Transmission of highly charged ions through microcapillaries

Nuclear Instruments and Methods in Physics Research B 164±165 (2000) 504±510

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Transmission of highly charged ions through microcapillaries K. T} okesi *, L. Wirtz, C. Lemell, J. Burgd orfer Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstr. 8-10, A-1040 Vienna, Austria

Abstract Transmission of highly charged ions through microcapillaries is studied theoretically by a classical trajectory Monte Carlo method. The interaction of highly charged ions with the internal surface of the capillary is treated within the framework of dielectric response theory. We analyze the distance of closest approach and the angular distribution of the highly charged ions. As a projectile we consider N6‡ with an energy of 2.1 keV/amu. We ®nd the resulting charge-state distribution of transmitted projectiles in good agreement with ®rst measurements. Moreover, our calculations indicate that grazing collisions with the microcapillary surface hold the promise of direct observation of charge transfer and hollow-atom formation at large distance from the surface. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: 34.50.Dy

1. Introduction Collisions between highly charged ions (HCIs) and solid surfaces are currently at the center of numerous experiments and theoretical investigations [1±9]. The main motivation of these e€orts is the study of the fundamental interaction mechanisms between HCIs and solid surfaces. The accurate knowledge of these processes is very important for the use of HCIs as a surface diagnostic tool as well as for surface modi®cations (see e.g. [10,11]). * Corresponding author. Permanent address: Institute of Nuclear Research, Hungarian Academy of Sciences, ATOMKI, H-4001 Debrecen, P.O. Box 51, Hungary. Fax: +43 1 58801 13699. E-mail address: [email protected] (K. T} okeÂsi).

From a number of experimental and theoretical studies the following scenario of the HCI-surface interaction has emerged: When a highly charged ion approaches a solid surface, one or more electrons can be resonantly captured at a characteristic distance (dc ) into a Rydberg state of the projectile with a high principle quantum number nc . As a result, a multiply excited Rydberg atom with inner shell vacancies, a so-called hollow atom, is created. For metal surfaces, charge transfer of the weakly bound conduction band electrons into a highly charged ion sets in at large distances (e.g. dc ˆ 18.3 a.u. for N6‡ on Ni) from the surface. These hollow atoms are often referred to as above-surface hollow atoms or hollow atoms of the ®rst generation (HA1). The classical over the barrier (COB) model [2,3] is widely used to estimate the distance dc where the ®rst resonant charge transfer can take place and the quantum number nc of the projectile

0168-583X/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 9 ) 0 1 0 8 0 - 0

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state into which the electron is captured. The COB model also gives information about the interaction time for an HCI above the target surface and the neutralization dynamics. The interaction time available between ®rst capture and impact on the surface is limited by the image acceleration and is typically of the order of tI < 10ÿ13 s. During such a short time interval, complete relaxation to the neutral ground state is impossible. Therefore, the ion reaches the surface in a multiply excited state. At or below the surface the outer charge cloud is replaced by a much tighter one with the radius of the order of the bulk value of the metallic screening radius, forming a hollow atom of the second generation (HA2). Therefore, close collisions at the surface and bulk penetration provide a much more ecient pathway to complete relaxation. In turn, however, the information on the above surface hollow atom is (to a large extent) erased as its direct observation is limited to the time interval tI . In particular, many properties of hollow atoms such as the much speculated existence of long-lived multiply excited resonances escape the observation. Very recently, interactions of HCI with internal surfaces in microcapillaries have been introduced as an alternative technique to study above surface processes [12,13]. Ions travelling approximately parallel to the capillary axis will be attracted by image forces toward the cylindrically shaped wall of the capillary (Fig. 1). While the dominant fraction of projectiles will either exit without experiencing charge transfer (trajectory type 1, Fig. 1) or undergo close collisions with the surface similar to conventional grazing surface collisions (trajectory type 2), trajectories of type 3 will un-

