1 Introduction

It is a pleasure to contribute to this issue dedicated to the life and work of Vincent McKoy at Caltech, USA. His contribution to the first modern computational approach, namely the Schwinger multichannel (SMC) method, for electron–molecule collisions is the basis of our understanding of this field today, beginning with his two seminal papers [1, 2] in 1981 and 1984. This method has its origins in the Schwinger variational (SV) method for solving problems involving potential scattering; in particular, the SV method is modified to treat the multicenter nature of polyatomic targets. As an illustrative example, in this report, the earlier measured elastic DCSs of \(\hbox {C}_{60}\) [3, 4] are re-normalized and compared with the theoretical results of Winstead and McKoy [6] that were obtained by using the SMC method. They found reasonably good agreement with the only previous high-level calculation by Gianturco and colleagues [7], which has prompted us to revisit the results from the previous \(\hbox {C}_{60}\) paper. The re-normalized elastic differential cross sections (DCSs) are subsequently used to place an absolute scale on the inelastic scattering cross sections of vibrational and electronic excitations, with the latter being compared to photoabsorption cross sections. Relating to low-energy electron spectroscopy, some coexisting solid-state phase features, which emerged in \(\hbox {C}_{60}\), are summarized for inelastic electron scattering processes.

As is commonly known, due to the establishment of a method of synthesizing \(\hbox {C}_{60}\) for mass production [8], there were commercially available sources of \(\hbox {C}_{60}\) in the early 1990s. That was 5 years after the first laboratory observation of \(\hbox {C}_{60}\) [9] and has boosted fullerene research. Although those results are not discussed in detail here, we note that: (1) natural fullerenes, \(\hbox {C}_{60}\) and \(\hbox {C}_{70}\), were discovered in shungite, an unusual carbon-rich rock found in Russia, and believed to be more than 600 million years old [10]. That discovery means that fullerenes can no longer be thought of as being purely artificial materials made in the laboratory using laser vaporization, carbon arcs, or combustion processes; (2) after successive observation efforts in the interstellar medium (ISM), or from one of the interstellar fine particles, fullerene was finally detected (see Fig. 1) in a planetary nebula by NASA’s Spitzer Space Telescope [11,12,13], and (3) more recently, the presence of \(\hbox {C}_{60}^{+}\) was confirmed using NASA’s Hubble Space Telescope in diffuse interstellar clouds (DICs), shedding light on the neutral and ionized clouds of the ISM [14]. Note that the DICs were discovered by Mary Lea Heger about 100 years ago, in 1919, and reported in the literature a few years later [15]. These observations were very similar to those determined in the latest laboratory experiments [16], although there remains a controversy over some issues. Nonetheless, the \(\hbox {C}_{60}\) molecule has been used by astronomers to probe and model the conditions of astrophysical environments, and ground-based quantitative and qualitative data are increasingly required now for AMO physics.

Fig. 1
figure 1

Buckyballs in Space (PIA13288, Image credit: NASA/JPL–Caltech) [11,12,13]

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In the early 1990s, electron inelastic scattering experiments from gas-phase \(\hbox {C}_{60}\) were performed to investigate electronic excitations [3, 17,18,19,20,21], electron attachment processes [22,23,24,25], and photoabsorption and photoionization data, as compiled in 1999 [26]. However, “elastic” electron scattering experiments from \(\hbox {C}_{60}\) have been limited to only two reports: one for electron impact energies below 11.5 eV [3] and another for energies above 100 eV [27]. In contrast, theoretical studies have been undertaken by several authors using the Born approximation for the intermediate- and high-energy region [28, 29], ab initio calculations [7, 30, 31], and a semiempirical approach, and they have been performed for some time now [32]. Below 10 eV, however, most of the calculations have been focused on finding the resonances expected from this typically high-symmetry and large molecule, and they are also related to studying the existence of very long-lived metastable anions produced by electron impact [22, 24]. Comparison between theoretical and experimental results had not previously been carried out for the angular distributions, because the differential cross sections were measured on a relative scale over a limited range of scattering angles.

As noted earlier, in this report we revisit the published results of crossed-beam experiments for elastic scattering of low-energy electrons from gas-phase \(\hbox {C}_{60}\). With the help of several calculations reported after the publication of the original data, the new normalized data are interpolated, and so can be visualized into 3D plots. These plots indicate more comprehensive and distinctive features of electron scattering from the cage-like fused-ring structure (truncated icosahedron) that is \(\hbox {C}_{60}\). Note that no re-measurements have been newly performed here for low-energy elastic electron scattering from \(\hbox {C}_{60}\). Such additional data would be required on an absolute scale, and over wide energy and angle ranges, to enable a valid comparison with theory, which is beyond the scope of this paper.

In the following section, we will outline our experimental method and data analysis, while in the results and discussion, the re-normalized elastic differential cross sections are presented in a 3D format, and some selected DCSs are also compared with the available theoretical results. Inelastic and elastic scattering data are revisited, for examining coexisting features of gas- and solid-state phases in the \(\hbox {C}_{60}\) atomic system. Finally, a relevant statement by V. McKoy is quoted as part of our concluding remarks.

2 Experimental method and data analysis

2.1 General experimental observations

The experimental apparatus has been described previously [33]. Briefly, the spectrometer consists of a hemispherical monochromator and a matching energy analyzer system, separated by an effusive gas oven (EELS). Electrons from the monochromator, collimated by an input lens, intercept an effusive molecular beam of \(\hbox {C}_{60}\) emitted from the oven. Scattered electrons are subsequently extracted by an achromatic lens system, before dispersion in the analyzer field. In short, the entire electron optics were designed by beam tracing methods, and they use computer-controlled voltages which follow the energy sweeps in order to keep the apparatus transmission constant. Experiments were performed with an overall energy resolution of about 50 meV for elastic scattering and inelastic electronic excitation, in order to increase sensitivity, while the energy resolution was about 34 meV for vibrational excitation. With the present energy resolution, rotational and some of the vibrational excitation modes could not be resolved and are included in what we designate as the “elastic” peak (see below).

