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Energy loss of ions in solids: Non-linear calculations for slow and swift ions
Nuclear Instruments and Methods in Physics Research B 195 (2002) 91–105 www.elsevier.com/locate/nimb
Energy loss of ions in solids: Non-linear calculations for slow and swift ions Nestor R. Arista Divisi on Colisiones At omicas, Centro At omico Bariloche and Instituto Balseiro, Comisi on Nacional de Energıa At omica, 8400 Bariloche, Argentina Received 28 September 2001; received in revised form 15 January 2002
Abstract The historical approach to describe the energy loss of swift ions in solids is based on the Bohr, Bethe and Bloch theories. As is well known, the central parameter in these theories is the ratio g ¼ Z1 e2 = hv, whose value is generally used to delimit the ranges of applicability of the Bohr (g > 1) and Bethe (g < 1) theories. The transition between these regimes can be obtained by changing the ratio Z1 =v, although not by simply changing v. In fact, this scheme breaks down at low velocities, where quantum and non-linear effects arise. This domain is characterized by the strong oscillatory Z1 dependence of the stopping powers. This paper proposes a self-consistent non-linear approach to calculate the energy loss of heavy ions on a wide range of velocities. The model is based on the transport cross-section approach and on a previous extension of the Friedel sum rule for moving ions. The purpose of this study is to develop a non-linear stopping power evaluation method that could be applied at finite ion velocities, bridging the current gap between the low- and high-energy models. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 34.50.Bw; 34.50.Fa
1. Introduction The study of the interaction of ions with matter has been guided for many years by the inspiring works of Bohr, Bethe and Bloch (BBB) [1–3], which provide the fundamental basis on which a large body of experimental and theoretical developments have grown. Several additional developments based on statistical approaches [4,5], and including also effective-charge models [6,7], have become useful to
E-mail address: [email protected] (N.R. Arista).
systematize a large amount of data covering nearly all possible projectile–target combinations and a wide energy range [8]. The developments in this area, and in particular the effective-charge concept, have been recently analyzed [9,10]. The models emerging from these studies have achieved a significant degree of development, and are currently used in computer simulations to represent ion interaction processes of interest for applications in materials studies and medical physics. However, when it comes to analyze the behavior of slow ions, or the final stages of the electronic slowing down process, it becomes evident that these models cannot describe in a satisfactory
0168-583X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 0 6 8 7 - 0
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way the physics of the energy loss phenomena in this range. Thus, one of the most relevant facts is the appearance of strong oscillatory effects in the dependence of the stopping powers with ion atomic number Z1 at low velocities [11–13]. These oscillatory effects were explained in terms of quantum scattering theory [14] based on transport cross-section (TCS) calculations for slow ions in an electron gas (representing the conduction or valence electrons in the solid). An essential difference between this method and the BBB framework is the evaluation of quantum effects in a nonperturbative way (by numerical solutions of the Schr€ odinger equation); this is usually referred to as a non-linear stopping power calculation, to distinguish it from other studies which include perturbative approximations. Further developments for slow ions included the treatment of the Z1 -oscillations using density functional (DF) methods [15]. This explains the behavior of the stopping powers in the range of low velocities (v < v0 , being v0 the Bohr velocity). A recent study [16] considers an extension of the DF method to velocities below v0 for light ions. But the question of extending the DF calculations to intermediate or large velocities, and to heavier ions, poses a much more complicated numerical problem. An alternative approach [17] which conserves the essential non-perturbative (or non-linear) characteristics of the quantum analysis, and also explains the oscillatory Z1 -dependence, is based on a self-consistent treatment of the screeningscattering problem by introducing a parametric scattering potential [18] which is self-consistently adjusted by an overall screening condition provided by the Friedel sum rule. The original formulation of this method was also restricted to low velocities (nearly static ions). A more recent development along this line provided an extension of the Friedel sum rule to moving ions [19]; this opened the possibility of carrying out calculations for various ions on a wide energy range. Previous calculations for He ions [20], protons and antiprotons [21], showed good agreement with experiments. The objective of this work is to explore the application of the self-consistent method based on
the extension of the Friedel sum rule to the case of swift ions, covering a wide range of Z1 -values of experimental interest, and trying to bridge the currently existing gap between the quantum nonlinear models applicable at low-energies and the perturbative and statistical (effective charge) models that apply at high energies. The present model is based on the free electron gas picture and therefore it is expected to be most appropriate to describe the energy loss in metallic targets. Previous authors have explored in a heuristic way the application of DF results based on free electron gas picture to non-metallic targets, obtaining generally good agreement with experiments. The applicability of free-electron-gas and related methods to non-metallic targets is a subject of current investigation. The scope of the present analysis is illustrated in Fig. 1, which shows the domains of applicability of the various theoretical approaches [22]. Here the ranges of applicability of the Bohr and Bethe theories are indicated, as well as the domains corresponding to stripped and dressed ions. An additional ‘‘non-linear quantum region’’ has been included in the lower part of this figure. This
Fig. 1. Characteristic regions for the description of the electronic stopping of heavy ions in solids [22]. The line v=v0 ¼ 2Z1 schematically separates the domains where classical and quantum perturbation theories apply, while the curve v=v0 ¼ 2Z12=3 indicates the range of velocities above which the ions become nearly fully stripped of electrons. The lower part of the graph shows the region where the non-linear quantum behavior dominates, characterized by the oscillatory dependence of the energy loss with the atomic number Z1 .
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covers the lower range of energies where the BBB scheme breaks down and one has to resort to full quantum mechanical analysis in order to explain the electronic stopping process. The content of this paper is the following. Section 2 describes the formulation, including the TCS approach, the extended Friedel sum rule (EFSR) method, and the stopping power integration over relative electron–ion velocities. Section 3 includes a study of the scattering potential and a discussion of the various ion models that were analyzed, and shows some preliminary calculations and comparisons between these models. Section 4 contains the main results of the present calculations, considering in particular the cases of slow and swift ions. Finally, Section 5 contains a brief summary of the work and several final remarks. 2. Formulation 2.1. Transport cross-section approach We start by studying the scattering of target electrons in the field of an incident ion with nuclear charge Z1 e, and consider the calculation of the TCS, which is defined as usual by Z ð1Þ rtr ð1 cos hÞ dr: This quantity may be calculated by classical and quantum mechanical methods. In the classical approach one should solve the scattering relation h ¼ hðbÞ, where b is the impact parameter, so that Eq. (1), with dr ¼ 2pb db, may be integrated if hðbÞ is known. In the quantum mechanical formulation, the TCS for the scattering of electrons with relative velocity vr and wave vector k ¼ mvr = h (respect to the scattering center), is given by 1 4p X rtr ðkÞ ¼ 2 ðl þ 1Þ sin2 ½dl ðkÞ dlþ1 ðkÞ ; ð2Þ k l¼0 where dl ðkÞ denotes the phase shift corresponding to the scattering of wave components with angular momentum l ¼ 0; 1; . . . For the present discussion it is illustrative to define also the ‘‘logarithmic number’’ Lðvr Þ, so that the TCS may be expressed as
rtr ðkÞ ¼
93
4pZ12 e4 Lðvr Þ: m2 v4r
ð3Þ
In this way one separates explicitly the Rutherford factor ð4pZ12 e4 =m2 v4r Þ corresponding to unscreened Coulomb interactions. A simple way to characterize the effects of screening is by assuming an effective scattering potential of the form: V ðrÞ ¼ ðZ1 e2 =rÞ expðarÞ, where a is the screening parameter. For this model potential one may calculate the TCS through various approaches, obtaining the following results for the logarithmic term [23]. (a) Classical calculation (g > 1): 2mv2r 1 LCL ðvr Þ ffi ln ð4Þ c : 2 Z1 e2 a (b) Quantum plane-wave (g < 1): 2mvr 1 LPW ðvr Þ ffi ln : 2 ha
approximation
(c) Semi-classical approach: 2mvr 1 LSCL ðvr Þ ffi ln Re½wðigÞ c : 2 ha
ð5Þ
ð6Þ
Here c ¼ 0:5772 and g is the Bloch parameter associated to the relative velocity vr , g ¼ Z1 e2 =hvr . The function wðxÞ in Eq. (6) is the digamma function, and the real part of wðigÞ may be calculated very accurately using the approximation [23] Re½wðigÞ ffi
1 lnð1 þ C2 g2 Þ c; 2
ð7Þ
with C ¼ expðcÞ ¼ 1:781. The assumptions used to derive these expressions [23] apply when the screening distance ks ¼ a1 is sufficiently large compared with the minimum impact parameter Z1 e2 =mv2r in the classical picture, and with the de Broglie length h=2mvr in the semiclassical and plane wave approximations, respectively. In other words, these expressions apply in general when the arguments of the logarithmic terms are large. The most comprehensive of these results is the semiclassical one, LSCL ðvr Þ, which may be considered the equivalent of the Bloch result for the TCS.
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an ion with velocity v, atomic number Z1 , carrying a number Ne of bound electrons, may be written in the form 1 2X ð2l þ 1ÞGl ðv; vF Þ ¼ Z1 Ne ; p l¼0
Fig. 2. Logarithmic functions related to the TCS of Eq. (3), versus the relative electron-ion velocity vr . The analytical results from Eqs. (4)–(6) are indicated as Cl, PW and SCL, respectively. The solid symbols are the results obtained from Eq. (2) using the phase shift values determined by numerical integrations of the radial Schr€ odinger equation. These calculations correspond to Z1 ¼ 5 and a ¼ 0:2 a.u.
