aa r X i v : . [ phy s i c s . p l a s m - ph ] A ug On the density limit in the helicon plasma sources
Igor A. Kotelnikov ∗ Budker Institute of Nuclear Physics, Novosibirsk, Russia andNovosibirsk State University, Novosibirsk, Russia
Existence of the density limit in the helicon plasma sources is critically revisited. The low-and high-frequency regimes of a helicon plasma source operation are distinguished. In the low-frequency regime with ω < √ ω ci ω ce the density limit is deduced from the Golant-Stix criterionof the accessibility of the lower hybrid resonance. In the high-frequency case, ω > √ ω ci ω ce , anappropriate limit is given by the Shamrai-Taranov criterion. Both these criteria are closely relatedto the phenomenon of the coalescence of the helicon wave with the Trivelpiece-Gould mode. Weargue that theoretical density limits are not achieved in existing devices but might be met in thefuture with the increase of applied rf power. PACS numbers: 52.35.Hr; 52.50.Qt; 52.50.Sw;
I. INTRODUCTION
One of the most challenging problems in the theoryof helicon plasma sources is a supposed existence ofthe plasma density limit [1–5]. For the helicon plasmasources, it is conventional to consider the density of theorder of cm − as very high, but preproduction ofplasma for fusion devices needs the density of the orderof cm − , at least. For this reason, in this paperwe critically revisit the foundations of the theory of he-licon heating, assuming that rf power is sufficiently largefor a plasma source to operate in the so called W-mode(helicon-Wave mode) as explained in Refs. [6, 7].We distinguish a low-frequency and a high-frequencyregimes of operation of a helicon source where the fre-quency ω is respectively smaller and larger than the hy-brid cyclotron frequency √ ω ce ω ci .The low-frequency regime ( ω ci < ω < √ ω ce ω ci ) ischaracterized by existence of the lower hybrid resonance.We reexamine the accessibility condition of the resonancein a radially inhomogeneous cylindrical plasma column,which is uniform along its axis. This condition is knownas the Golant-Stix criterion [8–10]. We provide a newderivation of this criterion, which reveals its connectionto the effect of the wave coalescence. We find that adensity limit indeed exist in this regime. However it hasnot a feature of a threshold since the limiting densitydepends on the value of the longitudinal refractive index N k = k k c/ω (the larger N k , the larger the limit), whereasthe spectrum of the plasma oscillations excited by an an-tenna is usually quite wide. Therefore, speaking abouta limiting density, we imply a value of N k , which corre-sponds approximately to the maximum in the absorptionspectrum of the antenna.In the high-frequency regime ( √ ω ce ω ci < ω < ω ce ), hy-brid resonances are not available, and the density limitoccurs because of the coalescence of the helicon andTrivelpiece-Gould (TG) waves. Corresponding density ∗ [email protected] limit is given by the Shamrai-Taranov criterion [3].The paper is organized as follows. The main equationsare reviewed in Secs. II and III. The Golant-Stix crite-rion is discussed in Sec. IV. A limiting plasma densityin the low-frequency mode of operation of the heliconsource is found in Sec. V. An optimal magnetic field isevaluated in Sec. VI. The high-frequency mode of the he-licon source is considered in Sec. VII, where a new sim-ple derivation of the Shamrai-Taranov criterion is given.Finally, in Sec. VIII we discuss how the low-frequencyregime matches the high-frequency regime of operation.
