IL NUOVO CIMENTO VOL. 110 A, N. 9-10 Settembre-Ottobre 1997
Fullerene-fullerene collisions
()
E. E. B. CAMPBELL, F. ROHMUNDand A. V. GLOTOV
Max-Born-Institut - Postfach 1107, D-12474 Berlin, Germany (ricevuto il 18 Luglio 1997; approvato il 15 Ottobre 1997)
Summary. — The results of experimental studies of fullerene-fullerene collisions are presented and compared with molecular dynamics simulations and statistical models. The similarities with and differences to nuclear heavy ion collisions are discussed. PACS 36.40 – Atomic and molecular clusters.
PACS 01.30.Cc – Conference proceedings.
1. – Introduction
Fullerenes are ideal model systems for investigating the dynamics of molecular, or cluster, systems with a large but finite number of degrees of freedom. They are attractive for a number of reasons: the beauty of C60makes it an aesthetically pleasing system to work with; the relative simplicity due to the high symmetry and presence of only one atomic element makes it attractive to theoreticians and the experimental accessibility and ease of production and handling means that fullerenes are beloved by experimentalists and are very suitable systems for developing new experimental methods.
Fullerenes show a range of properties which are similar to those of atomic metal clus-ters (or indeed also to phenomena which occur in nuclear systems on very different energy and time scales) such as thermal emission of electrons [1] (beta decay), black-body type radiation [2] (gamma-ray emission) and particle evaporation [3] (alpha decay). The bind-ing energies and ionisation potentials of the fullerenes and certain metal clusters are such that the three different decay or cooling mechanisms can compete on the microsecond timescale of a typical experiment. Recent experiments have been concerned with investi-gating the competition between these different mechanisms and in determining which are dominant in which excitation regime. This has relevance, e.g., for extracting the dissoci-ation energies for particle evapordissoci-ation and for determining the underlying physics of the thermal electron emission [4, 5].
(
)Paper presented at the 174. WE-Heraeus-Seminar “New Ideas on Clustering in Nuclear and Atomic Physics”, Rauischholzhausen (Germany), 9-13 June 1997.
Fig. 1. – Fusion and fragmentation reaction channels observed in fullerene-fullerene collisions
It is now well established that there are many analogies between nuclear physics and cluster physics concerning the structures of the systems (magic numbers) and collective excitations of the electrons in clusters compared with the collective excitations of neutrons versus protons in the nucleus, and very similar theoretical models can be applied in both fields to describe these phenomena. Much less is known about analogies in the dynamical behaviour of nuclei and clusters in collisions. In particular, it would be of great interest to know to what extent the models developed over the years for describing nuclear heavy-ion collisheavy-ions can be transferred to cluster-cluster collisheavy-ions and to larger macroscopic systems with a large but finite number of degrees of freedom such as liquid droplets [6] or even colliding galaxies. This has been the aim of our experimental studies on fullerene-fullerene collisions. Although fullerene-fullerenes may not be regarded as ideal cluster systems for these investigations due to their hollow cage-like structures, they are the only atomic cluster systems for which such experiments are possible at the present time.
Our experiments have concentrated on studying the fusion and subsequent fragmen-tation observed within a narrow collision energy window in fullerene-fullerene collisions. Figure 1 illustrates three of the reaction channels which we have observed. At rela-tively low collision energies it is possible to observe a metastable non-fragmented fused fullerene product (left-hand side), as the collision energy is increased slightly the fused compound cannot remain stable and will undergo evaporation of C2 molecules (centre) until at higher energies a stable (or metastable) fused compound is unable to form and the system undergoes multifragmentation. In the following paper we will discuss the exper-imental evidence for these different processes in relation to molecular dynamics simula-tions and statistical calculasimula-tions. We will show that there are a number of similarities but also some significant differences from the dynamics of heavy-ion collisions.
