Existence of a system of discrete volume-localized quantum levels for charged fullerenes
Rafael V. Arutyunyan, Petr N. Vabishchevich, Yuri N. Obukhov
aa r X i v : . [ phy s i c s . a t m - c l u s ] O c t Existence of a system of discrete volume-localized quantum levelsfor charged fullerenes
Rafael V. Arutyunyan, ∗ Petr N. Vabishchevich, † and Yuri N. Obukhov ‡ Nuclear Safety Institute, Russian Academy of Sciences,B. Tulskaya 52, 115191 Moscow, Russia
Abstract
In the framework of a simple physical model, we demonstrate the existence of a system of discreteshort-lifetime quantum levels for electrons in the potential well of the self-consistent field of chargedfullerenes and onion-like structures. For electrons, in the case of positively charged fullerenes andonion-like structures, combining analytic and numeric considerations we find that the energy ofthe volume-localized levels ranges from 1 eV to 100 eV. ∗ [email protected] † [email protected] ‡ [email protected] . INTRODUCTION Fullerenes represent one of allotropes of carbon, along with graphite, diamond, amorphouscarbon, nanotubes and graphene. Following the earlier theoretical predictions, the firstfullerene C molecule was experimentally discovered in the 1980-ies [1, 2] as a nanometer-size hollow spherical structure of 60 carbon atoms located at the vertices of a truncatedicosahedron. Subsequently, the production of fullerenes in large quantities was developedand the fullerene nanotubes and many other fullerenes were discovered, such as C , C andeven lager structures. This gave a start to an explosive growth of research in the area ofnanoscience, the historic development and the current status of which can be found in thenumerous reviews [3–7].During the recent time, the properties of charged fullerenes have been actively experi-mentally and theoretically investigated [8–28]. A considerable number of works are devotedto the study of their stability (lifetime), mechanisms for their charging and decay [29].The present paper is devoted to the discussion of the structure of the electronic spectrumof the charged fullerenes. Simple models are used to show the existence of the volume-localized discrete quantum levels for the usual fullerene and for the onion-like structures[30]. Here we confine our attention to the case of the positively charged fullerenes.Basic notations are as follows: m e and e are electron’s mass and the absolute value ofelectron charge, λ e = ~ m e c is the electron Compton length, ε is the electric constant ofvacuum; a = πε ~ m e e is the Bohr radius, α = e πε ~ c is the fine structure constant. II. DISCRETE VOLUME-LOCALIZED LEVELS: QUALITATIVE PRELIMINAR-IES
As a preliminary step, let us formulate the corresponding quantum-mechanical spectralproblem. With an account of the spherical symmetry of a fullerene, for the wave function weuse the standard ansatz ψ ( r, ϑ, ϕ ) = R ( r ) Y lm ( ϑ, ϕ ), with the spherical harmonics Y lm , andrecast the spherically symmetric Schr¨odinger equation [31] into a second order differentialequation d χdr − l ( l + 1) r χ + 2 m e ~ ( E − U ( r )) χ = 0 (1)2or the function χ ( r ) = rR ( r ), 0 ≤ r < ∞ , under the boundary conditions χ (0) = 0 , χ ( ∞ ) = 0 . (2)The form of solution is determined by the potential U ( r ).We will discuss the energy levels of an electron by starting from a simple model potential,and then move on to more complicate form of U . In the simplest model, one can describe acharged fullerene by the potential of a sphere with a constant surface charge density: U ( r ) = − Z Φ( r ) , Φ = e πε × R , r ≤ R, r , r > R, (3)where Z = N e is a positive charge, and R = R f is the fullerene radius. Our attention willbe mainly confined to the C fullerene, when R f = 6 . a .A first very approximate estimate of the energy levels can be obtained by replacing (3)with a spherical rectangular well of the depth U = Ze πε R f . Inside such a well (0 ≤ r ≤ R f ), anon-normalized solution of the Schr¨odinger equation (1) is described by the spherical Besselfunction χ = j l ( ξr/R f ) which satisfies the boundary condition at zero χ (0) = 0, whereasoutside the well ( R f < r < ∞ ) a solution that satisfies χ ( ∞ ) = 0 is given by the sphericalHankel function χ = h l ( iηr/R f ). The parameters η and ξ are algebraically related, ξ + η = 2 m e U R f ~ = 2 α Ze R f λ e , (4)and they determine discrete energy levels via E = − ~ η m e R f = − U + ~ ξ m e R f . (5)The values of parameters η and ξ are fixed by the continuity condition of the wave functionat r = R f . For l = 0, this yields η = − ξ cot ξ . (6)It is worthwhile to notice that the right-hand side of (4) is essentially greater than 1, andalready for a single charged C +160 we have m e U R f ~ = 13 .
