Origin of transversal displacement of the plasma flux moving in a curvilinear magnetic field
It is shown that transversal displacement of the plasma flux propagating across toroidal magnetic field, may be interpreted as a result of conversion (by means of Lorentz force) of the energy obtained by electrons in a field of polarization force and in a field of forces aroused due to magnetic fiel...
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| Cite this: | Origin of transversal displacement of the plasma flux moving in a curvilinear magnetic field / A.I. Timoshenko // Вопросы атомной науки и техники. — 2008. — № 6. — С. 186-188. — Бібліогр.: 17 назв. — англ. |
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Timoshenko, A.I. 2017-01-07T15:34:26Z 2017-01-07T15:34:26Z 2008 Origin of transversal displacement of the plasma flux moving in a curvilinear magnetic field / A.I. Timoshenko // Вопросы атомной науки и техники. — 2008. — № 6. — С. 186-188. — Бібліогр.: 17 назв. — англ. 1562-6016 PACS: 52.77.-J https://nasplib.isofts.kiev.ua/handle/123456789/110973 It is shown that transversal displacement of the plasma flux propagating across toroidal magnetic field, may be interpreted as a result of conversion (by means of Lorentz force) of the energy obtained by electrons in a field of polarization force and in a field of forces aroused due to magnetic field curvature, into kinetic energy of the transversal movement of the plasma as a whole. The same interpretation is valid also for the drift motion of a single particle. Показано, що поперечне зміщення плазмового потоку може бути інтерпретовано як результат перетворення (за допомогою сили Лоренца) енергії, отриманої електронами в полі поляризаційної сили і в полі сил, пов'язаних з кривизною магнітного поля, в кінетичну енергію поперечного руху всієї плазми. Така ж інтерпретація справедлива і для дрейфового руху окремої частинки. Показано, что поперечное смещение плазменного потока может быть интерпретировано как результат преобразования (посредством силы Лоренца) энергии, полученной электронами в поле поляризационной силы и в поле сил, связанных с кривизной магнитного поля, в кинетическую энергию поперечного движения всей плазмы. Такая же интерпретация справедлива и для дрейфового движения отдельной частицы. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Low temperature plasma and plasma technologies Origin of transversal displacement of the plasma flux moving in a curvilinear magnetic field Походження поперечного зміщення потоку плазми, що рухається вздовж криволінійного магнітного поля Происхождение поперечного смещения потока плазмы, движущегося вдоль криволинейного магнитного поля Article published earlier |
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Origin of transversal displacement of the plasma flux moving in a curvilinear magnetic field |
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Origin of transversal displacement of the plasma flux moving in a curvilinear magnetic field Timoshenko, A.I. Low temperature plasma and plasma technologies |
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Origin of transversal displacement of the plasma flux moving in a curvilinear magnetic field |
| title_full |
Origin of transversal displacement of the plasma flux moving in a curvilinear magnetic field |
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Origin of transversal displacement of the plasma flux moving in a curvilinear magnetic field |
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Origin of transversal displacement of the plasma flux moving in a curvilinear magnetic field |
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origin of transversal displacement of the plasma flux moving in a curvilinear magnetic field |
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Timoshenko, A.I. |
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Timoshenko, A.I. |
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Low temperature plasma and plasma technologies |
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Low temperature plasma and plasma technologies |
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2008 |
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Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Походження поперечного зміщення потоку плазми, що рухається вздовж криволінійного магнітного поля Происхождение поперечного смещения потока плазмы, движущегося вдоль криволинейного магнитного поля |
| description |
It is shown that transversal displacement of the plasma flux propagating across toroidal magnetic field, may be interpreted as a result of conversion (by means of Lorentz force) of the energy obtained by electrons in a field of polarization force and in a field of forces aroused due to magnetic field curvature, into kinetic energy of the transversal movement of the plasma as a whole. The same interpretation is valid also for the drift motion of a single particle.
Показано, що поперечне зміщення плазмового потоку може бути інтерпретовано як результат перетворення (за допомогою сили Лоренца) енергії, отриманої електронами в полі поляризаційної сили і в полі сил, пов'язаних з кривизною магнітного поля, в кінетичну енергію поперечного руху всієї плазми. Така ж інтерпретація справедлива і для дрейфового руху окремої частинки.
