10

NONNEUTRAL PLASMAS

Nonneutral plasmas are plasmas consisting of particles with a single sign of charge. These can be pure electron or positive ion plasmas (the latter may include several species), positron plasmas, as well as electron–antiproton plasmas. Nonneutral plasmas provide some unique research opportunities that are not available with “conventional” quasineutral plasmas (Davidson 1974; Davidson 1990; Dubin and O’Neil 1999). Due to strong the repulsion between like–charged particles, external fields are required to confine nonneutral plasmas. The confinement, which is usually provided by static electric and magnetic fields, can last a very long time – a few hours or even days. Since recombination cannot occur, nonneutral plasmas can be cooled to ultracryogenic temperatures (< 1 mK) where the kinetic energy of ions is much smaller than the energy of the mutual electrostatic interaction, and therefore the formation of liquid– and crystal–like states is possible.

10.1Confinement of nonneutral plasmas

Nonneutral plasmas can be confined by employing various electric and magnetic traps. Electrons can be localized directly on the surface of liquid helium, forming a two–dimensional system where the Wigner crystallization is observed. By using laser cooling of ions, strongly coupled nonneutral plasmas can be obtained and studied in Penning and Paul traps. Recently, the remarkable example of plasma condensation, the “crystalline beams”, was discovered by cooling the ions in a storage ring. Let us discuss briefly these methods of confinement.

10.1.1Electrons on a surface of liquid He

The interaction of a free electron with atoms of a liquid has two contributions: The long–range attraction due to the polarization and the short–range repulsion due to the exchange interaction. The former one causes the potential energy of the electron to decrease (as compared with the vacuum value) whereas the latter one increases the energy. Hence, both the sign and the magnitude of the mean potential energy in a dielectric medium, V0 (viz., the bottom of the conduction band for the electron embedded into the liquid), is determined by the competition of the polarization attraction and the exchange repulsion. For atoms or molecules with small polarizability, like He, Ne, H2, the exchange repulsion prevails and, hence, V0 > 0. This implies that an electron with a kinetic energy smaller than V0 cannot penetrate into the liquid. However, being outside the liquid, the electron is attracted to the surface by the image force. The corresponding potential is

376

CONFINEMENT OF NONNEUTRAL PLASMAS

377

Energy

eV .

.eV

Liquid

Gas

Fig. 10.1. Illustration of the electron energy spectrum near the surface of liquid He (Shikin

and Monarkha 1989).

	V (z) =		Q	,	Q =	e2(ε − 1)	,	(10.1)
		− z				4(ε + 1)

where ε is the dielectric permittivity of the liquid and the z–axis is pointed perpendicular to the surface. In the liquid (z < 0) we have V (z) V0. Thus, we have a one–dimensional potential well (see Fig. 10.1) where the electron can be trapped in the direction perpendicular to the surface, whereas along the surface it can move freely.

For liquid He, the potential barrier, V0 1 eV, is large compared to the electron binding energy. Therefore, with su cient accuracy one can set V0 = ∞ and thus employ Ψ|z=0 = 0 as the boundary condition for the electron wavefunction. In this case, it is easy to see that the problem of the energy spectrum of the trapped electron is reduced to that of the hydrogen atom (with the substitution e2 → Q),

	2k2	+	mQ2
En(k) =				, n = 1, 2, . . . ,	(10.2)
	2m		2 2n2

where k is the two–dimensional wavevector of the electron along the surface. Due to the low polarizability of the liquid He the binding energy of the electron in the ground state is low (E1(0) 8 K V0) and, hence, the assumption V0 = ∞ is well justified. Electrons are localized at 10−6 cm from the He surface and, therefore, the particular structure of V (z) at shorter (atomic) distances is not important. The properties of electrons localized on the surface of liquid He are discussed in more detail by Cole (1974), Shikin (1977), and Shikin and Monarkha (1989).

378	NONNEUTRAL PLASMAS
	Electrons	Gaseous helium

Liquid helium

Positively charged metal electrode

Fig. 10.2. Sketch of the experiment with electrons localized near the surface of liquid He.

There exists a simple method to create localized electrons on the surface of a liquid He (Shikin and Monarkha 1989). The source of electrons (e.g., corona discharges or radioactive isotopes) is located above the liquid surface, whereas the positively biased planar electrode is embedded into the liquid, close to the surface (see Fig. 10.2). The equilibrium number density of electrons on the surface, ns, is determined by the condition that the total electric field is equal to zero above the surface, which yields E = 2πens. By varying the magnitude of the bias on the electrode one can easily change ns in a very broad range, from 105 to 109 cm−2. The electron subsystem is not degenerate – even for ns = 109 cm−2 the Fermi energy is small enough, εF/k = π 2ns/km ≤ 10−2 K. Depending on the values of E and T , the coupling parameter of the two–dimensional electron subsystem, γ = π1/2e2n1s/2/kT , can be changed from 0 up to 102, thus covering the regions of an ideal gas, electron liquid and the Wigner crystal. The upper limit of γ is determined by two factors: the instability of the charged surface at high ns (and, hence, at large E ), and degeneration of electrons at low temperatures.