25. As a result, for highly chargedfullerenes ( Z ? e ) one can use an approximation of a very deep well U ≫ ~ m e R f , derivingthe energy levels from a condition of the vanishing of the wave function at the boundary: j l ( ξ ) = 0. 3 ABLE I. Electron energy levels for model potential (7). [Notation: n – level number, l – angularquantum number, i – radial quantum number]. n l i E (au) n l i E (au)1 0 1 − . − . − . − . − . − . − . − . − . − . − . − . The characteristic feature of the corresponding wave functions is that they obviouslydescribe the volume-localized states which are basically confined to the central part of thepotential, i.e., to the inner region of the fullerene r ≤ R f . The number of such statesincreases for the potential well becoming deeper, which happens when the charge Z of thefullerene grows.An interesting issue is actually how high is the value of an electric charge that a fullerenecan carry? In practice, one can multiply ionize C with the help of the highly chargedions, fast electrons, or photons [7]. Experimentally, charged fullerenes in the range of Z =0 , · · · , e , [13], and even up to Z = 10 e , [11], were produced in collisions of a beam of C with a beam of highly ionized Xe atoms; such charged fullerenes are stable on a time scaleof several µ s. The highest value Z = 12 e was observed for a charged fullerene (with thelifetime of of order of a µ s) ionized by intense short infrared laser pulses [12]. The theoreticanalysis of the Coulomb stability of highly charged fullerenes [14, 15] predicted the limitingvalue Z = 18 e on the basis of a conducting sphere model, whereas the existence of Z = 14 e was established theoretically [16–19] by means of the density functional theory. However,the predicted lifetime falls drastically –by ten orders– when Z increases from 11 to 14. III. ANALYTIC POTENTIAL FOR A CHARGED FULLERENE
The model above provides a rather simplified description in the sense that it does nottake into account the actual physical structure of a fullerene. A more realistic potential U ( r )can be constructed in the framework of the jellium model [32–37] as a sum of the positive4 ABLE II. Electron energy levels for charged fullerene potential (9) with Z = 1 e . [Notation: n –level number, l – angular quantum number, i – radial quantum number]. n l i E (au) n l i E (au)1 0 1 − . − . − . − . − . − . − . − . − . − . − . − . contribution of the carbon atom’s nuclei located on the spherical surface of the fullereneradius R f and the negative contribution of the electron clouds. The resulting potential isattractive and it has a cusp-shape form with the clear localization in the thin spherical shell.For C , the corresponding Lorentz-bubble potential reads U ( r ) = − ~ m e V ( r − R ) + d , (7)where the parameter V determines the depth, d the width, and R the position. In the self-consistent spherical jellium model based on the Kohn-Sham equations, these parameters arefixed [37] to the values V = 0 . , R = 6 . a , d = 0 . a . (8)In order to test the computational methods used in this paper, we found the energylevels E for different values of the angular quantum number l by integrating the Schr¨odingerequation (1) numerically. The results, see Table I, reproduce the findings of [37]. In contrastto the volume-localized feature of the wave functions for the model (3), the states for thepotential (7) mostly have a typical surface-localized behavior. At the center of a fullerene,the value U (0) = − .
016 au = − .
44 eV is only slightly below zero, and hence only fewdiscrete levels with the negative energy higher than that value correspond to the volume-localized states.Coming to the case of a charged fullerene, let us now modify the Lorentz-bubble potential(7) by including the contribution of the charged spherical surface (3). The generalization of5
IG. 1. Spectrum for charged fullerene potential (9) with Z = 1 e . the potential (7) for a charged fullerene model then reads U ( r ) = − ~ m e V ( r − R ) + d − Z Φ( r ) , (9)where Z is the charge of the fullerene.With such a modification, the central part of the potential deepens, so that U (0) = − .
17 au = − . Z = 1 e , whereas U (0) = − .
77 au = − . Z =5 e , and U (0) = − .
52 au = − . Z = 10 e . As a result, there are two types ofwave functions for the modified potential (9): the lower-energy states are distinctly surface-localized, whereas the higher energy levels correspond to the volume-localized quantumstates.We find the discrete quantum energy levels of an electron in the potential (9) by integrat-ing the Schr¨odinger equation (1) numerically. The corresponding results for different valuesof the fullerene charge Z are presented in Tables II-IV and Figs. 1-5. We limit ourselves tothe first 12 eigenvalues. 6 ABLE III. Energy levels for charged fullerene potential (9) with Z = 5 e . [Notation: n – levelnumber, l – angular quantum number, i – radial quantum number]. n l i E (au) n l i E (au)1 0 1 − . − . − . − . − . − . − . − . − . − . − . − . Z = 5 e . It is known that for the choice of parameters (8), the depth of the potential well (7)is too large and the calculated energy levels are not in agreement with the experimentalvalue of the electron affinity 2.65 eV for C . This issue was discussed in [23, 35–37] for7 IG. 3. Surface-localized wave functions for charged fullerene potential (9) with Z = 5 e .TABLE IV. Electron energy levels for charged fullerene potential (9) with Z = 10 e . [Notation: n – level number, l – angular quantum number, i – radial quantum number]. n l i E (au) n l i E (au)1 0 1 − . − . − . − . − . − . − . − . − . − . − . − . different model potentials in order to bring theory to a better agreement with experiment.By making use of the improved parameter set V = 0 . , R = 6 . a , d = 0 . a , onefinds an essentially shallower potential with the value at the center U (0) = − . IG. 4. Volume-localized wave functions for charged fullerene potential (9) with Z = 5 e . correctly reproduces the 2.65 eV detachment energy for the neutral C molecule [37]. Forsuch an improved parameter choice, the central part of the potential (9) for charged fullereneis shifted to U (0) = − . − . Z = 1 e , to U (0) = − .
