Two-channel approach to the average retarding force of metals for slow singly ionized projectiles
aa r X i v : . [ c ond - m a t . o t h e r] A ug Two-channel approach to the average retarding force of metals forslow singly ionized projectiles
I. Nagy
1, 2 and I. Aldazabal
3, 2 Department of Theoretical Physics, Institute of Physics,Budapest University of Technology and Economics,H-1521 Budapest, Hungary Donostia International Physics Center, P. Manuel de Lardizabal 4,E-20018 San Sebasti´an, Spain Centro de F´ısica de Materiales (CSIC-UPV/EHU)-MPC, P. Manuel de Lardizabal 5,E-20018 San Sebasti´an, Spain
Abstract
Based on the fundamental momentum-transfer theorem [Phys. Rev. Lett. , 11 (1965)] a novelcontribution to the retarding force of metallic systems for slow intruders is derived. This contribu-tion is associated with sudden charge-changing cycles during the path of projectiles. The sum of thenovel and the well-known conventional contributions, both expressed in terms of scattering phaseshifts, are used to discuss experimental data obtained for different targets. It is found that ourtwo-channel modeling, with two nonlinear channels, improves the agreement between several dataand theory and thus, as predictive modeling, can contribute to the desired convergence betweenexperimental and theoretical attempts on the retarding force. PACS numbers: 34.50.Bw . INTRODUCTION AND MOTIVATION According to a basic book on quantum mechanics by Landau [1] one of the most importantquantities in the interaction of heavy charged projectiles with fixed atoms is the averageenergy loss. This time-independent quantity, a kind of deposited energy, is an observableand due to conservation laws its measurement is feasible in experiments. Thus in thissubfield of nature (physics, human therapy) the real challenge resides in the convergence ofmeasurements and theories. Their interplay, a continuous one over a century, fertilizes thedevelopments on both side of approaches which can result in a transferable knowledge [2].The present contribution is dedicated to the case where singly ionized projectiles inter-act with constituents of metallic targets. The main challenge addressed here is to find areasonable combination of the quantum statistical and atomistic aspects of the energy lossprocess in real targets. As motivation, on which our new attempt is partially based, we startwith an established result. The well-known, conventional form [3–5] for the stopping power(written in Hartree atomic units, where e = m e = ~ = 1) of a homogeneous degenerateelectron gas (characterized by Fermi velocity v F ) for heavy intruders is given by dEdx = 2(2 π ) Z v F v e (cid:20) π Z − dx ( v − v e x ) v r σ tr ( v r ) (cid:21) dv e , (1)where v and v e ∈ [0 , v F ] are the projectile and system-electron velocities. One of theintegration-variables is x ≡ cos β , where β is the angle between v and v e . Thus v r =( v + v e − v e v x ). Clearly, in an interpretation based on independent-electron scattering offa heavy projectile moving with constant velocity v , the remaining task resides in a common two-body interaction V ( r ), in order to perform the statistical averaging over a Fermi-Diracdistribution with σ tr ( v r ) = 4 πv r ∞ X l =0 ( l + 1) sin [ δ l ( v r ) − δ l +1 ( v r )] . (2)In this scattering interpretation the analysis is based on the concept of asymptotic statesin the infinite past and future, i.e., involving large time differences. Sudden processes in time,like a local charge-change in metals, requires a refined approach on associated transitionamplitudes in time-dependent perturbation theory. In kick-like sudden [1] processes onemay use predetermined states as a complete set to treat the matrix elements in strong (buttransient in time) perturbations. Notice that precisely it is such a transient channel which2ould make difficulties in large-scale simulations, like in the orbital-based implementation ofTime-Dependent Density-Functional Theory (TDDFT).Such an implementation rests on averaging of quantum mechanical time-dependent en-ergy differences over certain time scales in order to define a