10.1.2Penning trap

The simple example of the Penning trap is shown in Fig. 10.3. The trap was proposed by Penning (1936) and since then it has been used extensively to confine charged particles. A conductive cylinder is divided into three sections: The central one is grounded, whereas the other two sections are biased positively (here, we suppose the ions to be charged positively). The external magnetic field B is parallel to the cylinder axis. The ions collect themselves in the region of the central grounded section, where the radial confinement is due to the magnetic field and the axial one is electrostatic. The radial confinement is associated with the azimuthal rotation of ions, which induces the radial Lorenz force balancing the centrifugal, electrostatic, and pressure forces pointed radially outwards.

Usually the size of the ion cloud is small compared to the trap dimensions, so that the potential well of the confinement is almost a parabolic one. The equilibrium shape of the cloud trapped in such a confinement is a rotational ellipsoid. The cloud rotates due to the pondermotive force, so that the analysis

CONFINEMENT OF NONNEUTRAL PLASMAS

379

E

B

E

Plasma

E

Fig. 10.3. Sketch of the Penning trap (Penning 1936; Dubin and O’Neil 1999).

should pe performed in the rotating reference frame. Thus, the e ective potential for charged particles confined in the Penning trap and rotating with frequency ω is (Dubin and O’Neil 1999)

	Φ(r, z) =	Miωz2				(z2 + β r2),					(10.3)
			2
where the parameters β and ωz are
β =	ω(Ωc − ω)		−	1	,		ω2	=	2kZiU	.	(10.4)
	ωz2			2
							z		Mi

Here Mi is the ion mass, U is the potential di erence, ωz is the eigenfrequency of the axial particle oscillations, k is a geometrical factor, and Ωc = eB/Mic is the ion cyclotron frequency. The parameter β determines the symmetry of Φ(r). For β = 1, the potential is spherically symmetric. For β > 1, the cloud is stretched along z–axis, and for β < 1 it is compressed.

The possibilities of the experimental investigation of strongly coupled nonneutral plasmas improved dramatically due to development of Doppler laser cooling (Chu 1999; Cohen–Tannoudji 1998; Phillips 1998). The main idea of laser cooling is easy to understand (Wineland et al. 1985; Dubin and O’Neil 1999). The laser beam which has a frequency slightly below the frequency of one of the electron transitions, is directed through the plasma (see Fig. 10.4). For ions having the velocity component directed opposite to the direction of the beam propagation, viz., k · v < 0 (where k is the wavevector of the laser beam and v is the ion velocity), the transition energy is shifted towards resonance due to the Doppler e ect. Hence, these ions absorb more e ectively than those moving in the direction of the beam. After the photon absorption, the ion velocity is decreased by ∆v = − k/Mi. The subsequent spontaneous reemission is spherically symmetric and, therefore, its contribution to the ion momentum is equal on average to zero. Thus, the net e ect of the absorption and reemission processes is cooling of the plasma.

The original idea of laser cooling was proposed by H¨ansch and Schawlow (1975), and Wineland and Dehmelt (1975). Three years later it had been implemented in experiments with ions of Mg+ by Wineland et al. (1978) and Ba+ by Neuhauser et al. (1978). The lower temperature limit (Doppler limit) which can

380	NONNEUTRAL PLASMAS
v	_
	hk
	a
	_
	v hk

b

c

Fig. 10.4. The sequence of processes that lead to the decrease of mean kinetic energy of ions during laser cooling: An ion moving with velocity v interacts with a photon having momentum k (a); the ion absorbs the photon and slows down by k/Mi (b); reemission of the photon in an arbitrary direction causes the ion velocity to decrease on average (Phillips 1998).

be achieved with the laser cooling is of the order of Γ (Letokhov et al. 1977, and Wieman et al. 1999), where Γ is the rate of spontaneous reemission from the excited state. This temperature is determined by the balance between the laser cooling and heating processes caused random phonon absorption and emission (Phillips 1998). The Doppler limit is typically about a few tenths of a mK for the allowed dipole transitions. By now, there have been a few methods developed to cool atoms and ions to temperatures much below the Doppler limit (Chu 1999; Cohen–Tannoudji 1998; Phillips 1998; Wieman et al. 1999). These are, e.g., sub– Doppler laser cooling and evaporative cooling (Wieman et al. 1999). The latter is analogous to the cooling of hot water in a tea–cup, due to the evaporation of more energetic molecules. After the evaporation, the energy of the remaining ions and atoms is redistributed due to collisions and the resulting temperature is decreased. Evaporative cooling allows one to decrease the temperature down to 50 nK or even lower (Wieman et al. 1999).

One of the most remarkable recent achievements of the laser cooling technique is the experimental observation of macroscopic quantum systems – the Bose–Einstein condensates predicted by Einstein in 1924. Examples of Bose– Einstein condensation are, e.g., superfluidity and superconductivity. The condensation of rarefied atomic gas was first achieved by the cooling of Rb atoms confined in a magneto-optical trap. The temperature in the experiment was low enough ( 200 nK) to ensure the overlapping of the de Broglie wave envelopes of individual atoms (Anderson 1995). Under these conditions most of the atoms condense in the ground state with zero momentum. Pre–cooling to tempera-