cross sections Our first venture in low energy electron scattering involved experiments with a pulsed electron beam of 2 keV electrons scattered by gas jets of He and Ar [13] with time of flight detection of ejected or secondary electrons at a fixed angle of 90◦ . The apparatus used for these experiments is shown in Fig. 1. This study was followed by a second study which included molecular nitrogen [14] and details of the experimental apparatus. The energy resolution of the method varied from 2 meV for a detected electron energy of 100 meV to 0.8 eV resolution at 100 eV detected energy. Experiments utilized a 200 ps pulse containing about 103 e/␮A of beam current. This work led to the use of a secondary electron source to measure the total electron impact cross section of He from 0.5 to 50 eV impact energy [15] utilizing target gas pressure variation in an absorption cell to allow analysis by means of the Beer-Lambert law. The experiment employed a 38 cm absorption cell with gas pressures varied from 1.7 to 7 mTorr with a claimed measuring accuracy for the cross section of 3%. This work was followed by a series of measurements

2. Pulsed electron beam time of flight spectroscopy The quest in our laboratory at Indiana to engage in low energy spectroscopy began using keV electron pulses with detection of the secondary electron spectrum by time of flight at a fixed ejection angle. In carrying out this work it was discovered that a high energy, 2 keV, pulse of electrons impinging on a solid surface produced an intense continuum of secondary electrons extending from near the elastic line to well below 1 eV. This continuum suggested that we might be able to carry out total electron impact cross section measurements using a pulsed electron beam of high energy to produce an electron pulse containing a wide continuum of energies which could then be transmitted through a cell filled with the gas to be measured. By varying the gas cell pressure and using time of flight detection of the transmitted electrons

Fig. 1. Time of flight apparatus used to measure total cross sections. Total flight distance from target to detector is ∼45 cm (Ref. [20], reproduced with permission).

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on N2 [16], O2 [17], SF6 [18], He [19], H2 [20], CH4 [21], CCl4, CCl3 F, CCl2 F2 , CClF3 and CF4 [22]. One problem in such measurements is that at high energy the scattering focuses into the forward direction so that the true attenuation of electrons at these energies is less than it should be. This is the reason that most of these measurements do not extend beyond 50 eV. We succeeded in devising a scheme, albeit a cumbersome one, to get around this problem. An absorption gas cell of variable length was used to measure the absorption of electrons of a fixed energy as a function of both cell length and gas pressure. By means of various models for the scattering dependence on the path length it proved possible to correct the total cross section for forward scattering into the detector [23]. 2.2. The angular dependence of secondary electrons An apparatus for the express purpose of measuring the angular dependence of secondary electron emission was constructed which consisted of a fixed time of flight detector (TOF) at a small scattering angle and a second TOF detector capable of scanning the angular range from 20◦ to 160◦ . See Fig. 2 for a cutaway view of the apparatus. The fixed angle detector could be set at 2◦ to either side of the forward direction and was used to measure the intensity of the elastic line for the purposes of normalization of the secondary spectra. Data from this instrument were reported for N2 [24] at a conference on modeling of radiation interactions with matter held at Argonne National Laboratory in December of 1983. A critical problem in this work was establishing the relative detection efficiencies of the microchannel plates used to detect the scattered and ejected electrons [25]. Extensive studies on He [26] and N2 [27] were reported at impact energies of 200, 500, 1000, and 2000 eV over angular range of 30–150◦ in steps of 15◦ , and for secondary electron energies from 2 eV to one half the incident energy. A final study of this type employed 800 eV electrons on CO with special attention to analysis of the myriad of auto-ionizing states between 1 and 3 eV in ejected electron energy [28].

2.3. Molecular fragmentation: mass spectroscopy Our foray into the field of mass spectroscopy started when we became aware of the needs of plasma modelers for cross sections for various types of electron induced molecular dissociation. We began this work by constructing a TOF apparatus with two ion detectors separated by 180◦ and both at right angles to the incident electron beam [29]. See Fig. 3 for a description of the apparatus. The new experiments employed a pulsed electron beam with extraction fields for ions so that positive ions would be accelerated toward one detector and negative ions would go to the other. The first measurements were made on the important etching gas CF4 [30] and showed that earlier measurements had employed overly optimistic error estimates. Another discovery of this early effort occurred in measurements of the Ar2+ /Ar+ ratio in the mass spectrum for atomic argon. Previous work claimed 15–20% uncertainty in the individual cross sections but 2–3% uncertainty in the cross section ratios. We were able to show that the assumed cancelation of errors when taking the ratio was not warranted and in fact a more realistic error estimate was in the neighborhood of 10–12% [31]. Recommended documentation for reporting values of the ratio Ar2+ /Ar+ in future experiments as well as a review of all previous studies of the ratio were published by the same authors [32]. 2.4. Molecular fragmentation: positive ion–ion coincidence and polar dissociation In the experiments reviewed here the full potential of the apparatus described above was utilized. These multi ion coincidence studies made it possible to estimate the neutral dissociation cross section for CF4 [33] for the first time. These first experiments reported positive ion–positive ion coincidence studies on CF4 with cross sections given for the break up reactions C+ + F+ , CF+ + F+ , CF2 + + F+ and CF3 + + F+ from 40 to

