Comment on "Plasma ionization by annularly bounded helicon waves" [Phys . Plasmas 13, 063501 (2006)]

0.09 MB·English

by  Robert W. Johnson

Comment on “Plasma ionization by annularly bounded helicon
waves” [Phys. Plasmas 13, 063501 (2006)]
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Robert W. Johnson
Alphawave Research, Atlanta, GA 30238, USA
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0 (Dated: January 22, 2009)
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n Abstract
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J The neoclassical calculation of the helicon wave theory contains a fundamental flaw. Use is

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2 madeof aproportional relationship between the magnetic field and its curlto derivethe Helmholtz

] equation describinghelicon wave propagation; however, by the fundamental theorem of Stokes, the
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- curl of the magnetic field must be perpendicular to that portion of the field contributing to the
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s local curl. Reexamination of the equations of motion indicates that only electromagnetic waves
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l propagate through a stationary region of constant pressure in a fully ionized, neutral medium.
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i PACS numbers: 52.25.Jm,52.50.Dg, 52.40.Fd
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[email protected]
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The neoclassical calculation of the helicon wave theory, presented for the case of an
annular waveguide in Reference [1] which follows Chen’s derivation [2] for a cylindrical con-
figuration, contains a fundamental flaw in its use of vector field theory [3, 4]. A proportional
relationship is derived between the magnetic field and its curl which contradicts the funda-
mental theorem of Stokes, thus the waves described cannot be physical. Reexamination of
the plasma equations of motion with consideration of the convective terms for the case of
vanishing mass flow, charge density, and pressure gradients leads to the familiar Ohm’s law
of conductive material, indicating that only the usual electromagnetic radiation propagates
through such regions. A similar discussion on the existence of whistler oscillitons [5, 6, 7, 8]
in geophysical plasmas is noted.
The model for infinite conductivity σ is given by the linearized system:
∇×E = −∂B/∂t , ∇×B = µ J , E = J×B /en , (1)
0 0 0
ˆ
where B is the applied magnetic field along some axis b and n is the plasma density
0 0
n = n = n , with solutions of the form B = B(r)ei(mθ+kz−ωt). The standard derivation
e i 0
of the helicon wave equation is here termeed “neoclassical” for its use of the quasineutral
approximation to justify its determination of the electric field from an equation of motion
rather than the remainder of Maxwell’s equations [9], as given by the substitution apparent
in Equation (9) of Reference [1],
iωB = ∇×E = ∇×(J×B )/en = (B ·∇)J/en = J(ikB /en ) , (2)
0 0 0 0 0 0
e e e e e
where we note that implicit use of a constant density has been made, as
∇×(J×B /n ) = [∇×(J×B )]/n −(J×B )×∇(1/n ) , (3)
0 0 0 0 0 0
with the result
∇×B = αB , (4)
e e
for α = (ω/k)(µ en /B ) = (ω/k)(ω2 /ω c2), leading to the Helmholtz wave equation
0 0 0 pe ce
(∇2 + α2)B = 0. However, for the real field B displaying circulation caused by the local
current J wehich from Equation (2) is in phase, by the fundamental theorem of Stokes [3, 10],
ˆ 2π ˆ
dl B(r)·(b×rˆ) rdθ B(r)·θ
(∇×B)·ˆb ≡ lim ∂A = lim 0 = 0 , (5)
A→0 H A r→0 H πr2
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where these r and θ refer to our local geometry, not the global one, as in Figure 1, the
component of the curl of B along B must vanish, thus Equation (4) cannot hold for both α
and B nontrivially real. As the wave in question propagates freely without attenuation, k
hence α is real, leading to an infinite wave number for finite ω or a zero frequency for finite k.
One might patch the theory by inclusion of the displacement term in the Maxwell-Ampere
equation, necessary when the rate of change of B is itself changing, ∂2B/∂t2 6= 0, and
essential to the propagation of electromagnetic waves, so that the equation ∇×(∇×B) =
µ ∇×(J+ǫ ∂E/∂t) leads to
0 0
∂E ∂2B ǫ ∂
∇2B+α αB+µ ǫ −µ ǫ = (cid:3)2 +α2 1− 0 B × B = 0 , (6)
0 0 o 0 0
(cid:18) ∂t (cid:19) ∂t2 (cid:20) (cid:18) en ∂t(cid:19)(cid:21)
0
where we recognize a modified Klein-Gordon equation for each component of B, admitting
α
∇2 +α2 +µ ǫ ω2 1+i ˆb × B = 0 . (7)
0 0 0
k
h (cid:16) (cid:17)i
e
For consistency, one should require of Equation (6) reduction to the standard wave equation
for conductors (cid:3)2B = µ σ∂B/∂t in the limit of vanishing applied magnetic field B → 0,
0 0
which admits attenuated solutions with finite ω and k. The first problem to arise is that
limB0→0α = ∞ratherthan0, andthesecond isthataninfiniteconductivity σ → ∞requires
∂B/∂t → 0, hence a static magnetic field.
Thedifficultyencounteredabovebytheneoclassicalmodelresultsfromthemisapplication
of the Ohm’s law equation, which itself is derived from the electron and ion equations of
motion. For a neutral, hydrogenic plasma of species s ∈ {e,i} with total particle density
n ≡ n +n = 2n , mass density ρ ≡ n m , mass flow velocity ρ V ≡ n m V ,
e i 0 m s s s m m s s s s
P P
free current density J ≡ n e V , and pressure p ≡ nT− ≡ n T− ≡ p , the
f s s s s s s s s s
P P P
equations of motion neglecting viscosity and gyromotion read
∂
n m +(V ·∇) V +∇p = n e (E+V ×B)+F , (8)
s s s s s s s s sk
(cid:20)∂t (cid:21)
where k 6= s for the friction term F = −F = m ν J /e which represents interspecies
ei ie e ei f
collisions. Fromthedefinitions ofthecurrent andflowvelocity wemayexchange theelectron
and ion velocities {V ,V } for the pair {V ,J } via
e i m f
ρ V m m V V J −m
m m e i e e f i
  = n    ⇒   = V +   . (9)
0 m
eρ
J −e e V V m m
f i i e
        
