Drift motion of charged particle in wave field of magnetic pumping under Сherenkov and cyclotron resonance conditions
The charged particle motion problem in electromagnetic field of magnetic pumping under Chrenkov and cyclotron resonance conditions is solved in drift approximation. The wave field is produced by alternating surface azimuthal current, modeling the current of solenoidal antenna, which use is considere...
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irk-123456789-1092332016-11-22T03:03:25Z Drift motion of charged particle in wave field of magnetic pumping under Сherenkov and cyclotron resonance conditions Yeliseyev, Yu.N. Stepanov, K.N. Фундаментальная физика плазмы The charged particle motion problem in electromagnetic field of magnetic pumping under Chrenkov and cyclotron resonance conditions is solved in drift approximation. The wave field is produced by alternating surface azimuthal current, modeling the current of solenoidal antenna, which use is considered within the frames of a developed ICR-method of isotope separation. The drift motion equations are derived and their three first integrals are found at arbitrary values of Larmor radius. It is shown that the increasing of a particle Larmor radius involves the increasing of radius of the Larmor center, i.e. involves drift of heated particles to plasma edge. During Larmor gyration these ions transit near to a system axis. В дрейфовом приближении решена задача о движении заряженной частицы в поле волны магнитной накачки в условиях черенковского и циклотронного резонансов. Поле волны создается поверхностным переменным азимутальным током, моделирующим ток соленоидальной антенны, использование которой рассматривается в рамках разрабатываемого ИЦР-метода разделения элементов и изотопов. Выведены уравнения дрейфового движения, справедливые при произвольной величине ларморовского радиуса, и найдены три их первых интеграла. Показано, что увеличение ларморовского радиуса частицы под действием волны накачки сопровождается увеличением радиуса ее ларморовского центра, т.е. дрейфом нагретых частиц на периферию плазмы. При ларморовском вращении эти ионы проходят вблизи оси системы. У дрейфовому наближенні вирішено задачу про рух зарядженої частки в полі хвилі магнітного накачування в умовах черенковського й циклотронного резонансів. Поле хвилі створюється поверхневим змінним азимутальним струмом, що моделює струм соленоїдальної антени, використання якої розглядається в рамках розроблюваного ІЦР-методу розділення елементів і ізотопів. Виведені рівняння дрейфового руху справедливі при довільній величині ларморовського радіуса, і знайдено три їх перших інтеграла. Показано, що збільшення ларморовського радіуса частки під дією хвилі накачування супроводжується збільшенням радіуса її ларморовського центра, тобто дрейфом нагрітих часток на периферію плазми. При ларморовському обертанні ці іони проходять поблизу осі системи. 2013 2013 Article Drift motion of charged particle in wave field of magnetic pumping under Сherenkov and cyclotron resonance conditions / Yu.N. Yeliseyev, K.N. Stepanov // Вопросы атомной науки и техники. — 2013. — № 1. — С. 84-86. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 52.40.Fd; 52.50.Qt http://dspace.nbuv.gov.ua/handle/123456789/109233 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Фундаментальная физика плазмы Фундаментальная физика плазмы |
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Фундаментальная физика плазмы Фундаментальная физика плазмы Yeliseyev, Yu.N. Stepanov, K.N. Drift motion of charged particle in wave field of magnetic pumping under Сherenkov and cyclotron resonance conditions Вопросы атомной науки и техники |
| description |
The charged particle motion problem in electromagnetic field of magnetic pumping under Chrenkov and cyclotron resonance conditions is solved in drift approximation. The wave field is produced by alternating surface azimuthal current, modeling the current of solenoidal antenna, which use is considered within the frames of a developed ICR-method of isotope separation. The drift motion equations are derived and their three first integrals are found at arbitrary values of Larmor radius. It is shown that the increasing of a particle Larmor radius involves the increasing of radius of the Larmor center, i.e. involves drift of heated particles to plasma edge. During Larmor gyration these ions transit near to a system axis. |
