Effect of AC electric field with the changeable in time frequency on trapped particles
The effect of alternating electric field on the particle detrapping in two kinds of magnetic configurations with different || modulation is considered here.
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2002
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| Zitieren: | Effect of AC electric field with the changeable in time frequency on trapped particles / A.Yu. Antufuev, A.A. Shishkin // Вопросы атомной науки и техники. — 2002. — № 4. — С. 56-58. — Бібліогр.: 3 назв. — англ. |
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irk-123456789-802472015-04-15T03:02:05Z Effect of AC electric field with the changeable in time frequency on trapped particles Antufuev, A.Yu. Shishkin, A.A. Magnetic confinement The effect of alternating electric field on the particle detrapping in two kinds of magnetic configurations with different || modulation is considered here. 2002 Article Effect of AC electric field with the changeable in time frequency on trapped particles / A.Yu. Antufuev, A.A. Shishkin // Вопросы атомной науки и техники. — 2002. — № 4. — С. 56-58. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 52.55.-s http://dspace.nbuv.gov.ua/handle/123456789/80247 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Magnetic confinement Magnetic confinement |
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Magnetic confinement Magnetic confinement Antufuev, A.Yu. Shishkin, A.A. Effect of AC electric field with the changeable in time frequency on trapped particles Вопросы атомной науки и техники |
| description |
The effect of alternating electric field on the particle detrapping in two kinds of magnetic configurations with different || modulation is considered here. |
| format |
Article |
| author |
Antufuev, A.Yu. Shishkin, A.A. |
| author_facet |
Antufuev, A.Yu. Shishkin, A.A. |
| author_sort |
Antufuev, A.Yu. |
| title |
Effect of AC electric field with the changeable in time frequency on trapped particles |
| title_short |
Effect of AC electric field with the changeable in time frequency on trapped particles |
| title_full |
Effect of AC electric field with the changeable in time frequency on trapped particles |
| title_fullStr |
Effect of AC electric field with the changeable in time frequency on trapped particles |
| title_full_unstemmed |
Effect of AC electric field with the changeable in time frequency on trapped particles |
| title_sort |
effect of ac electric field with the changeable in time frequency on trapped particles |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2002 |
| topic_facet |
Magnetic confinement |
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http://dspace.nbuv.gov.ua/handle/123456789/80247 |
| citation_txt |
Effect of AC electric field with the changeable in time frequency on trapped particles / A.Yu. Antufuev, A.A. Shishkin // Вопросы атомной науки и техники. — 2002. — № 4. — С. 56-58. — Бібліогр.: 3 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT antufuevayu effectofacelectricfieldwiththechangeableintimefrequencyontrappedparticles AT shishkinaa effectofacelectricfieldwiththechangeableintimefrequencyontrappedparticles |
| first_indexed |
2025-07-06T04:12:55Z |
| last_indexed |
2025-07-06T04:12:55Z |
| _version_ |
1836869407313231872 |
| fulltext |
EFFECT OF AC ELECTRIC FIELD WITH THE CHANGEABLE IN TIME
FREQUENCY ON TRAPPED PARTICLES
Alexander Yu. ANTUFYEV, Alexander A. SHISHKIN*
Department of Physics and Technology,
Kharkov “V.N.Karazin” National University, Kharkov-77, Ukraine,
*Institute of Plasma Physics, National Science Center "Kharkov Institute of Physics and
Technology", Kharkov-108, Ukraine
and
Department of Physics and Technology,
Kharkov “V.N.Karazin” National University, Kharkov-77, Ukraine
The effect of alternating electric field on the particle detrapping in two kinds of magnetic configurations with different |
B | modulation is considered here.
PACS: 52.55.-s
1. INTRODUCTION
The processes of detrapping / retrapping of the charged
particles under the high-frequency parallel electric field
can be considered as the resonance between the bounce
frequency of the trapped particle and the frequency of the
externally applied parallel electric field [1]. Principally
new effect studied here is the following. Linearly
increasing frequency is used for detrapping of charged
particle. The initial value of the frequency of the electric
field is close to the bounce frequency of the trapped
particle. Detrapping / retrapping processes of the charged
particle under the alternating parallel electric field is
studied by numerical integration of guiding center
equations in two types of magnetic configurations.
Outward shifted configuration and inward shifted
configuration [2] are considered and compared.
