Influence of polarization of a regular electromagnetic wave on stochastic acceleration of charged particles in an external magnetic field
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| Zitieren: | Influence of polarization of a regular electromagnetic wave on stochastic acceleration of charged particles in an external magnetic field / V.A. Buts, O.V. Manuilenko, A.P. Tolstoluzhsky, Yu.A. Turkin // Вопросы атомной науки и техники. — 2001. — № 3. — С. 86-88. — англ. |
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Buts, V.A. Manuilenko, O.V. Tolstoluzhsky, A.P. Turkin, Yu.A. 2015-03-29T18:39:09Z 2015-03-29T18:39:09Z 2001 Influence of polarization of a regular electromagnetic wave on stochastic acceleration of charged particles in an external magnetic field / V.A. Buts, O.V. Manuilenko, A.P. Tolstoluzhsky, Yu.A. Turkin // Вопросы атомной науки и техники. — 2001. — № 3. — С. 86-88. — англ. 1562-6016 PACS nambers: 52.35.Mw https://nasplib.isofts.kiev.ua/handle/123456789/79212 This work was supported partially by STCU grant # 253. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Influence of polarization of a regular electromagnetic wave on stochastic acceleration of charged particles in an external magnetic field Влияние поляризации регулярной электромагнитной волны на стохастическое ускорение заряженных частиц во внешнем магнитном поле Article published earlier |
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| title |
Influence of polarization of a regular electromagnetic wave on stochastic acceleration of charged particles in an external magnetic field |
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Influence of polarization of a regular electromagnetic wave on stochastic acceleration of charged particles in an external magnetic field Buts, V.A. Manuilenko, O.V. Tolstoluzhsky, A.P. Turkin, Yu.A. |
| title_short |
Influence of polarization of a regular electromagnetic wave on stochastic acceleration of charged particles in an external magnetic field |
| title_full |
Influence of polarization of a regular electromagnetic wave on stochastic acceleration of charged particles in an external magnetic field |
| title_fullStr |
Influence of polarization of a regular electromagnetic wave on stochastic acceleration of charged particles in an external magnetic field |
| title_full_unstemmed |
Influence of polarization of a regular electromagnetic wave on stochastic acceleration of charged particles in an external magnetic field |
| title_sort |
influence of polarization of a regular electromagnetic wave on stochastic acceleration of charged particles in an external magnetic field |
| 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. |
| publishDate |
2001 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Влияние поляризации регулярной электромагнитной волны на стохастическое ускорение заряженных частиц во внешнем магнитном поле |
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/79212 |
| citation_txt |
Influence of polarization of a regular electromagnetic wave on stochastic acceleration of charged particles in an external magnetic field / V.A. Buts, O.V. Manuilenko, A.P. Tolstoluzhsky, Yu.A. Turkin // Вопросы атомной науки и техники. — 2001. — № 3. — С. 86-88. — англ. |
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2025-11-27T02:56:54Z |
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2025-11-27T02:56:54Z |
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1850793043045646336 |
| fulltext |
INFLUENCE OF POLARIZATION OF A REGULAR
ELECTROMAGNETIC WAVE ON STOCHASTIC ACCELERATION
OF CHARGED PARTICLES IN AN EXTERNAL MAGNETIC FIELD
V.A. Buts, O.V. Manuilenko, A.P. Tolstoluzhsky, Yu.A. Turkin
NSC KIPT, 1 Akademicheskaya Str., Kharkov, 61108 Ukraine
e-mail: tolstoluzh@kipt.kharkov.ua
PACS numbers: 52.35.Mw
1 INTRODUCTION
One of the ways of realization of the "channel" of
direct field energy transmission to particles is creation
of conditions, at which the motion of particles in a field
of regular waves becomes chaotic. In the present report
the chaotic dynamics of charged particles in the field of
regular electromagnetic wave propagating at the angle
to the constant external magnetic field is considered.
The stochastic instability develops as a result of over-
lapping of nonlinear cyclotron resonances. We have
found conditions at which the particle, which is got into
regime of stochastic acceleration, remains in it without
reference to energy, which the particle is gained during
the interaction with wave. It is shown that these condi-
tions essentially depend on wave polarization. So if in
wave with circular or H-polarization the "unlimited"
stochastic acceleration is possible, in wave with E-po-
larization the "unlimited" stochastic acceleration is im-
possible.
