Evidence for classical star orbit in largeAlNclusters

Abstract

Mass spectra of Al/v clusters produced by a laser vaporization source have been recorded via standard time-of-flight spectroscopic methods. Regular signal oscillations are observed in the size range /V-200-900 when low-power near-threshold ionization is performed. This striking feature, which is a manifestation of the electronic shell structure of the aluminum clusters, is analyzed with the semiclassical theory of single-particle level density. The oscillation frequency is found to be well explained by the classical star orbit contribution. 03.65.Sq, 36.40.+d Most of the simple metal cluster properties reflect the electronic shell structure of the delocalized valence electrons. Large peaks and steps observed in cluster mass spectra or sudden discontinuities along the cluster size evolution of physical properties, such as the static polarizability or the ionization potential, are in many cases direct manifestations of this shell structure [1]. The underlying theoretical framework relevant to these clusters is similar to the shell theories applying to atoms and nuclei [2]. Except for the differences in the fundamental physical interaction or the potential shape, all these systems are well described in first approximation as a population of N e independent fermions moving in an effective one-particle spherical potential. In the spherical jellium model, worked out using the density-functional formalism [3,4], the discontinuities in the electronic properties arise after the filling of the Kohn-Sham subshells in J) by 2(2/4-1) valence electrons [5][6][7]. Among these filling numbers, which depend on the order of the («,/) levels, the characteristic sizes giving the most pronounced effects (magic numbers) correspond to the filling of successive groupings of subshells. For large sizes this bunching phenomenon is a general trend of systems whose classical motion is multiply periodic [8]. It has been proved by jellium calculations, and the related magic numbers have been observed via oscillations in mass spectra of Na", Cs"0, and Cs rt (S02) clusters [9][10][11][12]. However, as the cluster size increases the level density near the Fermi energy Ep becomes so large that the frontiers between adjacent bunches are blurred and cannot be determined unambiguously. Moreover, owing to the limits of selfconsistent jellium calculations for high structured state densities, to the unavoidable effective width of the levels principally due to the finite temperature of the clusters n(E) -2^<M,lO £ cos[d>(M,^)]exp[-X ? r(M,£ <7[13], and to the different simultaneous scales at which level clustering can occur, a more suitable approach is required.Recently, in a theoretical paper, Nishioka et al. have emphasized the fact that large simple metal clusters constitute the first real system models to exhaustively exemplify the semiclassical theory of single-particle quantum level density [13]. The interpretation of this theory has its foundations in early work by Gutzwiller [14]. It has been developed by Balian and Bloch in the case of empty...

Deformed harmonic oscillators for metal clusters: Analytic properties and supershells

“…The slope of N 1/3 i vs. i in this case is considerably lower (around 0.32) [1], indicating that closed orbits other than the triangular and squared ones should be present in the Balian and Bloch approach [17]. It will be interesting to examine if the 3-dim q-HO can provide any prediction for supershells in Al clusters, after choosing the value of the τ parameter in order to reproduce the right slope of N 1/3 i vs. i.…”

“…g) In the case of Al clusters, magic numbers have been determined experimentally in detail up to 1200 electrons [16], while additional experimental results up to 2700 electrons exist [17]. The slope of N 1/3 i vs. i in this case is considerably lower (around 0.32) [1], indicating that closed orbits other than the triangular and squared ones should be present in the Balian and Bloch approach [17].…”

Orbits in large aluminum clusters: Five-pointed stars

“…Experiments show that the oscillations in the mass spectra of large aluminum Al N [2][3][4] and sodium Na N [5] clusters ( N is the number of atoms in a cluster) differ significantly in shape. Whereas the oscillations for sodium proceed with beats, the aluminum clusters with N > 250 exhibit sinusoidal behavior with a frequency approximately twice that for sodium.…”

“…The spectra of the Al N clusters of smaller sizes represent an intricate pattern without any distinct period. In the literature, the cause of this distinction is discussed in terms of classical trajectories of electron motion in a self-consistent potential (the number of electrons in a cluster is N e = wN , where w is the valence of a metal).In [3], an attempt was undertaken to explain the experiment by invoking a spherical jellium model and the quasiclassical theory [6][7][8][9]. It was conjectured that, contrary to a hard potential of Na N clusters, in which a triangular trajectory and a square trajectory of a close frequency dominate (the oscillations with beats result from the interference of the relevant contributions), in a soft potential of Al N clusters with 250 < N < 900, the main contribution comes from a single trajectory shaped like a five-pointed star.…”

“…In [3], an attempt was undertaken to explain the experiment by invoking a spherical jellium model and the quasiclassical theory [6][7][8][9]. It was conjectured that, contrary to a hard potential of Na N clusters, in which a triangular trajectory and a square trajectory of a close frequency dominate (the oscillations with beats result from the interference of the relevant contributions), in a soft potential of Al N clusters with 250 < N < 900, the main contribution comes from a single trajectory shaped like a five-pointed star.…”