The oven [3] is made of solid Cu, with a 4-mm-long nozzle of 0.6 mm diameter drilled through its conical top. A small pipe pierces the oven, through which a sample gas is introduced from below, so that He can be admixed as a carrier gas from the normal gas inlet system, for setup and calibration purposes. Our experiments were performed at oven temperatures in the range from 440 to \(480^{\circ }\hbox {C}\). Note that for some of the experiments described later, the carrier gas was turned off.

The head pressure behind the nozzle was typically about 5.66–6.44 Torr, for He monitored by a Baratron gauge, wherein the condition for a “single collision” environment is also guaranteed in our experiment. Note that these experiments were started with the apparatus at room temperature, so that \(T \approx \) 300 K. Further note that the backing pressure of He was 1.52 Torr at the rear of the nozzle. The Baratron reading eventually reached up to about 5.66–6.44 Torr by heating the oven at \(T \approx \) 760 K, while its flow rate (4.478 \(\hbox {x10}^{-17}\) mol. \(\hbox {s}^{-1})\) was verified not to change by monitoring the flowmeter and variable leak valve installed in the inlet gas line. The peaking factor [3] was also considered for the gas beams emanating from the nozzle, when applying the pressure ratio formula for the conversion of the DCSs.

The fullerene used in the original measurements was obtained from Texas Fullerenes Corp., right after becoming commercially available in early 1992. It was specified as 80% pure with up to 20% admixture of \(\hbox {C}_{70}\). It also contained about 5% water [thermal analysis: Rigaku TAS 200]. Sublimation at lower temperatures most likely enhances the ratio of \(\hbox {C}_{60}\) to \(\hbox {C}_{70}\), with the three distinct \(\hbox {C}_{70}\) absorption peaks at 2.64 eV, 3.28 eV, and 5.25 eV only appearing at elevated oven temperatures. Thus, any contamination was readily detectable in our experimental arrangement and either vacuum sublimation or heating and pumping was found to eliminate the impurity spectrum as follows. Based on these sample characteristics, water and volatile contaminants were removed by baking at \(220^{\circ }\hbox {C}\) for several hours under a steady stream of He and with a baking pressure of 0.85 Torr. Sublimation, as observed by the onset of electron scattering processes, began at \(420^{\circ }\hbox {C}\), which agrees well with vapor pressure measurements but differs considerably from the \(300^{\circ }\hbox {C}\) reported for film deposition [34].

Note that the monochromator and analyzer (both within electric-field shielding boxes) are differentially pumped by turbomolecular pumps, in order to reduce the deleterious effects of corrosive gases and instabilities in the electron gun due to changes in the space charge distribution. However, all exposed surfaces, including the entrance optics of the monochromator and analyzer, received an ever-growing coat of \(\hbox {C}_{60}\) film. In other words, these measurements were subjected to the extremely “dirty” environment of vapor experiments. Errors in the elastic cross sections due to this ongoing coating were estimated to be at about 20–30%, but this should not influence the number or position of the peaks for the inelastic electron spectroscopy.

2.2 Data analysis

In the original report, we presented the data in a DCS versus Energy (eV) space format, viz. the elastic spectra that were plotted as a function of impact energy, 1.5 to 11.5 eV, and at specific scattering angles in the range \(30^{\circ }\) to \(90^{\circ }\), for each angle step of \(10^{\circ }\), as recorded at \(420^{\circ }\hbox {C}\). All were measured without the He carrier gas, and the separately recorded background was subtracted out. The intensity ordinate was calibrated only in terms of counts per sweep, for easier comparison of the “relative” intensities. In this report, those plots are smoothed and digitized by using the approximate curves of polynomials as determined by an iterative method based on a least-squares method. By doing so, our re-normalized cross sections are now visualized in the form of three-dimensional (3D) plots, viz.    in DCS-Energy (eV) and DCS-Scattered Electron Angle (\(\theta \)) space.

2.3 Re-normalization procedures

We estimated previously that the absolute elastic DCS at 7 eV and \(30^{\circ }\) was of the order of \(410 \times 10^{-16}\) \(\hbox {cm}^{2}\) \(\hbox {sr}^{-1}\), but that could be off by a factor of 2 or 3. Note that a lengthy estimate of the absolute size of the peak at 7 eV in the \(30^{\circ }\) trace can be found in the appendix of the original paper [3]. Thereafter, however, the theoretical results have claimed such a cross section to be two orders of magnitude smaller, and this would need to be confirmed by additional measurements. In fact, we noticed a similar issue in the available photoabsorption measurements. Prior photoabsorption measurements [26] have utilized vapor pressures between 4.6 and 12 times lower as we originally did, and as a result, the corresponding photoabsorption cross sections were “higher” in magnitude by approximately these factors.

So in the present re-estimate of the absolute elastic DCS at 7 eV and \(30^{\circ }\), we utilize the same data obtained from the determination of the elastic DCS on a simultaneous observation of the elastic peak and the vibrational peak at 180 meV. Independent measurements at \(T =\) 760 K gave intensities of 6360 : 41 counts for the He \(+ \hbox {C}_{60}\) mixture, and 4423 : 52 for pure \(\hbox {C}_{60}\), after the tail and background contributions were removed. Since He has no features at an energy loss of 180 meV, we can deduce the relative contributions to the elastic peak of the He \(+ \hbox {C}_{60}\) mixture as \(I_{He}= 2875\) counts, \(I_{C_{60}} = 3845\) counts (cf. Fig. 4). The observed intensities for He and \(\hbox {C}_{60}\), relative to their respective differential cross sections (DCSs), are given as a single formula in the relative flow method [38, 39]:

$$\begin{aligned} \frac{{DCS}_{C_{60}}}{{DCS}_{He}}=\frac{I_{C_{60}}}{I_{He}}\frac{Q_{He}}{Q_{C_{60}}}\sqrt{\frac{m_{He}}{m_{C_{60}}} }\frac{\chi _{He}}{\chi _{C_{60}}}, \end{aligned}$$
(1)