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The sum of Equations (8) gives the net force balance equation
∂V
m
ρ +C +∇p = J ×B , (10)
m + f
∂t
and their difference the generalized Ohm’s law equation
∂V 2m m ∂J
m e i f
n0(mi−me) + +C−+∇(pi −pe) = n0e(Vi +Ve)×B+2[n0eE−Fei] ,
∂t e(m +m ) ∂t
i e
(11)
where n0e(Vi + Ve) = 2n0eVm − n0(mi − me)Jf/ρm, and the convective terms C+,− ≡
n [m (V ·∇)V ±m (V ·∇)V ] are given by
0 i i i e e e
m m J J
e i f f
C = n (m +m ) (V ·∇)V + ·∇ , (12)
+ 0 i e (cid:20) m m e2 (cid:18)ρ (cid:19) ρ (cid:21)
m m
m m J J
e i f f
C− = n0(mi −me) (cid:20)(Vm ·∇)Vm − e2 (cid:18)ρ ·∇(cid:19) ρ (cid:21) (13)
m m
2n m m J J
0 e i f f
+ (V ·∇) + ·∇ V . (14)
m m
e (cid:20) ρ (cid:18)ρ (cid:19) (cid:21)
m m
With consideration now of a vanishing flow velocity V = 0, the generalized Ohm’s law
m
equation may be put into the form
memi∂Jf C− (mi −me)
+ + (J ×B) = E−ηJ , (15)
f f
e2ρ ∂t 2n e 2eρ
m 0 m
for resistivity η = m ν /n e2 where ν is the interspecies collision rate, when the electrons
e ei 0 ei
and ions have the same thermal energy T− = T− = T− . As the net force balance equation
e i 0
must also hold true, for a region of constant pressure ∇p = 0 we find that J ×B = C .
f +
Then follows
m m ∂J 1 n (m −m ) m m ∂J
e i f 0 i e e i f
e2ρ ∂t + 2n e (cid:20)C− + ρ C+(cid:21) = e2ρ ∂t = E−ηJf , (16)
m 0 m m
which, in terms of the conductivity σ ≡ 1/η and transit time τ ≡ 1/ν , reduces to
ei ei
m ∂J
i f
τ = σE−J . (17)
ei f
(m +m ) ∂t
i e
With an oscillating field E = Ee−iωt and current J = J e−iωt, one finds
f f
e m e
i
τ (−iω)J = σE−J , (18)
ei f f
(m +m )
i e
e e e
and bringing the factors of J together gives us
f
e m −1
i
J = σ 1−iωτ E , (19)
f ei
(cid:20) (m +m )(cid:21)
i e
e e
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which we compare to the usual complex conductivity of a material conductor [3],
n e2/m
0 e
J = E , (20)
f
(cid:18) ν −iω (cid:19)
e e
noting that Ohm’s law determines J in terms of E for a plasma just as in a normal con-
f
ductor. With substitution into the Maxwell equations, one achieves the usual modified
wave equations for a dispersive conducting medium, leading to the attenuated propaga-
tion of electromagnetic radiation, and with vanishing damping factor a plasma frequency of
ω = e n (m +m )/m m ǫ .
p 0 i e i e o
p
So, where have all the waves gone? They have disappeared along with the variations
in charge, mass, energy, and momentum density given by our assumptions of applicability.
Without a model allowing for the explicit determination of these quantities, one is left with
