Fusion mechanism in fullerene-fullerene collisions -- The deciding role of giant oblate-prolate motion
aa r X i v : . [ phy s i c s . a t m - c l u s ] M a r Fusion mechanism in fullerene-fullerene collisions–The deciding role of giant oblate-prolate motion
J. Handt and R. Schmidt ∗ Institut f¨ur Theoretische Physik, Technische Universit¨at Dresden, D-01062 Dresden, Germany (Dated: September 25, 2018)We provide answers to long-lasting questions in the puzzling behavior of fullerene-fullerene fusion:Why are the fusion barriers so exceptionally high and the fusion cross sections so extremely small?An ab initio nonadiabatic quantum molecular dynamics (NA-QMD) analysis of C +C collisionsreveals that the dominant excitation of an exceptionally “giant” oblate-prolate H g (1) mode plays thekey role in answering both questions. From these microscopic calculations, a macroscopic collisionmodel is derived, which reproduces the NA-QMD results. Moreover, it predicts analytically fusionbarriers for different fullerene-fullerene combinations in excellent agreement with experiments. PACS numbers: 36.40.-c
INTRODUCTION
After the pioneering C +C collision experiment ofE.E.B. Campbell et.al. in 1993 [1], cluster-cluster colli-sions became a versatile new field of research (for reviewssee [2–5]) with lasting interest [6–20]. In particular, fu-sion between fullerenes has been studied in great detail,both experimentally and theoretically [1, 21–37]. Fusionis a universal phenomenon in collisions between complexparticles covering many orders in size and energy fromheavy ion collisions in nuclear physics [38] to macroscopicliquid droplets [39, 40] or even colliding galaxies [41]. Itis a great challenge of ongoing interest to reveal universalsimilarities and basic differences of these mechanisms.Usually, the gross features of fusion can be understoodwith macroscopic arguments [10, 38, 40, 42] leading tothe general expression for the fusion cross section σ asfunction of the center mass energy E c.m. ≡ E of σ ( E ) = πR (cid:18) − V B E (cid:19) (1)with R = R + R the sum of the radii of the collid-ing partners and V B the fusion barrier (for a derivationsee e.g. [3, 38, 43]). This formula (known as “critical dis-tance model” in nuclear physics [38] or “absorbing spheremodel” in chemistry [43]) describes quantitatively the ex-perimental fusion cross section for atomic nuclei (with V B >
0) and liquid droplets (with V B = 0) [10, 39, 40]in the low energy range with E & V B [44]. It is expectedto hold also for collisions between metallic clusters [10](even with V B < V B of at least the sp bond breaking energy, or more perti-nently the Stone-Wales transformation energy [33, 34] ofa few eV [45] and, according to (1), a fusion cross sectionof the order of the geometrical one, πR ∼
150 ˚A [29].Experimentally, however, the fusion barriers are about one order of magnitude larger (around 80 eV) [29] andthe cross sections are even two magnitudes smaller (a few˚A ) [31].Up to now, there is no definite explanation for thesefindings, albeit some possible phenomenological reasonshave been discussed [3–5, 31, 32]. In addition, previous(at that time still approximate) microscopic QuantumMolecular Dynamics (QMD) calculations predicted thelarge fusion barriers [28]. From these studies it is alsowell known that only very few mutual initial orientationsof the colliding cages lead to fusion (without identifyingthem). Anyway, fullerenes typically do not fuse if theytouch, even at high impact energies, and the questionremains, why? In this work, we provide a clear answerto this longstanding question. MICROSCOPIC RESULTS