where Q (\(\propto P\): pressure) is the relative flow rate, m is the molecular mass, and \(\chi \) is the peaking factor. For further details on each factor, see the Appendix in the original report [3]. But in this report, the earlier vapor pressure measurements of \(\hbox {C}_{60}\) are reviewed [26]. We now employ a more reliable vapor pressure of \(2.99 \hbox { x}10^{-3}\) Torr, re-estimated by the improved vapor pressure curve of \(\hbox {C}_{60}\) [35] at 760 K. This is a crucial point, as Berkowitz indicated [26] also in the photoabsorption cross sections. The accuracy of the elastic DCS data depends intimately on the validity of the independently sourced \(\hbox {C}_{60}\) vapor pressure curves. The observed counts of elastically and inelastically scattered electrons were converted into absolute cross sections by the known differential elastic [36] and inelastic (\(2^{1}\hbox {P}\)) [37] cross sections (DCS) of He, as appropriate, and we have used the pressure ratio formula for the conversion of the elastic DCS developed at JPL and Caltech [38, 39]. As a result, the original elastic cross section is reduced by a factor of about “11.” The errors of 2 - 15% are estimated from the count rate fluctuations on the smoothed curves, and finally, a combination of all the uncertainty sources leads to total errors of the order of 20-35%.

3 Results and discussion

Some results have been described already in the original report [3], and so are only briefly summarized as follows.

3.1 Elastic spectra

Figure 2 shows our re-normalized cross sections visualized in the form of three-dimensional (3D) plots, viz. in DCS-eV and DCS-\(\theta \) space. By doing so, we illustrate the more comprehensive and distinctive features of electron scattering from a cage-like fused-ring structure (truncated icosahedron) that is \(\hbox {C}_{60}\). The impact electron energy ranges from 1.5 to 11.5 eV, in steps of 0.25 eV, and for scattering angles from \(30^{\circ }\) to \(90^{\circ }\) in steps of \(10^{\circ }\). The points are connected by straight-line segments corresponding to the present digitized DCSs, but applying our re-normalized absolute scale. In Fig. 3, we also compare a few selected angular distributions with the ab initio theoretical results available since the publication of the first experimental data at 4, 7, and 10 eV [3].

Fig. 2
figure 2

Re-normalized elastic DCS of \(\hbox {C}_{60}\), as visualized in the form of three-dimensional (3D) plots, viz. in DCS-eV and DCS-\(\theta \) space

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The known diameter of the \(\hbox {C}_{60}\) molecule is 7.1 Å, which is comparable to the de Broglie wavelength of an electron (8.8 Å) at 2 eV. In view of the electron scattering kinematical aspects, a clear diffraction pattern is expected to be produced due to interference phenomena in the scattering from the shell of the \(\hbox {C}_{60}\) cage, which is overlapped by another intermolecular multiple scattering feature from each C-atom. Chemical bonding usually results in a considerable anisotropic distortion of the electron charge distribution, and so there are many partial waves contributing to the forward scattering peak even in lower energy region of a few eV. As seen below, however, the dynamical scattering aspects complicate the situation more for such a low-energy region, where the interaction time is relatively long.

Previously, because no ab initio theoretical calculation was available for comparison, the general features were only briefly discussed in scattering simulations, by using a spherical potential of the Woods–Saxon type [3] and by adjusting the radius of \(\hbox {C}_{60}\) to be 3.5 Å. This was based on the idea of electron scattering and nuclear structure by Hofstadter [5]. Since then, however, there has been a remarkable improvement in the number of available theoretical treatments, as described below.

In Fig. 2, the regular positions of the maxima and minima in the spectra are overtly sketched in the present 3D plots. These plots give a perspective of the scattering phenomena which is much clearer than the previous presentation in simply the DCS versus eV space. Most of the elastic scattering stands out at around \(40^{\circ }\), like a mountain range, in the lower energy region, although we must note that the scattering angles were limited to the \(30^{\circ }\) to \(90^{\circ }\) range here. Two weak “bumps” are also visible with \(l= 5\) characteristics, around the 7eV region due to a shape resonance which can also be more clearly visualized in the 3D plots. This latter feature is hardly seen in the forward direction because of the effects of the many partial waves aggregating. Another feature is seen near the 2eV resonance that is predicted theoretically, which alters the general characteristics of the DCS for elastic scattering. A new maximum also appeared at around \(60^{\circ }\) and was clearly shifted from the main diffraction patterns below 4 eV.

As shown in Fig. 3, the calculations [6, 7, 39] indicate that the intensity of the elastic DCSs (of the order of \(\sim 300\hbox {--} 1000 \times \) \(10^{-16}\) \(\hbox {cm}^{2}\)/sr at \(0^{\circ }\) scattering angle) is peaked strongly in the forward direction and then declines to a minimum, like a valley, at around \(30^{\circ }\). As the electron scattering angle increases, further strong oscillatory structure is found right out to \(180^{\circ }\). Gianturco et al. [7, 30, 31] performed the first ab initio calculations, using a non-empirical model interaction constructed from first principles and solving the body-fixed coupled scattering equations for the collision processes. They provided, for the first time, both integral and differential elastic cross sections and suggested the presence of numerous metastable states. The energy range of their studies was covered from 0 to 30 eV. Their comparison was carried out very carefully with the original experimental results, which were only available at fixed angles and within the same range of collision energies. Two of the observed resonant features were identified as an \(h_{u}\)-symmetry shape resonance at around 2 eV, and a composite shape resonance of three different symmetry components (\(g_{u}, t_{1u}\), and \(t_{2g}\)) between 5 and 7 eV. Those features are more clearly confirmed in the present 3D plots, as described above. The magnitude of the calculated elastic cross sections was, as expected, fairly large for such a large molecule, being predominantly due to its large static dipole polarizability. Further, the question of the original absolute scale of our differential cross sections was addressed by their study.