It includes the other two cases as particular limits. Thus, for g 1, LSCL ðvr Þ ffi LCL ðvr Þ, whereas for g 1, LSCL ðvr Þ ffi LPW ðvr Þ. The behavior of these logarithmic functions is illustrated in Fig. 2 with explicit calculations for Z1 ¼ 5 and a ¼ 0:2. The solid circles show the results of numerical calculations using Eq. (2), with the phase shifts dl determined by numerical integrations of the radial Schr€ odinger equation. The figure also shows the previous approximating formulae, Eqs. (4)–(6). It may be observed that the SCL formula connects the CL and PW results in a continuous way, and it also approaches the exact quantum values at relatively high velocities. The negative values of LSCL at low velocities may be avoided by improving the approximation of Eq. (6) [23]. As the figure shows, the deviations of the approximations from the most accurate quantum calculations become more important at lower velocities. 2.2. Extended Friedel sum rule The generalization of the Friedel sum rule to finite (non-zero) velocities was developed in Ref. [19], so that only a brief description of the formulation is given here. The extended sum rule for
ð8Þ
where vF is the Fermi velocity of the solid. The functions Gl ðv; vF Þ take into account the contribution of each l-wave component to the screening charge, and may be expressed as integrals over a displaced Fermi sphere (DFS) of the corresponding phase-shift contribution, as follows: Z 1 ddl ðkÞ Gl ðv; vF Þ ¼ dX dk 4p DFS dk Z kmax ddl ðkÞ ¼ gðk; vÞ dk; ð9Þ dk kmin where kmin ¼ maxf0; v vF g and kmax ¼ v þ vF . The function gðk; vÞ takes into account the angular part of the integration over the DFS, and the expressions for the cases v < vF and v > vF were given in Ref. [19]. dl ðkÞ is the asymptotic phase shift of the electron wave function due to the scattering process, and is a function of the wave vector k corresponding to the relative electron–ion motion (k ¼ mvr =h, where ~ vr ¼ ~ ve ~ v). The derivative ½ddl ðkÞ=dk gives the contribution of each l-wave component to the accumulation of screening charge around the intruder ion. It may be shown from these expressions that one retrieves the usual Friedel sum rule in the low-velocity limit (v < vF ), and a perturbative form of the sum rule for high velocities (v > vF ) [19]. In the following we will use atomic units, so that the variables k and vr may be used indistinctly. Once the self-consistency condition of Eq. (8) is achieved (by varying the screening parameter a), we use the final phase shift values to calculate the TCS rtr , from the usual expression 1 4p X ðl þ 1Þ sin2 ½dl ðk; vÞ dlþ1 ðk; vÞ : rtr ðk; vÞ ¼ 2 k l¼0 ð10Þ We note that the phase shifts dl and the TCS rtr in Eq. (10) depend also on the ion velocity v, because of the optimization of the scattering po-
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distribution of electron velocities in a Fermi sphere (0 6 ve 6 vF ) and over the range of relative electron–ion velocities k ¼ j~ vr j (with jv ve j 6 vr 6 v þ ve ). This integration can be performed in a closed form using the expression [24,19] Z vF Z jvþve j 1 ve dve dk k 4 rtr ðk; vÞ Sðv; rs Þ ¼ 2 2 4p v 0 jvve j v2 v2e 1þ ; ð12Þ k2
Fig. 3. Calculations of the phase shifts dl for (i) Z1 ¼ 2, vr ¼ 1 and a ¼ 1 and (ii) Z1 ¼ 5, vr ¼ 5 and a ¼ 0:3, for a range of l values. The lines show the results obtained from the semiclassical approximation, Eq. (11), while the solid circles are the exact (numerical) values.
tential (through the adjustment of the screening parameter a ¼ aðvÞ) as described before. A couple of illustrative examples are shown in Fig. 3 which shows the calculated phase shifts for Z1 ¼ 2 and 5, relative velocities vr ¼ 1 and 5, and for two screening conditions: a ¼ 1 (strong screening) and a ¼ 0:3 (week screening). The circles are the values calculated by numerical integration of the Schr€ odinger equation whereas the lines are the values obtained from the semiclassical approximation [23] for dl , given by Z m 1 V ðrÞ dr dSCL ðkÞ ¼ h i ; ð11Þ l 2 h r0 k 2 ðl þ 1=2Þ2 =r2 1=2 where r0 ¼ ðl þ 1=2Þ=k. Clearly, the number of phase shifts required in the TCS calculation increases both with vr and with decreasing a. In both cases shown in Fig. 3, the semiclassical approximation gives an excellent description for l 1. This approximation was useful in order to speed up the calculations described next. 2.3. Stopping power integration After the TCS is evaluated for a wide range of relative velocities vr , we calculate the stopping power S ¼ hdE=dxi by integrating rtr over the
where rtr ðk; vÞ is calculated from Eq. (10) using the numerically obtained phase shift values. This yields the stopping power for any (non-relativistic) ion velocity v and for a given value of the Fermi velocity vF , related to the electron gas parameter rs by rs ¼ 1:919=vF ¼ 0:621n1=3 (in a.u.), where n is the electron gas density. Therefore, we characterize the final stopping value as a function of v and rs . The results of these calculations will be analyzed in the next sections.
3. The scattering potential Since we are dealing with partially stripped ions, the scattering potential in this study was assumed to be a sum of two components: (a) an ionic core potential Vcore ðrÞ, which includes the atomic screening provided by the bound electrons carried by the ion and (b) the outer screening potential Vs ðrÞ, which is produced by the target valence electrons in order to neutralize the net charge of the ion. The assumption of spherical symmetry of the potential is one of the main assumptions of this model, and may be consider to represent the spherical average of the real potential. A basic difference between both potentials, Vcore ðrÞ and Vs ðrÞ, is that while the ionic component remains essentially frozen when the ion velocity increases (besides the possible changes due to the increased stripping), the outer screening component changes with velocity since the electrons readjust themselves in a dynamical way to the field of the moving ion. The screening component Vs ðrÞ is adjusted for each velocity, in order to satisfy the extended sum rule. In these calculations this component was modeled through a Yukawa
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potential, using a single parameter a whose value is adjusted in a self-consistent way for each velocity v using this rule. The optimization method is the following: for each ion velocity an initial value of a is assumed and then many phase shifts dl ðkÞ are calculated (by numerical integration of the Schr€ odinger equation and using the semiclassical approximation for l 1); then Eqs. (8) and (9) are used to evaluate the extended sum rule. These calculations are repeated for each velocity by an iteration routine, varying the value of a until the sum rule (8) is precisely satisfied. In this way the changes in the screening conditions due to dynamical effects are taken into account by the variation of the screening parameter a. 3.1. Ion models In these calculations several ion models were tested. This includes in particular the following models: (a) Moliere and Thomas–Fermi (TF) potentials for neutral atoms [25]; (b) Brandt–Kitagawa (BK) model [7]; (c) Dedkov ion model [26]; (d) ‘‘Moliere-ion’’ model (to be defined next). The potential in all these cases is represented as Vtotal ðrÞ ¼ Vcore ðrÞ þ Vs ðrÞ ¼
N e e2 qe2 /core ðrÞ / ðrÞ; r r s
ð13Þ
where q is the charge of the ion and Ne ¼ Z1 q is the number of bound electrons; /core and /s are the core-screening and outer-screening functions respectively. The core-potential term is assumed to be a short-range potential, whereas the screening potential is adjusted by the self-consistent method (extended sum rule) and its range increases with ion velocity (dynamical screening effect). Some of the models (a)–(c) indicated above are well known and the first two have been widely used. The ‘‘Moliere ion’’ model is introduced and applied here based on the simplicity and convenience of the well-known Moliere potential for atoms. The modified potential for ions will be represented by Eq. (13) where Vcore ðrÞ ¼ ðNe e2 = rÞ/core ðrÞ, and where the core-screening function, /core ðrÞ /Mi ðrÞ, will be represented here by a sum of three terms,
/Mi ðrÞ ¼
3 X
Aj expðaj r=aTF Þ;
ð14Þ
j¼1 1=3
being aTF ¼ 0:8853=Z1 . In the case of neutral atoms (q ¼ 0) the normal values of the coefficients are used, namely [25]: Aj ¼ ð0:1; 0:55; 0:35Þ and aj ¼ ð6; 1:2; 0:3Þ; so that in this case one retrieves from Eq. (13) the Moliere-atom potential given by Vtotal ðrÞ ¼ ðZ1 e2 =rÞ/M ðrÞ, being /M ðrÞ the usual Moliere screening function. To represent ions with increasing degrees of ionization i ¼ q=Z1 , the coefficients in /Mi ðrÞ are modified as follows: first the value of the third coefficient A3 is decreased (until it eventually becomes zero, for i ¼ 0:35); for higher ionization (0:35 < i < 0:9) a similar reduction is done on the value of A2 and finally (if i > 0:9) the value of A1 is decreased (until complete ionization is reached). In any case (i.e. for a given velocity), the sum of these coefficients should correspond to the actual ionization degree q=Z1 , namely A1 þ A2 þ A3 ¼ Ne = Z1 ¼ 1 q=Z1 . A comparison between the charge densities corresponding to the TF, BK and Moliere potentials for neutral atoms is made in Fig. 4 for Z1 ¼ 80. As it may be seen, there are significant differences between the BK and the two other models; the BK model yields a higher concentration of electronic charge in a smaller volume. The same behavior was observed for all Z1 values. This will produce important differences in the stopping
Fig. 4. Comparison between the charge densities corresponding to the TF, BK and Moliere potentials for neutral atoms with atomic number Z1 ¼ 80.
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Fig. 5. Comparison between the present model (Moliere-ion) and Dedkov’s model for Z1 ¼ 80 and 20, and for two values of the ion charge in each case, q ¼ 10 and 60 (for Z1 ¼ 80), and q ¼ 5 and 10 (for Z1 ¼ 20).
coefficients, particularly for slow ions, as will be shown later. Fig. 5 shows comparisons between the Moliereion and Dedkov’s models for Z1 ¼ 80 and 20, and for two values of the ion charge q in each case. The differences between both models are not very significant, and so both models may be expected be equally appropriate for the present stopping power analysis.
Fig. 6. Stopping power calculations using the linear dielectric formulation (RPA) and the non-linear method proposed here, for bare ions with Z1 ¼ 1 and 7. The Bethe and Bloch limits for high velocities are indicated.
3.2. Preliminary calculations As a first test of the method, calculations for several simple cases were performed. These calculations were done by two different methods: (i) using the standard dielectric formalism [27] based on Lindhard’s (RPA) dielectric function eðk; xÞ, where the stopping power is represented in the form Z 1 2 dk 2 SðvÞ ¼ 2 jq ðkÞj pv 0 k ion Z kv 1 dx xIm ; ð15Þ eðk; xÞ 0 where qion ðkÞ is the form factor corresponding to the ion charge distribution [7] and (ii) following the quantum TCS approach with screening adjusted by the EFSR as described before.