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dergo only large-distance ``above surface'' collisions near the exit edge of the capillary. These ions can escape prior to hitting the wall and, hence, preserve the memory of the above surface hollow atom formation. In this work, transmission of HCI through microcapillaries is studied by a classical trajectory Monte Carlo method. The interaction between the HCI and the capillary surface is taken into account by the image interaction within the framework of dielectric response. We perform the simulations for a metallic microcapillary of Ni. As projectile, we use N6‡ with an energy of 2.1 keV/amu. 2. Theoretical framework The problem we are dealing with is the motion of a slow ion with charge q travelling with velocity ~ v in the microcapillary of radius q0 . We assume that the solid located outside the cylinder can be characterized by an isotropic dielectric constant …k; x† which is a function of the frequency x and wavenumber k of the electromagnetic disturbance. The electric potential is determined from the Poisson equation: r ÿ~ vt† : r2 U ˆ ÿ4pq d…~

…1†

The present boundary-value problem determining the interaction potential between the charge q and the capillary wall can be conveniently solved using cylindrical coordinates [14,15]. The ion is located at ~ vt ˆ ! r1 ˆ …q1 ; /1 ; z1 †. The potential is given by Z q 1 ‡ dx U< …q; /; z† ˆ j~ r ÿ~ r1 j 2p XZ dk Am eim/ eikz Im …kq†eÿixt ; if q < q0 ;  m

…2† and

Fig. 1. Sketch of the ideal microcapillary with typical ion trajectories. qc is the critical capture radius and dc is the critical capture distance.

Z 1 dx U> …q; /; z† ˆ 2p Z X dk Bm eim/ eikz Km …kq†eÿixt ;  m

if q > q0 : …3†

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Im and Km are the modi®ed Bessel functions of order m of the ®rst and second kind, respectively. The second term in Eq. (2) is the induced potential and the ®rst term is the electrostatic potential of the moving projectile which can be expanded as [15] Z XZ q q ˆ dx dk eim…/ÿ/1 † eik…zÿz1 †: j~ r ÿ~ r1 j p m  Im …kq< †Km …kq> †eÿixt ;

…4†

where q< (q> ) is the smaller (larger) of q and q1 . Am and Bm can be determined from the boundary conditions for the interface U< …q ˆ q0 † ˆ U> …q ˆ q0 †;

…5†

oU< oU> ˆ …k; x† : oq qˆq0 oq qˆq0

…6†

The resulting image potential is Z XZ q dx dk ei…kzÿxt‡m/† U…q; /; z† ˆ ÿ p m  Im …kq1 † vm …kq0 †Im …kq†d…x ÿ kv† ; …7† where

 Im …kq0 † vm …kq0 † ˆ ……k; x ˆ kv† ÿ 1† …k; kv† Km …kq0 † ÿ1 kq0 Imÿ1 …kq0 † ÿ mIm …kq0 † : ‡ kq0 Kmÿ1 …kq0 † ‡ mKm …kq0 †

…8†

The evaluation of Eq. (7) requires data for the dielectric constant …k; x† for the microcapillary material. Since detailed information about …k; x† exists only for a few solids and a rather restricted range of x and k values we use a plasmon-pole approximation. Following [16,17] we determine e…k; x† in terms of a sum of Drude-type functions which emphasize the in¯uence of single particle interactions for large momentum transfer and interactions with plasmons for small momentum transfer. In this approximation the energy loss function, i.e. the imaginary part of ÿeÿ1 …k; x† can be written as

Fig. 2. Energy loss function of Ni. Open circles: optical energy loss function derived from experimental data [18]; solid line: ®t to the experimental results using a sum of Drude-type functions.



 1 Im ÿ e…k; x† X Ai C i x : ˆ 2 2 2 2 2 i ‰…x0i ‡ k =2† ÿ x Š ‡ …Ci x†

…9†

To evaluate the constants (Ai , Ci and x0i ), we used the optical data for Ni as compiled by Palik [18] and ®t the limit e…k ˆ 0; x†. The resulting e…k; x† satis®es the generalized Thomas±Reiche± Kuhn sum rules [19] for response functions. Fig. 2 shows the energy loss function determined from experimental optical data and our ®t to the experimental results using a sum of Drude functions. From Eq. (7) we calculated ion trajectories using a standard 4th order Runge±Kutta method for the integration of the corresponding equations of motion.