75 au = − . Z = 5 e , and to U (0) = − . − . Z = 10 e . Notice that with an increase of theelectric charge Z , the form of the model potential changes very little when the parameters(8) are replaced by an improved parameter set. As a result, the numeric analysis of theelectronic spectrum for charged fullerenes with the improved set of parameters of the modelpotential confirms the existence of the volume-localized states, predicting slightly shiftedtheoretic values of the corresponding energy levels.The spherically symmetric analytical model potential (9) can be compared with the resultderived from DFT computations. Fig. 6 shows potential’s behavior along the radial directionthrough carbon’s atom (“atom” curve) and through the center between neighboring atoms(“middle” curve), respectively; averaging over the angles yields a better agreement. Fur-thermore, one can directly confirm the existence of the volume-localized states by making9 IG. 5. Spectrum for charged fullerene potential (9) with Z = 10 e . use of the Quantum Espresso package; the results of the corresponding DFT computationsare presented in Fig. 7, and it is satisfactory to see the consistency with the results in Fig. 4obtained in the framework of the model potentials approach. IV. ONION-LIKE FULLERENE
Onion (or onion-like) structures represent a highly interesting class of carbon systemswhich are obtained when fullerenes are concentrically enclosed one into another to forma double-, triple-, or in general a multi-layered object. Each (quasi-)spherical layer is afullerene ( C , C , C , C , · · · ), with the separation between shells equal to 0 .
335 nmwhich is slightly larger than the distance between the planar layers in graphite crystals. Theouter diameter of a typical 5-15 layered onion ranges between 4 to 10 nm, with the radiusof the innermost layer equal to R f , however smaller and much larger onion-like structuresare also observed. Following the first observation [38] of spherical multi-layered structures,10 IG. 6. Comparison of the analytical potential of the charged fullerene (9) for Z = 5 e with theDFT result: along the radial direction through carbon’s atom (“atom” curve) and through thecenter between neighboring atoms (“middle” curve). the concept of “carbon onion” was coined in 1992 when the formation of the onion-likespherical particles was demonstrated by heating of nanotubes with an electron beam [39–41]. Since then the physical and chemical characteristics of carbon onions was analyzedin numerous theoretical and experimental studies [42–45]; see the reviews [46–54] for thefurther information on the production, geometrical, physical and chemical properties, andapplications of onion-like carbon structures.In the context of the current investigation of the energy spectrum of charged carboncomplexes, the onion-like structures are qualitatively different from the usual fullerenes inthe sense that, in contrast to the latter case when the electric charge is smeared only overthe surface of fullerene’s hollow sphere, in the former case the electric charge is distributedin the volume of an onion sphere on its many internal layers.11 IG. 7. Wave functions obtained with the help of the DFT computations for a charged C fullerenewith Z = 5 e : both the surface-localized (solid and broken curves) and the volume-localized (twodot-style curves) states are found. For convenience of comparison, the wave functions are shownby normalizing their maximal values to unit. Accordingly, in a simplest model for the study of discrete volume levels of electronsin multi-layer onion-like charged fullerenes (somewhat similarly to the simplest model (5)and (6) of a rectangular spherical well for a fullerene), one can look for analytic estimatesby assuming a homogeneous density when the charges of consecutive layers of the onionstructure are proportional to the cube of the layer radius. In this case, the potential energyof an electron in an electrostatic field is as follows: U ( r ) = − Ze πε × R (cid:18) − r R (cid:19) , r ≤ R, r , r > R. (10)12ere Z is the total positive charge of an onion structure, and R = R on is its outer radius. Asa first step to understand the spectrum structure, we approximate the potential by extendingthe piece inside the sphere r ≤ R to all values of the radius: U ( r ) = − U + 12 m e ω r , (11)where we denoted U = 32 