Measurements of differential cross sections in the backscattering region

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Radiation Physics and Chemistry 76 (2007) 418–425 www.elsevier.com/locate/radphyschem

Measurements of differential cross sections in the backscattering region Brygida Mielewska Department of Physics of Electronic Phenomena, Gdan´sk University of Technology, 80-952 Gdan´sk, Poland Received 31 August 2005; accepted 17 January 2006

Abstract A review of experimental techniques which enable detection of electrons scattered in the backward direction, with an emphasis on recent developments in the magnetic angle-changing technique is presented. Results of measurements of differential cross sections for elastic electron scattering in the range of large scattering angles in rare gases and small molecules are discussed and compared with selected theoretical investigations. r 2006 Elsevier Ltd. All rights reserved. Keywords: Electron backscattering; Magnetic angle-changing technique; Elastic scattering

section at a given scattering angle y and electron energy E is given by

1. Introduction Studies of electron scattering from atomic and molecular targets particularly in the low electron energy region are a source of valuable information on electron–atom or electron–molecule interactions. Over the last decade, activity in this field has increased significantly, due to experimental advances, such as the use of an electronmirror spectrometer (Asmis and Allan, 1997) and the magnetic angle-changing technique (Read and Channing, 1996; Zubek et al., 1996). These new techniques allow measurements of scattering at large scattering angles, up to 1801. This paper reviews the latest developments and results of the measurements of differential cross sections in the backward direction from rare gas atoms and small molecules. In most experiments, differential cross sections (DCS) are measured in a cross-beam configuration, where an electron beam of a well-defined initial energy Ei collides with a target gas beam and scattered electrons of energy Es are detected, e ðE i Þ þ T i ! e ðE s Þ þ T f .

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Here Ti and Tf represent a target atom or molecule in the initial and final states, respectively. Differential cross Tel.: +48583472886; fax: +48583472821.

E-mail address: [email protected]. 0969-806X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2006.01.036

ds kf ðE; yÞi!f ¼ jf i!f ðE; yÞj2 , dO ki

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where ki and kf are the initial and final momentum of an electron, respectively, and f i!f is the scattering amplitude for the investigated process. If there is no change in the electron energy during the scattering process, the scattering is elastic and ki ¼ kf . Using the conventional spectrometric methods, differential cross sections for elastic scattering of electrons have been measured in the angular range from 101 to about 1301 and scarce experimental data have been obtained for angles above 1501. However, recently it has become possible to extend range of observed scattering angles up to 1801. An exact knowledge of DCS for electron scattering in a complete angular range allows more detailed comparison with theoretical calculations. Moreover, the integration of DCS over the angular range from 01 to 1801 allows an integral, s(E), and momentum transfer, sm(E), cross sections at a given electron energy E to be determined, from Z p ds ðE; yÞ sin y dy, (3) sðEÞ ¼ 2p dO 0 Z sm ðEÞ ¼ 2p 0

p

ds ðE; yÞð1  cos yÞ sin y dy. dO

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Both integral and momentum transfer cross sections are measured using independent experimental methods, transmission experiments and swarm method, and hence, comparison of results of these measurements with the integrated cross sections is particularly valuable. Recently, detailed review of experimental cross sections of atomic and small molecular targets have been given by Brunger and Buckman (2002), Karwasz et al. (2001), Zecca et al. (1996), McCarthy and Weigold (1995), and Trajmar and McConkey (1994). 2. Experimental techniques in the backscattering region