Показано, что поперечное смещение плазменного потока может быть интерпретировано как результат преобразования (посредством силы Лоренца) энергии, полученной электронами в поле поляризационной силы и в поле сил, связанных с кривизной магнитного поля, в кинетическую энергию поперечного движения всей плазмы. Такая же интерпретация справедлива и для дрейфового движения отдельной частицы.
|
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1562-6016 |
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https://nasplib.isofts.kiev.ua/handle/123456789/110973 |
| citation_txt |
Origin of transversal displacement of the plasma flux moving in a curvilinear magnetic field / A.I. Timoshenko // Вопросы атомной науки и техники. — 2008. — № 6. — С. 186-188. — Бібліогр.: 17 назв. — англ. |
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| first_indexed |
2025-11-27T01:27:22Z |
| last_indexed |
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| fulltext |
186 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2008. № 6.
Series: Plasma Physics (14), p. 186-188.
ORIGIN OF TRANSVERSAL DISPLACEMENT OF THE PLASMA FLUX
MOVING IN A CURVILINEAR MAGNETIC FIELD
A.I. Timoshenko
National Science Center "Kharkov Institute of Physics and Technology",
Kharkov, Ukraine, e-mail: timoshen@yandex.ru
It is shown that transversal displacement of the plasma flux propagating across toroidal magnetic field, may be
interpreted as a result of conversion (by means of Lorentz force) of the energy obtained by electrons in a field of
polarization force and in a field of forces aroused due to magnetic field curvature, into kinetic energy of the transversal
movement of the plasma as a whole. The same interpretation is valid also for the drift motion of a single particle.
PACS: 52.77.-J
1. INTRODUCTION
It is known that plasma jet (or plasmoid), after passing
through a quarter torus magnetic duct, shifts a little (~1
cm) in the B×R direction [1–3], where R is a radius of
the magnetic field B line of force, Fig.1.
It should be noted that this shift is opposite to the plasma
displacement predicted by a drift model [4–7] and it does
not agree with the flux-tube model [8,9], which foresees
only a radial displacement. An attempt to explain this
shift by a simple entrainment of unmagnetized ions by the
electrons with a speed of its electric (E×B) drift in the
radial polarization field RE , gives too much value
~ 25 cm [10,11]. Furthermore, if the ions are magnetized,
the model [10,11] gives no B×R displacement, while the
experiment confirms its existence [1,5]. This contradiction
have been overcame in [12,13], where the time scales
compared with the electron Larmor period have been taken
into account. During of such time intervals, the electron
component of the plasma drifts mainly with a mean
velocity e eV c F eB= perpendicular to some force Fe,
whereas the ions, having a much greater Larmor period,
pass only a small part of its Larmor orbit, deflecting in
the direction to which the force Fi acts on ions. Relative
displacement of the ion and electron components in the
plasma moving along a curvilinear magnetic field gives
rise to self-consistent polarization field E (see Fig. 2)
with the components E⊥ and RE (parallel to B×R and
to R, correspondingly) which, in the coordinate system
moving on the circular trajectory with the ions, depend
approximately on the time t in the following way
[12,13]:
( )
2 2
// //2 sini eM V mV eZ BE t t
eZ R eR M cη⊥
⎛ ⎞ ⎛ ⎞
≈ +⎜ ⎟ ⎜ ⎟
⎝ ⎠⎝ ⎠
, (1)
( )
2 2 2
// // //2 cosi e i
R
M V mV eZ B M VE t t
eZ R eR M c eZ Rη
⎛ ⎞ ⎛ ⎞
≈ + −⎜ ⎟ ⎜ ⎟
⎝ ⎠⎝ ⎠
, (2)
where m and e are the mass and charge of electron, M and
Z are the mass and degree of ionization of ion, //eV and //iV
are the electron and ion velocities parallel to magnetic field
B, D is a dielectric constant of magnetized plasma, and
// //i eV Vη = . Thus, the displacement of the centers of the
ion and electron distributions from a circular trajectory can
be easily calculated. In particular, taking into account that
ion component shifts in the B×R direction under direct
action of the polarization force eZE⊥ , and the electron one
moves owing to electric, centrifugal, gradient, and
polarization drift, we find for the average velocity V⊥
[12,13]:
2 2
//
, , 20
2t e e
i e R
cmV m c dEeZ cV V E dt E
M B eRB eB dt
η ⊥
⊥ ⊥ ⊥
⎛ ⎞
≈ = ≈ − + − ≈⎜ ⎟
⎝ ⎠
∫
2 2
//2i ec M V mV
B eZ R eR
η ⎛ ⎞
≈ +⎜ ⎟
⎝ ⎠
. (3)
Substituting well-known parameters of the vacuum-arc
plasma [2,8,14–16] into Eq. (3) and multiplying it by
//2 iR Vπ , we obtain for the displacement of the plasma
jet at the exit of quarter torus magnetic duct the value
~1 cm [12,13] that agrees with experimental results [1–3].