a theory of only electromagnetic waves. If one wishes to describe the behavior of other
types of waves in the plasma, one is forced to deal with the physics describing the oscillation
of the plasma medium. We note that the model above may be extended to incorporate
the effects of the gyromotion through the inclusion of the bound current J ≡ ∇ × M
b
~
for M ≡ n ~µ and the drift momentum ρ V ≡ ∇ × L for L ≡ n l , where ~µ
s s s m d g g s s s s
~ P P
and l are the magnetic moment and angular momentum per particle, respectively, to give
s
a total current J = J + J and momentum density ρ V = ρ (V + V ). As helicon
f b m m m d
antenna devices are in operation [11, 12] as prototypes for a high-efficiency ion source and
for an electric propulsion thruster, a better determination of what is happening within the
waveguide is warranted.
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[1] M. Yano and M. L. R. Walker, Physics of Plasmas 13, 063501 (pages 5) (2006).
[2] F. F. Chen, Plasma Phys. Control. Fusion 33, 339 (1991).
[3] D. Griffiths, Introduction to Electrodynamics (Prentice-Hall, Inc., Englewood Cliffs, NJ,USA,
1989), 2nd ed., ISBN 0-13-481367-7.
[4] L. H. Ryder, Quantum Field Theory (Cambridge University Press, 1985).
[5] J. F. McKenzie, E. M. Dubinin, and K. Sauer, Nonlinear Processes in Geophysics 12, 425
(2005), ISSN 1023-5809.
[6] F. Verheest, Nonlinear Processes in Geophysics 14, 49 (2007), ISSN 1023-5809.
[7] J. F. McKenzie, E. Dubinin, and K. Sauer, Nonlinear Processes in Geophysics 14, 543 (2007),
ISSN 1023-5809.
[8] F. Verheest, Nonlinear Processes in Geophysics 14, 545 (2007), ISSN 1023-5809.
[9] J. C. Maxwell, Royal Society Transactions 155 (1864).
[10] E. W. Weisstein, Curl, From MathWorld–A Wolfram Web Resource (2008).
[11] D. Palmer and M. L. R. Walker (44th Joint Propulsion Conference, Hartford, CT, 2008),
AIAA-2008-4926.
[12] D. Palmer, M. L. R. Walker, M. Manente, J. Carlsson, C. Bramanti, and D. Pavarin (5th
International Spacecraft Propulsion Conference, Crete, Greece, 2008), ISPC-0236-2008.
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a
B
θ r
A
FIG. 1: (Color online.) The definition of the curl of a vector field is the circulation per area,
(∇×B)·aˆ ≡ limA→0 ∂Adl B(r)·(aˆ×rˆ) /A. The curl of a real vector field must everywhere be
(cid:2)H (cid:3)
orthogonal to the local plane of circulation, hence that portion of the vector contributing to the
curl at that location.
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