Controlling the distribution function at stochastic acceleration of charged particles by a regular electromagnetic wave in an external magnetic field
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Cite this: | Controlling the distribution function at stochastic acceleration of charged particles by a regular electromagnetic wave in an external magnetic field / V.A. Buts, O.V. Manuilenko, A.P. Tolstoluzhsky, Yu.A. Turkin // Вопросы атомной науки и техники. — 2001. — № 5. — С. 92-94. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859775449785171968 |
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| author | Buts, V.A. Manuilenko, O.V. Tolstoluzhsky, A.P. Turkin, Yu.A. |
| author_facet | Buts, V.A. Manuilenko, O.V. Tolstoluzhsky, A.P. Turkin, Yu.A. |
| citation_txt | Controlling the distribution function at stochastic acceleration of charged particles by a regular electromagnetic wave in an external magnetic field / V.A. Buts, O.V. Manuilenko, A.P. Tolstoluzhsky, Yu.A. Turkin // Вопросы атомной науки и техники. — 2001. — № 5. — С. 92-94. — англ. |
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CONTROLLING THE DISTRIBUTION FUNCTION AT
STOCHASTIC ACCELERATION OF CHARGED PARTICLES
BY A REGULAR ELECTROMAGNETIC WAVE IN
AN EXTERNAL MAGNETIC FIELD
V.A. Buts, O.V. Manuilenko, A.P. Tolstoluzhsky, Yu.A. Turkin
Institute of Plasma Electronics and New Methods of Acceleration NSC KIPT
1 Akademicheskaya St., Kharkov, 61108 Ukraine
e-mail: tolstoluzh@kipt.kharkov.ua
PACS numbers: 52.50.Sw
1 INTRODUCTION
The possibility to control the particle distribution
function by the energy can be used, in particular, for de-
velopment of tunable plasma sources of charged parti-
cles and X-rays on the basis of plasma particle stochas-
tic acceleration (heating). In the present paper the re-
sults of one of perspective ways to realize such a control
with nonlinear dynamics of charged particles in electro-
magnetic fields are considered.
Chaotic dynamics of the charged particle ensemble
in a field of electromagnetic wave propagating at an an-
gle to the constant magnetic field is studied. The
stochastic instability develops as a result of overlapping
nonlinear cyclotron resonances. From the analysis of the
integral of motion, the set of equations has, it follows,
that the maximum and average energy, gained by the
particles interacting with a wave under conditions of de-
veloped dynamic chaos, is limited. It allows one to con-
trol the particle distribution function by the energy with
changing the angle between the wave vector (propaga-
tion vector) and the external magnetic field.
2 STATEMENT OF A PROBLEM. BA-
SIC EQUATIONS
We consider the motion of a charged particle in a
constant externally applied magnetic field
H Ho= { , , }0 0 and in the field of an electromagnetic
plane wave, propagating at the angle φ to the field
H :
E E ikr i t
H c k E ikr i t
o
o
= −
= −
Re{ exp( )},
Re{ [ ] exp( )}
α ω
ω
α ω
. (1)
Here Eo is the wave amplitude,
α α α α= { , , }x y zi is
the polarization vector of the wave, and the wave vector
( ) ( ){ }
k c N c N= ω φ ω φ/ sin , , / cos0 has only two
components k x and k z . It can be always obtained by
suitable choice of coordinates. N is the index of refrac-
tion. In dimensionless variables ( t t→ ω ,
r r c→ ω / , p p mc→ / ,
k kc→ / ω the equations of charged
particle motion can be reduced to the form:
( / ) Re( ) ( / )[ ]
/ Re{( ) }
p kp Ee ph
k pE e
i
h
i
= − + +
+
1 γ ω γ
γ
ψ
ψ
, (2)
/r p= γ , /ψ γ= −
kp 1 ,
where ψ = −
kr t ,
h H Ho= / , ω ωh oeH mc= / ,
E eE mco= α ω/ , γ = +( ) /1 2 1 2p is the particle en-
ergy,
p is its momentum. The set of Eqs. (2) possesses
the integral of motion:
p iEe r h k consti
h− + − =Re( ) [ , ]ψ ω γ . (3)
For subsequent analysis it is convenient in (2), (3) to
pass to new variables p pz⊥ , , , ,θ ξ η - guiding center
coordinates, by formulas:
p p p p p p
x p y p
x y z z
h h
= = =
= − = +
⊥ ⊥
⊥ ⊥
cos , sin , ,
/ sin , / cos
θ θ
ξ ω θ η ω θ
. (4)
and supposing that the amplitude of the electromagnetic
wave ε ωo oeE mc= / is rather small and taking into
account that the efficient interaction between the parti-
cle and the wave occurs if one of the resonance condi-
tions is fulfilled:
k p s sz z h+ − = = − −ω γ 0 2 1 0 1 2, ... , , , , , , ... (5)
Conditions of overlapping the nonlinear resonances.
