Excitation of high numbers harmonics by charge particles in a time-periodic electric field and a space-periodic potential
Regular and chaotic dynamics of a charged particle, which moves in an external time-periodic electrical field and in a field of space periodic potential, are investigated. We have obtained a system of integro-differential equations, which describes the non-linear self-consistent dynamics of excita...
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Buts, V.A. Marekha, V.I. Tolstoluzhsky, A.P. 2015-03-22T08:55:39Z 2015-03-22T08:55:39Z 2006 Excitation of high numbers harmonics by charge particles in a time-periodic electric field and a space-periodic potential / V.A. Buts, V.I. Marekha, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2006. — № 2. — С. 166-168. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 52.35.–g https://nasplib.isofts.kiev.ua/handle/123456789/78869 Regular and chaotic dynamics of a charged particle, which moves in an external time-periodic electrical field and in a field of space periodic potential, are investigated. We have obtained a system of integro-differential equations, which describes the non-linear self-consistent dynamics of excitation of electromagnetic waves by flows of charged particles. The analytical and numerical analysis of a full self-consistent set of equations is carried out. The results of this analysis qualitatively well agree with the experimental data. Исследована регулярная и хаотическая динамика заряженной частицы, которая движется во внешнем периодическом во времени электрическом поле и в поле периодического в пространстве потенциала. Получена система интегро-дифференциальных уравнений, которая описывает самосогласованную нелинейную динамику возбуждения электромагнитных волн потоками заряженных частиц. Проведен аналитический и численный анализ полной самосогласованной системы уравнений. Результаты этого анализа качественно хорошо согласуются с экспериментальными данными Досліджена регулярна та хаотична динаміка зарядженої частинки, що рухається у зовнішньому періодичному у часі електричному полі та у полі просторово-періодичного потенціалу. Одержана система інтегро-диференційних рівнянь, що описує нелінійну самоузгоджену динаміку збудження електромагнітних хвиль потоками заряджених частинок. Проведено аналітичний і числовий аналіз повної самоузгодженої системи рівнянь. Результати цього аналізу якісно добре узгоджуються з експериментальними даними. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Применение ускорителей в радиационных технологиях Excitation of high numbers harmonics by charge particles in a time-periodic electric field and a space-periodic potential Возбуждение высоких номеров гармоник заряженными частицами в периодическом во времени электрическом поле и в поле пространственно-периодического потенциала Збудження високих номерів гармонік зарядженими частинками у періодичному у часі електричному полі та у полі просторово-періодичного потенциалу Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Excitation of high numbers harmonics by charge particles in a time-periodic electric field and a space-periodic potential |
| spellingShingle |
Excitation of high numbers harmonics by charge particles in a time-periodic electric field and a space-periodic potential Buts, V.A. Marekha, V.I. Tolstoluzhsky, A.P. Применение ускорителей в радиационных технологиях |
| title_short |
Excitation of high numbers harmonics by charge particles in a time-periodic electric field and a space-periodic potential |
| title_full |
Excitation of high numbers harmonics by charge particles in a time-periodic electric field and a space-periodic potential |
| title_fullStr |
Excitation of high numbers harmonics by charge particles in a time-periodic electric field and a space-periodic potential |
| title_full_unstemmed |
Excitation of high numbers harmonics by charge particles in a time-periodic electric field and a space-periodic potential |
| title_sort |
excitation of high numbers harmonics by charge particles in a time-periodic electric field and a space-periodic potential |
| author |
Buts, V.A. Marekha, V.I. Tolstoluzhsky, A.P. |
| author_facet |
Buts, V.A. Marekha, V.I. Tolstoluzhsky, A.P. |
| topic |
Применение ускорителей в радиационных технологиях |
| topic_facet |
Применение ускорителей в радиационных технологиях |
| publishDate |
2006 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Возбуждение высоких номеров гармоник заряженными частицами в периодическом во времени электрическом поле и в поле пространственно-периодического потенциала Збудження високих номерів гармонік зарядженими частинками у періодичному у часі електричному полі та у полі просторово-періодичного потенциалу |
| description |
Regular and chaotic dynamics of a charged particle, which moves in an external time-periodic electrical field
and in a field of space periodic potential, are investigated. We have obtained a system of integro-differential equations,
which describes the non-linear self-consistent dynamics of excitation of electromagnetic waves by flows of
charged particles. The analytical and numerical analysis of a full self-consistent set of equations is carried out. The
results of this analysis qualitatively well agree with the experimental data.