Ze πε R on , ω = s Ze πε m e R . (12)The maximal specific charge Z/N tot of onion-like structures (where N tot is the total num-ber of atoms) before their decay would be smaller than that for C , but the absolute valueof the charge can be much larger. Accordingly, the depth of the potential well U of anelectron in the field of a positively charged onion structure then can reach the values oforder of 100 eV, thereby increasing the significance of the volume-localized quantum states.For the approximate potential (10), one can evaluate the energy spectrum analytically bymaking use of the well-known solution of the Schr¨odinger equation for the spherical oscillator[31]. For energy levels we find E = E + ~ ω (2 i + l − , (13) E = 32 r Ze πε R on s ~ m e R − r Ze πε R on ! , (14)whereas the wave functions of the corresponding stationary states are ψ nlm = const r l exp (cid:18) − λr (cid:19) Y lm ( θ, ϕ ) ×× F (cid:16) − i, l + 32 , λr (cid:17) , (15)where F is the degenerate hypergeometric function, λ = m e ω ~ = s Ze m e πε ~ R , (16)the radial quantum number i = 1 , , · · · , the angular quantum number l = 0 , , , · · · , and m = 0 , ± , · · · , ± l .As a particular application, let us consider a model of a 5-layer charged onion fullerenewith Z = 55 e and the size R on = 5 R f . The total charge arises from the assumption of ahomogeneous distribution of the electric charge on the inner layers proportionally to the13 ABLE V. Electron energy levels (13) for an onion-like fullerene potential (10) with Z = 55 and R on = 33 . a . [Notation: n – level number, l – angular quantum number, i – radial quantumnumber]. n l i E (au) n l i E (au)1 0 1 − . − . − . − . − . − . − . − . − . − . − . − . third power radius of the layer. The corresponding energy levels (13) for such onion modelare presented in Table V. The well-known degeneracy properties of an oscillator spectrumare manifest. It is instructive to compare these analytic estimates with the results of anumeric integration of the Schr¨odinger equation for a five-layered onion structure which aregiven in Fig. 8-11 and in Tables VI-VII for different values of the positive charge of thelayers.Theoretical analysis of the structure of discrete levels for electrons in the field of positivelycharged single- and multi-layer fullerenes suggests the existence of a system of quantumtransitions with emission of photons within a wide range of energies that depend on themagnitude of the charge and the lifetime of the levels in the range from 21 ns to 21 fs.In order to make estimates, let us recall that the characteristic lifetime for the spontaneousdipole transition is determined by the well-known expression for the photon emission rate P fi = ω πε ~ c | d fi | , (17)where ω is the emission frequency and d fi is the matrix element of the dipole transition froman initial ( i ) to a final ( f ) state. One can roughly estimate the order of magnitude of d fi fortransitions between the volume-localized discrete levels as eR f for charged fullerenes and as eR on for onion-like structures. These processes may occur during pulse charging of fullerenesor in the process of irradiation of already charged fullerenes by a flow of electrons and ions.Experimental confirmation of the existence of the volume-localized discrete levels is of greatinterest for the experimental research and practical problems including a development of the14ew sources of coherent radiation in a wide range of wavelengths.Simple numeric estimates are straightforward. Substituting the approximation | d fi | ∼ eR f , we recast (17) into P fi = 4(6 . α cλ e (cid:18) ∆ Em e c (cid:19) . (18)Here we assumed R f = 6 . a for the radius of the fullerene. As a result, we find for differentenergy transitions: P fi (∆ E = 1 eV) = 4 . × s − , P fi (∆ E = 10 eV) = 4 . × s − , P fi (∆ E = 100 eV) = 4 . × s − . More accurate estimates for transition rates in thesystem of discrete volume-localized levels of an electron in the field of a positively chargedfullerene can be derived from the evaluation of the value
IG. 8. Model of charged onion-like fullerene: contributions of layers and the resulting potential(19) with regular charge distribution (21).
Obviously, such a semi-empirical estimate is very approximate and needs to be furtherrefined on the basis of the microscopic calculations or the experimental measurements.
