Electron elastic scattering off A @C 60 : The role of atomic polarization under confinement
aa r X i v : . [ phy s i c s . a t o m - ph ] O c t Electron elastic scattering off A @C : The role of atomic polarization underconfinement V. K. Dolmatov, M. Ya. Amusia,
2, 3 and L. V. Chernysheva University of North Alabama, Florence, Alabama 35632, USA Racah Institute of Physics, Hebrew University, 91904 Jerusalem, Israel A. F. Ioffe Physical-Technical Institute, 194021 St. Petersburg, Russia (Dated: April 4, 2018)The present paper explores possible features of electron elastic scattering off endohedral fullerenes A @C . It focuses on how dynamical polarization of the encapsulated atom A by an incidentelectron might alter scattering off A @C compared to the static-atom- A case, as well as howthe C confinement modifies the impact of atomic polarization on electron scattering comparedto the free-atom case. The aim is to provide researchers with a “relative frame of reference” forunderstanding which part of the scattering processes could be due to electron scattering off theencapsulated atom and which due to scattering off the C cage. To meet the goal, the C cage ismodeled by an attractive spherical potential of a certain inner radius, thickness, and depth whichis a model used frequently in a great variety of fullerene studies to date. Then, the Dyson equationfor the self-energy part of the Green’s function of an incident electron moving in the combinedfield of an encapsulated atom A and C is solved in order to account for the impact of dynamicalpolarization of the encaged atom upon e + A @C scattering. The Ba@C endohedral is chosen asthe case study. The impact is found to be significant, and its utterly different role compared to thatin e + Ba scattering is unraveled. PACS numbers: 31.15.ap, 31.15.V-, 34.80.Dp, 34.80.Bm
I. INTRODUCTION
Electron elastic scattering off quantum targets is animportant fundamental phenomenon of nature. It hassignificance to both the basic and applied sciences andtechnologies. Yet, to date, the knowledge on the processof electron collision with such important quantum targetsas endohedral fullerenes A @C is far from complete. En-dohedral fullerenes are nanostructure formations wherean atom A is encapsulated inside the hollow interior of aC molecule. The authors are aware of only a handfulof works on this subject. These are the theoretical stud-ies of fast charged-particle ionization of A @C [1–3] andlow-energy electron scattering off A @C [4–6] calculatedin the framework of two different model approximations.Namely, in Refs. [4, 5] a static Hartree-Fock (HF) ap-proximation was employed. There, both the atom A andC were considered as nonpolarizable targets and theC cage was modeled by an attractive spherical poten-tial of a certain inner radius, thickness, and depth. InRef. [6], the authors kept the encaged atom “frozen”,modeled the C cage by the potential similar to thatused in Ref. [4, 5], but accounted for polarization of theC cage by incident electrons. The latter was evaluatedin a simplified manner by adding a static polarization po-tential − α/r ( α being the static polarizability of C )to the model C -potential. Note, a meager amount ofresearch on e + A @C collision is in contrast to the studyof photon- A @C collision, different aspect of which hasbeen intensely scrutinized in a great variety of theoret-ical works to date (see, e.g., Refs. [7–14] and referencestherein), including experimental studies [15, 16] (and ref-erences therein). Such disbalance in favor of the number and quality of studies of A @C photoionization versusresearch on electron- A @C collision is not accidental.Electronic collision with a multielectron target is a morecomplicated multifaceted process compared to photoniccollision with the same target. Therefore, the comprehen-sive description of electron scattering by a multielectrontarget is too challenging for theorists even with regard toa free atom, not to mention A @C targets.The present study does not aim at solving the difficultproblem of electron- A @C scattering in its entirety. In-stead, it focuses on the contribution