force-like quantity as stationaryobservable in stopping [6, 7]. In these large-scale simulations specification of the initial con-ditions is required to real-time propagation. For instance, in [7] two options were consideredfor helium in aluminum target. In the first one, the screened atom was included in thedetermination of the static initial state. In the second one, the initial condition was setup by adding an α -particle and thus producing a sudden change in the external potential.In both cases the authors control only the initial state and not the subsequent dynamicswhich is given by the time-dependent single-particle equations within TDDFT. Therefore,a smoothed evolving picture, without fast local charge-changing processes, is employed.First, as concretization of motivation, we integrate Eq.(1) by using models for the mo-mentum transfer cross section in order to get useful informations to phenomenology madein Section II after Eqs.(6). Namely, we take the form of σ tr ( v r ) = 4 πA α /v αr , in which α = 2and α = 4. By straightforward quadrature we obtain [5] from Eq.(1) for these models dEdx = 4 πA ( v F ) n v v F " − (cid:18) vv F (cid:19) = A π v F v " − (cid:18) vv F (cid:19) f or v ≤ v F ,dEdx = 4 πA ( v ) n v (cid:20) − (cid:16) v F v (cid:17) (cid:21) f or v ≥ v F .dEdx = 4 πA ( v F ) n v v F = A π v f or v ≤ v F ,dEdx = 4 πA ( v ) n v f or v ≥ v F . The above model results are, of course, in agreement with expected limits ( dE/dx ) = n vv F σ tr ( v F ) and ( dE/dx ) = n v σ tr ( v ), at v → v → ∞ , respectively. Earlier, carefultheoretical analysis [8] performed within an adiabatic framework on velocity-dependencestates that the next term beyond the v -proportional one is at least second order in velocity.Our closed expressions for v ≤ v F are in harmony with this important statement. Further-more, a certain weighted combination of our two expressions at v ≤ v F would result in analmost perfect v -proportionality. That, at this point simple mathematical, observation will3ecome a more transparent and physical one in Section II, where we extend the theory on theaverage retarding force beyond the common fixed-potential approximation by consideringphysically reasonable force matrix elements as independent channel contributions.The rest of this paper is organized as follows. Section II is devoted to the theory andthe discussion of the results obtained. The last Section contains a short summary and fewdedicated comments. As above, we use atomic units throughout this work. II. RESULTS AND DISCUSSION
We begin this section by outlining few important elements of stationary scattering theory.According to basic rules of quantum mechanics on expectation values of operators, oneshould consider the force matrix element [9] between orthonormal components of a scatteringstate to characterize the associated momentum transfer. Applying this quantum mechanicaltheorem, where σ tr ( v r ) ∝ P ∞ l =0 ( l + 1)[ I ( l, v r )] , one has [10–12] for the matrix elements I ( l, v r ) = (cid:20)Z ∞ drr R l ( r, v r ) ∂V ( r ) ∂r R l +1 ( r, v r ) (cid:21) = sin[ δ l ( v r ) − δ l +1 ( v r )] . (3)We stress that this remarkable identity rests on those states which refer to the scatteringSchr¨odinger wave equation with v r / V ( r ) external field. However, with partialwaves based on V ( r ), but with a net Coulomb field ∆ V c ( r ) = − q/r in space of V ( r ) we get I ( l, v r ) = (cid:20)Z ∞ drr R l ( r, v r ) ∂∂r (cid:16) − qr (cid:17) R l +1 ( r, v r ) (cid:21) = q cos[ δ l ( v r ) − δ l +1 ( v r )]2 v r ( l + 1) , (4)and with unperturbed ( u ) partial wave components the corresponding result becomes I ( u )2 ( l, v r ) = (cid:20)Z ∞ drr j l ( v r r ) ∂∂r (cid:16) − qr (cid:17) j l +1 ( v r r ) (cid:21) = q v r ( l + 1) . (5)Here we used the spherical Bessel functions of the first kind, i.e., the components of anunperturbed plane-wave (momentum) state, instead of self-consistent radial functions. Theseforms in Eqs.