Fig. 6 shows comparisons for bare ions with Z1 ¼ 1 and 7 (obviously, for bare ions there are no differences between the various ion models previously considered). The asymptotic limits from the Bethe and Bloch theories are also indicated in the figure. The figure illustrates the basic differences between linear (or perturbative) and non-linear calculations. We observe a significant reduction of the non-linear results as compared to the dielectric (RPA) values when the ion charge is increased from 1 to 7. The Bloch result describes very well this behavior for large velocities. Since the RPA values follow the Z12 scaling, the relative drop of the non-linear results indicates a ‘‘saturation’’ in the stopping process (i.e. the values increase by a much smaller proportion than expected from the Z12 scaling). In the case Z1 ¼ 7 one may observe that for high energies the present results approach
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Fig. 7. Stopping power calculations for partially stripped N3þ ions (with Z1 ¼ 7, Ne ¼ 4 and q ¼ 3), described according to the BK and Moliere-ion models. The dotted line shows the linear (Lindhard’s dielectric function) calculation for the BK ion model whereas the dashed line is the non-linear (extended sum rule) calculation for the same ion model. The solid line is the non-linear calculation using the Moliere-ion model.
first the Bloch limit (for v=Z1 < 1Þ merging finally with the linear results when v=Z1 > 1. This doublelimit behavior is similar to the one shown in Fig. 2 for a much simpler case. Fig. 7 shows calculations for partially stripped N ions (Z1 ¼ 7, Ne ¼ 4, q ¼ 3), obtained from different types of approaches: linear (dielectric) and non-linear calculations using the BK-ion model, and non-linear calculations using the Moliere-ion model. The ion potential form of Eq. (13) has been used in the non-linear calculations, using the Friedel sum rule to adjust the screening potential Vs ðrÞ. As it may be observed, both non-linear calculations are consistently smaller that the RPA values. In addition, there are significant differences at intermediate and low velocities between the BK and the Moliere ion model, as well as between the linear and non-linear calculations based on the BK-ion model. As it will be further illustrated in the next section, the non-linear calculations based on the BK-ion model yield also too small lowvelocity stopping power coefficients, S=v, for all Z1 . Fig. 8 shows the a values obtained from the application of the extended sum rule in the cases
Fig. 8. Values of the screening parameter a obtained from the application of the extended sum rule in the cases illustrated in Figs. 6 and 7 (i.e. Hþ , N7þ and N3þ ). The symbols are the selfconsistent a-values obtained by this method, while the dotted and dashed lines are the RPA values for the static pffiffiffiffi ða0RPA ¼ 1:563= rs Þ and high-velocity ðaRPA ðvÞ ffi xP =vÞ limits.
illustrated in Figs. 6 and 7 (i.e. Hþ , N7þ and N3þ ). The nearly constant values of a obtained by the self-consistent EFSR method (aEFSR ) at low velocities illustrate the ‘‘frozen-screening’’ limit for v < vF , but it should be noted that the limiting values of aEFSR required by the sum rule are generally different from the static RPA value [27] pffiffiffiffi given by a0RPA ¼ 1:563= rs (dotted line). On the other hand, for v > vF one observes a rapid convergence of the self-consistent aEFSR values to the high-velocity RPA limit, aRPA ðvÞ ffi xP =v (dashed line), wherepxffiffiffiP is the plasma frequency, related to rs by xP ¼ 3=rs3=2 . It should be remarked that this high-velocity limit, aEFSR ! xP =v, for v vF , is automatically provided by the EFSR without any external constraint. As a corollary, the EFSR results for the energy loss asymptotically approach the Bethe limit. Finally, Fig. 9 illustrates the differences in the stopping power results, calculated in a non-linear way according to the EFSR, using the Dedkov, the Moliere-ion, and the BK-ion models. These calculations correspond to C ions incident on amorphous carbon targets under conditions of charge equilibrium. The ions are assumed to have the average equilibrium charge (qs ) for each velocity as described in Section 4.
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Fig. 9. Non-linear stopping power calculations for C ions incident on amorphous carbon targets assuming different ion models: BK, Dedkov’s (D) and Moliere-ion (M). The target is represented as an electron gas with rs ¼ 1:6.
4. Non linear stopping power calculations Using the approach outlined before, further and more elaborate calculations were performed for various cases of experimental interest. We now consider the results obtained for slow and swift ions. 4.1. Slow ions As it was mentioned, one of the striking features of the low-energy stopping phenomena is the oscillatory dependence on the ion atomic number Z1 [11–13]. These oscillations are a clear signature of quantum effects, produced by strong perturbations of the scattered waves. They may be described rather accurately by the phase shift analysis and through TCS calculations. Fig. 10 shows results of calculations of the stopping coefficient S=v obtained by the previously described method, assuming neutral atoms and considering three types of potentials: Moliere, TF and BK. One observes a good agreement between the first two models. The results from the BK potential yield too small values and show some particular features which do not agree with experiments. The analysis of these potentials, and of the corresponding densities already shown in Fig. 4, indicates that the reason for this discrepancy lies in the much smaller sizes of the ions and atoms
99
Fig. 10. Quantum mechanical calculations of the low-velocity stopping coefficient, S=v, for neutral atoms, considering three types of descriptions: Moliere, TF and BK.
described by the BK model, a question that becomes critical at low velocities. The change in the interference pattern as the ion velocity increases is illustrated in Fig. 11, which shows calculations for v ¼ 0:2, 1, 2 and 3 a.u., using the Moliere atom model. As it may be observed, as the velocity increases the amplitude of the oscillations diminishes. The soft Z1 dependence obtained for v ¼ 3 a.u. resembles those expected from various statistical models [4–7]. In part (b) of this figure the values of S=v have been superposed in a linear scale to show more clearly how the amplitude of the oscillations decrease. By analyzing in detail the changes in the contributions to the TCS from the different l-values, one finds that the large amplitudes of the oscillations at low velocities arise from the fact that only a few phase shifts (typically l 1–4) are important, whereas at higher velocities the number of phase shifts that should be included in the sum of Eq. (10) increases steeply. It may be shown from qualitative arguments that for a screened potential of range ks (with ks ¼ 1=a in our case) one has to include a number of terms lmax > ks v to get a convergent behavior. The superposition of a large number of terms in Eq. (10) washes out the oscillatory Z1 dependence. The present calculations have been extended to the wide range of atomic numbers across the periodic table (1 6 Z1 6 92) for a velocity v ¼ 0:8 a.u. and for carbon targets. The results using both the Moliere and the TF models are shown in Fig. 12.
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Fig. 12. Calculations and experimental values [11–13] of the stopping power for a fixed ion velocity, v ¼ 0:8 a.u., in carbon targets (described as an electron gas with rs ¼ 1:6), as a function of projectile atomic number Z1 . The calculations correspond to Moliere ( ) and TF (- - -) potentials.
Fig. 11. Change in the Z1 -oscillation pattern with increasing ion velocity (v ¼ 0:2, 1, 2 and 3). The calculations have been performed assuming a Moliere potential, and for the electron density of carbon (rs ¼ 1:6). Panel (a) shows the variation in the stopping power values (on a logarithmic scale), whereas panel (b) shows the attenuation of the Z1 -oscillations, by representing the ratio S=v on a linear scale.
The comparison with the experiments is good for low and intermediate Z1 values, but it deteriorates in the upper Z1 range. Also illustrative are the growing differences obtained with the TF and Moliere models in this range. This indicates an increasing sensitivity to the way in which the projectile electronic density is described. These growing differences may be explained by analyzing the magnitude of the potentials for different Z1 values, as illustrated in Fig. 13 for Z1 ¼ 20 and 80. As it may be observed, for large Z1 values the Moliere potential decreases sharply with r, whereas on the other hand the TF potential decreases mildly. This long range behavior of the TF potential may be considered unrealistic; however, there is another important question that has not been taken into account in these calculations
Fig. 13. Variation of the Moliere (—) and TF (- - -) potentials, for Z1 ¼ 20 and 80, versus the radial distance r.
and may produce a similar effect, which is the charge state of each ion. The ionization would also increase the range of the potential and hence also the stopping (note that the mean charge is ex1=3 pected to increase roughly as Z1 in this range of velocities according to Bohr estimation [1]). Further studies along this line, considering the equilibrium charge state distribution of each particular ion, may be quite useful, as well as the use of more accurate electronic densities, like those provided by Hartree–Fock calculations. Nevertheless, it should be stressed that the overall agreement ob-
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tained from these non-linear calculations is satisfactory for many elements. The effect of average ion charge will be studied in some detail in the case of swift ions, where it becomes quite important and where some general estimations can be made. 4.2. Swift ions The previous results for slow ions include the calculation of the TCS for strongly screened systems (basically, neutral projectiles). This simplifying assumption applies only to slow projectiles. We now turn the attention to the analysis of calculations performed at intermediate and high energies. In this case, the projectiles get ionized and one has to resort to the use of model potentials that include a description of both the close-packed ion core as well as the extended screening charge distribution of target electrons that builds around it. This is represented by the two-component model for the self-consistent potential introduced by Eq. (13). The screening component of this potential, Vs ðrÞ, is adjusted according to the extended sum rule of Eq. (8), by varying the screening parameter a. The calculations presented here have been made using in Eq. (13) a screening function of the form /s ðrÞ ¼ expðarÞ, and the parameter a was adjusted for each ion velocity v. The required number of phase shifts was of the order of 10 at low velocities but it increases strongly with velocity. In fact, for v vF the value of a decreases asymptotically like the RPA value, a aRPA ¼ xP =v (dynamical screening limit, cf. Fig. 8), and so the maximum l-number (lmax v=aÞ increases like v2 . Hence, the convergence of the partial-wave series, Eq. (10), becomes slower. For the largest velocities considered here (20 a.u.), up to 600 phase shifts were included in the summation. However, the calculations converge rapidly when the high-l terms are approximated by the semiclassical values. These calculations have been performed for various ions incident on carbon targets. The choice of carbon here is based on the relatively small K-shell ionization contribution to the stopping power, which is not included in the present form of
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the non-linear model. Nevertheless, the K-shell contribution has been included here in a simplified way using the Bloch approximation for the stopping logarithm, with an excitation energy corresponding to the K shell of carbon. In this approximation, shell corrections have been neglected, and the Bloch function was set to zero in the lower velocity region where it turns negative. Two types of non-linear stopping power calculations were performed using the following approaches: (i) direct TCS calculations assuming a ‘‘frozen’’ Moliere atom potential, followed by stopping integrations using Eq. (12) and (ii) selfconsistent calculations using Moliere-ion potentials with variable external screening adjusted according to the extended sum rule method. Some further calculations have also been made using Dedkov’s model, treated with the same variable screening and sum rule methods, but no significant differences have been obtained as compared to those based on the Moliere ion model. An additional problem in the calculation for ions is that it requires an explicit specification of the charge of the ions for each velocity. The ions are assumed to have reached a charge-state equilibrium [28]; however, these charge states cannot be specified accurately, since the experiments so far give only the charge states after emerging from a solid foil, and, among the different models proposed, some of them predict differences between the charge states inside the solid and those of the emerging ion beams [28]. Due to this currently unavoidable uncertainty, the present calculations have been done considering two possible equilibrium charge states inside the foil: (i) the ‘‘effective charge’’ given by the usual fitting [28]: Zeff ¼ Z1 ½1 expðc1 v=Z1b1 Þ , with c1 ¼ 0:95, b1 ¼ 2=3 and (ii) the ‘‘average emerging charge’’ for solid foils, qs , which may be well represented by the Nikolaev– 1=c c Dmitriev fitting [29]: qs ¼ Z1 =½1 þ ð1=uÞ 1 1 , with c1 ¼ 0:6 and u ¼ v=v1 Z 0:45 , being v1 ¼ 1:645 a.u. It should be noted, however, that neither of these approaches is completely justified a priori, since, on one side, the effective charge Zeff is empirically obtained [8] assuming a Z 2 scaling which is not consistent with the lines of the present nonlinear description, and, on the other, the average emerging charge may be different from the actual
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equilibrium charge of the ions inside the solid, as indicated before. Hence, the calculations based on these charge values will be considered as exploratory. Based on these premises, Figs. 14–17 show results of calculations for C, Cl, Ni and U ions incident on carbon targets, described as an electron gas with rs ¼ 1:6 (corresponding to the density of amorphous carbon and considering four valence electrons per atom). The calculations are compared with experimental results from various sources [30,36]. The figures illustrate several aspects. The lower curve (dotted line) shows the results obtained from the standard Moliere-atom model without screening adjustment. As it may be observed, the ‘‘frozen’’ Moliere-atom model yields too small stopping power values. This is expected, since this model does not take into account the expansion of the screening cloud around the ions, whose size grows with ion velocity (ks ¼ a1 v=xP , for large v). The effect of the expanding screening cloud – which is automatically taken into account by the extended sum rule – is essential to obtain the right behavior of the stopping power at intermediate and high energies.
