Z 1 -oscillation in the retarding force of metals for slow ions: Comparative study of theoretical modelings
aa r X i v : . [ c ond - m a t . o t h e r] D ec Z -oscillation in the retarding force of metals for slow ions:Comparative study of theoretical modelings 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 (Dated: December 8, 2020)
Abstract
Theoretical calculations, applying elements of the orbital-based density functional method forself-consistent screening in an electron gas to the stopping power of metals for slow projectiles arereviewed, by focusing on differences in modelings. New, two-channel-based results on the averageretarding force are presented. All theoretical results are compared with pioneering experimentaldata, obtained for carbon target, on Z -oscillation in stopping at random collisional condition. PACS numbers: 34.50.Bw . INTRODUCTION AND MOTIVATION According to the profound classification of Hawking [1], a modeling of reality in theoreticalphysics is good if it contains few adjustable elements, agrees with and explain existingobservations, and makes detailed predictions about future observations. For instance, withinour personal selection, in the famous treatment of Landau on extended, interacting Fermisystems the determination of the low-energy excitation spectrum requires the introductionof just one adjustable element [2, 3], the quasiparticle effective mass ( m ∗ ).Hawking adopted a view that we can call model-dependent realism, which provides aframework, generally of mathematical nature, with which to interpret physical phenomena.But different theories can successfully describe the same phenomenon [1]. Moreover, manytheories that proven successful were later replaced by other ones based on new concepts .Motivated by such a profound view on model-dependent realism [1], here we turn to aparticular phenomenon well-known in different subfield of nature. It is the energy loss ofcharged projectiles [4] in matter, and the characterization of the slowing down by an averageretarding force experienced by heavy intruders. Evidently, the phenomenon prescribes aquantum (wave) mechanical consideration of electrons in the target material. Furthermore,a retarding-force interpretation of the energy loss per unit path length ( dE/dx ) for ionsrequires the calculation of expectation values of the force operator (gradient of an externalpotential) considering complete sets for electron states in order to quantify an observable.Pioneering experiments [5, 6] with slow ( v = 0 .
41) ions with Z ∈ [1 ,
20] on solid carbontarget ( Z = 6, with four valence electrons) at random collisional condition revealed apronounced Z -oscillation in the observable stopping power (we use Hartree atomic unitsthroughout this work). For an account on Z -oscillation in gases we refer to [7].Our present, quantum mechanical, understanding of Z -oscillations in metallic targetswith slow charged projectiles dates back to [8–10], as we will outline in the next Section.New elements to these single-channel theoretical results (obtained for a Kohn-Sham-typedegenerate electron gas model) were already considered and quantified in [11, 12] and, quiterecently, in [13] via two (phase-shift-based) nonlinear channels to the average retarding force.For a discussion of statistical, quasiclassical theoretical attempts [14, 15] on the electronicstopping power we refer to a summary [16]. Our work here is dedicated to a comparativestudy on different modelings [8–13] of the observed Z -oscillations in carbon.2 I. COMPARISON AND DISCUSSION
We follow here a time-ordered path in order to outline the above-selected [8–13] modelingson low-velocity ( v ) stopping power. In order to concentrate on concepts, and associated newadjustable elements, we will avoid extensive (repeated) mathematical details (where it ispossible without distorting the reasoning) by referring for them to the original papers.In a scattering interpretation the energy loss per unit path length is a simple product dEdx = n [( vv F ) σ tr ( v F )] (1)where σ tr is the transport (momentum transfer) integrated cross section. The electronnumber density (a part of constituents of metals) is n and the velocity in the last occupiedone-electron momentum state (plane wave) is v F . In this semiclassical interpretation σ tr canbe based on a probability (instead of a probability amplitude) and is defined by σ tr = 2 π Z π dθ sin θ (1 − cos θ ) dσ ( θ ) (2)in terms of the differential cross section dσ ( θ ). Quantum statistics for fermions is encodedin v F ≡ (3 π n ) / . The product vv F is a simple arithmetic average of energy change ( ω ) intwo-particle (binary) collision, i.e., < ω > = v [(2 v F + 0) /