Motivated by our recent findings of the dominatingrole of the A g (1) breathing mode in C -laser interac-tion [46, 47], we have reanalyzed fullerene-fullerene col-lisions with the help of the ab initio nonadiabatic quan-tum molecular (NA-QMD) method [48–51]. For systemsas large as we are investigating here, NA-QMD is nu-merically more efficient than its ab initio QMD approx-imation [49–51]. In extension to previous studies [3, 27–29, 31] we include a normal mode analysis [52] of thevibrational kinetic energy. This method decomposes thetotal kinetic vibrational excitation energy E vib into theindividual contributions of all 174 eigenmodes of C asfunction of time t according to E vib ( t ) = X i =1 m c r i = X ν =1 m c X i =1 ˙ r i b iν ! with the atomic carbon mass m c , the atomic velocities˙ r i (in the molecular center of mass system without ro-tational components) and the normal mode eigenvectors b iν . FIG. 1. (color online) Normal mode analysis of centralC +C collisions at an impact energy of E c.m. = 104 eV.Shown are the vibrational kinetic energies E vib of all 174 in-ternal normal modes as function of time t for (a) typical scat-tering and (b) typical fusion events. Distances of closest ap-proach R ret are indicated by vertical, dashed lines. The dom-inating role of the H g (1) mode (black thick lines) is apparent. In fig. 1, such an analysis, is shown for two cen-tral C +C collisions with the same impact energy of E c.m. = 104 eV but for different initial orientations ofthe clusters, leading in one case to scattering (fig. 1(a))and, in the other, to fusion (fig. 1(b)). In both cases, theextraordinary dominance of the H g (1) mode is obvious.During approach, this mode is very strongly excited toa much higher degree than any other vibrational mode.Its excitation energy of several eV is huge as compared toa single quantum of this mode of ~ ω mode ∼
34 meV [53]and, thus, its amplitude is extremely large, as comparedto a typical displacement of the elementary excitation(“giant” H g (1) mode). Consequently at the distance ofclosest approach R ret , a highly deformed oblate-oblate configuration of the double cluster system is formed (seeabove illustrations in fig. 1). This clearly distinguishablestate accommodates practically the whole impact energyinto deformation (potential) energy, which is quantita-tively shown in fig. 2(a), also for other impact energies.Up to this stage of the collision, there is no appreciabledifference to the other collision systems (nuclei, droplets),where at this “critical distance” the system loses immedi-ately its memory and the energy is dissipated into inter-nal degrees of freedom (DOF) leading to a hot compound.The fundamental difference to the other systems con-sists in the specific properties of the oblate-prolate modein fullerenes and its special role in collisions. First of all,among all vibrational modes, the H g (1) mode in C hasthe largest oscillation period of T = 122 fs [53]. Thisis comparable with a typical collision time and there-fore, once excited, this mode will not lose immediatelyits memory, as clearly seen in figs. 1(a) and 1(b). Sec-ond, the oblate-prolate mode is the only eigenmode whichcan couple directly to the relative motion via its elon-gated prolate-prolate configuration, provided it survivesthe dissipative coupling to all the other internal modes.This is exactly what happens in C +C collisions andresults in the majority of cases in scattering, like a fis-
10 12 14 16 18
R (a.u.) U ( e V ) (a) E ret t (fs) E k i n ( e V ) E d ( ∞) ∆ E (b) FIG. 2. (color online) Typical NA-QMD results for centralC +C collisions at impact energies of E c.m. = 40 eV (red),60 eV (blue), and 104 eV (black), indicated as dotted hori-zontal lines, and obtained by ensemble averaging over 20 dif-ferent initial orientations. (a): The potential (deformation)energy of the relative motion U ( R ) in the entrance channelas function of the distance R . The locations and lengths ofthe vertical arrows denote the distances of closest approach R ret and the total deformation energy E ret ≡ U ( R ret ) storedat R ret , respectively. The dashed line is the harmonic fit U ( R ) = k (cid:0) R − R (cid:1) with k = 0 .