Fig. 15. Non-linear calculations of the energy loss of Cl ions in carbon targets, and comparisons with experimental results [32,33]. ( ): frozen Moliere-atom model; (- - - and —): Moliere-ion model, with q ¼ Zeff and q ¼ qs , respectively. The lines denoted with a and b are, in each case, the results without and with the K-shell energy loss term.
Fig. 16. Non-linear calculations of the energy loss of Ni ions in carbon targets and comparisons with experimental fitting results [34]. ( ): frozen Moliere-atom model; (- - - and —): Moliere-ion model, with q ¼ Zeff and q ¼ qs , respectively. The lines denoted with a and b are, in each case, the results without and with the K-shell energy loss term.
Fig. 14. Non-linear calculations of the energy loss of C ions in carbon targets (with rs ¼ 1:6), and comparisons with experimental results [12,30,31]. The dotted line shows the results for a frozen Moliere-atom model. The dashed and solid lines, for q ¼ Zeff and q ¼ qs , respectively, correspond to the Moliere-ion model with external screening adjusted using the extended sum rule. The lines denoted with aand b are, in each case, the results without and with the K-shell energy loss term.
The calculations using either q ¼ Zeff or q ¼ qs are shown by dashed and solid lines, respectively. The lines denoted a and b are the calculations without and with inclusion of the carbon K-shell contribution to the energy loss. We observe that the differences due to the ion charge assumption (either q ¼ Zeff or q ¼ qs ) are very small for C ions, but they become increasingly important for heavier ions. We find that a better
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Fig. 17. Non-linear calculations of the energy loss of U ions in carbon targets and comparisons with experimental results [35,36]. ( ): frozen Moliere-atom model; (- - - and —): Moliere-ion model, with q ¼ Zeff and q ¼ qs , respectively. The lines denoted with a and b are, in each case, the results without and with the K-shell energy loss term.
agreement with experiments is obtained by the q ¼ qs assumption. The comparisons with the experimental results in the first three cases are good, while the results for U ions are still not satisfactory. In all cases, the results for q ¼ Zeff are too low as compared with the experiments, whereas those for q ¼ qs appear to be more appropriate. It should be noted that the energy losses due to charge exchange or projectile excitations have been ignored in these calculations. In conditions of dynamical equilibrium one may consider different processes of electron capture by the moving ions, followed either by direct ionization or by excitation and subsequent ionization, so that cycles of electron capture and loss are completed. Such cyclic processes also contribute to the equilibrium stopping power, but a quantitative study is still missing. According to previous estimations [37] for the case of Ni projectiles in carbon, charge exchange processes may be expected to yield a maximum contribution of the order of 5% for energies around the stopping power maximum. A contribution of this order may be considered compatible with the present estimations, but more extensive calculations of these effects would be useful for an accurate description. It may also be observed that in the case of C ions (where the uncertainty between the q ¼ Zeff
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or q ¼ qs assumptions is small) the current predictions overestimate the experimental values (even though projectile inelastic processes are ignored). This may indicate a lack of accuracy of the method. It is finally illustrative to analyze the present results in terms of the magnitude of the effective Bloch parameter, g ¼ q=v, which provides a measure of the expected non-linearities in the energy transfer in terms of the ion charge q (instead of the conventional definition in terms of the nuclear charge Z1 e). This parameter may be considered useful when the collision radius (rcol maxfq=mv2 , h=2mvg) is larger than the ion radius, a condition that applies in both cases illustrated here. The values of g depend also on the charge-equilibrium model used. Fig. 18 shows the values of Zeff ðvÞ and qs ðvÞ (right scales) and the corresponding values of g (left scales) for C and U ions. The range of
Fig. 18. Values of the effective charge Zeff and average charge qs (right scales) and of the corresponding values of g ¼ q=v (left scales) for C and U ions. The dashed lines show the values of Zeff and geff ¼ Zeff =v; the solid lines show the values of qs and gs ¼ qs =v.
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energies corresponds to velocities larger that 1 a.u. (25 keV/u) since the charge equilibrium models are not considered accurate at low velocities, and the usefulness of the parameter g should be revised. In the case of C there is little difference between the models and we find maximum g values of about 1.5. In the case of U, we observe important differences, and the maximum values fall in the range gmax 4–7 at low velocities. On one side, the figure illustrates the growing importance of non-linearities for slow ions (i.e. the region of large Z1 -oscillations described before), but it also shows that the non-linear domain extends over a wide range of energies.
5. Summary and prospect The main purpose of this work was to propose and explore a different approach to the study of the energy loss of ions in solids, including the treatment of non-linear effects in an extended energy range. The use of the phase shifts sum rule to optimize different model potentials provides a method to extend in a satisfactory way the range of applicability of non-linear (i.e. non-perturbative) quantum calculations, covering a wider range of ion energies, and making contact with the classical and perturbative formulations in the corresponding domains where each of them apply. The new approach is based on the following concepts: (a) the TCS method, as a practical way to calculate non-linear stopping power values; (b) a simultaneous treatment of scattering and screening effects through a self-consistent potential and (c) an extension of the Friedel sum rule to finite velocities, as a means to adjust this screening potential. This is complemented with the study of specific models to describe the ion potential. In the low-velocity range the present model appears as a convenient alternative to the more complex DF method, and explains in a satisfactory way the strong oscillatory effects observed in that range. The agreement with experiments is good at low energies for the elements in the first half of the
periodic table (up to about Z1 ¼ 40), and it deteriorates progressively for heavier elements, where simultaneously, the dependence on the ion model becomes larger. The case of slow very heavy ions remains to be studied in more detail. Additionally, the role played by the charge states of slow ions should be estimated. The main difficulty in producing reliable results for swift ions arises from the uncertainty in the values of the equilibrium ion charge inside the solid, since, as is well known, the empirical values of the effective charge (Zeff ) differ from the observable mean charge (qs ) of ions emerging from solids. The present study indicates that, in the cases where the differences between these charge values are significant (i.e. for heavy ions), the non-linear calculations based on the mean charge values (qs ) yield a better agreement with the experiments. Recent calculations by Maynard et al. [38], although based on a different approach, have led to similar conclusions. The question of understanding these charge differences, a very basic aspect of ion–solid interactions, is however a longstanding and still open one [39,40], so that more conclusive comments cannot be made here. But it seems clear that the non-linear stopping approach may lead to new views of this problem. This is a point of considerable interest requiring more extensive studies. The present calculations have been performed assuming charge-equilibrium conditions; in this case, the strong non-linear effects are partially attenuated by the strong screening provided by bound electrons, so that the possibility of obtaining large values of g ¼ q=v is limited by the partial neutralization of the ion. A different situation, which may be still more favorable to study large non-linear effects is that of channeling experiments using bare or highly charged ions [41], where it is possible to achieve even larger g values. In this case, strong non-linear effects in the stopping power should be expected. Some of the results obtained in previous experiments have shown unexpected departures from the Z1 -scaling predicted by perturbation theories [42]. The possibility of explaining these effects by non-linear calculations is also a question that should be explored.
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Acknowledgements Financial support from ANPCYT (PICT 0303579) is gratefully acknowledged. The author thanks P. Sigmund for a critical reading of the manuscript and many useful comments.
References [1] N. Bohr, Kgl. Danske Videnskab. Selskab., Mat. -Fys. Medd. 18/8 (1948) 1. [2] H. Bethe, Ann. Phys. (Leipzig) 5 (1930) 325. [3] F. Bloch, Ann. Phys. (Leipzig) 16 (1933) 285. [4] J. Lindhard, M. Scharff, Phys. Rev. 124 (1961) 128. [5] O.B. Firsov, Soviet Phys. JETP 9 (1959) 1076. [6] B.S. Yarlagadda, J.E. Robinson, W. Brandt, Phys. Rev. B 17 (1978) 3473. [7] W. Brandt, M. Kitagawa, Phys. Rev. B 25 (1982) 5631. [8] J. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, Pergamon, New York, 1985. [9] P. Sigmund, Phys. Rev. A 56 (1977) 3781. [10] P. Sigmund, A. Schinner, Nucl. Instr. and Meth. B 174 (2001) 535. [11] J.H. Ormrod, H.E. Duckworth, Can. J. Phys. 41 (1963) 1424; J.H. Ormrod, J.R. Macdonald, H.E. Duckworth, Can. J. Phys. 43 (1965) 275. [12] B. Fastrup, P. Hvelplund, C.A. Sautter, Kgl. Danske Videnskab. Selskab., Mat. -Fys. Medd. 35 (1966) 10; P. Hvelplund, B. Fastrup, Phys. Rev. 165 (1968) 408. [13] W.N. Lennard, H. Geissel, D.P. Jackson, D. Phillips, Nucl. Instr. and Meth. B 13 (1986) 127. [14] J. Briggs, A. Pathak, J. Phys. C 6 (1973) L153; J. Briggs, A. Pathak, J. Phys. C 7 (1974) 1929. [15] P.M. Echenique, R.M. Nieminen, J.C. Ashley, R.H. Ritchie, Phys. Rev. 33 (1986) 897; P.M. Echenique, M.E. Uranga, in: Interaction of Charged Particles with Solids and Surfaces, in: A. Gras-Martı, H.M. Urbassek, N.R. Arista, F. Flores (Eds.), NATO ASI Series, Vol. B 271, Plenum, New York, 1991, p. 39. [16] A. Salin, A. Arnau, P.M. Echenique, E. Zaremba, Phys. Rev. B 59 (1999) 2537. [17] J. Calera-Rubio, A. Gras-Martı, N.R. Arista, Nucl. Instr. and Meth. B 93 (1994) 137. [18] B. Apagyi, I. Nagy, J. Phys. C 20 (1987) 1465.
