The threshold displacement energy of buckminsterfullerene C 60 and formation of the endohedral defect fullerene He@C 59
M. H. Stockett, M. Wolf, M. Gatchell, H. T. Schmidt, H. Zettergren, H. Cederquist
aa r X i v : . [ phy s i c s . c h e m - ph ] J u l The threshold displacement energy of buckminsterfullerene C and formation of theendohedral defect fullerene He@C a, ∗ , Michael Wolf a , Michael Gatchell a,b , Henning T. Schmidt a , Henning Zettergren a ,Henrik Cederquist a a Department of Physics, Stockholm University, Roslagstullsbacken 21, SE-106 91 Stockholm, Sweden b Institut f¨ur Ionenphysik und Angewandte Physik, Universit¨at Innsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria
Abstract
We have measured the threshold center-of-mass kinetic energy for knocking out a single carbon atom from C − incollisions with He. Combining this experimental result with classical molecular dynamics simulations, we determine asemi-empirical value of 24.1 ± cage. We report the first observation of an endohedral complex with an odd number of carbon atoms,He@C − , and discuss its formation and decay mechanisms.
1. Introduction
Irradiation by electrons, ions, or atoms may displaceindividual atoms from their original positions in solids [1],small aggregates of matter [2], or free molecules [3]. Suchdefects can for example be used to tailor nanostructuredmaterials with new intriguing functionalities [4, 5, 6], andserve as reactive sites for efficient molecular growth pro-cesses inside molecular clusters [2, 7, 8], for example indifferent planetary atmospheres. Introductions of defectsmay also be a limiting factor in high-resolution transmis-sion electron microscopy, as material modifications dur-ing image capture may yield images that do not repre-sent the original sample material [9, 10]. In addition,collisions with H and He atoms leading to atom displace-ment are thought to be important mechanisms for destruc-tion of large molecules in the interstellar medium [11, 12].The key intrinsic (projectile independent) target propertyquantifying the effect of these types of radiation damage isthe so-called threshold displacement energy, T d , the mini-mum energy transfer to a single atom required to perma-nently displace it from its initial position [1].The threshold displacement energy is distinct from re-lated quantities such as the dissociation energy or the va-cancy energy in that it includes the energy barrier betweenthe parent and product states. While independent of pro-jectile, T d depends somewhat on the angle between themomentum imparted to the primary knock-on atom andthe inter-atomic bonds [13, 14]. Nevertheless, typical val-ues of T d are widely used to model radiation damage acrosswide ranges of energy. In recent breakthrough experiments[15, 16], the threshold displacement energy for single-layer ∗ Corresponding author
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[email protected] (Mark H. Stockett) graphene was reported to be T d = 23 . T d have also been deduced for gas-phase PolycyclicAromatic Hydrocarbon (PAH) molecules [17].Here, we determine the C → C +C threshold dis-placement energy for isolated C , which has a perfectlysymmetric icosahedral structure where the C atoms havea different hybridization than graphene or PAHs. Withall atoms equivalent, C is the ideal prototype for dis-placement studies of molecules. Previously reported val-ues of T d for C from electron microscopy experimentsand from Molecular Dynamics simulations have spanned awide range from 7.6-15.7 eV [18] and 29.1 eV [13], respec-tively. Models of radiation damage to fullerenes have gen-erally assumed a value in the middle of this range, around15 eV, equivalent to the vacancy energy [19, 20, 2].In addition to the threshold displacement energy forC , we also report the first observation of the endohe-dral defect fullerene complex He@C − formed in collisionswhere the He is captured following C displacement. Endo-hedral complexes B@C n , where an atom or molecule B istrapped inside the fullerene cage, has been of interest sincethe dawn of the fullerene era, both from pure and appliedperspectives [21, 22, 23, 24, 25]. The present observationof He@C − provides insight into the formation and stabil-ity of endohedral complexes with odd numbers of carbonatoms.In typical fullerene fragmentation experiments, energydeposited through interactions with e.g. photons, elec-trons, or fast ions is converted into internal vibrationalenergy. For C , where C → C +C is the lowest energydissociation channel [26, 27], this leads to the well-knownstatistical product distribution dominated by fragmentswith even numbers of carbon atoms, C − n , n = 1 , , ... .In collision experiments like those presented here, productswith odd numbers of C atoms like C have occasionally Preprint submitted to Elsevier