(4-5) are based on the fact that in cases with abrupt perturbations the originalstationary system has no time [1] to relax to the stationary state of a new Hamiltonian.We will consider these amplitudes as the proper ones when there is a sudden change inthe self-consistent V ( r ), as in the case of charge-changing ( q = 1) processes generated bythe binary interaction with fixed lattice ions. This charge-change results in an excess barefield ∆ V c ( r ) = − /r . The square of [ I ( l, v r ) − I ( u )2 ( l, v r )] can characterize, in a quantum4echanical interpretation, an extra (kick-like) momentum transfer due to the sudden changein the external field. That square is, in fact, a regularized transition probability. Such aregularization is needed since both I and I ( u )2 would give divergent results after l -summation.This regularized channel gives (at q = 0) a novel form for the associated cross section σ (2) tr ( v r ) = 4 πv r (cid:18) qv r (cid:19) ∞ X l =0 l + 1 sin (cid:20) δ l ( v r ) − δ l +1 ( v r )2 (cid:21) , (6)to which a simple trigonometrical identity (1 − cos α ) = 4 sin ( α/
2) is employed.Before our quantitative analysis, we continue with phenomenology. There are importantdifferences between Eq.(6) and Eq.(2), i.e., between σ (2) tr ( v r ) and the conventional one givenby Eq.(2) and denoted from here by σ (1) tr ( v r ). The kinematical prefactors show that thenew term ( ∝ v − r ) vanishes faster at large scattering wave number v r . Thus at v r ≤ δ ≃ π for a dominating phase shift, the new term can become the dominating one.A combination of the v − r and v − r dependences in [ σ (1) tr ( v r ) + σ (2) tr ( v r )] with the integratedcharacteristics found with separated model cross sections in the Introduction, signals that avelocity-proportionality in the stopping power holds, practically up to v ≃ v F from below.Now, we turn to the quantitative part of this Section. We will determine numerically thetwo quantities, denoted by Q (1) ( v F ) and Q (2) ( v F ), by which the low-velocity stopping powerof metals (a system of an electron gas and lattice ions) takes a friction-like form1 v dEdx = (cid:2) Q (1) ( v F ) + Q (2) ( v F ) (cid:3) , (7)where the two coefficients (when q = 0) are given by the following expressions Q (1) ( v F ) = 43 π v F ∞ X l =0 ( l + 1) sin [ δ l ( v F ) − δ l +1 ( v F )] , (8) Q (2) ( v F ) = 43 π q ∞ X l =0 l + 1 sin (cid:20) δ l ( v F ) − δ l +1 ( v F )2 (cid:21) . (9)Our summation of two channel cross sections in Eq.(7) resembles, mathematically, to thewell-known [13] rule in potential scattering with a spin-orbit interaction term where we sumthe direct (non-spin-flip) and spin-flip partial differential cross sections for electron scatteringfor any spin orientation before scattering. There, the integrated cross sections, needed toobservables, are obtained by integrating over all scattering angles. Remarkably, the spin-flippart depends on an amplitude difference, similarly to our regularized difference.5 ABLE I: Partial contributions, Q (1) ( v F ) and Q (2) ( v F ) at q = 1, to Eq.(7). Phase shifts, based onthe orbital version of DFT [19–21], are employed. See the text for further details. r s = 1 . r s = 2 r s = 3 Z Q (1) Q (2) Q (1) Q (2) Q (1) Q (2) .
305 0 0 .
255 0 0 .
162 02 0 .
755 0 .
069 0 .
427 0 .
134 0 .
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912 0 .
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439 0 .
247 0 .
117 0 . .
112 0 .
235 0 .
557 0 .
323 0 .
191 0 . .
417 0 .
298 0 .
749 0 .
374 0 .
307 0 . .
692 0 .
366 0 .
874 0 .
413 0 .
346 0 . .
777 0 .
369 0 .
825 0 .
449 0 .
275 0 . .
631 0 .
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483 0 .
167 0 . .
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512 0 .
085 0 . .
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035 0 . .
815 0 .
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690 0 .
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146 0 .
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846 0 .
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360 0 . .
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502 1 .
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437 0 .
297 0 . .
975 0 .
480 1 .
219 0 .
404 0 .
234 0 . .
386 0 .
458 1 .
364 0 .
386 0 .