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[19] A.F. Lifschitz, N.R. Arista, Phys. Rev. A 57 (1998) 200. [20] A.F. Lifschitz, N.R. Arista, Phys. Rev. A 58 (1999) 2168. [21] N.R. Arista, A.F. Lifschitz, Phys. Rev. A 59 (1999) 2719. [22] J.U. Andersen, G.C. Ball, J.A. Davies, J.S. Forster, J.S. Geiger, H. Geissel, V.A. Ryabov, Nucl. Instr. and Meth. B 90 (1994) 162. [23] L. de Ferrariis, N.R. Arista, Phys. Rev. A 29 (1984) 2145. [24] P. Sigmund, Phys. Rev. A 26 (1982) 2497. [25] M.A. Kumakhov, F.F. Komarov, Energy loss and ion ranges in solids, Gordon and Breach, New York, 1981. [26] G.H. Dedkov, Soviet Phys. Uspekhi 38 (8) (1995) 877. [27] J. Lindhard, A. Winther, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 34 (4) (1964). [28] H.-D. Betz, Rev. Mod. Phys. 44 (1972) 465. [29] V.S. Nikolaev, I.S. Dmitriev, Soviet Phys. JETP 20 (2) (1965) 409; Phys. Lett. 28A (1968) 277. [30] D.C. Santry, R.D. Werner, Nucl. Instr. and Meth. B 53 (1991) 7. [31] Zheng Tao, L. Xiting, Z. Yongjum, X. Zonghuang, S. Dingyu, W. Xuemei, Z. Qiang, Nucl. Instr. and Meth. B 135 (1998) 169. [32] W. Booth, I.S. Grant, Nucl. Phys. 63 (1965) 481. [33] H. Paul, Compilation of stopping power data; Available from
Energy loss of ions in solids: Non-linear calculations for slow and swift ions Nestor R. Arista Divisi on Colisiones At omicas, Centro At omico Bariloche and Instituto Balseiro, Comisi on Nacional de Energıa At omica, 8400 Bariloche, Argentina Received 28 September 2001; received in revised form 15 January 2002
Abstract The historical approach to describe the energy loss of swift ions in solids is based on the Bohr, Bethe and Bloch theories. As is well known, the central parameter in these theories is the ratio g ¼ Z1 e2 = hv, whose value is generally used to delimit the ranges of applicability of the Bohr (g > 1) and Bethe (g < 1) theories. The transition between these regimes can be obtained by changing the ratio Z1 =v, although not by simply changing v. In fact, this scheme breaks down at low velocities, where quantum and non-linear effects arise. This domain is characterized by the strong oscillatory Z1 dependence of the stopping powers. This paper proposes a self-consistent non-linear approach to calculate the energy loss of heavy ions on a wide range of velocities. The model is based on the transport cross-section approach and on a previous extension of the Friedel sum rule for moving ions. The purpose of this study is to develop a non-linear stopping power evaluation method that could be applied at finite ion velocities, bridging the current gap between the low- and high-energy models. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 34.50.Bw; 34.50.Fa
1. Introduction The study of the interaction of ions with matter has been guided for many years by the inspiring works of Bohr, Bethe and Bloch (BBB) [1–3], which provide the fundamental basis on which a large body of experimental and theoretical developments have grown. Several additional developments based on statistical approaches [4,5], and including also effective-charge models [6,7], have become useful to
E-mail address: [email protected] (N.R. Arista).
systematize a large amount of data covering nearly all possible projectile–target combinations and a wide energy range [8]. The developments in this area, and in particular the effective-charge concept, have been recently analyzed [9,10]. The models emerging from these studies have achieved a significant degree of development, and are currently used in computer simulations to represent ion interaction processes of interest for applications in materials studies and medical physics. However, when it comes to analyze the behavior of slow ions, or the final stages of the electronic slowing down process, it becomes evident that these models cannot describe in a satisfactory
0168-583X/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 0 6 8 7 - 0
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way the physics of the energy loss phenomena in this range. Thus, one of the most relevant facts is the appearance of strong oscillatory effects in the dependence of the stopping powers with ion atomic number Z1 at low velocities [11–13]. These oscillatory effects were explained in terms of quantum scattering theory [14] based on transport cross-section (TCS) calculations for slow ions in an electron gas (representing the conduction or valence electrons in the solid). An essential difference between this method and the BBB framework is the evaluation of quantum effects in a nonperturbative way (by numerical solutions of the Schr€ odinger equation); this is usually referred to as a non-linear stopping power calculation, to distinguish it from other studies which include perturbative approximations. Further developments for slow ions included the treatment of the Z1 -oscillations using density functional (DF) methods [15]. This explains the behavior of the stopping powers in the range of low velocities (v < v0 , being v0 the Bohr velocity). A recent study [16] considers an extension of the DF method to velocities below v0 for light ions. But the question of extending the DF calculations to intermediate or large velocities, and to heavier ions, poses a much more complicated numerical problem. An alternative approach [17] which conserves the essential non-perturbative (or non-linear) characteristics of the quantum analysis, and also explains the oscillatory Z1 -dependence, is based on a self-consistent treatment of the screeningscattering problem by introducing a parametric scattering potential [18] which is self-consistently adjusted by an overall screening condition provided by the Friedel sum rule. The original formulation of this method was also restricted to low velocities (nearly static ions). A more recent development along this line provided an extension of the Friedel sum rule to moving ions [19]; this opened the possibility of carrying out calculations for various ions on a wide energy range. Previous calculations for He ions [20], protons and antiprotons [21], showed good agreement with experiments. The objective of this work is to explore the application of the self-consistent method based on
the extension of the Friedel sum rule to the case of swift ions, covering a wide range of Z1 -values of experimental interest, and trying to bridge the currently existing gap between the quantum nonlinear models applicable at low-energies and the perturbative and statistical (effective charge) models that apply at high energies. The present model is based on the free electron gas picture and therefore it is expected to be most appropriate to describe the energy loss in metallic targets. Previous authors have explored in a heuristic way the application of DF results based on free electron gas picture to non-metallic targets, obtaining generally good agreement with experiments. The applicability of free-electron-gas and related methods to non-metallic targets is a subject of current investigation. The scope of the present analysis is illustrated in Fig. 1, which shows the domains of applicability of the various theoretical approaches [22]. Here the ranges of applicability of the Bohr and Bethe theories are indicated, as well as the domains corresponding to stripped and dressed ions. An additional ‘‘non-linear quantum region’’ has been included in the lower part of this figure. This
Fig. 1. Characteristic regions for the description of the electronic stopping of heavy ions in solids [22]. The line v=v0 ¼ 2Z1 schematically separates the domains where classical and quantum perturbation theories apply, while the curve v=v0 ¼ 2Z12=3 indicates the range of velocities above which the ions become nearly fully stripped of electrons. The lower part of the graph shows the region where the non-linear quantum behavior dominates, characterized by the oscillatory dependence of the energy loss with the atomic number Z1 .
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covers the lower range of energies where the BBB scheme breaks down and one has to resort to full quantum mechanical analysis in order to explain the electronic stopping process. The content of this paper is the following. Section 2 describes the formulation, including the TCS approach, the extended Friedel sum rule (EFSR) method, and the stopping power integration over relative electron–ion velocities. Section 3 includes a study of the scattering potential and a discussion of the various ion models that were analyzed, and shows some preliminary calculations and comparisons between these models. Section 4 contains the main results of the present calculations, considering in particular the cases of slow and swift ions. Finally, Section 5 contains a brief summary of the work and several final remarks. 2. Formulation 2.1. Transport cross-section approach We start by studying the scattering of target electrons in the field of an incident ion with nuclear charge Z1 e, and consider the calculation of the TCS, which is defined as usual by Z ð1Þ rtr ð1 cos hÞ dr: This quantity may be calculated by classical and quantum mechanical methods. In the classical approach one should solve the scattering relation h ¼ hðbÞ, where b is the impact parameter, so that Eq. (1), with dr ¼ 2pb db, may be integrated if hðbÞ is known. In the quantum mechanical formulation, the TCS for the scattering of electrons with relative velocity vr and wave vector k ¼ mvr = h (respect to the scattering center), is given by 1 4p X rtr ðkÞ ¼ 2 ðl þ 1Þ sin2 ½dl ðkÞ dlþ1 ðkÞ ; ð2Þ k l¼0 where dl ðkÞ denotes the phase shift corresponding to the scattering of wave components with angular momentum l ¼ 0; 1; . . . For the present discussion it is illustrative to define also the ‘‘logarithmic number’’ Lðvr Þ, so that the TCS may be expressed as
rtr ðkÞ ¼
93
4pZ12 e4 Lðvr Þ: m2 v4r
ð3Þ
In this way one separates explicitly the Rutherford factor ð4pZ12 e4 =m2 v4r Þ corresponding to unscreened Coulomb interactions. A simple way to characterize the effects of screening is by assuming an effective scattering potential of the form: V ðrÞ ¼ ðZ1 e2 =rÞ expðarÞ, where a is the screening parameter. For this model potential one may calculate the TCS through various approaches, obtaining the following results for the logarithmic term [23]. (a) Classical calculation (g > 1): 2mv2r 1 LCL ðvr Þ ffi ln ð4Þ c : 2 Z1 e2 a (b) Quantum plane-wave (g < 1): 2mvr 1 LPW ðvr Þ ffi ln : 2 ha
approximation
(c) Semi-classical approach: 2mvr 1 LSCL ðvr Þ ffi ln Re½wðigÞ c : 2 ha
ð5Þ
ð6Þ
Here c ¼ 0:5772 and g is the Bloch parameter associated to the relative velocity vr , g ¼ Z1 e2 =hvr . The function wðxÞ in Eq. (6) is the digamma function, and the real part of wðigÞ may be calculated very accurately using the approximation [23] Re½wðigÞ ffi
1 lnð1 þ C2 g2 Þ c; 2
ð7Þ
with C ¼ expðcÞ ¼ 1:781. The assumptions used to derive these expressions [23] apply when the screening distance ks ¼ a1 is sufficiently large compared with the minimum impact parameter Z1 e2 =mv2r in the classical picture, and with the de Broglie length h=2mvr in the semiclassical and plane wave approximations, respectively. In other words, these expressions apply in general when the arguments of the logarithmic terms are large. The most comprehensive of these results is the semiclassical one, LSCL ðvr Þ, which may be considered the equivalent of the Bloch result for the TCS.
