Particle detrapping under AC electric field effect as the resonance process
Detrapping / retrapping processes of the particle under the AC parallel electric field is studied by the analytical methods and the numerical integration of guiding center equations. It is shown that these processes can be considered as the resonance between the bounce frequency of the trapped parti...
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irk-123456789-823592015-05-29T03:02:08Z Particle detrapping under AC electric field effect as the resonance process Antufyev, A.Yu. Shishkin, A.A. Magnetic Confinement Detrapping / retrapping processes of the particle under the AC parallel electric field is studied by the analytical methods and the numerical integration of guiding center equations. It is shown that these processes can be considered as the resonance between the bounce frequency of the trapped particle and the frequency of AC parallel electric field. 2000 Article Particle detrapping under AC electric field effect as the resonance process / A.Yu. Antufyev, A.A. Shishkin // Вопросы атомной науки и техники. — 2000. — № 3. — С. 13-15. — Бібліогр.: 5 назв. — рос. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/82359 533.9 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Magnetic Confinement Magnetic Confinement Antufyev, A.Yu. Shishkin, A.A. Particle detrapping under AC electric field effect as the resonance process Вопросы атомной науки и техники |
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Detrapping / retrapping processes of the particle under the AC parallel electric field is studied by the analytical methods and the numerical integration of guiding center equations. It is shown that these processes can be considered as the resonance between the bounce frequency of the trapped particle and the frequency of AC parallel electric field. |
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Antufyev, A.Yu. Shishkin, A.A. |
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Antufyev, A.Yu. Shishkin, A.A. |
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Antufyev, A.Yu. |
| title |
Particle detrapping under AC electric field effect as the resonance process |
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Particle detrapping under AC electric field effect as the resonance process |
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Particle detrapping under AC electric field effect as the resonance process |
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Particle detrapping under AC electric field effect as the resonance process |
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Particle detrapping under AC electric field effect as the resonance process |
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particle detrapping under ac electric field effect as the resonance process |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Magnetic Confinement |
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Particle detrapping under AC electric field effect as the resonance process / A.Yu. Antufyev, A.A. Shishkin // Вопросы атомной науки и техники. — 2000. — № 3. — С. 13-15. — Бібліогр.: 5 назв. — рос. |
| series |
Вопросы атомной науки и техники |
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AT antufyevayu particledetrappingunderacelectricfieldeffectastheresonanceprocess AT shishkinaa particledetrappingunderacelectricfieldeffectastheresonanceprocess |
| first_indexed |
2025-07-06T08:51:53Z |
| last_indexed |
2025-07-06T08:51:53Z |
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1836886957778534400 |
| fulltext |
UDC 533.9
Problems of Atomic Science and Technology. 2000. N 3. Series: Plasma Physics (5). p. 13-15
13
Particle Detrapping under AC Electric Field Effect as the Resonance
Process
A.Yu. Antufyev and A.A.Shishkin*
Plasma Physics Chair , Department of Physics and Technology,
Kharkov “V.N.Karazin” National University, Kharkov-77, 61077, UKRAINE,
* Institute of Plasma Physics, National Science Center, “Kharkov Institute of Physics and
Technology”, Academicheskaya str. 1, Kharkov –108, 61108, UKRAINE
and
Plasma Physics Chair, Department of Physics and Technology,
Kharkov “V.N.Karazin” National University, Kharkov-77, 61077, UKRAINE
Detrapping / retrapping processes of the particle under the AC parallel electric field is studied by the
analytical methods and the numerical integration of guiding center equations. It is shown that these processes can
be considered as the resonance between the bounce frequency of the trapped particle and the frequency of AC
parallel electric field.
1. Introduction
Trapped particles in the toroidal magnetic trap can
be the reason of the enhanced transport coefficients
(neoclassical transport) in plasma due to the large (in
comparison with the passing particles) deviation of the
trapped particles from the initial magnetic surface due to
drifts in the inhomogeneous magnetic field. The
reduction of the neoclassical transport in the stellarator
type devices is the subject of the numerous efforts.
Among them the application of the AC electric field
launched from outside to convert the trapped particles
into the passing particles can be very important [ 1-3 ].
Detrapping of the particles under the AC electric field
can be also very helpful if one should like to solve the
task to control electric field in plasma by loss cone
particle injection into the helical field [4]. That is why
the process of conversion is under study in this paper.