191 0 . We stress at this point that we employ to summation in Eq.(7) an a priori unit-weightassumption. In reality, i.e., at channeling-like conditions in metals, the impact parameter-dependent closest approach of intruders and lattice ions [7, 14–16] may influence that as-sumption. In more simple terms, our present weighting would refer to random-collisionsituations. Nonequal weighting might be based on certain probabilistic inputs [17] to sum two nonlinear channel. But, such inputs need, in our modeling, an additional justification,since one can not apply stationary linear-response ideas to a sudden effect.6able I contains our numerical results for Q (1) ( v F ) and Q (2) ( v F ) at selected values ofthe r s Wigner-Seitz radius and at q = 1. The partial phase shifts, calculated by DFTat the Fermi momentum, are taken from earlier works [19–21]. Both Q (1) and Q (2) areoscillating functions. But Q (1) ( r s ) has a strong, direct density-dependence via v F ∝ r − s inEq.(8). Thus, at metallic densities, Q (2) ( r s ) in the sum [ Q (1) + Q (2) ] can make an importantmodulation in the Z -oscillation of Q (1) , especially around its minima. For Z = 1, we takeour values for Q (1) obtained within the explicit version [5] of DFT. There a single Eulerequation is solved in an iterative self-consistent way. That calculation does not consider adoubly-populated weakly bound (extended) state around an embedded proton in an electrongas, in harmony with experimental facts, obtained by positive muons, on the nonexistenceof muonium in metals.Despite this, there is a perfect agreement [5] with Q (1) results obtained from the implicit,orbital-based, DFT. This agreement signals that it is the short-range part of proton-screeningwhich needs a nonlinear treatment. In simple terms, that range is the most important oneto determine the first few phase shifts. Our Q (2) = 0 values for Z = 1 are in accord with ascreened-proton picture without bound state, where there is no charge-changing cycle, thus q = 0 during the motion of the projectile. Since the experimental data, obtained with lowvelocity proton projectiles for Al and Ni targets, are in very reasonable harmony [15, 16, 18]with nonlinear theory [19–21] based solely on Q (1) ( v F ), we have a transferable knowledge inthis case. A desired convergence between the two sides of understanding is achieved.For all other Z ≥ v ≤ v F , the q = 1 value as the most plau-sible one. This conservative value seems to be a realistic one with singly ionized intruders.Higher q values might have relevance when there is a large electronic overlap between cloudsof colliding atoms. We believe that such a partial channel with q > transition from our discrete- q modeling of charge-changing cycles to the pioneering [22] quasiclassicalwork where the retarding force (the observable) is related to an electron-density-flux con-structed from the statistical Thomas-Fermi theory of atoms. With a transition-study onemight arrive at a deeper understanding of an integrated (classical trajectory Monte Carlo)approach [23] for energy loss and capture processes.There is certain contradiction (c.f., Fig. 2, below), for Al target between low-velocityexperiment [15] and TDDFT [6, 7, 24] results in the case of He + projectiles. In this case our7ovel result, based on [ Q (1) + Q (2) ], is in harmony with [7] for the off-channeling situation.For the channeling simulation our Q (1) also gives a very reasonable agreement with [6, 7].We stress that [24] uses α -particle ( He ++ ) as projectile and an atom-centered optimizedGaussian basis set to model the energy transfer. The observed agreements (see, the discus-sion at Fig. 2) are especially remarkable in the light of careful experiment [25] performed onelectron emission from aluminum. There a perfect linearity in the velocity of helium ions,with v ≤ .
6, was obtained, and thus a quantitative agreement with [6] was concluded.Notice, in the spirit of discussion made already in [7] for proton and helium intruders,that we can image experimental situation where, at very low ion velocities, only the neutralscreened-atom gives contribution via its Q (1) , and the Q (2) channel becomes active only froman intermediate velocity below the Fermi velocity. Such a modeling would fit to the exper-imental [15] suggestion on two, both linear in ion-velocity, parts on the whole kinematicalrange v ≤ v F . Of course, the acceptance of such a suggestion presupposes that the under-lying experiment-evaluation method behind data is a well-justified one for the whole rangeof ion velocity. For these simple intruders, a partial support to such a view could be basedon surface-scattering experiment [26] performed with singly ionized ions ( Z ∈ [1 ,
20] and atprojectile velocity v = 0 .
5) scattered from an aluminum surface at variable scattering angles.There, the challenging problem of inhomogeneity in the electron density profile of the elec-tron salvage in front of a metal surface was studied, with an accompanied phenomenologicalrefinement of factors in ( dE/dx ) = ( n v )[ v F σ (1) tr ( v F )]. We will return to this experiment, atthe discussion of Fig. 3 which is devoted to an important comparison for Z > Q (1) + Q (2) ] > Q (1) values for the average retarding forcein metals, we turn to a brief discussion of data [14] obtained for an other free-electron-likematerial, Mg. For this metal v F ≃ . He + and proton projectileswas performed for v ≥ v F . It was found that the ratio ( R ) of stopping powers with theseintruders becomes about two, in contrast to a ratio of about unity which is based on Q (1) values of self-consistent DFT. Our novel approach would result in a ratio ( R >
2) which isnot in contradiction with experimental suggestion. As support, we note that in [26], i.e.,in surface experiment, the helium per proton stopping ratio was found to be always higherthan unity, even for r s ( z ) >
3. Clearly, the desired convergence of theories and experimentsrequires further studies for Mg ( r s ≃ .
7) and, say, for Ca ( r s ≃
3) as well, within large-scaleTDDFT simulations with proton and helium intruders at v ≤ v F .8 Z R a nd R FIG. 1: Illustrative dimensionless ratios, R (dashed curve) and R (solid curve), defined in thetext, as a function of Z . The density parameter is r s = 1 .