Исследована регулярная и хаотическая динамика заряженной частицы, которая движется во внешнем периодическом во времени электрическом поле и в поле периодического в пространстве потенциала. Получена система интегро-дифференциальных уравнений, которая описывает самосогласованную нелинейную динамику возбуждения электромагнитных волн потоками заряженных частиц. Проведен аналитический и численный анализ полной самосогласованной системы уравнений. Результаты этого анализа качественно хорошо согласуются с экспериментальными данными
Досліджена регулярна та хаотична динаміка зарядженої частинки, що рухається у зовнішньому періодичному у часі
електричному полі та у полі просторово-періодичного потенціалу. Одержана система інтегро-диференційних рівнянь,
що описує нелінійну самоузгоджену динаміку збудження електромагнітних хвиль потоками заряджених частинок.
Проведено аналітичний і числовий аналіз повної самоузгодженої системи рівнянь. Результати цього аналізу якісно добре
узгоджуються з експериментальними даними.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/78869 |
| citation_txt |
Excitation of high numbers harmonics by charge particles in a time-periodic electric field and a space-periodic potential / V.A. Buts, V.I. Marekha, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2006. — № 2. — С. 166-168. — Бібліогр.: 11 назв. — англ. |
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2025-11-25T21:12:25Z |
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| fulltext |
EXCITATION OF HIGH NUMBERS HARMONICS BY CHARGE PARTI-
CLES IN A TIME-PERIODIC ELECTRIC FIELD AND A SPACE-PERI-
ODIC POTENTIAL
V.A. Buts, V.I. Marekha, A.P. Tolstoluzhsky
NSC KIPT, Kharkov, Ukraine
E-mail: vbuts@kipt.kharkov.ua
Regular and chaotic dynamics of a charged particle, which moves in an external time-periodic electrical field
and in a field of space periodic potential, are investigated. We have obtained a system of integro-differential equa-
tions, which describes the non-linear self-consistent dynamics of excitation of electromagnetic waves by flows of
charged particles. The analytical and numerical analysis of a full self-consistent set of equations is carried out. The
results of this analysis qualitatively well agree with the experimental data.
PACS: 52.35.–g
1. INTRODUCTION
It is known that in a vacuum an oscillator effectively
radiates high numbers of harmonics only if it has a large
energy. So, for the synchrotron radiation the maximum
of radiation is obtained at harmonics with number
3~ γν [1]. Many authors (see, for example [2-6] and
referenced therein) studied the radiation of relativistic
particles in a periodically inhomogeneous medium. The
interest to such radiation is conditioned by the fact that
due to the Doppler effect there is a possibility to excite
effectively the short-wave radiation λ γ~ /d 2 .
It is possible to name one more mechanism for
short-wave radiation generation which does not require
the use of high-energy beams. This is the radiation of
high numbers harmonics by nonrelativistic oscillators
moving in the periodically inhomogeneous medium as
well as by charged particles moving in an external time-
periodic electrical field, and in the field of an external,
space-periodic, potential [7-9].
Meaning features of the radiation mechanism, which
we investigate, one can expect the creation of sources of
an intensive coherent X-radiation with the wavelength
/dλ β≈ .
In this paper we have investigated, analytically and
numerically, the capabilities of excitation of high num-
bers harmonics by ensembles of charged particles. We
carried out the analytical and numerical analysis of a
full self-consistent set of equations. The analytical re-
sults are in good agreement with the numerical results,
and qualitatively well agree with the data of experi-
ments [10].
2. MOTION OF A CHARGED PARTICLE
IN A FIELD OF STRATIFIED -
INHOMOGENEOUS POTENTIAL AND
IN A PERIODIC ELECTRICAL FIELD
Let a charged particle to move in the external time-
periodic electrical field ( )0,0, zE E
r
( )( ) sinz ext extE t E tω= − Ч Ч and in the field of periodic poten-
tial ( )zkcosgU)z(U z ⋅⋅+= 0 .