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an ion with velocity v, atomic number Z1 , carrying a number Ne of bound electrons, may be written in the form 1 2X ð2l þ 1ÞGl ðv; vF Þ ¼ Z1 Ne ; p l¼0
Fig. 2. Logarithmic functions related to the TCS of Eq. (3), versus the relative electron-ion velocity vr . The analytical results from Eqs. (4)–(6) are indicated as Cl, PW and SCL, respectively. The solid symbols are the results obtained from Eq. (2) using the phase shift values determined by numerical integrations of the radial Schr€ odinger equation. These calculations correspond to Z1 ¼ 5 and a ¼ 0:2 a.u.
It includes the other two cases as particular limits. Thus, for g 1, LSCL ðvr Þ ffi LCL ðvr Þ, whereas for g 1, LSCL ðvr Þ ffi LPW ðvr Þ. The behavior of these logarithmic functions is illustrated in Fig. 2 with explicit calculations for Z1 ¼ 5 and a ¼ 0:2. The solid circles show the results of numerical calculations using Eq. (2), with the phase shifts dl determined by numerical integrations of the radial Schr€ odinger equation. The figure also shows the previous approximating formulae, Eqs. (4)–(6). It may be observed that the SCL formula connects the CL and PW results in a continuous way, and it also approaches the exact quantum values at relatively high velocities. The negative values of LSCL at low velocities may be avoided by improving the approximation of Eq. (6) [23]. As the figure shows, the deviations of the approximations from the most accurate quantum calculations become more important at lower velocities. 2.2. Extended Friedel sum rule The generalization of the Friedel sum rule to finite (non-zero) velocities was developed in Ref. [19], so that only a brief description of the formulation is given here. The extended sum rule for
ð8Þ
where vF is the Fermi velocity of the solid. The functions Gl ðv; vF Þ take into account the contribution of each l-wave component to the screening charge, and may be expressed as integrals over a displaced Fermi sphere (DFS) of the corresponding phase-shift contribution, as follows: Z 1 ddl ðkÞ Gl ðv; vF Þ ¼ dX dk 4p DFS dk Z kmax ddl ðkÞ ¼ gðk; vÞ dk; ð9Þ dk kmin where kmin ¼ maxf0; v vF g and kmax ¼ v þ vF . The function gðk; vÞ takes into account the angular part of the integration over the DFS, and the expressions for the cases v < vF and v > vF were given in Ref. [19]. dl ðkÞ is the asymptotic phase shift of the electron wave function due to the scattering process, and is a function of the wave vector k corresponding to the relative electron–ion motion (k ¼ mvr =h, where ~ vr ¼ ~ ve ~ v). The derivative ½ddl ðkÞ=dk gives the contribution of each l-wave component to the accumulation of screening charge around the intruder ion. It may be shown from these expressions that one retrieves the usual Friedel sum rule in the low-velocity limit (v < vF ), and a perturbative form of the sum rule for high velocities (v > vF ) [19]. In the following we will use atomic units, so that the variables k and vr may be used indistinctly. Once the self-consistency condition of Eq. (8) is achieved (by varying the screening parameter a), we use the final phase shift values to calculate the TCS rtr , from the usual expression 1 4p X ðl þ 1Þ sin2 ½dl ðk; vÞ dlþ1 ðk; vÞ : rtr ðk; vÞ ¼ 2 k l¼0 ð10Þ We note that the phase shifts dl and the TCS rtr in Eq. (10) depend also on the ion velocity v, because of the optimization of the scattering po-
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distribution of electron velocities in a Fermi sphere (0 6 ve 6 vF ) and over the range of relative electron–ion velocities k ¼ j~ vr j (with jv ve j 6 vr 6 v þ ve ). This integration can be performed in a closed form using the expression [24,19] Z vF Z jvþve j 1 ve dve dk k 4 rtr ðk; vÞ Sðv; rs Þ ¼ 2 2 4p v 0 jvve j v2 v2e 1þ ; ð12Þ k2
Fig. 3. Calculations of the phase shifts dl for (i) Z1 ¼ 2, vr ¼ 1 and a ¼ 1 and (ii) Z1 ¼ 5, vr ¼ 5 and a ¼ 0:3, for a range of l values. The lines show the results obtained from the semiclassical approximation, Eq. (11), while the solid circles are the exact (numerical) values.
tential (through the adjustment of the screening parameter a ¼ aðvÞ) as described before. A couple of illustrative examples are shown in Fig. 3 which shows the calculated phase shifts for Z1 ¼ 2 and 5, relative velocities vr ¼ 1 and 5, and for two screening conditions: a ¼ 1 (strong screening) and a ¼ 0:3 (week screening). The circles are the values calculated by numerical integration of the Schr€ odinger equation whereas the lines are the values obtained from the semiclassical approximation [23] for dl , given by Z m 1 V ðrÞ dr dSCL ðkÞ ¼ h i ; ð11Þ l 2 h r0 k 2 ðl þ 1=2Þ2 =r2 1=2 where r0 ¼ ðl þ 1=2Þ=k. Clearly, the number of phase shifts required in the TCS calculation increases both with vr and with decreasing a. In both cases shown in Fig. 3, the semiclassical approximation gives an excellent description for l 1. This approximation was useful in order to speed up the calculations described next. 2.3. Stopping power integration After the TCS is evaluated for a wide range of relative velocities vr , we calculate the stopping power S ¼ hdE=dxi by integrating rtr over the
where rtr ðk; vÞ is calculated from Eq. (10) using the numerically obtained phase shift values. This yields the stopping power for any (non-relativistic) ion velocity v and for a given value of the Fermi velocity vF , related to the electron gas parameter rs by rs ¼ 1:919=vF ¼ 0:621n1=3 (in a.u.), where n is the electron gas density. Therefore, we characterize the final stopping value as a function of v and rs . The results of these calculations will be analyzed in the next sections.
3. The scattering potential Since we are dealing with partially stripped ions, the scattering potential in this study was assumed to be a sum of two components: (a) an ionic core potential Vcore ðrÞ, which includes the atomic screening provided by the bound electrons carried by the ion and (b) the outer screening potential Vs ðrÞ, which is produced by the target valence electrons in order to neutralize the net charge of the ion. The assumption of spherical symmetry of the potential is one of the main assumptions of this model, and may be consider to represent the spherical average of the real potential. A basic difference between both potentials, Vcore ðrÞ and Vs ðrÞ, is that while the ionic component remains essentially frozen when the ion velocity increases (besides the possible changes due to the increased stripping), the outer screening component changes with velocity since the electrons readjust themselves in a dynamical way to the field of the moving ion. The screening component Vs ðrÞ is adjusted for each velocity, in order to satisfy the extended sum rule. In these calculations this component was modeled through a Yukawa
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potential, using a single parameter a whose value is adjusted in a self-consistent way for each velocity v using this rule. The optimization method is the following: for each ion velocity an initial value of a is assumed and then many phase shifts dl ðkÞ are calculated (by numerical integration of the Schr€ odinger equation and using the semiclassical approximation for l 1); then Eqs. (8) and (9) are used to evaluate the extended sum rule. These calculations are repeated for each velocity by an iteration routine, varying the value of a until the sum rule (8) is precisely satisfied. In this way the changes in the screening conditions due to dynamical effects are taken into account by the variation of the screening parameter a. 3.1. Ion models In these calculations several ion models were tested. This includes in particular the following models: (a) Moliere and Thomas–Fermi (TF) potentials for neutral atoms [25]; (b) Brandt–Kitagawa (BK) model [7]; (c) Dedkov ion model [26]; (d) ‘‘Moliere-ion’’ model (to be defined next). The potential in all these cases is represented as Vtotal ðrÞ ¼ Vcore ðrÞ þ Vs ðrÞ ¼
N e e2 qe2 /core ðrÞ / ðrÞ; r r s
ð13Þ
where q is the charge of the ion and Ne ¼ Z1 q is the number of bound electrons; /core and /s are the core-screening and outer-screening functions respectively. The core-potential term is assumed to be a short-range potential, whereas the screening potential is adjusted by the self-consistent method (extended sum rule) and its range increases with ion velocity (dynamical screening effect). Some of the models (a)–(c) indicated above are well known and the first two have been widely used. The ‘‘Moliere ion’’ model is introduced and applied here based on the simplicity and convenience of the well-known Moliere potential for atoms. The modified potential for ions will be represented by Eq. (13) where Vcore ðrÞ ¼ ðNe e2 = rÞ/core ðrÞ, and where the core-screening function, /core ðrÞ /Mi ðrÞ, will be represented here by a sum of three terms,
/Mi ðrÞ ¼
3 X
Aj expðaj r=aTF Þ;
ð14Þ
j¼1 1=3
being aTF ¼ 0:8853=Z1 . In the case of neutral atoms (q ¼ 0) the normal values of the coefficients are used, namely [25]: Aj ¼ ð0:1; 0:55; 0:35Þ and aj ¼ ð6; 1:2; 0:3Þ; so that in this case one retrieves from Eq. (13) the Moliere-atom potential given by Vtotal ðrÞ ¼ ðZ1 e2 =rÞ/M ðrÞ, being /M ðrÞ the usual Moliere screening function. To represent ions with increasing degrees of ionization i ¼ q=Z1 , the coefficients in /Mi ðrÞ are modified as follows: first the value of the third coefficient A3 is decreased (until it eventually becomes zero, for i ¼ 0:35); for higher ionization (0:35 < i < 0:9) a similar reduction is done on the value of A2 and finally (if i > 0:9) the value of A1 is decreased (until complete ionization is reached). In any case (i.e. for a given velocity), the sum of these coefficients should correspond to the actual ionization degree q=Z1 , namely A1 þ A2 þ A3 ¼ Ne = Z1 ¼ 1 q=Z1 . A comparison between the charge densities corresponding to the TF, BK and Moliere potentials for neutral atoms is made in Fig. 4 for Z1 ¼ 80. As it may be seen, there are significant differences between the BK and the two other models; the BK model yields a higher concentration of electronic charge in a smaller volume. The same behavior was observed for all Z1 values. This will produce important differences in the stopping
Fig. 4. Comparison between the charge densities corresponding to the TF, BK and Moliere potentials for neutral atoms with atomic number Z1 ¼ 80.