of electron scatteringonly off the encapsulated atom A to the entire collisionprocess. The significance of the present study is that itprovides an important frame of reference for (future) un-derstanding of which part of electron- A @C scatteringcould be due to scattering off the encapsulated atom A (unraveled in the present work) and which is due to other“facets” of the entire A @C system. Research results,thus, provide a relative rather than absolute knowledge.To meet the goal, the C cage is modeled, as in Refs. [4–6, 17], by a spherical potential of a certain inner radius,thickness, and depth. Polarization of the C cage byincident electrons will, thus, be ignored (being not thesubject of the focused study). This is in contrast to ac-counting for polarization of the encapsulated atom in thepresent study. The neglect by polarizability of C byincident electrons should not be over-dramatized. Theeffect of polarizability is electron-energy-dependent andmay either enhance or decrease scattering cross section,at certain electron energies. Therefore, when scatteringoff C is dramatically decreased, or where scattering ofthe encapsulated atom A is dramatically increased, a rel-ative role of scattering off the atom A will (might) besignificant. Furthermore, A @C has the hollow interiorwhich is not totally occupied by the atom (i.e., not to-tally filled in with charge density). As such, it acts asa resonator relative to incident electronic waves. There-fore, at wave frequencies, matching resonance frequen-cies of the A @C -resonator, there will be a significantincident-electron-density build-up in the hollow interiorof A @C . Obviously, this build-up of electron densitywill be positioned near the encapsulated atom A . There-fore, the effect of atomic polarization on electron scatter-ing might become comparable or even more importantthan the C polarization effect, at resonance frequencies.As such, the impact of atomic polarizability on e + A @C scattering it cannot be dropped out of the considerationat all. It is, therefore, indisputably needed (and interest-ing, and does make sense) to study how polarization ofthe encapsulated atom can affect the scattering processeven in the neglect by polarization of the fullerene cageby incident electrons. To account for atomic polariza-tion under confinement, the authors employ the Dysonformalism for the self-energy part of the Green’s func-tion of a scattered electron [18, 19], adapt it to the caseof the electron motion in a combined field of the encap-sulated multielectron atom A and the model static C cage, solve the generalized Dyson equation, and, thus,calculate the electron elastic-scattering phase shifts andcorresponding cross sections for the e + A @C scatter-ing reaction. The study is restricted to electron elasticscattering at low electron energies ǫ < ∼ A @C scattering. This is because it falls into a main-stream of intense modern studies where numerous aspectsof the structure and spectra of atoms under various kindsof confinements (impenetrable spherical, spheroidal, di-hedral, Debye-like potentials, etc.) are being attackedfrom many different angles by research teams world-wide(see, e.g., numerous review articles in Refs. [20–22]).Such studies are interesting from the view point of basicscience. Results of the present study add new knowledgeto the collection of atomic properties under confinementas well, particularly revealing the impact of atomic polar-ization under confinement on electron-atom scattering.Atomic units are used throughout the paper unlessspecified otherwise. II. THEORYA. e + A @C scattering in the framework of staticC
1. Model static HF approximation