2 STATEMENT OF A PROBLEM. THE BA-
SIC EQUATIONS
We consider the motion of a charged particle in con-
stant externally applied magnetic field
H Ho= { , , }0 0
and in the field of an electromagnetic plane wave, prop-
agating 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 - wave amplitude,
α α α α= { , , }x y zi -
polarization vector of the wave, wave vector
{ }
k c N c N= ⋅ ⋅ω φ ω φ/ sin , , / cos0 has only two
variebles choice of coordinates. N - is the index of re-
fraction. In dimensionless variables ( t t→ ω ,
r r c→ ω / ,
p p mc→ / ,
k kc→ / ω ) the equa-
tions of 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 H eH mco h o= =/ , /ω ω ,
E eE mco= α ω/ , γ = +( ) /1 2 1 2p - particle energy,
p - its momentum. The set of Eqs. (2) possesses the in-
tegral of motion:
p iEe r h k consti
h− + − =Re( ) [ , ]ψ ω γ . (3)
For subsequent analysis, it is convenient in (2), (3) to
transform 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)
Suppose that the amplitude of the electromagnetic field
ε ωo oeE mc= / is sufficiently small and taking into
account that the particle will interact efficiently with the
wave if it fulfills one of the resonance conditions:
k p s sz z h+ − = = − −ω γ 0 2 1 0 1 2, ... , , , , , , ... (5)
after averaging over the fast time scale one can find fol-
lowing equations, which describe particle motion in the
case of an isolated resonance:
( ) cos( ) /p k v W pz z s o s⊥ ⊥= −1 ε θ
cos( ) / , cos( ) / ,p k W Wz z s o s s o s= =ε θ γ γ ε θ γ (6)
/ sin( )θ ω γ ε θs z z h o s sk v s F= − + +1 ,
where F s k v p sJ Js z z y s x s= − − ′ +⊥( ) / ( ( ) / ( ))1 α µ µ α µ
+ ′ − −
− ′ + −
⊥
⊥
⊥k v s
J sk
p
p
J
v J k k v J
x
h
y
s
x y
s
z z s x y h z z s
2
1
ω
α
µ
µ
α
γ
µ
α µ α ω µ
( ) ( ( )
( )) ( / )( ) ( ) ,
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 - the Bessel function,
′J s ( )µ - its derivative over the argument.
3 CONDITIONS OF NONLINEAR RESO-
NANCES OVERLAPPING
Lets 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 , (7)
a closed set of equations for θ s and ~γ s can be obtained
from (3):
~ cos( ) /γ ε θ γs s o s sW= 0 , ( ) ~ /θ γ γs z s sk= −2
01 . (8)
Equations (8) are the equations of the mathematical pen-
dulum. It is easy to find the width of the nonlinear iso-
lated resonance from these equations:
∆ ~ / ( )γ εs o s zW k= −2 1 2 . (9)
One ought to notice, that when we derived the equations
(8) we neglected the addend ε θ εo s s oF sin( ) ∝ , as-
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №3.
Серия: Ядерно-физические исследования (38), с. 86-88.
86
suming it small. It is never correct to do this, when Fs
includes the components, which are proportional to
1 / p⊥ and, under condition that p⊥ tends to zero, it is
necessary to take it into account. The distance between
adjacent resonances was found with the help of the reso-
nant conditions (5) and the approximate integral of mo-
tion (7):
δ γ ω~ / ( )s h zk= −1 2 (10)
Relations (9), (10) allow one to put down the general-
ized Chirikov’s criterion:
ε ωo h s zW k> −2 216 1/ ( ) (11)
of the development of the local instability of charged
particles motion under their interaction with the electro-
magnetic wave in an external magnetic field.
The width of the nonlinear resonance ∆ ~γ s under
fulfilment the condition (11) becomes greater than the
distance between the neighboring resonances δ γ~s . The
overlapping of the nonlinear resonances takes place.
Then the dynamics of particles becomes chaotic.
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 space ( γ , ,p pz⊥ ) particle mo-
tion is determined by approximate integral (7) and res-
onace conditions (5), which in the plane ( p pz⊥ , ) be-
come:
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= , (12)
p s p k s kh z z h z⊥ + − − =2 2 2 2 21 0ω ω( / ( )) ,
s kh z
2 2 21ω = − .
Particle, remaining on the integral (7) diffuses on
resonances (12). To estimate the resonance width with
number which greatly exceeding s∗ , we find constant
a , which is the part of (7). By substituting in integral
(7) value of energy (5) we take:
a s k p k sz zs z h( ) ( )∗
∗
∗= − −1 2 ω . (13)
Value p
zs∗ we determine from the first equation (12):
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 (14)
Substituting p
zs∗ from (13) into (12) we find:
a s s k ph z
s
( ) ( ) ( )∗ ∗
⊥ ∗= ± − − −
2 2 2 2 21 1ω (15)
Let us suppose now, that the particle has got in the
vicinity with number n s> > ∗ . To find width of this
resonance it is necessary to calculate pzn and p n⊥ .
Using resonance condition and first equation from
(12) we find:
p a s k n kzn z h z= + −∗( ( ) ) / ( )ω 1 2 , p n
kn
h
z
⊥ =
ω
. (16)
4 INFLUENCE OF ELECTROMAGNETIC
WAVE POLARIZATION ON CHARGED
PARTICLES’ ACCELERATION REGIME
Estimate width of the nonlinear (9) for wave with
circular polarization (
k = {sin , , cos }φ φ0 ,
α φ φ= −{ cos , , sin }i ) when n s> > ∗ :
∆ ~ / sinγ ε ω φn o hc n= 4 1
1 3 3 , ( c1 0 411= . ) (17)
and separation between adjacent resonances (10):
δ ω φγ n n h, / sin+ =1
2 . (18)
Using (17) and (18) we can write the criterion of devel-
opment of the stochastic instability of charged particles
motion as:
16 11
1 3ε φ ωo hc n / sin / > . (19)
From (19) it is seen, that the more n , the better ful-
filled inequation (18). It is connect to that width of non-
linear resonace (18) increase with growth of number n .