5, which corresponds to a high-densitydegenerate electron gas. This would refer to the plasma frequency of Au..
At this point, i.e., before the presentation and discussion of our illustrative Figures, wewould like to mention a very recent attempt [27] where a two-channel modeling was presentedfor the spectral line-width in plasma environments. The authors of that insightful workdemonstrated that the commonly used expression for the line-width neglects a potentiallyimportant contribution from electron-capture processes. Their numerical value signals thata proper sum of two contributions can be about twice of the conventional estimation. Inthe field of high-energy-density plasmas, our q -mediated enhancement in stopping powermay contribute to the proper determination of the ignition threshold [28] in a deuterium-tritium-alpha energy deposition process. There, via a plausible postulation, the theoretical underprediction of stopping data was associated [28] with ion-ion nuclear scattering.Now, we illustrate our novel results by three Figures. In Fig.1, for r s = 1 . R = [ Q (1) ( v F , Z ) /Q (1) ( v F , Z = 1)] / and R = { [ Q (1) ( v F , Z ) + Q (2) ( v F , Z )] /Q (1) ( v F , Z = 1) } / , i.e., ratios of nonlinear quan-tities. One might consider [18, 29] these ratios as a kind of effective charge. This Figurereflects, in a highly phenomenological manner, that the so-called Z oscillations may getimportant modulations especially around the minima of the conventional R ratio. Noticethat the such-defined ratios are square roots of physical magnitudes. This mathematicaloperation has a smoothing character (c.f., Fig. 3) with renormalized oscillating functions.9ig. 2 is devoted to ( dE/dx ) quantities, in atomic units, obtained for Al ( r s ≃ .
13) withhelium projectiles. The velocity-range, in atomic units, is v ∈ [0 , . Z = 2) linear-response (first-order Born) approximation, where theelectron gas dielectric function at the RPA level, i.e., without static or dynamic local-fieldcorrections [30], is used. In such an approximate theory, the stopping is proportional to Z .The corresponding form, employed in a cornerstone work [20] as well, is given by (cid:18) dEdx (cid:19) RP A = v π ( Z (cid:20) ln (cid:18) a + 23 (cid:19) − a − a + 2 (cid:21) (cid:18) a a − (cid:19) ) , (10)where a = πv F is an abbreviation. Its value is about a ≃ .
83 for Al. This expression is aparticular realization of the modeling made in the Introduction with σ tr ( v r ) ∝ (1 /v r ).Our present results (at Z = 2) correspond to the green solid curve [ Q (1) + Q (2) ], andthe green dashed curve [ Q (1) ]. The black solid and red dash-dotted curves with filled circlesare taken from Fig. 5b of [7]. They refer to off-channeling and channeling conditions,respectively. Notice that an earlier TDDFT result of [6] (not shown here) agrees preciselywith the black curve. There is a fortuitous similarity between the RPA result for thehomogeneous electron gas and results plotted via curves by solid green and dash-dotted redwith dots. Neither the screening-treatment nor the scattering-description of RPA is correctto a nonlinear situation. For proton, where a very reasonable agreement [15, 16, 18] betweennonlinear Q (1) ( v F ) and data was found, the above Eq.(10) with Z = 1 would give a serious underestimation [20].Experimental [15] data, used already in TDDFT to comparison [7], are plotted here bya dashed magenta curve with filled triangles. This curve signals a two-slope behavior withlinearities in projectile velocity. Remarkably, a quite similar, i.e., two-slope, behavior wasfound in [16] for Ni ( r s ≃ .
8) with singly-ionized He + intruder. There, a comparison withTDDFT results [31] is made, by using 1 .
