Unique role of orbital angular momentum in subshell-resolved photoionization of C₆₀

Matthew A McCune¹, Mohamed E Madjet² and Himadri S Chakraborty¹

¹ Department of Chemistry and Physics, Northwest Missouri State University, Maryville, MO 64468, USA
² Institute of Chemistry and Biochemistry, Free University, Fabeckstrasse 36a, D-14195 Berlin, Germany

E-mail: himadri@nwmissouri.edu

Received 28 July 2008, in final form 29 July 2008
Published 1 October 2008

Angular momentum is an important determinant of the static and dynamical properties of electrons in atoms, molecules and mesoscopic systems. For the systems of nanometric dimensions, angular momentum often induces novel effects in the electronic structure. For example, tubular-shaped image states have been predicted for the single-walled carbon nanotubes which are formed in the potential isolated from the tube due to the electron's high (transverse) angular momentum [1]. Lower angular momentum image states for both single- and multi-walled carbon nanotubes have also been predicted [2] and probed by the femtosecond time-resolved photoemission [3]. In the dynamical regime, for example, innershell photoionizations, the angular momentum of the ejected electron forms a repulsive potential barrier that influences the cross section just above the ionization threshold by inducing so-called shape resonances. Beyond the threshold, however, the energetic photoelectron circumvents the barrier. On the other hand, the photoionization of large systems, like C₆₀, shows strong modulations in the cross section [4, 5] due to the dominant electron ejections from regions that see rapid changes in the electronic potential. Since these systems can support electrons over a greater range of angular momentum than atomic systems, the potential can be significantly modified by the repulsive barrier of a fast-rotating electron. The modification may sensitively alter the cross section's behavior for these systems, creating an effect that goes beyond the `ordinary' effect of angular momentum. Since with the increase of the ejected electron's linear momentum its de Broglie wavelength shortens (to enable the electron further explore the potential shape), the effect is expected to enhance with greater photon energies.

In this communication, considering photoionization from various subshells of C₆₀ we show that the subshell's angular momentum plays a distinct role to determine the oscillatory structure in the cross section. The effect can often be so strong that the shape of the cross sections can be highly altered. To interpret the result we analyze Fourier transforms of the cross sections to the reciprocal (radial) coordinate space.

A detailed description of our method of calculation is published elsewhere [6]. The ground state of C₆₀ is constructed in the local density approximation (LDA) by homogeneously smearing sixty C^4 +  ions into a jellium hull. This disregards the motion of the 120 tightly bound atomic 1s electrons of binding energy about –290 eV and treats the dynamics of the remaining 240 valence electrons, four (2s²2p²) from each carbon atom, self-consistently. The known value of 3.54 Å is used for the C₆₀ radius R. The width (Δ) of the hull is determined as 1.50 Å as required for the charge neutrality and the experimental ionization potential. The model gives rise to a ground-state potential having distinct edges at the positions 2.79 Å and 4.29 Å of inner (R_i) and outer (R_o) radii, respectively, and a relatively flat interior, indicating considerable delocalization of the electrons. The ground configuration of the 16 occupied subshells in this potential is 1s²1p⁶1d¹⁰1f¹⁴1g¹⁸1h²²1i²⁶1j³⁰2s²2p⁶1k³⁴2d¹⁰2f¹⁴1l¹⁸2g¹⁸2h¹⁰ in the harmonic oscillator notation. The levels group into the bands of radial symmetries, σ and π, corresponding to the principal quantum numbers n  =  1 and 2, respectively. 180 electrons occupy ten (ℓ  =  0–9) angular momentum subshells of the σ-band while the remaining 60 fill in six (ℓ  =  0–5) subshells of the π-band. Although limited to the spherical geometry, our ground-state results in many ways resemble results from quantum chemical calculations [7, 8] and spectroscopic measurements [9, 10].

Following equation (13) of [6], the photoionization cross section , corresponding to a dipole transition nℓ → kℓ ' is given as

where the induced density δV includes, besides the dipole interaction, terms representing electron correlations in the time-dependent LDA (TDLDA). Obviously, when the correlation terms are omitted, δV equals the dipole operator to yield the independent particle LDA result.

Figure 1 presents cross sections for three selected subshells calculated in the LDA. For the 1s subshell we also include the corresponding TDLDA result to show that at energies above 50 eV there is no essential difference between the LDA and the TDLDA. As known, only at lower energies are the oscillations obliterated by the plasmon resonances to render the TDLDA (which describes the plasmons) very different from the LDA [6]. Anyway, comparing results between two σ subshells, 1s and 1l of lowest and highest angular momentum respectively, remarkable differences in the structures are noted above 50 eV. While both show smaller oscillations, 1s exhibits sharp minima at about 60 eV and 260 eV to deviate from 1l by more than two orders of magnitude around these energies. The minima result from a beating-type oscillation that is very prominent in 1s. The larger 1l cross section is partly due to a higher oscillator strength available to the 1l channel. But that alone can not explain the large differences which must be due to the alteration in oscillations from 1s to 1l. On the other hand, the 2h cross section, the only π subshell shown, differs distinctly from both 1s and 1l, and features a minimum at about 115 eV where it deviates the most.

