Electronic stopping of protons at intermediate velocities

Nuclear Instruments and Methods in Physics Research B69 (1992) 1(1-17 North-Holland

[AN B

Beam Interactions with Materials&Atoms

Electronic stopping of protons at intermediate velocities G. Schiwietz and P .L. Grande '

Hal-Meiuur-Institut Berlin GmbH Deparurneiet Pi, Glienicker Straße 100, D-1000 Berlin 39, Geranany

Received 311 October 1991 and in revised form 23 December 1991

At intermediate velocities, near the stopping power maximum, most theories fail in the description of electronic energy transfer mechanisms. Only theories which go beyond perturbation theory are applicable in this velocity regime. A single-center coupledchannel code which includes dynamic curved projectile trajectories and which is h-used on an expansion of the time-dependent electronic wave function in terms of atomic wave functions is applied to the calculation of stopping powers. These time-consuming calculations may serve as a benchmark test for other models and allow for an accurate determination of the importance of different processes leading to the energy loss of ions in gases or insulators . Improved stopping, ionization and total reaction cross sections are computed for protons penetrating atomic H targets at energies of 10 to 500 keV/u. The results are compared to experimental data and to the predictions of the first-order plane-wave Born approximation .

1. Introduction The theoretical treatment of the slowing down of ions in matter has been greatly improved over the last decades. The free electron gas treatment for conduction band electrons has reached a particularly high level of sophistication . Reliable theoretical models have been developed for high incident velocities r (vP >> ru, where r u is the Bohr velocity) [1-41 and for low projectile energies (rP << r ) [5,61. For intermediate velocities, the stopping power has to be calculated for each projectile charge state separately [7,8] and nonperturbative models should be applied, since each target electron undergoes multi-step interactions with the projectile [9,10). This paper will focus attention on atomic treatments of the energy transfer process, which may be applied to all electron ; of insulators or of gas atoms, as well as to inner-shell electrons of conductors. Table 1 summarizes the approximations and range of validity of various models for ion-atom collisions. All models discussed in the following are restricted to the independent-electron approximation, as far as stopping power calculations arc concerned . However, this approximation is expected to be of minor importance, since: either the cross section for multi-electron transitions or the corresponding energy transfers arc small . One of the most serious approximations to the collision dynamics is the use of classical equations to describe electronic motion instead of a full quantum n CNPq fellow. on leave from Universidade Federal do Rio Grande do Sul, Brazil. 0168-583X/92/$05.00 0

treatment. Nevertheless, classical models may yield helpful analytical formulas to estimate cross sections or stopping powers . This holds true for the Bohr model [11], which describes the interaction of a moving point charge with a quasi-free electron at rest, or for the Firsov model [12], which describes electron capture and loss at low velocities on the basis of statistical assumptions. The binary encounter approximation (BEA) [13-7] goes beyond the Bohr model, in the sense that the target electron velocity distribution is incorporated in the model . For some simple cases, analytical formulas may also be derived from this model. In the classical trajectory Monte Carlo (CTMC) model the initial velocity as well as the coordinate space distribution of the electron are represented by a statistical ensemble [18-21] and all three-body forces between projectile, active electron and target core arc integrated numerically . This treatment was also extended to the four-body problem [21] and is quite reliable for energies near or somewhat above the stopping power maximum . Multiple interactions of the electron with the projectile and target core influence the collision dynamics in this energy regime . The main disadvantage of the CTMC is the neglect of molecular effects, such as quasi-molecular promotion, which are fully incorporated in quantum mechanical molecular orbital (MO) theories [22]. Such effects arc important e .g . for the description of electron capture at low energies. Finally, the dipole excitation or ionization cross sections, which arc dominant for low energy transfers at high incident velocities, are strongly underestimated in any of the classical collision models [23,21].