Subsequently, Winstead and McKoy [6] reported cross sections for elastic scattering of low-energy electrons by \(\hbox {C}_{60}\), calculated within the fixed-nuclei-static-exchange approximation in the framework of their SMC approach. They took advantage of the high parallel efficiency of the SMC method, which permitted the use of a quite large basis set to describe the high symmetry of \(\hbox {C}_{60}\). There was reasonably good agreement between the high-level calculations, considering the limitations of them both. It was noted that the resonance positions are shifted downward, compared to those of the static-exchange resonance energies, once the target polarizability is properly accounted for. In addition, the structure of the cross section was analyzed with reference to a simple spherical-shell model which demonstrated many features of the static exchange results. Both theoretical results closely emphasized the importance of the polarizability. As was the case with the results of Gianturco et al. [7, 30, 31], the original DCSs were also two orders of magnitude larger than those from the SMC, and McKoy [6] suggested again that additional measurements on an absolute scale, and over wider energy and angular ranges, were needed.

Fig. 3
figure 3

Elastic DCSs for electron scattering by \(\hbox {C}_{60}\) at 4, 7, and 10 eV

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Present re-normalized angular distributions (\(\bullet \)) are compared with theoretical DCSs [6, 7, 39] now available in the literature. See also legend in figures.

Although simple, but nonetheless useful, semiempirical approaches have been applied for elastic scattering by Amusia and co-workers [32], in which the complicated and highly polarized \(\hbox {C}_{60}\) target was modeled by a rectangular (using radial coordinates) square well potential, in combination with a static polarization potential. More recently, those calculations were extended to use the Dirac partial wave methodology by Dubey and Jose [40], which qualitatively supports the other theoretical results as shown in Fig. 3. All the available theories are more oriented to the traditional physics approach, i.e., considering the concept of scattering from a model potential. For more details on those four aforementioned theoretical approaches, see their original papers.

In this report, reanalysis is performed on the previous data, in detail over the whole of parameter-space as mentioned above, and the resulting re-normalized DCSs are compared with those theoretical results in Fig. 3. Finally, we observe that quantitatively good agreement between the experimental and theoretical results is found. The relative flow technique we used to set the absolute DCS scale was originally developed at JPL, Caltech, and has yielded reliable data, specifically, elastic scattering DCSs, for many molecules [41, 42]. Furthermore, the elastic DCSs can be measured “simultaneously” with the inelastic DCSs, i.e., they can serve as a calibration standard for the absolute determination of the inelastic DCS, as discussed shortly below.

According to Inokuti [43], for cross section data to be applicable to any applied or practical problems, they must fulfill the threefold requirement that the data be correct, absolute, and comprehensive. This article shows a typical computational and experimental collaboration for \(\hbox {C}_{60}\), which relies on the fact that when it is experimentally difficult to obtain cross sections, the computational approach may be applied to estimate that cross section, or vice versa.

3.2 Inelastic spectra

Since the first inelastic electron spectroscopy was performed for gas-phase \(\hbox {C}_{60}\) by Burose et al. [18], three measurements [3, 19, 20] were subsequently carried out at around the same time. These were specifically for “low energy electron inelastic” scattering from molecular \(\hbox {C}_{60}\) under single collision conditions and below 100 eV.

3.2.1 Vibrational excitation

Jiggling \(\hbox {C}_{60}\) in space was detected using NASA’s Spitzer Space Telescope [11], by analyzing the infrared (IR) light from Tc 1, a planetary nebula consisting of material shed by a dying star. \(\hbox {C}_{60}\) jiggles, or vibrates, in a variety of ways – 174 modes to be exact. Four of those modes (526 \(\hbox {cm}^{-1}\), 574 \(\hbox {cm}^{-1}\), 1182 \(\hbox {cm}^{-1}\), and 1429 \(\hbox {cm}^{-1})\) cause the molecules to either absorb or emit infrared light, and all four modes were identified by the Spitzer Telescope. Thus, the ground-based data are increasingly required now for AMO physics and relevant future modeling studies.

Fig. 4
figure 4

Vibrational electron energy loss spectrum (EELS) of \(\hbox {C}_{60}\) at an impact energy of 7 eV. The broken curve is the vibrational component, riding on the estimated tail of the elastic peak. The gas-phase result is compared with the solid film high resolution EELS spectrum at 3.7 eV for \(90^{\circ }\) from ref. [35]. Some peaks are not identified (indicated as “?”). The normalized absolute scale is derived for 7 eV and at a scattered electron angle of \(50^{\circ }\)

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A typical energy-loss spectrum for vibrational excitation of gas phase \(\hbox {C}_{60}\) is compared with the solid film EELS spectrum [45] in Fig. 4. The broken grey lines in that figure represent the vibrational component, as obtained by subtraction of the tail of the elastic peak. We combined the traces from the \(30^{\circ }\), \(40^{\circ }\), \(50^{\circ }\), and \(60^{\circ }\) observations, because they show very little angular dependence. A weak shape resonance is expected at 7 eV, where a broad bump is seen with a \(\ell =5\) character as in Fig. 2. The present measurements, covering several months of observation but still retaining a combined resolution of 34 meV and an incident electron beam of intensity of about \(10^{-9}\hbox {A}\) (\(\sim 17\,\hbox {meV}\) and \(\sim 10^{-10}\hbox {A}\) at best), were usually operationally obtained. Note that in order to maintain “single-collision” conditions, the pressure of the gas target must be kept at a maximum of \(10^{-3}\)–\(10^{-4}\) Torr. This situation is in sharp contrast with the high-resolution EELS (HREELS) of a solid surface [8, 44], which has a much higher atomic-target density, and hence readily accomplishes an energy resolution of just several meV.

IR absorption lines are expected at 65.3 meV, 70.7 meV, 145 meV, and 174.4 meV [45]. As compared with our high-resolution spectra in Fig. 4, two dotted peaks contained these features and some unresolved optically forbidden transitions (Raman absorption is expected to occur at 34, 54, 61.5, 88, 96, 136, 155, 177, 182, and 195 meV) [46, 47]. Figure 4 contains a further feature at around 350 meV (the compound peak of 330 meV and 380 meV at \(50^{\circ }\)) and another at 380 meV, which are shifted from the small bands of the solid film at 260 meV and 370 meV, respectively. In terms of an “electron-molecule collision,” these would be identified as “a harmonic” phenomenon, as expected often with shape resonances (i.e., assigned to the \(g_{u}, t_{1u}\), and \(t_{2g}\) symmetry) [6, 30]. In addition, there may be a small “superelastic peak” at around −80 meV, on the left tail of the peak extending from −60 meV to −150meV. A very faint combination band may also exist at 450 meV, but this has not been observed with the solid film. On the other hand, the combination band found at 250 meV with the solid is absent from our observations.

