Journal of Electron Spectroscopy and Related Phenomena 161 (2007) 2–5
Identification of higher order contributions in three-dimensional (e,2e) cross-sections for helium A. Dorn ∗ , M. D¨urr, B. Najjari, N. Haag, C. Dimopoulou, D. Nandi, J. Ullrich Max-Planck-Institut f¨ur Kernphysik, D-69117 Heidelberg, Germany Available online 16 February 2007
Abstract Fully differential cross-sections for single ionization of helium by 102 eV and by 1 keV electron impact have been obtained using an advanced reaction microscope. The data cover a large range of emission angles for a low-energy (E ≤ 15 eV) electron and different scattering angles for the fast electron. Significant electron emission out of the projectile scattering plane in between the binary and the recoil lobes is observed. The experimental data are compared with theoretical predictions from a three-Coulomb wave function model. © 2007 Elsevier B.V. All rights reserved. Keywords: Ionization; Electron impact; Kinematically complete
1. Introduction Electron impact ionization of atoms and molecules plays an important role in various areas like plasma and atmospheric physics and, therefore, the accurate knowledge of the respective cross-sections as well as the detailed understanding of the underlying reaction mechanisms are essential. The total ionization cross-section is dominated by collisions with strongly asymmetric kinematics where the projectile transfers only small amounts of energy and momentum to the target. This situation is considered to be experimentally well studied and theoretically essentially understood. In particular for higher impact energies of a few hundred eV the fact that the final state or post collision interaction of the two continuum electrons can be neglected allows the use of perturbative models as the first and second Born approximations. For lower energies down to about 100 eV more sophisticated treatments as, e.g., the three-Coulomb wave function (3C) [1], as well as its advanced version, the DS3C model [2] and distorted wave approaches [3,4] have proven to be applicable. More recently, numerically exact approaches which deliver accurate results for the low-energy regime have been developed, as the convergent close coupling (CCC) [5] and the exterior complex scaling technique (ECS) [6]. On the other hand the strongly asymmetric kinematics, so far, was studied exten-
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sively only for the coplanar scattering plane where the ejected electron is detected in the plane spanned by the incoming and the scattered projectile. Electron emission out of the scattering plane was not investigated except in one experiment by Beaty et al. [7] who for a particular, limited out of plane geometry did not find any peculiarity. Here we present a kinematically complete study for 102 eV and 1 keV electron impact on helium covering a large part of the solid angle for a slowly ejected electron (E2 < 15 eV) [8]. The resulting three-dimensional ionization cross-section images show electron emission out of the scattering plane which is a signature of higher order projectile target interactions and which cannot be reproduced by the applied 3C model. 2. Experiment The experiment was performed using a reaction microscope [9], which can detect one or several electrons in coincidence with the residual ion emerging from an ionizing collision. As depicted in Fig. 1 a well focused (1 mm), pulsed electron beam (pulse length ≈ 1.5 ns, repetition rate = 200 kHz, ≈104 electrons/pulse), produced by a standard thermo cathode gun, crosses and ionizes a supersonic He jet (1 mm diameter, 1012 atoms/cm3 ). Using parallel electric (1 V/cm) and magnetic (6 G) fields, the fragments in the final state are projected onto 2D position and time sensitive multi-hit channel plate detectors equipped with a delay-line read-out. In this way a large part of the full solid angle is covered, 100% for the detection of target ions
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Fig. 1. Scheme of the advanced reaction microscope.
and 80% for electrons below 15 eV. From the hitting positions and the time-of-flight the momentum vectors of the particles can be calculated. While the electric field is generated by means of two electrode arrays above and below the collision region, the magnetic field is produced by a pair of Helmholtz-coils. Different from all previous designs, the present reaction microscope has been decisively improved in order to meet the special requirements of electron impact ionization experiments reaching (i) maximum flexibility in the choice of the electron beam energy and extraction field strengths, (ii) additional redundancy in the data and (iii) substantially improved momentum resolution compared to ion-impact measurements. This is achieved by guiding the electron projectile beam (defining the z-direction, see coordinate frame in Fig. 1) exactly parallel to the electric and magnetic extraction fields, requiring a central bore (5 mm diameter) in the forward electron detector to allow for the passage of the non deflected electrons. Note, due to the jet velocity transversal to the extraction y direction, ions have an offset momentum of kR ≈ −6 a.u. hitting the ion detector located off the projectile beam axis. Thus, (i) any beam energy can be realized, with the present gun between 30 eV and 2 keV, aiming to reach eV beam energies with meV energy resolution from a photo-cathode in the near future. Moreover, (ii) scattered projectile electrons with a transverse momentum of 0.2 a.u. ≤ k1⊥ ≤ 1.2 a.u. are detected as well, such that a triple coincidence of all final state continuum particles delivers superior background suppression and, due to over-determined kinematics, optimal control by securing rather than relying on momentum conservation. This is demonstrated in Fig. 2 where it is shown for electron impact single ionization of helium, that the sum momentum of the three measured longitudinal momentum components of the final state particles corresponds to the incoming projectile momentum. Finally, (iii) from the scattered electron position and time-of-flight its scattering angle and the momentum transfer can be directly obtained with a resolution that is a factor of two or three better than that achievable by reconstruction from the recoil-ion and the ejected electron momenta via momentum conservation. For the present experiments as for typical ion-impact data the recoil-ion momentum resolution is kR⊥ , kRz ≈ (0.25, 0.15) a.u. For all electrons, including the scattered ones, the transversal resolu⊥ ≈ 0.1 a.u. The longitudinal resolution for the slow tion is k1,2 ionized electrons is k1z ≈ 0.02 a.u., while for the scattered pro-
Fig. 2. Sum of the longitudinal momentum components of the three final state continuum particles. The incoming projectile energy was E0 = 102 eV corresponding to the momentum of k0 = 2.75 a.u.
