CONFINEMENT OF NONNEUTRAL PLASMAS

381

Fig. 10.5. Side view of the plasma cloud confined in an asymmetric Penning trap and consisting of 8 · 104 ions 9Be+ (Dubin and O’Neil 1999).

tures 10 K was achieved with Doppler laser cooling, and then the evaporation method was employed. The condensate can be visualized both in real space, due to a sudden increase of the atom density in the middle of the trap, where the confining potential has a minimum, and in momentum space, because of the steep peak in the velocity distribution at zero value. The first observations of the Bose–Einstein condensates triggered enormous theoretical and experimental activity. Many theoretical hypotheses have been confirmed – among these, e.g., the shape of the condensate wavefunction and its dependence on the mutual atom interaction, the temperature dependence of the condensed atom fraction and the thermal capacity, etc.

Let us focus again on the properties of nonneutral plasmas. The shape of the plasma cloud can be determined experimentally by measuring light scattering or the ion fluorescence (Dubin and O’Neil 1999; Bollinger et al. 2000). Figure 10.5 shows the side view of a small plasma cloud ( 8·104 of Be+ ions) in the Penning trap. Ions are crystallized into a bcc lattice. This can be seen both with the analysis of Bragg scattering and by direct observation of the luminescence (the latter employs the stroboscopic e ect, which allows us to “freeze” the rotation of the cloud at frequency 102 kHz).

10.1.3Linear Paul trap

The Paul trap is a quadrupole consisting of four parallel cylindrical electrodes, as shown in Fig. 10.6 (Raizen et al. 1992). Two diagonal electrodes are grounded, and a r.f. voltage is applied to two other electrodes. Each cylinder consists of two or three segments, so that a positive potential at edges of the trap together with

382	NONNEUTRAL PLASMAS

b

Fig. 10.6. Schematics (a) and side view (b) of the Paul trap (Raizen et al. 1992).

the r.f. field provides the axial confinement. Similar to the case of the Penning trap, the potential in the central region can be well approximated by a parabolic profile, Eq. (10.3). Now, the parameter β is given by the following expression:

			β =	ω2		1					(10.5)
					r	−		,
				ω2			2
					z
where		Z2U 2							2kZiUdc
ω2	=					ω2 =					(10.6)
		i	rf		,					.
		2M 2r4Ω2							Mi
r						z
		i	0

Here Udc is the magnitude of the positive bias, Urf and Ω are the amplitude and frequency of the r.f. voltage, respectively, r0 is the distance measured from the central axis of the trap to the electrode surface, and k is some constant coe cient which takes into account the particular geometry. In Paul traps one can obtain long quasicrystalline structures (see the discussion in Section 10.2).

10.1.4Storage ring

Figure 10.7 shows the ion storage ring PALLAS which was employed to investigate strongly coupled OCP plasmas (Sch¨atz et al. 2001; Schramm et al. 2002).

CONFINEMENT OF NONNEUTRAL PLASMAS

383

Fig. 10.7. Axial and radial cross–section of the r.f. quadrupole storage ring PAL-

LAS (Schramm et al. 2002).

Basically, the storage ring is a rf quadrupole trap which is very similar to the Paul trap. Sixteen segmented drift tubes enable the manipulation of ions along the orbits. The radial confinement is provided by the r.f. field with amplitude Urf = 200 V and frequency Ω = 2π · 6.3 MHz applied between the quadrupole electrodes. In the storage ring, the ions are confined in a parabolic potential well,

	Φ(r) =	Miωr2	r2	,	(10.7)
		2

where ωr is given by Eq. (10.6) with r0 = 2.5 mm.

In order to accumulate ions in the storage ring, a weak collimated beam of 24Mg atoms is used. The atoms are ionized by a focused electron beam. Periodically, the trapped ions cross the oppositely directed laser beams, as shown in Fig. 10.7. Both lasers are tuned to frequencies close to the 3s2S1/2−3p2P3/2 transition in the 24Mg+ ion, which corresponds to the wavelength 280 nm and provides the conventional Doppler laser cooling. The resonance ion fluorescence is detected by a photomultiplier and a CCD matrix.

After the accumulation of a su cient number of ions with velocities 1000 m s−1, the resonant light pressure of the (first) laser beam directed along the ion motion is employed to accelerate the ensemble. The frequency ω1 grows continuously, until the ions occur in resonance with the (second) decelerating beam. A fixed value of frequency ω2 is chosen to provide the necessary stationary velocity of the ion beam (in experiments by Sch¨atz et al. 2001 the velocity was 2800 m s−1). The longitudinal velocity distribution in the ion beam is substantially narrowed, because of the dispersive nature of the force produced by the laser beams. The decay of the ion motion in the transverse direction is provided by the mutual Coulomb collisions, and also because of the small noncollinearity of the ion and laser beams.

Apparently, the first reported crystallization of ions in a storage ring was observed by Dement’ev et al. (1980) and Budker et al. (1976) in experiments with