384	NONNEUTRAL PLASMAS

a proton beam cooled by electrons. Recently, the phase transition of a “gaseous” ion beam into a one–dimensional “crystallized” beam was also observed in the storage ring PALLAS (Sch¨atz et al. 2001; Schramm et al. 2002). The obtained results are discussed in the next section.

10.2Strong coupling and Wigner crystallization

In the limit of high densities and low temperatures, the electrons behave very much like a degenerate ideal Fermi gas. This is because the mean energy of the Coulomb interaction (V rs−1) does not increase with the density as fast as the Fermi energy (εF rs−2). In metals V and εF are about the same, so that the electron subsystem is a weakly nonideal degenerate plasma. In accordance with the prediction made by Wigner 1934, as the density decreases the mean kinetic (Fermi) energy can become much smaller than the mean potential energy. Then the electron localization and the formation of lattices becomes energetically favorable. The electrons perform small oscillations around the equilibrium (note that usually the zero–point vibrations can be neglected). The melting of a Wigner crystal can be achieved only by increasing the density, since the temperature is small compared to the Fermi energy. In the classical limit, electron crystallization is also possible. The properties of classical electron crystals are well described in the framework of the OCP model (see discussion in Chapter 5).

For electrons localized on the surface of liquid He, it is rather easy to achieve the conditions necessary for crystallization (Shikin and Monarkha 1989). For typical surface densities of electrons, ns = 108–109 cm−2, the corresponding Fermi energy (εF/k < 10−2 K) is much smaller than both the potential energy and the temperature. The surface electrons in experiments obey classical statistics, so that the strength of the mutual interaction between the electrons is characterized by the coupling parameter γ = π1/2e2n1s/2/kT . The first experimental evidence of Wigner crystallization was presented by Grimes and Adams 1979, who observed a series of resonances in the r.f. energy absorption during the excitation of electrons on the surface of liquid He (see Fig. 10.8). These resonances only appear when the temperature is below a certain critical value T which is a function of the electron density, T n1s/2. The explanation of these results was given by Fisher et al. (1979): It turned out that at low temperatures electrons form a triangular lattice with the period d = 21/23−1/4n−s 1/2 ≥ 2 ·10−5 cm. Each electron causes surface distortion, so that the motion in the external electric field drives capillary waves (ripplons). The electron–ripplon interaction induces coupling of the electron and ripplon oscillations and, hence, causes resonance absorption of the r.f. radiation when the capillary–wave wavenumber is equal to a reciprocal vector of the electron lattice.

The Wigner crystal melts and the resonances disappear as the temperature increases. With good accuracy, the coupling parameter remains constant on the melting line, γ = 137 ± 15 (see Fig. 10.9). Similar to the melting of solids, electron lattices start melting at high temperatures because the formation of dislocations becomes thermodynamically favorable, which eventually causes destruction

STRONG COUPLING AND WIGNER CRYSTALLIZATION

385

.

.

. 2

MHz

Fig. 10.8. Resonance absorption of the electromagnetic waves by a Wigner crystal (Grimes and Adams 1979). The resonance emerges at T = 0.457 K, when the two–dimensional electron system crystallizes.

of the lattices. This dislocational mechanism of melting is confirmed by the values of the melting temperature and the shear modulus obtained from the MD simulations by Morf (1979). The corresponding coupling parameter on the melting line is 120 < γ < 140.

Let us now discuss the properties of nonideal plasmas where the ions are strongly correlated, so that liquid–like (short–range order) and crystalline (long– range order) structures have been predicted and observed with the increase of the coupling parameter γ. The properties of such ordered structures can vary dramatically. They are determined not only by the value of γ, but also by the parameters of the trap and the size of the structure. Based on the spatial scales of the plasma, one can distinguish three di erent regimes (Dubin and O’Neil 1999): macroscopic plasmas, mesoscopic plasmas, and Coulomb clusters. The size of the macroscopic plasmas is so large that the surface e ects have practically no influence on the average physical properties. The microscopic structure inside such plasmas coincides with that of infinite homogeneous plasmas. For the strongly correlated case, when ions form crystalline structures, the bulk properties prevail over the surface properties when the number of ions is fairly large, Ni ≥ 105. However, when the correlation is not very strong (γ ≤ 10), the surface e ects can be neglected at Ni ≥ 103 (Dubin and O’Neil 1999). This is because the correlation length in a liquid phase is of the order of one or two interparticle distances, and the surface e ects do not penetrate beyond this length. When

386	NONNEUTRAL PLASMAS

cm

Crystall

Liquid

.

.

.

.

.

.

Fig. 10.9. Phase diagram of a two–dimensional electron Wigner crystal localized on the surface of liquid He (Grimes and Adams 1979). The linear fit corresponds to a constant value of the coupling parameter γ = 137.

