Diffraction of Fast Electrons on the Fullerene C60 ... - APS Link Manager

VOLUME 81, NUMBER 13

PHYSICAL REVIEW LETTERS

28 SEPTEMBER 1998

Diffraction of Fast Electrons on the Fullerene C60 Molecule Leonid G. Gerchikov St. Petersburg Technical University, St. Petersburg, Russia 195251

Peotr V. Efimov, Valerii M. Mikoushkin, and Andrey V. Solov’yov A. F. Ioffe Physical-Technical Institute of the Academy of Sciences of Russia, 194021 St. Petersburg, Russia (Received 21 April 1998) In this paper we present results of experimental and theoretical studies of both elastic and inelastic scattering of fast electrons on the fullerene C60 in the gas phase. We report the first experimental observation of the electron diffraction on C60 , manifesting itself in both elastic and inelastic scattering processes. Diffraction patterns of the cross sections carry important information about the characteristics of the target fullerene, such as its size, distribution of the charge density near the cluster surface, frequencies of plasmon resonance excitations, etc. [S0031-9007(98)06890-2] PACS numbers: 61.48. + c, 34.80. – i

In this paper we report the first experimental observation of the diffraction of fast electrons on the C60 molecule, recently predicted in [1]. The diffraction phenomena arise both in the electron elastic and inelastic scattering processes. The detailed theoretical treatment of these phenomena in electron scattering on metal clusters and fullerenes has been recently given in [1–3]. The diffraction features of the electron scattering cross sections have not been studied in the earlier performed energy loss experiments on fullerenes in the gas phase [4]. There were no experiments performed on the electron elastic scattering on fullerenes so far. For the understanding of the experimental results obtained, let us explain first the physical nature of the diffraction phenomena arising in elastic scattering of an electron on C60 [1]. Because of the spherical-like form of the C60 molecule, the charge densities of electrons and ions near the surface of the fullerene are much higher than in the outer region. These densities are characterized by the radius of the fullerene R and the width of the fullerene shell, a ø R. The de Broglie wavelength of a fast electron in collision with fullerene is small compared to R. At the same time the characteristic scattering length, 1yq, where q is the transferred momentum (we use the atomic system of units, me h¯ jej 1), can be larger than the average interatomic distance in C60 , which is approximately as large as a. Under these conditions the electron scattering amplitude becomes determined by the charge density of the entire fullerene rather than the charge density of a single carbon atom. In this limit, the scattering amplitude and the corresponding cross section possesses a series of diffraction maxima and minima. The positions of these minima and maxima are mainly determined by the radius of the molecule. In this case the process of electron elastic scattering on C60 becomes qualitatively similar to the diffraction of an electron at a shell. The cross section of elastic scattering of a fast electron on C60 in the Born approximation (see, e.g., [5]) reads as 0031-9007y98y81(13)y2707(4)$15.00

ds 4 (1) 4 Fsqd2 . dV q Here Fsqd is the form factor of C60 , q jp 2 p 0 j is the momentum transfer, and p, p 0 are the momenta of the electron in the initial and the final states, respectively. For the description of the fullerene’s form factor let us express Fsqd as a product of the form factor of the atomic concentration nsqd and the form factor of a single carbon atom FA sqd: X expsiq ? rj d FA sqdnsqd , (2) Fsqd FA sqd j

where the summation is performed over all coordinates, rj , of carbon atoms in the fullerene. The applicability of this approximation has been examined in [1] for metal clusters. For fullerenes, the applicability of relationship (2) has been verified by performing ab initio many-body calculations [6]. The form factor of an isolated carbon atom, FA sqd, calculated in the Hartree-Fock approximation decreases as a function of q on the scale q , 1. The form factor of atomic concentration, nsqd, characterizing the geometry of C60 , varies more rapidly. Indeed, for the momenta transfer smaller than an atomic unit, q ø 1, the model of the homogeneous distribution of carbon atoms over the fullerene sphere can be used. In this model nsqd N sinsqRdyqR oscillates with the period q 2pyR ø 1. Namely, these oscillations form the diffraction pattern of the differential cross section (1). The angular dependence of the cross section (1) for the projectile electron energy ´ 809 eV is shown in Fig. 1 by a solid line. It possesses a series of diffraction maxima and minima [1]. Experimental measurements performed in our present work prove this behavior (see Fig. 1 and discussion below). Diffraction phenomena take place also in the electron inelastic scattering process on C60 [1,3]. The plasmon modes of different multipolarity can be excited when an electron collides with C60 . The probability of excitation © 1998 The American Physical Society

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FIG. 1. Experimental (full and open circles) and theoretical (solid curve) angular dependencies of the differential elastic scattering cross section in collision of a 809 eV electron with the C60 molecule. Full and open circles correspond to the two independent sets of measurements. The dashed line is the differential cross section for the mixture containing 60% of C60 and 40% of equivalent isolated carbon atoms. The scale for this curve is given in the right hand axis.