August 1, 2018 een observed [21, 20, 26, 28]. These products are fin-gerprints of non-statistical fragmentation, where carbonatoms are displaced in billiard-ball like collisions [3]. Thisprocess takes place on timescales that are too short (sub-femtoseconds) for local excitations to distribute over thewhole molecular system. The exceptionally low yield ofC relative to statistical fragmentation products has so farprecluded systematic experimental studies of this mecha-nism.Here, we have developed a refined approach to deter-mine displacement energies for free molecules in collisionswith particles that improves upon earlier such methodsin an important way, namely by eliminating the (usuallydominant) contribution of statistical fragmentation fromthe product distribution. This is achieved by colliding Hewith C − ions and measuring the threshold behavior forthe formation of negatively charged fragments e.g. C − and C − . Because the electron affinity of C (2.664 eV[29]) is much lower than any of the dissociation energiesof the system (which are >
10 eV [26, 27], see also Tables1 and 2 below), any trajectories depositing enough en-ergy to induce statistical unimolecular dissociation mostlikely lead to electron loss and thus do not contribute tothe negative ion product spectrum. In this way, we selectthose trajectories where essentially all the excitation en-ergy is transferred to the primary knock-on atom, and aslittle as possible to the other atoms in the C cage or tothe electronic degrees of freedom. By eliminating the ma-jor source of background from the measurement, greatersensitivity to minute cross-sections for non-statistical frag-mentation is achieved, without which the present resultsfor C would not be possible. We combine experimentalmeasurements of the threshold center-of-mass energy fornon-statistical fragmentation of C − in collisions with Hewith classical Molecular Dynamics (MD) simulations of theknockout process and a statistical model of electron lossfrom C − and C − to determine the threshold displacementenergy T d . Because the extra electron in C − has only asmall effect on the binding between C atoms, this resultapplies to neutral as well as anionic C .
2. Methods
Experiments were performed using the ElectrosprayIon Source Laboratory (EISLAB) accelerator mass spec-trometer, which has been described previously [30, 31].Continuous beams of C − were produced by means of Elec-troSpray Ionization, with tetrathiafulvalene (TTF) addedas an electron donor [32] to a 1:1 mixture of methanol anddichloromethane containing fullerite. In the ion source,the C − ions undergo many low-energy collisions and areassumed to equilibrate to roughly room temperature. Fol-lowing production of the ions by ESI, an ion funnel wasused to collect and focus the ions. Two octupole ion guides transported the ions through two stages of differen-tial pumping. A quadrupole mass filter was used to mass-select C − ions, which were then accelerated to 3–15 keVand passed through a 4 cm long collision cell containing thetarget gas (He). This C − kinetic energy range correspondsto center-of-mass energies, E CM , of 20–80 eV for collisionswith He in the cell. An Einzel lens and two pairs of electro-static deflector plates served as a large-angular-acceptancekinetic-energy-per-charge analyzer after the gas cell. In-tact C − and daughter anions were detected on a 40 mmmulti-channel plate (MCP) with a position-sensitive re-sistive anode. The deflection voltage combined with theposition on the MCP (along the direction of deflection) ofeach detected anion was used to calculate its kinetic energyper charge [31].Total destruction cross sections ( i.e. the sum of thecross sections for all processes – fragmentation and/or electron-loss or ionization processes – that change the mass-to-charge ratio of C − in collisions with He under the presentexperimental conditions) were determined by measuringthe attenuation of the parent C − beam as a function oftarget gas density. The attenuation is fit to a single expo-nential decayΓ( p ) = Γ e − pσl/k B T (1)where Γ( p ) is the parent C − count rate as a function ofthe He pressure p in the gas cell, Γ is the count rate at p = 0, σ is the total C − destruction cross section to bedetermined, l is the length of the gas cell (4 cm), and k B is Boltzmann’s constant. The temperature T is 300 K.In order to determine the absolute cross section forthe formation of a specific daughter ion such as C − , wedistribute the total cross section σ according to the cor-responding number of product ions N i ( p ) relative to thenumber of intact parent ions N par ( p ) detected in the samespectrum: σ i = σ N i ( p ) N par ( p )( e pσl/k B T − , (2)where N par ( p )( e pσl/k B T −
1) corresponds to the total num-ber of parent ions destroyed as this spectrum was recorded.