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uniform magnetic field applied along axis of the system. An electron ‘‘delay line’’ between the monochromator and the scattering region is included to distinguish the forward and backscattered electrons using a pulsed electron beam and the time-of-flight technique. Electrons scattered inelastically at 01 traverse the apparatus and reach the detector, while those scattered at 1801 are reflected by potential barrier at the monochromator exit and arrive at the detector with a time delay. The electron-mirror spectrometer has been used in the measurements of DCSs for excitation of four n ¼ 2 states in helium and brought results extending those from conventional electrostatic spectrometer (01–1361). The magnetic angle-changing technique (Read and Channing, 1996; Zubek et al., 1996) is applicable both for measurements of elastic and inelastic electron collisions. It has been also used for processes induced by photons (e.g., photodissociation, photoionisation). In this technique a magnetic angle-changer which consists of two or three pairs of concentric solenoids generates a static magnetic field localized in the scattering region. The localization is achieved by adjusting currents in the solenoids, in such a way that at least a dipole moment of the magnetic field equals zero. The magnetic field decreases very rapidly with an increasing radial distance from the interaction region and does not perturb operation of electrostatic lenses of the spectrometer. The magnetic field applied in the spectrometer deflects the incident electron beam and scattered electrons. Fig. 3 shows the trajectories of the incident and elastically scattered electrons in the localized magnetic field. Due to the conservation of azimuthal component of the angular momentum, the incident electrons traverse the scattering center, the scattered electrons stay in the scattering plane and move radially away from the scattering region. The angle of deflection depends on the electron energy E and the current in the inner coil I1, which can be described by

During the 80 years of history of electron scattering experiments, only few methods have been developed which allowed detection of backscattered electrons. The first one, a magnetic field immersed spectrometer (Gagge, 1933), applied a uniform magnetic field and was capable of measurements over the full angular range, from 01 to 1801. The principle of the operation of this spectrometer is illustrated in Fig. 1. The magnetic field (perpendicular to the plane of the figure), which is produced by Helmholz coils, deflects electrons leaving the gun to move along circular path. The gun can be rotated about the axis of the gas beam between two extreme positions corresponding to 01 and 1801 shown in the figure. Scattered electrons pass a set of slits where the last slit S2 and a Faraday cup are mounted on a movable drawer. By varying the distance between slits S1 and S2 it is possible to collect inelastically scattered electrons of selected energies. The energy and angular resolutions of this technique were rather low due to the effects of the magnetic field. An electron-mirror spectrometer developed by Asmis and Allan (1997) enables measurements of a ratio of DCS at 01 to that at 1801 at a given incident energy in an inelastic process. A schematic diagram of the spectrometer together with the diagram of applied potentials is shown in Fig. 2. An electron beam is selected in a trochoidal monochromator and analyzed by two analyzers with a

cI 1 a ¼ A arcsin pffiffiffiffi , E

Fig. 1. Schematic diagram of magnetic field immersed spectrometer; after Gagge (1933).

where A and c are constants for a given coils geometry (Read and Channing, 1996). It is seen that higher energy of deflected electrons requires larger currents. However, the strength of the magnetic field produced by the solenoids is limited by the physical size of the coils and by the risk of overheating in the vacuum environment. In one of the first sources of localized magnetic field in order to deflect 11 eV electrons by an angle of 701, current of 3.5 A was applied to a set of coils (Zubek et al., 1999). The subsequent modifications of this method provided magnetic fields sufficient to deflect electrons of energy up to 120 eV. The first attempt to increase energy range with the use of the magnetic angle-changing technique involved design of an iron-cored solenoid system (Cubric et al., 2000). In this source, space between two pairs of coaxial coils was filled with a ferromagnetic core with magnetic permeability of 1000. This increased the magnetic field

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Fig. 2. Electrode configuration and potential characteristics of electron-mirror spectrometer; after Asmis and Allan (1997).

Fig. 3. Electron trajectories of incident and elastically scattered electrons of the energy of 10 eV deflected in the localized magnetic field. The solenoids are shown in the figure by rings surrounding the scattering region. Electron trajectories computed with the use of CPO-3D program.

strength by a factor of 5, the energy of deflected electrons by a factor of 25, while preserving good field localization. In this way deflection by 901 of electrons of 120 eV energy was achieved at the inner coil current of 0.69 A. Another version of the magnetic angle changing device was developed by Allan (2000). In his system two pairs of solenoids are wound from a single piece of copper tubing providing additional water cooling, as shown in Fig. 4a. The main advantages of this design include: (i) open selfsupporting structure, (ii) application of only one current source for both solenoids, (iii) circulation of cooling water permitting high currents up to about 8 A. This coil system generates magnetic field capable of deflecting 100 eV electrons by an angle of 551. An interesting modification of the magnetic angle changer was suggested by Cho et al. (2003). In this case, the system consisting of two pairs of coaxial solenoids was shielded by m-metal in order to minimize the effect of the external stray magnetic fields. With the use of such a self-

