Two features of dynamics of the charged particles in external magnetic field and in field of electromagnetic wave
It has shown that regimes with superdiffuse in space of velocity are inherent for moving of charged particles in external magnetic field and in fields of electromagnetic waves. The regimes with dynamic tunneling were finding too.
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2006
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| Cite this: | Two features of dynamics of the charged particles in external magnetic field and in field of electromagnetic wave / V. A. Buts // Вопросы атомной науки и техники. — 2006. — № 6. — С. 166-168. — Бібліогр.: 7 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-822902025-02-09T16:50:55Z Two features of dynamics of the charged particles in external magnetic field and in field of electromagnetic wave Buts, V.A. Plasma electronics It has shown that regimes with superdiffuse in space of velocity are inherent for moving of charged particles in external magnetic field and in fields of electromagnetic waves. The regimes with dynamic tunneling were finding too. 2006 Article Two features of dynamics of the charged particles in external magnetic field and in field of electromagnetic wave / V. A. Buts // Вопросы атомной науки и техники. — 2006. — № 6. — С. 166-168. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 52.20.-j; 05.45.-a https://nasplib.isofts.kiev.ua/handle/123456789/82290 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Plasma electronics Plasma electronics Buts, V.A. Two features of dynamics of the charged particles in external magnetic field and in field of electromagnetic wave Вопросы атомной науки и техники |
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It has shown that regimes with superdiffuse in space of velocity are inherent for moving of charged particles in external
magnetic field and in fields of electromagnetic waves. The regimes with dynamic tunneling were finding too. |
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Buts, V.A. |
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Buts, V.A. |
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Two features of dynamics of the charged particles in external magnetic field and in field of electromagnetic wave |
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Two features of dynamics of the charged particles in external magnetic field and in field of electromagnetic wave |
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Two features of dynamics of the charged particles in external magnetic field and in field of electromagnetic wave |
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Two features of dynamics of the charged particles in external magnetic field and in field of electromagnetic wave |
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Two features of dynamics of the charged particles in external magnetic field and in field of electromagnetic wave |
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two features of dynamics of the charged particles in external magnetic field and in field of electromagnetic wave |
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Two features of dynamics of the charged particles in external magnetic field and in field of electromagnetic wave / V. A. Buts
// Вопросы атомной науки и техники. — 2006. — № 6. — С. 166-168. — Бібліогр.: 7 назв. — англ. |
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Вопросы атомной науки и техники |
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166 Problems of Atomic Science and Technology. 2006, 6. Series: Plasma Physics (12), p. 166-168
TWO FEATURES OF DYNAMICS OF THE CHARGED PARTICLES
IN EXTERNAL MAGNETIC FIELD AND IN FIELD
OF ELECTROMAGNETIC WAVE
V. A. Buts
National Science Center “Kharkov Institute of Physics and Technology”,
61108, Kharkov, Ukraine, e-mail: vbuts@kipt.kharkov.ua
It has shown that regimes with superdiffuse in space of velocity are inherent for moving of charged particles in external
magnetic field and in fields of electromagnetic waves. The regimes with dynamic tunneling were finding too.
PACS: 52.20.-j; 05.45.-a
1.INTRODUCTION
The main tendency of the development of high fre-
quency electronics is contain in using more and more in-
tense electromagnetic fields. The dynamic of charged
particles in such fields can significantly change. Some
peculiarities of this dynamic were described in the works
[1,2]. Below we shall show that at enough large strength
of the fields the regimes with super-diffuse are more
character regimes for moving charged particles in external
magnetic field and in field of electromagnetic wave. Ex-
cept it, below, we shall show that there are regimes with
dynamic tunneling. At these regimes the super-diffuse
take place. At the dynamic tunneling the particles very
quickly overpass through stochastic layers. It is necessary
to note that at usual conditions the stochastic layers pos-
sess property of the “stickiness”. It means that the time
which the particles spend into these layers or in their vi-
cinity significantly large then the time that these particles
spend in another phase space. The regime of the dynamic
tunneling can very useful at the cascade acceleration of
the charged particles.
2. BASIC EQUATIONS
Let’s look at the charged particle which is moving in
external magnetic field 0H , which is directed along z
axis and in field of the electromagnetic field with arbi-
trary polarization. This wave has following components:
( )r r rr
ε ω= −Re exp ;E ikr i t
( ){ }0 , ,x y zE E iα α α≡
r
(1)
( )Re expcH kE ikr i tω
ω
= −
r rr r r ,
where { }, ,x y ziα α α α≡
r - polarization vector.
Without any restriction we can consider that the wave
vector k
r
has only two components xk and zk .