Let us suppose that during the particle interaction
with the wave its energy varies a little, i.e.
γ γ γ γ γ= + < <0 0s s s s
~ , ~ , where γ 0s meets the reso-
nant condition (5) and in view of the approximate inte-
gral of motion:
p k a constz z− = =γ , (6)
a closed set of equations for θ s and ~γ s can be obtained
from (2):
~ cos( ) /γ ε θ γs s o s sW= 0 , ( ) ~ /θ γ γs z s sk= −2
01 , (7)
W p sJ p J p Js x s y s z z s= − ′ +⊥ ⊥α µ µ α µ α µ( ) / ( ) ( ) ,
µ ω= ⊥k px h/ , J s ( )µ is the Bessel function,
′J s ( )µ is its derivative over the argument.
Equations (7) are the equations of the mathematical
pendulum. It is easy to find the width of the nonlinear
isolated resonance from these equations:
∆ ~ / ( )γ εs o s zW k= −2 1 2 . (8)
From resonant conditions (5) and approximate inte-
grals (6) we find the distance between the neighbouring
resonances:
δ γ ω~ / ( )s h zk= −1 2 . (9)
mailto:tolstoluzh@kipt.kharkov.ua
From expressions (8), (9) it is possible to write
Chirikov’s generalized criterion:
ε ωo h s zW k> −2 216 1/ ( ) , (10)
of development of a local instability of charged particle
motion during particle interaction with the electromag-
netic wave in the external magnetic field.
Let us suppose, that the particle is in resonance with
the number s s= * and the amplitude of the field is
those, that the stochastic instability of charged particle
motion takes place. In the space ( γ , ,p pz⊥ ) the parti-
cle motion is determined by approximate integral (7)
and resonance conditions (5), which in the plane (
p pz⊥ , ) become:
p
s
k
p k s
k
s kh
z
z
z h
z
h z
⊥
−
−
+
−
−
− −
=
2
2 2
2
2
2
2 2 2
1
1
1
1
1
ω
ω
ω
( )
(
( )
)
( )
, k z
2 1≠ ,
p s p sh z h⊥ − = −2 2 22 1ω ω , k z
2 1= , (11)
p s p k s kh z z h z⊥ + − − =2 2 2 2 21 0ω ω( / ( )) ,
s kh z
2 2 21ω = − .
The particle remaining on the integral (6) diffuses on
resonances (11). To estimate the resonance width with a
number which is greatly exceeding s∗ , we find the con-
stant a , which is the part of (6). By substituting in inte-
gral (6) the value of energy (5) we obtain:
a s k p k sz zs z h( ) ( )∗
∗
∗= − −1 2 ω . (12)
We find the value p
zs∗ from the first equation (11):
p s k
k k
s
k
p
zs
h z
z z
h
z s
∗
∗ ∗
⊥ ∗=
−
±
− −
− −
ω ω
1
1
1 1
1
2 2
2 2
2
2 . (13)
Substituting p
zs∗ from (13) into (12) we find:
a s s k ph z
s
( ) ( ) ( )∗ ∗
⊥ ∗= ± − − −
2 2 2 2 21 1ω (14)
Let us suppose now, that the particle has got in the
vicinity of the resonance with the number n s> > ∗ . To
find the width of this resonance it is necessary to calcu-
late pzn and p n⊥ . Using the resonance condition and
the first equation from (10) we find:
p a s k n kzs z h z= + −∗( ( ) ) / ( )ω 1 2 , p n
kn
h
z
⊥ =
ω
. (15)
3 CONTROLLING THE DISTRIBUTION
FUNCTION
For the wave with a polarization
α φ φ= −{ cos , , sin }0 (E-wave) the width of the non-
linear resonance decreases with the growth of n :
( )∆ ~ sin*γ ε φn a s c n= − −4 0 0
1 3 3 , (16)
(constant c0=0.477) and criterion (10) can be represen-
ted as:
16 11 3 2ε φ ωo o hc a s n( ) sin //∗ − > (17)
From (17) it is seen, that at high n the criterion ceases
to be fulfilled. It means, that the diffusion of particles
into the high-energy region due to development of
stochastic instability of charged particle motion is im-
possible, because of overlapping the nonlinear cyclotron
resonances (5).
