Estimation of dimensions of interaction space necessary for particle acceleration in stochastic field
It is considered electron acceleration in the field having time dependence, which differs from sinusoidal one with phase jumps introduction. The connection is found between electron energy and characteristic dimension of device, in which such energy may be achieved with considered acceleration metho...
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| Date: | 2014 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2014
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| Cite this: | Estimation of dimensions of interaction space necessary for particle acceleration in stochastic field / V. Ostroushko // Вопросы атомной науки и техники. — 2014. — № 5. — С. 140-142. — Бібліогр.: 2 назв. — англ. |
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| author | Ostroushko, V. |
| author_facet | Ostroushko, V. |
| citation_txt | Estimation of dimensions of interaction space necessary for particle acceleration in stochastic field / V. Ostroushko // Вопросы атомной науки и техники. — 2014. — № 5. — С. 140-142. — Бібліогр.: 2 назв. — англ. |
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| description | It is considered electron acceleration in the field having time dependence, which differs from sinusoidal one with phase jumps introduction. The connection is found between electron energy and characteristic dimension of device, in which such energy may be achieved with considered acceleration method or with collision heating. It is shown that the acceleration in the field with jumps of phase requires much greater dimension of space for electron motion than the heating to the same energy in sinusoidal field in presence of elastic collisions.
Рассмотрено ускорение электрона в поле с зависимостью от времени, отличающейся от синусоидальной введением скачков фазы. Установлена связь между энергией электронов и характерным размером устройства, в котором такая энергия может быть достигнута, при рассмотренном способе ускорения и при столкновительном нагреве. Показано, что ускорение в поле со скачками фазы требует значительно большего размера области движения электронов, чем нагрев до такой же энергии в синусоидальном поле при наличии упругих столкновений.
Розглянуто прискорення електрона у полi iз залежнiстю вiд часу, яка вiдрiзняється вiд синусоїдальної введенням стрибкiв фази. Встановлено зв'язок мiж енергiєю електронiв та характерним розмiром пристрою, у якому така енергiя може бути досягнута, при розглянутому способi прискорення та при зiткненному нагрiваннi. Показано, що прискорення у полi зi стрибками фази потребує значно бiльшого розмiру областi руху електронiв, нiж нагрiвання до такої саме енергiї у синусоїдальному полi за наявностi пружних зiткнень.
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ESTIMATION OF DIMENSIONS OF INTERACTION SPACE
NECESSARY FOR PARTICLE ACCELERATION IN
STOCHASTIC FIELD
V.Ostroushko∗
National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received Month 24, 2014)
It is considered electron acceleration in the field having time dependence, which differs from sinusoidal one with
phase jumps introduction. The connection is found between electron energy and characteristic dimension of device,
in which such energy may be achieved with considered acceleration method or with collision heating. It is shown
that the acceleration in the field with jumps of phase requires much greater dimension of space for electron motion
than the heating to the same energy in sinusoidal field in presence of elastic collisions.
PACS: 29.17.+w, 52.80.Pi
1. INTRODUCTION
One of the methods of electron energy increase in
high frequency low-pressure gas discharge is so called
collisionless heating, which recently attracts some at-
tention ([1], [2]). In fact, in this method, electrons
are subjected to the force, which time dependence is
complicated, but determined completely, regardless
of electron motion, and another forces are absent (in
particular, collisions are absent). So, electrons are
accelerated in a given field, as it takes place in accel-
erators, and the question arises naturally about the
use of such devices as charged particle accelerators.
In the present work, it is made the estimation of de-
vice dimension necessary for electron acceleration up
to a given energy in the field with phase jumps simi-
lar to one described in [1], [2]. As the effectiveness of
such field use is substantiated there with aid of anal-
ogy with collision heating, the relevant dimension is
also estimated here in the case of collision heating,
and for the case of phase jumps the statistical char-
acteristics of the jumps are taken according to the
different variants of ones of collisions.