The nonrelativistic equations of electron motion in
such a field can conveniently be presented as:
( ) ( )2
0sin sin ,
,
z
z
dV d
d d V
τ ε τ ω ς
ς τ
м = − Ω −п
н
=по
(1)
here: /z zV Vс= , zk c tτ = Ч , 2/ zeE mc kε = , 2 2
0 /eg mcω = ,
/ext zk cωΩ = , zk zς = .
System (1) is reduced to the equation of mathemati-
cal pendulum with external periodic force
( ) ( )2
0 sin sinς ω ς ε τ+ = − Ω&& . (2)
The radiation power of harmonic oscillator ( )sin osz a tω=
can be expressed by the formula (see, for example, [1]):
2 4 2 33osW t e a cω∂ ∂ ≈ (3)
and radiation takes place at a frequency of oscillator
osω . The higher harmonics are small.
In our case, under the approximation 2
0ε ω> > , for
n>>1 from (1) it is possible to obtain the maximum am-
plitude of particle displacement at n -th Fourier har-
monics ( )2 2 1
0 ( )n za J n kω µ − −= Ω (µ∼n, 2µ ε= Ω ).
The formula (3) will become as:
( ) ( ) ( )
22 2 2 2 23ext nW t e c eg mc n Jω µ∂ ∂ ≈ (4)
So, one can see that conditions of maximum radiation
extkc mβ ω ω= = in this case completely coincide with
the condition of oscillator radiation in a periodically in-
homogeneous dielectric [7], i.e. in both cases the maxi-
mum radiation corresponds to the same frequency. In
the cases 2
0ω εі the role of the periodical potential is
more significant.
3. LIMITATIONS ON ENERGY AND
WAVELENGTH OF RADIATION
Let's define the conditions, under which the process
of radiation is possible. For particle radiation it is neces-
sary, at least, that its energy should exceed the energy of
radiated quantum
2 / 2mV ωΕ = і h . (5)
From (5) we find the condition for the length of radiated
wave
22 /h mcλ βі , (6)
here с – velocity of light; m – particle mass; β=V/c – its
dimensionless velocity (Fig.1).
___________________________________________________________
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2006. № 2.
Series: Nuclear Physics Investigations (46), p.166-168.166
mailto:vbuts@kipt.kharkov.ua
As it is clear from the figure, the wavelengths, ac-
cessible to the radiation, are below the point of intersec-
tion of curves and above the line 22 /h mcλ β= .
Fig.1. Dependence of radiated wavelength
22 /h mcλ β= (firm line) and dependence /dλ β=
(dash line) versus β for d = 6⋅10-8 cm
4. RADIATION OF OSCILLATORS FLOW
It is particularly interesting to investigate the in-
duced radiation of an ensemble of such oscillators, not
the radiation of one oscillator. The fullest description of
the self-consistent process of interaction of charged par-
ticles with the exciting field implies the simultaneous
solution of the Maxwell equations for the electromag-
netic field and equations of charged particles motion in
the exited fields.
0
rot , rot 4
( ) sin ,ext
B t c E E t c H j
dp dt eE e c v B F t e U dr dt v
∂ ∂ ∂ ∂ π
ω
= − = −
= + + − =ґ С
r r r r v
r r r rr r r r , (7)
where extω , extF are the frequency and the amplitude
of the external force, which acts upon the oscillator
(produces oscillator). Let's suppose that the oscillations
of the oscillator occur along the axis Z.
While investigating the elementary mechanism of
the oscillator radiation it was found out, that a direction-
al diagram corresponds to a dipole radiation, i.e. the ra-
diation is directed in a transverse direction with respect
to direction of the oscillator oscillation. Therefore, we
will search for such solution for an exited wave
E Re A( t )exp( ikx )=
rr
. (8)
We will study a time evolution of the electromagnet-
ic field, in which the only E E Hx z y, , components are
nonzero. Let's substitute the field expressions (8) into
the set of equations (7). By averaging the obtained equa-
tions on space phase of perturbation, we get the follow-
ing set of equations for the fields and characteristics of
the oscillator:
Re exp( )x xdp d ixτ ε= − Re exp( ),z yh ixν
( ) ( )0Re exp( ) Re exp sin sinz z x ydp d ikx v h ix f w zτ ε τ κ= + − Ω − ,
2 2, , / 1 ,x z x zdx d v dz d v p p pτ τ ν= = = + +r r (9)
( )
2
2
0
2 2 exp( )z y b z od d ih ix dx
π
ε τ ω π ν= − − −т ,
,y zdh d iτ ε= ( )
2
2
0
2 2 exp( )x b x od d ix dx
π
ε τ ω π ν= − −т .