In the present work, the C cage is modeled by aspherical potential U c ( r ) defined as follows: U c ( r ) = (cid:26) − U , if r ≤ r ≤ r + ∆0 otherwise. (1)Here, r , ∆, and U are the inner radius, thickness, anddepth of the potential well, respectively.Next, the wavefunctions ψ nℓm ℓ m s ( r , σ ) = r − P nl ( r ) Y lm ℓ ( θ, φ ) χ m s ( σ ) and binding energies ǫ nl of atomic electrons ( n , ℓ , m ℓ and m s is the standard setof quantum numbers of an electron in a central field, σ isthe electron spin variable) are the solutions of a systemof the “endohedral” HF equations: (cid:20) − ∆2 − Zr + U c ( r ) (cid:21) ψ i ( x ) + Z X j =1 Z ψ ∗ j ( x ′ ) | x − x ′ |× [ ψ j ( x ′ ) ψ i ( x ) − ψ i ( x ′ ) ψ j ( x )] d x ′ = ǫ i ψ i ( x ) . (2)Here, Z is the nuclear charge of the atom, x ≡ ( r , σ ), andthe integration over x implies both the integration over r and summation over σ . Eq. (2) differs from the ordinaryHF equation for a free atom by the presence of the U c ( r )potential in there. This equation is first solved in orderto calculate the electronic ground-state wavefunctions ofthe encapsulated atom. Once the electronic ground-statewavefunctions are determined, they are plugged back intoEq. (2) in place of ψ j ( x ′ ) and ψ j ( x ) in order to cal-culate the electronic wavefunctions of scattering-states ψ i ( x ) and their radial parts P ǫ i ℓ i ( r ).Corresponding electron elastic-scattering phase shifts δ ℓ ( k ) are then determined by referring to P kℓ ( r ) at large r : P kℓ ( r ) → r π sin (cid:18) kr − πℓ δ ℓ ( k ) (cid:19) . (3)Here, k and k ′ are the wavenumbers of the incident andscattered electrons, respectively, and P kℓ ( r ) is normalizedto δ ( k − k ′ ). The total electron elastic-scattering crosssection σ el ( ǫ ) is then found in accordance with the stan-dard formula for electron scattering by a central-potentialfield: σ el ( k ) = 4 πk ∞ X ℓ =0 (2 ℓ + 1) sin δ ℓ ( k ) . (4)This approach solves the problem of e +A@C in astatic approximation, i.e., without accounting for polar-ization of the A @C system by incident electrons.In the literature, some inconsistency is present inchoosing the magnitudes of ∆, U and r of the model E l e c t r on e l a s t i c - sc a tt e r i ng c r o ss s e c t i on Electron energy (eV)
MHF, from Ref. [17] ∆ = 1.9, R = 5.8, U = 0.302 ∆ = 2.9102, R = 5.262, U = 0.2599 ∆ = 1.25, R = 6.01, U = 0.422 e + C FIG. 1. (Color online) e + C elastic-scattering cross section(in units of a , a being the first Bohr radius of the hydrogenatom) calculated both with the use of different values of theparameters r , ∆, and U of the spherical potential U c ( r )(present work) and in the framework of ab initio MHF [17],as marked. potential U c ( r ) for C : r = 5 .
8, ∆ = 1 . U = 0 . r = 6 .
01, ∆ = 1 . U = 0 .
422 [2, 9], or ∆ = 2 . r = 5 . U = 0 . e + A @C scattering, we performed the corresponding calculationsof e + C scattering. Calculated results are plotted inFig. 1 against calculated data obtained in the frameworkof the sophisticated ab initio static-exchange molecular-Hartree-Fock (MHF) approximation [17].One can see that it is the set of parameters proposedin Ref. [17] which leads [17] to the overall qualitative andsemi-quantitative agreement between some of the mostprominent features of e + C elastic scattering predictedby the model spherical-potential approximation and abinitio MHF. Correspondingly, in the present work, as inRef. [17], the U c ( r ) potential is defined by ∆ = 2 . r = 5 . U = 0 .
2. Multielectron approximation: a polarizable atom A In order to account for the impact of polarization of anencapsulated atom A by incident electrons on e + A @C elastic scattering, let us utilize the concept of the self-energy part of the Green’s function of an incident electron[18, 19].In the simplest second-order perturbation theory inthe Coulomb interelectron interaction V between the in-cident and atomic electrons, the irreducible self-energypart of the Green’s function Σ( ǫ ) of the incident electron (a) (b) (c) (d) FIG. 2. The irreducible self-energy part Σ( ǫ ) of the Greenfunction of a scattering electron in the second-order pertur-bation theory in the Coulomb interaction, referred to as theSHIFT approximation (see text). Here, a line with a right ar-row denotes an electron, whether a scattered electron (states | ǫ ℓ i and | ǫ ′ ℓ ′ i ) or an atomic excited electron (a state | m i ), aline with a left arrow denotes a vacancy (hole) in the atom(states h j | and h i | ), a wavy line denotes the Coulomb inter-electron interaction V . ε '' l '' E n l ' < 0 ε ' l ε ' l ε ' l ε l ε l