II. DISPERSION EQUATION
We consider a simple plasma consisting of the elec-trons and a single kind of ions. In the approximation ofcold collisionless plasma, both plasma species are char-acterized by the two quantities each, the Langmuir fre-quency ω ps = p πe s n s /m s and the Larmor frequency Ω s = e s B/m s c with the subscript s = e standing forthe electrons and s = i for the ions. Assuming that themagnetic field B is directed along the axis z of the axialsymmetry of the plasma column, the permittivity tensorreads ε ↔ = ε ig − ig ε
00 0 η , (1)where ε = ε + + ε − − ω p (cid:0) ω + Ω e Ω i (cid:1) ( ω − Ω e ) ( ω − Ω i ) , (2a) g = ε + − ε − − ω p ω (Ω e + Ω i )( ω − Ω e ) ( ω − Ω i ) , (2b) η = 1 − ω p ω , (2c) ε ± = 1 − ω p ( ω ∓ Ω e ) ( ω ∓ Ω i ) , (2d) ω p ≡ ω pe + ω pi . (2e)These expressions are derived from the well-known for-mulas ε = 1 − X s ω ps ω − Ω s ,g = X s Ω s ω ω ps ω − Ω s ,ε ± = ε ∓ g, using the quasi-neutrality condition ω pe Ω i + ω pi Ω e = 0 . (3)Since Ω e < , below we also use alternative notations ω ce = − Ω e = | e | Bm e c , ω ci = Ω i = e i Bm i c for the cyclotron frequencies when it is more convenientto operate with designedly positive frequencies.A dispersion equation is obtained from the wave equa-tion ǫ ↔ · E + N ( N · E ) − N E = 0 , written in the Fourier domain with N = c k /ω denotingthe vector of the refractive index. The same equation inthe matrix form reads ε − N k − ig N ⊥ N k ig ε − N N k N ⊥ η − N ⊥ E x E y E z = 0 , where N ⊥ = N x = N sin θ , N k = N z = N cos θ , and N y is assumed to be zero. Equating the determinant of thisequation to zero yields the dispersion equation A N ⊥ − B N ⊥ + C = 0 , (4)where A = ε, B = ε + ε − + ηε − εN k − ηN k , C = η (cid:16) N k − ε + (cid:17) (cid:16) N k − ε − (cid:17) . It is quadratic regarding N ⊥ and consequently has twosolutions N ⊥± = ( B ± p B − AC ) / A . (5)However at a given k and θ the same dispersion equationyields 5 eigenfrequencies ω ( j ) as shown in Fig. 1. The twosolutions (5) mean that only 2 eigenmodes at most cansimultaneously propagate at a given frequency ω . Theydiffer by the magnitude of k and their polarizations. Wewill focus on the eigenfrequency ω (2) which is the secondby the magnitude of ω ; it is shown in purple color inFig. 1. kc /ω ce - ω () j / ω c e Helicons(Whistlers)LHW(TG)Alfven (cid:2)(cid:3)(cid:2)(cid:4) (cid:2)(cid:3)(cid:4) (cid:4) (cid:4)(cid:2) (cid:4)(cid:2)(cid:2)(cid:2)(cid:3)(cid:2)(cid:2)(cid:4)(cid:2)(cid:3)(cid:2)(cid:4)(cid:2)(cid:3)(cid:4)(cid:4)(cid:4)(cid:2)(cid:4)(cid:2)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:3) (cid:4) (cid:2) (cid:5) (cid:6) (cid:5) (cid:3) (cid:2) (cid:4) (cid:6) (cid:2) (cid:5)(cid:5) (cid:7)(cid:5) (cid:5)(cid:7)(cid:8)(cid:9)(cid:10)(cid:6)(cid:11)(cid:12)(cid:13) (cid:14)(cid:15) (cid:16)(cid:11)(cid:17)(cid:9)(cid:8)(cid:13)(cid:6)(cid:8)(cid:12)(cid:6)(cid:8)(cid:5)(cid:18)(cid:11)(cid:10)(cid:12)(cid:19) (cid:3) (cid:20)(cid:6)(cid:11)(cid:12)(cid:13)(cid:19) (cid:2) (cid:21)(cid:7)(cid:22)
Figure 1. (Color on-line) Dispersion of the coldplasma waves at fixed angleof propagation: θ = π/ , ω p /ω ce = 3 . The terms he-licons (whistlers), Trivelpiece-Gould (TG) and low hybridresonance (LHR) designatedifferent parts of the samebranch ω (2) of the dispersionrelation (purple curve). Figure 2. (Color online)Dispersion of the cold plasmawaves at fixed k k = ω ce /c , ω p /ω ce = 10 . The heliconsand TG modes are separatedby a coalescence point. Otherbranches: Alfven wave (bluelower curve); slow extraordi-nary, upper hybrid wave (yel-low); ordinary wave (green);fast extraordinary wave (up-per blue). In the region of interest ω ∼ √ ω ce ω ci near