We shall measure the time in 1ω− , velocities in c , wave
vector in / cω , impulse in mc . Except it, it is useful to
introduce dimensionless amplitude 0 /eE mcωε = . The
equations of motion in these variables will look as:
( ) [ ] ( )1 Re Re ;
/ ; / 1,
i iHkp kP e pe p e
r p kp
ψω
ε ε
γ γ γ
γ ψ γ
Ψ
= − + +
= = −
r rr
r r rrr r&
rr r r& &
(2)
where 0 0, / ; / ;
.
Ht e H H eH mc
kr
τ ω ω ω
ψ τ
≡ ≡ ≡
= −
rr
rr
The equations (2) have following integral of move-
ment:
( ) [ ]Re i
Hp i e re k constψε ω γ− + − =
rrr rr . (3)
For further it is convenient to introduce new dependent
variables ||, , ,p p θ ξ⊥ η :
||
cos , sin ,
, sin, cos
x y z
H H
p p p p p
p pp x y
θ θ
ξ η θ
ω ω
⊥ ⊥
⊥ ⊥
= =
= = − = +
. (4)
3. SUPER-DIFFUSE
The system (2) was studied in works [3, 4] for the cases
when amplitude of the wave was enough small ( 1ε << )
and for relativistic case ( 1p⊥ >> ). The main peculiarity
of the slow dynamic of the charged particles is fact that
this motion are described by mathematical pendulum. If
the amplitude of the wave is enough large the investiga-
tion of the system (2) can be fulfilled only by using nu-
merical methods. Such investigations were fulfilled. The
significant result of these investigations is that all nonlin-
ear cyclotron resonances are strongly overlapped. The
dynamic of the charged particles are chaotically. It’s very
significantly that all moments from second moment are
increasing in the time. Moreover the higher moments are
always larger and grow more quickly than the more low
moments.
It’s well known that for unrestricted large second mo-
ments are correspond on microscopic level to the proc-
esses type “Levi flight”. In this case the usual definition
of the diffuse coefficients is not correct because this coef-
ficient is proportional to second moment. In this case it’s
necessary to receive equation for distribution function.
This equation must be or integro-differential or differen-
tial with fractional derivatives. We first of all should like
to know expression for distribution function for large ve-
locities (expression for tails of the distribution function)
and asymptotic behavior of this function at a large time.
mailto:vbuts@kipt.kharkov.ua
167
In this section we'll use these restrictions. It will permit us
to receive not only equation for function of distribution
but permit to receive its analytical expression. Let’s the
moving of the charged particle in external electromag-
netic fields will define by function ( )f v . This function
determines a density of probability for particles to dis-
place from initially located place in the space of velocity
on the value v . The value v is determining dimensionless
velocity, which is normalized at initial value of the veloc-
ity ( 0/v v v≡ ). This function obeys such expression:
( ) 1f v dv
∞
−∞
=∫ . (5)
The fact that the all moments from second moment are
infinitely can be expressed by expression:
( )n nv v f v dv
∞
−∞
= = ∞∫ , 2n ≥ . (6)
We shall consider that a casual jump in velocity space
has equal probability for direction along velocity axis. It
means that the function ( )f v must be even. Namely the
tails of the distribution function define super-diffuse be-
havior. The details of the function ( )f v at small value
velocity don’t play significant role. The wide class of the
functions ( )f v will give the same asymptotic dynamic.
That is why we shall choice this function in the form that
permits us to receive analytical results with smaller ef-
forts. Using these reasons, we write this function in the
form:
( ) 1/ 22
( 1/ 2) 1( )
( ) 1
f v
v
α
α
π α +
Γ +
=
Γ +
. 0 1α< < (7)
It’s easy to see that this function obey all our require-
ments. The combination of the gamma-function Euler is
chosen only for simplicity of the further calculation.
Let’s notice that similar form for the density of the prob-
ability was used in work [5]. Using the usual ideology for
receiving kinetic equations (see, for example, [6]) we can
write:
[ ]( , 1) ( , ) ( , ) ( , ) ( )n v n n v n n v v n n v n f v dv
∞
−∞
′ ′ ′+ − = − −∫ , (8)
where ( , )n v n - density of the particles.
If the moments are restricted then we can resolve the
expression inside integral and being limited only by sec-
ond moments, we shall receive usual diffuse equation
with diffusion coefficients 2 / 2D v= . If the second
moment is unlimited then the equation will remain inte-
gro-differential. In common cases to solve these equa-
tions are difficult problem. But we are interested with
expressions for the density of the particles only for large
value of velocity. In this case we can to receive enough
simple equations. Except it we shall find their analytical
solution. For this purpose, using (8), we shall receive the
equation for Fourier-images of density of particles:
( )1k
k k
n f n
t
∂
= −
∂
, (9)
where
12 ( )
( )kf k K k
α
α
αα
−
= ⋅
Γ
; ( )K kα - Macdonald function.