Fig. 3. Sum energy spectrum of the final state electrons for the projectile energy of E0 = 102 eV. Below Esum = 47 eV the spectrum is multiplied by the factor 100.
jectiles at E0 = 102 eV we estimated k1z ≈ 0.08 a.u. As can be seen in Fig. 3, this results in an energy resolution of the final state electrons which is sufficient to discriminate between the ground state and excited states of the residual ion. For E0 = 1 keV, the longitudinal momentum of the fast scattered projectile is not resolved due to its very short time of flight to the detector. This is no major restriction since, for the small scattering angles considered here, k1z = k0 − dk0z can be obtained from the energy loss of the projectile with dk0z = (EI + E2 )/vp (EI : ionization potential, E2 : ejected electron energy), with an uncertainty below 0.04 a.u. 3. Results In Fig. 4 experimental triple differential cross-sections for E0 = 102 eV, E2 = 10 eV and two different projectile scattering angles of θ 1 = 10◦ (a) and θ 1 = 15◦ (b) corresponding to momentum transfers of |q| = 0.67 a.u and 0.82 a.u., respectively, are shown. Due to the central bore in the electron detector the crosssection is not recorded inside a cone with 30◦ opening angle around the projectile axis. The 3D representation nicely reveals
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A. Dorn et al. / Journal of Electron Spectroscopy and Related Phenomena 161 (2007) 2–5
Fig. 4. Triple differential cross-sections for single ionization of helium by 102 eV electron impact and an ejected electron energy of E2 = 10 eV. (a) experimental cross-section for θ 1 = 10◦ projectile scattering angle (|q| = 0.67 a.u.). (b) θ 1 = 15◦ (|q| = 0.82 a.u.). The directions of the incoming and the scattered projectile as well as the momentum transfer are indicated by arrows. (c) 3C calculation for θ 1 = 10◦ . (d) 3C calculation for θ 1 = 15◦ . The axes show the cross-section in atomic units. The binning used in the representation of the cross-sections is 10◦ in the polar as well as in the azimuth angles.
the over-all shape and structures of the electron emission patterns. Clearly the binary lobes pointing to the right, originating from a binary knock-out of target electron, due to post collision interaction with the scattered projectile are tilted backwards with respect to the momentum transfer axis. The same is true for the recoil lobes pointing to the bottom left side of the diagrams, originating from backscattering of the ejected electron within the ionic potential. The strong sensitivity of the binary to recoil peak height ratio to the magnitude of |q| becomes obvious in comparing Fig. 4a and b. These features which are well known already from earlier experiments where electron emission was measured inside the scattering plane (dotted plane indicated in Fig. 4a) are well reproduced by the 3C calculations in Fig. 4c and d. This model was used in its standard form [1] with a two parameter ground-state wave function (type d in [10]). On the other hand the experimental and theoretical results diverge for emission angles in between the binary and the recoil lobes. The experimental cross-section does not show the deep minimum predicted by 3C theory. In particular the large scattering angle case exhibits a cross-section ridge bridging the minimum in a plane perpendicular to the incoming projectile beam. Since this feature is not axially symmetric with respect to the momentum transfer direction it can be attributed to a higher order effect in the interaction between the projectile and the target. While at low projectile energies (here E0 = 102 eV) such higher order contributions are to be expected we examine in the next step fast electron impact at E0 = 1 keV, a situation where a first-order
Fig. 5. Experimental triple differential cross-sections for single ionization of helium by 1 keV electron impact and an ejected electron energy of Eb = 10 eV. (a) θ a = 3.2◦ and (b) θ a = 6.7◦ .
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description should be more adequate. The resulting experimental cross-sections for E2 = 10 eV and two different projectile scattering angles θ 1 = 3.2◦ (|q| = 0.5 a.u.) and θ 1 = 6.7◦ (|q| = 1.0 a.u.) are shown in Fig. 5a and b, respectively. Now the final state interaction of the electrons is strongly reduced and indeed the observed binary and recoil lobes are essentially axially symmetric with respect to the q-direction. Nevertheless we recover the cross-section ridge between both lobes, albeit smaller relative to the binary peak height compared to the low-energy (E0 = 102 eV) case. In particular at |q| = 1 a.u. where the recoil lobe is small this structure becomes distinct and comparable in magnitude with the recoil lobe. Thus, second-order contributions can be identified even for kinematical situations which are strongly dominated by first-order collisions. The reason is that electron emission perpendicular to the scattering plane due to a first-order process is strongly suppressed and therefore even small second-order contributions can easily be identified. In conclusion we have measured fully differential crosssections for single ionization of helium for slow and for fast electron impact covering a large part of the full solid angle for slow electron emission. The resulting 3D cross-section pat-
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terns show the well known behaviour within the scattering plane which, at the higher projectile energy, is consistent with a pure first-order ionization mechanism. On the other hand the electron emission perpendicular to the scattering plane is dominated by higher order effects. Even at 1 keV the corresponding features can be clearly identified demonstrating the significance of the perpendicular geometry, which is essentially free from first-order contributions, for the testing of state of the art theories. References [1] [2] [3] [4] [5] [6] [7]
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