1992 - Elsevier Science Publishers B .V. All rights reserved

G. Schiwietz, P.L. Grande / Electronic stopping ofprotons Table 1 Approximations and estimated range of validity (in parentheses) for different nonrelativistic ion-atom collision models applicable to the computation of electronic stopping cross sections . The mean electron orbital velocity is denoted by r Single-step models Multi-step models Solutions in all orders Simplified models of the interaction Classical models"

Firsov model (vp « r  ), Bohr model

Quantum mechanical Bethe-Bloch theory models " (rP > r . ZP l r P < 0.4)

BEA

CCMC (r, = r  )

coupled-channel theories PWBA (Z P / rP < 0.4), second-order models, MO-model (rp « r  ) PSS-method (Z p / rp < 0.5) (arbitrary v P, Z p)

" For the description of the electronic motion . Thus, only quantum mechanical models yield accurate results for either low or high incident energies . For example, the quantum mechanical plane-wave Born approximation (PWBA) is reliable for excitation, ionization, as well as electron capture and loss, at high velocities or for low projectile charges [7,24-26] . Since these treatments rely on first-order perturbation theory, they arc applicable if the reaction probabilities are low and if the target wave function is only slightly disturbed during the collision . The Bethe theory yields the exact asymptotic high energy limit of the PWBA for excitation and ionization, though its range of validity is shifted somewhat towards higher energies as compared to the PWBA . The Bethe-Bloch theory provides an analytical expression for the stopping power of ions at high incident energies [2,3]. Models which go beyond first-order perturbation theory incorporate more than a single interaction between projectile and active electron, i .e ., they are able to describe multi-step processes. It should be emphasized, that all quantum models listed in table 1 ar,; three-body theories. Thus, they include the (screened) Coulomb interaction between target core and electron . This is not the case for the classical BEA and Bohr model, which are two-body theories (where the target only determines the binding energy and in the BEA the initial election velocity) . When the ratio of the projectile charge Z P and the projectile velocity exceeds about 0 .4 a.u . (atomic units : e = h/2,rr = m,. = 1 a.u.) first-order perturbation theory is no longer valid [27]. Two so-called polarization effects then come into play: i) Electrons may be excited or ionized via multi-step processes. The long range part of the projectile-target interaction leads to a considerable deformation of the target electron wave function in a first step (a positive incident ion attracts the electron) and this changes the ionization probabilities at small internurlear distances. Such effects may be described in second- or higherorder perturbation theory . Stopping cross sections have recently been calculated with such treatments for harmonic oscillator targets [28] . ii) The projectile-target interaction adds to the attractive target core electron potential and introduces time-dependent transition energies. This leads to the

so-called binding effect in the case of positively charged ions and may be accounted for in a perturbed stationary state (PSS) model [29]. The first polarization effect leads to an increase and the second to a decrease of excitation or ionization cross sections in the case of positively charged incident particles. For antiprotons or other negative incident particles the situation is reverse . The corresponding sign-of-charge dependence of the stopping cross section is known as Barkas effect [30]. When Zp /o P exceeds about 0.6 a .u . an infinite number of projectileelectron interactions should be taken into account . In principle, this is done in numerical solutions of the time-dependent Schrödinger equation. The most reliable of these models seem to be coupled-channel calculations . The first stopping power calculations of this type were only recently performed [9,10], since the computation of stopping cross sections requires the treatment of many final electronic states and is very time consuming. The ingredients of this coupled-channel model will be described in the following and a comparison with experimental results and predictions of first-order perturbation theory will be discussed in the last section of this paper . If not otherwise indicated, atomic units will be used throughout this paper. 2. Atomic orbital method (AO) In a full quantum mechanical description, the ionatom collision process is described by the Schrödinger equation H((R), (r})V`((R), (r) , t) = i -V`((R) , (r), f),

(1)

where (r) represents all electronic coordinates and (R) is the set of projectile and target nuclear coordinates. Under the assumption of classical nuclear trajectories R(t), the electronic system obeys the time-dependent Schrödinger equation [31,32]

12

G. Schiwietz, P.L. Grande / Electronic stopping ofprotons

and, self-consistently, the nuclear trajectories arc obtained through the Hamilton equations with the averaged Hamiltonian H_((R }) = «,((r) , t) I H((R), (r)) IIP,((r), t)). In this work, eq. (2) as well as the classical motion of the nuclei are solved numerically for each impact parameter b. The many-body Hamiltonian in eq . (2) is treated in the framework of the independent particle model(IPM) for one active electron [33] . Therefore, any dynamic correlation effects during the collision, as well as static Pauli correlation, are neglected in the present model. In this case, eq. (2) may be solved by expanding OJt) in terms of unperturbed eigenfunctions of the target (single center atomic wave function 0,). Thus eq. (2) may be replaced by a set of coupled first-order differential equations for the coefficients a,(t)= «, IOJI)) originating from the expansion d a.(t)= ~ai(t)e ' Vn~ (R(t)), i with the internuclear distance R and t

di

(4)