15 as multiplying factor for the simulation-basedresults. As we already discussed above, we can image such a two-slope behavior within thepresent theoretical framework with certain, presumably closest-approach-dependent [32],finer tuning of our two nonlinear channels. A complete convergence is still not achieved.The two black squares, for α -projectile, are taken from Fig. 5b of [24], for our velocityrange. They are based on TDDFT with an optimized, localized atomistic, Gaussian basisset. We speculate that, for extended systems with slow ions, the screening action of themetallic electron gas needs a further consideration. Moreover, singly ionized He + intruder,10 v (a.u.) d E / dx ( a . u . ) FIG. 2: (Color online) Stopping power, dE/dx , for helium projectiles as a function of the velocity v ∈ [0 , . Z = 2. See the text for further details.. instead of He ++ , might be more close to the experimental situation.In Fig. 3 we plot the observable quantities, (1 /v )( dE/dx ), as a function of Z . Theexperimental data (black dots and triangles) are used earlier [20] to a comparison with Q (1) ,which is denoted here by a dashed curve. It was stated in this pioneering work that there isa substantial disagreement with data in magnitude, particularly around the minimum. Ournew result, [ Q (1) + Q (2) ], is denoted by a solid curve. Notice that data symbols, withouterror bars, refer to v = 0 .
411 (dots) and v = 0 .
826 (triangles). The target is the frequentlyused prototype of free-electron metal, aluminum. By inspection, one can observe an essen-tial improvement in agreement between data and the novel approach. Here we return toexperiment in [26], i.e., to the above-mentioned surface experiment. There, although withsomewhat smaller deviations from the conventional Q (1) [ v F ( z )]-type scaling, also a system-atic upward enhancement in stopping power was established. In the present two-channel11 Z v d E __ dx ( a . u . ) FIG. 3: Results for the friction-like coefficients (1 /v )( dE/dx ) are plotted in atomic units. Thedashed and dotted curves refer to Eq. (8) and Eq. (9), respectively. The sum of Eq.(8) and Eq.(9),the solid curve, refers to our novel approach. The experimental data points (for Al) are the samewhich were used in a pioneering work leaded by Echenique [20]. They are denoted by solid circlesand triangles. . modeling, such an enhancement can be associated with a Q (2) -proportional contribution. III. SUMMARY AND COMMENTS
In this theoretical paper we have investigated the problem of average retarding forceof metallic targets for slow projectiles. Beyond the well-known contribution, establishedfor electron-slow-intruder scattering in a degenerate electron gas, a novel contribution isderived which is associated with charge-change cycles due to lattice ions. Our main closedresult is given by Eq.(6), which is, in the terminology of this sub-field of physics, a nonlinearform, similarly to the more conventional one in Eq.(2). These forms are implemented here bystandard phase shifts obtained by applying the orbital-based DFT to screening in an electrongas. For helium projectiles, we made comparisons with selected results of experiments [15]and large-scale simulations [6, 7] in TDDFT for a free-electron-like metal, Al. With ournovel contribution to the retarding force, the agreement with these results is improved.As we discussed in Section I, our novel channel describes a sudden perturbation whichis not explicit in recent TDDFT simulations. There a smoothed evolving picture is em-12loyed, without fast local charge-changing processes. As Fig. 2 signals our green curvesbracket the TDDFT outputs obtained for channeling and off-channeling conditions withinan evolving picture. We believe that further efforts in TDDFT are needed to tackle explic-itly charge-changing processes. However, by construction, TDDFT simulations are able tomodel the lattice-related details of realistic targets. In order to get further informations onthe capabilities of different theoretical methods the problem of projectiles in alloys could bean important one. There, different lattice ions could influence (presumably, due to differ-ent closest approaches) the charge-changing fast processes giving an opportunity to see theadvantage of our modeling over those based on a smoothed picture in the time domain.A challenging problem in ratio-data-interpretation [14] for Mg is discussed as well. Ourtwo-channel-based result is in reasonable agreement with data at around v = v F ≃ . Z ∈ [1 ,
18] in Al is made, and improved agreement is found.Based on these agreements, we suggest further efforts within TDDFT along these lines.The percentage differences (about 400% and 40%, roughly) of the conventional Q (1) (dashedcurve) and the new [ Q (1) + Q (2) ] (solid curve) results in Fig. 3 for Z = 12 in comparisonwith experimental data on pure Al target heralds that alloy-targets (see, above) could berelevant candidates to understand differences between theoretical attempts discussed in thiswork.Notice that a recent adiabatic modeling [29], motivated by experiment in [18], also resultsin remarkable deviations from a simple modeling with Q (1) ( v F ). There the density inhomo-geneity was considered, via lattice-atom-volume averaging of Q (1) [ v F ( r )], as a modulatingeffect. Such an averaging was applied successfully [33] for stopping of swift Z = ± r s ≃ . r s ≃
3) is in accord with this.Thus, at this moment, we have two, i.e., q -dependent and inhomogeneity-dependent,effects which result in enhancement in the electronic stopping power beyond the conventional,i.e., Q (1) -dependent, theoretical estimation. Both seem to be, a priori , relevant in reality.13heir proper weights and interplay need future investigations. Cases with self-irradiatedcondition [34, 35] could be especially important in this ( q = 0) respect. For instance, Niions in Ni target [35]. In such a symmetric case we can image (for a metal) even q = 2 toour Q (2) channel. For r s ≃
2, Ni ion with its Z = 28 represents the second minimum in Z -oscillation [19, 21]. In our modeling we get [ Q (1) + Q (2) ] ≃ [0 .