ε l ε ' l ε l Σ ~ Σ Σ Σ ++ Σ ~=~ Σ FIG. 3. The matrix element of the reducible self-energy part˜Σ( ǫ ) of the Green’s function of a scattering electron, whereΣ is the irreducible self-energy part of the Green’s functiondepicted in Fig. 2. This approximation is referred to asthe SCAT approximation (see text). Note, when calculat-ing h ǫ ℓ | ˜Σ | ǫ ℓ i analytically, the summation over unoccupied dis-creet and integration over continuum excited states (markedas ǫ ′′ ℓ ′′ ) along with the summation over the occupied states inthe atom marked as E nℓ ′ must be performed. is depicted with the help of Feynman diagrams in Fig. 2.The diagrams of Fig. 2 illustrate how a scattered elec-tron “ ǫ ℓ ” perturbs (read: polarizes) a j -subshell of theatom by causing j → m excitations from the subshelland then gets coupled with these excited states itself viaboth the Coulomb direct [diagrams (a) and (b)] and ex-change [diagrams (c) and (d)] interactions. Numericalcalculation of electron elastic-scattering phase shifts inthe framework of this approximation is addressed by thecomputer code from Ref. [19] labeled as the “SHIFT”code. Correspondingly, the authors refer to this approxi-mation as the “SHIFT” approximation everywhere in thepresent paper.A fuller account of electron correlation (read: polar-ization) in e + A @C elastic scattering is determined bythe reducible ˜Σ( ǫ ) part of the self-energy part of the elec-tron’s Green function [19]. The matrix element of thelatter are represented diagrammatically in Fig. 3.The above diagrammatic equation for ˜Σ can be writtenin an operator form as follows:ˆ˜Σ = ˆΣ − ˆΣ ˆ G (0) ˆ˜Σ . (5)Here, ˆΣ is the operator of the irreducible self-energy partof the Green’s function calculated in the framework ofSHIFT (Fig. 2), ˆ G (0) = ( ˆ H (0) − ǫ ) − is the HF oper-ator of the electron Green’s function, and ˆ H (0) is theHF Hamiltonian operator of the electron + A @ C sys-tem. Clearly, the equation for the matrix elements of˜Σ accounts for an infinite series of diagrams by couplingthe diagrams of Fig. 2 in various combinations. Numer-ical calculation of electron elastic-scattering phase shiftsin the framework of this approximation is addressed bythe computer code from Ref. [19] labeled as the “SCAT”code. Correspondingly, the authors refer to this approxi-mation as the “SCAT” approximation everywhere in thepresent paper. SCAT works well for the case of electronscattering off free atoms [19]. This gives us confidencein that SCAT is a sufficient approximation for pinpoint-ing the impact of correlation/polarization on e + A @C scattering as well.In the framework of SHIFT or SCAT, the electronelastic-scattering phase shifts ζ ℓ are determined as fol-lows [19]: ζ ℓ = δ HF ℓ + ∆ δ ℓ . (6)Here, ∆ δ ℓ is the correlation/polarization correction termto the calculated HF phase shift δ HF ℓ :∆ δ ℓ = tan − (cid:16) − π D ǫℓ | ˜Σ | ǫℓ E(cid:17) . (7)The mathematical expression for D ǫℓ | ˜Σ | ǫℓ E is cumber-some. The interested reader is referred to [19] for de-tails. The matrix element D ǫℓ | ˜Σ | ǫℓ E becomes complexfor electron energies exceeding the ionization potential ofthe atom-target, and so does the correlation term ∆ δ ℓ and, thus, the phase shift ζ ℓ as well. Correspondingly, ζ ℓ = δ ℓ + iµ ℓ , (8)where δ ℓ = δ HF ℓ + Re∆ δ ℓ , µ ℓ = Im∆ δ ℓ . (9)The total electron elastic-scattering cross section σ el isthen given by the expression σ el = ∞ X ℓ =0 σ ℓ , (10)where σ ℓ is the electron elastic-scattering partial crosssection: σ ℓ = 2 πk (2 ℓ + 1) cosh 2 µ ℓ − cos 2 δ ℓ e µ ℓ . (11) III. RESULTS AND DISCUSSION: e + Ba@C SCATTERING
3. Preliminary notes
In the aims of the present paper, the authors focus onelectron scattering off Ba@C , as the case study. This isbecause the Ba atom is a highly polarizable atom. It isexpected to retain its high polarizability under the C confinement as well. This provides one with a betteropportunity to learn how atomic polarization can alterelectron scattering off A @C compared to scattering offthe static target.Furthermore, the study focuses on the electron energyregion of up to ǫ ≈ λ > A exceeds greatly the bondlength D ≈ .