To show possibility of controlling the particle distri-
bution function by energies with changing the angle φ
between the wave vector
k and the external magnetic
field, for the ensemble of 1000 particles uniformly dis-
tributed on the phase ψ , the equations (2) were numeri-
cally solved. The refraction index N = 2 . The angle
φ is chosen so, that k z > 1 ( φ π= 01. ). The initial en-
ergy of all particles was identical and was equal to
γ = 10001. . The initial value of the longitudinal pulse
for all particles was chosen as zero. Polarization of the
wave
α = { , , }0 0i , wave amplitude Eo =0.25, cy-
clotron frequency ω h =0.5. Dynamics of the particle
distribution function by energy in time, the time depen-
dence of average particle energy and dispersion was in-
vestigated. Calculations was checked with the help of
integral (6), which was conserved with precision no
more than 10-6.
Let us begin the analysis of results from Fig. 1, whe-
re for φ π= 01. the resonant conditions (11) are repre-
sented (marked as s), integral (6) (marked as I) and the
projection of hyperboloid ( γ 2 2 21= + +⊥p pz ) on the
plane (γ, pz) at p⊥ = 0 (marked γ m ). γ m is the mini-
mum value of the energy, which the particle can have at
given pz , i.e. on the plane (γ, pz) none can get to the
area below γ m . From Fig. 1 it is seen, that the resonant
interaction of particles with the wave is possible only
for resonances with the numbers s = 0, ±1, ±2, ±3.
Fig. 1. Resonant conditions (11) marked s, integral
(6) marked I and projection of hyperboloid
γ 2 2 21= + +⊥p pz on the plane ( γ , pz ) at p⊥ = 0
marked γ m .
In an initial point of time all particles are in the point
A. They are moving according to the integral I. At the
chosen wave amplitude E0=0.25, as noted below, the
resonances s = 0, ±1, ±2 are overlapped, and dynamics
of particles is stochastic, that leads to their diffusion by
energy.
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №5.
Серия: Ядерно-физические исследования (39), с. 93-94.
93
However, due to that particles are moving according
to the integral I, which is above γ m only between the
points B and C, the particles are distributed in the seg-
ment [B, C]. It is shown below, that this distribution is
uniform by energy, and its settling time is small.
The resonances with the numbers s and s+1 are
overlapped, if the sum of their half-widths (8) is more
than the distance between them (9). Amplitude of the
field, at which it occurs, is:
( )ε ωo h z s s s sk p J p J= − ′ + ′⊥ ⊥ + +
2 2 1 2
1 1
1 2 2
41/ * ( ) ( )/ / , (18)
where: p s ks o o h z⊥
−= − + − −γ γ ω2 2 2 2 2 11 1( )( ) ,
is determined from conditions of crossing of the integral
I and resonance s. As the value p s⊥ does not depend on
the sign of s, and the module | |J s
′ is contained in (17),
one can limit oneself to values s ≥ 0 . We find from
(18) ε ∗ =( , ) .0 1 0 027 , ε ∗ =( , ) .1 2 0102 i.e., the reso-
nances with the numbers s = 0, ±1, ±2 are overlapped,
and dynamics of particles, which in the initial point of
time were in the point A (Fig. 1), must be chaotic.
Numerical results of analysis of particle motion, (in
the initial point of time particle was in the point A
(Fig. 1) and had initial phase ψ = 0 ) are given in
Fig. 2-4. In Fig. 2 the time dependence of the particle
energy is represented.
Fig. 2. Time dependence of the particle energy.
One can see, that particle dynamics is irregular, and
its energy varies in finite limits γ min = 1 ,
γ max .= 1764 . From integral (5) and determination
γ = + +⊥( ) /1 2 2 1 2p pz , we will find γ min and γ max
values of the particle energy (point B and C in Fig. 1):
γ γ γmax ( ( ) )( )= + − + −k k kz o z o z
2 2 2 21 1 1 , (19)
γ γ γmin ( ( ) )( )= − − + −k k kz o z o z
2 2 2 21 1 1 .