28 + ( q × . q = 2 and v = 1, one arrives at [( Q (1) + Q (2) ) /Q (1) ] ≃
11, thus the correspondingstopping power would change steeply to about ( dE/dx ) ≃ eV /A . For transition metals,that show a high electronic stopping power [35], the spin-flip process needs a thorough inves-tigation. The electron spin direction is no longer conserved during electron-atom collision.One has to consider the total angular momentum j = ( l + s ) operator in order to constructa complete set of spin-angle functions which are needed to expansions. We left this excitingsub-problem in stopping theory with a new (spin) degree of freedom to future studies.Notice that at high ion velocity, our new term would scale as ( q/v ) with respect to theconventional, i.e., Bethe-like [1], leading one [7, 24]. There, a term with [ q ( v ) /v ] can give a slowly vanishing enhancement. Thus the high-velocity limit, first of all under self-irradiatedcondition [35], also requires further investigations. The precise relevance of permutation-based, similarity-aided level crossing [22, 36] behind higher q ( v )-values seems to be an otherinteresting sub-problem in stopping theory. The Bethe-limit, especially for metals with theirdense electron gas, is not a simple cumulative sum of isolated atomic contributions [37].Based on the established capability of our new modeling for metals, we believe thatthe two-channel approach developed here can find application in other important fields aswell. For instance, in the friction problem of diatomic molecules during their dissociativeadsorption on metallic surfaces. There, based on an empirically motivated local-density-friction approximation, a local Q (1) [ v F ( r )] is employed [38]. We argue here that transientelectronic processes, due to dissociation in an electron gas, could be related to Q (2) inEq.(6). For instance, the case of N, with its Z = 11, might be a good candidate, asour Table I suggests. We stress, however, that at high target-temperatures, the couplingto phonon modes, i.e., to quanta of lattice vibrations, can open a new [39] channel toinelastic processes, beyond the friction-like channel discussed in our present study for coldmetals. Still, as Figure 1 of [39] signals, the proper magnitude of this latter channel couldbe important. Indeed, the so-far neglected [39] charge-transfer-type [related to Q (2) ( v F )]processes, especially with highly reactive molecules, may have impact on conclusions.14e close with few general comments. The wave functions of the conventional orbital-based DFT for embedded Z are used [19–21] here to calculate the induced electron density.That is the basic variable of the underlying variational theory. The phase shifts are, there-fore, auxiliary quantities [19]. Their sums over angular momentum quantum numbers alwayssatisfy the associated neutrality condition of a self-consistent orbital-based approximation,i.e., the Friedel sum rule and the Levinson theorem [13] for local interactions. Since theseare satisfied by construction for any form of a local many-body term in the Schr¨odinger-likeequations, the physical quality of DFT results needs further, i.e., energetic, justifications.But, in accord with closely related statements [19, 21], the highly improved quantitativeagreements with experimental facts justify, a posteriori , our phase-shift-based two-channelmodeling with a novel term for the retarding force. Generally, and in accord with Landaubasic attempt [36] for Fermi liquids, a modeling is good if it contains few adjustable elements,agrees with several observations, and makes controllable predictions. We stress, finally, thatthe truly exciting theoretical problem of interparticle-interaction, i.e., correlated motion ofelectrons, is considered in stopping calculation only at the mean-field level. However, at leastfor a prototype two-particle correlated model system, recent exact result [40] for the energyshift in time-dependent (passing) perturbations indicates that proper independent modes ,rather than effective single-particle states, could pave the path to future developments. Acknowledgments
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