.the number of particles is low enough, so that the plasma shape and size play an important role, but on the other hand is large enough (Ni ≥ 102) in order to employ major methods of statistical mechanics, the plasma is referred to as a mesoscopic plasma. And finally, when the number of particles is very low (Ni ≤ 10), the ions form, at low temperatures, simple geometrical configurations

– the Coulomb clusters. The structure of clusters is determined by the confining fields and the number of particles. In early experiments with strongly coupled nonneutral plasmas confined in the Penning and Paul traps, the number of ions was relatively low, 102 < Ni < 104 (Bollinger and Wineland 1984; Gilbert et al. 1988; Raizen et al. 1992). Therefore, most of the numerical simulations have been performed for a mesoscopic plasma confined in a parabolic potential well. Figure 10.10 shows the results obtained with Monte Carlo simulations for a cloud of 400 ions confined in the Penning trap (Dubin 1996). The angular velocity of the cloud rotation, ω, was tuned to provide the spherical symmetry of the e ective potential Φ (see Eq. (10.3) with β = 1). For a weak correlation, γ 1, the ion density slowly falls o and approaches zero at the cloud surface. As γ increases, the density decrease near the surface becomes steeper, approaching a step function shown by the dashed line. In addition, the emerging oscillatory structure suggests local ordering (the spatial scale characterizing decay of the oscillations is a measure of the correlation length). The formation of the oscillatory structure can be considered as a precursor of crystallization.

Similar behavior was observed in the one–component plasma model by Ichimaru et al. (1987). By increasing γ, the oscillation amplitude grows until the minimum ion density between the neighboring peaks becomes equal to zero, so that the ion cloud is separated into a set of concentric shells. The distance between the consecutive shells as well as the interparticle distance within each shell is close to the Wigner–Seitz radius rs. Thus, the number of ions in the shell is

STRONG COUPLING AND WIGNER CRYSTALLIZATION

387

5

4

3

2

1

0

n(r,γ)/n 0

				γ = 150
				γ = 10
				Cold
				fluid
				γ = 1
0	2	4	6	8 r/rs

Fig. 10.10. Equilibrium ion density versus the radius in a spherically symmetric cloud of nonneutral plasma containing 400 ions at di erent values of the coupling parameter

γ (Dubin 1996). The dashed line corresponds to the low–temperature limit of the mean field theory with ni(r) = ni and rs = (4πni/3)−1/3 .

proportional to its surface, Ni2/3, and the number of shells is proportional to

Ni1/3.

For β = 1 the cloud is a spheroid consisting of concentric shells. Atβ → ∞ the cylindrical concentric shells are formed (Rahman and Schi er 1986), whereas for β → 0 ions arrange themselves into a two–dimensional lattice in the xy–plane (Rafac et al. 1991; Bedanov and Peeters 1994).

Totsuji et al. (2002) have investigated with MD simulations the nonneutral plasma containing a fairly large number of ions, 5 · 103 < Ni < 1.2 · 105, and confined in a spherically symmetric parabolic well. It turns out that the shell structure is energetically favorable only for Ni < 104, whereas for larger number of particles the formation of the bcc bulk lattice surrounded by a few spherical shells near the surface provides the energy minimum (see Fig. 10.11).

In experiments by Gilbert et al. (1988) and Bollinger et al. (2000), the shell

	structure and the bcc lattice have been observed for N						4
						i	= 1.5 · 10 and Ni =
	2 · 10	5	, respectively. A laser cooled plasma of Be	+
					ions at temperature 10 mK

was studied in Penning traps. For typical ion density ni 4 · 108 cm−3, this corresponds to the coupling parameter γ > 200.

In experiments by Sch¨atz et al. 2001, three laser beams crossing the plasma cloud at di erent angles were used. The beams induced luminescence, which allowed them to obtain a three–dimensional visualization of the cloud. Figure 10.12 shows the image of the structure which contains 1.5 · 104 ions and consists of 11 shells. Both the number of shells and the distances between them agree very

388	NONNEUTRAL PLASMAS

y/r

50

a

25

s 0

-25
-50		-25	0		25	50
-50
			x/rs
56	b
4
3
2
1
0			20 r/r
	0	10		30	40	50
			s

Fig. 10.11. Cross–section of the bcc lattice containing 120 032 ions (Totsuji et al. 2002). The equatorial slice at |z| < 1.19rs (a) and the radial distribution normalized to the mean ion density (b) are shown.

well with the results of numerical simulations. Wineland et al. (1985), used the Bragg scattering of the cooling laser light along with the direct observation of the luminescence. Figure 10.13 shows the di raction image of a spherically symmetric plasma cloud of about 7.5 · 105 Be+ ions. The image was obtained by employing Bragg scattering from di erent crystalline planes. The light spots in Fig. 10.13 represent very well the rectangular reciprocal lattice, which clearly suggests the formation of the bcc bulk lattice with (110) plane perpendicular to the laser beam.