12 a.u. and R = 18 . E kin as func-tion of time t for scattering events. The mean values of thefinal kinetic energies ∆ E ≈
17 eV and of the correspondingenergies dissipated into internal DOF E d ( ∞ ) = E c.m. − ∆ E are indicated by double arrows. sion process with the prolate-prolate configuration at thescission point [42] (note the pronounced excitation of themode in fig. 1(a) at t ∼
165 fs). Only strong coupling tothe internal DOF can prevent this mechanism, allowingfor fusion (note the strongly damped prolate oscillationof the mode in fig. 1(b) at t ∼
165 fs). Thus, the compet-itive coupling of the oblate-prolate mode to the relativemotion and to all the other (bath-like) vibrational DOFdetermines the reaction channel !The strength of the bath coupling is solely determinedby the amount of energy stored in the mode at R ret . Thiscoupling dominates, if the energy exceeds a certain limitwhich generally can happen only at appropriate large im-pact energies. This explains (at least preliminarily andqualitatively) the high fusion barriers. At a fixed impactenergy just above the barrier E c.m. & V B (as in fig. 1),only very rare and specific initial orientations of the clus-ters can lead to high H g (1) excitation energies , namely,those with the principal axes of the H g (1) mode alignedto the collision axis, ensuring maximal energy transfer,which is the case in the example shown in fig. 1(b) (notethat E vib of the mode during approach in the case offusion (fig. 1(b)) is twice as large as compared to scat-tering (fig. 1(a))). This, finally, explains the low fusioncross sections and completes the present, new picture ofthe fusion mechanism.It modifies also the hitherto existing interpretation ofscattering, as a “bouncing off” mechanism [3, 4, 22, 28,31], like in collisions between two soccer balls. Instead,“fission” via the prolate-prolate configuration stronglysuggests, that the final kinetic energy of the fragmentsis largely independent on impact energy. This is nicely FIG. 3. (color online) Phenomenological collision model withtwo DOF, the distance between the centers R ( t ) and the di-ameter D ( t ) of the fullerenes: Two collinear colliding springs(with spring constants k mode and masses m ) interact duringapproach via a third, massless spring (spring constant k andinitial length R c ) located at the collision center. confirmed in the calculations and shown in fig. 2(b).The collision scenario presented in figs. 1 and 2 ischaracteristic for fullerene-fullerene reactions and qual-itatively observed in our NA-QMD calculations also forthe other combinations and finite, small impact param-eters. Despite its microscopic complexity, the mecha-nism is nevertheless simple and can be understood anddescribed by ordinary macroscopic concepts, as will beshown in the following. COLLISION MODEL
The basic idea is to reduce drastically the 360-dimensional scattering problem to a one-dimensional onewith only two, but relevant collective DOF, treated ex-plicitly in a time-dependent fashion: the distance be-tween the centers R ( t ) (relative motion) and the diame-ters of the fullerenes D ( t ) (aligned along their principalaxes of the oblate-prolate mode). Both are coupled viathe contact distance R c = R − D (see fig. 3). The cou-pling to the other internal DOF will be treated implicitlyin the exit channel only. The macroscopic model is de-signed as follows:(i) In the entrance channel, the system consists of twocollinear colliding springs with initial lengths D ( t = 0) = D , spring constants k mode and masses m = M at theends (describing the fullerenes with mass M and their H g (1) modes). Tightly located in between there is athird, massless spring with initial length R c and constant k , describing the repulsive potential U during approach(remember fig. 2(a) and see fig. 3 and fig. 4(a)).(ii) In the exit channel, the massless repulsive spring isreplaced by a “dissipative” potential U d , which describesthe coupling to all other internal vibrational DOF, andhence, controls the reaction channel (see fig. 4(b)).The coupled Newton equations in the entrance channel E ret U(R c ) R cret R c E c.m. R c0 E ret U d (R c ) E d ( ∞) exit channel (R. > 0)entrance channel (R. < 0) R c R cret FIG. 4. (color online) Schematic plot of the potential energyin the entrance channel U ( R c ) (left) and, for the case of scat-tering, in the exit channel U d ( R c ) (right) as function of thecontact distance R c . Dotted blue lines with directional arrowsindicate a typical trajectory (idealized as a straight line in theexit channel) with impact energy E c.m. at the initial distance R c and potential energy E ret at the distance of closest ap-proach R ret c . The asymptotic value of the potential energy inthe exit channel U d ( R c → ∞ ) ≡ E d ( ∞ ) is also indicated. read µ ¨ R = − dUdR c (cid:12)(cid:12)(cid:12)(cid:12) R c = R − D µ mode ¨ D = − dV mode dD + 12 dUdR c (cid:12)(cid:12)(cid:12)(cid:12) R c = R − D (2)with the reduced masses µ = M and the mass con-stant µ mode = M . The harmonic potentials are given by U ( R c ) = k ( R c − R c ) with k = 0 .
