Transition of the electronic response from molecular-like to jellium-like in cold, small sodium clusters

Abstract

Absolute photoabsorption cross sections for Na + n (2≤n≤21) were measured in the visible energy range. The cluster ions were produced in a gas aggregation source and thus have a canonical distribution of internal energy corresponding to a temperature of ∼ 105 K. The spectra for n≤9 and 11 exhibit between two and six absorption lines, and are in qualitative agreement with ab inito quantum chemical calculations. For n=15 and 21, the position of the resonances can be explained as excitations of a nearly free electron gas in a spheroidal container. An evolution is thus observed from molecular-like transitions to a giant collective resonance of the electron cloud. The integrated oscillator strength is 0.95 per 3s-electron in the energy range covered for n≥4, showing that the main excitations of the valence electrons have been found. 36.40.Cg; 36.40.Gk; 36.40.Mr PACS: I IntroductionAlkali metal clusters have been studied intensively in recent years [1][2][3][4][5][6][7]. Two reviews have appeared summarizing the current theoretical [5] and experimental [6] knowledge. Sodium in particular serves as a prototype case for the free electron model, since its Fermi-surface deviates less than 0.1 % from the spherical shape [7,8]. Therefore, the transition from the atom, via the molecule, the cluster, the small particle, to bulk matter can be well studied in this system.In this paper we address the question up to which cluster size the molecular description is appropriate, and when one can change to the computationally less demanding solid state models. For a small numbers of atoms per cluster, the modern methods of quantum chemistry are effective in describing the electronic excitation of the valence electrons [9]. The quantum chemical methods solve the Schrödinger equation using the Born-Oppenheimer approximation. This is the more accurate method, in principle. But the computer time needed increases dramatically with increasing cluster size, making calculation for n > 21 impractical today. For larger alkali clusters a description in terms of solid state or nuclear physics concepts yields good results for the abundance spectra as well as for the electronic properties [4,5]. The latter models neglect the core potentials of the ions and their geometric arrangement, assuming instead, that an interacting electron gas is confined by a structureless background. It is computationally and conceptually interesting to see where the two concepts merge.Optical spectra of small, neutral sodium clusters have been reported earlier [10][11][12]. The neutral clusters are not mass selected in these experiments. After photo excitation and ensuing fragmentation, the neutral clusters are ionized, thus fragmentation from larger clusters onto the species under investigation has to be considered carefully [11]. In order to avoid this problem, several groups have studied charged clusters, which allow for a mass selection prior to photofragmentation, making the interpretation of the data much easier [13][14][15][16]. The evolution of the plasmon resona...

Classification of Phase Transitions in Small Systems

“…A common way to investigate if a transition in a finite system is a precursor of a phase transition in the corresponding infinite system is to study the particle number dependence of the appropriate thermodynamic potential [4]. However, this approach will fail for all system types where the nature of the phase transition changes with increasing particle number which seems to be the case, e.g., for sodium clusters [5] or ferrofluid clusters [6]. For this reason a definition of phase transitions for systems with a fixed and finite particle number seems to be desirable.…”

Doubly charged sodium clusters: plasmon response and photoproduction

“…This resonance takes nearly all the oscillator strength of the valence electrons. For sufficiently small and cold clusters the resonance is interpreted as a strong electronic transition in a molecule composed of several sodium atoms [3][4][5]. For increasing cluster size, the number of absorption lines increases until they coalesce into broad structures.…”

“…The focus of preceding optical experiments on neutral [1,2,[7][8][9], and singly positively charged sodium clusters [4,5,[10][11][12][13] (which allow a better mass selection prior to the photon interaction) has been the size and temperature dependence of the plasmon resonance. Besides the experimental work on the stability of highly charged clusters against fragmentation [14,15], no experimental data have been available on the optical properties of the doubly or higher charged clusters.…”

Tricritical points in a Vicsek model of self-propelled particles with bounded confidence

“…These phenomena can be but are not always precursors of phase transitions in the corresponding infinite system. For example, there are even equilibrium systems such as sodium clusters [60] where the nature of the phase transition seems to change with increasing particle number. In spite of that, it has been shown that phase transitions can be classified at fixed and finite particle number, avoiding the thermodynamic limit.…”