Condition (19) is criterion of the particle caption into
regime of "unlimited" stochastic acceleration. At given
field amplitude, the number of resonance, the particle is
in, and cyclotron frequency, (19) allows to estimate the
angle range φ , at which "unlimited" stochastic acceler-
ation is possible. From (19) it is seen, that transversal
propagation of wave with respect to magnetic field is
optimal for stochastic acceleration.
For wave with linear polarization of type:
α = { , , }0 0i (M-wave) condition (19) does not vary. If
polarization has type
α φ φ= −{ cos , , sin }0 (Е-wave),
the width of nonlinear resonance decreases with growth
of number n :
( )∆ ~ sin*γ ε φn a s c n= −4 0 0
1 3 3 , ( c0 0 477= . ), (20)
and criterion (11) is possible to represent as:
16 11 3 2ε φ ωo o hc a s n( ) sin //∗ − > . (21)
From (21) it is seen that at big n criterion (21) ceas-
es to be fulfilled. It means, that the diffusion of particles
in high-energy region through development of stochas-
tic instability of charged particles motion due to over-
lappings of nonlinear cyclotron resonances (5) is impos-
sible. The strong dependence from polarization of wave
is explained by structure of function Ws , with which the
width of nonlinear resonance is determined. If α y ≠ 0 ,
at s >> 1 the width of nonlinear resonance is determined
by the second term in Ws , which at s >> 1 gives:
W ss ∝ 1 3/ . In the case E - wave, when α y = 0 , the
width of nonlinear resonance is determined by the first
and last terms in Ws , which combination at s >> 1
gives: 31−∝ sWs .
87
5 NUMERICAL ANALYSIS RESULTS
In order to demonstrate possibility of "unlimited"
stochastic acceleration of charged particles by electro-
magnetic wave with polarization
α φ φ= −{ cos , , sin }i ,
propagating on the angle φ to
Ho , it was numerically
solved equations (2) for assembly out of 400 particles,
uniformly distributed on phase ψ , c by identical val-
ues γ o =1.794 and pzo =0.5. Refractive index N=1.
Angle φ π= 0 3. is chosen so, that all particles in initial
point of time are in resonance s∗ = 3 . Amplitude of
field Eo = 0 2. , cyclotron frequency ω h = 0 5. . The
similar calculations were carried out and for the wave
with polarization
α φ φ= −{ cos , , sin }0 .
The precision of calculations was checked with the
help of integral (5) and did not exceed 10-4. At the cho-
sen parameters the particles in initial point of time are
moving nonregularly. Dependence amplitude of the
field E∗ at which the resonances with the numbers s
and s+1 are overlapped, from the numbers of resonance
s, for the wave with circular polarization
α φ φ= −{ cos , , sin }i is given on Fig. 1 (а), and for
Е-wave with polarization
α φ φ= −{ cos , , sin }0 on
Fig. 1(b).
Fig. 1. Dependence of amplitude of overlapping reso-
nances E∗ from resonance number s .
It is clear, that for the circular polarized wave with
the growth of particle energy the condition of resonance
overlapping is fulfill better and better.
Maximum value of the number of resonance for
Е-wave is s=16 ( γ 16 11≈ ), above this criterion of
overlapping nonlinear cyclotron resonances of first or-
der is inadequate. On Fig. 2 there is projection of particle
trajectory with initial phase ψ = 0 to plane ( γ , pz ) under
interaction with circular polarized wave
α φ φ= −{ cos , , sin }i , which is approximate integral (7).
Fig.2. Projection of particle trajectory to plane ( γ , pz )
.
One can see, that approximate integral is conserved.
On Fig. 3-4 the time dependences of particle energy and
maximum Lyapunov index for wave with polarization
α φ φ= −{ cos , , sin }i are represented. It is seen that
particle dynamics is nonregular.
The similar dependences, but for particle moving in
the field of E-wave are shown on Fig. 5-6.
Fig. 3. Time dependence of particle energy.
Fig. 4. Time dependence of Lyapunov index.
Fig. 5. Time dependence of particle energy.
Fig. 6. Time dependence of Lyapunov index
From these figures one can see, that at the chosen
parameters of problem the particle dynamics with initial
phase ψ =0 is chaotic. Particles with other initial phases
behave similarly.
On Fig. 7-8 the time dependences of average energy
< >γ and S o
2 2= < − >( )γ γ (average on ensemble) are
represented at particles motion in the field of wave with
circular polarization (а) and in the field of E-wave (b).
Fig. 7. Time dependence < >γ .
Fig. 8. Time dependence S o
2 2= < − >( )γ γ .
It is seen that the rate of energy gain by particles in
case (a) considerably exceeds the one in case (b). In
case (b), average energy is γ ∝ 5 . The close value for
average energy gives criterion (19) γ 16 6∝ . The dif-
fusion coefficient, in case (a) more than on the order ex-
ceeds diffusion coefficient obtained in case (b).
This work was supported partially by STCU grant
# 253.
88
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