| format |
Article |
| author |
Yeliseyev, Yu.N. Stepanov, K.N. |
| author_facet |
Yeliseyev, Yu.N. Stepanov, K.N. |
| author_sort |
Yeliseyev, Yu.N. |
| title |
Drift motion of charged particle in wave field of magnetic pumping under Сherenkov and cyclotron resonance conditions |
| title_short |
Drift motion of charged particle in wave field of magnetic pumping under Сherenkov and cyclotron resonance conditions |
| title_full |
Drift motion of charged particle in wave field of magnetic pumping under Сherenkov and cyclotron resonance conditions |
| title_fullStr |
Drift motion of charged particle in wave field of magnetic pumping under Сherenkov and cyclotron resonance conditions |
| title_full_unstemmed |
Drift motion of charged particle in wave field of magnetic pumping under Сherenkov and cyclotron resonance conditions |
| title_sort |
drift motion of charged particle in wave field of magnetic pumping under сherenkov and cyclotron resonance conditions |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2013 |
| topic_facet |
Фундаментальная физика плазмы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/109233 |
| citation_txt |
Drift motion of charged particle in wave field of magnetic pumping under Сherenkov and cyclotron resonance conditions / Yu.N. Yeliseyev, K.N. Stepanov // Вопросы атомной науки и техники. — 2013. — № 1. — С. 84-86. — Бібліогр.: 6 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT yeliseyevyun driftmotionofchargedparticleinwavefieldofmagneticpumpingundersherenkovandcyclotronresonanceconditions AT stepanovkn driftmotionofchargedparticleinwavefieldofmagneticpumpingundersherenkovandcyclotronresonanceconditions |
| first_indexed |
2025-07-07T22:44:22Z |
| last_indexed |
2025-07-07T22:44:22Z |
| _version_ |
1837029930158784512 |
| fulltext |
84 ISSN 1562-6016. ВАНТ. 2013. №1(83)
DRIFT MOTION OF CHARGED PARTICLE IN WAVE FIELD OF
MAGNETIC PUMPING UNDER CHERENKOV AND CYCLOTRON
RESONANCE CONDITIONS
Yu.N. Yeliseyev, K.N. Stepanov
Institute of Plasma Physics NSC “Kharkov Institute of Physics and Technology”,
Kharkov, Ukraine
E-mail: eliseev2004@rambler.ru
The charged particle motion problem in electromagnetic field of magnetic pumping under Chrenkov and cyclo-
tron resonance conditions is solved in drift approximation. The wave field is produced by alternating surface azi-
muthal current, modeling the current of solenoidal antenna, which use is considered within the frames of a devel-
oped ICR-method of isotope separation. The drift motion equations are derived and their three first integrals are
found at arbitrary values of Larmor radius. It is shown that the increasing of a particle Larmor radius involves the
increasing of radius of the Larmor center, i.e. involves drift of heated particles to plasma edge. During Larmor gyra-
tion these ions transit near to a system axis.
PACS: 52.40.Fd; 52.50.Qt
Interaction of particles with a wave is a basis for
calculation of effect of a selective heating of ions in a
developed ICR-method of isotope separation [1]. In this
paper the solution of a problem about a charged particle
motion in the homogeneous magnetic field and in the
vortex electromagnetic field of a wave of magnetic
pumping of small amplitude under Cherenkov and cy-
clotron resonance conditions is presented in the drift
approximation.
1. WAVE OF MAGNETIC PUMPING
The wave is produced by the azimuthal surface cur-
rent 0 ( ) cos( )zj j r a k z tϕ = δ − −ω modeling the current
in solenoidal antenna. Its usage is considered within the
method of ICR-separation [1]. The wave field inside the
solenoid ( r a< ) has components , ,r zE H Hϕ [2]:
1
0
1
( )sin( ),
( ) cos( ),
( )sin( ) .
z
z
z z
z
z
r z
E CI r k z t
k c
H C I r k z t
k
k c
H C I r k z t
ϕ = − Λ −ω
Λ
= Λ −ω
ω
= Λ −ω
ω
(1)
Here 0
0 0 1 12
1
4, ( ) ( )
( )
E
C E j aK a I a
I a c
πω
= = Λ Λ
Λ
– is the
amplitude of azimuthal electric field near the solenoid
( r a= ),
2
2 2
2 0, 1.z
z
k ck N
c
ω
Λ = − > = >>
ω
2. DERIVATION OF DRIFT EQUATIONS
We derive the equations of particle drift motion in
the field (1) using the method [3]. Introducing a com-
plex variable exp( )u x iy r i= + = ϕ we write equations
of motion in this variable:
0 0
exp( ) exp( ) , (2)
.