V,~ '(R(t))=«,IV,(R(t),r)10,), (5) where E, is the energy associated with the target wave function fir,. For bare incident ions, the active-electron-projectile interaction Vn,, is just the Coulomb potential. However, in the case where the projectile carries electrons, we use the hydrogenlike screened potential Zn _n n 1 ) V,(R - r) = - [ + n. Z~ff + IR-rl 1 R -r 1 ( Xcxp(-2Z~ff IR-rI)J,

(6)

where Zn is the projectile nuclear charge, nn is the number of projectile electrons and Z,. ff is the effective projectile charge as seen by the electrons which arc attached to the incident ion. It is pointed out that the dynamic interaction between the bound projectile electrons and the target electrons is not included in the present model. This, so-called dynamic screening or anti-screening, results in an enhancement of the ionization and excitation cross sections at high incident energies [34-36]. Considering the angular and radial parts of the atomic target-centered wave function 46,, we have evaluated the matrix elements (5) using the multipole expansion 1,+t,

V41(R) = _ E W,'':'ntG,:'(R)Y,.M(R), (7)

with - (4Tr(21,+ 1)(21,+ 1) ~ 21 + 1

W~ :m - II and

1, ~ (0

1, 0

L)1, 0 -M,

G,'(R) == 1'dr r=X,*X,ƒt.(r, R), where

min(r, R) t.

rnax(r, R)

(

1, m,

L -M )

(7a)

(7b)

(7c)

for an unscreenc i ;projectile. In the case of a screened projectile, as given by eq. (6), f(r, R) can be written in terms of modified spherical Bessel functions of the third kind and its derivative [37,38]. 1 and m are the quantum numbers associated with angular momentum and angular-momentum projection and X,, X, arc the radial wave functions of the states i and j respectively. The symbols (" ") in eq . (7a) represent the Wigner "3j" symbol as described in ref. [39]. The cross section for exciting or ionizing the active target electron to a state i is given by u, = lim f'db 21rbIa,(t, b)I `,

(8)

and the electronic stopping cross section can be directly computed as S,

= F_ Q,

A E

(9)

since each excited or continuum state i corresponds to a well defined energy transfer AEi = E, - E,, (E,, is the initial state energy). The treatment of continuum states, the adopted basis set, computation time and other details of the calculation may be foundelsewhere [9,10]. 3. Results and discussion In order to estimate the reliability of the singlecenter coupled-channel stopping power theory, results for different reaction processes may be compared to experimental data obtained under single-collision conditions. In previous papers, we have investigated multiple differential ionization cross sections at intermediate to high energy H++ He collisions (20 to 5000 kcV). It was shown that the present model yields accurate results for electron energy spectra [9], for the angular distribution of ejected electrons [40], as well as for energy transfer at small impact parameters [10] . In the present work an improved (coupled-channel) treatment of collisions between neutral atoms is introduced .

G. Schiwietz, P.L. Grande / Electronic stopping ofprotons

10-16

ö Û

10

10 16

10 2 101

Projectile Energy (keV)