12 a.u. (see fig. 2(a))and V mode ( D ) = k mode ( D − D ) with the spring con-stant k mode = µ mode ω = 0 .
51 a.u., obtained fromthe experimental frequency of the H g (1) mode in C ( ω mode = 273 cm − [53]). The equations of motion(EOM) (2) can be solved analytically (see appendix).They describe the collision up to the distance of clos-est approach, i.e., the classical returning point R ret c . Atthis point the potential U ( R c ) for R c > R ret c is replacedby the “dissipative” one U d ( R c ) which, in dependenceon E c.m. , controls the outcome. It is therefore repulsive(leading in any case to scattering) or attractive (leadingusually to fusion, but not necessarily always). Thus, ithas the general form U d ( R c ) = (cid:0) E ret − E d ( ∞ ) (cid:1) f ( R c − R ret c ) + E d ( ∞ ) (3)where the form factor f must fulfill the conditions, f (0) =1 (ensuring the continuity of the potential at R ret c ) and f ( ∞ ) = 0 (making sure that the maximal amount ofdissipated energy cannot exceed E d ( ∞ ), in the case ofscattering). In fact, the concrete radial dependence ofthe potential (3) is not relevant, and thus, we choose asimple exponential form of f ( x ) = exp (cid:0) − x ∆ (cid:1) with ∆ = (cid:12)(cid:12) R ret c − R c (1 − E d ( ∞ ) E ret ) (cid:12)(cid:12) , which guarantees also continuityof the force at R ret c in the case of scattering.With this, all model parameters ( k , k mode , ∆ E ) arefixed and the EOM (2) can be easily solved numerically. R modelNA-QMDE c.m. = 40 eV E c.m. = 60 eV E c.m. = 150 eV E k i n t (fs) E v i b t (fs) t (fs)scattering scattering fusion FIG. 5. Comparison between NA-QMD (dashed lines) andmodel calculations (solid lines) for central C +C collisionsat three impact energies E c.m. : distance between the centers R in a.u. (top), kinetic energy of the relative motion E kin (middle), and vibrational kinetic energy of the H g (1) mode E vib (bottom) in eV as function of time t . The results are shown in fig. 5 and compared with NA-QMD calculations (for movies see ).The model reproduces nearly precisely the microscopiccalculations for both scattering ( E c.m. = 40 ,
60 eV) andfusion ( E c.m. = 150 eV). The ongoing oscillations of somequantities in the exit channel are the natural consequenceof the absence of a damping mechanism for the H g (1)mode in the spring model. The most impressive result,however, concerns the fusion barrier predicted by themodel of V B = 85 eV, which is in excellent agreementwith former (extremely expensive) QMD calculations [28]of V B = 80 eV. ANALYTICAL SOLUTION FOR THE FUSIONBARRIER
The simplicity of the EOM (2) and the transparencyof the ansatz (3) encouraged us to further elaborate themodel and to demonstrate its predictive power. As men-tioned already, the EOM (2) can be solved analytically,and, from this solution one can derive an approximate ex-pression for the energy stored during approach E ret (seeFig 4(a)), resulting in a linear dependence on the impactenergy of E ret = α ( κ ) E c.m. with κ the ratio of the springconstants κ = k mode k = M ω k (4)and the universal function α ( κ ) of α ( κ ) = 14( κ + 1) " κ − − √ κ + 1 p κ + 1 + √ κ + 1 × sin ( π κ + 1 + √ κ + 1 √ κ ) − κ − √ κ + 1 p κ + 1 − √ κ + 1 (5) (see appendix). The difference ( E ret − E d ( ∞ )) determinesthe (positive or negative) slope of the potential U d (3) atthe returning point R ret c . Taking an idealized trajectoryas shown in fig. 4 (i.e., neglecting the acceleration of themode near the barrier), the fusion condition simply reads E ret ! = E d ( ∞ ) at E c.m. = V B . With E d ( ∞ ) = E c.m. − ∆ E (see fig. 2(b)), the fusion barrier becomes V B = ∆ E − α ( κ ) . (6)For C +C collisions, this approximate expression gives V B = 82 eV, which is very close to the exact modelvalue ( V B = 85 eV) obtained in the upper dynami-cal calculations. To obtain a first insight about thequalitative trends for the other combinations (C +C ,C +C ), we ignore subtleties and use the same k and∆ E values as obtained already for C +C by NA-QMD fine tuning. The internal spring constant for C ,however, is carefully chosen and fixed again by exper-iment. From the (partly) non-degenerate five H g (1)modes in C ( E ′ , E ′′ , A ′ ) an experimental mean value of ω mode (C ) = 261 cm − has been reported [54] giving thespring constant k mode (C ) = 0 .