2.1 Magnetic field model
The components of the magnetic field are as follows:
( ) ( )( ) ( )ϕϑε mlaralmRBB l
lmr −−= − sin1
0 (1)
( ) ( )( ) ( )ϕϑεϑ mlaralmRBB l
lm −−= − cos1
0 (2)
( ) ( ) ( ){ }ϕϑεϑϕ mlarRrBB l
lm −++= coscos10 (3)
where 0B is the magnetic field at a circular axis of the
torus with major radius R and minor radius a. The
coordinates ϕϑ ,,r are connected with circular axis of the
torus: radial variable r, toroidal and poloidal angles ϑ , ϕ
; m is the number of magnetic field periods along the
torus, l is the number of the helical winding poles; lmε
are the coefficients of harmonics of the magnetic field.
The magnetic field absolute value is given by ϕBB ≈ .
The vertical magnetic field is added to the main magnetic
field.
.cos
;sin
)(
)(
ϑ
ϑ
ϑ ⊥
⊥
=
=
BB
BB
vert
vertr
(4)
Inward shifted configuration and outward shifted
configuration are obtained by varying of the vertical
magnetic field ⊥B .
2.2 Electric field model
Electric field which acts on the charged particle is taken
in such form:
( ) ( )δϕϑϕ +Ω−= tmlEE lm coscos (5)
where lmE is electric field amplitude, Ω is the frequency
and δ is the phase of the electric field oscillation. During
the process of detrapping the frequency of the electric
field Ω is linearly increasing from toroidal
bounceω to helical
bounceω ,
where toroidal
bounceω is the bounce frequency of toroidally
trapped particle and helical
bounceω is the bounce frequency of
helically trapped particle.
2.3 Guiding Center equations
Guiding center equations in coordinates ϕϑ ,,r can be
reduced to the following form:
( ) ( )ϑϕϑϕ
υυ
υ BB
eB
Mc
BE
B
c
B
Br r ∇
+
−−= ⊥
3
22
||
2|| 2
2
; (6)
( ) ( ) rBB
eB
Mc
B
Br ∇
+
+= ⊥
ϕ
ϑ υυ
υϑ 3
22
||
|| 2
2 ; (7)
( ) ( ) ( ) rBB
eB
Mc
B
B
rR ∇
+
−=− ⊥
ϑ
ϕ υυ
υϕϑ 3
22
||
|| 2
2
cos ; (8)
( )
∂
∂+
∂
∂−−=
ϑ
ϑµϕϑ
υ
υ ϕ B
r
BrrR
M
eE
2
cos1
||
|| ; (9)
∂
∂+
∂
∂=
⊥
⊥ ϑ
ϑ
υ
µυ B
r
Br
2 . (10)
where ||υ is the parallel velocity of the charged particle,
⊥υ is its perpendicular velocity and B2
⊥= υµ is the
adiabatic invariant.
Runge-Kutta method is used for numerical integration of
guiding center equations.
3. ANALYSIS OF THE TRANSITION
PROCESS
Using guiding center equations and expressions for
magnetic field components we obtain magnetic surface
56 Problems of Atomic Science and Technology. 2002. № 4. Series: Plasma Physics (7). P. 56-58
function Ψ . Here is the expression for the function 0Ψ
without vertical magnetic field:
( )
−
+=Ψ ϕϑε ml
a
r
m
R
R
mrB
l
lm cos
2
2
0
0 . (11)
For further studies we consider l=2 case in order to
simplify the expressions.
Using the equation 0=Ψ∇B , considering only linear
perturbing terms, we can write ( ) ( ) 01010 =Ψ+Ψ∇+ BB ,
where 0B is the main magnetic field, 1B is the small
adding to the main magnetic field due to the vertical
magnetic field, and 1Ψ is the small adding to the function
Ψ due to the vertical magnetic field. Full function Ψ
takes the following form:
( )
( ) ( )[ ]ϕϑϑ
ϕϑε
mArrAB
m
a
r
m
R
R
mrB
l
lm
−+−+
+
−
+=Ψ
⊥ coscos1
2cos
2
2
2
0
(12)
where 2
2
22 2 RamA mε= . Using the equation for the
contour of the magnetic surface and considering A<<1,
we found the expression for the shift of the magnetic axis
∆ :
+=∆ ⊥ 220
2
am
R
R
mBB mε (13)
The modulation of the magnetic field can take the
following form:
( )[ ] ,cos12cos
cos1
2
2
0
0
0
+−
+
++=
ϑσϕϑε
ϑ
m
a
r
R
r
BB
m
(14)
where ( ) 0
22 rmR−∆=σ , 0r is averaged radius of the
shifted magnetic surface.