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Fig. 5. Comparison between the present model (Moliere-ion) and Dedkov’s model for Z1 ¼ 80 and 20, and for two values of the ion charge in each case, q ¼ 10 and 60 (for Z1 ¼ 80), and q ¼ 5 and 10 (for Z1 ¼ 20).
coefficients, particularly for slow ions, as will be shown later. Fig. 5 shows comparisons between the Moliereion and Dedkov’s models for Z1 ¼ 80 and 20, and for two values of the ion charge q in each case. The differences between both models are not very significant, and so both models may be expected be equally appropriate for the present stopping power analysis.
Fig. 6. Stopping power calculations using the linear dielectric formulation (RPA) and the non-linear method proposed here, for bare ions with Z1 ¼ 1 and 7. The Bethe and Bloch limits for high velocities are indicated.
3.2. Preliminary calculations As a first test of the method, calculations for several simple cases were performed. These calculations were done by two different methods: (i) using the standard dielectric formalism [27] based on Lindhard’s (RPA) dielectric function eðk; xÞ, where the stopping power is represented in the form Z 1 2 dk 2 SðvÞ ¼ 2 jq ðkÞj pv 0 k ion Z kv 1 dx xIm ; ð15Þ eðk; xÞ 0 where qion ðkÞ is the form factor corresponding to the ion charge distribution [7] and (ii) following the quantum TCS approach with screening adjusted by the EFSR as described before.
Fig. 6 shows comparisons for bare ions with Z1 ¼ 1 and 7 (obviously, for bare ions there are no differences between the various ion models previously considered). The asymptotic limits from the Bethe and Bloch theories are also indicated in the figure. The figure illustrates the basic differences between linear (or perturbative) and non-linear calculations. We observe a significant reduction of the non-linear results as compared to the dielectric (RPA) values when the ion charge is increased from 1 to 7. The Bloch result describes very well this behavior for large velocities. Since the RPA values follow the Z12 scaling, the relative drop of the non-linear results indicates a ‘‘saturation’’ in the stopping process (i.e. the values increase by a much smaller proportion than expected from the Z12 scaling). In the case Z1 ¼ 7 one may observe that for high energies the present results approach
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Fig. 7. Stopping power calculations for partially stripped N3þ ions (with Z1 ¼ 7, Ne ¼ 4 and q ¼ 3), described according to the BK and Moliere-ion models. The dotted line shows the linear (Lindhard’s dielectric function) calculation for the BK ion model whereas the dashed line is the non-linear (extended sum rule) calculation for the same ion model. The solid line is the non-linear calculation using the Moliere-ion model.
first the Bloch limit (for v=Z1 < 1Þ merging finally with the linear results when v=Z1 > 1. This doublelimit behavior is similar to the one shown in Fig. 2 for a much simpler case. Fig. 7 shows calculations for partially stripped N ions (Z1 ¼ 7, Ne ¼ 4, q ¼ 3), obtained from different types of approaches: linear (dielectric) and non-linear calculations using the BK-ion model, and non-linear calculations using the Moliere-ion model. The ion potential form of Eq. (13) has been used in the non-linear calculations, using the Friedel sum rule to adjust the screening potential Vs ðrÞ. As it may be observed, both non-linear calculations are consistently smaller that the RPA values. In addition, there are significant differences at intermediate and low velocities between the BK and the Moliere ion model, as well as between the linear and non-linear calculations based on the BK-ion model. As it will be further illustrated in the next section, the non-linear calculations based on the BK-ion model yield also too small lowvelocity stopping power coefficients, S=v, for all Z1 . Fig. 8 shows the a values obtained from the application of the extended sum rule in the cases
Fig. 8. Values of the screening parameter a obtained from the application of the extended sum rule in the cases illustrated in Figs. 6 and 7 (i.e. Hþ , N7þ and N3þ ). The symbols are the selfconsistent a-values obtained by this method, while the dotted and dashed lines are the RPA values for the static pffiffiffiffi ða0RPA ¼ 1:563= rs Þ and high-velocity ðaRPA ðvÞ ffi xP =vÞ limits.
illustrated in Figs. 6 and 7 (i.e. Hþ , N7þ and N3þ ). The nearly constant values of a obtained by the self-consistent EFSR method (aEFSR ) at low velocities illustrate the ‘‘frozen-screening’’ limit for v < vF , but it should be noted that the limiting values of aEFSR required by the sum rule are generally different from the static RPA value [27] pffiffiffiffi given by a0RPA ¼ 1:563= rs (dotted line). On the other hand, for v > vF one observes a rapid convergence of the self-consistent aEFSR values to the high-velocity RPA limit, aRPA ðvÞ ffi xP =v (dashed line), wherepxffiffiffiP is the plasma frequency, related to rs by xP ¼ 3=rs3=2 . It should be remarked that this high-velocity limit, aEFSR ! xP =v, for v vF , is automatically provided by the EFSR without any external constraint. As a corollary, the EFSR results for the energy loss asymptotically approach the Bethe limit. Finally, Fig. 9 illustrates the differences in the stopping power results, calculated in a non-linear way according to the EFSR, using the Dedkov, the Moliere-ion, and the BK-ion models. These calculations correspond to C ions incident on amorphous carbon targets under conditions of charge equilibrium. The ions are assumed to have the average equilibrium charge (qs ) for each velocity as described in Section 4.
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Fig. 9. Non-linear stopping power calculations for C ions incident on amorphous carbon targets assuming different ion models: BK, Dedkov’s (D) and Moliere-ion (M). The target is represented as an electron gas with rs ¼ 1:6.
4. Non linear stopping power calculations Using the approach outlined before, further and more elaborate calculations were performed for various cases of experimental interest. We now consider the results obtained for slow and swift ions. 4.1. Slow ions As it was mentioned, one of the striking features of the low-energy stopping phenomena is the oscillatory dependence on the ion atomic number Z1 [11–13]. These oscillations are a clear signature of quantum effects, produced by strong perturbations of the scattered waves. They may be described rather accurately by the phase shift analysis and through TCS calculations. Fig. 10 shows results of calculations of the stopping coefficient S=v obtained by the previously described method, assuming neutral atoms and considering three types of potentials: Moliere, TF and BK. One observes a good agreement between the first two models. The results from the BK potential yield too small values and show some particular features which do not agree with experiments. The analysis of these potentials, and of the corresponding densities already shown in Fig. 4, indicates that the reason for this discrepancy lies in the much smaller sizes of the ions and atoms
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Fig. 10. Quantum mechanical calculations of the low-velocity stopping coefficient, S=v, for neutral atoms, considering three types of descriptions: Moliere, TF and BK.
described by the BK model, a question that becomes critical at low velocities. The change in the interference pattern as the ion velocity increases is illustrated in Fig. 11, which shows calculations for v ¼ 0:2, 1, 2 and 3 a.u., using the Moliere atom model. As it may be observed, as the velocity increases the amplitude of the oscillations diminishes. The soft Z1 dependence obtained for v ¼ 3 a.u. resembles those expected from various statistical models [4–7]. In part (b) of this figure the values of S=v have been superposed in a linear scale to show more clearly how the amplitude of the oscillations decrease. By analyzing in detail the changes in the contributions to the TCS from the different l-values, one finds that the large amplitudes of the oscillations at low velocities arise from the fact that only a few phase shifts (typically l 1–4) are important, whereas at higher velocities the number of phase shifts that should be included in the sum of Eq. (10) increases steeply. It may be shown from qualitative arguments that for a screened potential of range ks (with ks ¼ 1=a in our case) one has to include a number of terms lmax > ks v to get a convergent behavior. The superposition of a large number of terms in Eq. (10) washes out the oscillatory Z1 dependence. The present calculations have been extended to the wide range of atomic numbers across the periodic table (1 6 Z1 6 92) for a velocity v ¼ 0:8 a.u. and for carbon targets. The results using both the Moliere and the TF models are shown in Fig. 12.
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Fig. 12. Calculations and experimental values [11–13] of the stopping power for a fixed ion velocity, v ¼ 0:8 a.u., in carbon targets (described as an electron gas with rs ¼ 1:6), as a function of projectile atomic number Z1 . The calculations correspond to Moliere ( ) and TF (- - -) potentials.
Fig. 11. Change in the Z1 -oscillation pattern with increasing ion velocity (v ¼ 0:2, 1, 2 and 3). The calculations have been performed assuming a Moliere potential, and for the electron density of carbon (rs ¼ 1:6). Panel (a) shows the variation in the stopping power values (on a logarithmic scale), whereas panel (b) shows the attenuation of the Z1 -oscillations, by representing the ratio S=v on a linear scale.
The comparison with the experiments is good for low and intermediate Z1 values, but it deteriorates in the upper Z1 range. Also illustrative are the growing differences obtained with the TF and Moliere models in this range. This indicates an increasing sensitivity to the way in which the projectile electronic density is described. These growing differences may be explained by analyzing the magnitude of the potentials for different Z1 values, as illustrated in Fig. 13 for Z1 ¼ 20 and 80. As it may be observed, for large Z1 values the Moliere potential decreases sharply with r, whereas on the other hand the TF potential decreases mildly. This long range behavior of the TF potential may be considered unrealistic; however, there is another important question that has not been taken into account in these calculations
Fig. 13. Variation of the Moliere (—) and TF (- - -) potentials, for Z1 ¼ 20 and 80, versus the radial distance r.