The coordinates 0( , )r r r τ=r r r
and pulses 0( , )p p r τ=r r r
are the Lagrangian coordinates and pulses of particles.
The integration in the right part of these equations for
fields is the integration over the initial values of oscilla-
tor coordinates. The set of equations (9) is written in the
dimensionless variables:
, , , / , , ,z
p eEkct kr r p k k kc
mc mckc
τ κ ε ω→ → → → = =
rr rr r r
,eHh
mckc
=
rr
0
exteE
f
mckc
= , zegk
w
mcω
= ,
2
2
2
4
( )
b
b
e n
m kc
π
ω = ,
/V kν ω=
rr , where m, e − mass and charge of
electrons, nb − density of oscillators.
5. RESULTS OF THE NUMERICAL ANALYSIS
The numerical analysis of the self-consistent set of
equations (9) confirmed the presence of an instability in
the considered system. The values of dimensionless
parameters were the following: bω =0.3, w=0.02, 0f
=0.02, n =5. Under these conditions the excitation of
10-th harmonics by oscillators in the periodic potential
with k5=κ was observed. The results of simulation
are presented in Figs.2,3.
Fig.2. Dependence of amplitude zε on time
Fig.3. Power spectrum zε
The dark filling inside the envelope amplitude
corresponds to the high frequency oscillations.
Spectrum of the field zε has a maximum at the
frequency, which equal 1, and that corresponds to the
10-th harmonic of the oscillation frequency of the
oscillators. The results of numerical analysis
qualitatively well agree with the data of experiments
[10].
6. CONDITIONS OF COLLECTIVE
RADIATION OF ENSEMBLE OF
OSCILLATORS
At the particles motion in the fields of complicated
configuration the development of stochastic instability
is possible. The presence of such instability in beam
systems is similar to the appearance of thermal spread
of particles over velocities. The thermal spread can es-
sentially limit the minimal wavelength, which can be
excited by a beam in induced way.
We will see that, formally, the conditions of devel-
opment of stochastic instability are fulfilled. However,
the particle dynamics remains practically regular.
The presence of a perturbation results in a formation
of a stochastic layer on a phase plane nearby a separa-
trix – an area in which particle motion is irregular and
158
the separatrix is splitted. The width of such splitted sep-
aratrix is proportional to perturbation [11]. The numeri-
cal analysis of the Eq. (2) confirms this fact (Fig.4).
Fig.4. Poincare map of mathematical pendulum: ε
=0.005, δ =0.025
By passing to a moving frame by the variable trans-
formation 2( / ) sin( )ξ ς ε δ δ τ= + , the next equation is
obtained for ξ from Eq.(2)
2( / ) sin( ) 0n
n
J nξ ε δ ξ δ τ
Ґ
= − Ґ
+ + =е&& , (10)
where 2( / )nJ ε δ is the Bessel function of order n,
0extδ ω ω= .
Eq.(10) describes the phase change of the particle ξ
which the high number of waves excites. An interaction
of the particle with one of these waves will be most
effective under fulfillment of the resonance condition
nξ δ=& . The typical feature of such interaction is a
large number of nonlinear resonances. A parameter,
determining the degree of influence of the nearby
resonances on each other, is a parameter /rezK ω= ∆ ∆ ,
a ratio of the resonance width to the distance between
resonances, having a simple physical sense of degree of
resonances overlapping [11]. In our case, the distance
between the resonances is ω δ∆ = , and the width of
resonance with the number of n is equal
22 ( / )rez nJ ε δ∆ = . (11)
When condition K >1 is fulfilled all the nearby
resonances are overlapped and, from the formal point of
view, a chaotic motion must be realized in the whole
region of phase plane of Eq.(10). However, when K
>> 1, the chaotic motion begins regularizing itself. It is
due to the fact that at the parameter values 2/ε δ >>1 (
δ <<1) the resonances with the numbers of n 2/ε δЈ
remain indistinguishable. Motion of particles here can
be described as a motion in a new effective resonance
and such motion will be regular practically in the whole
region of phase plane except for the region near the split
separatrix. The numerical analysis of the Eq.(2) also
confirms this fact (see Fig.4).