V. NUMERIC ANALYSIS OF ONION-LIKE FULLERENE MODEL
The analytic model which we considered in the previous section, treats a charged onionfullerene as a homogeneously charged solid sphere. In other words, in such an approach theinternal structure is not taken into account. In order to improve the picture, we replacethe model (10) with a potential that explicitly describes an onion fullerene as a multi-layerstructure with the concentric charged spheres enclosed one into another. For concreteness,let us consider a simple model of an onion-like fullerene with five layers. We construct thecorresponding potential in the form of a superposition of the five Coulomb-like contributions16
IG. 9. Wave functions for charged onion-like fullerene model potential (19) with regular chargedistribution (21). (3) coming from each of the charged spherical shell U ( r ) = X k =1 U k ( r ) , (19)where U k ( r ) = − N k e πε × kR on , r ≤ k R on , r , r > k R on . (20)Here Z k = N k e is the charge on a layer number k = 1 , , ...,
5. The choice of the values ofthe electric charge on shells is a highly nontrivial issue which basically should be determinedby the physical procedure used to charge the layers. Below, we analyze the two options.17
ABLE VI. Electron energy levels for charged onion-like fullerene model potential (19) with regularcharge distribution (21). [Notation: n – level number, l – angular quantum number, i – radialquantum number]. n l i E (au) n l i E (au)1 0 1 − . − . − . − . − . − . − . − . − . − . − . − . . Regular charge distribution Suppose, one can arrange a charged onion structure in such a way that it closely re-produces a model of a uniformly charged solid sphere. We will call this a regular chargedistribution. In this case, the values of the charge on the shells are proportional to thesquare of layer’s radius: N = 1 , N = 4 , N = 9 , N = 16 , N = 25 . (21)The resulting potential U ( r ) and its constituents (15) are shown in Fig. 8. The spectrumis presented on Fig. 9 and Table VI. The numeric computations should be compared to theestimates obtained on the basis of analytic model which we studied in the previous section.The data in Table V and Table VI are in a very good agreement.Although the total charge of such onion fullerene Z = 55 e is large, the resulting specificcharge (per total number of atoms) is actually smaller than that of the maximally charged C . Moreover, even with such a large total charge, this onion fullerene model satisfiesthe simple criterion of stability since the electric field on its boundary does not exceed thecritical value for carbon. B. Irregular charge distribution
Obviously, it is a highly nontrivial technical problem to produce a regularly charged onionfullerene. It is more likely that in practice, in the course of an actual experimental laboratorysetup, an onion structure can be only charged in an irregular way. In that case, one shouldexpect to find that the charges on each shell would have more or less arbitrary values. Inorder to analyze the corresponding spectrum numerically, let us choose N = 3 , N = 12 , N = 27 , N = 48 , N = 75 . (22)The resulting potential U ( r ) and its constituents (15) are shown in Fig. 10. The spectrumis presented on Fig. 11, and the first energy eigenvalues for the irregular choice are listed inTable VII.For such an irregularly charged onion fullerene, the total charge is Z = 165 e , and thevalue of the electric field strength on the outer boundary E = 7 . × V/m is well belowthe stability limit. As a certain self-consistency test, one can verify that, keeping the same19
IG. 11. Wave functions for charged onion-like fullerene model potential (19) with irregular chargedistribution (22). total charge, the mostly positive area is at the surface (the outmost shell) of an onion-likefullerene which results in more shallow well with a smaller depth of the potential at thecentre.
VI. DISCUSSION AND CONCLUSION
In the framework of a simple physical model, we demonstrate the existence of a system ofdiscrete short-lifetime quantum levels for electrons in the potential well of the self-consistentCoulomb field of charged fullerenes and onion-like structures. For electrons, in the case ofpositively charged fullerenes and onion-like structures, the energy of the volume-localizedlevels ranges from 1 eV to 100 eV.We use an idealized spherically symmetric model potentials in our study. As one knows,geometrically a C fullerene is a truncated icosahedron with carbon atoms located in its20 ABLE VII. Electron energy levels for charged onion-like fullerene model potential (19) withirregular charge distribution (22). [Notation: n – level number, l – angular quantum number, i –radial quantum number]. n l i E (au) n l i E (au)1 0 1 − . − . − . − . − . − . − . − . − . − . − . − .