44 ˚ A between the carbon atoms in C .Correspondingly, the incoming electrons will “see” theC cage as a homogeneous rather than “granular” cage.This makes it appropriate to model the C cage by anon-granular, i.e., “smooth” potential. Moreover, as wasshown in Refs. [23, 24], a low-energy electron motion inthe field of C is insensitive to details of the “smooth”potential, i.e., to whether the potential is the poten-tial with round borders and unparallel walls or simply asquare-well potential, as long as λ ≫ D . This addition-ally justifies the use of the square-well potential, Eq. (1),in the present study. Second, correlation/polarization ef-fects, which are of the primary concern of this paper,are expected, as usually, to be most strong primarily atlow-energy electron collisions with multielectron targets.Third, at these low energies, earlier, there were predictedspectacular confinement-induced resonances in e + A @C scattering [4, 5], similar to those predicted in e +C scat-tering [5, 17, 25]. The presence of the confining C cagein this case, as well as in the case of scattering off emptyC , results in the emergence of positive quasi-discretestates in the field of A @C or C . When quasi-discretestates are present, then, in accordance with the knownfact, “... resonance scattering occurs because a positivelevel with ℓ = 0, though not a true discrete level, is quasi-discrete: owing to the presence of the centrifugal poten-tial barrier, the probability that a particle of low energywill escape from this state to infinity is small, so that thelifetime of the state is long” [26].It is interesting to explore how the predicted reso-nances in e + A @C scattering can be altered by theeffects of polarization of the encapsulated atom A by in-cident electrons.Next, the calculations of electron scattering offBa@C and free Ba, performed in the present work inthe framework of both SHIFT and SCAT, accounted forthe monopole, dipole, quadrupole, and octupole pertur-bations of the valence 6 s and 5 p subshells of free andencaged Ba by incident electrons. Finally, in view of lowvalues of the electron energies, only the s -, p -, d -, f -, and g -partial electronic waves have been accounted for in thecalculations. The contribution of other partial electronicwaves is negligible, at given energies.
4. Results and discussion
Corresponding calculated HF, SHIFT, and SCAT datafor the real parts of phase shifts δ ℓ ( ǫ ), partial σ ℓ ( ǫ ) andtotal σ el ( ǫ ) cross sections for electron elastic scatteringoff Ba@C are displayed in Fig. 4 (the imaginary parts µ ℓ of the phase shifts, when present, are small at thechosen energies and present little interest for discussion).As an important finding, one learns from Fig. 4 thatthe C confinement does not at all “extinguish” the pos-sibility for the encapsulated Ba atom to be strongly po-larized by incident electrons. On the contrary, the polar-ization impact is found to affect dramatically both theelectron scattering phase shifts and partial σ ℓ as well astotal σ el cross sections. All this is obvious from the com-parison of calculated results obtained in the frameworkof HF, on the one hand, and SHIFT and SCAT, on theother hand, approximations. As another important re-sult, one finds that accounting for only the second-order(with respect to the Coulomb interaction) correlation ef-fects, as in SHIFT, is important but far insufficient for thecalculation of low-energy electron scattering off Ba@C .Indeed, the correlation impact beyond the second-orderapproximation, i.e., accounted for in the SCAT approxi-mation, is utterly significant – the lower the energy, thestronger the impact.For the next, it is both interesting and necessary tobring to the attention of the reader that the empty staticC cage can only support the s -, p -, and d -bound states[5, 17]. In contrast, the “stuffed” C , i.e., Ba@C , wasfound to lose a s -bound state, but acquire, instead, a f -bound state, in the static HF approximation [5]. Now,it follows from the present study that by “unfreezing”the encapsulated Ba atom, i.e., by making it polarizable,the lost s -bound state is returned to the Ba@C system,and the latter keeps the former p -, d - and f -bound states.Moreover, the Ba@C system is found to start support-ing a second p -bound state as well. All this becomes clearby noting the calculated SCAT values of the s -, p -, d -,and f -phase shifts at ǫ → δ SCAT s → π , δ SCAT p → π , δ SCAT d → π , and δ SCAT f → π . The zero-energy values ofthese phase shifts satisfy the generalized version of Levin-son theorem for scattering in a potential field [26] which(the generalized theorem) could be derived by the directnumerical calculation and written as follows: δ ℓ ( ǫ ) | ǫ → → ( N n ℓ + q ℓ ) π. (12)Here, N n ℓ is the number of closed subshells with given ℓ in the ground-state configuration of an atom, whereas q ℓ is the number of additional empty bound states withthe same ℓ in the field of A @C . Given that, for theground-state of the encapsulated Ba atom, N n s = 6, N n p = 4, N n d = 2, and N n f = 0, one finds that, inaccordance with the SCAT values of the phase shifts, q n s = 1, q n p = 2, q n d = 1, and q n f = 1. This translatesinto one s -, two p -, one d -, and one f -bound states sup-ported (one at a time) by Ba@C . Note that calculatedSHIFT phase shifts, in contrast to SCAT data, point tothe existence of neither s - nor second p -bound state inthe field of Ba@C . This discrepancy between the cal-culated SHIFT and SCAT data speaks, in general, to theimportance of a fuller account of electron correlation inthe calculation of e +A@C scattering.The discovered in the framework of SCAT emergenceof a s -bound state and a second p -bound state in thefield of Ba@C has profound consequences for both thecorresponding partial and total electron-scattering crosssections. Namely, because the phase shift δ SCAT s , on theway to its value of 7 π at ǫ = 0, passes through the valueof modulo π/ ǫ ≈ .