Classical molecular dynamics simulations of collisionsbetween C and He were carried out using the LAMMPSsoftware package [33]. We used the reactive Tersoff poten-tial [34, 35] to describe the bonds between C atoms andthe Ziegler-Biersack-Littmark (ZBL) potential [36] for theHe-C interaction. For each collision energy 10,000 ran-domly oriented trajectories were simulated (with randomorientations of the C molecule) and followed for 500 fswith a time step of 5 × − s. One frame from a trajec-tory leading the formation of the He@C − reaction productis shown in Figure 1. The full video of this trajectory isavailable in the electronic version of this article.2 igure 1: Image from a video of a MD trajecotry leading to He@C − .The C atom (black) at the right of the frame has been knocked outby the He atom (blue), which has been captured by the C cage.The full video is available in the electronic version of this article. Charge C ho -C pn -C -1 2.59 3.56 3.370 7.50 6.69 6.93 Table 1: Electron binding energies (in eV) for fullerene anions andneutrals calculated with DFT at the B3LYP/6-31+G(d) level. Seetext for isomer naming convention.
Density Functional Theory (DFT) was used to calcu-late electron detachment and dissociation energies of C − and C − . These calculations were performed using Gaus-sian09 [37] at the B3LYP/6-31+G(d) level. Following theremoval of a single carbon atom from C , there are twoclosed-cage structures of C to which the system may re-lax. These differ in the bonding between the three C atomswhich were adjacent to the knocked out C; the naming hereis adopted from Ref. [8]. The ho -C − isomer, in which thecage-closing leads to the formation of a hexagonal ( h ) andan octagonal ( o ) ring at the vacancy cite, is 1 eV lower inenergy than pn -C − , where a pentagonal ( p ) and a nonag-onal ( n ) ring are formed. For C − , the lowest energy neu-tral isomer from Ref. [38], labeled C v -C , was used as astarting point for the calculations.
3. Results − Electron binding energies for both neutral and anionicC and C (both ho and pn isomers) are presented inTable 1. The present value for the electron affinity of C ,2.59 eV, is close to measured values ( e.g. are much higher at 3.56 eV for ho -C and 3.37 eV for pn -C .DFT-calculated dissociation energies for the main de-cay channels are given in Table 2. Our value for the C − → C → C → C → Charge C +C C +C C +C-1 11.6 10.1 4.540 12.6+1 11.8 Table 2: Dissociation energies (in eV) for lowest energy isomers ofC and C , ho -C and C v -C , in various charge states calculatedwith DFT at the B3LYP/6-31+G(d) level. C − +C dissociation energy of 10.1 eV is similar to the 10-11 eV measured for neutral and cationic C [27, 26]. Thedissociation energy for C → C +C, which we calculatefor anionic, neutral and cationic species, is around 12 eVwhich is significantly higher than for C -loss from C inall cases and in agreement with previous calculations (forneutrals and cations [8]). The dissociation energy calcu-lated, however, does not consider any energy barrier be-tween the the initial and final states. Given that C isalmost never observed in fullerene fragmentation experi-ments, even when employing excitation energies far in ex-cess of all dissociation energies, it is possible that muchhigher barriers are present for C → C +C than forC → C +C fragmentation processes. In contrast tothe dissociation energy, the threshold displacement energy T d reflects the barriers for the C → C +C process.Our calculated dissociation energy for C − → C − +C israther low at 4.54 eV and close to the previously calculated5.4 eV for C +59 [8]. This is also likely to contribute to thelow observed yield of C in experiments – the productsare fragile and may rather easily undergo delayed fragmen-tation processes. Internal energies upwards of 40 eV areknown to be required to observe C -loss from fullereneson microsecond timescales [38]. Formation of C in a sta-tistical process would involve similar if not higher internalenergies, most of which would remain with the C leadingto