Ordered crystalline structures have also been observed in Paul traps by Raizen et al. (1992), Drewsen et al. (1998), and Hornekaer et al. (2001). The

STRONG COUPLING AND WIGNER CRYSTALLIZATION

389

Diagonal cooling

Probe

Perpendicular cooling

Fig. 10.12. Experimentally observed ordered structure containing about 1.5 ·104 ions and consisting of 11 shells plus the central string. The image is obtained by employing three crossing laser beams (Gilbert et al. 1988).

Fig. 10.13. Bragg scattering image of an ionic crystal in the Penning trap, as obtained by video strobing at the frequency of the crystal rotation (Bollinger et al. 2000).

simplest examples are the string–like crystals. By changing the parameters of the trap (e.g., the amplitude of the r.f. field or the potential di erence between the electrodes) one can obtain two– or three–dimensional structures. Figure 10.14 shows a zigzag–like structure formed by 11 ions (10 199Hg+ ions and one unidentified ion which did not produce fluorescence). The ions do not form a helix but remain in a plane, because of the weak azimuthal asymmetry of the confinement. Well–ordered structures containing more than 105 Mg+ ions have been investigated by Drewsen et al. (1998). The structures are stretched significantly along the trap axis and have up to 10 concentric cylindrical shells around the central string (see Fig. 10.15). The transitions associated with the appearance of new

Fig. 10.15.

390	NONNEUTRAL PLASMAS

50 µm

Fig. 10.14. Image of a zigzag crystal containing 10 199Hg+ ions and one unidentified ion which does not fluoresce (Raizen et al. 1992).

Experimental image (a) and results of the MD simulations (b) of a stretched crystal containing 3500 ions (Drewsen et al. 1998).

shells along the axis can be seen, which is fully in agreement with the results of the MD simulations performed in the same work. Drewsen et al. 1998 studied the structural transitions occurring in strongly coupled nonneutral plasmas with an increase of the temperature (i.e., decrease of γ). As γ decreases the shell structure becomes less pronounced, and at the lowest achieved value γ 4 the shells practically disappear. Interesting investigations of structural properties of two–component ionic crystals have been performed by Hornekaer et al. (2001), where 24Mg+ and 40Ca+ ions were used. Equations (10.3), (10.5), and (10.6)

STRONG COUPLING AND WIGNER CRYSTALLIZATION

391

	100
rate (kHz)	80
	60
Fluorescence	40
	20

0

–150

cold beam (q = 0.2) crystallizing beam (q = 0.33) crystalline beam (q = 0.31)

phase transition

–100

–50

0

Relative detuning (Γ/2)

Fig. 10.16. Fluorescence intensity of the ion beam versus the detuning frequency of the cooling lasers ∆ω (frequency is in units of the line half–width Γ = 2π ·42.7 MHz). The beam contains 1.8 · 104 ions Mg+. The arrow indicates inhomogeneity of the curve at q = 0.33 suggesting the transition to a crystalline state (Sch¨atz et al. 2001).

suggest that the axial confinement in the Paul trap does not depend on the ion mass, whereas in the radial direction the confinement is mass–dependent. Therefore, the lighter Mg+ ions concentrate in the vicinity of the trap axis and form an inner structure consisting of cylindrical shells, which resembles very much the structures observed in one–component crystals. The outer part of the cloud consists of Ca+ ions and has a spheroidal shape which is practically una ected by the Mg+ ions – only a few shells close to the axis acquire the cylindrical form. Discussion of earlier work on the centrifugal separation of ions (mostly in Penning traps) is presented in the review by Dubin and O’Neil (1999).

The phase transition of gaseous–like ion beams into one–, two–, or three– dimensional crystalline structures has been recently observed in the r.f. quadrupole storage ring PALLAS (Sch¨atz et al. 2001; Schramm et al. 2001, 2002); the setup is described in section 10.1.4. Typical behavior of a cold ion beam of about 1.8 · 104 particles is shown in Fig. 10.16 for di erent values of the stability parameter, q = 2eUrf /mΩ2r02 (Sch¨atz et al. 2001). The fluorescence grows as the detuning of the cooling lasers, ∆ω = ω2 − ω1, decreases. This growth, however, is limited because in close proximity of the resonance, the light absorption can be e ective only for one of two photons having almost the same frequencies and moving in the opposite directions. The velocity distribution of ions broadens and, therefore, the fluorescence decreases as the resonance is