23 eV), σ SCATs becomes large,at lower energies, in contrast to the predictions obtainedwith the help of inferior HF and SHIFT. Similarly, theincrease of δ SCATp to 6 π at ǫ = 0 results in large σ SCATp aswell, at low energies, again, in contrast to the predictionsby HF and SHIFT.Talking about the d - and g -partial cross sections, onecan see that each of them is dominated by the strongresonance. Its emergence clearly follows from the behav-ior of the d - and g -phase shifts. Indeed, the phase shiftsfirst tend to increases towards modulo π with decreasingenergy, but, because the field turns out to be not strongenough, before that value is reached, they sharply de-crease, passing through the value of modulo π/
2. This isa typical behavior of phase shifts upon electron scatteringon quasibound states [27]. Next, note how the resonancesin the d - and g -partial cross sections become higher, nar-rower, and shift towards lower energy as more correlationis accounted for in the calculation (compare calculatedHF versus SHIFT versus SCAT results). We thus findthat the inclusion of more correlation into the calcula-tion of e + Ba@C scattering increases the strength ofthe field of the Ba@C -scatterer, since the above notedchanges in the resonances are typical for electron scatter-ing in a field of increasing strength [27].Next, note that no f -resonance emerges in e + Ba@C scattering calculated in either of the three approxima-tions utilized in this paper. This is in contrast to elec-tron elastic-scattering off empty static C . There, thesharp f -resonance was predicted to emerge at low ener-gies [5, 17, 25] (this is the extremely narrow resonancenear zero plotted in Fig. 1). The reason for the absenceof the f -resonance in e + Ba@C scattering is interest-ing. It is directly associated with that a noticeable part ofthe valence electronic charge-density of encapsulated Bais drawn into the C shell [5]. Therefore, the field insidethe “stuffed” C becomes more attractive than in emptyC . It turns the f -state into a bound state, thereby elim-inating the emergence of a f -resonance in e + Ba@C scattering. Now, as it has been shown in the paragraphabove, the inclusion of correlation into the calculationof e + Ba@C scattering increases the field of Ba@C .Therefore, the f -state remains bound. This is why the σ s σ f σ p σ g σ d g-resonanced-resonance σ tot Electron energy (eV) E l e c t r on e l a s t i c - sc a tt e r i ng c r o ss s e c t i on δ s δ f δ p δ g δ d FIG. 4. (Color online) Main panels: Calculated partial σ ℓ ( ǫ ) and total σ el ( ǫ ) cross sections (in units of a ) for electron elasticscattering off Ba@C , obtained in the frameworks of the model static HF (dashed line), multielectron SHIFT (dash-dottedline) and SCAT (solid line) approximations. Insets: Real parts δ ℓ ( ǫ ) of the phase shifts ζ ℓ ( ǫ ) (in units of radian) calculated inHF (dashed line), multielectron SHIFT (dash-dotted line) and SCAT (solid line) approximations. f -resonance does not take place in e + Ba@C scatteringeven if polarization of the encapsulated Ba atom by in-cident electrons is accounted for in the calculation. Ourgeneral prediction is that there will be no f -resonances onquasibound states in electron scattering off any A @C system in case where there is a noticeable transfer of elec-tronic charge-density from the encapsulated atom to theC shell.Furthermore, it is interesting to compare how muchdifferently polarization of the Ba atom by incident elec-trons affects electron elastic scattering off free Ba versusBa@C . The corresponding calculated HF and SCATtotal electron elastic-scattering cross sections are de-picted in Fig. 5.The calculated data reveal a spectacular difference be-tween the role