2] = vv F .The wave character of system electrons appears in the quantum mechanical descriptionof σ tr by using a complex scattering amplitude [ f ( θ, v F )] to dσ ∝ | f ( θ, v F ) | . Thus, in termsof Bessel phase shifts in partial waves for regularized (screened) interaction one arrives at σ tr ( v F ) = 4 πv F ∞ X l =0 ( l + 1) sin [ δ l ( v F ) − δ l +1 ( v F )] . (3)Pioneering papers [8, 9] are based on the above-outlined kinetic framework. Their remarkablecontribution (an adjustable element) was to apply phase shift outputs of the self-consistentprocedure of auxiliary-orbital-based (Kohn-Sham) DFT for screening of an embedded barecharge Z , independently of incoming-ion charge states. In such a buildup-procedure ofcomplete screening, where we respect the impact of a metallic medium in all one-electronstates a priori , one has the Friedel sum rule of neutrality with charge Q on [17] impurity Q ≡ Z = 2 π ∞ X l =0 (2 l + 1) δ l ( v F ) , (4)and the Levinson theorem of scattering theory with local [18] interaction is satisfied.3otice, that the possible role of relative (r) kinematics within an independent-particle-scattering picture beyond the impurity limit ( v →
0) was investigated in thesis-papers [19,20] for v ∈ (0 , v F ), by considering prefixed (at v →
0) DFT potentials [8–10] as plausible ones.Notable reduction in ratios, obtained at v →
0, of maximum/minimum in Z -oscillationwas observed for v → v F . As expected on physical grounds, and due to mathematicsin σ tr ( v r ) = (4 π/v r ) P ∞ l =0 ( l + 1) sin [ δ l ( v r ) − δ l +1 ( v r )], boundary values in oscillations arethe most sensitive to tuning the relative (r) wave number. The comparison with surface-experimental data [21], which are almost free from complications due to lattice-ions, gavean improved character for v = 0 . v F ≃ .
87) in Z -oscillation.Of course, as we know from an insightful book [22], the above-discussed (mean-field)consideration of screening is one possibility. An alternative way uses the consideration thatthe matrix elements of the external potential which connect occupied states described bySlater determinant has no effect on the many-body wave function since their effect cancels outin a determinant wave function. This fact suggests starting from the Schr¨odinger equationin which the potential is replaced by a truncated potential which is defined like the Bethe-Goldstone potential, except that it is concerned with one particle only, and therefore simpler.However, such a pseudopotential is nonlocal since in its definition for a given value of thespace coordinate the value of one-particle wave function at other space-points enter.We are not aware of such an implementation for screening to stopping calculations viaphase shifts, but we should keep in mind this alternative way for cases of partially-ionizedprojectiles since there are similar values in σ tr for small phase shift differences or differencesclose to π . With nonlocal [18] interaction the modified Levinson theorem becomes[ δ l (0) − δ l ( ∞ )] = π ( n b + n p ) (5)where n b is the number of true bound states and n p is the number of ”bound states” ex-cluded by the Pauli exclusion principle for fermions. For instance, in electron-helium atomscattering in vacuum n b = 0 and n p = 1, thus δ → π since there is no stable negative ion.In orbital-based construction of DFT with Z = 2 one has n b = 1 and n p = 0 thus δ → π forvanishing scattering wave number, i.e., at very low density [23] of the electron gas. However,a similar tendency to π does not imply, to our best knowledge, the same finite values for thescattering length and thus for cross sections related to observables.Results based on [8, 9] for (1 /v )( dE/dx ), obtained within an orbital-based (Kohn-Sham)4reatment for screening and scattering of an embedded Z are plotted in Fig. 1 by dashedcurve for r s ≃ . r s = [3 / (4 πn )] / = (9 π/ / /v F . Thisparameter represents the valence electron density of carbon target . It was pointed out inthe pioneering [9] that the data (solid circles) and theory (dashed curve) show the sameoverall trend with good quantitative agreement for Z <