of plasmon with the given angular momentum oscillates as a function of the electron scattering angle, because of the electron diffraction at the fullerene edge. The oscillatory structure is determined by the radius of C60 and also by the angular momentum of the excited plasmon. Plasmon modes carrying different angular momenta provide dominating contributions to the differential cross section at different electron scattering angles, which lead to the significant angular dependence of the differential electron energy loss spectrum. In the Born approximation, the differential inelastic scattering cross section of a fast electron in collision with a cluster in the vicinity of a plasmon resonance reads as [1–3] 2 4p 0 X ds 2 jl sqRd s2l 1 1d Im al sD´d . (3) d´0 dV pp l R 2l q4 Here D´ p 2 y2 2 p 02 y2, ´0 p 02 y2 is the electron energy loss, l is the angular momentum, jl sxd is the spherical Bessel function of the order l, Im al sD´d is the imaginary part of the multipole dynamic polarizability of the fullerene. Note that only terms corresponding to the multipole plasmon excitations of the electrons should be included to the sum over l in (3), namely, l 1, 2, 3 for C62 [1,3]. Higher multipole excitations are formed by single electron transitions rather than collective electron excitations and thus do not contribute much to the energy loss spectrum in the vicinity of the plasmon resonance. The frequency dependence of the multipole polarizabilities has the resonance behavior pin the vicinity of the Mie frequencies vl , where vl lsl 1 1dNe ys2l 1 1dR 3 for the fullerene C60 , Ne 240 is the number of the delocalized electrons. The polarizabilities al svd can be 2708

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described either in the plasmon resonance approximation (see, e.g., [6]) or calculated in the random phase approximation [7–9]. Because of the resonance behavior of the polarizabilities, the cross section (3) as a function of ´ exhibits resonances in the vicinity of vl . The amplitudes of these resonances are determined by the diffraction factor jl2 sqRdyR 2l q4 . Namely, these diffraction factors determine the relative significance of the multipole plasmon modes in various ranges of electron scattering angles and thus the resulting shape of the differential energy loss spectrum. To illustrate this effect we present in Fig. 2 the angular dependence of the multipole plasmon excitation cross sections dsl ydV. Experimental study, which we report in this paper, proves the diffraction behavior of the partial inelastic scattering cross sections (see Fig. 2 and discussion below). Note that the partial cross sections, dsl ydV, presented in Fig. 2 have been calculated in the second Born approximation. The first Born approximation (3) is well applicable in the region of small transferred momenta, q , 1 or scattering angles u , 10± at ´ 809 eV. In the region q . 1, the process of elastic scattering on the fullerene shell with the subsequent excitation of surface multipole plasmons becomes dominating. This process is described by the formulas of the second Born

FIG. 2. Theoretical (solid line) and experimental (dashed line) angular dependences of the partial cross sections dsl ydV for (a) l 1, (b) l 2, and (c) l 3.



approximation, which are rather straightforward and thus we do not present them in this paper. Our experiment has been carried out by using an energy analyzer of mirror type to measure the EELS at different scattering angles of the incident electron beam [10]. A cylindrical cell has been used to produce the gas target of the C60 molecules. The soot, containing 15%–20% of C60 and about 1.5%–2% of C70 molecules [11], was loaded into the cell. Fullerenes in the gas cell were evaporated by heating at temperatures 300– 315 ±C. The organic impurities were removed by the many hours of heating at temperatures 200–250 ±C. The low relative energy resolution D´y´ 1% of the analyzer was chosen to provide high sensitivity of the device for the incident electron energy ´ 809 eV and to reduce measurement time. The latter was important because of the rather fast decrease of the density of the target. At the same time the chosen resolution provides a reliable separation of the elastic scattering peak from the inelastic one, corresponding to the excitation of the ss 1 pd plasmon. The differential elastic and inelastic scattering cross sections have been obtained by integrating over the square of these two peaks. The results of the two sets of measurements of angular dependence of the differential elastic scattering cross section are shown in Fig. 1 using open and solid circles. The cross section dependence obtained theoretically is shown in this figure by a solid line. Experimental data have been normalized to the theoretical cross section at the second diffraction maximum (u 5±). We report quite good agreement of the experimental and the theoretical results in position of the first and the second maxima. The entire pattern of the differential cross section obtained theoretically is very similar to that from the experiment. In the vicinity of diffraction maxima at u , 10± the cross section greatly exceeds the elastic scattering cross section on the equivalent number of isolated atoms because of the coherent interaction of the projectile electron with the fullerene sphere. In the region u . 10±, where q . 1, the projectile electron scatters on individual carbon atoms of the fullerene rather than on the entire fullerene sphere. Therefore diffraction features of the cross section in the region u , 10± are much more pronounced than in the region u . 10±. Note that in the region u , 10±, where q , 1, theoretical cross section has zeros while the experimental one does not. The presence of zeros at q ø pkyR , 1, where k is an integer, in theoretical curve is the consequence of the coherent scattering of electron on the fullerene sphere. However in experiment, zeros in the cross section can disappear because of various reasons. For example, this can occur due to the presence of carbon atoms or some other impurities in the gas cell. In Fig. 1 we illustrate the differential cross section for the mixture containing 60% of C60 and 40% of isolated carbon atoms by the dashed line (the scale for this curve is given in the