60 equivalent vertices. Such a geometrical shape is very close to a sphere and it is possibleto use the spherical harmonics to describe the electronic structure of a neutral C in termsof free electrons on a sphere of radius R . The corresponding symmetries of the icosahedrongroup are Γ = A g + T g + 2 T u + T g + 2 T u + 2 G g + 2 G u + 3 H g + 2 H u , and with the helpof a simple H¨uckel scheme one can calculate the eigenvalues and degeneracies of this system[57]. (In fact, the first such calculation by Bochvar and Gal’pern [58] predicted the existenceof stable carbon structures with the icosahedron geometry a decade before the experimentaldiscovery of fullerenes). The H¨uckel scheme calculation is perfectly consistent with thedensity functional theory computation [59] that explicitly demonstrates the correspondenceof A g , T u , H g , T u + G u to l = 0 , , , s, p, d, f ) states, respectively; see also [60] forthe higher quantum angular numbers up to l = 9.It is worthwhile to mention that the electronic structure of fullerenes (and, in general,interaction of electrons and photons with fullerenes and fullerene-like systems) is widelystudied by means of the simple model potentials. In particular, along with the square-welltype potentials one considers the Dirac bubble and Gaussian-type potentials, as well asmodels the fullerene cage by a spherical jellium shell [61–68]. The previous discussions weremostly focused on the neutral systems, whereas here we have analysed the case of positivelycharged fullerenes and onion-like structures.The results obtained provide a consistent qualitative picture both for the chargedfullerenes and for the onion-like structures. In order to refine our very approximate findings,21ne certainly needs a further investigation on the basis of the microscopic calculations, aswell as the experimental measurements.An experimental confirmation of the existence of the volume-localized discrete levelswould be of great interest for the experimental research and practical problems including adevelopment of the new sources of coherent radiation in a wide range of wavelengths. ACKNOWLEDGMENTS
We thank P. S. Kondratenko and participants of the seminar of the Theoretical PhysicsLaboratory, Institute for Nuclear Safety (IBRAE) for the fruitful and stimulating discussions.We are grateful to A. V. Osadchy for the help with making the DFT computations andproducing the corresponding Figures 6 and 7. [1] W. H. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl, and R. E. Smalley, C : Buckminster-fullerene, Nature , 162 (1985).[2] W. Kr¨atschmer, L. D. Lamb, K. Fostiropoulos, and D. R. Huffman, Solid C : a New Formof Carbon, Nature , 354 (1990).[3] A. V. Eletskii and B. M. Smirnov, Fullerenes and carbon structures, Phys. Usp. , 935 (1995).[4] A. Hirsch and M. Brettreich, Fullerenes: Chemistry and Reactions (Wiley-VCH Verlag: Wein-heim, 2005).[5] K. D. Sattler, ed.,
Handbook of nanophysics. Clusters and fullerenes (CRC Press: Boca Raton,2011).[6] Yu. Gogotsi, ed.,
Nanomaterials Handbook , 2nd ed. (CRC Press: Boca Raton, 2017).[7] E. E. B. Campbell,
Fullerene Collision Reactions (Kluwer Academic Publ.: N.Y., 2004).[8] P. P. Radi, M.-T. Hsu, M. E. Rincon, P. R. Kemper, and M. T. Bowers, On the structure,reactivity and relative stability of the large carbon cluster ions C +62 , C +60 and C +58 , Chem. Phys.Lett. , 223 (1990).[9] N. Troullier and J. L. Martins, Structural and electronic properties of C , Phys. Rev. A ,1754 (1992).[10] K. Yabana and G. F. Bertsch, Electronic structure of C in a spherical basis, Physica Scr. , 633 (1993).[11] A. Brenac, F. Chandezon, H. Lebius, A. Pesnelle, S. Tomita, and B. A. Huber, Multifrag-mentation of Highly Charged C60 Ions: Charge States and Fragment Energies, Physica Scr. T80B , 195 (1999).[12] V. R. Bhardwaj, P. B. Corkum, and D. M. Rayner, Internal Laser-Induced Dipole Force atWork in C Molecule, Phys. Rev. Lett. , 203004 (2003).[13] J. Jensen, H. Zettergren, H. T. Schmidt, H. Cederquist, S. Tomita, S. B. Nielsen, J. Rangama,P. Hvelplund, B. Manil, and B. A. Huber, Ionization of C and C molecules by slow highlycharged ions: A comparison, Phys. Rev. A , 053203 (2004).[14] H. Zettergren, H. T. Schmidt, H. Cederquist, J. Jensen, S. Tomita, P. Hvelplund, H. Lebius,and B. A. Huber, Static over-the-barrier model for electron transfer between metallic sphericalobjects, Phys. Rev. A , 032710 (2002).[15] H. Zettergren, J. Jensen, H. T. Schmidt, and H. Cederquist, Electrostatic model calculationsof fission barriers for fullerene ions, Eur. Phys. J. D , 63 (2004).[16] S. D´ıaz-Tendero, M. Alcam´ı, and F. Mart´ın, Coulomb Stability Limit of Highly Charged C q +60 Fullerenes, Phys. Rev. Lett. , 013401 (2005).[17] S. D´ıaz-Tendero, M. Alcam´ı, and F. Mart´ın, Structure and electronic properties of highlycharged C and C fullerenes, J. Chem. Phys. , 184306 (2005).[18] R. Sahnoun, K. Nakai, Y. Sato, H. Kono, Y. Fujimura, and M. Tanaka, Theoretical investiga-tion of the stability of highly charged C molecules produced with intense near-infrared laserpulses, J. Chem. Phys. , 184306 (2006).[19] R. Sahnoun, K. Nakai, Y. Sato, H. Kono, Y. Fujimura, and M. Tanaka, Stability limit ofhighly charged C cations produced with an intense long-wavelength laser pulse: Calculationof electronic structures by DFT and wavepacket simulation, Chem. Phys. Lett. , 167(2006).[20] G. S. Iroshnikov, Calculation of the cross section for charge transfer in fullerene-fullerenecollisions, JETP , 707 (2006).[21] G. S. Iroshnikov, Calculation of the cross section for charge transfer in C +70 + C fullerene-fullerene collisions, JETP , 706 (2007).[22] H. Kono, K. Nakai, and N. Niitsu, Ab initio molecular dynamics of highly charged fullerenecations in intense near-infrared laser fields, Proc. of SPIE , 672606 (2007).