Thus, as an estimated value of measure that shows
the degree of an electron motion chaotization, one can
take the ratio of the region of phase plane occupied by
the split separatrix to the region of phase plane at which
motion of particles is regular.
REFFERENCES
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sian).
3. V.A. Buts, A.V. Shchagin // UFZh. 1998, v.43, №9,
p.1172-1174 (in Russian).
4. V.G. Baryshevsky, I.Ya. Dubovskaya // The totals of
science and engineering. Series: Beams of charged
particles and solid. 1992, №4 (in Russian).
5. V.L. Ginzburg // UFN. 1996, v.166, №10 (in Rus-
sian).
6. M.A. Piestrup, P.P. Finman // IEEE Journal of Quan-
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8. V.A. Buts // Radiotekhnika. 1997, v.9, p.9-12. (in
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9. V.A. Buts // ZhTF. 1999, v.69, №5, p.132-134. (in
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10. V.A. Buts, E.A. Kornilov. // Problem of Atomic Sci-
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and new methods of acceleration. 2003, №4(3),
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11. G.М. Zaslavsky, R.Z. Sagdeev. Introduction into
nonlinear physics. M.: “Nauka”, 1988, p.368 (in Rus-
sian).
ВОЗБУЖДЕНИЕ ВЫСОКИХ НОМЕРОВ ГАРМОНИК ЗАРЯЖЕННЫМИ ЧАСТИЦАМИ В ПЕРИОДИЧЕСКОМ
ВО ВРЕМЕНИ ЭЛЕКТРИЧЕСКОМ ПОЛЕ И В ПОЛЕ ПРОСТРАНСТВЕННО-ПЕРИОДИЧЕСКОГО ПОТЕН-
ЦИАЛА
В.А. Буц, В.И. Мареха, А.П. Толстолужский
Исследована регулярная и хаотическая динамика заряженной частицы, которая движется во внешнем периодическом
во времени электрическом поле и в поле периодического в пространстве потенциала. Получена система интегро-диффе-
ренциальных уравнений, которая описывает самосогласованную нелинейную динамику возбуждения электромагнитных
волн потоками заряженных частиц. Проведен аналитический и численный анализ полной самосогласованной системы
уравнений. Результаты этого анализа качественно хорошо согласуются с экспериментальными данными
ЗБУДЖЕННЯ ВИСОКИХ НОМЕРІВ ГАРМОНІК ЗАРЯДЖЕНИМИ ЧАСТИНКАМИ У ПЕРІОДИЧНОМУ
У ЧАСІ ЕЛЕКТРИЧНОМУ ПОЛІ ТА У ПОЛІ ПРОСТОРОВО-ПЕРІОДИЧНОГО ПОТЕНЦИАЛУ
В.О. Буц, В.І. Мареха, О.П. Толстолужський
Досліджена регулярна та хаотична динаміка зарядженої частинки, що рухається у зовнішньому періодичному у часі
електричному полі та у полі просторово-періодичного потенціалу. Одержана система інтегро-диференційних рівнянь,
що описує нелінійну самоузгоджену динаміку збудження електромагнітних хвиль потоками заряджених частинок.
Проведено аналітичний і числовий аналіз повної самоузгодженої системи рівнянь. Результати цього аналізу якісно добре
узгоджуються з експериментальними даними.
___________________________________________________________
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2006. № 2.
Series: Nuclear Physics Investigations (46), p.166-168.166
EXCITATION OF HIGH NUMBERS HARMONICS BY CHARGE PARTICLES IN A TIME-PERIODIC ELECTRIC FIELD AND A SPACE-PERIODIC POTENTIAL
V.A. Buts, V.I. Marekha, A.P. Tolstoluzhsky
E-mail: vbuts@kipt.kharkov.ua
PACS: 52.35.–g
5. RESULTS OF THE NUMERICAL ANALYSIS
REFFERENCES
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