Los Angeles , California 90095 - 1594 UNIVERSITY OF CALIFORNIA • LOS ANGELES Instability Driven Radial Transport in a Helicon Plasma

Known for their ability to produce high densities at low power, helicon discharges have found many practical uses. However, with light gases it has been found that the plasma density saturates, then falls, as the magnetic field is increased. This can be explained by the onset of a drift-type instability, whose threshold agrees well with linear theory. Measurements of radial and axial particle fluxes show that the enhanced losses occur via a new mechanism: The fluctuation-induced radial transport stops short of the boundary and serves only to move plasma into flux tubes of larger cross section, where it is lost by axial flow. 52.25.Fi, 52.25.Gj, 52.35.-g, 52.50.Dg Typeset using REVTEX 1 Helicons are cylindrically bounded whistler waves lying on the fast R-wave dispersion branch of electromagnetic plasma waves [1–4]. Radio-frequency (rf) plasma sources based on helicons are noted for their unusually high ionization efficiency. These sources are used in plasma processing of semiconductors, ionospheric plasma research, ion lasers, general plasma physics experiments, and plasma thruster research [5–10]. Sakawa et al [11] have reported that the plasma density n in this discharge does not monotonically increase with magnetic field strength in light gases. This has also been observed in several other helicon experiments [12–15]. Furthermore, the possibility of ion mass effects in these discharges has been borne out by the work of Scime et. al [6]. Efforts to increase the maximum attainable plasma density in these sources are driven by their applicability to space propulsion. Arguably the most important characteristic of magnetized plasma sources is the scaling of equilibrium density with magnetic field strength, n0(B0), and understanding the physics behind this trend is critical. Investigation of the n0(B0) scaling was performed on the apparatus shown in Fig. 1, with magnetic field strength ≤ 1.5 kG and 2 kW of 13.56 MHz power into a right-helical [16] antenna. Gas fill pressure was kept constant at 8 mTorr, and neutral depletion effects were determined to be negligible. Characterization of fluctuations was done by measuring the magnitude and phase of density and floating potential fluctuations (ñ, φ̃) with a tripleprobe technique employed in magnetic fusion [17,18]. Ẽ was determined from the relation Ẽ = −ikθφ̃; where kθ is the azimuthal wavenumber calculated from the phase difference between the two probe tips measuring φ̃. Density fluctuations were measured by biasing the third tip well into the ion saturation regime. Radial profiles of the azimuthal wavenumber at various B0 revealed a discrete azimuthal mode structure. All measurements were taken 30 cm downstream of the antenna mid-plane, unless specified otherwise. Measurements of n0(B0) are given in Fig 2b for all gases used in this experiment. This figure also shows a spectrogram of the density fluctuations ñ(B0) in neon. Note that the saturation in plasma density and onset of strong, low-frequency fluctuations in ñ occur at the same magnetic field, which will be referred to as Bcrit. This correlation was seen in 2 all gases used in the experiment, and Bcrit increased with increasing ion mass. Heavier gases (neon, argon) showed a saturation, while the light gases (hydrogen, helium) suffered a decrease in plasma density beyond Bcrit. Fig. 2c shows the magnitude of the dominant frequency component of ñ fluctuations in helium, illustrating the onset of strong low-frequency fluctuations that continue to grow past Bcrit. The instability threshold can be predicted from a two fluid model employing the usual linear perturbation technique [19] to describe the fluctuating quantities ñ and φ̃ in a slab geometry. Collision rates were dominated by neutrals. Ions were assumed to be at room temperature, thus eliminating magnetic viscosity effects; electron inertia was neglected; fluctuations were assumed to be electrostatic (β 1 for this experiment) and were Fourieranalyzed in the θ (y) and z directions; wave-particle resonance effects were neglected; and the density n0(x) and space potential Vs(x) profiles were allowed to vary. Perturbations had the form exp[i(mθ + kzz − ωt)], and were assumed to be absolute, with kz real and ω = ωR+ iγI complex. The detected fluctuations had ω small enough to justify the ordering ω, ωEB(x), ω∗(x) Ωc (1) where the E0x × B0z drift, electron diamagnetic drift, and ion cyclotron frequencies are defined respectively by ωEB(x) = kyvE(x) = ky V ′ s (x) B0 ω∗(x) = ky KTe eB0 n0(x) n0(x) ; Ωc = eB0 Mi . (2) In the above equations, ky is the azimuthal wavenumber; x corresponds to the radial coordinate in a slab geometry; and ′ denotes ∂/∂x. Coupling the linearized equations of motion and continuity for each species in a singly ionized plasma via the plasma approximation