the lowerhybrid frequency ω LH (see below) the larger root N ⊥ + corresponds to a lower hybrid wave, which in last yearsis also called the Trivelpiece-Gould (TG) wave [11]. Thesmaller root N ⊥− corresponds to the helicons which werecalled whistlers in the past. However, all these waves lieon the same curve on the graph of the frequency ω (2) ( k, θ ) versus the wave number k if a value of the angle θ betweenthe direction of the wave vector k and magnetic field B isfixed as shown in Fig. 1. The lowest frequency portion ofthe purple curve, where ω (2) ∝ k , corresponds to the fastmagnetosonic wave. When ω (2) > ω ci , it transforms intothe helicons with the dispersion ω (2) ∝ k cos θ . Finally,as we approach the lower hybrid frequency ω LH , which ina dense plasma is approximately equal to ω LH ≈ √ ω ce ω ci the helicons are transformed into the Trivelpiece-Gouldwaves; in this region ω ≈ ω LH and the frequency is almostindependent of k . III. SINGULAR POINTS
Standard approach to the study of wave propagationin a cold magnetized plasma includes a search of singu-lar points. Most textbooks distinguish two kinds of suchpoints, namely the plasma (hybrid) resonances, where N → ∞ , and the cutoffs, where N → ; see e.g.[10, 12, 13]. This exhausts the list of singular points in acold plasma at a fixed angle θ between the wave vector k and the magnetic field B . When considering the propaga-tion of waves in a cylinder with a radially inhomogeneousprofile of the plasma density, we should rather assumethat a fixed quantity is the longitudinal component ofthe wave vector k k = k cos θ , or, equivalently, the longi-tudinal component of the refractive index N k = k k c/ω .For a fixed N k , one more type of the singular points ap-pears, namely the point of coalescence, where the tworoots (5) of the dispersion equation merge. The coales-cence point of the helicon and TG waves is readily seenin Fig. 2. In contrast to Fig. 1, the curve ω (2) ( k k , k ⊥ ) isnot a monotonically rising function of k ⊥ , and, at a fixedplasma density, it assumes a maximal value at the coa-lescence point rather than at the lower hybrid resonanceas in Fig. 1.Below we briefly review all three types of the singularpoints in a simple plasma in order to introduce notationsrequired for further treatment. A. Hybrid resonances
For a fixed N k , the resonant points are determined bythe condition A = 0 . (6)When it is satisfied, the larger of the two solutions (5) ofthe dispersion equation (4) tends to infinity, N ⊥ + → ∞ .Equation (6) determines the so-called plasma or hybridresonances . Since it is linear with respect to the squareof the total plasma frequency ω p = ω pe + ω pi , it has aunique solution ω p, res = (cid:0) ω ce − ω (cid:1) (cid:0) ω − ω ci (cid:1) ω ce ω ci − ω . (7)As a negative value of ω p has no meaning, the right-hand-side of Eq. (7) must be positive. It occurs in the rangeof frequencies ω ci < ω < √ ω ce ω ci , (8)which corresponds to the lower hybrid resonance (LHR).Another range of frequencies ω > ω ce (9)corresponds to the upper hybrid resonance . There are noplasma resonances if ω < ω ci or √ ω ce ω ci < ω < ω ce . (10)Below we will say that Eq. (7) determines the reso-nant electron density n res , which can be expressed fromEq. (7) using the definition of ω p . Note also that so far allthe formulas were exact and, in particular, we have notneglected the small ratio m e /m i . However in the rangeof frequencies ω ∼ √ ω ce ω ci Eq. (7) can be simplified tothe following expression ω pe, res ≈ ω ce ω ω ce ω ci − ω . (11) Taking into account the quasi-neutrality condition (3), itcan be easily reduced to the form ω LH ≈ ω pi + 1 ω ce ω ci , (12)known in the literature. It approximately determines thefrequency of the lower hybrid resonance and can be foundin many textbooks while an exact expression for ω LH canbe readily derived from Eq. (7). B. Cutoffs