2. Resonance Phenomena in the Motion of
Particle under AC Electric Field
If the particle with the mass M , electric charge q
and magnetic moment µ moves in the magnetic field
zB,0,0=B , where ( )kzBB Bz cos10 ε−= , and under the
electric field zE,0,0=E , where
( ) tlzEE Ez Ω−= sincos10 ε , the position of the particle is
described with the variable ξ , where kz≡ξ , that
satisfies the following equation
τξεξξε
εµ
ξ sincos1sin
2 2
02
2
2
0
−
Ω
=
+
Ω
+
k
l
M
qkEkB
EB
B &&&
(1)
Here derivative is taken on tΩ≡τ . The equation above
can be treated as the nonlinear oscillator equation. The
solution of (1) takes the form
+−+
Ω
+
Ω
= t
Bk
B
bbb )]
4
1
)1((
4
1
cos{[
0
2
2
2
00 ε
µ
ωω
ξ
ω
ξξ
.sin
)1(
}
22
2
0 τ
ω
εαχ
Ω−
−Ω++
b
E (2)
Here
2
2
0 kB B
b
εµ
ω = and
2
0
Ω
=
M
keE
α ;
0ξ and
0χ are
integration constants.
The expression (2) means that the process of
detrapping or more exactly the conversion of the
trapped particle into passing under the effect of the
externally applied electric field can be treated as the
resonance phenomenon in the case when the bounce
frequency
bω of the trapped particle is equal to the
frequency of the external electric field Ω .
For the new variable x , where xεξ = , and the
parameter of smallness 1<<ε , equation (1) takes the
form
],sin)1(~
xx)1(
2
x
6
[xx 2
0
2
2
3
2
2
2
2
2
τεα
ε
µ
ωω
ε
ω
E
B
bbb
Bk
−+
++−
Ω
=
Ω
+ &&& (3)
where αεα ~3= .
Taking into account the difference between
bω and Ω ,
namely ∆=Ω− εω 22
b
, it is possible to consider the cases
of the exact resonance and the resonance with deviation.
The solution is taken in the form
),,(u),,(u)cos( x 2
2
1 τθετθετθ aaa +++= , (4)
Bogolyubov-Mitropol'skiy method is used to obtain the
equations for the amplitude a and phase θ as the slowly
changed variables
,cos)1(
2
~
θε
αε
τ Ed
da
−−=
].sin)1(
2
~
))1(
4
1
(
4
[
2 0
2
22
2
2
θε
α
ε
µ
ε
ε
τ
θ
E
B
a
Bk
a
d
d
−+
++Ω+−+
Ω
∆
=
(5)
Problems of Atomic Science and Technology. 2000. N 3. Series: Plasma Physics (5). p. 13-15
14
The “phase portrait” of particle detrapping / retrapping
process is described with the expression
).sin)1(
2
1~
16
1
(
4
422
2
θεαε
ε
ψ EB aaAa −−+
Ω
∆
−= (6)
where
).1(
4
1
0
2
2
BB Bk
A ε
µ
+
Ω
−= (7)
In the case of the exact resonance ( 0=∆ ) the amplitude
( ) 3/1α≅a . (8)
It means the larger deviation of the particle from the
state of trapped one to the state of the passing one.
In the case of the resonance with the deviation there
exist three singular points
,
)1(~ 2
1 ∆
Ω−
= Ea
εαε 1sin 1 −=θ ;
,
2
)1(~2 2
22 ∆
Ω−
+
Ω
∆
= E
BA
a
εαε
ε
1sin 2 =θ ; (9)
,
2
)1(~2 2
23 ∆
Ω−
−
Ω
∆
= E
BA
a
εαε
ε
1sin 3 −=θ .
The positions of these points are shown on Fig.1.
The half-width δ of the resonance is evaluated from (6)
and
.
1
)1(22
2
2
B
b
E
A−
Ω
−
=
ω
εα
δ
(10)
As one can see the half-width δ of the resonance is
proportional to the square root of the AC electric field
amplitude and reverse proportional to the square root of
the deviation 22
bω−Ω .
-1 6
X (arb. units)
-3.5
3.5
Y
(a
rb
. u
ni
ts
)
3
.
2 1
Fig.1. Phase portrait of the particle detrapping under
AC electric field
3. Numerical Study of Particle Detrapping
The process of detrapping of particle is
demonstrated for the more complicated magnetic
geometry that is corresponding to the practical tasks.
The heliotron /torsatron configuration is taken with the
parameters of Heliotron DR [5]: the number of the
helical winding poles 2=l , the magnetic field number
15=m , the magnetic field at the circular axis
TB 05.00 = , the large radius of torus cmR 90= , the
small radius of torus cmah 5.13= .