103

Fig. 1 . Upper part: Experimental ionization cross section (open squares [41], open triangles 1421) in comparison with coupled-channel (solid line) and PWBA results [24) (dotted line) for the collision of hydrogen atoms . Lower part: averaged experimental excitation, ionization and electron capture cross sections (dashed lines [501) and total reaction cross sections (squares) in comparison with coupled-channel results (solid line) for protons on atomic hydrogen . Fig. 1 (upper part) displays experimental and theoretical results for projectile ionization in H ° + H ° collisions as a function of the projectile energy . The experimental results were obtained from final projectile charge-state distributions measured under single-collision conditions by McClure and by Hill et al . [41,42] and corrected for a small contribution due to electron capture (H° + H ° - H + + H - ). The figure displays two theoretical curves, namely PWBA results [24] and results of our coupled-channel calculation (AO) . In both theories, only single-electron processes were taken into account. The AO results are in agreement with the experimental data at projectile energies below 20 keV and the PWBA results seem to agree with the experimental data at energies below 10 keV. This is, however, only an accidental agreement, since the PWBA underestimates the experimental data at 1 keV by an order of magnitude (not shown in the figure), whereas coupled-channel calculations (with a small size basis set [43]) are in reasonable agreement with measured values down to 1 keV. At energies above 20 keV, two-electron processes start to occur, as was shown by Bates and Griffing [34] . These processes, double-ionization or simultaneous excitation-ionization of both

13

collision partners, dominate the projectile ionization cross sections at energies above 70 keV . Bates and Griffing performed very elaborate PWBA calculations for this di-electronic contribution in H n + Hu collisions. Presently, there are some approximate PWBA treatments available, which may also be applied to heavier systems [36]. It is noted that the di-electronic processes only have a small influence on stopping cross sections [7], since these processes are dominated by low energy transfers and the H ° charge-state fraction is quite low at energies above 70 keV [44]. Nevertheless, the di-electronic contribution is more important for H (' + H O than for any heavier collision system, since this contribution is roug-ty proportional to the number of electrons, whereas ionization, due to the static screened nucleus, is roughly proportional to the square of the nuclear charge (ZP). In most of our previously investigated cases, electron capture was of minor importance . Since charge transfer is accurately described only in a two-center theory [45-48], it may be argued that single-center theories fail in the case of high capture probabilities. However, a hypothetical single-center calculation with a complete (infinite) basis set inust provide the exact solution of the time-dependent Schrbdinger equation, including charge transfer [49]. With such a basis one could project the time-dependent wave function, in the limit of large times, onto bound projectile states and compute all the capture probabilities. The question is left open, to what extent a large basis set (about 100 gerade states [9] are used in this work) is able to mimic capture processes . A rough estimate shows that the projectile H(Is) state may be described up to internuclear distances of about 5 a.u . with the target-centered basis states of our calculation . Fig . 1 (lower part) displays experimental excitation, capture and ionization cross sections [50], as well as total reaction cross sections (the sum of the above) for H + + H collisions, as a function of the incident energy, compared to the results obtained in our model . The experiments were all performed with atomic hydrogen targets and the data are subject to an uncertainty of about 15% in absolute value . The total reaction cross section, in an atomic-orbital expansion, may be defined as atot = 2arf db b(1 - Pia;tial(b)), where Pmitial is the probability for staying in the initial state (elastic scattering) at the impact parameter b. It is evident from the figure that our atomic-orbital coupled-channel results (AO) are in excellent agreement with the experimental data. Furthermore, it is emphasized that resonant charge transfer dominates the total cross section at low speeds. Thus, we conclude that the process of electron capture is sufficiently well included in out calculation . At energies above 90 keV, only single center effects occur and our partial results for excitation and ionization cross sections match the experi .

14

G. Schtwietz, P.L . Grunde / Fiectronic stopping of protons

mental data (not shown in the figure) . However, within our model, it seems impossible to distinguish between electron capture and ionization at energies below 90 keV or between capture and excitation at energies below 50 keV in H + + H collisions . At these low velocities, it is interesting to investigate the energy loss spectra in detail . Fig . 2 displays energy loss spectra calculated with the coupled-channel code for an impact parameter of 0.1 a.u. in H'+ H collisions at low incident energies. Data points for energy transfers below the hydrogen binding energy (1,, = 13.6 eV) were obtained from the population of bound target-centered states per energy interval. This interval is proportional to ZPZ2/?13 for large principal quantum numbers :r. The arrows in the figure indicate the energy transfers corresponding to electrons captured into high projectile Rydberg states or into low continuum states of the projectile . As can be seen from fig. 1, electron capture is the dominant contribution to the electronic energy loss at projectile energies of 10 and 30 keV. The experimental data on charge transfer in H + + H collisions show that resonant electron capture (H++H(is)-H(ls)+H+) exceeds 90% of the total capture cross section at low incident energies . Thus, the electron capture contribution to the energy transfer should appear at energies below those indicated by the arrows. In the case of pure electron capture the contribution to a certain final energy transfer may be estimated. For a given projectile state n, the energy transfer in atomic units can be written as

where the projectile binding energy is denoted I . Thus, under the assumption of dominant resonant

Fig . 2 . Coupled-channel results for the energy loss spectra of protons incident on atomic hydrogen at energies of 10 and 30 keV for an impact parameter of 0.1 a.u .