The values γ min and γ max , obtained in (19), are
identical with the values obtained numerically.
In Fig. 3 the spectrum of the particle energy is repre-
sented.
Fig. 3. Spectrum of the particle energy.
In Fig. 4 the time dependence of the maximum Lya-
punov index L is shown. The value of the maximum
Lyapunov index is 0.096 that indicates on the local in-
stability of particle motion. The calculations which were
carried out for other initial phases ψ have shown, that
at the chosen field amplitude the particles dynamics is
stochastic, and the particles are not captured in one of
resonances s = 0, ±1, ±2.
Fig. 4. Time dependence of the maximum Lyapunov
index L.
The function of particle distribution by energy in the
point of time t T= 250 ( T - wave period) is repre-
sented in Fig. 5.
Fig. 5. Function of particles distribution by energy.
The value of the minimum energy is 1.0, and that of
the maximum one is 1.764. It is identical with those val-
ues of γ min and γ max , which are obtained analytically
and numerically for one particle.
In Fig. 6 the time dependence of the average energy
< >γ (average over the ensemble) is shown.
Fig. 6. Time dependence of the average energy <γ>
(average over the ensemble).
It is seen, that for the time of about 30 periods, <γ>
results to the stationary value: ( γ min + γ max )/2=1.382.
The time dependence of the dispersion σ 2 is repre-
sented in Fig. 7.
Fig. 7. Time dependence of the dispersion
σ γ γ2 2= < − < > >( ) .
One can see that after reaching the stationary
regime, the energy spreading does not change.
At the angle φ=0.375π kz<1, integral (5) is nonclosed
and can not limit the energy gaining by particles during
stochastic acceleration that is confirmed by the numeri-
cal result.
Thus, we have showed, that there are possibilities to
control the process of fast heating of particles (about
hundreds periods of field) and to obtain the particle dis-
tribution function of a high quality in the given range of
energies.
Work was supported partially by STCU grant # 253.
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №5.
Серия: Ядерно-физические исследования (39), с. 94-94
94
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| id | nasplib_isofts_kiev_ua-123456789-78997 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-02T09:00:29Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Buts, V.A. Manuilenko, O.V. Tolstoluzhsky, A.P. Turkin, Yu.A. 2015-03-24T16:13:56Z 2015-03-24T16:13:56Z 2001 Controlling the distribution function at stochastic acceleration of charged particles by a regular electromagnetic wave in an external magnetic field / V.A. Buts, O.V. Manuilenko, A.P. Tolstoluzhsky, Yu.A. Turkin // Вопросы атомной науки и техники. — 2001. — № 5. — С. 92-94. — англ. 1562-6016 PACS numbers: 52.50.Sw https://nasplib.isofts.kiev.ua/handle/123456789/78997 Work was supported partially by STCU grant # 253. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Controlling the distribution function at stochastic acceleration of charged particles by a regular electromagnetic wave in an external magnetic field Управление функцией распределения заряженных частиц по энергии при их стохастическом ускорении полем регулярной электромагнитной волны во внешнем магнитном поле Article published earlier |
| spellingShingle | Controlling the distribution function at stochastic acceleration of charged particles by a regular electromagnetic wave in an external magnetic field Buts, V.A. Manuilenko, O.V. Tolstoluzhsky, A.P. Turkin, Yu.A. |
| title | Controlling the distribution function at stochastic acceleration of charged particles by a regular electromagnetic wave in an external magnetic field |
| title_alt | Управление функцией распределения заряженных частиц по энергии при их стохастическом ускорении полем регулярной электромагнитной волны во внешнем магнитном поле |
| title_full | Controlling the distribution function at stochastic acceleration of charged particles by a regular electromagnetic wave in an external magnetic field |
| title_fullStr | Controlling the distribution function at stochastic acceleration of charged particles by a regular electromagnetic wave in an external magnetic field |
| title_full_unstemmed | Controlling the distribution function at stochastic acceleration of charged particles by a regular electromagnetic wave in an external magnetic field |
| title_short | Controlling the distribution function at stochastic acceleration of charged particles by a regular electromagnetic wave in an external magnetic field |
| title_sort | controlling the distribution function at stochastic acceleration of charged particles by a regular electromagnetic wave in an external magnetic field |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/78997 |
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