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
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
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
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
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). 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=
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
Of these, β
2≈kωc/(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 iν 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
2≈kωc/(-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.
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); 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
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.)
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
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
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
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
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.