To understand the result we first assess the origin of the oscillations. In the acceleration gauge the LDA radial dipole matrix element from equation (1) can be written as which embodies the notion that the electron in a radial potential V(r) describing the hull requires a recoil force –∇V(r) to ionize. For a perfect square-well potential with edges at R_i and R_o the gradient of the potential is the Dirac δ-functions at these positions. Approximating the radial continuum wave by the asymptotic form cos(kr – ℓ 'π/2) of the spherical Bessel function, the matrix element simply becomes the sum of two purely oscillatory terms in the photoelectron linear momentum (k), indicating electron emissions from only the edges:

where a_o and a_i are the values of the radial bound wave ψ_nℓ(r) at R_i and R_o respectively; note that the anti-symmetric nature of the δ-functions is reflected in the negative sign between the terms in equation (2).

The physical potential [6] calculated in the LDA is however not quite the square-well type, but one of somewhat softer edges with a flat interior. Therefore, the gradient of the real potential will feature optimum structures of finite widths at the edges instead of the zero-width δ-functions. As a result, the matrix element of a real potential can still be approximated by equation (2) but only after multiplying with a non-oscillatory function A(k), as the semi-classical study [11] of the photoionization of metal clusters suggests. More generally, if the softnesses at the potential edges are unequal, a lesser production of photoelectrons will occur at the softer edge. This we can account for by introducing parameters h_i and h_o in equation (2) that are proportional, respectively, to the height and depth of the structure in ∇V at R_o and R_i. Squaring the modulus of the matrix element we then obtain a simple form of the subshell cross section as

where B  =  a²_o  +  a²_i. Evidently, equation (3) delineates four frequencies of oscillations in each subshell cross section. These frequencies are the diameters 2R_o, 2R, 2R_i and the width Δ of the C₆₀ hull. Indeed, our calculations showed excellent agreement with observed oscillations in the photo-intensity ratio of the two outermost occupied levels of C₆₀ that showed four frequencies [4, 12]. But why do the oscillations change so strongly from the lowest to the highest angular momentum states as seen in figure 1?

To answer this question we return to the notion of electron–photon dipole interaction in the acceleration gauge that directly involves ∇V(r), where V(r) is the actual (effective) potential that the bound electron experiences in the radial direction. For an electron in a subshell of angular momentum ℓ, V(r) is the sum of the attractive well V₀(r) and a centrifugal repulsive term ℓ(ℓ  +  1)/2r². This repulsion, stronger at lower r, can modify the shape of the effective potential such that its inner edge significantly softens if ℓ is large. This is seen in the lower panel of figure 2 for three σ-subshells. While for 1s, ℓ  =  0, the potential remains unmodified and hence symmetric at the edges, for an intermediate ℓ  =  5 (1h) and the highest ℓ  =  9 (1l) significant modifications are noted. In fact, the larger the angular momentum, the greater the softening at R_i. On the other hand, the outer edge of the potential, while becoming more attractive as ℓ increases, retains its general shape. The top panel of figure 2 displays how the effect translates into the corresponding derivatives of the potentials. As seen, the depth of the derivative structure at R_i progressively diminishes with the increasing ℓ, but the height of the structure at R_o remains almost unaltered. We emphasize that the sizes of these structures, their depth or height, are measures of the amount of photoelectrons emanated from the edges that quantum-interfere to yield oscillations in the cross section. Therefore, considering the parameters h_i and h_o in equation (3) we can scrutinize the relative alteration of the strength of various oscillation components as a function of ℓ.

This is done by determining the Fourier transforms (FT) of the cross sections from 50 eV to 1 keV photon energy range. Before applying the transform, the photon energy scale is converted to the k_nℓ-scale of the electron ionized from the nℓ subshell of threshold E_nℓ by in atomic units. The non-oscillatory background of the cross section is then neutralized by dividing the cross section with a fit of the background. The fast Fourier transform algorithm is used after suitably windowing this oscillatory part for smoother transform curves. The FT magnitudes for the selected three subshells, 1s, 1h and 1l, are presented in figure 3, upper panel. While all three curves depict four frequencies by producing four peaks, the relative strength of the peaks are very different from one to another; in fact the difference between the 1s and 1l result is spectacular. In general, for a given FT curve the height of each peak corresponds to the strength of respective oscillation component in the cross section. This means that the height is proportional to the amplitude of the respective cosine term in equation (3). Also, the bound waves of all occupied subshells extend over the same region in space and are either symmetric (σ) or anti-symmetric (π) roughly about the center of the potential; see figure 2, lower panel, in which 1s and 2h waves illustrate this point. Hence, we can approximate |a_i|  =  |a_o| in equation (3) for all the subshells. As a result, the strength of each peak in a FT curve primarily depends on h_i and h_o, the depth and the height of the derivative structures in the upper panel of figure 2. To obtain a relative picture of the strength of various frequency components, therefore, in the upper panel of figure 3, we choose a scale in which the 2R_o peaks of the curves are kept at the same height, since h_o is roughly identical for all the subshells (see figure 2).