55 a.u. For the (slightly)asymmetric C +C collision a reasonable mean valueof k mode (C ) and k mode (C ) of k mode (C / C ) =0 .
53 a.u. is used.With these parameters, the predicted fusion barriersfrom (6) are compared with high precision QMD val-ues [28], in table I (first two rows). The analytical modelreproduces the right trend and delivers absolute valueswithin 10% accuracy. Obviously, the H g (1) frequencies ω mode and fullerene masses M , eq. (4) determine the fu-sion barriers in fullerene-fullerene collisions.This is strongly supported by comparing the predic-tions with experimental data. In this case, the finitetemperature ( T ≈ ω mode . Using an arbitrarycommon scaling factor for k mode of 0 .
85, the predictedfusion barriers by eq. (6) are in beautiful agreement withthe experiment (fourth and fifth row in table I).
CONCLUSION
To summarize, ab initio
NA-QMD studies have finallycleared up the fusion mechanism in fullerene-fullerenecollisions. The excitation of a “giant” H g (1) mode ex-plains both large fusion barriers and small fusion crosssections. This microscopic picture is non-ambiguouslyconfirmed by a macroscopic spring model which depicts TABLE I. Fusion barriers V B in eV for various fullerene-fullerene combinations as predicted by our former QMD cal-culations [28] (first and third rows) and the present analyticalmodel (second and fourth rows). The experimental values [29]are presented in the last row. T = 0 K T = 2000 KQMD model QMD model exp.C +C
80 82 60 65 60 ± +C
94 87 70 69 70 ± +C
104 93 75 73 76 ± clearly the physics, reproduces the NA-QMD results andthe experimental fusion barriers quantitatively .We note finally, that a “giant” vibrational excitationof the A g (1) breathing mode in C has been found re-cently in a time-resolved laser experiment [46]. The gen-eral investigation of large amplitude motion in fullerenes,including laser-induced fission [47], could become an in-teresting new field of research. APPENDIX: ANALYTICAL SOLUTION IN THEENTRANCE CHANNEL
Inserting the harmonic potentials U ( R c ) and V ( D ),the EOM (2) can be written as µ ¨ R = − k ( R c − R c ) (cid:12)(cid:12) R c = R − D µ mode ¨ D = − k mode ( D − D ) + k (cid:0) R c − R c (cid:1)(cid:12)(cid:12) R c = R − D (A.1)which can be solved by making an exponential ansatz e ı Ω t . Doing so, the fundamental eigenfrequencies Ω / ofthe system readΩ / = ω q κ + 1 ± p κ + 1 | {z } f / ( κ ) (A.2)with the frequency ω = q kµ and the force constant ratio κ as defined in eq. (4).With the initial conditions R (0) ≡ R = D + R c , ˙ R (0) = v c.m. ,D (0) = D , ˙ D (0) = 0the solution of (A.1) is given by R ( t ) = v c.m. ω X i =1 , a i sin (Ω i t ) + R , (A.3) D ( t ) = v c.m. ω X i =1 , b i sin (Ω i t ) + D (A.4)with the amplitudes a / = f / ( κ ) (1 ∓ κ √ κ +1 ) and b / = ∓ f / ( κ ) 1 √ κ +1 . With the analytical solution (A.3), (A.4) the potentialenergy at the returning point E ret can be calculated. E