Fig. 4. Modifications of magnetic angle changing technique: (a) watercooled magnetic angle-changer; after Allan (2000), (b) conical solenoid system; after Linert et al. (2004a).

shielding device, the operation of the electron spectrometer became relatively insensitive to the variation in the ratio of the inner-to-outer coil current. Using this method, measurements of differential cross sections for elastic scattering in krypton were carried out. The inner coil current of 1.2 A provided an angular deflection of 901 at an incident electron energy of 15 eV. Linert et al. (2004a) developed a novel conical–solenoid system, shown in Fig. 4b, in which the dipole and octupole

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moments are cancelled out (cancellation of the quadrupole and 16-pole follows from the cylindrical symmetry of the system of currents). This shape provides not only a very rapid decrease of the magnetic field but also an open structure of coils and a high efficiency in pumping the target gas away from the scattering region. Moreover, relatively low currents (less than 1 A) are required to deflect electron beams of energies below 15 eV. The conical solenoid system was used in the electrostatic spectrometer (Zubek and King, 1994) which consisted of a hemispherical electron monochromator and a rotatable analyzer covering, without application of the magnetic field, an angular range from 101 to 1201 with respect to the incident electron beam. This spectrometer has been used in the measurements of absolute differential cross sections for elastic electron scattering in argon (Mielewska et al., 2004) and molecular oxygen (Linert et al., 2004a, b), which are discussed in more detail in the following sections. The conical–solenoid system has been also used in a new 2D Toroidal Energy and Angle-Resolving Electron Spectrometer (TEARES) (Siggel-King et al., 2004, 2005) to study He photoionization. Here, the measurements were performed with the use of photon beam of 80 eV from the Daresbury SRS beamline MPW6.1. A deflection angle up to 1001 for electrons of energy of 56 eV was achieved with the inner coil current of 5 A. Recently, application of the magnetic angle-changer in low-energy superelastic electron scattering from the 41P1 excited state of calcium is reported (Hussey and Murray, 2005). 3. Differential cross sections in elastic electron scattering In this section, recent measurements of DCS for elastic scattering in rare gases and small molecules are discussed, with a particular emphasis on the region of large scattering angles. The absolute DCSs presented here have been measured with the use of a relative flow method (Srivastava et al., 1975; Nickel et al., 1989). In this method helium serves as a reference gas together with theoretical DCSs for elastic scattering calculated by Nesbet (1979) in the energy range below 19 eV and by Saha (1989) for higher energies. 3.1. Rare gases Argon has been extensively studied both theoretically and experimentally since the first measurements of elastic DCSs performed by Ramsauer and Kollath (1932a, b). More experiments in the low-energy range were recently performed by Srivastava et al. (1981), Furst et al. (1989) and Gibson et al. (1996). None of these measurements, however, provided cross sections above 1401 (except at 167.51 by Ramsauer and Kollath, 1932a, b). In Fig. 5a, recent DCSs of Mielewska et al. (2004) for elastic electron scattering measured above 1301 at 10 eV are compared with other available experimental and theoretical results. The theoretical calculations include the adiabatic exchange approximation of McEachran and Stauffer (1983) which

