Highly transverse velocity distribution of convoy electrons emitted by highly charged ions

Nuclear Instruments and Methods in Physics Research B 205 (2003) 830–834 www.elsevier.com/locate/nimb

Highly transverse velocity distribution of convoy electrons emitted by highly charged ions M. Seliger b

a,*

}kesi b, C.O. Reinhold c, J. Burgdo €rfer , K. To

a

a Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstr. 8-10/136, A-1040 Vienna, Austria Institute of Nuclear Research of the Hungarian Academy of Sciences (ATOMKI), P.O. Box 51, H-4001 Debrecen, Hungary c Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6373, USA

Abstract We present a theoretical study of convoy electron emission resulting from highly charged ion (HCI) transport through carbon foils. Employing a classical transport theory we analyze the angular and energy distribution formed by multiple scattering of electrons in the solid. We find that the convoy electron distribution becomes highly transverse at intermediate foil thicknesses representing an oblate spheroidal distribution due to the stepwise excitation of the HCI. The calculated convoy electron spectra are found to be in good agreement with recent measurements. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 34.10.+x; 34.50.Fa Keywords: Convoy electron emission; CTT

1. Introduction The angular and energy distributions of electrons emitted in ion–atom and ion–solid collisions have been studied extensively [1,2] during the last few decades. In ion–atom collisions one of the most prominent features of these emission spectra is that in the forward direction they exhibit a cuspshaped peak at the energy corresponding to the same velocity as the incident ion. In ion–solid collisions this feature is commonly referred to as convoy electron peak (CEP) [1]. Its origin is far more complex than in ion–atom collisions as a *

Corresponding author. Fax: +43-1-58801-13699. E-mail address: [email protected] (M. Seliger). URL: http://dollywood.itp.tuwien.ac.at/~marek/.

multitude of collision processes open alternative pathways to populating these low-lying continuum states. Very recently, measurements of the CEP for relativistic projectiles have become available [3]. Due to the vanishingly small cross section for electron capture to continuum (ECC) [4] at very high velocities, loss from the projectile by stepwise excitation (electron loss to the continuum, ELC) [5] and multiple scattering of liberated electrons are the main source of the CEP in this case. Extending our previously developed classical transport theory (CTT) [6,7] for evolution of hydrogenic projectile states during transmission through solids, we investigate in this communication the convoy electron emission by (moderately) relativistic projectiles. In line with the experiment [3], we focus on Ar17þ ions with an energy of 390 MeV/amu traversing thin self-supporting

0168-583X/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0168-583X(03)00587-1

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amorphous carbon foils of thickness varying from 25 to 9190 lg/cm2 . This collision system has several attractive features: In the relativistic velocity regime the average distance between two collisions, the mean free path, is long. The thinnest carbon foils available have a thickness of the order of the collisional mean free path, thus providing a testing ground for the limit of single collisions. By varying the foil thickness we have the opportunity to follow the time evolution of the projectile for up to several hundred collisions. In the present case we find that multiple scattering leads initially to the build-up of a transient population of Rydberglike states of the HCI. Ionization of such states gives rise to a highly transverse angular distribution. Comparison with recent experimental data shows excellent agreement and confirms this scenario. Atomic units ðjej ¼ me ¼ h ¼ 1, c ¼ 137Þ will be used unless otherwise stated.

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stochastic force (second term in (1)) acting on the electron in addition to the Coulomb force. The very high projectile velocity (vp ¼ 97 a.u.) allows us to treat the electron–solid interaction in the impulsive momentum transfer approximation, i.e. momenta D~ pi are transferred instantaneously at collision times ti reducing the transport problem to a random walk of the projectile electron along Kepler orbits subject to a stochastic sequence of momentum transfers. The probability distribution of D~ pi is determined from the relativistic differential inverse mean free path, whereas the flight times Dti between two collisions are obtained from the corresponding integral inverse mean free path. The collisions described by D~ pi and Dti are the scattering of the electron at the screened ionic cores and at target electrons in the solid. Details are discussed elsewhere [8,9].