further C-loss. For non-statistical knockout fragmenta-tion, however, much of the collision energy is carried awayby the knocked out C atom, leaving colder C . In thecase of C − anions, the dissociation energy is comparableto the electron binding energy, and these decay channelscan be expected to compete. Experimental total destruction cross sections for C − +Hecollisions are determined by measuring the attenuation ofthe C − beam as a function of He target density [31] andthe results are shown for a range of different center-of-mass collision energies in Figure 2. Also shown in Figure2 are the single carbon knockout cross section from ourMD simulations. These simulations indicate that knock-out only gives a small contribution to the total C − de-struction cross section. Furthermore, the simulations donot include secondary fragmentation or electron emissionprocesses and should thus be considered upper limits forC − formation.3 (cid:2) (cid:3) (cid:4)(cid:4) (cid:5) (cid:6) (cid:7) (cid:8) (cid:9) (cid:10) (cid:3) (cid:11) (cid:5) (cid:12) (cid:13)(cid:14) (cid:15) (cid:13)(cid:16) (cid:5) (cid:8) (cid:17) (cid:18) (cid:5) (cid:19) (cid:14)(cid:13)(cid:18)(cid:20)(cid:21)(cid:16) (cid:5) (cid:1)(cid:7)(cid:11)(cid:9)(cid:7)(cid:2)(cid:15)(cid:3)(cid:22)(cid:15)(cid:23)(cid:24)(cid:4)(cid:4)(cid:5)(cid:25)(cid:11)(cid:7)(cid:2)(cid:26)(cid:27)(cid:5)(cid:25) (cid:1)(cid:23) (cid:5)(cid:12)(cid:7)(cid:28)(cid:19)(cid:14) (cid:16)(cid:14) (cid:13)(cid:14)(cid:14) (cid:13)(cid:16)(cid:14)(cid:29)(cid:3)(cid:9)(cid:24)(cid:30)(cid:5)(cid:12)(cid:25)(cid:31) (cid:19)(cid:1) !(cid:14)(cid:15) " (cid:1) (cid:5)(cid:1) (cid:16)$(cid:15) "(cid:1)" !(cid:14)(cid:15) " (cid:1) (cid:5)(cid:1) !(cid:14) " (cid:1) (cid:5) (cid:15) " Figure 2: Absolute total destruction cross section for C − in collisionswith He determined from beam attenuation measurements (filled cir-cles). Carbon knockout cross sections (C − +He → C − +C+He) fromMD simulations, electron emission cross sections (C − +He → C + e − +He) from the statistical model described in the text, and thesum of these two channels (open diamonds, squares, and circles, re-spectively). Our MD simulations model only the nuclear scatter-ing part of the He-C interaction and does not includethe scattering of the He atom on the electrons in C or delayed decay processes such as unimolecular dissocia-tion ( i.e. statistical fragmentation) or thermionic electronemission. Depending on the kinetic energy of the incidentC − ions, product ions travel for 10–40 µ s between the gascell and the mass analyzing system [30, 31], giving ampletime for such delayed processes to occur. Nevertheless,we can use the fullerene excitation energies from simu-lations to reproduce the measured total C − destructioncross section, which is presumably dominated by electronloss, using a statistical model. All parameters used in thismodel are taken from the present DFT calculation (Tables1 and 2) or from the literature as detailed below. None ofthese parameters are adjusted to obtain agreement withexperiment. The same model will also be used in Section3.4 to estimate the fraction of C − that is produced suffi-ciently cold to survive the flight from the gas cell to theend of the deflection field in the kinetic-energy-per-chargeanalyzer without losing the electron.Here, we will first consider the survival probability ofcollisionally excited C − ions flying through the energy an-alyzer. First, the excitation energy deposited in the systemis extracted from the MD simulations, taking the nuclearscattering on 60 carbon atoms into account. To this weadd the internal energy of C − before the collision, whichis taken to be 0.44 eV for ∼