In present work we intend to show: 1) velocity (3)
can be derived in a different way, without use of the field
R0
V//
Bϕ = B0R0/R
Z
ϕ
Experiment Drift model
Hydrodynamic
model
B0
Fig.1
Fig. 2. Lorenz force LeF , acting on the electrons, bal-
ances the centrifugal and polarization force - ReE
much sooner than Lorenz force LiF , acting on ions,
would balance the centrifugal force acting on it
RiF
eV
- RE LeF
LiF
iV
ReE
B
В×R
R
E⊥
187
E⊥ ; 2) plasma transversal drift in the B×R direction has
a clear physical sense. Furthermore, the frameworks of
applicability of the model proposed in [12,13] and the
condition of plasma movement along the toroidal
magnetic field are appreciated.
2. ENERGY OF RADIAL DISPLACEMENT
OF ELECTRONS
Note that in absence of any force, the leading centers of
electrons would follow the magnetic field lines.
Appearance of polarization field RE [see Eq. (2)] and the
forces concerned with a magnetic field curvature, cause a
drift in the B×R direction and the mean radial
displacement Rd [12,13] of electron leading centers
(along with ions) from a circular trajectory with a radius
R0 (see Fig. 3):
2 2 2
//
, .
2i e
R i R e
V mVM c Md d
eZ B eZ R eR
η ⎛ ⎞⎛ ⎞≅ ≈ +⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠
. (4)
Since the deflection Rd is proportional to the force
2 2
//2i eM V mVF
Z R R
= + , each of electrons obtains the energy
2 22 2
//
,
1 1 2
2 2
i e
R e
M M V mV cW Fd
Z Z R R eB
η
⊥
⎛ ⎞ ⎛ ⎞= ≅ +⎜ ⎟ ⎜ ⎟
⎝ ⎠⎝ ⎠
, (5)
which is delivered through the Lorentz force and field ⊥E
to the mass ZM (and to electron mass, which may be
neglected), increasing the kinetic energy 2 2MV Z⊥
of
plasma transversal motion. Equating this energy to the right
member of Eq. (5) and solving it for the velocity V⊥
, we find
2 2
//2i eV mVc MV
eB Z R R
η
⊥
⎛ ⎞
= +⎜ ⎟
⎝ ⎠
that coincides with the expression
(3). From this it follows: 1) there is no necessity to know the
field ⊥E for determining of transversal displacement of
central part of the plasma jet, it is enough to have only the
field RE and the mean radial displacement Rd , which can
be easily calculated from expression
2
, , 0 0
t i
R i R e R
V eZd d dt E dt
R M
τ ⎛ ⎞
≈ = +⎜ ⎟
⎝ ⎠
∫ ∫ and Eq. (2) [12,13];
2) the identity of expressions for the velocity V⊥
obtained in
a different ways is an additional evidence of the full self-
consistency of the fields RE and ⊥E [see Eqs. (1), (2)] with
a drift motion of electrons and direct motion of ions.
Note that the same interpretation is applicable to a
single particle moving across curvilinear magnetic field.
The particle (or its leading center) having a charge q and
mass M, when flying into the toroidal magnetic field,
experience (along with transversal drift motion) an
average radial shift from a circular trajectory (see Fig. 3).
This shift is 2
//Rd Rρ≈ (if // Rρ << ), where
// //cMV qBρ = [4]. As this shift is proportional to the
centrifugal force
2 2
//
R
c MVd M
qB R
⎛ ⎞
≈ ⎜ ⎟
⎝ ⎠
, the particle obtains
perpendicular energy 22 2
//1
2 2R
MVM cFd
qB R
⎛ ⎞⎛ ⎞
= ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠
, which
transforms by means of the Lorentz force (see the electron
trajectory on Fig. 2) into kinetic energy 2 2MV⊥
of a drift
motion. Solving equation 22 22
//
2 2
MVMV M c
qB R
⊥ ⎛ ⎞⎛ ⎞
= ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠
for the
velocity V⊥
, we find 2
//V cMV qBR⊥ = , i.e. the velocity of
centrifugal drift. It is clear that this simple example can be
easily extended on a case of arbitrary force.