FULLERENE-FULLERENE COLLISIONS 1193
2. – The experiment
The experimental set-up has been described in detail before [7, 8] and will be only briefly described here. A positively charged fullerene ion beam is produced by evapo-rating fullerene powder from an oven at a temperature of about500
o
C and ionising by electron impact. The ions are extracted into the main scattering chamber by a pulsed electric field. They collide with neutral fullerene vapour in a scattering cell and the posi-tively charged products from the collisions are detected by a reflectron time-of-flight mass spectrometer. The pressure of fullerenes in the scattering cell is such that we can be sure of single collision conditions. A retarding field energy selector in front of the channel plate ion detector can be used to determine the kinetic energies of the reflected ions. The re-flectron can be rotated around the scattering cell which allows the determination of the angular distribution of fragment ions [9]. For all the results discussed in this paper the reflectron was placed in the line of the projectile ion beam with an acceptance angle of
1:2
.
3. – Fusion and fragmentation reactions
The absolute cross-section for fusion in C+60 + C60 collisions is shown in fig. 2(a) as a function of centre-of-mass collision energy. The details of how we extract the abso-lute cross-sections from the time-of-flight data are given in [8]. In order to extract these cross-sections we considered both the non-fragmented C+120which we detected in the mass spectrometer as well as all fragments from this species that could be observed with the detector at00:6
. There is a fairly narrow energy window in which the fusion reaction can be seen. The signal can be observed for collision energies beyond about 60 eV, rises to a maximum at about 140 eV and then decreases abruptly and has practically disappeared for collision energies beyond 200 eV [10]. The picture is qualitatively very similar to what is seen in nuclear heavy ion collisions (HIC) [11]. In fig. 2(b) we show a schematic picture of the reaction channels which occur in HIC as a function of impact parameter and colli-sion energy. The behaviour can be described in terms of a simple absorbing sphere model. The effective potential energy of the colliding system,Ve (R) is given by
Ve (R )=V(R )+ `2~2 2R2 =V(R )+ b2E R2 ; (1)
where the second term gives the centrifugal energy which arises for finite impact param-eters withRthe distance between the collision partners,`the angular momentum,bthe
impact parameter,the reduced mass andEthe collision energy. For fusion to occur, the
collision energy must be equal to or exceed the effective potential energy at the critical distance for the reaction given, for simplicity, by the sum of the radii of the two collision partners (R12)
Ve (R12)E:
(2)
Thus, at threshold, one obtains the expression
V(R12)+ b2maxE
R212 =E;
Fig. 2. – a) Experimentally determined fusion cross-section for C+ 60+ C
60. b) Schematic picture of reactions occuring in nuclear heavy ion collisions as a function of collision energy and impact parameter. Adapted from [11].
whereV(R12)is the potential barrier for the reaction and bmax is the maximum impact
parameter that can lead to reaction. This then leads to the simple expression for the reaction cross-section fus=b2max=R212 1, V(R12) E (4)
which is plotted as a full line in fig. 2(b). Equation (4) is only valid if the fused compound is stable against centrifugal fragmentation, i.e. if the angular momentum is smaller than a critical angular momentum`crit[12]. Thus for energies larger than the critical energy
Ecrit=V(R12)+ ~2`2crit 2R212 ;
FULLERENE-FULLERENE COLLISIONS 1195
Fig. 3. – Fusion cross-sections as a function of 1/(collision energy). a) C+ 70+ C 70, b) C + 60=70 + C70=60, c) C+ 60 + C
60. Symbols: experimental values. Full lines: linear fits according to the absorbing sphere model (eq. (4)) multiplied by an average fusion probability . Dashed lines: calculations in high energy range beyond the onset of centrifugal fragmentation (eq. (6)).
