Variation of the ℏω with the particle number and the appearance of “kinks” for atomic clusters

Abstract

ABSTRACT:The dependence of the harmonic oscillator (HO) energy level spacinghω on the particle number N is studied analytically for atomic (metal) clusters on the basis of their electronic densities, parametrizing Ekardt's results (for sodium clusters) by means of a Fermi distribution. An interesting feature of such an approach is that it leads, under the assumptions made, to "kinks," that is, to "marked discontinuities in the slope" ofhω at the closed shells. These discontinuities diminish as N increases. Key words: atomic (metal) clusters; sodium clusters; jellium model; harmonic oscillator; Fermi distribution T he dependence of the harmonic oscillator (HO) energy-level spacinghω on the particle number N (that is, on the number of the valence electrons of the atoms in the jellium model) for atomic (metal) clusters has attracted theoretical interest [1 -5]. Related experience from nuclear [6,7] or hypernuclear [8,9] physics has been very valuable, and the methods used in these fields have been adjusted to the cluster case. Two general approaches have been followed.In the first one [1 -3], which has been mainly discussed in the case of large values of N, the energy scale is related to the size of the cluster, as this is determined by the mean-square radius of the Correspondence to: M. E. Grypeos; e-mail: grypeos@auth.gr. electronic density distribution. It has been assumed that this density has a constant value equal to the bulk conduction electron density ρ B , and the size of the (spherical) cluster is normalized so that it contains N valence electrons, that is, R 0 = r s N 1/3 where r s = (3/4πρ B ) 1/3 is the Seitz-Wigner radius. The expression for large N which results in this way is where r s is in atomic units andhω in eV. An improved expression has also been derived by taking into account the so-called spill-out [3,10,11] of the electrons beyond the boundary R 0 . This spill-out

A Simplified Analytic Treatment of Shell-Effects in Metal Clusters

“…As regards the ionization potential, however, expression (7) for IP should lead to reasonable estimates provided that the parameters involved are suitably determined. In view of this remark w r e considered the experimental values of IP for sodium clusters which are quoted in ref.…”

“…In view of this remark w r e considered the experimental values of IP for sodium clusters which are quoted in ref. [9] and we fitted the theoretical results on the basis of expression (7) to the experimental values taking Vo,r 0 , and a' as adjustable parameters. The best fit values of these parameters turn out to be V 0 =3.950eV, r 0 =3.945A and a=4.60lA.…”

“…The latter expression for < r 2 >/?isessentially exact (that is if the usually negligible exponential terms are omitted).It is noted that the "half-way" radius c of the distribution is given explicitly in terms of the particle number Ν (see formula (4) of [7]).…”

“…We focus our attention to the estimates of the energy per particle E/N and to the ionization potential (IP). In deriving the approximate expression, for these quantities, use is made of the expression of the oscillator energy level spacing hujin terms of Ν [6] and in particular of an appropriate expression derived recently [7] by means of a Fermi distribution for the average electronic density. The approximate expression for E/N and IP are given in the section 2, while the third section is devoted to some numerical,results and comments.…”