La subsunción real del consumo bajo el capital es la subsunción real del

Experiments with a number of helicon reactors over the years have shown that a density peak is present at low magnetic fields (<5 mT) as the magnetic field is increased. While undoubtedly seen before in early helicon experiments, Chen seems to be the first to have explicitly reported such peaks in the literature [84]. Using a helicon reactor with a 2 cm diameter glass tube, a density peak was seen between 0.5₋3 mT at 0.5 Pa with argon gas for an input power of 1600 W. The maximum density of this peak was about 6_×1018m−3, some 30₋40% larger than the density before or after the peak. A number of other researchers have reported similar density peaks occurring for a similar range of magnetic

§1.4 Helicon Waves 31

fields [73, 85, 86, 87, 88, 89]. Carter [89] observed a density peak at 1 mT, about 1 mT wide, with maximum densities of 3.5_×1018m−3 at 0.13 Pa for an input power of 1000 W. Shinohara [87] studied a number of different helicon antennas under a range of operating conditions, and observed density peaks with most of the antennas. Wang [85] investigated different background gases and excitation frequencies, and found that the magnetic field at the maximum of the density peak was approximately proportional to the applied frequency. Wang also measured the antenna loading resistance, and found peaks well correlated with the measured density peaks. However these low-field peaks were not seen in all helicon reactors [90] (see also Fig. 7.8 in Boswell’s original thesis [75]). At low magnetic fields, the condition ω << ωce is no longer valid, and the helicon dispersion relation accounting for finite electron mass needs to be used [78]. This is repeated below, by writing Eqn. 1.59 in a slightly different form as

kkz− ω ωce k2 = µ0n0qω B0 (1.77)

Chen [68] has studied these density peaks using the HELIC computational code of Ar- nush [91, 92], and peaks in the antenna loading resistance are indeed observed if reflection at axial boundaries is considered. This led to the proposal that constructive interference between waves from the antenna and reflected waves are responsible for the low-field peak. The code used is a collisional code for helicon waves that treats the plasma as a cold di- electric tensor (somewhat similar to that used in Section 1.4.2), and is two-dimensional, allowing non-uniform radial density profiles, most antenna geometries, but restricted to a uniform magnetic field. By then applying conducting or insulating axial boundary con- ditions, the antenna loading resistance can be found by solving Maxwell’s equations. For a loop antenna, the results show that the power absorption is peaked under the antenna in the near field, while the radial power absorption is found to be peaked at the plasma boundary, due to the TG mode. The resistance peaks in the code are seen to be approx- imately proportional to the magnetic field at the peaks, and thus shift with increasing applied field [68].

Since the code treats the plasma as “fixed”, it does not model plasma transport, and thus requires the density as an input. It is therefore not able to predict a density peak. However, from power balance arguments, it is expected that the density is related to the antenna resistance, since a larger resistance means more power is deposited within the plasma, and thus the density would increase [68]. m= 0 antennas16 were seen to produce the largest resistance peaks, and require reflecting end plates for this to occur. Cho [69] has written a similar independent code, and revisited the low-field peak problem, since as pointed out, it is unclear that wave reflection at axial boundaries should be most effective at low fields. His calculations show that the plasma resistance can be large for system eigenmodes that occur near the magnetic field where the helicon and TG waves are coupled,

and depending on the antenna geometry, can be peaked a low fields [69]. Similarly to Chen [68],m= 0 antennas require reflecting end plates, butm= 1 antennas can produce peaks regardless of reflection. A resonance occurs when a EM wave matches a particular system eigenmode, and the plasma resistance can then be a local maximum in phase space. When conducting end plates exist, interference causes a wave with a particular wavelength to be most strongly excited17. For an antenna of fixed length however, the antenna length itself gives a particular wavelength, which can subsequently be enhanced by wave reflection.

m= 1 antennas most strongly excite waves with a wavelength equal to twice the antenna length (or a harmonic), thus causing a resistance peak at the corresponding magnetic field and density [69]. Results show that this peak moves to higher fields as the density increases, and that the peak density is proportional to the applied excitation frequency, confirming experimental results [69].

Sato [88] has recently performed detailed low-field experiments in a reactor with a uni- form magnetic field, and suggested Landau damping as a possible cause. A phased helical antenna was used, with rf signals being phased temporally, thus allowing the direction of rotation of the rf excitation fields to be controlled. Density peaks were observed for powers of between 200₋2000 W, with the peak moving to higher magnetic fields as the density increased. At these powers, the measured densities were in the range 1_×1017₋5_×1017m−3 for a pressure of 0.05 Pa. They found that the magnetic field direction giving a density peak varied according to the rotation direction of the rf fields, and the low-field peaks must be being induced through an electron interaction [88]. At low magnetic fields the TG wave propagates within the plasma, unlike at high fields where it is confined to the plasma edge. Thus mode conversion from helicon to TG was felt to be inappropriate un- der these conditions [72, 88]. Landau damping was investigated and thought to contribute to the power transfer, however no fast electrons were measured. The phase velocity of TG waves was seen to be close to the electron thermal speed, and thus Landau damping could be very effective under these conditions. The density peaks were explained as a local matching between the wavelength of the EM fields of the antenna, and a wave generated by the dispersion relation (which depends on the density and applied magnetic field) [88]. At very low magnetic fields, no TG wave is present as the magnetic field is too small so that it is cut off (that is, from the dispersion relation it is prohibited from propagating). When the magnetic field is too large though, the wave phase velocity becomes too high so that electrons cannot absorb energy by Landau damping, and inefficient wave excitation occurs due to a mismatch of the imposed wavelength from the antenna [88]. However this hypothesis has not been verified.

Low-field helicons are potentially attractive for processing or propulsion applications, since the lower required magnetic fields can result in more economical systems and the resulting densities are within a convenient range. Chen [58, 90] has performed studies with low-field helicons in a processing context, and has recently started using permanent

17_{It is as if the reflecting end plate creates a mirror image of the antenna, with the antenna “length”}