Fig. 3 . Upper part : Theoretical electronic stopping cross section results for protons incident on atomic hydrogen . AO corresponds to a single-center expansion (this work) and AO + represents a two-center coupled-channel calculation [48,58]. PWBA stopping cross sections (dotted lines) were calculated according to the description of Bates and Griffing [24] . Middle part: Theoretical electronic stopping cross section results for neutral hydrogen projectiles on atomic hydrogen . Lower part : Equilibrium mean stopping cross section per atom for hydrogen beams penetrating a molecular hydrogen target . Experimental values: closed triangles [631, open triangles [62], closed circles [601, closed squares [59] and crosses [61] . The arrow indicates the maximum contribution due to the electron-electron interaction in H" +H" collisions. charge transfer, the most probable transferred energy should be about 19 and 8 eV for incident energies of 30 and 10 keV, respectively . The coupled-channel result for the most probable transferred energy at 30 keV is about 19.5 eV (see fig. 2), which is in good agreement with the above estimate for electron capture (a similar agreement was also found for 40 keV). This is an indication for a satisfying description of electron capture with a finite number of target-centered basis states. However, at the projectile energy of 10 keV the coupled-channel result for the most probable transferred energy is about 12.7 eV, which exceeds the estimated value by about 5 eV . This is a sign for breakdown of the present single-center expansion at low incident energies. It is pointed out that this breakdown only affects the energy-loss distribution and not

G. Schiwietz, P.L. Grande / Electronic stopping ofprotons the total reaction cross section (see fig. 1). Hence, we estimate the stopping power calculations (with the present basis set) to be valid at energies higher than 20 keV in the case of H + + H collisions . For lower velocities, either two-center theories are necessary to describe the collision dynamics, or the number of basis states must be increased . Fig. 3 displays theoretical results for electronic stopping cross sections in H + + H collisions (upper part) and H ° + H collisions (middle part). Coupled-channel results (AO and AO + ) are compared to predictions of the first-order perturbation theory (PWBA) for ionization and excitation. The AO + model is a two-center theory with additional united-atom orbitals, added in order to simplify the description of quasi-molecular excitation processes [48]. This model has proven to yield reliable results for electron capture and excitation cross sections down to very low incident energies. The AO + stopping cross sections plotted in the figure were derived from partial excitation and electron capture cross sections and total ionization cross sections [48,54] . First, it is noted that the single-center AO results exceed the AO + coupled-channel data at energies below 25 keV . This is due to the above discussed failure of the single-center expansion with a finite (incomplete) basis set . Furthermore, all coupled-channel results arc significantly higher than the PWBA values at energies below 150 keV. For H + + H collisions this difference is mainly due to charge transfer, and for H" + H collisions it comes from multi-step processes, e .g . within a single collision an electron may be excited first and then ionized in a second step . At energies around 500 keV, the AO data are about 5% lower than the PWBA results . This is a signature of the binding effect . "." % -ich may also be described by a perturbed stationary state theory [29]. The lower part of fig. 3 shows mean stopping cross sections for a charge-state equilibrated hydrogen beam penetrating a molecular hydrogen target . The experimental data were measured independently at different laboratories [59-62]. The theoretical curves for H, were obtained by weighting the corresponding theoretical results for atomic hydrogen targets (upper and middle part of the figure) with the experimentally determined charge-state fractions for hydrogen in H 2 gas [44]. For energies below 30 keV the AO + results were used for the H + stopping contribution . The agreement between experimental data and coupledchannel results is, in general, 5% or better. This demonstrates that the coupled-channel models allow for an accurate description of the different physical mechanisms leading to energy loss of light ions in gases. There are, however, two competing effects which have been neglected in the treatment described above . One is the di-electronic enhancement of ionization and

