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Helicons and their Surface Mode

By Charles R. Legendy

The “second solution” of the helicon equation, the surface mode, has attracted much attention in recent years, since it arguably holds the key to the operation of the helicon plasma source.

The properties of the surface mode have been described in detail in the original helicon papers, back in the mid-sixties. However, these descriptions have been overlooked in many of the subsequent helicon-related publications. This probably happened because at the time helicons were just a curiosity, and not all their details were of interest. Whatever the reason, the outcome is that by now these descriptions of the surface mode are buried in the old papers, and are apparently unknown to many current contributors to the helicon literature. In fact, some of the old results are now (unfortunately) re-invented and published as if they were new.

I am writing this brief note in order to call attention to the original helicon papers. Incidentally, one message from these papers is, as will be seen, that the most crucial properties of the surface mode do not require that the frequencies and magnetic fields be within the Trivelpiece-Gould (TG) range; in other words, the terms “surface mode” and “TG mode” cannot be used interchangeably.

The purpose of the website format of this note is to make the old papers in question available on the Internet, since they have become difficult to access in some libraries.

The calculations in question (Legendy, 1964; Legendy, 1965) were carried out in connection with solid-state plasmas; but the assumptions are no different from the uniform fixed-ion plasma approximation often used today when exact results are sought.

KMT (1965) and Legendy (1964)

Shortly after the original announcement by (Bowers et al, 1961), the helicon equations and their main consequences were independently derived by “KMT” (Klozenberg, McNamara and Thonemann, 1965) and me (Legendy, 1964  -html-). The KMT paper included rigorous discussion of the parameters including both gas and solid-state plasmas, and contained extensive computer simulation. Their main calculations relied on the usual simplifying assumptions shared by solid-state work, and carried a collision term and an electron inertia term. My paper skipped the discussion of the parameters (since it had already been provided by others) and the calculations only carried the collision term. My emphasis was on finding exact solutions to the helicon boundary-value problem for cylindrical and plane configurations; and in particular on introducing the surface mode.

The practical limitation of KMT was that their cylindrical solution ignored the driving field (the field set up by the external antenna); and this made it ill-suited for matching theoretical results to some experiments. Their antenna had to be envisioned at one end of a long cylindrical sample, because the assumptions required the fields to go to zero at r→∞, which was not always desirable since in practice the antenna is often placed around the cylinder. The currently used cylindrical solution (Chen and Boswell, “Helicons – The Past Decade,” 1997; see equation (3)) has the same form as mine (see my page A1721). For a number of years after its publication, my 1964 paper was principally cited as the source of this cylindrical solution. In retrospect, though, the paper’s main contribution was probably its description of the surface mode.

KMT also made explicit reference to the surface mode; but their description of it (see their Section 5 and the last few lines of their Appendix A) was obscure and barely noticeable. I believe that one cannot shed light on the properties of the surface mode without illustrating it in the simple case of the infinite plane interface (see my Sections 4B, 5B, and 6) where the calculations feature sinusoids and exponentials instead of Bessel functions. In addition, the infinite plane treatment makes it possible to show that the maximum current density in the surface mode is quite sensitive to the angle between the plasma surface and the external magnetic field, and sharply peaks at surfaces tangential to the field (see my equation (6.5) on page A1722).

The surface mode solution of the helicon equation

The helicon equation in uniform plasmas with negligible ion motion leads to two modes; their different dispersion relations are conveniently described in terms of a quadratic equation (reproduced here using the notation of KMT, 1965, and Chen and Boswell, 1997) and its roots. The equation is:

δβ2 – kβ + kw2= 0,


δ =(ω-iν)/ωc ,


ω is the helicon frequency, ν the electron collision frequency, ωcthe electron cyclotron frequency, ωp the plasma frequency, and k the component of the wave number along the applied magnetic field. The roots β1 and β2 of the equation are the wave numbers of the two helicon solutions.

Noting that in the region of interest δ «1, and also 4δkw2/k2«1, the two roots are approximately:

β1 kw2/k, β2k/δ.

Of these, β 2kωc/(ω-iν) corresponds to the surface mode.

When the denominator “ω – iν” of β2 is dominated by the ω term, we get the Trivelpiece Gould mode; when it is dominated by the term, we get the surface mode described by KMT and me.

When I speak of “surface mode” in this writing, I mean the latter mode, the one corresponding to β 2kωc/(-iν). Its salient properties are that (1) it is most significant in surfaces parallel to the B0 field, (2) in such surfaces its resistive loss occurs mainly close to the surface (hence the term surface mode), (3) in such surfaces, also, the resistive loss per unit area goes to a nonzero limit even when the resistivity is (formally) allowed to go to zero, and (4) when surface loss is substantial, energy is transferred at a substantial rate from the propagating mode into the surface mode.