Within the relatively poor resolution of 34 meV, for our experiments, there seems to be no shift in frequency from the photoabsorption data of gas-phase \(\hbox {C}_{60}\). As shown in Fig. 4, however, the peaks in the high-resolution spectra from solid \(\hbox {C}_{60}\) film on Si (100) correspond more or less to the Raman absorptions. For further details, Grenterblum et al. [43] have given a detailed analysis of the single vibrational features. In this report, the absolute scale is placed to the vibrational loss peaks (see middle graph in Fig. 4), using the re-normalized elastic DCS at 7 eV and \(50^{\circ }\). We base the determination of the inelastic DCSs on a simultaneous observation of the elastic peak and the main vibrational peak at 180 meV, which gave intensities of 4423:52 counts for pure \(\hbox {C}_{60}\) after tail and background removal.

Note that the reduction in the stretching frequency of the terminally bound \(\hbox {C}_{60}\) on film (i.e., weak van der Waals aggregates of \(\hbox {C}_{60})\), from the value observed for the gas phase, can be explained in terms of the \(\hbox {C}_{60}\) - substrate bonding nature [48]. In addition, there are 6 degrees of freedom of motion involved in the lattice vibration, as well as the rotation even in the condensed system of \(\hbox {C}_{60}\), and thus, as might have been supposed, the unusual carbon-rich rock has been used as lubricators.

However, in contrast to our experiment, as indicated in the lower plot of Fig. 7, almost no excitation of purely vibrational states was observed below the first electronic transition at 2.2 eV when using the constant residual energy loss mode [19, 21]. Those workers [19,20,21] speculated that very long-lived metastable states of \(\hbox {C}_{60}^{-}\) are produced by direct interaction of electrons from 0.2 to 8 eV, which survive for a while with the trapped electrons, and finally decay by releasing an electron, “like elastic scattering,” only after the \(\hbox {C}_{60}^{-}\) ions have thermally drifted away from the collision region. The efficiency of the energy migration increases with the density of the internal modes (the rotation and vibration motions), and hence, the energy migration becomes significant in large molecules [48]. Note that, as shown in Fig. 7, the impact energy \((E_{0})\) was swept from 0.08 to 10 eV, for residual energies \(\varDelta E_{r}\) of 0.08 to 1 eV and of 20 eV. Therefore, their impact energies were supposed to cover the incident energies of 3.7 and 7 eV of Fig. 4. Further note their energy loss regions are truncated below 1 eV, and hence, no evidence is presented for vibrational excitations.

3.2.2 Electronic excitation

The \(\hbox {C}_{60}\) molecule, the largest known molecule in space, has been used by astronomers to prove and model the conditions of astrophysical environments, where “ultraviolet radiation” is, in general, high and “electron” density is estimated to be of the order of 0.037/cm\(^3\) [49]. Being such a large molecule, another more interesting aspect of \(\hbox {C}_{60}\) is the possibility of the existence of collective “plasmon” states in “gas-phase \(\hbox {C}_{60}\)” which are excited by fast charged particles. Refer to Li et al. [50], wherein historical perspectives on this are well summarized. \(\hbox {C}_{60}\) is a typically high symmetry-large “molecule,” with 240 valence electrons occupying the \(\sigma \) and \(\pi \) orbitals of a cage-like fused-ring structure composed of twenty hexagons (\(\textit{sp}^{2}\)) and twelve pentagons (\(\textit{sp}^{3})\). The loosely bound electrons in the valence and conduction bands can “probably” oscillate “collectively” around the quasi-stationary sets of nuclei, producing oscillating polarization that can be described by a quasi-particle, called a “plasmon,” in analogy to those in the “condensed-phase” of C-atoms. In the free electron gas model, simple considerations give a plasmon excitation energy of \(E_{p} = (4\pi \textit{ne}^{2}\hslash ^{2}/m)^{1/2}\), where n is the electron density and the other symbols have their usual meaning. Using the density of solid \(\hbox {C}_{60}\), 1.678 \(\hbox {g/cm}^{3}\), the plasmon energy is calculated to be 21.5 eV.

Since the mass production of \(\hbox {C}_{60}\) became feasible, the study of collective excitations, referred to as “plasmonics,” in nanometer-sized structures, i.e., carbon nanotubes and graphene, has become of great importance in many contemporary applications [51].

3.2.3 VEELS spectra from graphite, C\(_{\mathbf {60}}\) film on C, and gas-phase C\(_{\mathbf {60}}\)

In the original paper [3], we discussed in detail the \(\uppi \) and \(\uppi + \upsigma \) plasmons (or interband plasmons) of the \(\hbox {C}_{60}\) film in high-energy EELS as well as in the photoabsorption and emission experiments [3]. However, one experiment [53] was missed in that study and so we revisit those discussions of the high-energy EELS data below.

In Figs. 5 and 6, the valence electron energy loss spectra (VEELS), for energy loss processes less than 50 eV, corresponding to the UV and VUV region, are overviewed for diamond, graphite [54], \(\hbox {C}_{60}\) film on a supporting C-film [53], and gas-phase \(\hbox {C}_{60}\) [3, 19, 52]. In diamond, two broad and intense energy losses are centered at 23.5 eV and 33.3 eV, which are assigned as a possible surface plasmon \((E_{s}=\sqrt{E}_{p})\) and a bulk plasmon \((E_{\mathrm{p}})\), respectively. On the other hand, plasmon losses in graphite [54, 55], at 7 and 27 eV, are assigned as \(\uppi \) and \(\upsigma + \uppi \) plasmons. The graphite species, being a model system for this class of materials, has been most extensively studied with respect to its electronic structures, both experimentally and theoretically [56].