For the 1s subshell the angular momentum is zero and therefore h_i  =  h_o. This results in the identical strength of the 2R_i and 2R_o frequency peaks, as seen in the upper panel of figure 3. This is also true for the 2R and Δ peaks of 1s which are of nearly equal strength. However, the latter pair is roughly twice as tall as the former pair which is due to an extra factor of 2 with each of the terms corresponding to the 2R and Δ frequencies in equation (3). For the 1h subshell the situation is different: the relatively softer inner edge of the 1h radial effective potential due to the finite angular momentum ensures h_i < h_o (see figure 2) which translates into considerable weakening of the 2R_i, 2R and Δ peaks. Quantitatively, since from equation (3) the 2R_i peak strength is proportional to (h_i)² while the strengths of both 2R and Δ peaks to h_i, a 50% reduction in the 2R_i peak height of 2h from the corresponding result of 1s should result in about 70% reduction of the other two peaks. This is what is seen by comparing the 1s and 1h FT curves. For the highest angular momentum subshell 1l, however, the effect is huge. Not only do the peaks 2R and Δ strongly diminish (by roughly 70% from the 1s result), but also the peak corresponding to the frequency 2R_i almost disappears—a result that would not have been uncovered without the Fourier analysis. In essence, going from 1s to 1l, the gradual weakening of the electron emissibility at R_i alters the strength of frequency components, with the only exception of the frequency 2R_o, and this reflects in the evolution of the corresponding cross section structure.

State-selective measurements of photoelectron intensities can be carried out via the standard technique of electron spectroscopy. However, to ensure enough molecular character of the measured data, the photon energy must not intrude on the carbon K-shell continuum. Also, accurate measurements can be made relatively easily for the ratio of two subshells. Hence, we examined if the cross section ratios for energies not exceeding 290 eV can produce FT results capable of illustrating the effect. The results of three ratios are shown in figure 3, lower panel; for a ratio, two subshells separated by unit angular momentum are considered so that their ℓ-behaviors remain roughly identical. Barring some offsets in the positions, the relative strengths of the peaks are reasonably well reproduced. This indicates that the new effect predicted can be and should be observed for C₆₀ if inner subshell measurements are made.

Note also in equation (3) that the first three oscillations depend on the final angular momentum ℓ ' through a phase shift. Since the initial and final angular momenta are connected through the dipole selection, ℓ '  =  ℓ ± 1, it has already been shown [13] that these oscillations between two subshells of angular momentum differing by unity are out-of-phase with each other; this trend is also seen in the experiment for two outer subshells [4]. However, the term oscillating with the frequency Δ in equation (3) is independent of the angular momentum. Thus, for subshells of a given radial symmetry (π or σ) the beating oscillations must be in-phase. On the other hand, going from a σ (n  =  1) to a π (n  =  2) subshell, the respective symmetric and anti-symmetric radial waves ensure that the product a_ia_o in equation (3) reverses the sign. Consequently, the Δ oscillation is out-of-phase between a σ and a π cross section. This effect is seen between 1s and 2h cross sections in figure 1, in which the out-of-phase beating oscillation further increases their differences.

In summary, we show using Fourier analysis that the orbital angular momentum of the bound electron critically controls the photoelectron creation at the inner edge of the C₆₀ hull. The ramification is a systematic variation in the photoionization cross sections of various subshells. This variation significantly amplifies between the highest and the lowest angular momentum electrons. Fourier transforms of cross section ratios in the range 50–290 eV indicate that the effect should be discernible if subshell-differential measurements can be made via photoelectron spectroscopy. Although the jellium model is an approximation, the predicted effect should be real, at least in the contribution to the cross section from the delocalized electrons. Finally, C₆₀ is merely the laboratory to study the phenomenon. But the effect should be ubiquitous in nanoparticles of other symmetries and structures that support clouds of delocalized electrons of high angular momenta contained within sharp boundaries.

Acknowledgment

The work is supported by the NSF. MEM acknowledges support from the Cluster of Excellence `Unifying Concepts in Catalysis' coordinated by the Technische Universität Berlin and funded by the Deutsche Forschungsgemeinschaft.

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