7. However, as was emphasizedin the other pioneering work [8], experimental values tend to increase more rapidly as Z increases. According to the qualitative discussion of physics in [8], in a kinetic picture, theincrease can be due to several effects. For instance, the ionic radius increases with Z andtherefore heavier ions see a larger effective electron density than the lighter ones.The closely related problem of charge inhomogeneity under random collisional conditionin carbon, was addressed and quantified in [11]. There, as a necessary extension of [15] withhidden details [16], an averaging procedure was defined based on a statistical local-densityapproximation, with [ dE/dx ]( r ) local inputs constrained by Eq.(4), via r s ( r ) as follows h dEdx i ≡ N a Z r a dEdx ( r ) 4 πr dr. (6)Here N a is the number of atoms per unit volume and r a is the atomic cell radius. The dash-dotted curve in Fig. 1 shows the such-obtained averaged results. There is a remarkableimprovement especially at around the experimental minimum and beyond it for higher Z .However, in a more recent analysis [24] on averaging procedures it was pointed out thatone of the shortcomings of the above approach is the fact that it includes in the integrationall the atomic electrons ( r min = 0) without consideration of the important binding effectsthat will tend to cancel the contribution of inner shell(s) to the stopping in the case of slowions. In the case of carbon with four valence electrons this shell is the doubly occupied 1 s shell. Clearly by excluding this (compact in its extension) shell from averaging one goes backto the dashed curve. In our opinion, this alarming observation on the realistic role of target-atom inner shells remains valid for metallic targets with heavier [25] atomic constituents aswell. Intuitively, swift projectiles [26] without bound electrons around them, say Z = ± n . However, as many-body methods [2, 17, 27] indicate, the excitation5 v d E __ dx ( a . u . ) FIG. 1: Stopping power of carbon ( Z = 6) target ( r s ≃ .
6) for ions with atomic number Z .Transmission data (denoted by solid circles), obtained at random collisional condition, are takenfrom [5, 6] for v = 0 .
41. The dashed curve corresponds to the conventional (single-channel) resultobtained in DFT at embedding condition [8, 9]. The dash-dotted, and dash-dot-dot-dashed curves(with new elements beyond this conventional description) refer to [11] and [12], respectively. Thesolid curve is based on Eqs.(7-9), and refers to our two-channel approach developed recently [13].. spectrum above such a surface may get modulations (due to dynamical exchange-correlation[28] between electrons) beyond the one prescribed by the imaginary part of the noninter-acting ( m ∗ = m ) density-density response function [which is ∝ ω , independently of thecharacteristics ( v F ) of the filled ground-state], i.e., of the Lindhard function of the time-dependent linearized Hartree-type approximation for screening. In [10] we made an attemptto combine static screening nonlinearities with many-body modulation in the electron-holeexcitation spectrum (which should depends on m ∗ , according to Landau’s physics-based[2, 27] intuitive treatment) responsible for dissipation. Only small changes in stopping,depending slightly on Z ∈ [5 , r s = 2.Therefore, it was a surprise when a sophisticated approach [12] based on time-dependentdensity-functional theory (TDDFT), for the prototype model system of an embedded Z and an interacting electron gas, within its local-density approximation gave a much higher modulating effect [a density-gradient-dependent additive term to Eq.(1)] in Z -oscillation6f results plotted by dashed curve. This prediction is exhibited in Fig. 1 by a dash-dot-dot-dashed curve. The theoretical enhancement was interpreted tentatively as excitoniceffect in electron-hole pair excitation and good agreement with experimental data on carbontarget (see, Fig. 1) at random collisional condition was concluded. Later the above-outlinedadjustable element of modeling was reviewed [30] within the physically more consistent time-dependent current density functional theory (TDCDFT). Briefly, this new approximation,due to its tensorial character, can satisfy important conservation laws and associated sumrules. Owing to such a physical consistency, the re-adjusted numerical results show onlyvery small changes to pioneering DFT results, i.e., to the dashed curve in Fig. 1.Viewing metals as systems of degenerate electron gases (with corresponding r s parame-ters) and fixed lattice ions, a recent theoretical attempt [13], with the goal to extend [8–10],considered the role of sudden charge-change cycles, as an adjustable new element to the aver-age retarding force experienced by projectiles in their slowing down under random collisionalconditions. There, the force interpretation was implemented by using quantum mechani-cal calculations for matrix elements, instead of direct probability interpretation behind adifferential cross section needed in the classical definition in Eq.