Hydrodynamic model of collective electronic oscillation in spherical geometries
Creators: name: Marek, Michalewicz
Publisher: name: Zenodo
Year: 1995

description: Nearly sixty years ago Felix Bloch wrote a seminal paper [1] laying the ground­ work for a treatment of normal modes of electron gas in a continuum held formulation. It was later to be called "the hydrodynamic model of the electron gas” [2]. The earlier formulations were applied to problems of atomic physics such as the stopping power, photoabsorption and scattering cross section [1-4]. The idea of treating the ground- state properties of an inhomogeneous electron gas system as being determined by the ground state electronic density no(r) is the basis of the Thomas-Fermi method [5, 6]. The hydrodynamic model derived by Bloch [1] may be considered an extension of the static Thomas-Fermi method to the electronic systems in which the total den­sity n(r, t) is allowed to deviate slightly from the Thomas-Fermi equilibrium density no(r). The hydrodynamic model was later commonly applied to problems of metal, semiconductor and surface physics [7-9]. It proved especially useful in the treatment of the point charge interacting with the metal surface.<br> A local charge interacting with the metal electrons may be considered as inter­ acting with the electron density fluctuations, plasmons, represented by a Boson field [10], In metals the hydrodynamic model serves to construct the normal modes of collective electronic oscillations and to formulate the interaction part of the Hamilto­ nian as was done for the semi-infinite metal surface by Manson and Ritchie and by Mahanty and co-workers [11-13].<br> The self-energy formalism utilizing the plasmon field description was used to study the tunelling potential experienced by a charge in the model of the scanning tunelling microscope (STM) [14-16]. The scanning tip in the model of the STM was treated as a spherical particle. This study exposed some new and interesting features of the dispersive hydrodynamic model in spherical geometry. It was observed that the dispersive hydrodynamic model admits only a finite number of surface plasmons on the sphere, with eigenenergies wi in the interval 1/3 &lt; (omega_l/omega_P)^2&lt; 1, where omega_P = ....... is the plasma frequency.<br> Kubo considered the question of validity of the traditional band structure and electron gas description for the small metallic clusters [17,18]. It is known that such systems exhibit quantum size effects [19-21]. The electronic structure and collective electron dynamics in small spherical systems was in recent years studied by a variety of theoretical approaches [22-32]. Rom the practical point of view the properties of minute metallic spheres are of substantial importance in catalysis [33], colloid science, aerosol physics and environmental sciences [34]. Arrays of metallic micro-hemispheres have been fabricated for opto-electronic applications [35].<br> The hydrodynamic model is a dynamic, long-wavelength description of the collec­tive electronic modes. In the last thirteen years progress has been made in extending the density functional theory to the time dependent phenomena both in practical<br> terms within the so-called time dependent local density approximation (TDLDA) [39,22-25], as well as at the fundamental level of the theory [36-38]. It is still an open challenge to further understand the connection between the dynamic linearized \ hydrodynamic model and the time dependent density functional theory. Some rela­tive merits and differences between the hydrodynamic model and the time-dependent local-density-approximation method were discussed in Ref. [48], but this topic de­ serves further focussed studies. As a step in this direction in Section 2 we present the general derivation of the hydrodynamic model from the density functional formalism, in the spirit of the original work of Ying and co-workers [41,42].<br> The exciting recent development was the discovery of the new form of molecular carbon, (760 [43]. It was suggested that such a molecular system could be treated with­ in the framework of the hydrodynamic model [44]. Collective electronic excitations<br> on the novel fullerenes play important role in cohesion of C_60 crystal [45]. The total electron density in C^o molecule was treated as a spherical shell of finite thickness and normal modes of such a system were analyzed on the basis of the hydrodynamic model [46]. Section 3 contains general derivation of the hydrodynamic dispersion re­ lation for the surface plasmons in the metallic shell of finite thickness. The dispersion relation for the solid sphere and the void are obtained as the limiting cases. A greatly refined experimental technique allowed vom Felde, Fink and Ekardt to observe the dispersion of multiple surface plasmons on minute potassium spheres using the electron-energy-loss-spectroscopy [47]. We have applied the hydrodynamic model to analyze their experiment [48]. The main results of that analysis and the final conclusions will be outlined in Section 4.<br> Systems like small metallic particles containing from ~ 102 &lt; N &lt;~ 104 and the carbon cluster consisting of 60 atoms are perhaps the best candidates for the application of the hydrodynamic model. New experimental advances paved the way<br> to a detailed tests of the validity of hydrodynamic models for such systems and for the assumptions which lead to particular choice of parameters. In this report we present a selection of examples of application of the dispersive hydrodynamic model<br> to the systems recently studied experimentally which have spherical symmetry. We apply the abrupt jellium boundary conditions throughout this work. A discussion of refinements of the hydrodynamic model to include the diffuse electron density surface profile and the studies of non-spherical systems will be given in the concluding section.<br> It should be understood that the list of references is to a certain degree arbitrary and reflects the particular interests of the author.