the fusion cross-section would be expected to decrease as a function ofEaccording to
fus=
}2`2crit 2E
:
(6)
This is indicated by the dashed line in fig. 2(b). In order to see how well our experimen-tal data for fusion between two fullerenes agree with this simple model we have plotted the fusion cross-sections for three different collision systems (C+60+ C60, C+60
=70
+ C70=60
and C+70+ C70) as a function of 1=Ein fig. 3. In the low collision energy range close to
the barrier we have reasonably good agreement with the model as long as we include a multiplication factor in eq. (4). This can be regarded as an average probability for fu-sion to occur in the reactions after all the energetic and impact parameter constraints have been taken into account by the model. Detailed quantum molecular dynamic (QMD) simulations [13] have shown that this is due to a ”bouncing off ” mechanism which is a
TABLEI. – Summary of quantities extracted from the experimental dependence of the fusion cross-section on the collision energy using the absorbing sphere model (ASM) and the steric model (SM). The values of the fusion barriers (units of eV) are compared with molecular dynamics calculations V(R12) QMD from ref. [13]. Coll. System V(R12) ASM V(R12) SM V(R12) QMD P `crit=~ bcrit=A˚ C+ 60 +C 60 67(7) 60(1) 60 0.023 24500 5.14 C+ 60=70 +C 70=60 84(9) 70(7) 70 0.08 23000 4.75 C+ 70 +C 70 86(12) 76(4) 75 0.18 25900 4.72
consequence of the strongly directional covalent bonding in the fullerene cage. Trajecto-ries at the same collision energy and impact parameter can lead to very different results depending on the relative orientation of the hexagons and pentagons in the two colliding fullerenes. This is one of the main differences between fullerene collisions and HIC. A sec-ond important difference seen from the QMD calculations is that the total kinetic energy of the system after fullerene fusion is less than the impact energy [10]. In nuclear fusion the total kinetic energy in the fused system is generally larger then the impact energy due to the gain of binding energy [11]. This would also be the situation for fullerenes if the most energetically stable C+120isomer was formed in the collisions, however, this is not the case and the typical product is a highly deformed ”peanut” isomer (see fig. 1). This isomer corresponds to a fairly stable local minimum on the potential energy surface with the binding energy per atom close to or less than that of C60[14]. This effect is again pri-marily a consequence of the strong, directed covalent bonds in the fullerenes. It reduces the final vibrational energy and thus stabilizes the fused compound against evaporation. This is the main reason why we can clearly observe fusion products on thes timescale of
our experiments.
Comparison of our data with the absorbing sphere model allows us to extract the ener-getic barrier for the fusion reaction as well as the average fusion probability for all three collision systems. A better fit to the experimental data in the barrier region is found by using an extension to the absorbing sphere model which takes into account the steric ef-fects [8-13]. This model can reproduce the curvature of the experimental data points near the threshold if the fusion probability is assumed to be approximately proportional to the collision energy [8]. The values for the fusion thresholds extracted with both models are given in table I along with predictions from molecular dynamics simulations. Excellent agreement is found if the calculations assume an initial temperature of 2000 K which is a realistic value under our experimental conditions [8]. The average fusion probability is also given in table I and is seen to increase more rapidly than the geometrical cross-section. The reason for this is not yet clear.
Assuming that the maximum of the measured cross-section corresponds to the critical impact energyEcrit (eq. (5)), we can extract the critical angular momentum and impact
parameter for each collision system. The values are also given in table I and are ap-proximately the same for the three sets of collision partners. According to eq. (6) the cross-section would be expected to decrease linearly towards zero as the collision energy is increased to infinity. As can be seen from fig. 3 the experimental values decrease much more rapidly than this as the collision energy is increased.
col-FULLERENE-FULLERENE COLLISIONS 1197
Fig. 4. – Average fusion product size as a function of collision energy for the different collision systems. Symbols: experimental values. Full lines: calculations assuming successive evaporation of C2molecules. Arrows mark the position of
E
crit(see text).
lision energy dependence of the average product cluster size detected in the experiment. This is plotted in fig. 4. There is a linear fall in product size with increasing collision en-ergy up to a value which is dependent on the size of the collision partners. For moderate excitation energies the most likely fragmentation channel of fullerenes is the metastable evaporation of C2molecules as can be seen, e.g., in the results of MD simulations shown in fig. 1. We can model this behaviour by assuming a simple Arrhenius expression for the evaporation rate constant and using a Monte Carlo procedure to simulate the effect of successive C2loss on the experimental timescale [8]. In this way we can reproduce the ex-perimental results in the linear regime very satisfactorily by using a dissociation energy for C2loss of 6.4 eV, as shown by the full lines in the figure. This gives a lower limit to the dissociation energy since we have not taken any competing energy loss mechanisms such as radiative decay [4] into consideration, however, it is similar to values often quoted for dissociation energies of smaller fullerenes with similar internal energies [15].