3. Results and discussion In our simulations, a Ni capillary with a nominal radius q0 of 125 nm (2360 a.u.) and length L ˆ 1:5 lm  28; 400 a.u. was used. Since ensembles of experimentally available capillaries do not form ideal cylinders but have ``bumps'', bottlenecks and inhomogeneities, these deviations from the ideal geometry are taken into account within a

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statistical ensemble. The realistic capillary is simulated by allowing the cylindrical radius to vary randomly by 5% around its nominal value. The cylindrical walls are assumed to form terraces with lengths randomly distributed between 20 and 50 nm (400±1000 a.u.). Furthermore, fringe-®eld e€ects due to the edge at the capillary end are taken into account. The potential for the in®nitely extended cylinder (Eqs. (7) and (8)) is replaced near the exit plane by a three-dimensional wedge potential [20]. To estimate corrections due to fringe-®elds, the cylindrically shaped surface is replaced by a planar surface (i.e., a Cartesian wedge). This is justi®ed since the relevant ensemble of trajectories we analyze in the following is near the exit surface close to within  20 a.u. of one of the walls (forming the wedge) while the opposite wall is at that point about 250 nm ( 4700 a.u.) apart. Note, however, that such an approximation does not hold in the interior of the capillary where the distances to the opposing walls can be comparably large. As soon as the ion approaches the wall within the critical distance dc for capture, the electronic and ionic dynamics has to be treated self-consistently as charge transfer in¯uences the ion trajectory. The electronic dynamics is simulated within a Monte Carlo approach: electron capture, resonant ionisation, radiative and non-radiative decay are followed as a random event-by-event sequence with rates taken from the COB model for capture and loss [2,3] modi®ed for the cylindrical geometry. Auger and radiative rates are taken from atomic structure codes [21±27]. Details of the simulation will be given elsewhere [28]. Note that for any event the in¯uence of previous events along the history of one trajectory (ionic position, energy gain, charge state, shell occupation) is taken into account through variation of relevant parameters which determine the probability for subsequent events. Within the independent-particle model, simultaneous multiple capture and loss is included. The in¯uence of charge transfer processes on trajectories leaving the capillary is illustrated in Fig. 3 where we compare representative trajectories with frozen charge states as used in a preliminary investigation [29] with the present self-consistent

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Fig. 3. Comparison between ionic trajectories of incident N6‡ with frozen charge state q ˆ 6 and dynamically evolving charge state reaching q ˆ 2.

solution. The particular initial conditions were selected to highlight this e€ect. For the same incident charge state, the scattering angle for the frozen charge state is generally larger than for the case of dynamically evolving projectile charges due to the larger image acceleration in the absence of the neutralization process. For the same reason, the dynamical evolution of the charge state reduces the number of close ¯y-by events and increases the average distance of closest approach to the exit edge. In order to study the hollow-ion formation in microcapillary transmission, we have performed a classical trajectory Monte Carlo simulation with an ensemble of 5  107 primary trajectories. The spatial distribution of the ensemble is uniform across the opening of the capillary cylinder. As a ®rst test case we analyze the charge-state distribution of outgoing Nq‡ ions for which experimental data are available [13]. To simulate the spread of the incident beam, we use a Gaussian angular distribution of hi with a full width at half maximum (FWHM) of 2 , which is the estimated experimental value [30]. It is important to realize that the charge-state distribution of the ensemble reaches its asymptotic stable limit only after the

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ion is about 100 lm, i.e., almost macroscopic distances, downstream from the exit surface when the Auger relaxation is complete. This is in sharp contrast to conventional specular re¯ection surface scattering processes where, as a result of close collisions, the ®nal charge state is reached on a receding branch of the trajectory within a few angstrom above the surface. Fig. 4 shows the ®nal charge-state distribution of incident 2.1 keV/amu N6‡ ions transmitted through a Ni microcapillary. The agreement with the data is remarkably good. Within the subensemble of exiting ions, the incident charge state strongly dominates ( 99%). The ratio of the incident charge-state fraction to all other chargestates is directly related to the ratio of the entire capillary cross-section of the cylinder to the area subtended by a ring of thickness dc (Fig. 1) and therefore provides a direct test for the distance of ®rst capture. The present simulation gives overall a good agreement for ionic charge states on an absolute scale indicating that the COB model describes the complex multielectron transfer process above the surface quite accurately. Since the ®nal charge-state distribution depends also on a multitude of Auger relaxation processes, the distribution of electrons among n shells of the hollow atom also appears to be predicted reasonably well.