300 K C as calculated byYoo et al. [39]. Finally, we add a small energy contri-bution from electronic stopping obtained by scaling theresults by Schlath¨olter et al [40] to the collision velocity inthe present experiment. Adding these three contributions,we arrive at a total internal energy E int for each individual (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:8) (cid:9) (cid:10) (cid:2) (cid:11) (cid:9) (cid:12) (cid:9) (cid:13) (cid:6) (cid:14) (cid:15) (cid:2) (cid:11) (cid:16) (cid:17) (cid:6) (cid:18)(cid:15)(cid:3)(cid:9)(cid:19)(cid:13)(cid:6)(cid:20)(cid:21)(cid:16)(cid:16)(cid:6)(cid:7)(cid:18) (cid:22)(cid:23) (cid:6)(cid:17)(cid:24)(cid:25) (cid:26) (cid:27) (cid:28) (cid:29)(cid:30)(cid:30)(cid:6)(cid:3)(cid:31)(cid:30)(cid:25)(cid:6)(cid:3)(cid:31)(cid:27)(cid:27)(cid:6)(cid:3)(cid:31)(cid:18) (cid:22)(cid:23) (cid:6) (cid:27)(cid:24)(cid:6)(cid:3)(cid:31) !(cid:3)"(cid:22) !(cid:3)"(cid:22) (cid:22) (cid:22) Figure 3: Negatively charged product distributions followingC − +He collisions recorded at several values of E CM . The dashedlines are to guide the eye. The measured energy loss (the differencein kinetic energy between the incident C − ion and the fragment an-ion is due to the combination of the change in mass and the energytransfer to the internal degrees of freedom of the molecular systemand is given on the lower horizontal scale in units of E CM . The ex-pected kinetic energies of C − (blue) and C − (black) fragments withthe same velocity as incident C − are indicated with arrowheads onthe upper horizontal axis. trajectory in the MD-simulation, an internal temperature T [ K ] = 1000 + ( E int [ eV ] − . /C [41]. The effective tem-perature is then T eff = T − E b / C , where C = 0 .
138 K/eVis the heat capacity of C [41], and E b = 2 .
664 eV isthe electron binding energy of C − (the electron affinity ofC ) [29]. We calculate the survival probability for C − ions as a function of their internal energy E int by apply-ing an Arrhenius expression for the electron detachmentrate k = νe − E b /k B T eff [32]. Here the pre-exponential fac-tor ν is taken to be 10 s − [42], and k B is Boltzmann’sconstant. This model gives a maximum internal energy of16–17 eV for C − to survive our experimental timescalesof 10-40 µ s, depending on the acceleration energy.By combining the statistical model for electron lossand the MD simulations for knockout, we obtain the totalC − destruction cross section (knockout plus electron loss)shown in Figure 2. The agreement with the correspondingmeasured quantity is satisfactory, validating our compu-tational approach, and indicating that electron emissionis indeed the dominant C − destruction mechanism in thisenergy range. In particular we see an onset of the exper-imental total C − destruction cross section around 16 eVwhich is consistent with the results from the statisticalmodel - with center-of-mass collision energies below 16 eV,the system cannot be sufficiently heated to emit an elec-tron on the experimental timescale. In Figure 3 we show distributions of negatively chargedreaction products of C − +He collisions for center-of-mass4ollision energies ranging from E CM = 61 eV to E CM =77 eV, which corresponds to kinetic energies of 11–14 keVkeV for the incident C − ions in the laboratory referenceframe. The collision products contributing to the energyspectra in Figure 3 are those which are sufficiently coldto retain their negative charge through the analyzing sys-tem after the gas cell. The spectra are measured undersingle-collision conditions as has been confirmed by mea-suring the C − beam attenuation as a function of pres-sure in the gas cell. The distributions are plotted on an E loss = E parent − E product scale – the difference in kineticenergy of the parent ion C − and the product ion – in unitsof the center-of-mass collision energy E CM . The measuredkinetic energy loss is due not only to the difference in massbetween the incident C − ion and the product anion butalso to the energy transferred to the internal degrees offreedom of the molecular system. Capture of the He atomby the fullerene cage also reduces the kinetic energy. Theenergy loss due only to mass loss from C − without energytransfer would be 3 N Closs × E CM where N Closs is the numberof C atoms lost. The expected positions of C − and C − fragments with the same velocity as the incident C − ionswould thus be E loss = 3 × E CM