To estimate a range of validity of the model [12,13]
and this analysis too, let us take into consideration that the
ion may drift perpendicular to some force Fi, if this force
is equalized mainly by the Lorentz force (as it is shown
on Fig. 2 for the electron trajectory). This condition is
satisfied, when the deflection 2
//Rd Rρ≅ of a single ion
is lesser than a charge separation parameter δ in a
plasma. This parameter is 02RE enδ π= [12,13], if the
plasma density profile is Gaussian. Then, accounting
Eq. (2), we can write 2 2
0 // 02 2R iE en MV Ze n Rδ π π= = >
2
// Rρ> , and find an estimation: ZMncB 02π> , or
00.1B n A Z> (in Gauss), where A is a relative atomic
weight, and 0n is the plasma density on the axis of the
plasma flux. So, if 0 0.1B n A Z< , the ion centrifugal
drift in a plasma is impossible, because the centrifugal
force is balanced by a polarization force during much lesser
time than the ion Larmor period. For a typical vacuum-arc
plasma density in filters, 12 3~ 10n cm− [2,3,7,10,11], ions
may experience the centrifugal drift, if the magnetic field
strength would exceed 5105 ⋅ G. At the usual 150−600 G
in filtering devices, the drift of ions may take place, if the
plasma density would not exceed 5 310 cm− . So, the range of
applicability of this consideration includes all practically
important cases.
3. DISCUSSION
As it follows from Eq. (5), on entering at the
curvilinear path, the part W⊥
of the initial plasma kinetic
energy //W begins to circulate in the cross-section of the
plasma flux. This perpendicular energy is approximately
equal to ( )2
// //W R Wηρ⊥ ≈ [see Eq. (5)] and appears as a
result of centrifugal acceleration of the ions. In the first
stage, the energy W⊥
is absorbed by electrons during of its
Fig. 3. Radial oscillation of the plasma jet (or a single
particle) flying into the curvilinear magnetic field. Rd
is a mean deflection of the jet (or the particle leading
center) from the circular trajectory
B R
В×R
Rd
//V
R0
188
displacement in the radial polarization field RE . At the
second stage, this energy is passed to the ions through the
Lorentz force and positive polarization field ⊥E [see
Eq. (1) and Fig. 2)], which accelerate ions in the B×R
direction. As soon as transversal velocity of the ions passes
via a maximum, field ⊥E becomes negative [see Eq. (1)].
In this field electrons experience electric drift opposite to
R, taking away the energy W⊥
from the ions. This last drift
returns the ion component (through the field - RE , see
Fig. 2) on the circular trajectory. At the moment of return,
the ions (and plasma as a whole) fully lose its transversal
energy and then a cycle repeats itself again.
Thus, the transversal movement of the plasma in the
B×R direction is only a link in the chain of
transformations of the energy ( )2
// //W R Wηρ⊥ ≈ .
Since always
//W W⊥ < , the inequality
// 1Rηρ << is a
condition of plasma movement along the toroidal
magnetic field. In reality, at the point of entry, plasma
always moves straightforward, on tangent to the magnetic
field line, and the question only consist in: how long this
motion does continue (see Fig. 3). Evidently, its duration
is proportional to
//W W⊥
, which grows with the radial
displacement Rd or ratio //MV qB . In more accurate
approximation, the coefficient at //W in Eq.
( )2
// //W R Wηρ⊥ ≈ is modified and asymptotically approaches
to one, if the ratio //MV qB increases. Nevertheless, the
approximate condition
// 1Rηρ < shows too that vacuum-arc
plasma may be guided by essentially weak magnetic field (of a
few dozens of Gauss [2]) and, in this case, the plasma loss
in curved magnetic filters occurs due to a large
characteristic radius of the plasma flux rather [17] than
because of its radial displacement.
REFERENCES
1. V.S. Voitsenya, A.G. Gorbaniuk et al. On polarization of a
plasma propagating in a curved magnetic field// Sov. Phys.-
Tech. Phys. 1965, v.35, N7, p.1330-1332 (in Russian).
2. V. Zhitomirsky, L. Kaplan et al. Ion current distribution
in a filtered vacuum arc deposition system// Surf. and
Coat. Techn. 1995, v.76/77, p.190-196.
3. A. Anders, S. Anders et al. Transport of vacuum arc
plasmas through magnetic macroparticle filters// Plasma
Sourses Sci. Technol. 1995, v.4, p.1-12.