(2).We believe that, within the wave mechanical description of one-electron states, such amatrix-element-based [13] treatment with an additional sudden time-dependent process isa physically consistent one. Clearly, and in agreement with an enlightening analysis [31] ofan esteemed expert in the field of stopping, the major (from stopping point of view) changefrom a free Fermi gas occurs because the strong Coulomb field around lattice nuclei allowan increase in the rate of local (sudden) processes of electron jumping into and away frombound states on the projectile. Band-related further, presumably l -dependent, effects arenot addressed here in our projectile-centered construction of one-particle scattering states.Technically, we used earlier established works [32–34] for screened interaction energy inorder to model a new (regularized Coulombic) adjustable dissipative channel which couldmodel the sudden ionizing role of lattice ions in the many-body system of electrons at randomcollisional condition in transmission. Concretely, in [13] we derived1 v dEdx = (cid:2) Q (1) ( v F ) + Q (2) ( v F ) (cid:3) , (7)7here the two coefficients ( 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)We used phase shifts values from earlier works [8–10] in this two-channel modeling and theresults ( q = 1) obtained from Eq.(7) is plotted by solid curve in Fig. 1. The agreementwith data, especially beyond the first maximum in oscillation, is very reasonable. Thus,we believe, that the underlying complete set of scattering states generated in DFT viaan iterative procedure by embedding a bare charge Z is, a posteriori , the proper relaxedone even for singly ionized external projectiles. This was expected on physical grounds byconsidering the immediate screening action of the charged many-body system of metals.In order to provide a template-like figure to future new developments, we do not plotdata (for instance for Al target as in [13] for Z ≤
20, at random collisional condition) inour second, illustrative Fig. 2. This shows, in its clinical form, our prediction with a newadjustable element to characterize an observable. We suggest further attempts. Especially,experiments with Z ∈ [20 ,
40] on Al at random collisional condition are recommended.We motivated our two-channel approach for an average retarding force by referring tosudden local charge changing processes behind q = 1 at random collisional situation in met-als. Can one characterize such an average within the framework of an effective single-channelmodeling via ¯ σ tr ( v F ) in Eq.(1)? The rest of this comparative study on capabilities of ad-justable elements is motivated by such a question. Our discussion below on this (challenging)question is based on quantum physics but has an intuitive character.We return to an important message put forward in [31]. For slow moving ions there isstationary flow (at far distances) of electrons. It was pointed out that a total balance inthat flow could replace the concept of local capture-loss processes. But no quantification ofthis idea was presented. Later, in a combined paper, we quantified [35] such an idea basedon conservation laws, considering the case of slow He ions where charge-change may occur.We applied, as constraint to a one-parametric model potential, the form derived in [36] forthe dipolar backflow ( D b ) amplitude in terms of phase shifts solely at the Fermi energy D b = 1 π ∞ X l =0 (2 l + 1) sin (2 ¯ δ l ) + 4 π ∞ X l =0 ( l + 1) sin ¯ δ l sin ¯ δ l +1 sin (¯ δ l − ¯ δ l +1 ) (10)8
10 20 30 40Z v d E __ dx ( a . u . ) FIG. 2: Illustrative theoretical stopping power at r s = 2 for slow ions with atomic number Z . Thedashed and dotted curves refer to Eq.(8) and Eq.(9), respectively. Their sum is plotted by solidcurve. See the text, and [13], for further details on our force-interpretation behind channels.. For the very clear, pedagogically enlightening discussion of conservation laws behind theabove form we refer to the original detailed research paper. This expression reduces to thefamiliar Friedel sum rule of charge-screening ( D b ⇒ Z ) for small phase shifts, i.e., in thefirst-order Born limit where ( Z /v F ) << He + andproton [described solely by its Q (1) ] stopping ratio R in metals at low ion velocities. At r s = 1 .