The critical energyEcritwhich we defined as the maximum in the fusion cross-section
is also indicated on fig. 4. For C+60+ C60this lies at the energy where the experimental data deviate from linear behaviour but is still within the linear regime for the two larger
Fig. 5. – Average excitation energy as a function of fragment temperature as determined by a maximum entropy model [16] for C+
60and C + 120.
collision systems. Comparison with fig. 3 shows that the experimental data lie on the straight line given by eq. (6) as long as the fragmentation of the fused compound can be modelled in terms of C2evaporation. The reason for the deviation from the simple model, where the fusion cross-section is limited by centrifugal fragmentation, is the onset of a second fragmentation mechanism where larger ring and chain fragments are produced and emitted in all directions. They can therefore not be detected in our experiments where the detector samples fragments which fall within the angular range of00:6
.
This hypothesis is supported by applying a statistical maximum entropy model to the fused compound. Such a model has recently been applied to C60[16] and is able to explain the onset of the very typical bimodal fragmentation pattern observed for C+60for excita-tion energies beyond approximately 85 eV. The appearance of small fragment ions in the form of rings and chains is related to the onset of a liquid-gas phase transition in the ex-cited fullerene [16]. The relationship between the average excitation energyhEiand the
temperature of the fragments is illustrated in fig. 5. The bimodal fragment distribution onsets for temperatures slightly over 4000 K which, for C+60, occurs at an excitation energy of 85 eV. The model can easily be extended to C+120and other larger fullerenes by making some simple assumptions about the binding energies and number of isomers involved [17]. The results for C+120are also plotted on fig. 5. The phase transition occurs at a similar tem-perature but, due to the larger number of degrees of freedom, the corresponding average excitation energy is shifted to about 150 eV. This is in excellent agreement with the en-ergy where deviation from the C2evaporation behaviour is observed (fig. 4). The onset of a phase transition where the fullerene structure is lost in the fused compound can also be confirmed by molecular dynamics simulations. Figure 6 shows the results of four trajec-tories calculated for C60+ C70collisions at collision energies of 160 eV (upper) and 170 eV (lower), which lie approximately at the onset of the phase transition in this system. After 1 ps some very weird forms can be seen in the calculations and it is obvious that one can no longer discuss the fused compound in terms of a stable (or metastable) fullerene structure.
FULLERENE-FULLERENE COLLISIONS 1199
Fig. 6. – The results of four QMD trajectory calculations after 1 ps for C60 + C70at a collision energy of 160 eV (upper) and 170 eV (lower).
4. – Conclusion
A fusion reaction can be observed in fullerene-fullerene collisions within a narrow col-lision energy window lying between 60 and 200 eV. The results can be described in terms of a simple phenomenological absorbing sphere model. The main difference observed in these collisions compared to heavy-ion collisions is that the strongly directional covalent bonding in the fullerenes leads to a very low probability of fusion even after the energetic and impact parameter constraints have been taken into account by the model. Moreover, there is a ”bouncing off ” mechanism observed which seems to be typical for fullerene collisions and the total kinetic energy in the fused system after collision is less than the impact energy. The fragmentation of the fused compound has been studied by combining molecular dynamics simulations with statistical models. At moderate excitation energies the fused compound can stabilise by evaporating C2 molecules. However, at a given ex-citation energy (which depends on the total number of degrees of freedom in the fused compound), there is the onset of a phase transition which leads to the production of large ring and chain fragments. The transverse momenta imparted to these products on frag-mentation remove them from the acceptance angle of our ion detector so that the fusion cross section which we detect experimentally falls much more quickly with increasing col-lision energy than predicted by the simple model.
We would like to thank O. KNOSPEand, in particular, R. SCHMIDTfor encouraging us to try these experiments in the first place and for their continuing support with quantum molecular dynamics simulations and many fruitful discussions. We would also like to thank K. HANSEN for his help with the Monte Carlo calculations, R. D. LEVINEand T. RAZ
for their help with the maximum entropy calculations and, last but not least, I. V. HER -TELfor his constant encouragement of this project. Financial support from the Deutsche Forschungsgemeinschaft through Sfb 337 ”Energie- und Ladungsaustausch Molekularer Aggregate” is gratefully acknowledged.
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