ret is given by E ret ≡ U ( R ret c ) = k (cid:0) R ret c − R c (cid:1) (A.5)with R ret c ≡ R c ( t ret ) = R ( t ret ) − D ( t ret ).The returning time t ret defined by ˙ R ( t ret ) = 0 is approxi-mated by t ret ≈ π . The approximation is justified sincethe second term of the sum in eq. (A.3) dominates for theparameter range of κ used here ( a a ≪ E ret = α ( κ ) E c.m. with the impact energy E c.m. = µ v and the coeffi-cient α ( κ ) as defined before in eq. (5).We thank Sebastian Schmidt (ETH Zurich) for manyuseful comments and critical reading of the manuscript.We gratefully acknowledge the allocation of computer re-sources from ZIH and MPI-PKS, Dresden and appreciatethe support of the DFG through Einzelverfahren. ∗ Corresponding author; [email protected][1] E. E. B. Campbell, V. Schyja, R. Ehlich, and I. V. Hertel,Phys. Rev. Lett., , 263 (1993).[2] R. Schmidt and H. Lutz, Comments At. Mol. Phys., ,461 (1995).[3] O. Knospe and R. Schmidt, in Theory of Atomic andMolecular Clusters , edited by J. Jellinek (Springer, 1999)p. 111.[4] E. E. B. Campbell and F. Rohmund, Rep. Prog. Phys., , 1061 (2000).[5] E. E. B. Campbell, Fullerene Collision Reactions (Kluwer Academic Publishers, Dordrecht, 2003).[6] R. Schmidt, G. Seifert, and H. O. Lutz, Physics LettersA, , 231 (1991).[7] G. Seifert, R. Schmidt, and H. O. Lutz, Physics LettersA, , 237 (1991).[8] O. Knospe, R. Schmidt, E. Engel, U. Schmitt, R. Drei-zler, and H. Lutz, Physics Letters A, , 332 (1993).[9] R. Schmidt and H. Lutz, Physics Letters A, , 338(1993).[10] R. Schmidt and H. O. Lutz, Phys. Rev. A, , 7981(1992).[11] R. Schmidt, J. Schulte, O. Knospe, and G. Seifert,Phys. Lett. A, , 101 (1994).[12] J. Schulte, O. Knospe, G. Seifert, and R. Schmidt,Phys. Lett. A, , 51 (1995).[13] F. Zhang, F. Spiegelmann, E. Suraud, V. Frayss´e,R. Poteau, R. Glowinski, and F. Chatelin, Physics Let-ters A, , 75 (1994).[14] H. Shen, P. Hvelplund, D. Mathur, A. B´ar´any, H. Ced-erquist, N. Selberg, and D. C. Lorents, Phys. Rev. A, , 3847 (1995).[15] B. Farizon, M. Farizon, and M. J. Gaillard,Int. J. Mass Spectrom., , 259 (1999). [16] H. Br¨auning, R. Trassl, A. Diehl, A. Theiß, E. Salzborn,A. A. Narits, and L. P. Presnyakov, Phys. Rev. Lett., , 168301 (2003).[17] J. Rogan, R. Ram´ırez, A. H. Romero, and M. Kiwi, Eur.Phys. J. D, , 219 (2004).[18] O. Kamalou, B. Manil, H. Lebius, J. Rangama, B. Hu-ber, P. Hvelplund, S. Tomita, J. Jensen, H. Schmidt,H. Zettergren, and H. Cederquist, Int. J. Mass Spec-trom., , 117 (2006).[19] D. Alamanova, V. G. Grigoryan, and M. Springborg, J.Phys.: Condensed Matter, , 346204 (2007).[20] J. Jakowski, S. Irle, B. G. Sumpter, and K. Morokuma,J. Phys. Chem. Letters, , 1536 (2012).[21] C. Yeretzian, K. Hansen, F. Diederich, and R. L. Whet-ten, Nature, , 44 (1992).[22] D. L. Strout, R. L. Murry, C. Xu, W. C. Eckhoff, G. K.Odom, and G. E. Scuseria, Chem. Phys. Lett., , 214(1993).[23] B. L. Zhang, C. T. Wang, C. Chan, and K. M. Ho,J. Phys. Chem., , 3134 (1993).