Fig. 5
figure 5

Typical VEELS spectra from allotropes of carbon: namely diamond, graphite, condensed-phase \(\hbox {C}_{60}\) on C-film, and gas-phase \(\hbox {C}_{60}\) at \(\theta = 3^{\circ }\) and \(5^{\circ }\)

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A molecule of \(\hbox {C}_{60}\) represents an intermediate form between diamond and graphite. Kuzuo et al. [53] observed, for the first time, two interband plasmon peaks (see arrow lines in Fig. 5) at 6.4 and 25.5 eV in “solid \(\hbox {C}_{60}\)” (thickness of 50 to 100 nm) on a supporting film of carbon. They used a Transmission Electron Microscope (TEM-JEM 1200EX, energy resolution: 20–70 meV, and incident energy of 60 keV) equipped with double-focus Wien filters, and their spectra show differences when compared to that of graphite (see Fig. 5). Indeed, we find that there are altogether 9 structures (see vertical lines) originating from the interband transitions of valence electrons to the \(\uppi ^{*}\) and \(\upsigma ^{*}\) bands. The former shows typical \(\uppi -\uppi ^{*}\) interband transitions, between the well-separated molecular levels having \(\uppi \)-electron character, as shown in Fig. 5 . Note that those plasmons are analyzed more rigorously in the single scattering distributions (SSD), viz. when the multiple scattering phenomenon, inevitable in solids, is removed from the observed spectrum by applying the Fourier-Log deconvolution method [57]. Furthermore, Kuzuo et al. [53] confirmed those features in terms of the real and the imaginary parts of the dielectric functions derived from a Kramers–Kronig analysis [55]. Note that some authors restrict the term plasmon to the case where \(\varepsilon _{1}\), of \(\varepsilon (\omega ) = \varepsilon _{1} + i\varepsilon _{2}\) passes through zero, also called the plasmon “of pure breed.” This criterion is satisfied sufficiently in graphite, but an “almost” zero crossing is probed in solid \(\hbox {C}_{60}\).

In Fig. 5, spectra, very similar to that from solid-state \(\hbox {C}_{60}\), are also presented for the electron energy loss from “gas-phase \(\hbox {C}_{60}\),” with an energy resolution of 1.2 eV [52]. Those spectra are reminiscent of molecular aggregation effects, which could occur even in dense gas-phase \(\hbox {C}_{60}\), when compared to the other three spectra for a free \(\hbox {C}_{60}\) molecule in Fig. 6. However, the features of the electron energy loss spectra for the dense-phase \(\hbox {C}_{60}\) case are observed to vary clearly along the dotted curve [55], when the scattering angles increase from \(3^{\circ }\) to \(5^{\circ }\) (only depicted two angles from \(3^{\circ }\) to \(13^{\circ }\) by one degree step) as shown in Fig. 5.

Fig. 6
figure 6

Wide-range energy loss spectra for a free \(\hbox {C}_{60}\) molecule at the pseudo-photoabsorption impact energy of \(\sim \) 1 keV and at 100 eV

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The existence of collective states, plasmons, in metals is generally accepted, although some interpretation problems still persist [58]. The broadening in the energy loss spectra, such as here, has been explained conceptually as being due to the interactions between \(\hbox {C}_{60}\) molecules in the condensed-phases (and in liquids). However, many kinds of condensed-phase effects are involved [59, 60] in the connection of atomic and molecular physics and condensed-phase physics, including the occurrence of molecular-aggregation effects. When compared with experimental results for gas-phase \(\hbox {C}_{60}\), we wish to note again that readers should be aware of the requirement that it is necessary to ensure only “single collisions” occur with gas targets, which depends strongly on the gas pressures as mentioned earlier.

The original gas-phase data spectrum, under single-collision conditions, is very similar to that of Burose et al. [18] (1.1 keV, \(\theta =0^{\circ }\) ) except that they observed a plateau over 10–18 eV energy loss, whereas the original data drop continuously, as shown in Fig. 6. Beyond the ionization energy—note the clear threshold at 7.58 eV [61]—there is a broad feature at 9–12 eV energy loss, part of which has also been seen in the scattering from solids at 9.3 and 10.2 eV. Photoelectron measurements [62] on the valence band of solid \(\hbox {C}_{60}\) show equivalent (but much clearer) peaks at 9 eV, 11.2 eV (broad), 12.4, 13.7, and 15.6 eV, which are strongly correlated with the highest occupied \(\upsigma ^{*}\) orbitals of molecular \(\hbox {C}_{60}\). Keller and Coplan [17] confirmed also a narrow peak at 6 eV, in the gas-phase experiment, and a very broad peak at 22 eV. They explained these features as arising due to plasmon resonances. This latter feature probably corresponds to the peak observed at 18 eV for the 1.1 keV and 0° spectrum of Burose et al. [18]. Those pseudo-photoabsorption spectra supported a “giant” plasmon excitation at 20 eV in the VUV photoionization of gas-phase \(\hbox {C}_{60}\), as observed by Hertel et al. [63]. However, in contrast to our discussion of plasmons that was based on a model originally applied to metals, the more traditional molecular orbital approaches have also been proposed, according to a large concentration of orbitals with binding energies between \(\sim \)13–35 eV, containing \(\sim \)200 of the 240 valence electrons. The transition probabilities, from these orbitals to the ionization continuum, would therefore generate a superposition of many transitions composing this broad resonance [64].

The only remaining feature in our spectrum is the small peak at 28.3 eV, the position of the “\(\upsigma + \uppi \) plasmon” as observed in the solid and gas phases between \(\sim 25.5\,\hbox {eV}\hbox { and }\sim 29.2\,\hbox {eV}\) using high-energy EELS (see Fig. 5). However, those peaks are much wider (\(\approx \) 8–10 eV), while ours are smaller and narrower, being a few eV wide. The debate on the molecular or condensed-phase nature of the various plasmon modes, as well as the complete deconvolution of single particles and collective structures, will have to await even more detailed research.