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take into account dipole polarization and exchange treated exactly, the R-matrix calculations of Fon et al. (1983) and Bell et al. (1984), the multiconfigurational self-consistentfield method of Saha (1991), which includes dynamical polarization and electron correlation effects, the polarizedorbital method of Dasgupta and Bhatia (1985) and the relativistic calculations of McEachran and Stauffer (1997) and Sienkiewicz and Baylis (1987). The experimental data of Furst et al. (1989) and Gibson et al. (1996) agree well with each other, particularly in the range from 151 to 1101. The results obtained by Mielewska et al. (2004) at 1301 are consistent with those of Gibson et al. (1996). It is worth noting that the theoretical results tend to overestimate the experimental DCSs in the region close to 1801 by about 9% in the case of calculations of Bell et al. (1984) and by about 100% in the case of calculations of McEachran and Stauffer’s (1983). Cho et al. (2003, 2004a) reported measurements of elastic DCS in krypton in the range from 151 to 1801 at energies from 10 to 100 eV. In Fig. 5b, the results obtained at 10 eV are compared with experimental data of Srivastava et al. (1981), Weingartshofer et al. (1974), Ramsauer and Kollath (1932b) and several theoretical calculations (Bell et al., 1998; Sienkiewicz and Baylis, 1992; Gianturco and Rodriguez-Ruiz, 1994; McEachran and Stauffer, 2003; Fon et al., 1984). Good agreement for backward scattering is found between the experimental data and the relativistic calculations of Sienkiewicz and Baylis (1992) which included polarization effects. The most recent relativistic calculations of McEachran and Stauffer (2003), which include dynamic distortion and absorption effects, show a very good agreement with experimental results for small-to-mid scattering angles but at 1801 their DCS lies approximately 50% above the experimental data. The theoretical treatment of Gianturco and Rodriguez-Ruiz (1994) reproduces well the measured angular dependence of the data of Cho et al. (2003) in the range from 1301 to 1801, but the agreement in the forward scattering and mid-angle region is unsatisfactory. Importance of absorption effects in electron–krypton scattering in the energy range up to 100 eV was analyzed thoroughly by Cho et al. (2004a). In Fig. 5c, their experimental and theoretical results at 20 eV are compared with other experimental (Srivastava et al., 1981; Danjo, 1988; Wagenaar et al., 1986) and theoretical results (Fon et al., 1984; Sienkiewicz and Baylis, 1992; McEachran and Stauffer, 2003). The best agreement in the region close to 1801 is seen between the experiment and calculations of McEachran and Stauffer (2003), which take into account the absorption effects via a complex optical potential calculated as a sum of real potential VR(r) of static and polarization interaction and of imaginary potential iVI(r) responsible for inelastic processes. 3.2. Molecular targets Absolute differential cross sections for elastic scattering in the backward direction have been measured for small

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(c) Fig. 5. Absolute differential cross sections for elastic electron scattering in rare gases; (a) argon, incident electron energy of 10 eV, (b) krypton, incident electron energy of 10 eV, (c) krypton, incident electron energy of 20 eV.

molecules, oxygen (Linert et al., 2004b), water (Cho et al., 2004b) and methane (Allan and Zˇivanov, 2005). Scattering of electrons from molecular oxygen at low electron energies and for scattering angles below 1501 have been investigated since 1970s (Trajmar et al., 1971; Shyn and Sharp, 1982; Woeste et al., 1995) but the first absolute cross sections measured with the use of relative flow method were obtained by Sullivan et al. (1995) and Green et al. (1997). Most recently Linert et al. (2004b) have measured DCSs up to 1801 in energy range from 7 to 20 eV. In Fig. 6a and b the experimental DCSs for energies of 7 and 20 eV, respectively, are compared with the results of R-matrix calculation of Woeste et al. (1995) and of Machado et al. (1999), which combines the Schwinger method together with the distorted-wave approximation and uses a complex optical potential. At 7 eV (Fig. 6a) the results of Linert et al. (2004b) are in a very good agreement with the data of Sullivan et al. (1995) and Green et al. (1997) in the common angular range 90–1301 although in the range of large scattering angles do not support those of Shyn and Sharp (1982). The available theoretical calculations predict different shapes of the cross sections with the most significant differences at forward angles. In the case of the backward scattering, the results of Woeste et al. (1995) follow well the experimental data, up to 1601 but the most recent calculations of Machado et al. (1999) predict at 1801 value of DCS approximately 30% higher than the experimental one. For 20 eV (Fig. 6b), agreement among the experimental results (Linert et al., 2004a, b; Sullivan