ci
z r
ci ci
r
u i u
H Hei E i u z i
M B B
vez H
M c
ϕ
ϕ
+ ω =
⎛ ⎞
= ⋅ ϕ −ω +ω ⋅ ϕ⎜ ⎟
⎝ ⎠
= −
&& &
& &
&&
Here 0 / 0ci eB Mcω = > – is the cyclotron frequency
of a particle. We search he solution of equations (2) in
the form
exp( ) exp( ) exp( )u r i R i i= ϕ = ⋅ θ + ρ⋅ ϑ (3)
Particle motion is described by cylindrical coordinates
of the Larmor center R , θ , coordinates of a particle
on the Larmor circle ρ , 0 сtϑ = ϑ −ω ( 0ϑ - initial phase
of Larmor rotation) and longitudinal variables
0 0
t
zz z v dt= + ∫ , zv .
According to (3) vectors exp( )r iϕ , ( )R exp i⋅ θ ,
0exp[ ( )]cii tρ⋅ ϑ −ω form a triangle. Taking this fact into
account we apply the Graf summation theorem for Bes-
sel functions [4], having in our case the form
1
1 0
( )exp( )
( ) ( )exp{ [ ( ) ]}, (4)
p
p p ci
p
I r i
I R I i p p t
=+∞
+
=−∞
Λ ϕ =
= Λ Λρ θ− ϑ −θ + ω∑
to the terms in the right-hand sides of equations (2) and
express them in variables 0, , , , , zR z vθ ρ ϑ .
We find the approximate solution of equations (2)
supposing that a wave amplitude is small
( 0 / ( )ci cieE M rω << ω ) and a particle moves under Che-
renkov or cyclotron resonance conditions with a pump-
ing wave
1, 2, 3,...,, .ciz z ci nk v n = ± ± ± Δω << ωω = + + ω + Δω
(5)
In the absence of wave the coordinates
0 0, , , , , zR z vθ ρ ϑ are the integrals of motion. In the pres-
ence of wave of small amplitude the right-hand sides of
equations (2) contain small fast and slow oscillating
summands. Neglecting the fast oscillating summands
and taking into account only the slow oscillating ones
(i.e. using the method of averaging), we obtain the equa-
tions of particle drift motion:
ISSN 1562-6016. ВАНТ. 2013. №1(83) 85
( ) ( ) ( )
0
( )
( ) sin ,
( )
( ) ( )
cos , (6)
( ) ( )
( )
( ) sin ,
( )
( )
( ) cos
,
,
ci n
n n
ci
ci n n
n
ci
ci n
n n
ci
ci n
n n
ci
z
z
z
n dIeR C I R
M R d
dI R dIeR C
M d R d
n dIe C I R
M d
dIe dC I R
M d d
z v
k cev C
Mc
ω Λρρ
=− Λ Ψ
ω ω Λρ
ω Λ Λρ
θ = Λρ Ψ
ω ω Λ Λρ
ω Λρ
ρ = − Λ Ψ
ω ω Λρ
ρϑ =
⎡ ⎤ω Λρ
− Λ Λρ Ψ⎢ ⎥
ω ω Λρ Λρ⎢ ⎥⎣ ⎦
=
= −
ω
&
&
&
&
&
&
( )
( ) sin .
( )
n
ci n n
dI
I R
d
Λρ
ω ρ Λ Ψ
Λρ
Here 0( ) ( )n ci zn n t k zΨ ≡ ϑ −θ + ω− ω − – is a resonance
phase slowly changing under conditions (5). Using the
equations for θ and ϑ in (6) we obtain the equation for
nΨ in the form
( )
( ) ( )
( )
( )
( )1
( )
( ) ( )
cos . (7)
( ) ( )
ci n
n
ci
n ci z z
n n
n
n dIe d
C I R
M d d
n k v
dI R dI
R d R d
ω Λρ
− Λ Λ Λρ +
ω ω Λρ Λρ Λρ
Ψ = ω− ω − −
⎧ ⎡ ⎤
⎨ ⎢ ⎥
⎣ ⎦⎩
Λ Λρ ⎫ρ
+ Ψ⎬
Λ Λρ ⎭
&
The equations for R , ρ , nΨ , zv (6), (7) form the
closed set of equations. The phases θ and ϑ are deter-
mined by the solution of this closed set.
