approached. The increase of the stability parameter q (or the confining potential) causes a noticeable change of the ion beam properties. For q = 0.33 one can see an abrupt decrease in the fluorescence intensity at a certain value of ∆ω (indicated by the arrow in the figure), followed by a narrow asymmetric peak. Such behavior can be interpreted as a result of the balance between the laser cooling and the heating of ions in a r.f. field. The e ciency of the heating rapidly falls o as the ions become more ordered. The slight decrease of the stability parameter (q = 0.31) and, hence a decrease of the r.f. heating, causes the phase transition to occur at larger values of ∆ω. Because of the noise, this makes detection of the transition very di cult. Another indication of the phase transition is a sudden decrease of the beam diameter. For the experimental conditions corresponding to Fig. 10.16, the formation of the rotating ion string (viz., one–dimensional ion crystal) with the interparticle distance a 20 µm was observed. The longitudinal temperature measured in the reference frame moving together with the beam was about 3 mK, which corresponds to the coupling parameter γ > 500. By changing the parameters of the plasma and the trap (first of all, the number of ions), two– dimensional zigzag and three–dimensional helix structures have been obtained and investigated by Schramm et al. (2002). It was shown that the stability range of the crystalline structures formed in the storage ring is substantially smaller than that obtained in stationary ionic crystals and in the MD simulations.

10.3Melting of mesoscopic crystals

The nonneutral plasmas form crystalline states at su ciently low temperatures. As the temperature grows the crystal melts, yet keeping the correlation and the short–range order, and when the temperature is high enough the plasma is in a disordered gaseous state. Both the numerical simulations and experiments suggest that the stable crystalline state of the macroscopic plasma is the bcc lattice, and melting occurs at γ 173. In mesoscopic plasmas, di erent types of ordered structures can be formed. As discussed in the previous section, ions confined in a parabolic potential well (typical for ion traps) form clouds with a shell structure and a well–defined surface. In the surface layer and in each shell ions arrange themselves into a triangle lattice. As the number of ions in the cloud increases, the transition to a bcc lattice is observed in the bulk region, which is typical of infinite plasmas.

The melting of such finite–size crystalline structures has been studied by Schi er (2002) with MD simulations. In finite systems the transitions between ordered and disordered states do not occur abruptly – they develop continuously in a certain range of parameters (e.g., temperature or density). Figure 10.17 shows two outer spherical shells of the cloud containing 104 ions at two di erent temperatures. Also, the radial density distribution and the pair correlation function g(r) are shown, as obtained from simulations at three di erent temperatures corresponding to the crystalline, liquid, and gaseous states.

The potential energy and the heat capacity per particle versus the temperature are shown in Fig. 10.18. One can see the well–defined melting transition,

kT

2.5

2.0

1.5

C

10

4

Temperature, 1000·γ-1

Fig. 10.18. Total energy U and thermal capacity C per particle for an ensemble of charges

(Schi er 2002). The calculations are for an infinite system (◦) and for systems containing 104 ions (•). Temperature is in units of γ−1.

both in a finite system of 104 ions and in an infinite Coulomb system, although in the former case the melting temperature is somewhat smaller and the transition itself is smeared out. Based on the analysis of the di usion coe cients performed by Schi er (2002), a significant increase in the ion di usion within each shell, as compared with the di usion between the neighboring shells, has been revealed in the vicinity of the melting transition.

Schi er (2002) also addressed the question: Which factor plays the more important role in the decrease of the melting temperature observed in mesoscopic crystals – the finite size or di erent types of ordered structures? Figures 10.19 and 10.20 show the heat capacity and the melting temperature calculated for a system of 100, 1000 and 10 000 ions, as well as for an infinite system. The height of the heat capacity peak decreases and its maximum shifts towards smaller temperatures as the number of ions decreases. The obtained results exhibit a smooth dependence on the size, regardless of the particular crystalline structure. Figure 10.20 shows the melting temperature, Tm, versus the number of ions in the outmost shell, Ns, normalized to the total ion number. One can see that ∆Tm = Tm(∞) − T (N ) Ns. Hence, one can claim that the decrease of the melting temperature is almost solely due to the finiteness of the system.

3.5Infinite Matter

3.0ions

Specific heat

Temperature 1000 γ−1

Thermal capacity of nonideal plasmas near the melting point (Schi er 2002). The calculations are for 100 ( ), 1000 ( ), and 10 000 (•) ions as well as for an infinite

system (dashed line).

10.4Coulomb clusters

In this section we discuss the properties of small (Ni 10) systems of charged particles. At su ciently low temperatures they form ordered structures called “Coulomb clusters”. Although the properties of such clusters and mesoscopic crystalline systems are similar, the Coulomb clusters exhibit a number of quite interesting features. Below we discuss the clusters trapped in a parabolic potential well, since this confinement is typical for most experiments and numerical (MD and MC) simulations. The equilibrium configurations of particles are determined by the minimum of the potential energy. Taking into account Eq. (10.3), the energy is

one can reduce the potential energy to the form which depends only on the number of particles and the asymmetry parameter β. In a spherically symmetric

Fig. 10.21. Configurations of Coulomb clusters in a spherically symmetric parabolic po-

tential well (Rafac et al. 1991).

Fig. 10.22. Dimensionless energy of the electrostatic interaction for N –particle clusters

(Tsuruta and Ichimaru 1993).

In anisotropic confinement the equilibrium configuration is determined by the value of β. For β 1, the ion cloud is stretched along the z–axis, forming one– dimensional ordered structures. For β 1 ions form two–dimensional structures in the xy–plane. The ground state of two–dimensional Coulomb clusters is well studied (Lozovik 1987; Lozovik and Mandelshtam 1990, 1992; Rafac et al. 1991; Bedanov and Peeters 1994). For Ni ≤ 5 the ions lie on a circle, for 6 ≤ Ni ≤ 8 one of the ions is located in the center, and for Ni = 9 two ions are inside the ring. For Ni = 15 the second (inner) ring is completed (5, 10) and for Ni = 16 one ion again appears in the center (1, 5, 10). The two–dimensional ring structures are completely analogous to the three–dimensional shell structures in spheroidal nonneutral plasmas.