et al., 1995; Shyn and Sharp, 1982; Trajmar et al., 1971) improves and the theoretical cross section of Machado et al. (1999) stays in a very good agreement with the experimental data. Water has been extensively studied as a molecular target of a paramount biological importance. There is a wealth of information available in the literature on the elastic scattering of electrons in the energy range from 0.5 to 400 eV and covering angular range from 101 to 1561 (Danjo and Nishimura, 1985; Shyn and Cho, 1987; Johnstone and Newell, 1991). Recently, Cho et al. (2004b) have obtained DCSs up to 1801 in the energy range from 4 to 50 eV using the magnetic angle-changing technique. In Fig. 7, their DCS at 6 eV is compared with other experimental and theoretical results. Here, the agreement among experimental data is reasonable within the stated uncertainties up to about 1201. Above 1201, the results of Shyn and Cho (1987) differ from those of Cho et al. (2004b) by almost 100%. The results of theoretical calculations (Okamoto et al., 1993; Machado et al., 1995), particularly those of Gianturco et al. (1998) (the coupled-channel calculations in the body-fixed frame with the exact static-exchange potential) and the most recent results of the R-matrix calculation (Faure et al., 2004) follow the experimental results over the entire angular range. Methane being a relatively simple molecule of spherical symmetry is often considered as a convenient testing system for experimental and theoretical studies of polyatomic targets. While there are ample measurements and

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Fig. 6. Absolute differential cross sections for electron elastic scattering in molecular oxygen at incident electron energies of (a) 7 eV and (b) 20 eV.

Fig. 7. Absolute differential cross sections for electron elastic scattering in water vapour at incident electron energy of 6 eV.

Fig. 8. Absolute differential cross sections for electron elastic scattering in methane at incident electron energy of 5 eV.

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calculations of differential cross sections in methane reported over the past 30 years (Rohr, 1980; Tanaka et al., 1982; Sohn et al., 1986; Shyn and Cravens, 1990; Boesten and Tanaka, 1991; Bundschu et al., 1997; Gianturco et al., 1995; Machado et al., 1998), the elastic scattering in the full angular range have only recently been determined by Allan and Zˇivanov (2005). In Fig. 8, DCSs for elastic scattering at 5 eV are compared with available experimental and theoretical results. The overall agreement among the experimental data is fairly good. The calculations of Machado et al. (1998), who used the Schwinger variational iterative method, are in very good accord with the experimental results, except from 301 to 901, where they slightly overestimate the measured DCS, and in the backscattering region, where they are approximately 25% lower than the results of Allan and Zˇivanov (2005). On the other hand, the theoretical cross section of Jain (1986), in which an optical spherical potential has been used, agrees well with the experimental results at backward angles but at forward and mid-angles it deviates from all experimental cross sections.

4. Conclusions Recent progress in experimental techniques, and particularly in the development of the magnetic angle-changing technique accelerates studies of electron scattering off gaseous targets in the backward hemisphere. Since the first report of this method in 1996, many technical and experimental difficulties such as overheating and limitations in the energy of the deflected electron beam have been overcome. Application of magnetic angle-changer facilitates measurements of differential cross sections for elastic and inelastic collision processes in atomic and molecular targets and also studies of photoionization and photodissociation processes. The differential cross sections for rare gas atoms and small molecules presented in this paper are, to the best of author’s knowledge, the first ones reported in the angular range from 1301 to 1801. These DCSs measured with the use of magnetic anglechanging technique overlap well with those obtained previously at lower scattering angles using conventional spectrometric methods. Comparison of these DCSs with the results of theoretical calculations reveals substantial discrepancies that exist above 1301 for low electron energies (o15 eV). It is particularly interesting to point that in case of argon, all available theories overestimate DCS at 1801 even though their agreement with the experimental cross sections at lower angles is satisfactory. It was stressed by Gianturco and Rodriguez-Ruiz (1994) that at low electron energies and high scattering angles the DCS depend strongly on the applied treatment of the polarization-correlation interactions. Discrepancies that still exist among theories and experiments warrant further investigations.