[24] S. G. Kim and D. Tom´anek, Phys. Rev. Lett., , 2418(1994).[25] D. H. Robertson, D. W. Brenner, and C. T. White,J. Phys. Chem., , 15721 (1995).[26] Y. Xia, Y. Xing, C. Tan, and L. Mei, Phys. Rev. B, ,13871 (1996).[27] F. Rohmund, E. E. B. Campbell, O. Knospe, G. Seifert,and R. Schmidt, Phys. Rev. Lett., , 3289 (1996).[28] O. Knospe, A. V. Glotov, G. Seifert, and R. Schmidt,J. Phys. B, , 5163 (1996).[29] F. Rohmund, A. V. Glotov, K. Hansen, and E. E. B.Campbell, J. Phys. B, , 5143 (1996).[30] Y. Xia, Y. Mu, C. Tan, Y. Xing, and H. Yang, Nucl.Instr. and Meth. in Phys. Res. Sec. B, , 356 (1997).[31] A. Glotov, O. Knospe, R. Schmidt, and E. E. B. Camp-bell, Eur. Phys. J. D, , 333 (2001).[32] E. E. B. Campbell, A. V. Glotov, A. Lassesson, andR. D. Levine, C. R. Physique, , 341 (2002).[33] Y. Zhao, R. E. Smalley, and B. I. Yakobson, Phys. Rev.B, , 195409 (2002).[34] S. Han, M. Yoon, S. Berber, N. Park, E. Osawa, J. Ihm,and D. Tom´anek, Phys. Rev. B, , 113402 (2004).[35] N. Kaur, K. Dharamvir, and V. Jindal, Chem. Phys., , 176 (2008). [36] J. Jakowski, S. Irle, and K. Morokuma, Phys. Rev. B, , 125443 (2010).[37] Y. Wang, H. Zettergren, P. Rousseau, T. Chen,M. Gatchell, M. H. Stockett, A. Domaracka, L. Adoui,B. A. Huber, H. Cederquist, M. Alcam´ı, and F. Mart´ın,Phys. Rev. A, , 062708 (2014).[38] R. Bock, Heavy Ion Collisions , Vol. 1-3 (North-Holland,Amsterdam, 1980).[39] G. Brenn and A. Frohn, Exp. Fluids, , 441 (1989).[40] A. Menchaca-Rocha, F. Huidobro, A. Martinez-Davalos,K. Michaelian, A. Perez, V. Rodriguez, and N. Cˆarjan,J. Fluid Mech., , 291 (1997).[41] C. Struck, Physics Reports, , 1 (1999).[42] R. W. Hasse and W. D. Myers, Geometrical Relationshipsof Macroscopic Nuclear Physics (Springer-Verlag Berlin,1988).[43] R. D. Levine and R. B. Bernstein,
Molecular ReactionDynamics and Chemical Reactivity (Oxford UniversityPress, New York, 1987).[44] We do not consider here the high energy range, where thecross sections generally decreases with E c.m. , see [10].[45] M. S. Dresselhaus, G. Dresselhaus, and P. C. Eklund, Science of Fullerenes and Carbon Nanotubes (AcademicPress, San Diego, 1996).[46] T. Laarmann, I. Shchatsinin, A. Stalmashonak,M. Boyle, N. Zhavoronkov, J. Handt, R. Schmidt, C. P.Schulz, and I. V. Hertel, Phys. Rev. Lett., , 058302(2007).[47] M. Fischer, J. Handt, G. Seifert, and R. Schmidt, Phys.Rev. A, , 061403(R) (2013).[48] T. Kunert and R. Schmidt, Eur. Phys. J. D, , 15(2003).[49] M. Fischer, J. Handt, and R. Schmidt, Phys. Rev. A, , 012525 (2014).[50] M. Fischer, J. Handt, and R. Schmidt, Phys. Rev. A, , 012526 (2014).[51] M. Fischer, J. Handt, and R. Schmidt, Phys. Rev. A, , 012527 (2014).[52] G. P. Zhang and T. F. George, Phys. Rev. Lett., ,147401 (2004).[53] Z.-H. Dong, P. Zhou, J. M. Holden, P. C. Eklund, M. S.Dresselhaus, and G. Dresselhaus, Phys. Rev. B, , 2862(1993).[54] X. Q. Wang, C. Z. Wang, and K. M. Ho, Phys. Rev. B,51