The main magnetic field ( )Φ∇=B is modeled with the
use of the magnetic field potential
+−−=Φ ∑
n
n
hmn rmnar
m
R
RB ϑεϕϑεϕ sin)sin()/( 0,1,0
(11)
where
0B is the magnetic field at the circular axis, R
and ha are the major and minor radii of the helical
winding; ϕϑ,,r are the coordinates connected with the
circular axis of the torus, r is the radial variable, ϑ and
ϕ are the angular variables along the minor and major
circumference of the torus, ϑ increases in the direction
opposite to the main normal of the circular axis of the
torus; metric coefficients are the following: 1=rh ,
rh =ϑ
, ϑϕ cosrRh += ; m is the number of the
magnetic field periods along the torus, l is the helical
winding pole number. The index n assumes the values
ln = , 1−l , 1+l ;
mn,ε are the coefficients of the
harmonics of the magnetic field. The coefficients
mn,ε
are chosen in such way that the magnetic surfaces and
particle orbits are like those which are obtained in the
paper [5].
The AC parallel electric field which effect on the
particle is chosen in a such form
)cos()]cos(1[
~~
0 EtmlEE φϕϑ +Ω−−= , (12)
where
bω=Ω , 2/πφ =E .
The test particle (electron with the energy =W 750 eV
and VV /||
=0.001) is the helically trapped one after the
start, then it becomes the blocked particle and comes
back to the helically trapped state again (Fig.2).
75 90 105
R(cm)
-15
0
15
Z(
cm
)
Fig. 2. The projection on vertical plane of the trajectory
of electron with the starting point 0r =2 cm, under
0ϑ =π and
0ϕ =0 (without AC electric field)
The corresponding velocity change on time is shown on
Fig.3 (top).
Problems of Atomic Science and Technology. 2000. N 3. Series: Plasma Physics (5). p. 13-15
15
0 60
-1
1
0 30
-1
1
Fig.3 The test particle velocity on time without AC
electric field (solid line) and the dependence of AC
electric field (dashed line) on time
This test particle is moving without the AC electric
field.
In the case of the AC electric field is applied the
trajectory of the particle changes remarkably. The
particle starts as the passing during some time, then
becomes the helically trapped and transforms into the
passing particle (Fig.4 and 5).
75 90 105
R(cm)
-15
0
15
Z(
cm
)
Fig.4. The projection on vertical plane of the trajectory
of electron from Fig.2 with the AC electric field.
One can see that point of the trajectory (Fig.4) where
the helically trapped particle transforms into the passing
one. This conversion takes place when the frequency of
the AC electric field is close to the bounce frequency of
the helically trapped particle (Fig.3 down). From Fig.5
one can see that the particle starts the motion as the
passing one, then becomes the helically trapped and
transfers into passing one.
0 30
-1
1
Fig. 5 The dependence of the test electron velocity
under the AC electric field
4. Conclusions
4.1. Detrapping of the charged particle under AC
electric field can be considered as the resonance process
when the bounce frequency of the trapped particle is
equal to the frequency of the AC electric field.
4.2. With the use of the analytical methods for the
simplified model of the magnetic field it is shown that
the equation of the particle motion under the AC electric
field is the equation of the non-linear oscillator with the
external force. The resonance separatrix in the phase
plane separates the states of the deeply trapped particles,
barely trapped and “near passing” ones.
4.3. Particle conversion from the helically trapped
state into passing ones in the magnetic field with the
toroidal and helical contributions is shown by the
numerical integration of the guiding center equations in
the presence of AC electric field. The reduction of the
deviation of the particle from the initial magnetic
surface is noticeable.
References
[1] Dobrowolny, M., Pogutse, O.P. Influence of High-
Frequency Electric Fields on Equilibrium and Stability
of Toroidal Plasmas. Phys.Rev. Lett. 25 (1970) 1608.
[2] Demirkhanov, R.A., Stotland, M.A., Khil’, S.V.
Conversion of trapped charged particles into trapped
particles in a high- frequency electric field. Sov. Phys.-
Tech. Phys.17 (1973) 1128.
[3] Voitsenya , V.S., Voloshko, A.Yu., Kalinichenko,
S.S., Solodovchenko, S.I., Shtan’, A.F. Effect of
alternating electric fields on the diffusion of
collisionless plasma in the Saturn stellarator. Sov. J.
Plasma Phys. 3 (1977) 659.
[4] Motojima, O., Shishkin A.A., Inagaki, S., Watanabe,
K. Possible control scenario of radial electric field by
loss cone particle injection into helical device. Nucl.
Fusion 40 (2000) 833.
[5] Morimoto, S., Obiki, T., Lin, H., Hartwell, G.J,
Schneider, T.A., Knowlton, S.F. Gandy, R.F. Magnetic
and Drift Measurements in HELIOTRON DR and
Compact Auburn Torsatron.Transactions of Fusion
Technology, 27 (1995) 202.
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