3. INTEGRALS OF DRIFT MOTION
Combining the equations for ρ& and R& we find one
first integral of the equations set (6):
2 2
1
2
ci
P
R C
M
ϕ−ρ = =
ω
. (8)
It determines the form of drift trajectories in Rρ plane.
Fig. 1. Drift trajectories in Rρ plane. Line R a+ ρ =
corresponds to a radius of solenoid, line R = ρ is the
asymplote of hyperbolas
The trajectories have the form of hyperbola (Fig. 1)
along which the ions drift under action of the wave field
of magnetic pumping (1) under cyclotron resonance
conditions (5). As it is seen from integral (8), the form
of drift trajectories does not depend on the number of
cyclotron harmonic n on which the resonance is real-
ized. This is a consequence of axial symmetry of a wave
field (1) and conservation of the generalized angular
momentum Pϕ .
If the values of radius of Larmor center R and Lar-
mor radius ρ satisfy the condition R a+ρ < , then the
particle, moving on Larmor trajectory, remains inside
the cylinder r a= , on which the solenoidal antenna is
placed, and interacts with a wave of magnetic pumping.
If coordinates of a particle satisfy the condition
R a+ ρ ≥ , then particle falls on the antenna during
Larmor rotation and stop the interaction with a wave.
Combining the equations for R& and zv& we find one
more first integral of the equations set (6), determining
drift trajectories in zRv plane:
( ) 2
2/ 2z c zv k n R C− ω = . (9)
Fig. 2. Drift trajectories in 2
zR v plane. Along y axis
the value 2R varies. Along x axis the longitudinal
velocity varies in frame of reference, moving with a
wave ( )/z zv k−ω . The cyclotron resonances (5) take
place only if the approximate equality is fulfilled
( )/ /z z ci zv k n k−ω ≈ − ⋅ω . Integer numbers n along x
axis specify the number of resonance cyclotron har-
monic. Positive n correspond to a normal Doppler ef-
fect, negative n – to anomalous Doppler effect
At 0zk ≠ the trajectory has the form of parabola in
zRv plane. More accurately, they are the segments of
parabola placed near velocity values, where the reso-
nance condition (5) is fulfilled. In Fig. 2 the trajectories
are presented in axes 2
zR v . In these axes the trajectories
are the straight line segments.
At 0zk = the trajectories degenerate into the lines
zv const= . In this case drift equations for R , ρ and
nΨ form the closed set.
The third first integral is a Hamiltonian of a particle
in the magnetic field and in the field of a pumping
0,0 0,2 0,4 0,6 0,8 1,0
0,0
0,2
0,4
0,6
0,8
1,0
R
a
a
ρ
R a+ ρ =
Radius of solenoid
R = ρ
86 ISSN 1562-6016. ВАНТ. 2013. №1(83)
wave. Its form can be determined by combining the
equations (6) or by averaging the exact expression of
Hamiltonian. Finally, we obtain its form:
0 1 3H H H C≡ + = . (10)
Here
2 2 2
0 2 2
ci zM p
H
M
ω ρ
= + (11)
is the well-known unperturbed (without pumping wave)
Hamiltonian of a particle and
( ) ( )1 cosci
n n nH e CI R I
ω ′= Λ ρ Λρ Ψ
ω
(12)
is the addition describing the interaction of a particle
with a pumping wave. It should be noted, that expres-
sions (10) – (12) determine Hamiltonian function rela-
tive to canonically conjugated variables ( ),RJ θ ,
( ),Jρ −ϑ , ( ),zp z , where ( ) 21 2R ciJ M R= ω ,
( ) 21 2 ciJ Mρ = ω ρ are the known adiabatic invariants
of a particle in magnetic field.
CONCLUSIONS
The drift equations are derived and its integrals are
found for particle moving in a homogeneous magnetic
field and in a vortex wave field of magnetic pumping
under Cherenkov and cyclotron resonance conditions.
The drift equations (6), (7) and integrals (8)-(12) are
obtained at arbitrary value of particle Larmor radius ρ .