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right hand axis of Fig. 1). The differential electron elastic scattering cross section on single carbon atoms does not have diffraction oscillations and thus it forms the smooth background removing zeros in the angular dependence of the cross section. The zeros could also disappear due to the shape vibrations of the fullerene, which depend on the temperature of the gas cell. The presence of the higher nonspherical fullerene molecules in the gas cell also is a factor acting against the zeros in the angular dependence of the cross section. So, one can state that the differential electron elastic scattering cross section carries a lot of important information both on the structure of a single fullerene molecule and the properties of the target gas cell. Further experiments performed with various fullerene gas cells at different temperatures and different energies of the collision should provide more information showing how the electron diffraction manifests itself in all these circumstances. Experiments with fullerene beams are also desirable, in particular for studying vibrational dynamics of fullerenes and for investigation of temperature effects. Now let us consider the EELS spectra measured in the vicinity of (s 1 p)-plasmon resonance. The position of the broad and intensive (s 1 p)-plasmon peak centered at the energy D´ , 25 eV depends significantly on the electron scattering angle and it turns out to be higher than the dipole (s 1 p)-plasmon resonance energy, v1 19 eV, obtained from a photoabsorption study [12]. This discrepancy arises due to a simple reason. By photons, one can excite only the dipole plasmon mode (l 1), while during electron scattering higher multipole plasmon excitations can be excited as well. To illustrate this feature we plot in Figs. 3a and 3b differential EELS measured at the electron scattering angles u 4±, 13±, correspondingly. Each of these spectra was fitted by the sum of three resonance plasmon contributions with the angular momenta l 1, 2, 3. The EELS with the lowest average energy of the (s 1 p) peak and the sharpest left shoulder of the resonance was analyzed first (u 4, Fig. 3a). The dipole mode dominates over this spectrum. Fitting this spectrum, we have obtained the dipole plasmon resonance energy rather accurately. Note that the value obtained, v1 18 eV, is 1 eV less than obtained from the photoabsorption experiment [13]. The value v1 18 eV has been used to fit all other measured spectra. Shifts of the resonance frequencies for the plasmon excitations with l 2 and 3 relative v1 were estimated according to the Mie theory and then v1 , v2 , v3 varied with the purpose to achieve a better fit of the experimental data. Each multipole contribution has been fitted by the corresponding Gaussian profile with the dispersion mainly determined by the natural width and the absolute energy resolution. Knowledge of the apparatus function allowed us to estimate the natural width as G 8 eV. Our result coincides with the value given in [13]. 2709



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figure shows that various multipole modes are characterized by different oscillatory patterns. As a consequence of these oscillations the position of the maximum in the total (s 1 p)-plasmon spectrum depends significantly on the electron scattering angle. The disagreement of the (s 1 p)-plasmon resonance energies reported in various experimental works dealing with the fullerite C60 may be a result of domination of various multipole plasmon excitations at different experimental conditions. The analysis performed in our paper shows that diffraction of fast electrons on fullerenes and other clusters provides an efficient tool for studying properties of these species. This work was made possible by financial support of INTAS Grant No. N 96-477 and of the Volkswagen Foundation.

FIG. 3. Differential electron energy loss spectrum (EELS) measured at two angles of the scattered electrons, u 4± (a) and 13± (b), are shown by solid lines. The result of fitting procedure of the (s 1 p)-plasmon peak by the sum of three partial multipole plasmon contributions is shown by the dashed line. Partial contributions to the EELS are shown by thin solid lines, marked by the value of the corresponding angular momentum.

In Figs. 3a and 3b we show examples of our fitting procedure performed for the electron scattering angles u 4±, 13±. These angles have been chosen to illustrate the domination of the dipole and the quadrupole plasmon modes in the differential EELS at different angles of the scattered electron. The energies and the widths of the multipole plasmon resonances derived from the analysis of EELS measured at different angles of the scattered electron do not deviate from their averaged values more than 0.5 eV. These data, being the result of the direct measurements, gave the statistical error about 0.2 eV for the following values: v1 18.1 eV, v2 24.3 eV, v3 29.0 eV, and G 8 6 1 eV. Angular dependence of the partial cross sections dsl ydV has been obtained by integrating over the square of the corresponding multipole plasmon resonance. These dependencies are shown in Fig. 2 by solid line (theory) and dashed line (experiment). Theoretical and experimental data are in satisfactory agreement. This

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