ni = ne results in a quartic dispersion relation 0 = ω̂ C2 s ( −ω∗νe + ik zv th ω̂νe + ik2 zv 2 th ) + (k x + k 2 y)(ω̂ + iνi) Ωc(Ωc + v′ E) 3 −x ′ 0/n0(ω̂ + iνi) Ωc(Ωc + v′ E) + kyn ′ 0/n0 Ωc + v′ E − k 2 z ω̂ + iνe + i2kxkyv ′ E(ω̂ + iνi) 2 Ωc(Ωc + v ′ E) 2 − 2k yv ′ E(ω̂ + iνi) Ωc(Ωc + v′ E) 2 , (3) where νj is the collision rate for each species, vth the electron thermal velocity, Cs the ion sound speed, and ω̂ = ω − ωEB the doppler shifted eigenfrequency. The problem was re-cast in cylindrical geometry, and radial eigenfunctions of ñ and φ̃ were found as well. Equation 3 yields the local frequency and growth rate for a given set of measured equilibrium profiles n0(x), Vs(x), and parameters B0, Te, and Mi. The perpendicular flux driven by the fluctuations, Γ⊥ =< ñṽ >, was computed using ñ from the ion continuity equation along with ṽ from the ion equation of motion, giving a function that depended on the magnitude of φ̃ and θnφ, the phase angle between ñ and φ̃. Fig. 2c shows the variation of the predicted growth rate γ with B0, along with the measured magnitude of the low frequency fluctuations in φ̃ and ñ in helium. Measured fluctuation frequencies were consistent with predicted values. The maximum predicted growth rate is in excellent agreement with the measured maximum in |φ̃| and onset of strong fluctuations in |ñ|. The model predicted instability onset for all gases used in the experiment, as shown in Fig. 3. Note that the predicted growth rate in Fig. 2c decreases after reaching a maximum, while ñ/n0 continues to grow. This is due to the linear nature of the model, which correctly predicts behavior up to just beyond the onset, but not in the turbulent saturated regime. The triple probe used in this experiment allowed direct measurement of the radial flux Γ⊥ driven by the instability at discreet radii [18]. This is compared in Fig. 2d to the predicted radial flux calculated from the measurement of |φ̃| and θnφ. Note that the instability onset(Fig. 2c), saturation (before decrease) in n0(B0) (Fig. 2b), predicted maximum growth rate(Fig. 2c), and maximum in Γ⊥ (Fig. 2d) are all correlated to the same Bcrit ≈ 760G for helium. The behavior of Γ⊥ with radius in helium, given in Fig. 4, shows that Γ⊥(r) is restricted 4 to the interior of the plasma column. This localization was observed over the entire range of B0 in Fig. 2d where Γ⊥(r) was significant. The measurement of ñ(B0) in Fig. 2c shows that the discharge has entered a strongly turbulent regime above Bcrit. Direct correlation of n0(B0) with instability characteristics becomes extremely difficult. The fluctuation-driven radial flux decreases instead of remaining constant and is localized in radius. This appears to be inconsistent with the density decrease at the highest magnetic fields, unless their is another loss mechanism. Axial loss was thus investigated by measuring radial profiles of the ion axial flow velocity in terms of the mach number M [20], where v||i = MCs, at the antenna mid-plane. These are shown in Fig. 5 and indicate a greatly enhanced axial loss above Bcrit. An explanation can be formulated by considering both the axial and radial losses. Consider dividing the plasma column in to two regions, a central tube of radius 2cm and an outer shell extending from r = 2cm [where Γ⊥(B0) was measured] to the outer boundary. Before the instability onset, axial losses from the outer shell dominate. In the region of B0 where the instability is growing and Γ⊥ is strong, axial losses in the central region are starting to become significant, while radial losses are greatly enhanced, thus transporting particles to the outer shell region. These particles, along with the background ones, are then lost axially over a larger cross-sectional area than that of the central region and hence generate a greater loss rate. At the highest B0, Fig. 5 shows that axial losses are much larger than at lower fields and that they occur mostly in the outer shell. We have found that the saturation of n0(B0) correlates with the onset of a low-frequency instability whose characteristics agree with theory in the linear regime. At fields well beyond the threshold, ñ continues to increase and n0 to decrease with B0, but the < ñφ̃ > correlation shows that the radial flux maximizes at r ≈ 2cm, falling to ≈ 0 at the edge, as if there were an internal transport barrier. The loss is then dominated by axial loss to the endplates at the acoustic velocity, the radial transport serving only to move plasma into flux tubes of larger cross-section. At the highest magnetic field of 1500G, the radial flux appears to be small everywhere, but we suspect that the < ñφ̃ > correlation technique fails in this highly 5 turbulent regime. Thus, the n0 vs. B0 behavior of helicon discharges is in good agreement with linear instability theory in regions in which that theory is applicable. The authors wish to thank G. Tynan for useful discussions.