Fig. 2. Schematic of the pulsed electron beam time of flight apparatus. The second drift tube moves in a horizontal plane about the scattering center and the electron beam direction is 69.8◦ with respect to the rotation axis. CFTD = constant fraction timing discriminator; STOP and START are the stop and start inputs to the time to amplitude converter (TAC); MCPHA = multichannel pulse-height analyzer; AMP = amplifier (Ref. [27], reproduced with permission).

R.A. Bonham / Journal of Electron Spectroscopy and Related Phenomena 155 (2007) 1–6

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Fig. 3. Schematic of apparatus. Acronyms are: MCP: microchannel plate detector; TOF: time of flight; EGUN: electron gun; AMP: preamplifier; CFD: constant fraction discriminator; PDG: programmable delay gate; TDC: time-to-digital converter; MEM: CAMAC memory; CON: CAMAC control module; e TOF tube: electron TOF tube. Note, the electron detector was placed 45 in. out of the plane of the figure in the forward direction and normal to the ion TOF tubes (Ref. [37], reproduced with permission).

500 eV in electron impact energy [34]. This study was followed by second study entitled Covariant mapping mass spectroscopy using a pulsed electron ionizing source: which was the first reported electron impact study of its type [35] and 8 previously unobserved break up channels for CF4 were reported. This experiment was somewhat analogous to Codling’s experiments with laser pulses but closer to Eland’s experiments using syn-

chrotron radiation [36]. By employing a TOF detector at 45◦ to the incident electron beam to detect scattered electrons and by studying mass spectra of the gases He, Ar, Kr, N2 , and CF4 it proved possible to obtain absolute efficiencies for the microchannel detectors for ions [37]. As a by-product of this study the first estimate of the total neutral dissociation cross section for CF4 was reported. A final experiment with the apparatus which utilized both negative and positive ion detection, usually termed polar dissociation, was carried out on CF4 where the only negative ion channels observed above 18 eV were those involving F− [38]. All the electron impact cross sections available up to 1994 for the important etching gas CF4 have been critically reviewed [39]. 2.5. Molecular fragmentation: electron ion coincidence experiments

Fig. 4. Cross-sectional view of the apparatus showing the position of the five electron detectors. The plane of this view is perpendicular to the view shown in Fig. 3 (Ref. [40], reproduced with permission).

Five TOF electron detection tubes were placed at scattering angles of 28◦ , 45◦ , 71◦ , 112◦ , and 135◦ in a plane perpendicular to the plane containing the incident electron beam and the positive and negative ion detectors in order to discover the angular dependence of electrons exciting a particular break up process (Fig. 4). In the case of N2 at incident energies of 24.5, 33.1, and 33.6 eV the angular dependence of the elastic scattering, N + N+ dissociative ionization, and N2 + ionization channels were observed [40]. Since the total cross section was known it was possible to determine the angular dependence of electron collisions leading to neutral dissociation plus excitation. For CF4 at impact energies of 22, 25, and 34 eV it was possible to observe elastic scattering, CF3 + + F, CF2 + + 2F (or F2 ), and CF+ + 3F (or F + F2 ) ionization channels [41]. This allowed the determination of elastic, total ionization, and total neutral dissociation cross sections for the impact energies studied. Note that CF4 is a special case since no excitation without dissociation has ever been observed.

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3. Summary The motivation for the work reviewed in this paper came primarily from the needs of theorists engaged in modeling radiation damage and plasma processing. A review of the new experimental methods developed in my laboratory for carrying out these tasks has been presented in chronological order of development. Ref. [39] contains values for all cross sections for the important etching gas, CF4 , known at the time the paper was written (1994). The work described here made substantial contributions to this pool of knowledge. In addition, data has been reviewed here on the many other molecules that we have studied. In particular, the work on the ratio of Ar2+ /Ar+ , described in Refs. [31,32], established the parameters that were important for judging the correct uncertainty estimate for measuring ion ratios by mass spectroscopy. In Ref. [33] the first estimate for the total neutral dissociation cross section for the etching gas CF4 was given which was made possible by our measurement of multi ion dissociation channels coupled with total cross section data measured by ourselves and others.

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

Acknowledgments

[25] [26] [27]

The author wishes to thank his many collaborators and the many organizations, both governmental and private that have supported the work reviewed here.

[28] [29] [30]

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[31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

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