4. N.A. Khizhnyak. Motion of plasmoid in a magnetic
field of toroidal solenoid// Sov. Phys.-Tech. Phys. 1965,
v.35, N5, p.847-857.
5. V.S. Voitsenya, A.G. Gorbaniuk et al. Motion of
plasmoid in a curvilinear magnetic field// Sov. Phys.-Tech.
Phys. 1967, v.37, N2, p.262-273 (in Russian).
6. E.N. Sokolenko, N.A. Khizhnyak. Mass ion separation
in the plasma flow moved along the toroidal magnetic
field// Probl. Atom. Sci. Techn, Ser. “Plasma Phys.” (5).
2000, N3, p.84-86.
7. X. Shi, Y.Q. Tu et al. Simulation of plasma flow in toroidal
solenoid filters// IEEE Trans. Plasma Sci. 1996, v.24, N6,
p.1309-1318.
8. I.I. Aksenov, V.G. Padalka et al. Study on motion of
plasma flux in a curvilinear plasma-optics system// Sov. J.
Plasma Phys. 1980, v.6, N2, p.312-317 (in Russian).
9. I.I. Aksenov, A.N. Belokvostikov et al. Plasma flux
motion in a toroidal plasma guide// Plasma Phys. Control.
Fus. 1986, v.28, N5, p.761-770.
10. B.A. Alterkop, E. Gidalevich et al. Vacuum arc plasma jet
propagation in a toroidal duct// J. Appl. Phys. 1996, v.79. N9,
p.6791-6802.
11. B.A. Alterkop, V.N. Zhitomirsky et al. Propagation of
vacuum arc plasma beam in a toroidal filter// IEEE Trans.
Plasma Sci. 1996, v.24, N6, p.1371-1377.
12 A.I. Timoshenko, V.S. Taran et al. Transversal
displacement of the plasma flux during its motion in a
toroidal magnetic field// AIP Conf. Proc. / New York:
“Melville”, 2008, v.993, p.121-124.
13. A.I. Timoshenko, V.S. Taran et al. A methode of
matching of ion and electron motion equations for plasma
moving in a toroidal magnetic field // Ukr. J. Phys. 2008,
v.53, N4, p.345-350.
14. A. Anders, Yu. G. Yushkov. Ion flux from vacuum
arc cathode spots in the absence and presence of magnetic
field// J. Appl. Phys. 2002, v. 91, N8, p. 4824-4832.
15. A. Anders. Ion charge state distribution of vacuum arc
plasmas: The origin of species// Phys. Rev. E. 1997, v.55,
N1, p.969-981.
16. A. Anders. A Periodic table of ion charge-state
distributions observed in the transition region between
vacuum sparks and vacuum arcs// IEEE Trans. Plasma
Sci. 2001, v.29, N2, p.393-398.
17. A.I. Timoshenko. Plasma loss in a curved magnetic
filters// Probl. Atom. Sci. Techn. Ser. “Pl. El. New Meth.
Accel.” (6). 2008, N4, p.274-279 (in Russian).
Article received 22.09.08.
ПРОИСХОЖДЕНИЕ ПОПЕРЕЧНОГО СМЕЩЕНИЯ ПОТОКА ПЛАЗМЫ, ДВИЖУЩЕГОСЯ
ВДОЛЬ КРИВОЛИНЕЙНОГО МАГНИТНОГО ПОЛЯ
А.И. Тимошенко
Показано, что поперечное смещение плазменного потока может быть интерпретировано как результат
преобразования (посредством силы Лоренца) энергии, полученной электронами в поле поляризационной силы и в поле
сил, связанных с кривизной магнитного поля, в кинетическую энергию поперечного движения всей плазмы. Такая же
интерпретация справедлива и для дрейфового движения отдельной частицы.
ПОХОДЖЕННЯ ПОПЕРЕЧНОГО ЗМІЩЕННЯ ПОТОКУ ПЛАЗМИ, ЩО РУХАЄТЬСЯ
ВЗДОВЖКРИВОЛІНІЙНОГО МАГНІТНОГО ПОЛЯ
О.І. Тимошенко
Показано, що поперечне зміщення плазмового потоку може бути інтерпретовано як результат перетворення (за
допомогою сили Лоренца) енергії, отриманої електронами в полі поляризаційної сили і в полі сил, пов'язаних з
кривизною магнітного поля, в кінетичну енергію поперечного руху всієї плазми. Така ж інтерпретація справедлива і для
дрейфового руху окремої частинки.
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