5, which correspond to a comparatively high density, the ratios are similar and allratios are below the first-order Born value, i.e., four. At r s = 2 our R NA ratio is in accordwith TDDFT calculation in [37]. At low density, r s = 3, our ratio is in nice agreementwith experimental prediction [38] for v ≤ . .
7, but is essentially larger than R ENAR ≃ . .
7) value is due to underestimation of proton stopping in TDDFTsimulation. This fact was stated in [39] devoted to He/H anomaly in Au. For completeness,we note that at r s = 3 our two-channel-based He stopping is roughly equivalent to protonstopping calculated (in DFT) at r s = 1 .
5. We stress that this (separate) statement is also in9
ABLE I: Ratios ( R i ) of He and H stopping powers for the metallic range of r s in a time-orderedrepresentation. The correspondences are i = EN AR [9], i = N ES [35], and i = N A [13]. The3rd column refers to the amplitude of the dipolar backflow ( D b ) with DFT phase shifts based oncomplete screening (Friedel sum rule) of an embedded charge Z = 2. The 5th column showsthe dipolar backflow amplitude of a forced calculation [35] at the D b ≡ Z = 2 constraint in thelongitudinal current of the slowly moving charge. See the text for further details. r s R ENAR D ENARb R NES D NESb R NA . .
44 1 .
12 2 .
45 2 2 . . .
59 0 .
80 2 .
14 2 2 . . .
81 0 .
27 1 .
77 2 2 . harmony with experimental data [38] obtained on gold target for low ion velocity, v ≤ . s -like state to define r s . In more quantitative terms, we have[ Q (1) + Q (2) ]( r s = 3) | Z =2 ≃ .
38 and Q (1) ( r s = 1 . | Z =1 ≃ .
31 based on Table I of [13].Quite independently of r s we have a reasonable similarity between R NES and R NA val-ues, which may indicate the applicability of an adjustable element discussed in [31], andquantified in [35]. Here we found by comparison that, in a statistical sense for randomcollisional situations, we can model (reinterpret) the local capture-loss processes by usinga constraint in the electronic flow around a slow heavy ion which represents a longitudinalcurrent as well, not only a static charge. Clearly, with moving charges in a charged many-body system one has to satisfy a fundamental constraint prescribed by continuity equationbetween induced charges and the longitudinal current. Remarkably, a recent large-scaleorbital-based TDDFT simulation emphasized [40] that application of current-based (notonly charge-based) implicit version could be a promising way for future more sophisticatedextensions within the framework of first-principles methods. In the orbital-free, i.e., explicit,version of TDDFT an attempt to include current via a dynamical kinetic-energy functionalhas already been presented recently [41, 42]. These modelings of reality within TDDFTwould fit, of course, to Hawking profound classification outlined in the Introduction.10 II. SUMMARY
We have reviewed earlier approaches and presented new results on the average retardingforce of metallic targets for slow projectiles, using pioneering experimental data on Z -oscillation in carbon at random collision condition. All theoretical approaches, selected toour comparative work, are based on different adjustable elements. These are analyzed indetails in order to arrive at established conclusions. We pointed out challenging problemswhich need future developments in the important field of ion stopping in metals. Acknowledgments
This work was supported partly by Project PID2019-105488GB-I00 of the Spanish Min-istry of Science, Innovation, and Universities, MICINN. [1] S. Hawking and L. Mlodinov,
The Grand Design (Bantam Books, London, 2010), Chap. 3.[2] L. P. Kadanoff and G. Baym,
Quantum Statistical Mechanics (Perseus Books, Cambridge,1962), Chap. 11.[3] A. B. Migdal,
Qualitative Methods in Quantum Theory (Benjamin, London, 1977), Chap. 5.[4] L. D. Landau and E. M. Lifshitz,
Quantum Mechanics (Pergamon Press, London, 1958).[5] J. H. Ormrod and H. E. Duckworth, Can. J. Phys. , 1424 (1963).[6] J. H. Ormrod, J. R. Macdonald, and H. E. Duckworth, Can. J. Phys. , 275 (1964).[7] Yu. A. Belkova and Ya. A. Teplova, Bull. Russian Acad. Sci. , 153 (2010).[8] M. J. Puska and R. M. Nieminen, Phys. Rev. B , 6121 (1983).[9] P. M. Echenique, R. M. Nieminen, J. C. Ashley, and R. H. Ritchie, Phys. Rev. A , 897(1986), and references therein.[10] I. Nagy, A. Arnau, and P. M. Echenique, Phys. Rev. A , 987 (1989).[11] J. Calera-Rubio, A. Gras-Marti, and N. R. Arista, Nucl. Instrum. Methods B , 137 (1994).[12] V. U. Nazarov, J. M. Pitarke, C. S. Kim, and Y. Takada, Phys. Rev. B , 121106(R) (2005).[13] I. Nagy and I. Aldazabal, Phys. Rev. A , 032814 (2020), and references therein.[14] O. B. Firsov, Soviet Phys. JETP, , 1076 (1959).[15] J. Lindhard and M. Scharff, Phys. Rev. , 128 (1961).