We are interested with large value of the velocity
( 1v >> ). In Fourier-images it corresponds for small
«wave numbers» ( 1k << ). In this case we use the asymp-
totic expression for Macdonald function. The equation
(9) became significantly simple:
2
2
(1 )
(1 )2
k
k
kn n
t
α
α
α
α
Γ −∂
= −
∂ Γ +
. (10)
The solving of this equation looks like:
2
2
(1 )
( ) exp (0)
(1 )2k k
k
n t t n
α
α
α
α
Γ −
= −
Γ +
. (11)
To the equation (10) in usual space there corresponds
the equation:
2 1
( 1/ 2) ( )
( )
n n v dv
t v v α
α
α π
∞
+
−∞
′ ′∂ Γ +
=
∂ Γ ′−∫ . (12)
Its solution is possible to present as:
( , ) ( , ) ( ,0)n v t G v v t n v dv
∞
−∞
′ ′ ′= −∫ . (13)
The integral in equation (12) should be understood in
sense of the main meaning. In this case this integral is
definition of the fractional derivative [7]. Thus, the inte-
gro-differential equation (12) is equivalent to the differen-
tial equation with fractional derivative. The Green func-
tion in equation (13) has enough bulky expression. We
shall not write out it. We shall use the fact that as in usual
diffuse equations at large time ( t → ∞ ), the structure of
Green function becomes smooth and it is possible to take
out from integral:
( , ) ( , )n v t N G v t→ ⋅ ( ,0)N n v dv
∞
−∞
= ∫ . (14)
At the large time ( t → ∞ ), as it follow from expres-
sion (11) for Fourier-image of the density the contribution
will be given only harmonics with 0k → . This fact essen-
tially simplifies Green function. Finally, at t → ∞ and
1v >> the expression for density of particles gets a simple
form:
2 1
( 1/ 2)( , )
( )
tn v t N
v α
α
π α +
Γ +
=
⋅Γ
. (15)
The more significant result, which are follow from this
expression is that the density of the particles with growth
of velocity is decreasing not as exponent (as it take place
at the conventional diffuse), but under power law.
4. DYNAMIC TUNNELING
The short-cat equation of the system (2) in nonrelativis-
tic case ( 1p⊥ << ) and for 0zk = we can represent in
form:
0( ( ) )si
s
dP Re iJ e
d
θµ ε
τ
⊥ ′= (16)
2
02
11 Re( ( ) )sis H
s
H
d s s J e
d
θθ ω
µ ε
τ γ ω µ
= − −
.
The significant difference of this system from investi-
gated in works [3,4] is the fact that the phase portrait is
not mathematical pendulum portrait. This portrait has or
only one special point that is “center” or has two such
168
points and one point that is type “saddle”. In last case this
phase portrait is topology equivalent to the phase portrait
of Duffing oscillator ( 3 ( )x x x f tα β+ ⋅ + ⋅ =&& ). The dy-
namic tunneling regime for Duffing oscillator is well in-
vestigated. We have studied system (16) and Duffing os-
cillator for behavior of the higher moments. It’s hap-
pened, that in this regime the higher moments, as well as
in section 3, has unrestricted growing, i.e. for these re-
gimes the super-diffuse is characteristic.
In figures 1-3 potential Duffing oscillator, its phase
portrait and characteristic realization are submitted. From
the realization it is visible, that in a regime of dynamic
tunneling the particle quickly jumps from one potential
hole in another. Thus, in this regime the particles do not
wander in the stochastic sea. REFERENCES
1. A.V. Buts, V.A. Buts. Dynamic of the charged parti-
cles in a field intense transversal electromagnetic
wave // JETPh. 1996, v.110, N. 3(9), p.818-831.
2. V.A. Buts, V.V. Kuzmin. Dynamic of the particles in
a field’s large intensity // Modern radioelectronics-
Uspekhi. 2005, 11, p. 5-20.
3. V.A. Buts, V.A. Balakirev, A.P. Tolstoluzhskiy,
Yu.A. Turkin. Chaotization of the moving beams of
the phased oscillators // JETPh. 1983, v.84, N 4,
p. 1279-1289.
4. V.A. Buts, V.A. Balakirev, A.P. Tolstoluzhskiy,
Yu.A. Turkin. Dynamic of the charged particles in
fields of the two electromagnetic waves // JETPh.
1989, v. 95, N 4, p. 1231-1245.
5. K.B. Chukbar // JETPh Letters. 1993, v.58, p. 87.
6. A.J. Lichtenberg, M.A. Lieberman. Regular and
Stochastic Motion // Springer-Verlag. New York:
Heidelberg Berlin, 1983, p. 499.
7. S.G. Samko, A.A. Kilbas, O.I. Marichev. Integrals
and Derivatives With Fractional Order and Some
Their Application // Science and Technik. Minsk,
1987, p.687.
.
, ,
.
.
.
, , -
.
.
Fig.1. The potential of Duffing oscillator
Fig.3. Localization of a particle. One can
see quick jumps from left hole to right
and back
Fig.2. Phase portrait of Duffing oscillator
|