The cutoff points N ⊥ = 0 are found from the equation C = 0 . (13)Since C = η (cid:16) N k − ε + (cid:17) (cid:16) N k − ε − (cid:17) , the first of the cutoffs is found from the equation η = 0 .It occurs at the density such that ω p = ω = ω p, cut . (14)This density is usually very low. For example, for thefrequency ω/ π = 13 . MHz, which is often used inthe helicon plasma sources, it is as low as n e = 2 . × cm − .Two more cutoffs are found by equating N k to ε ± ,which yields ω p = (cid:16) − N k (cid:17) ( ω ± ω ce ) ( ω ∓ ω ci ) . For any sign of the factor (1 − N k ) only one of them hitsthe interval ω ci < ω < √ ω ce ω ci . In case of slow wavewith N k > and ω ci < ω < ω ce , ω p = (cid:16) N k − (cid:17) ( ω ce − ω ) ( ω + ω ci ) = ω p, cut . (15)In a typical helicon plasma source, the first cutoff (14)falls on the periphery of a plasma column, where thedensity is low, while the second cutoff (15) is located ina more dense plasma core. C. Coalescence points
The monographs on plasma physics usually describethe wave dispersion in a cold magnetized plasma at afixed angle of propagation θ = arccos( N k /N ) . In thiscase, as seen in Fig. 1, the dispersion curves are mono-tonically rising functions of the wavenumber k , they donot intersect and do not touch each other (except for thecase θ = 0 ). However, at a fixed value of N k , there ap-pear the points of coalescence, where the roots of Eq. (4) Ω p (cid:144) Ω ce2 - N ¦ H a L Ω p (cid:144) Ω ce2 N ¦ H b L Ω p (cid:144) Ω ce2 N ¦ H c L Ω p (cid:144) Ω ce2 N ¦ H d L Figure 3. (Color online) To the derivation of the Golant-Stixcriterion. The solid line shows N ⊥ for a propagating solution N ⊥ > of the dispersion equation (4), and the dashed lineshows − N ⊥ for an evanescent solution N ⊥ < ; ω/ √ ω ci ω ce =0 . , ω ce /ω ci = 1836 . , (a) N k = 0 . , (b) N k = 1 . , (c) N k = 5 / , (d) N k = 2 . . The vertical dashed line indicatesthe position of the lower hybrid resonance. merge as shown in Fig. 2. The coalescence points arefound from the equation B − AC = 0 , (16)which means that N ⊥ + = N ⊥− . Equation (16) isquadratic with respect to ω p and, therefore, has two solu-tions, which we denote as ω p, coal and ω p, coal ; we do notgive here explicit expressions for these quantities becausethey are too complex.A solution of the dispersion equation (4) for severalvalues of the refractive index N k and a fixed frequency ω is shown in Fig. 3 depending on the dimensionless param-eter ω p /ω ce . Figures 3,a and 3,b show that at a relativelylow value of N k a zone of opacity, where N ⊥ is negativeor complex for both branches (5), is located between thetwo zones of transparency, where N ⊥ > for at leastone of the two branches. The lower hybrid resonance islocated in the upper density zone of transparency. Theopaque zone makes it inaccessible for a wave propagat-ing inward radially from outside of the plasma column.A low density zone of transparency matches a vacuum re-gion (where ω p /ω ce = 0 ) if N k < (Fig. 3,a). However,the vacuum region becomes opaque if N k > (Fig. 3,b);in this case, the low density zone of transparency beginsfrom the first cutoff (14). As N k increases, the opaquezone, located between the low density and the high den-sity zones of transparency, gradually shrinks. Its bound-aries are the points of coalescence, where ω p is eitherequal to ω p, coal or ω p, coal . IV. THE GOLANT-STIX CRITERION