3. Convoy electron emission 2. Brief outline of the theoretical description The initial state of the electron is represented within the CTT by a probability density in phase space which is initially given by a microcanonical distribution with the binding energy of the projectile electron in the ground state (Ar17þ (1s)). The time evolution is given by a reduced Liouville equation which is solved by test particle discretization, i.e. a Monte Carlo method [6–8]. The dynamics of each test particle is governed by a Langevin equation ZP~ r X ~ p_ ¼  3 þ D~ pi dðt  ti Þ; ð1Þ r i where ZP is the projectile charge. Here and in the following ð~ r;~ pÞ denote the position and the momentum of the electron in the projectile frame whereas primed variables as ð~ r0 ;~ p0 Þ correspond to the target frame. Outside the solid the time evolution of the electron attached to the projectile is determined only by the central Coulomb force represented by the first term of (1) which in a classical picture can be characterized by Kepler orbits with constant energy. Inside the solid the interaction with target electrons and nuclei results in a stochastic series of collisions described by a

Convoy electron emission is a subset of all ionization events converting Ar17þ into Ar18þ . Here we investigate the energy and angular distribution of emitted electrons. Fig. 1 displays the contour lines of the longitudinal ðv0k Þ and the transverse ðv0? Þ velocity distribution of convoy electrons for nine different propagation distances ranging from the single-collision regime to the multiple scattering regime involving up to hundreds of collisions. The direct ionization from the ground state results in a near-isotropic distribution with a weak enhancement in the transverse direction (d ¼ 25 lg/cm2 ). For longer propagation distances (d J 200 lg/cm2 ) the distribution becomes narrow due to emission from higher excited states and highly transverse reflecting the multistep ionization of convoy electrons through a sequence of intermediate excited and Rydberg states. Finally, for very thick foils (d J 3000 lg/cm2 ) the distribution increasingly resembles the Newton circle however shifted toward lower parallel velocities v0k < vp that is due to the slowing down of convoy electrons in inelastic electron–electron collisions representing post-ionization effects. The build-up of a highly transverse two-dimensional velocity distribution P ð~ vÞ at intermediate

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Fig. 2. Multipole expansion parameter of the CEP bk ðvÞ for a foil thickness of d ¼ 25 lg/cm2 as a function of the electron velocity in the rest frame of the projectile ion. Sign of bk changed as indicated in order to separate even and odd multipoles.

5 0 x x x 85 90 95 100 85 90 95 100 85 90 95 100 105

parallel velocity (a.u.) Fig. 1. Contour lines (0.1,0.2,. . .,0.9) of the velocity distribution (in a.u.) of simulated convoy electrons emitted by an Ar17þ (390 MeV/amu) in transport through a carbon foil. The foil thickness in units of lg/cm2 is denoted for every graph in the top right and the intensities have been normalized to one. The horizontal direction corresponds to the velocity parallel to the projectile ion. The initial projectile ion velocity of v ¼ 97 a.u. is denoted by a cross.

the significant velocity region for the CEP (up to v 6 1 a.u.). This multipole expansion features a number of interesting properties: r0 ðvÞ (not shown) represents the overall intensity at the given radial velocity v with the well-known cusp singularity proportional to 1=v factorized out. The first order-term

distances can be most conveniently parametrized in terms of a multipole expansion of the CEP [10] in the rest frame of the projectile ion ! 1 X drð~ vÞ r0 ðvÞ ¼ 1þ P ð~ vÞ ¼ bk ðvÞPk ðcos hÞ ; d~ v v k¼1 where Pk ðcos hÞ are the Legendre polynomials of the order k as a function of the polar angle h. The anisotropy coefficients bk ðvÞ are given by Z v drð~ vÞ dðcos hÞPk ðcos hÞ bk ðvÞ ¼ ðk þ 1=2Þ r0 ðvÞ d~ v ð3Þ with r0 ðvÞ ¼ vb0 ðvÞ parameterizing the angle-integrated probability. Fig. 2 shows the multipole expansion for electrons emitted in the transport through a layer of 2500 a.u. of carbon (d ¼ 25 lg/ cm2 ) while in Fig. 3 we show the average bk ðvÞ in

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Foil Thickness [µg/cm ] Fig. 3. Average of bk ðvÞ for the CEP within v 6 1 a.u. as a function of the target thickness. Sign of hbk i changed as indicated in order to separate even and odd multipoles.