Acknowledgements I am very grateful to Prof. Mariusz Zubek and Prof. George C. King for years of fruitful collaboration and for their support and encouragement in the course of studies with the magnetic angle-changing technique. Also, I would like to thank Irek Linert for his help with the experimental work and Prof. Michael Allan and Prof. Robert McEachran for providing their DCS in digital form. I am grateful to East West Task Force (EWTF) of European Physical Society for financial support in the CEPAS conference participation. Part of this work has been carried out within the programme, Electron Induced Processes at the Molecular Level (EIPAM) sponsored by European Science Foundation. It has been also partly supported by Polish Ministry of Science and Information Society Technologies. References Allan, M., 2000. J. Phys. B: At. Mol. Opt. Phys. 33, L215–L220. Allan, M., Zˇivanov, S., 2005. 14th International Symposium on ElectronMolecule Collisions and Swarms- satellite of ICPEAC 27-30 July, 2005, Campinas, SP-Brazil, p. 94. Asmis, K.R., Allan, M., 1997. J. Phys. B: At. Mol. Opt. Phys. 30, 1961–1974. Bell, K.L., Scott, N.S., Lennon, M.A., 1984. J. Phys. B: At. Mol. Opt. Phys. 17, 4757–4765. Bell, K.L., Berrington, K.A., Hibbert, A., 1988. J. Phys. B: At. Mol. Opt. Phys. 21, 4205–4216. Boesten, L., Tanaka, H., 1991. J. Phys. B: At. Mol. Opt. Phys. 24, 821–832. Brunger, M.J., Buckman, S.J., 2002. Phys. Rep. 357, 215–458. Bundschu, C.T., Gibson, J.C., Gulley, R.J., Brunger, M.J., Buckman, S.J., Sanna, N., Gianturco, F.A., 1997. J. Phys. B: At. Mol. Opt. Phys. 30, 2239–2259. Cho, H., Gulley, R.J., Buckman, S.J., 2003. J. Korean Phys. Soc. 42, 71–75. Cho, H., McEachran, R.P., Tanaka, H., Buckman, S.J., 2004a. J. Phys. B: At. Mol. Opt. Phys. 37, 4639–4645. Cho, H., Park, Y.S., Tanaka, H., Buckman, S.J., 2004b. J. Phys. B: At. Mol. Opt. Phys. 37, 625–634. CPO-3D programme, available at /http://www.electronoptics.com.S Cubric´, D., Ward, R., King, G.C., Read, F.H., 2000. Rev. Sci. Instrum. 71 (9), 1–3. Danjo, A., 1988. J. Phys. B: At. Mol. Opt. Phys. 21, 3759–3766. Danjo, A., Nishimura, H., 1985. J. Phys. Soc. Jpn. 54, 1224–1227. Dasgupta, A., Bhatia, A.K., 1985. Phys. Rev. A 32, 3335–3343. Faure, A., Gorfinkel, J.D., Tennyson, J., 2004. J. Phys. B: At. Mol. Opt. Phys. 37, 801–807. Fon, W.C., Berrington, K.A., Burke, P.G., Hibbert, A., 1983. J. Phys. B: At. Mol. Phys. 16, 307–321. Fon, W.C., Berrington, K.A., Hibbert, A., 1984. J. Phys. B: At. Mol. Phys. 17, 3279–3294. Furst, J.E., Golden, D.E., Mahgerefteh, M., Zhou, J., Mueller, D., 1989. Phys Rev. A 40, 5592–5600. Gagge, A.P., 1933. Phys. Rev. 44, 808–814. Gianturco, F.A., Rodriguez-Ruiz, J.A., 1994. Phys. Rev. A 47, 1075–1086. Gianturco, F.A., Rordiguez-Ruiz, J.A., Sanna, N., 1995. Phys. Rev. A 52, 1257–1265. Gianturco, F.A., Meloni, S., Paioletti, P., Lucchiese, R.R., Sanna, N., 1998. J. Chem. Phys. 108, 4002–4012. Gibson, J.C., Gulley, R.J., Sullivan, J.P., Buckman, S.J., Chan, V., Burrow, P.D., 1996. J. Phys. B: At. Mol. Opt. Phys. 29, 3177–3195.

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