23] M. E. Madjet, H. S. Chakraborty, J. M. Rost, and S. T. Manson, Photoionization of C : amodel study, J. Phys. B: At. Mol. Opt. Phys. , 105101 (2008).[24] E. M. Nascimento, F. V. Prudente, M. N. Guimar˜aes, and A. M. Maniero, A study of theelectron structure of endohedrally confined atoms using a model potential, J. Phys. B: At.Mol. Opt. Phys. , 015003 (2011).[25] R. G. Polozkov, V. K. Ivanov, and A. V. Solov’yov, Photoionization of the fullerene ion C +60 ,J. Phys. B: At. Mol. Opt. Phys. , 4341 (2005).[26] R. G. Polozkov, V. K. Ivanov, A. V. Verkhovtsev, A. V. Korol, and A. V. Solov’yov, Newapplications of the jellium model for the study of atomic clusters, J. Phys.: Conf. Ser. ,012009 (2013).[27] M. L¨uders, A. Bordoni, N. Manini, A. Dal Corso, M. Fabrizio, and E. Tosatti, Coulombcouplings in positively charged fullerene, Phil. Mag. B , 1611 (2009).[28] D. Liu, N. Iwahara, and L. F. Chibotaru, Dynamical Jahn-Teller effect of fullerene anions,Phys. Rev. B , 115412 (2018).[29] Y. Wang, M. Alcam´ı, and F. Mart´ın, Stability of charged fullerenes, in: Handbook ofnanophysics. Clusters and fullerenes , Ed. K. D. Sattler (CRC Press: Boca Raton, 2011) 25.[30] R.V. Arutyunyan, Theoretical investigation of electronic properties of highly chargedfullerenes, Preprint No. IBRAE-2018-08 (Inst. Nucl. Safety IBRAE, Moscow, 2018) 16 p.[31] L. D. Landau and E. M. Lifshitz,
Quantum mechanics. Non-relativistic theory , 3rd ed. (Perg-amon Press: Oxford, 1977).[32] A. Rubio, J. A. Alonso, J. M. L´opez, and M. J. Stott, Collective electronic excitations inmetal coated C , Phys. Rev. B , 17397 (1994).[33] V. K. Ivanov, G. Yu. Kashenock, R. G. Polozkov, and A. V. Solov’yov, Photoionization crosssections of the fullerenes C and C calculated in a simple spherical model, J. Phys. B: At.Mol. Opt. Phys. , L669 (2001).[34] V. K. Ivanov, G. Yu. Kashenock, R. G. Polozkov, and A. V. Solov’yov, Method for calculatingphotoionization cross sections of fullerenes in the local density and random phase approxima-tions, JETP , 658 (2003).[35] A. K. Belyaev, A. S. Tiukanov, A. I. Toropkin, V. K. Ivanov, R. G. Polozkov, and A. V.Solov’yov, Photoabsorption of the fullerene C and its positive ions, Physica Scr. , 048121(2009).
36] A. V. Verkhovtsev, R. G. Polozkov, V. K. Ivanov, A. V. Korol, and A. V. Solov’yov,Hybridization-related correction to the jellium model for fullerenes, J. Phys. B: At. Mol.Opt. Phys. , 215101 (2012).[37] A. S. Baltenkov, S. T. Manson, and A. Z. Msezane, Jellium model potentials for the C molecule and the photoionization of endohedral atoms, A @ C , J. Phys. B: At. Mol. Opt.Phys. , 185103 (2015).[38] S. Iijima, Direct observation of the tetrahedral bonding in graphitized carbon black by HREM,J. Crystal Growth , 675 (1980).[39] D. Ugarte, Curling and closure of graphitic networks under electron-beam irradiation, Nature , 707 (1992).[40] D. Ugarte, How to fill or empty a graphitic onion, Chem. Phys. Lett. , 99 (1993).[41] D. Ugarte, Onion-like graphitic particles, Carbon , 989 (1995).[42] M. S. Zwanger, F. Banhart, and A. Seeger, Formation and decay of spherical concentric-shellcarbon clusters, J. Crystal Growth , 445 (1996).[43] S. Tomita, M. Fujii, S. Hayashi, and K. Yamamoto, Electron energy-loss spectroscopy ofcarbon onions, Chem. Phys. Lett. (1999) 225.[44] A. V. Okotrub, L. G. Bulusheva, A. I. Romanenko, V. L. Kuznetsov, Yu. V. Butenko, C. Dong,Y. Ni, and M. I. Heggie, Probing the electronic state of onion-like carbon, AIP ConferenceProceedings , 349 (2001).[45] V. D. Blank, V. D. Churkin, B. A. Kulnitskiy, I. A. Perezhogin, A. N. Kirichenko, S. V.Erohin, P. B. Sorokin, and M. Yu. Popov, Pressure-Induced Transformation of Graphite andDiamond to Onions, Crystals , 68 (2018).[46] O. A. Shenderova, V. V. Zhirnov, and D. W. Brenner, Carbon Nanostructures, Critical Re-views in Solid State and Materials Sciences , 227 (2002).[47] B.-S. Xu, Prospects and research progress in nano onion-like fullerenes, New Carbon Materials , 289 (2008).[48] Y. V. Butenko, L. ˇSiller, and M. R. C. Hunt, Carbon onions, in: Handbook of nanophysics.Clusters and fullerenes , Ed. K. D. Sattler (CRC Press: Boca Raton, 2011) 34.[49] C. Chang, B. Patzer, and D. S¨ulzle, Onion-Like Inorganic Fullerenes, in:
Handbook ofnanophysics. Clusters and fullerenes , Ed. K. D. Sattler (CRC Press: Boca Raton, 2011) 51.[50] C. He and N. Zhao, Production of Carbon Onions, in:
Handbook of nanophysics. Clusters and ullerenes , Ed. K. D. Sattler (CRC Press: Boca Raton, 2011) 24.[51] J. Bartelmess and S. Giordani, Carbon nano-onions (multi-layer fullerenes): chemistry andapplications, Beilstein J. Nanotechnol. , 1980 (2014).[52] V. Georgakilas, J. A. Perman, J. Tucek, and R. Zboril, Broad Family of Carbon Nanoal-lotropes: Classification, Chemistry, and Applications of Fullerenes, Carbon Dots, Nanotubes,Graphene, Nanodiamonds, and Combined Superstructures, Chem. Rev. , 4744 (2015).[53] O. Mykhailiv, H. Zubyk, and M. E. Plonska-Brzezinska, Carbon nano-onions: Unique car-bon nanostructures with fascinating properties and their potential applications, InorganicaChimica Acta , 49 (2017).[54] Y. V. Butenko, L. ˇSiller, and M. R. C. Hunt, Carbon onions, in: Nanomaterials Handbook ,Ed. Yu. Gogotsi (CRC Press: Boca Raton, 2017) 392.[55] E. W. M¨uller, Field ionization and field ion microscopy, in: “Advances in Electronics andElectron Physics” , Ed. L. Marton (Academic Press: New York, 1960) vol. 13, p. 83.[56] R. G. Forbes, Field electron and ion emission from charged surfaces: a strategic historicalreview of theoretical concepts, Ultramicroscopy , 1 (2003).[57] P. W. Fowler and J. Woolrich, π -Systems in three dimensions, Chem. Phys. Lett. , 78(1986).[58] D. A. Bochvar and E. G. Gal’pern, On hypothetic systems: carbon-dodehaedron s -icosahedronand carbon- s -icosahedron, Doklady Akad. Nauk USSR , 610 (1973).[59] S. Saito, A. Oshiyama, and Y. Miyamoto, Electronic structures of fullerenes and fullerides,in: “Computational Approaches in Condensed-Matter Physics” , edited by S. Miyashita, M.Imada, and H. Takayama (Springer: Berlin, 1992) 22.[60] A. V. Verkhovtsev, A. V. Korol, and A. V. Solov’yov, Quantum and classical features of thephotoionization spectrum of C , Phys. Rev. A , 043201 (2013).[61] W. Jask´olski, Confined many-electron systems, Phys. Rept. , 1 (1996).[62] L. L. Lohr and M. Blinder, Electron photodetachment from a Dirac bubble potential. A modelfor the fullerene negative ion C , Chem. Phys. Lett. , 100 (1992).[63] M. J. Puska and R. M. Nieminen, Photoabsorption of atoms inside C , Phys. Rev. A ,1181 (1993).[64] M. Ya. Amusia, A. S. Baltenkov, and B. G. Krakov, Photodetachment of negative C − ions,Phys. Lett. A , 99 (1998).
65] J. P. Connerade, V. K. Dolmatov, P. A. Lakshmi, and S. T. Manson, Electron structure ofendohedrally confined atoms: atomic hydrogen in an attractive shell, J. Phys. B: At. Mol.Opt. Phys. , L239 (1999).[66] J. P. Connerade, V. K. Dolmatov, and S. T. Manson, A unique situation for an endohedralmetallofullerene, J. Phys. B: At. Mol. Opt. Phys. , L395 (1999).[67] G. Schrange-Kashenock, 4 d → f resonance in photoabsorption of cerium ion Ce andendohedral cerium in fullerene complex Ce @ C +82 , J. Phys. B: At. Mol. Opt. Phys. , 185002(2016).[68] Z. Felfli and A. Z. Msezane, Simple method for determining binding energies of fullerenenegative ions, Eur. Phys. J. B , 78 (2018)., 78 (2018).