of polarization in electron scattering off Baand Ba@C . Namely, it appears that the effects of polar-ization in e + Ba@C scattering act oppositely to the ef-fects in e +Ba scattering. Thus, whereas σ SCATel ( e +Ba) ≪ σ HFel ( e + Ba) at ǫ < ∼ e + Ba@C scattering in about the same energyregion: σ SCATel ( e + Ba@C ) ≫ σ HFel ( e + Ba@C ). Al-ternatively, whereas σ SCATel ( e + Ba) ≫ σ HFel ( e + Ba) at ǫ > ∼ . σ SCATel ( e + Ba@C ) ≪ σ HFel ( e +Ba@C ) in there. It is, thus, found in the presentstudy that the effects of atomic polarization in electron scattering off the free and encapsulated inside C atomsmay follow opposite routes. This is an interesting obser-vation.Lastly, note that there are energy regions, specifically,0 . < ∼ ǫ < ∼ . ǫ > ∼ . σ SCATel ( e +Ba@C ) ≪ σ SCATel ( e + Ba). This means that the gas-medium of big-sized A @C s can be more transparent toincident electrons than the gas-medium of smaller-sizedisolated atoms A themselves. This counter-intuitive ef-fect was earlier unveiled in Ref. [4] in the framework ofthe static HF approximation, but appears to retain itsplace even if the encapsulated atom is polarizable, as isshown in the present paper. IV. CONCLUSION
The present work has provided a deeper insight intopossible features of low-energy electron elastic scatter-ing off A @C fullerenes. This has been achieved bystudying the dependence of e + Ba@C elastic scatteringwith account for polarization of encapsulated Ba by in-cident electrons. It has been demonstrated that the po-larization effect results in dramatic differences betweenelectron scattering off Ba@C evaluated with and with-out inclusion of polarization into the calculation. It has e + Ba@C versus e + Ba scattering H F S C A T e + Ba HFSCAT e + Ba@C E l e c t r on e l a s t i c - sc a tt e r i ng c r o ss s e c t i on Electron energy (eV)
FIG. 5. (Color online) Calculated total electron elastic-scattering cross sections σ el ( ǫ ) (in units of a ) for electronscattering off Ba@C , obtained in the frameworks of themodel static HF (dashed line) and multielectron SCAT (solidline) approximations, as well as off free Ba (HF, dash-dot-dot;SCAT, dash-dot), as marked. been found that a fuller account for correlation effects in e + A @ C scattering is utterly important. Furthermore,it has been unraveled in the present study that the im-pact of polarization on electron scattering off A @C maybe both qualitatively and quantitatively different thanthat in the case of electron scattering by the free atom A . For instance, it has been demonstrated that wherepolarization significantly enhances the e + Ba@C scat-tering cross section, it significantly diminishes the e + Bascattering cross section and vice verse. This leads to thepossibility for electron scattering off A @C to becomesignificantly weaker than in the case of electron scatteringby the isolated atom A , in certain energy regions. Thiscounter-intuitive effect has been found to be stronger andoccur in a broader energy region than when polarizationis ignored.Lastly, the present study provides researchers withbackground information which is useful for future stud-ies of electron scattering by A @C , particularly aimedat elucidating of a possible significance of a simultaneouspolarization of both the C cage and encaged atom byincident electrons. This will make the A @C more at-tractive, so that predicted in the present study featuresof e + A @C may appear at different energies, or disap-pear at all, some actual bound states may be convertedto resonances, etc. Such effects, however, are subject toan independent study. V. ACKNOWLEDGEMENTS
V.K.D. acknowledges the support by NSF Grant No.PHY-1305085.
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