As trapped or blocked particle alters the direction of its
motion, the trajectory has reflection points. While altering
the direction of the particle motion the longitudinal
velocity changes its sign and equals zero at the reflection
point. Thus the analysis of the equation (9) for ||υ is
important for understanding the condition of particle
transition from the trapped state to the passing one. Now
we pass to the analysis of the equation (9). Using the
expressions (6), (7) for r and ϑr we can write the
expression for ||υ in such form:
( ) ( ) ∗⊥∗⊥⊥
∗
−∇
+
+
+
+=
G
B
BF
eB
Mc
B
cF
M
eE
2
2
||
4
22
||
2
||3||
24
2 υ
υ
υυυ
υυ
ϕ
ϕ
(15)
where
( )
( ) ;cos21
4cos21
220
4
2
2
222
2
−+−
−
+−≡
⊥
∗
ϕϑε
εϕϑε
m
a
r
R
BB
a
r
m
Rm
a
r
m
BF
m
mmo
(16)
( )
+−−≡ ⊥
∗ B
m
B
m
a
rBG m
0
220 sin2 ϕϑε . (17)
As one can see, the expressions for ∗F and ∗G contain
harmonics ( )ϕϑ m−sin and ( )ϕϑ m−cos . Therefore it is
possible to choose the form of the electric field model to
contain harmonics ( )ϕϑ m−sin or ( )ϕϑ m−cos .
4. Particle orbits in Inward Shifted and Outward
Shifted configurations
For the further numerical study we take the magnetic
configuration with the following parameters: R = 1000
cm, a = 200 cm, 0B = 3T, l = 2, m =10. The
configuration with such parameters is chosen to
understand the physics of considered phenomena in the
magnetic systems with large aspect ratio.
Inward shifted configuration.
Fig.1 presents the magnetic field modulation in this
configuration with 016,00 −=⊥ BB . When the proton
Fig. 1 Magnetic field modulation in INWARD SHIFTED
configuration
with the energy 1 keV, 2,0|| =υυ moves in such
magnetic configuration it is trapped on the toroidal
inhomogenity of the magnetic field (blocked particle).
The high-frequency externally applied electric field with
linearly increasing frequency is used to detrap the
particle. The motion of the particle changes considerably.
Fig. 2 Illustration of the transition of the test particle
from the blocked state into the passing one in INWARD
SHIFTED configuration.
On the Fig.2 one can see the parallel velocity of the test
particle while transiting from the blocked state into the
passing one. The vertical cross-section of the orbit of the
passing particle is shown on Fig.3.
57
Fig. 3 Transit particle orbit in INWARD SHIFTED
configuration.
Fig. 4 Magnetic field modulation in OUTWARD
SHIFTED configuration.
Outward shifted configuration.
On the Fig.4 it is shown the magnetic field modulation,
corresponding to the outward shifted configuration with
019,00 =⊥ BB . When the test particle (1keV proton
with 2,0|| =υυ ) moves in the magnetic field of a such
configuration, it is helically trapped. But, as it is seen
from the Fig.5, when the electric field with linearly
increasing frequency (dashed line on the Fig.5) acts on
the test particle it becomes the passing one. The Fig.6
shows the vertical cross-section of the test particle orbit.
In real stellarator type devices including the helical
devices the magnetic axis is shifted from the circular axis
of the torus. This shift is increased under the effect of the
finite β , where β is the ratio of the plasma gas kinetic
pressure to the magnetic pressure ( 28 BnTπβ = ). The
displacement of magnetic axis ∆ is the measure of β .
CONCLUSION
1. The principally new thing considered here is the
application of the alternating electric field with
variable in time frequency with the goal to force the
transition of the charged particle from the trapped
state into the passing one.
Fig. 5 Illustration of the transition of the test particle
from the trapped state into the passing one in OUTWARD
SHIFTED configuration.
Fig. 6 Test particle orbit while transiting from the
trapped state into the passing one in OUTWARD
SHIFTED configuration.
2. The process of transition is studied in two
configurations with different magnetic field
modulation. It is shown that particle transition from
the trapped state into one takes place in both
configurations.
3. Magnetic field model with l=2 is considered in
numerical studies and it is shown that the AC electric
field has to contain l=1 harmonics.
ACKNOWLEDGEMENTS
This work is carried out under Science and Technology
Center in Ukraine Project #2313.
REFERENCES
[1] A.Yu.Antufyev and A.A.Shishkin, Particle
Detrapping under AC Electric Field Effect as the
Resonance Process. Problems Of Atomic Science And
Technology. 2000. N3. Series: Plasma Physics p.13-15.
[2] Mynick H.E., Chu T.K., Boozer A.H., Class of Model
Stellarator Fields with Enhanced Confinement. Phys.
Rev. Lett., 1982, 48, N5, p. 322-325.
[3] Е.Д. Волков, В.А. Супруненко, А.А. Шишкин,
СТЕЛЛАРАТОР, Киев, Наукова думка, 1983.
58
Acknowledgements
This work is carried out under Science and Technology Center in Ukraine Project #2313.
References
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