and may produce a similar effect, which is the charge state of each ion. The ionization would also increase the range of the potential and hence also the stopping (note that the mean charge is ex1=3 pected to increase roughly as Z1 in this range of velocities according to Bohr estimation [1]). Further studies along this line, considering the equilibrium charge state distribution of each particular ion, may be quite useful, as well as the use of more accurate electronic densities, like those provided by Hartree–Fock calculations. Nevertheless, it should be stressed that the overall agreement ob-
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tained from these non-linear calculations is satisfactory for many elements. The effect of average ion charge will be studied in some detail in the case of swift ions, where it becomes quite important and where some general estimations can be made. 4.2. Swift ions The previous results for slow ions include the calculation of the TCS for strongly screened systems (basically, neutral projectiles). This simplifying assumption applies only to slow projectiles. We now turn the attention to the analysis of calculations performed at intermediate and high energies. In this case, the projectiles get ionized and one has to resort to the use of model potentials that include a description of both the close-packed ion core as well as the extended screening charge distribution of target electrons that builds around it. This is represented by the two-component model for the self-consistent potential introduced by Eq. (13). The screening component of this potential, Vs ðrÞ, is adjusted according to the extended sum rule of Eq. (8), by varying the screening parameter a. The calculations presented here have been made using in Eq. (13) a screening function of the form /s ðrÞ ¼ expðarÞ, and the parameter a was adjusted for each ion velocity v. The required number of phase shifts was of the order of 10 at low velocities but it increases strongly with velocity. In fact, for v vF the value of a decreases asymptotically like the RPA value, a aRPA ¼ xP =v (dynamical screening limit, cf. Fig. 8), and so the maximum l-number (lmax v=aÞ increases like v2 . Hence, the convergence of the partial-wave series, Eq. (10), becomes slower. For the largest velocities considered here (20 a.u.), up to 600 phase shifts were included in the summation. However, the calculations converge rapidly when the high-l terms are approximated by the semiclassical values. These calculations have been performed for various ions incident on carbon targets. The choice of carbon here is based on the relatively small K-shell ionization contribution to the stopping power, which is not included in the present form of
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the non-linear model. Nevertheless, the K-shell contribution has been included here in a simplified way using the Bloch approximation for the stopping logarithm, with an excitation energy corresponding to the K shell of carbon. In this approximation, shell corrections have been neglected, and the Bloch function was set to zero in the lower velocity region where it turns negative. Two types of non-linear stopping power calculations were performed using the following approaches: (i) direct TCS calculations assuming a ‘‘frozen’’ Moliere atom potential, followed by stopping integrations using Eq. (12) and (ii) selfconsistent calculations using Moliere-ion potentials with variable external screening adjusted according to the extended sum rule method. Some further calculations have also been made using Dedkov’s model, treated with the same variable screening and sum rule methods, but no significant differences have been obtained as compared to those based on the Moliere ion model. An additional problem in the calculation for ions is that it requires an explicit specification of the charge of the ions for each velocity. The ions are assumed to have reached a charge-state equilibrium [28]; however, these charge states cannot be specified accurately, since the experiments so far give only the charge states after emerging from a solid foil, and, among the different models proposed, some of them predict differences between the charge states inside the solid and those of the emerging ion beams [28]. Due to this currently unavoidable uncertainty, the present calculations have been done considering two possible equilibrium charge states inside the foil: (i) the ‘‘effective charge’’ given by the usual fitting [28]: Zeff ¼ Z1 ½1 expðc1 v=Z1b1 Þ , with c1 ¼ 0:95, b1 ¼ 2=3 and (ii) the ‘‘average emerging charge’’ for solid foils, qs , which may be well represented by the Nikolaev– 1=c c Dmitriev fitting [29]: qs ¼ Z1 =½1 þ ð1=uÞ 1 1 , with c1 ¼ 0:6 and u ¼ v=v1 Z 0:45 , being v1 ¼ 1:645 a.u. It should be noted, however, that neither of these approaches is completely justified a priori, since, on one side, the effective charge Zeff is empirically obtained [8] assuming a Z 2 scaling which is not consistent with the lines of the present nonlinear description, and, on the other, the average emerging charge may be different from the actual
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equilibrium charge of the ions inside the solid, as indicated before. Hence, the calculations based on these charge values will be considered as exploratory. Based on these premises, Figs. 14–17 show results of calculations for C, Cl, Ni and U ions incident on carbon targets, described as an electron gas with rs ¼ 1:6 (corresponding to the density of amorphous carbon and considering four valence electrons per atom). The calculations are compared with experimental results from various sources [30,36]. The figures illustrate several aspects. The lower curve (dotted line) shows the results obtained from the standard Moliere-atom model without screening adjustment. As it may be observed, the ‘‘frozen’’ Moliere-atom model yields too small stopping power values. This is expected, since this model does not take into account the expansion of the screening cloud around the ions, whose size grows with ion velocity (ks ¼ a1 v=xP , for large v). The effect of the expanding screening cloud – which is automatically taken into account by the extended sum rule – is essential to obtain the right behavior of the stopping power at intermediate and high energies.
Fig. 15. Non-linear calculations of the energy loss of Cl ions in carbon targets, and comparisons with experimental results [32,33]. ( ): frozen Moliere-atom model; (- - - and —): Moliere-ion model, with q ¼ Zeff and q ¼ qs , respectively. The lines denoted with a and b are, in each case, the results without and with the K-shell energy loss term.
Fig. 16. Non-linear calculations of the energy loss of Ni ions in carbon targets and comparisons with experimental fitting results [34]. ( ): frozen Moliere-atom model; (- - - and —): Moliere-ion model, with q ¼ Zeff and q ¼ qs , respectively. The lines denoted with a and b are, in each case, the results without and with the K-shell energy loss term.
Fig. 14. Non-linear calculations of the energy loss of C ions in carbon targets (with rs ¼ 1:6), and comparisons with experimental results [12,30,31]. The dotted line shows the results for a frozen Moliere-atom model. The dashed and solid lines, for q ¼ Zeff and q ¼ qs , respectively, correspond to the Moliere-ion model with external screening adjusted using the extended sum rule. The lines denoted with aand b are, in each case, the results without and with the K-shell energy loss term.
The calculations using either q ¼ Zeff or q ¼ qs are shown by dashed and solid lines, respectively. The lines denoted a and b are the calculations without and with inclusion of the carbon K-shell contribution to the energy loss. We observe that the differences due to the ion charge assumption (either q ¼ Zeff or q ¼ qs ) are very small for C ions, but they become increasingly important for heavier ions. We find that a better
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Fig. 17. Non-linear calculations of the energy loss of U ions in carbon targets and comparisons with experimental results [35,36]. ( ): frozen Moliere-atom model; (- - - and —): Moliere-ion model, with q ¼ Zeff and q ¼ qs , respectively. The lines denoted with a and b are, in each case, the results without and with the K-shell energy loss term.
agreement with experiments is obtained by the q ¼ qs assumption. The comparisons with the experimental results in the first three cases are good, while the results for U ions are still not satisfactory. In all cases, the results for q ¼ Zeff are too low as compared with the experiments, whereas those for q ¼ qs appear to be more appropriate. It should be noted that the energy losses due to charge exchange or projectile excitations have been ignored in these calculations. In conditions of dynamical equilibrium one may consider different processes of electron capture by the moving ions, followed either by direct ionization or by excitation and subsequent ionization, so that cycles of electron capture and loss are completed. Such cyclic processes also contribute to the equilibrium stopping power, but a quantitative study is still missing. According to previous estimations [37] for the case of Ni projectiles in carbon, charge exchange processes may be expected to yield a maximum contribution of the order of 5% for energies around the stopping power maximum. A contribution of this order may be considered compatible with the present estimations, but more extensive calculations of these effects would be useful for an accurate description. It may also be observed that in the case of C ions (where the uncertainty between the q ¼ Zeff
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or q ¼ qs assumptions is small) the current predictions overestimate the experimental values (even though projectile inelastic processes are ignored). This may indicate a lack of accuracy of the method. It is finally illustrative to analyze the present results in terms of the magnitude of the effective Bloch parameter, g ¼ q=v, which provides a measure of the expected non-linearities in the energy transfer in terms of the ion charge q (instead of the conventional definition in terms of the nuclear charge Z1 e). This parameter may be considered useful when the collision radius (rcol maxfq=mv2 , h=2mvg) is larger than the ion radius, a condition that applies in both cases illustrated here. The values of g depend also on the charge-equilibrium model used. Fig. 18 shows the values of Zeff ðvÞ and qs ðvÞ (right scales) and the corresponding values of g (left scales) for C and U ions. The range of
Fig. 18. Values of the effective charge Zeff and average charge qs (right scales) and of the corresponding values of g ¼ q=v (left scales) for C and U ions. The dashed lines show the values of Zeff and geff ¼ Zeff =v; the solid lines show the values of qs and gs ¼ qs =v.
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energies corresponds to velocities larger that 1 a.u. (25 keV/u) since the charge equilibrium models are not considered accurate at low velocities, and the usefulness of the parameter g should be revised. In the case of C there is little difference between the models and we find maximum g values of about 1.5. In the case of U, we observe important differences, and the maximum values fall in the range gmax 4–7 at low velocities. On one side, the figure illustrates the growing importance of non-linearities for slow ions (i.e. the region of large Z1 -oscillations described before), but it also shows that the non-linear domain extends over a wide range of energies.
5. Summary and prospect The main purpose of this work was to propose and explore a different approach to the study of the energy loss of ions in solids, including the treatment of non-linear effects in an extended energy range. The use of the phase shifts sum rule to optimize different model potentials provides a method to extend in a satisfactory way the range of applicability of non-linear (i.e. non-perturbative) quantum calculations, covering a wider range of ion energies, and making contact with the classical and perturbative formulations in the corresponding domains where each of them apply. The new approach is based on the following concepts: (a) the TCS method, as a practical way to calculate non-linear stopping power values; (b) a simultaneous treatment of scattering and screening effects through a self-consistent potential and (c) an extension of the Friedel sum rule to finite velocities, as a means to adjust this screening potential. This is complemented with the study of specific models to describe the ion potential. In the low-velocity range the present model appears as a convenient alternative to the more complex DF method, and explains in a satisfactory way the strong oscillatory effects observed in that range. The agreement with experiments is good at low energies for the elements in the first half of the
periodic table (up to about Z1 ¼ 40), and it deteriorates progressively for heavier elements, where simultaneously, the dependence on the ion model becomes larger. The case of slow very heavy ions remains to be studied in more detail. Additionally, the role played by the charge states of slow ions should be estimated. The main difficulty in producing reliable results for swift ions arises from the uncertainty in the values of the equilibrium ion charge inside the solid, since, as is well known, the empirical values of the effective charge (Zeff ) differ from the observable mean charge (qs ) of ions emerging from solids. The present study indicates that, in the cases where the differences between these charge values are significant (i.e. for heavy ions), the non-linear calculations based on the mean charge values (qs ) yield a better agreement with the experiments. Recent calculations by Maynard et al. [38], although based on a different approach, have led to similar conclusions. The question of understanding these charge differences, a very basic aspect of ion–solid interactions, is however a longstanding and still open one [39,40], so that more conclusive comments cannot be made here. But it seems clear that the non-linear stopping approach may lead to new views of this problem. This is a point of considerable interest requiring more extensive studies. The present calculations have been performed assuming charge-equilibrium conditions; in this case, the strong non-linear effects are partially attenuated by the strong screening provided by bound electrons, so that the possibility of obtaining large values of g ¼ q=v is limited by the partial neutralization of the ion. A different situation, which may be still more favorable to study large non-linear effects is that of channeling experiments using bare or highly charged ions [41], where it is possible to achieve even larger g values. In this case, strong non-linear effects in the stopping power should be expected. Some of the results obtained in previous experiments have shown unexpected departures from the Z1 -scaling predicted by perturbation theories [42]. The possibility of explaining these effects by non-linear calculations is also a question that should be explored.
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Acknowledgements Financial support from ANPCYT (PICT 0303579) is gratefully acknowledged. The author thanks P. Sigmund for a critical reading of the manuscript and many useful comments.
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