Fig. 4. Final charge-state distribution of incident 2.1 keV/amu N6‡ ions transmitted through a Ni microcapillary. Open circles: experiment [13]; solid circles: simulation.

There are, however, a few noticeable discrepancies for charge states q ˆ 3 and 4 which correspond to electron con®gurations with a net capture of two or three electrons, respectively. Their apparent enhanced stability could possibly be due to the presence of extremely long-lived metastable con®gurations which are currently not yet included in the simulation. Possible candidates for such metastable multiply excited states could include high-spin states or high angular-momentum states [31] or, alternatively, planetary type con®gurations [32]. This aspect will be explored in the future in more detail. From our simulations we can also extract the angular (h) distribution as well as the distance of closest approach (b) distribution relative to the nearest exit edge. To the extent that the h and b distributions are correlated, measurements of the charge-state distributions in coincidence with the scattering angle would provide the means to resolve subsequent stages of the hollow-atom formation above the surface. In order to demonstrate the angular separation we performed our simulation with an FWHM of the angular distribution of the incident beam of 0.2 . Note that for higher incident charge states the angular separation can be clearly seen even with much less stringent requirements on the initial beam divergence. Fig. 5 displays the emerging two-dimensional correlation patterns between the scattering angle and the distance of closest approach b, where panel (a) represents the asymptotic observable distributions while (b) gives the distributions in close proximity to the exit surface. Similar to grazing incidence surface collisions [4,33] the image acceleration of the HCI toward the wall manifests itself as a pronounced shift in the angular distribution towards larger scattering angles. We indeed observe a banana-shaped h ÿ b correlation pattern with a very well-localized angular distribution. It is noteworthy that the angular distribution for each charge state is much narrower (FWHM 6 0:05 ) than the initial angular divergence of the beam (FWHM ˆ 0:2 ). This is a direct consequence of the intrinsic high selectivity among all initial conditions which lead to transmission through the microcapillary in a given charge state. Low charge states are strongly

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tribution for the charge state q ˆ 5 is not due to a wider angular range for single capture in distant ¯y-by events but is due to the Auger decay of hollow atoms of (initially) lower charge states. To illustrate this e€ect we show in Fig. 5(b) the h ÿ b correlation pattern for the same charge states but during ¯ight as the ions pass through the exit surface. The di€erence between this distribution and the one in Fig. 5(a) is exclusively due to the Auger cascade while the angular and distance of closest approach distribution pattern is already essentially frozen out. This simulation suggests that di€erent stages of the hollow atom/ ion formation for Nq‡ should be directly accessible in future experiments provided that an angular resolution of the order of h 6 0:05 can be achieved. Even more detailed information could be extracted when the decay of the multiply excited state by photon or electron emission could be measured in coincidence with the scattering angle.

4. Conclusion

Fig. 5. Two-dimensional correlation pattern between the scattering angle and the distance of closest approach b calculated (a) at macroscopic distance (after Auger relaxation), (b) at the exit surface of the capillary.

correlated with close ¯y-by's (small bs) and larger scattering angles (h ' 1:25 ) while the e€ective single capture channel (q ˆ 5) extends from the threshold angle hc ˆ 1:13 for single capture to larger angles. Notice that below hc no capture takes place since such angles would correspond to b > dc . The comparatively broad angular dis-

We have presented Monte Carlo simulations of the transmission of multiply charged ions passing through a microcapillary target. The simulation treats the ionic and electronic degrees of freedom simultaneously. We ®nd the fraction of transmitted projectiles which are partially neutralized in good agreement with recent experiments. Moreover, the charge-state-dependent correlation between scattering angle and distance of closest approach predicts that angular resolved charge state distributions may provide direct information on the evolution of the charge cloud of a hollow atom at large distances from the surface.

Acknowledgements The work was supported by the Austrian Fonds zur F orderung der wissenschaftlichen Forschung (P12470-TPH).

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