and E loss = 6 × E CM , re-spectively, and are indicated with arrowheads on the upperaxis of Figure 3.The energy spectra are dominated by intact C − ionsat E loss = 0. Four daughter anion peaks are observed: twojust below the expected C − energy ( E loss = 3 × E CM ),and two just below the expected C − energy ( E loss = 6 × E CM ). Based on comparisons with our MD simulations( vide infra ), we assign in both cases the higher energypeaks to endohedral complexes. The daughter ion peakwith the highest kinetic energy (smallest energy loss) is thefirst observation of an endohedral defect fullerene complexwith an odd number of C atoms, namely He@C .The observed C − and He@C − products are most likelydue to secondary losses of loosely bound C atoms fromC − and He@C − , respectively. We have calculated theC − → C − +C dissociation energy to be 4.54 eV (see Table2), and that for the endohedral complex is likely compa-rable. Direct statistical dissociation C − → C − +C has adissociation energy of 10.1 eV according to our calcula-tions and is not competitive with electron emission (2.664eV [29]) and thus should not contribute significantly to thenegative product spectrum.As can be seen in Figure 3, the peaks in the productdistributions assigned to endohedrals remain roughly atthe same E loss values, while the C − and C − peaks shiftto greater E loss values with decreasing E CM . In the toppanel of Figure 4, we compare the measured and simu-lated mean energy loss for the observed product ions as afunction of E CM . For the experimental data, these valuesare extracted by fitting Gaussian peak shapes to each ofthe four daughter ions; two Gaussians are used for the tailof the parent C − peak. The peaks assigned to endohe-dral complexes are shifted by between 0.25 to 0.5 units of E CM relative to the expected E loss values for C − and C − (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:8) (cid:9) (cid:10)(cid:10) (cid:7) (cid:11) (cid:1) (cid:12) (cid:13) (cid:7) (cid:14) (cid:15)(cid:16)(cid:17) (cid:7) (cid:12)(cid:3)(cid:2)(cid:18)(cid:3)(cid:4)(cid:19)(cid:9)(cid:20)(cid:19)(cid:13)(cid:21)(cid:10)(cid:10)(cid:7)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:11)(cid:3)(cid:22)(cid:14)(cid:23)(cid:24) (cid:25)(cid:24) (cid:16)(cid:24) (cid:26)(cid:24) (cid:27)(cid:24)(cid:28)(cid:3)(cid:29)(cid:12) (cid:25)(cid:15)(cid:19) (cid:28)(cid:3)(cid:29)(cid:12) (cid:25)(cid:27)(cid:19) (cid:12) (cid:25)(cid:15)(cid:19) (cid:12) (cid:25)(cid:27)(cid:19) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:2) (cid:8) (cid:9) (cid:10) (cid:5) (cid:2) (cid:10) (cid:11) (cid:6) (cid:5) (cid:12) (cid:13)(cid:14)(cid:13)(cid:13)(cid:14)(cid:15)(cid:13)(cid:14)(cid:16)(cid:13)(cid:14)(cid:17)(cid:13)(cid:14)(cid:18)(cid:19)(cid:14)(cid:13) (cid:8) (cid:20)(cid:2)(cid:10)(cid:5)(cid:2)(cid:21)(cid:22)(cid:23)(cid:24)(cid:22)(cid:25)(cid:4)(cid:11)(cid:11)(cid:8)(cid:26)(cid:10)(cid:2)(cid:21)(cid:27)(cid:12)(cid:8)(cid:28)(cid:2)(cid:29)(cid:30)(cid:16)(cid:31) (cid:31)(cid:13) (cid:31)(cid:31) (cid:17)(cid:13) (cid:17)(cid:31) (cid:13) (cid:31) !(cid:2)"(cid:20) (cid:31) !(cid:2)"(cid:20) (cid:31)(cid:18)(cid:22) (cid:20) (cid:31) (cid:20) (cid:31)(cid:18)(cid:22) Figure 4: Top: Mean energy losses of reaction products in units of E CM ; filled symbols: experiment, open symbols: MD simulations.The uncertainties in the mean position are smaller than the symbols.Bottom: Stacked area plot showing the relative intensities of theproduct peaks as a function of E CM . with the same velocity as the incident C − ion at 3 × E CM and 6 × E CM , respectively, which are indicated by horizon-tal lines in the upper panel of Figure 4. This is differentfrom the situation when atoms like He are captured by in-tact C , where the increase in mass induces a well-definedenergy loss equal to E CM [43]. For He@C − formation,the knocked out carbon atom carries away some energyand energy losses smaller than E CM are therefore possi-ble. Knockout collisions without capture giving C − andC − have a broader range of possible energy