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P1 ðcos hÞ ¼ cos h describes, to leading order, the forward–backward asymmetry in the spectrum. The term P2 ðcos hÞ represents the intensity distribution of a Hertz dipole associated with dipole transitions. The key point is now that due to multiple scattering as well as ‘‘hard’’ collisions high-order anisotropies are introduced well beyond the dipole limit. With increasing foil thickness, the expansion coefficients of higher-order even multipoles, bk ðvÞ, increase. For intermediate propagation distances, d < 1000 lg/cm2 , the distribution is dominated by a large number of even multipoles which in both magnitude and sign closely resemble the expansion of spheroidal harmonics, Smn ðcos h; cÞ, in terms of the Legendre polynoms Pk ðcos hÞ, with the parameter c being a measure for the ratio of the focal length to the wave length in the spheroidal wave function [11]. As shown by Szab o et al. [12] S00 ðcos h; cÞ describes the differential cross section dr=dv of a Rydberg electron emitted in a single collision with a gaseous target quite well. This function represents a strongly transverse, oblate spheroidal distribution. In view of the build up of excited states our present results are easily understood. With increasing path length, multiple collisions (mostly with ionic cores) sequentially build up transient population of highly excited bound states of the HCI. Subsequent collisions, in turn, result in bound to continuum transitions and hence convoy electron emission. As the magnitude of average momentum transfer jDpi j is, in general, large compared to the characteristic momentum distribution, pn ¼ ZP =n, for large principal quantum numbers n, the resulting angular distribution deviates strongly from the ‘‘soft’’ or dipole limit giving rise to the oblate distribution. It is worth noting that the present calculation of the angular distribution of the CEP is solely based on classical dynamics while the calculation of ELC emission from Rydberg atoms in single collisions was based on a quantum treatment in the first Born approximation [12]. It is well known that classical calculations fail to converge to the dipole limit [13] for soft collision jDpj ! 0. The fact that jDp=pn j > 1 is a prerequisite for a strongly oblate spheroidal distribution is also partly responsible for the good agreement between the CTT and the quantum

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calculation since the relevant ionization process is far from the dipole (or Bethe) limit. For longer propagation distances, odd multipoles become sizable, signaling the shift of the distribution to lower velocities in the lab frame. In general, the multipole parameters bk depend on v. Fig. 2 indicates, however, that odd multipoles increase linearly with v while even multipoles approach a finite limit as v ! 0.

4. Comparison with experiments In the experimental setup [3] a small aperture after the foil confines the acceptance angle to Dh ¼ 1° centered around the initial beam direction

Fig. 4. Comparison of the recent theoretical calculation (solid line) with experimental data (symbols) for convoy electrons emitted by the transport of an Ar17þ ion (390 MeV/amu) through a carbon foil of following thickness: (a) 25 lg/cm2 (2500 a.u.); (b) 530 lg/cm2 ; (c) 9190 lg/cm2 . Theoretical data have been convoluted with the experimental resolution (Dh ¼ 1°, DE ¼ 9 keV). Intensity has been normalized to one.

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guiding the electrons towards a magnetic spectrometer where a semiconductor detector counts all electrons in DE ¼ 9 keV centered around the nominal kinetic energy. The three different frames in Fig. 4 pertain to the three different regimes accessible: Fig. 4(a) displays the convoy peak in the (approximate) single-collision regime with a cusp shape dominated by b2 . Fig. 4(b) is a much narrower CEP reflecting the oblate spheroidal distributions discussed in the previous section. Finally Fig. 4(c) displays a drastically broadened and shifted CEP that is subject to a large number of slowing collisions subsequent to ionization. For all three cases we find excellent agreement with experimental data confirming, among other properties, the highly anisotropic emission pattern at intermediate propagation distances. A more complete analysis and a more comprehensive comparison between the experiment and theory is in progress [14]. In summary, study of convoy electron emission induced by moderately relativistic HCIs allows the study of the evolution of the CEP from the near single-collision limit to the true multiple scattering regime involving up to hundreds of electron–ion core and electron–electron collisions. The current CTT based on a Monte Carlo solution of a microscopic Langevin equation is capable to describe this complex evolution process, including the transient build-up of a high degree of anisotropy, quite well.

Acknowledgements We thank Y. Yamazaki and his colleagues for providing us with data prior to publication. The

work was supported by the Austrian FWF and EU under contract no. HPRI-CT-2001-50036. K.T. also acknowledges support from OAD A-19/2001. C.O.R. acknowledges support by the DCS, OBES, USDOE, managed by UT-Battelle, LLC, under contract DE-AC05-00OR22725.

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