The opaque zone, described in Sec. III C, disappearswhen ω p, coal = ω p, coal (17)and the two coalescence points merge as shown in Fig. 3,c.Equation (17) has a unique solution with respect to N k and determines a critical value N k , crit = ω ce ω ci ω ce ω ci − ω (18)of the longitudinal refractive index. The opaque zone isabsent and, hence, the lower hybrid resonance is accessi-ble, if N k > N k , crit . (19)Note that the expression (18) is exact and obtained with-out any simplifying assumption. Although it is rathersimple, its derivation is somewhat cumbersome and wasperformed using the Wolfram Mathematica [14].The condition (19) is equivalent to the Golant-Stix cri-terion, which specifies the conditions of the penetrationinto the plasma of an electromagnetic wave with a fre-quency of the order of the lower hybrid resonance fre-quency. The criterion was previously derived by V.E.Golant in Ref. [8]. He discarded some small terms andwrote his criterion in the form N k > ω pe, res ω ce (20)(see Eq. (12) in [8]), where the resonant value of theelectron plasma frequency ω pe, res was determined fromthe approximate equation − ω pe ω ce + ω pi ω = 1 + ω pe ω pe, res . Its solution coincides with approximate Eq. (11). Sur-prisingly, but the substitution of the approximate ex-pression (11) to approximate inequality (20) recovers theexact criterion (18).A subtle derivation of a criterion similar to (20) canbe found in the textbook [10] (see §4-12 and Eq. (103)there). For this reason, the authorship of the criterion(20) is also attributed to R.H. Stix.Before concluding this section, it should be emphasizedthat the Golant-Stix criterion refers to the case when ω < √ ω ce ω ci , so that a lower hybrid resonance can exist inthe plasma column. As is clear from Eq. (18), the criticalvalue N k , crit of the square of the refractive index formallybecomes negative if ω > √ ω ce ω ci . It means that theopaque zone cannot shrink to zero and, hence, the highdensity transparent zone is inaccessible in if ω > √ ω ce ω ci .The high-frequency regime of the helicon plasma sourcesoperation is considered in Sec. VII. V. LIMITING DENSITY
Equation (18) determines a critical value of the lon-gitudinal wave number k k , crit = ( ω/c ) N k , crit . Since N k , crit > , an antenna must launch a slowed wave. Theslower the wave, the higher the density of the plasma thatcan be heated. Indeed, ω p, res = (cid:0) ω ce − ω (cid:1) (cid:0) ω − ω ci (cid:1) ω ce ω ci − ω ≈ ω ce ω ci k k , crit c . (21)In practical units, the resonant electron density is n e, res [ cm − ] = 2 × AZ − ( λ k [ cm ]) − , (22)where A it the atomic weight of the plasma ions, Z istheir charge state, n e is expressed in cm − , and the wavelength λ k = 2 π/k k in cm.In the vacuum region the wave is evanescent since N ⊥ , vac = 1 − N k , crit = − ω ω ce ω ci − ω ≈ − ω p, res ω ce < . Therefore, for achieving a higher density an antenna mustbe placed as closer to the plasma as possible in orderto reduce the wave attenuation in the opaque area atthe plasma periphery; this might be a difficult technicalproblem.Rarefied periphery of the plasma is also opaque to thewave up to the first cutoff, i.e. at ω p < ω . The cutoff density is substantially smaller than the reso-nant one since ω ω p, res ≈ ω ci ω ce N − k , crit < Zm e m i . Therefore one can hope that external rf field can tunnelthrough the peripheral opaque zone to the plasma core.Equations (18) and (21) in principle solve the problemof optimization of a helicon plasma source at a given fre-quency of the RF field and a given magnetic field. Equa-tion (18) yields the required wavelength, and Eq. (21)gives the maximal density of the plasma, which can beheated at such parameters.