15

excitation cross sections (see upper part of fig . 1) . the other is the influence of the molecular target structure on stopping cross sections . Di-electronic processes in H" + H" collisions were treated using first-oricr perturbation theory by Dalgarno and Griffing ;7], and their maximum contribution of 13% to the stopping cross section is indicated by an arrow at 55 keV in the lower part of fig. 3 . The corresponding curve is not plotted in fig. 3, since there are no coupled-channel results for di-electronic stopping cross sections available. The molecular effect is of similar importance and is rather difficult to estimate . The ionization of projectile electrons in H" + H, collisions will be influenced by the nonspherical screening of each of the target hydrogen atoms, as well as by the coherent excitation due to the simultaneous interaction with botri target atoms, which will slightly increase the stopping cross section . The ionization of target electrons dominates the stopping power at energies above 100 keV and will be reduced due to the increased ionization potential in H, (15.8 eV compared to 13 .6 eV for H) . Using PWBA calculations with an effective target charge of 1 .078, we estimated that the largest reduction of stopping cross sections due to molecular effects is of the order of 10% at projectile energies around 50 keV and the reduction at 1 MeV is only 0.3%. Altogether, with a proper treatment of tire different di-electronic processes and molecular target effects, stopping cross sections could be calculated for a proton beam penetrating molecular hydrogen targets with uncertainties of about 2% or less . 4. Conclusions In this work, the first coupled-channel calculations are presented for stopping cross sections of neutral collision systems . Approximations involved in different atomic stopping power models and their range of validity are discussed . Spec~al attention is given to coupledchannel calculations within the single-center atomicorbital expansion method . Comparison is made with first-order perturbation theory and experimental data for stopping powers, total reaction cross sections and ionization cross sections in H" + + H collisions. The coupled-channel theory is in good agreement with experimental data, as long as single-electron processes are considered . Furthermore, it is shown that the charge-transfer process may be simulated in a singlecenter expansion . However, for incident energies below 25 keV in H + + H collisions, the energy loss spectra indicate a breakdown of the single-center treatment with only 100 states . It is found that the difference between experimental stopping power data and coupled-channel results is 5% at the stopping power maxi mum and less at other energies . This remaining dis-

16

G. Schiwietz, P. L. Grande / Electronic stopping of protons

crepancy exceeds the numerical uncertainty of our results and is assigned to di-electronic processes and effects of the molecular target structure.

Acknowledgements The authors would like to thank B. Sulik and M. Briere for helpful discussions and critical reading of the present manuscript.

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G. Schiwietz, P.L. Grande /Electronic stopping ofprotons [52] M.B . Shah and H.B. Gilbody, J. Phys. B14 (1981) 2361 ; M.B . Shah, D.S . Elliott and H.B . Gilbody, J. Phys. B20 (1987) 2481 . [53] J.T. Morgan, J. Geddes and H.B . Gilbody, J. Phys . B6 (1973) 2118. [54] T. Kondow, R.J . Girnius, Y.P . Chong and W.L . Fite, Phys. Rev. A10 (1974) 1167 . [55] R.F. Stebbings, R.A. Young, C.L . Oxleyand H. Ehrhardt, Phys. Rev. 138 (1965) 1312. [56] R.A . Young, R.F. Stebbings and J.W. McCowan, Phys. Rev. 171 (1968) 85 .

[571 [581 [591 [60] [611 [621 [631

17

IT Park, J.E . Aldag and J.M . George, Phys. Rev. Lett . 34 (1975) 1253. W. Fritsch, private communication . E. Bonderup and P. Hvelplund, Phys . Rev. 4 (1971) 562. S.D . Warshaw, Phys. Rev. 76 (1949) 1759. R.A. Langley, Phys. Rev. B12 (1975) 3575 . H.K. Reynolds, D.N .F . Dunbar, W.A. Wenzel and W. Whaling, Phys. Rev. 92 (1953) 742. J.A. Phillips, Phys. Rev. 90 (1953) 532, dand tprojectiles were used for incident energies below 40 keV.