Relation of the surface mode to the TG mode

It is worth emphasizing that the surface mode does not require the contributions of finite mass. My calculations neglected the electron mass throughout the 1964 paper (meaning that the paper left out the TG mode altogether); KMT carried the electron mass term; but in the surface mode discussion they, too, neglected it through the assumption “Ωe >> ν >> ω” (Ωe=electron cyclotron frequency).

In other words, contrary to the currently fashionable terminology which uses “surface mode” and “TG mode” interchangeably, the surface mode is alive and well when the combination of frequency and magnetic field is outside the Trivelpiece-Gould range. In the Trivelpiece-Gould mode, by definition, the displacement current is significant and so are the electric fields. In typical helicon experiments at Cornell, including ones where the surface mode accounted for most of the energy loss, the ratio of displacement current to conduction current, as well as the ratio of electric field energy to magnetic field energy, was of the order of 10-16.

Please don’t get me wrong. I believe (as do many others) that the “second solution” of the helicon equation is the one responsible for the ion producing effect of helicons. Its intense currents intuitively translate to ionizing collisions in gaseous plasmas. But I dispute the contention that high-efficiency ion production only arises as one enters the “TG frequency range”. Such a contention is inconsistent with the fact that in the original experiments at Boswell’s lab, where the optimal parameters for ion production were first developed (see Boswell's Thesis, 1970; Boswell and Chen, 1997), there was no evidence of a dramatic increase in the ionization rate in runs where the frequency-magnetic-field combination approached the TG range.

Anomalous behavior of the surface mode

One of the most counterintuitive results of my calculations had to do with the behavior of the helicon field (by which I meant both the propagating mode and the surface mode) in those plasma samples which had significant surface area parallel to the external magnetic field. As the resistivity decreased (ωc/ν → ∞) the surface layer in such samples became thinner; but the currents in the surface layer became proportionally stronger, in such a way that the total resistive loss in these surfaces did not go to zero in the zero-resistivity limit. In practice, the nonphysical behavior disappears as the effects of finite electron mass and finite cyclotron radius become significant.

All the same, in our experimental setup at Cornell these electron effects were negligible; and in samples aligned with the field the surface loss often dominated the energy picture. The finding suggested that there was more to helicons than met the eye; but at the time we could not foresee just what.

An alternative way to look at the surface anomaly in the idealized helicon plasma is to ask whether an arbitrarily shaped finite helicon sample can have modes of undamped oscillation. At the time of the original work the question bothered me because, unlike vacuum waves, helicons waves appeared to be an asymmetrical disturbance. It was not clear to me that they contained enough wave components to be confined to the inside of a plasma sample. In the end I was able to answer the question, though; and I came up with a formal proof that the helicon equation, with fixed plasma density and zero resistivity, led to oscillatory solutions in finite samples (Legendy, 1965  -html-); but there was a catch. The shape of the sample had to be such that there was no surface loss!

The proof consisted of inventing a linear operator whose eigenvalues were frequencies. The definition of the operator combined a set of boundary conditions along closed surfaces with the effects of an applied magnetic field and with Maxwell’s equations. By showing that the operator was self-adjoint, I showed, within the assumptions of the proof, the existence of a complete set of orthogonal eigenfunctions which satisfied the helicon equation and had real eigenfrequencies. The sample shape could be almost arbitrary; but the proof could only be completed when the amount of boundary surface tangential to the magnetic field lines was a “set of measure zero”. The interpretation of the result was that when there was a finite surface area subject to surface loss, the oscillations were damped (even though the conductor was “perfect”); and accordingly the frequencies could not be real.

My fellow graduate student at Cornell, John Goodman, also became interested in the curious predictions regarding helicon surface loss.He first performed an early test in a plate sample oriented parallel to the field, a sample so chosen that according to the calculations most of the loss in it would be surface loss (Goodman and Legendy, 1964).Subsequently John carried out a very careful experimental series using cylindrical samples (Goodman, 1968) and achieved strikingly close agreement between experiment and theory. It can be said that, after Goodman’s experiments, the existence of the solid-state surface mode, as I described it, can be considered proven.

Incidentally, in retrospect it appears possible that the original helicon experiment (Bowers et al, 1961) would have failed if its cylindrical sodium sample had been aligned parallel to the magnetic field. Surface loss would have dulled the oscillations, and the helicon phenomenon would possibly never have been noticed. The clean oscillatory ringing of the sample was due to the fact that the cylinder axis was at right angles to the external magnetic field! (The odd positioning was due to the space constraints of fitting the elongated sample, immersed in a helium Dewar, between the pole pieces of the magnet.)

