STRONGLY COUPLED DUSTY PLASMAS AND PHASE TRANSITIONS 459

is 10−3–10−2 s. This time is short enough, and hence the particle interaction with external fields can be expressed in terms of e ective constant charge. A time-averaged charge calculated from this model is in most cases close to the values obtained for levitating particles from the balance between gravity and electrostatic forces. The charge fluctuations with amplitudes comparable to the charge itself can be one of the reasons preventing the formation of highly ordered structures of dust particles.

When a particle gets into the track region, the cascade electrons having a mean energy of 100 eV could cause charges which are su cient for dust system crystallization. To investigate the charging process, a numerical model based on a system of equations describing two–dimensional space–time track evolution has been developed. The system of equations included the kinetic equation for electrons, the continuity equation for heavy components (ions, atoms, etc.), the Poisson equation, and equations describing chains of plasma–chemical reactions. It follows from the calculations that a particle can collect no more than 10 electrons from one track. This means that the influence of many tracks is required in order to produce the large charge on the particle.

In order for the charging rate in the track regions to prevail over that due to drift flows, an ionizing flux of a value of approximately 1013 cm−2 s−1 is required. Such a flux can be obtained in the beam of a charged particle accelerator. Experiments using a circular continuous beam with an aperture of 15 mm, current of 1A, and proton energy of 2 MeV were performed. In the absence of an external electric field, the stratification (i.e., formation of regions free of dust particles) of the dust component in neon was observed.

11.3.3Dust clusters in plasmas

Dust clusters in plasmas constitute ordered systems of a finite number of dust particles interacting via the pairwise repulsive Debye–H¨uckel potential and confined by external forces (e.g., of electrostatic nature). Such systems are sometimes also called Coulomb or Yukawa clusters. The di erence between dust clusters and dust crystals is conditional: both systems in fact consist of a finite number of particles. The term “dust clusters” is usually reserved for systems with number of particles N 102 −103, while for larger formations the term “dust crystals” can be used. A more precise definition of clusters would be the ratio of the number of particles in the outer shell to the total number of particles in the system. For crystals, this ratio should be small. Similar systems are met, for instance, in the usual singly charged plasmas in Penning or Paul traps (Dubin and O’Neil 1999; Gilbert et al. 1988), where the vacuum chamber is filled with the ions, as well as in colloidal solutions (Grier and Murray 1994). The distinctions between systems are due to di erent types of interaction potential and di erent forms of confining potential.

Historically, clusters consisting of repulsive particles in an external confining potential were first investigated with the use of numerical modeling (mostly by Monte Carlo and molecular dynamics methods). Taking into account the pos-

Fig. 11.15. Video images of experimentally found (Juan et al. 1998) (a) typical dust cluster structures consisting of di erent number of particles (the scales are not the same for the pictures, typical interparticle spacing is between 0.3 and 0.7 mm); (b) typical shell configurations of several dust clusters composed of di erent numbers of particles.

sibility of applying the simulation results to dust clusters, we mention here the works of Candido et al. (1998); Lai and I (1999); Totsuji (2001); Totsuji et al. (2001); and Astrakharchik et al. (1999a,b). Most simulations were performed for two–dimensional clusters in an external harmonic (parabolic) potential. Such a configuration is usually realized in ground–based experiments with dusty plasma in gas discharges. The simulations show that for a relatively small number of particles in the cluster the “shell structure” is formed with the number of parti-

cles Nj in the jth shell ( j Nj = N ). At zero temperature, unique equilibrium configuration (N1, N2, N3, . . .) exists for a given particle number N . Such configurations form an analog to Mendeleyev’s Periodic Table, the structure of which depends on the shape of the interaction potential, confining potential, and their relative strengths. At finite temperatures, metastable states with energies close to the ground state can also be realized.

The first experimental investigation of dust clusters was reported by Juan et al. (1998). The experiment was performed in the sheath of an r.f. discharge. A hollow coaxial cylinder 3 cm in diameter and 1.5 cm in height was put on the bottom electrode to confine the dust particles. Clusters with a number of particles from a few up to 791 were investigated. A photo images of typical cluster structures with di erent numbers of particles are exemplified in Fig. 11.15(a). Figure 11.15(b) shows a series of typical shell configurations observed for dust clusters. For a large number of particles, the inner particles form a quasi–uniform hexagonal structure, while near the outer boundary particles form several circular shells. The mean interparticle separation increases up to about 10% from the center to the cluster boundary.