transfers andare thus shifted by larger energies on average. Mean en-ergy losses from our MD simulations are in good agreementwith the experimental values for He@C − and C − . It isimportant to note that the MD simulations reproduce thedifferent dependencies of the center-of-mass collision en-ergy for both these fragments – the energy loss increaseswith decreasing collision energy for the C − fragment whilethe energy loss of the endohedral complex He@C − remainsclose to constant in both the experiment and in the simu-lations.The energy shifts of the experimentally observed He@C − and C − products relative to the expected energy of C − are similar to those of C − and He@C − , respectively. This5 (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:8) (cid:5) (cid:9) (cid:10) (cid:11)(cid:12)(cid:11)(cid:13)(cid:11) (cid:7) (cid:14)(cid:5)(cid:15)(cid:16)(cid:5)(cid:17)(cid:18)(cid:19)(cid:20)(cid:18)(cid:21)(cid:22)(cid:23)(cid:23)(cid:7)(cid:3)(cid:15)(cid:5)(cid:17)(cid:24)(cid:25)(cid:7)(cid:8)(cid:5)(cid:9)(cid:10)(cid:12)(cid:11) (cid:13)(cid:11) (cid:26)(cid:11) (cid:27)(cid:11) (cid:28)(cid:11)(cid:11)(cid:3) (cid:16)(cid:29)(cid:21)(cid:30) (cid:3) (cid:16)(cid:29)(cid:3)(cid:31) ! "(cid:21)(cid:30) ! " (cid:21)(cid:30) $ % & (cid:7) (cid:8) (cid:28)(cid:11) (cid:18) (cid:28)(cid:26) (cid:7) ’ ( (cid:12) (cid:7) (cid:10) (cid:11))(cid:11)(cid:28))(cid:11)(cid:12))(cid:11) (cid:3)(cid:31) (cid:21)(cid:30) Figure 5: Above: experimental and simulated cross sections for de-tection of daughter anions arising from non-statistical fragmentationin C − +He collisions. For the simulations, C − daughters predictedto be lost to electron emission by our statistical model are excludedfrom the cross section ( cf. Figure 5). The solid lines are fits to Eq.(3) which gives the threshold energy E th . Below: MD-simulatedmean He-C energy transfers h ∆ E He i in collisions leading to C fragments cold enough to survive the experiment. The solid lineis a power law fit; dashed lines illustrate the determination of thethreshold displacement energy T d . is consistent with our view that these products originatefrom C − → C − +C as a second step after knockout. OurMD simulations do not include statistical fragmentationprocesses such as C − → C − +C and we are thus unable tocompare simulation with experiment in this case.In the bottom panel of Figure 4, one can see that therelative intensities of the four anion fragment peaks (againextracted using Gaussian fits) vary little as a function ofthe collision energy down to 45 eV, beyond which we can-not resolve He@C − from the tail of the C − peak. Thissupports our interpretation that all four products arisefrom a common mechanism with a single threshold energy. As discussed above, the four different reaction chan-nels leading to the production of He@C − , C − , He@C − ,or C − fragments are all non-statistical in nature and canbe considered to be due to a common mechanism. Accord-ingly, we deduce the total absolute cross section for non-statistical fragmentation from our mass spectra by sum-ming the intensities of all four anion fragment peaks andrelating this sum to the C − total destruction cross sec-tion as described in Section 2.1. The resulting experimen-tal cross section for C − +He collisions yielding negativelycharged fragments is given in the upper panel of Figure 5. Also included in Figure 5 are results from our MD sim-ulations combined with our statistical model for delayedelectron emission from C − products. In the upper panelwe give the cross section for single carbon knockout and inthe lower panel the average energy transfer to the molec-ular system h ∆ E He i in collisions leading to C knockout.Included in both of these quantities are only those tra-jectories which leave the resulting C sufficiently cold tosurvive the timescale of our experiment. To calculate thesurvival probability of C − , we first determine the inter-nal energy of C at the end of each MD trajectory. Theheat capacity for C is found by scaling that of C bythe number of degrees of freedom, and we use a value of3.56 eV for the electron binding energy (Table 1). Due tothe higher electron binding energy of C − compared to C − ,our statistical model gives a higher internal energy cutoffof 24-25 eV for C − to survive until detection in the exper-iment. The effects of these considerations are negligiblefor E CM <