16] P. Sigmund, Bull. Russian Acad. Sci. , 153 (2008).[17] G. Mahan, Many-Particle Physics (Plenum Press, New York, 1981).[18] P. G. Burke,
Potential Scattering in Atomic Physics (Plenum Press, New York, 1977).[19] R. Vincent and I. Nagy, Phys. Rev. B , 073302 (2006).[20] R. Vincent and I. Nagy, Nucl. Instrum. Methods B , 182 (2007).[21] H. Winter, J. I. Juaristi, I. Nagy, A. Arnau, and P. M. Echenique, Phys. Rev. B , 245401(2003), and references therein.[22] R. Peierls, Surprises in Theoretical Physics (Princeton University Press, Princeton, 1979).[23] I. Nagy and A. Arnau, Phys. Rev. B , 9955 (1994).[24] E. A. Figueroa and N. R. Arista, J. Phys.: Condens. Matter , 015602 (2010).[25] F. Matias, P. L. Grande, M. Vos, P. Koval, N. E. Koval, and N. R. Arista, Phys. Rev. A ,030701(R) (2019).[26] I. Nagy, Phys. Rev. A , 014901 (2002), and references therein.[27] D. Pines, The Many-Body Problem (Benjamin, New York, 1961), pp. 260-278.[28] I. Nagy, J. L´aszl´o, and J. Giber, Z. Phys. A , 221 (1985).[29] P. M. Echenique, I. Nagy, and A. Arnau, Int. J. Quantum Chem. , 521 (1989).[30] V. U. Nazarov, J. M. Pitarke, Y. Takada, G. Vignale, and Y.-C. Chang, Phys. Rev. B ,205103 (2007).[31] J. Lindhard and A. Winter, Nucl. Phys. A166 , 413 (1971).[32] G. D. Gaspari and B. Gy¨orffy, Phys. Rev. Lett. , 801 (1972).[33] L. B¨onig and K. Sch¨onhammer, Phys. Rev. B , 7413 (1989).[34] J.-M. Tang and D. J. Thouless, Phys. Rev. B , 14179 (1998).[35] I. Nagy, Ch. Eppacher, and D. Semrad, Phys. Rev. B , 5270 (2000), and references therein.[36] W. Zwerger, Phys. Rev. Lett. , 5270 (1997).[37] M. Ahsan Zeb, J. Kohanoff, D. S´anchez-Portal, and E. Artacho, Nucl. Instrum. Methods B , 59 (2013).[38] S. N. Markin, D. Primetzhofer, M. Spitz, and P. Bauer, Phys. Rev. B , 205105 (2009).[39] M. Ashan Zeb, J. Kohanoff, D. S´anchez-Portal, A. Arnau, J. I. Juaristi, and E. Artacho, Phys.Rev. Lett. , 225504 (2012).[40] A. Schleife, Y. Kanai, and A. A. Correa, Phys. Rev. B , 014306 (2015).[41] Y. H. Ding, A. J. White, S. X. Hu, O. Certik, and L. A. Collins, Phys. Rev. Lett. , 145001 , 144302(2018)., 144302(2018).