Finally, we wish to note that “polymerization and depolymerization” reactions of \(\hbox {C}_{60}\) molecules, coalescence, like the peanut-shaped \(\hbox {C}_{120}\) [65], are induced by the incident electrons and the subsequently produced secondary electrons in condensed-phase \(\hbox {C}_{60}\) [66], and thus, the relevant electronic properties might be changed from those of molecular \(\hbox {C}_{60}\) [67]. For more instructive information, see Fig. 1 in reference [68]. It may also reflect on the shifting observed in the vibrational excitation spectra of the \(\hbox {C}_{60}\) film as discussed above.

3.2.4 Interband transitions of valence electrons to the \(\uppi ^*\) band

Fig. 7
figure 7

EELS spectra of \(\hbox {C}_{60}\) for interband transitions of valence electrons to the \(\pi \rightarrow \pi *\) band

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The spectra measured for gas-phase \(\hbox {C}_{60}\) [3, 19, 20] are compared in Fig. 7, see the lowest three curves, for impact energies below 10 eV. A typical interband excitation pattern is clearly visible, which represents the well-separated molecular levels having \(\pi \)-electron character, i.e., individual excitations of the free molecular nature of \(\hbox {C}_{60}\). The widths of the peaks are narrower, \(\approx \) 70% less for gas phase \(\hbox {C}_{60}\) than for the film \(\hbox {C}_{60}\) [53] in Fig. 5, where some bulk effects (the interaction between the \(\hbox {C}_{60}\) molecules) may be superimposed. As discussed above, with reference to Figs. 5 and  6, these features were very similar to those in Fig. 2 of Burose et al. (10–20 eV, \(\theta = 0^{\circ }\)) [18] in which a third peak was observed at \(\sim 6.1\) eV and assigned to a collective excitation in the \(\uppi \)-electron system (i.e., a \(\uppi \)-plasmon) [43, 53]. The spectrum implies that almost all of the 60 \(\uppi \)-electrons are strongly localized in the cage and contribute to the \(\uppi -\uppi ^{*}\) individual excitations.

The traces in our data at 50 and 100 eV and for \(5^{\circ }\) (upper curves in Fig. 7) contain evidence of a broader band at \(\sim 6\,\hbox {eV}\) which clearly surpasses the noise level. Smaller and fainter structures can also be seen for each one electron \(\pi \rightarrow \pi ^{*}\) individual transition at 2.2 eV, 3.7 eV, and 4.8 eV. Of perhaps more interest to us is the small structure at 7.9 eV, which is \(\sim 0.3\,\hbox {eV}\) higher than the first ionization energy of 7.58 eV. This corresponds to structures observed clearly at 7.87, 8.12, 8.29, 8.45, and 9.2 eV in the photoexcitation measurements [69]. They were assigned to Feshbach resonances associated with core-excited Rydberg states converging to the relevant ionization limits. Optically forbidden features are more clearly observed for the transitions at 2.26 eV and 2.98 eV by Bulliard et al. [19] and Abouaf et al. [20], who used the constant residual energy loss mode for their measurements. It is also noted that there is similarity in the electronic excitation spectra between \(\hbox {C}_{60}\) and benzene [70].

Abouaf et al. [20] performed an angular analysis of the scattered electrons, for characterizing optically allowed and forbidden transitions using the electron crossed-beam spectrometer. Their 6.1 eV peak showed: (i) different angular characteristics, compared with other individual excitations and (ii) a position shift in the peak from 6.1 eV to 6.4 eV, as the scattering angle varies from \(5^{\circ }\) to \(15^{\circ }\), at \(E_{\mathrm{r}} = 20\,\hbox {eV}\), while the peak position of the \(8^{1}\hbox {T}_{1u}\) electronic state does not change at 5.8 eV. These features support the previous assignment for the existence of a \(\uppi \)-plasmon state at 6.1 eV.

In this report, supplementary verifications are performed for those observations, in which two angular features, i.e., (i) and (ii) above, satisfy the conventional evaluation criteria for testing bulk plasmon excitations [51] in solids. One is for the dispersion relation and another for the differential cross section. Note that the dispersion relation is clearly merged for the position shifts of the \(\uppi + \upsigma \) plasmons of gas-phase \(\hbox {C}_{60}\) in Fig. 5, because 210 of the \(\uppi + \upsigma \) electrons contribute to the collective motions.

3.2.5 Comparison with the photoabsorption cross section

Returning to the cross section study, we now place an absolute scale to the electronic excitations of gas-phase \(\hbox {C}_{60}\), as shown in the upper two curves in Fig. 7. The determination of the electronic excitation DCSs is based on a simultaneous observation of the inelastic peaks for \(\hbox {C}_{60}\) and He, i.e., we are now using a He mixture, in the energy loss region from 0 to 50 eV and at impact energies of 50 and 100 eV. The observed counts of the scattered electrons were subsequently converted into absolute cross sections using the ratio of the scattering intensities and the known differential inelastic cross sections of the He \(2^{1}\hbox {P}\) electronic state [37]. Here, we have also used the pressure ratio formula for the conversion of the relative DCS to an absolute scale using the revised sublimation vapor pressure [35].

To our knowledge, there are no absolute DCSs for electronic-state excitation of \(\hbox {C}_{60}\) by electron impact currently available in the literature. In contrast, the absolute photoabsorption cross sections of \(\hbox {C}_{60}\) in the gas phase has been reported in an ultraviolet–visible range, below 6.5 eV [69, 71,72,73,74,75,76]. In what follows, we employ the data of Smith [75] and Coheur [76], as was recommended by Berkowitz [26]. Both of them found good agreement between their absolute intensities in the gas phase and in the solution spectra. Other data [71,72,73,74], however, are excluded because they are all larger in value, even though they agree well with each other. In order to compare with the absolute photoabsorption cross sections of Smith [75], the pseudo-photoabsorption spectrum of the \(\hbox {6}^{1}\hbox {T}_{\mathrm{1u}}\) state for 1000 eV and \(1.5^{\circ }\) in Fig. 6 [17] is transformed to a relative experimental generalized oscillator strength, \({GOS}_{{exp}}\) (\(K^{2})\), using the standard formula [77] as a function of the momentum transfer squared, \(K^{2}\). Though only one experimental point is available, the relative GOS (RGOS) is fitted using a semi-theoretical formula proposed by Vriens [78] for dipole allowed transitions as:

$$\begin{aligned} {GOS}_{exp}\left( K^{2} \right) =\frac{1}{\left( 1+x \right) ^{6}}\left[ \sum \limits _{m=0}^\infty \frac{f_{m}x^{m}}{\left( 1+x \right) ^{m}} \right] \, \end{aligned}$$
(2)

where \(x=K^{2} \big / \alpha ^{2}\), \(\mathrm {\alpha =}\sqrt{B} \big / R +\sqrt{B-E} \big / R \), and B and E are the binding and the excitation energies of the target electron being excited, respectively. In Eq. (2), \(f_{m}\) are constants to be determined in a least-square fitting for the experimental \({GOS}_{exp}(K^{2})\), and R is the Rydberg energy (13.61 eV). The coefficient \(f_{0}\) in Eq. (2) corresponds to the experimental optical oscillator strength (OOS). When the Born approximation is satisfied in the collisional system, it is well known that the experimental GOS as a function of \(K^{2}\) shows the same behavior along the unique curve independent of the impact energy [79], which gives an OOS of 3.8 atomic unit (a.u.). for the \(6^{1}\hbox {T}_{\mathrm{1u}}\) state [26] at the limit of \(K^{2}\rightarrow 0\). The Keller et al. [17] RGOS is found to be 3.1 a.u. at a \(K^{2}\) of 0.051 au, which corresponds to a normalized absolute DCS of \(1.9 \times 10^{-14}\) \(\hbox {cm}^{2}\) /sr for the \(6^{1}\hbox {T}_{\mathrm{1u}}\) state at 1000 eV and \(1.5^{\circ }\). Note that the elastic DCS value employed to make the normalization was \(3.7\times 10^{-13} \hbox {cm}^{2}/\hbox {sr}\). Our GOS (3.1au at 0.059 au) determined from our DCS at 100 eV and \(5^{\circ }\) does follow almost the same GOS curve obtained for \(1.5^{\circ }\) at 1000 eV [17] and reaches about the same OOS (3.9 a.u.) as \(K^{2}\rightarrow 0\). This result is illustrated in Fig. 8. Considering the present acceptable error range of 25–30%, this observation is supportive of our absolute DCS values for the electronic excitations.

Fig. 8
figure 8

Result of our GOS versus \(K^{2}\) analysis for electron impact excitation of the \(6^{1}\hbox {T}_{\mathrm{1u}}\) and \(8^{1}\hbox {T}_{\mathrm{1u}}\) electronic states of gas-phase \(\hbox {C}_{60}\). See also legend on the figure, where the (– \(\bullet \) – \(\bullet \) –) curves are the relevant fits to our data

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Further, Keller’s GOS for the \(8^{1}\hbox {T}_{\mathrm{1u}}\) state, obtained from the normalized absolute DCS for the \(6^{1}\hbox {T}_{\mathrm{1u}}\) state, is fitted again by Vriens’ formula, which reaches to a slightly larger OOS of 5.7 a.u. than the value of 5.0 expected from photoabsorption data [26]. Nonetheless, given the uncertainties on those data, this level of accord is actually very good. Our GOS of 4.4 au at 0.061 au for the \(8^{1}\hbox {T}_{\mathrm{1u}}\) state is again very close to that of Keller’s (4.4 au at 0.051 au), and the present fitting curve converges to an OOS of 5.9 a.u. Although an impact energy of 100 eV might be below the limit of validity for this approach, this analysis is again supportive of our absolute DCS values for the electronic excitation of the \(8^{1}\hbox {T}_{\mathrm{1u}}\) state. Furthermore, the discrepancy of \(\sim 14\) %, observed between the pseudo-photoabsorption and photoabsorption OOSs, may, at least in part, be the result of a contribution from the \(\uppi \) plasmon at 6.1 eV. If so, as seen in in the \(\uppi + \upsigma \) plasmon at 28.3 eV, the plasmon excitations might be much smaller in a free \(\hbox {C}_{60}\) molecule compared to that in the condensed-phase \(\hbox {C}_{60}\) we discussed earlier. Note that it is a general characteristic of collective excitations, namely longitudinally collective oscillations of the free electron gas density, that they are very intense when initiated by electron scattering but are quite weak when optically excited.

4 Conclusions

In this special issue paper, we revisited “elastic scattering,” and the vibrational and electronic-state excitation DCSs of \(\hbox {C}_{60}\) by electron impact, although the re-measurements requested by Prof. F. H. Gianturco and the late Prof. V. McKoy are yet to be done for the elastic scattering case. However, the previous inconsistency between experimental and theoretical absolute elastic DCS values was solved by re-normalizing the original cross sections [3] in this report. On doing so, we now obtained qualitatively and quantitatively good agreement between experimental and theoretical results. Furthermore, in considering vibrational and electronic-state excitations, we observed that more interesting physics and chemistry are involved, as well as more debate being needed in some aspects of the behavior of this cage-like fused-ring structure (truncated icosahedron) of molecular \(\hbox {C}_{60}\). Finally, we note that none of the recent electron scattering cross section studies on condensable targets (e.g., Refs. [80,81,82,83]) have attempted to employ the relative flow technique, as developed at JPL, to set the absolute scale of their data. Therefore, the implementation of that approach, such as we have done here, when studying condensable targets must be done very carefully with due caution needing to be exercised. The benefits of employing the relative flow technique with condensable targets are manifest, but rely intimately on the accuracy of the available pressure versus temperature vapor curves.

As a great theoretical quantum chemist, the late Prof. V. McKoy had devoted his life to our community, particularly, to the most important aspects of quantitative research. He always encouraged our community’s experimentalists, using statements like [84]; “the elastic scattering is of primary importance simply because it is, at low impact energies, the largest cross section by far, often an order of magnitude larger than all inelastic cross sections combined. Although the concept of elastic scattering is easy to grasp, in practice the term elastic is used in several qualified senses, particularly in connection with studies. Because it is important to be aware of these shades of meaning, we devote this section to a discussion of them... ”Footnote 1