60 eV but there are significant corrections forhigher energies. For comparison to experiment, we do notneed to consider the competitive C − → C − +C channel, asC − is included in our total non-statistical product crosssection.The experimental and simulated cross sections are qual-itatively similar, albeit with an offset in the observed thresh-old energy for knockout. Such an offset has been observedpreviously in comparison between experimental and simu-lated knockout cross sections [17], and may be due to theapproximate, classical nature of the Tersoff potential usedto model the C-C bonds. To determine the threshold en-ergy for carbon knockout, we model the experimental andMD cross section assuming that the primary process is anelastic binary collision between a He projectile with ki-netic energy equal to the He-C − center-of-mass E CM anda free C atom at rest. This is justified in that the knockoutprocess happens on an ultrafast timescale on which the re-maining 59 C atoms are standing still. For such collisions,Chen et al. [44] give the following expression, based onLindhard scattering theory [45], for the cross section σ KO leading to energy transfer above a given fixed value (herethe displacement energy T d ): σ KO = A/E CM π arccos − ( p E th /E CM ) − , (3)where E th is the minimum center-of-mass energy (for theHe-C system) required to transfer T d . We take A and E th as fit parameters, yielding E expth = 35 . ± E MDth = 41 . ± . E th . There is some (barely sig-nificant) deviation close to threshold, which probably dueto the fact that T d is not single-valued, but rather variessomewhat with respect to the angles between the impartedmomentum and the molecular bonds [13, 14], as well aswith the stretching of the bonds due to molecular vibra-tions [15].6 igure 6: Histogram of displacement energies from projectile-freeMD simulations. The solid line is a fit to the sum of two Gaussiansdescribing a bimodal distribution. The threshold energy E th is projectile dependent [12];to obtain the intrinsic threshold displacement energy T d weneed the energy transferred from the projectile (He) to thetarget at threshold, which we extract from our MD simula-tions. From a power-law fit to h ∆ E He i = c × E CM P we ob-tain the mean energy transfer at the experimental knock-out threshold E CM = E expth . This is our semi-empiricalvalue for the C → C +C threshold displacement en-ergy, T SEd = 24 . ± . c and P , and E expth . Our semi-empirical value of T d is much higher than generally assumed previously forfullerenes (around 15 eV [18, 19, 20, 2]), and is similar tothose measured for the planar sp − hybridized carbon sys-tems graphene (23.6 eV [15, 16]) and PAHs (23 . ± . T MDd = 26 . ± . ± ±
4. Conclusions
This is, to our knowledge, the first determination ofa threshold displacement energy for free fullerenes or forany type of fullerene based material. This intrinsic mate-rial property is a key parameter for modeling radiationdamage in many contexts, for example during electronmicroscopy imaging [15] or gas-phase reactions in the in-terstellar medium [12]. Finally, the surprising observa-tion of the endohedral defect fullerene complex He@C − is a remarkable testament to the intriguing complexity offullerene reactions, which continue to fascinate more than30 years after their discovery.
5. Acknowledgments
This work was performed at the Swedish National In-frastructure, DESIREE (Swedish Research Council Con-tract No. 2017-00621). It was further supported by theSwedish Research Council (grant numbers 2014-4501, 2015-04990, 2016-03675, 2016-04181, 2016-06625). See Supple-mental Material for videos of selected MD simulations.
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