The found integrals (8)-(12) make it possible to inte-
grate on time the equations of drift motion (6), (7), to
build trajectories in a phase space , , ,z nR vρ Ψ and thus
to solve the motion problem completely.
As it is seen from integrals (8), (9) the form of drift
trajectories does not depend on amplitude and the distri-
bution of the wave field on radius. The drift velocities,
of course, depend on these factors.
The integrals (8), (9) coincide with correspondent
integrals of the drift motion of a particle in the field of a
running along magnetic field potential wave having
azimuthal number 0m = [3].
As results from the integral (8), the increasing of a
particle Larmor radius ρ involves the increasing of
radius of the Larmor centre R , i.e. pumping wave in-
volves drift of heated particles outside, to plasma edge.
This peculiarity explains the observed radial drift of
particles interacting with a pumping wave in numerical
calculations [4, 5].
REFERENCES
1. D.A. Dolgolenko, Yu.A. Muromkin // Uspekhi Fizi-
cheskikh Nauk. 2009, v. 179, p. 369-382.
2. M.P. Vasil’yev, L.I. Grigor’yeva, et. al. // Journal of
Technical Physics. 1964, v. 34. p. 1231.
3. Yu.N. Yeliseyev, K.N. Stepanov // Ukrainian Journal
of Physics. 1983, v. 28, p. 683-692.
4. H. Bateman, A. Erdelyi. Higher transcendental func-
tions, v. 2. M.: «Nauka», 1974, 295 p.
5. V.I. Volosov, V.V. Demenev, et. al. // Plasma Phys.
Rep. 2002, v. 8, p. 559-564.
6. K.P. Shamrai, E.N. Kudryavchenko // Problems of
Atomic Science and Technology. 2008, v. 14, № 6,
p. 183-185.
Article received 10.10.12
ДРЕЙФОВОЕ ДВИЖЕНИЕ ЗАРЯЖЕННОЙ ЧАСТИЦЫ В ПОЛЕ ВОЛНЫ МАГНИТНОЙ
НАКАЧКИ В УСЛОВИЯХ ЧЕРЕНКОВСКОГО И ЦИКЛОТРОННОГО РЕЗОНАНСОВ
Ю.Н. Елисеев, К.Н. Степанов
В дрейфовом приближении решена задача о движении заряженной частицы в поле волны магнитной на-
качки в условиях черенковского и циклотронного резонансов. Поле волны создается поверхностным пере-
менным азимутальным током, моделирующим ток соленоидальной антенны, использование которой рас-
сматривается в рамках разрабатываемого ИЦР-метода разделения элементов и изотопов. Выведены уравне-
ния дрейфового движения, справедливые при произвольной величине ларморовского радиуса, и найдены
три их первых интеграла. Показано, что увеличение ларморовского радиуса частицы под действием волны
накачки сопровождается увеличением радиуса ее ларморовского центра, т.е. дрейфом нагретых частиц на
периферию плазмы. При ларморовском вращении эти ионы проходят вблизи оси системы.
ДРЕЙФОВИЙ РУХ ЗАРЯДЖЕНОЇ ЧАСТКИ В ПОЛІ ХВИЛІ МАГНІТНОГО НАКАЧУВАННЯ
В УМОВАХ ЧЕРЕНКОВСЬКОГО Й ЦИКЛОТРОННОГО РЕЗОНАНСІВ
Ю.М. Єлісеєв, К.М. Степанов
У дрейфовому наближенні вирішено задачу про рух зарядженої частки в полі хвилі магнітного накачу-
вання в умовах черенковського й циклотронного резонансів. Поле хвилі створюється поверхневим змінним
азимутальним струмом, що моделює струм соленоїдальної антени, використання якої розглядається в рам-
ках розроблюваного ІЦР-методу розділення елементів і ізотопів. Виведені рівняння дрейфового руху спра-
ведливі при довільній величині ларморовського радіуса, і знайдено три їх перших інтеграла. Показано, що
збільшення ларморовського радіуса частки під дією хвилі накачування супроводжується збільшенням раді-
уса її ларморовського центра, тобто дрейфом нагрітих часток на периферію плазми. При ларморовському
обертанні ці іони проходять поблизу осі системи.
|