VI. OPTIMAL MAGNETIC FIELD
We can change a statement of the problem to beginwith, so to speak, the antenna. Suppose that the fre-quency and wavelength are fixed by the antenna systemdesign; it means that N k = k k c/ω is a given parameter.Rewriting Eq. (18) in the form ω ci ω ce = ω N k N k − (23) then determines the magnitude of the magnetic field. Inpractical units, B ∗ [ kG ] = 15 . r AZ N k q N k − f [ GHz ] , (24)where f = ω/ π is the linear frequency, expressed ingigahertz. For f = 13 . MHz and N k ≫ it yields B ∗ = 208 G.Meaning of B ∗ can be understood as follows. If B B ∗ TG branch propagates till the lower hybrid resonance at ω p = ω p, res , and the helicon branch can penetrate intoeven more dense plasma where it can deposit energy dueto particle collisions.Thus, B ∗ is a minimal magnetic field required to switchon the mechanism of plasma heating due to lower hybridresonance. Some experiments (see in particular [15–17])demonstrate that helicon discharge is most easily firedwhen B ∼ B ∗ in that sense that required rf power sup-ply is minimal. However, the helicon plasma sources areknown to effectively operate even at smaller magneticfield. In other words, a frequency in the range (10) canalso be effectively used in such sources. We proceed tothe analysis of this range in the next Section. VII. HIGH-FREQUENCY HELICON SOURCES Ω p (cid:144) Ω ce2 N ¦ H a L Ω p (cid:144) Ω ce2 N ¦ H b L Ω p (cid:144) Ω ce2 N ¦ H c L
10 20 30 40 50 60 Ω p (cid:144) Ω ce2 N ¦ H d L Figure 4. (Color online) To the derivation of theShamrai-Taranov criterion. Solid line shows an exact so-lution of Eq.(4), and dashed line is approximate solution(28). Accuracy of the approximation improves as N k grows; ω/ √ ω ci ω ce = 4 : (a) N k = 2 , (b) N k = 4 , (c) N k = 8 , (d) N k = 16 . Blue and purple curves are respectively the heliconand TG waves. For a frequency in the range (10), the helicon and TGwaves can propagate in the low density zone of trans-parency as shown in Fig. 4. In this case, the maximalplasma density, that can be heated by these waves atgiven ω and k k , is defined by ω = ω p, coal . (25)It is limited by the coalescence of the helicon and TGwaves. The effect of coalescence can be understood usingthe Appleton-Hartree-Booker simplified dispersion rela-tion of the helicon waves (see eg. [13, 18]) ω = ω ce kk k c ω pe + k c , (26)where k = q k ⊥ + k k . In the limit k ⊥ → ∞ it alsodescribes TG waves. Same k þ Coalescense pointGenerator FrequencyEigenfrequency for low n Eigenfrequency for high n k ¦ Ω Figure 5. (Color online) Eigenfrequency of the helicon andTG waves vs. k ⊥ for different values of the plasma densityand a fixed k k . The eigenfrequencies (26) are drawn in Fig. 5 for var-ious values of ω pe at a fixed value of k k . The uppercurves correspond to the periphery of the plasma columnwhere ω pe is small whereas the bottom curves representthe column core where ω pe is larger. A horizontal linerepresents frequency ω of the antenna so that propagat-ing waves correspond to the intersection point of the solidcurves with this line. The horizontal line crosses the up-per curves only once (eg., blue upper curve in the figure),which means that only one branch of the waves can prop-agate at the plasma periphery, namely the TG wave. Asecond intersection point with smaller k ⊥ , which corre-sponds to the helicon wave, appears closer to the centerplasma (dark-yellow curve). Even closer to the plasmacolumn core, the dispersion curves pass below the hori-zontal line, which means that a sufficiently dense plasmais not transparent to the waves with given ω and k k .This conclusion is qualitatively confirmed by Fig. 4 whichdraws both exact and approximate solutions of the dis-persion equation.Equation (26), rewritten in the form k c − ω ce N k kc + ω pe = 0 , (27)has a solution k ± c = ω ce N k ± q ω ce N k − ω pe . (28) It is real, and therefore describes two propagating waves,when ω pe < ω ce N k = ω pe, max . (29)The condition (29) poses the density limit in the high-frequency regime of operation of a helicon plasma source.In practical units n e, max [ cm − ] = 0 . × AZ − ( ω ci ω ce /ω )( λ k [ cm ]) − . (30)The condition (29) was first obtained by K. P. Shamraiand V. B. Taranov in Ref. [3]. In comparison with theaccurate criterion ω p < ω p, coal , (31)it provides a reasonable accuracy only if N k ≫ . In-deed, as seen from Fig. 4, Eq. (28) becomes accurate atsufficiently large N k .A dense plasma, ω pe > ω pe, max , is opaque for bothbranches of the waves (the helicons and TG modes) since k ⊥± are complex there. In a non-uniform plasma, merg-ing (degeneration) of the two different wave branches isexpected to be followed by the mutual linear conversion ofthese waves (see e.g. [10]). Thus, a helicon wave convertsinto a TG wave near the surface ω pe = ω max and viceversa. Note that the smaller of the two wave branches(28), which corresponds to the minus sign, is the heliconwave; the second, signed by + , is the TG wave. The he-licons also have a lower limit on the density defined by k ⊥ = 0 , ie, k = k k so the rarefied plasmas are opaque forhelicons as seen in Fig. 4. Substituting k = k k in (27)yields the cutoff plasma frequency ω pe, min = ω ( ω ce − ω ) N k . (32)Thus, the helicon wave can propagate in the finite densityrange ω pe, min < ω pe < ω pe, max . (33)It shrinks to zero width at ω = ω ce . VIII. DISCUSSION
We have derived two expressions for the limitingplasma density in a helicon plasma source. Accordingto Eq. (21), obtained for the case ω < √ ω ce ω ci , at agiven k k maximal density is defined by the relation ω pe, max ≈ ω ce ω ci k k c . (34)In case ω > √ ω ce ω ci , according to Eq. (29) ω pe, max ≈ ω ce ω k k c . (35)The expressions (34) and (35) match each other at ω = √ ω ce ω ci and, thus, provide a smooth objective func-tion for choosing optimal parameters of a helicon plasmasource. If we assume that ω and k k are fixed by rf systemdesign, the only remaining parameter will be the mag-netic field B . Starting an optimization procedure from alow magnetic field, we see that, according to Eq. (35), in-creasing B would increase allowed plasma density. How-ever, increasing B above the limit ω = √ ω ce ω ci revertsthe density scaling from Eq. (35) to Eq. (34). Then, thelimiting density becomes insensitive to B . From this rea- soning, we see that the condition ω ≈ √ ω ce ω ci mightbe optimal for the operation of a helicon source as wesuggested in Sec. VII. Some experiments [15–17] indicatethat a helicon discharge is fired at a minimal rf powerprovided that magnetic field is within some range aroundthe value given by Eq. (24) although the range of allowedmagnetic fields widens as rf power increases.To the best of our knowledge, the density limits givenby Eqs. (34) and (35) are not achieved in existing heliconplasma sources. However the results instantly grow withthe increase of applied rf power so one may suppose thatthese limits might be met in the future. [1] F. F. Chen, “Helicon plasma sources,” in High DensityPlasma Sources , edited by O. A. Popov (Noyes Publica-tions, Park Ridge, NJ, 1995) Chap. 1, pp. 1–75.[2] A. R. Ellingboe and R. W. Boswell,Physics of Plasmas , 2797 (1996).[3] K. P. Shamrai and V. B. Taranov,Plasma Sources Science and Technology , 474–491 (1996).[4] T. Lafleur, C. Charles, and R. W. Boswell,Journal of Physics D: Applied Physics , 055202 (2011).[5] S. Shinohara, T. Tanikawa, T. Hada, I. Fu-naki, H. Nishida, T. Matsuoka, F. Otsuka,K. Shamrai, T. Rudenko, T. Nakamura, et al. ,Fusion Science and Technology , 164 (2013).[6] A. R. Ellingboe and R. W. Boswell, Physics of Plasmas (1996).[7] P. Chabert, N. Braithwaite, and N. S. J. Braithwaite, Physics of Radio-Frequency Plasmas (Cambridge Univer-sity Press, 2011).[8] V. Golant, Sov. Phys.-Tech. Phys. , 1980 (1972).[9] T. H. Stix, The Theory of Plasma Waves (McGraw-Hill,New York, 1962).[10] T. H. Stix,
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