Energy transfer from the propagating mode into the surface mode

To demonstrate that the theory predicts energy transfer between the modes it is enough to construct a thought experiment which makes the need for such transfer obvious.

First, it is noted that, in plasma samples of constant density, it is easy to design experiments where an antenna maintains a constant-amplitude helicon field, and most of the resistive loss is from the surface mode. In such an experiment, by necessity, the surface mode loses energy faster than the propagating mode.

Let us assume, then, that contrary to the claim there is no energy transfer between the modes. In that way the energy lost from the surface mode cannot be replenished from the propagating mode.Now, if it is possible to position the antenna in such a way that the energy cannot be replenished from the antenna either, then nothing will counteract a steady decrease in the amplitude of the surface mode.

And, in fact, it is possible to avoid replenishment of the surface loss from the antenna by placing the antenna at one end of a long cylindrical sample, far away from the region under consideration (as in KMT), and utilize the fact that the surface mode does not propagate well along the field lines.

The result of such a setup will be that the amplitude of the surface mode will decrease faster than the amplitude of the freely propagating mode.

But the boundary conditions require that the amplitudes of the two modes maintain a constant ratio. This reduces the assumption of no energy transfer between the modes to an absurdity; energy must be transferred between the modes (somehow – the reasoning does not tell us how).

In other words, the boundary conditions force a steady energy flow out of the propagating mode and into the surface mode.


R. W. Boswell and F. F. Chen, “Helicons – the Early Years,” IEEE Trans. Plasma Science 25, 1229-1244 (1997)

R. Bowers, C. R. Legendy, and F. E. Rose, “Oscillatory Galvanomagnetic Effect in Metallic Sodium,” Phys. Rev. Letters 7, 339-341 (1961)

F. F. Chen and R. W. Boswell, “Helicons – the Past Decade,” IEEE Trans. Plasma Science 25, 1245-1257 (1997)

J. M. Goodman, “Helicon Waves, Surface-Mode Loss, and the Accurate Determination of the Hall Coefficients of Aluminum, Indium, Sodium, and Potassium,” Phys. Rev. 171, 641-658 (1968)

J. M. Goodman and C. R. Legendy, “Joule loss in a ‘perfect’ Conductor in a Magnetic Field,” MSC Report #201, Cornell University, Ithaca, NY (1964)

J. P. Klozenberg, B. McNamara and P. C. Thonemann, “The Dispersion and Attenuation of Helicon Waves in a Uniform Cylindrical Plasma,” J. Fluid Mech. 21, 545-563 (1965)

C. R. Legendy, “Existence of Proper Modes of Helicon Oscillations,” J. Math. Phys. 6, 153-157 (1965)

C. R. Legendy, “Macroscopic Theory of Helicons,” Phys. Rev. 135, A1713-A1724 (1964)

A. W. Trivelpiece and R. W. Gould, “Space Charge Waves in Cylindrical Plasma Columns,” J. Appl. Phys. 30, 1784-1793 (1959)

Acknowledgements and Copyright notes

A copy of Prof. Roderick Boswell's doctoral dissertation has been kindly provided to us by the Library of Flinders University. It is the only extant corrected version; and the opportunity to photocopy it is gratefully acknowledged.

Copyright to several papers included on this website is held by the publishers of the journals in which they appeared; their permission to reproduce the papers is acknowledged with thanks. The publications in question are the following:


R. Bowers, C. Legendy, and F. Rose, Oscillatory Galvanomagnetic Effect in Metallic Sodium, Phys. Rev. Lett. 7, 339–341 (1961).Copyright (1961) by the American Physical Society. Reprinted by permission of the American Physical Society.

C. R. Legéndy, Macroscopic Theory of Helicons, Phys. Rev. 135, A1713–A1724 (1964). Copyright (1964) by the American Physical Society.Reprinted by permission of the American Physical Society.

C. R. Legéndy, “Existence of Proper Modes of Helicon Oscillations,” J. Math. Phys. 6, 153-157 (1965). Copyright (1965) by the American Institute of Physics.Reprinted by permission of the American Institute of Physics.

J. M. Goodman, Helicon Waves, Surface-Mode Loss, and the Accurate Determination of the Hall Coefficients of Aluminum, Indium, Sodium, and Potassium, Phys. Rev. 171, 641-658 (1968). Copyright (1969) by the American Physical Society.Reprinted by permission of John M. Goodman and the American Physical Society.