2. MOTION IN THE FIELD WITH FULLY
ACCIDENTAL PHASE JUMPS
In this case, the values of phase before and after jump
are assumed to be independent: at t ∈ (0, t1) the elec-
tron velocity time dependence is
v(t) = v0 + ucos (ωt+ φ0) , (1)
at once after t1 the dependence is
v(t) = v1 + ucos [ω (t− t1) + φ1] , (2)
and the values of φ0 and φ1 are independent. Due to
continuity of velocity at t = t1, the equality
v1 = v0 + ucos (ωt1 + φ0)− ucosφ1 (3)
takes place. With integration of velocity over time
interval (0, t1), one comes to the equality
z1 = z0 + v0t1 + (u/ω) [sin (ωt1 + φ0)− sinφ0] , (4)
which connects the coordinates z0 and z1 at time in-
stants 0 and t1. From the equalities (3) and (4), aver-
aging the products and squares of their different parts
over φ0 and φ1 independently, denoting such aver-
ages by angle brackets with index φ, and taking into
account the equality 2⟨sin (α+ φ0) sinφ0⟩φ = cosα
(for a fixed α), one gets the equalities⟨
v21
⟩
φ
= v20 + u2,
⟨z1v1⟩φ = (z0 + v0t1) v0 + u2(2ω)−1sin (ωt1),⟨
z21
⟩
φ
= (z0 + v0t1)
2
+ (u/ω)2 [1− cos (ωt1)].
The next averaging is carried out over the values
of t1, with use of statistical characteristics of Poisson
process having frequency ν. Taking into account the
equality
∫∞
0
νexp(−νt+ ı̇ωt) dt = ν(ν − ı̇ω)−1, one
comes to the equalities
⟨z1v1⟩ = z0v0 + v20ν
−1 + u2ν
[
2
(
ν2 + ω2
)]−1
,⟨
z21
⟩
= z20 + 2z0v0ν
−1 + 2v20ν
−2 + u2
(
ν2 + ω2
)−1
(angle bracket without indexes means the last averag-
ing). For the velocity and coordinate after the great
numberN of phase jumps, when full velocity becomes
much greater than oscillation velocity,
⟨
v2N
⟩
≫u2,
one gets the relationships
⟨
v2N+1
⟩
−
⟨
v2N
⟩
= u2,
⟨zN+1vN+1⟩ − ⟨zNvN ⟩≈ν−1
⟨
v2N
⟩
,⟨
z2N+1
⟩
−
⟨
z2N
⟩
≈2ν−1⟨zNvN ⟩.
Taking the sums over jumps, one obtains
∗author E-mail address: ostroushko-v@kipt.kharkov.ua
140 ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2014, N5 (93).
Series: Nuclear Physics Investigations (63), p.140-142.
⟨
v2N
⟩
≈u2N, ⟨zNvN ⟩≈(2ν)−1u2N2,⟨
z2N
⟩
≈
(
3ν2
)−1
u2N3≈
(
3ν2u4
)−1 ⟨
v2N
⟩3
. (5)
3. MOTION IN SINUSOIDAL FIELD
WITH ELASTIC COLLISIONS
In this case, the phases in (1) and (2) are not inde-
pendent, but connected with the equality
φ1 = ωt1 + φ0. (6)
As collisions are elastic, at the instant of colli-
sion the absolute value of velocity is not changed,
but the motion direction is changed, so that
the equality v1 + ucosφ1 = −v0 − ucos (ωt1 + φ0)
is kept. Taking into account the equality (6),
from the equalities v1 = −v0 − 2ucos (ωt1 + φ0)
and (4), with taking of products and squares
of their different parts and averaging over
φ0, one gets the equalities
⟨
v21
⟩
φ
= v20 + 2u2,
⟨z1v1⟩φ = − (z0 + v0t1) v0 − u2ω−1sin (ωt1),⟨
z21
⟩
φ
= (z0 + v0t1)
2
+ (u/ω)2 [1− cos (ωt1)].
The next averaging, over the value of t1, ac-
cording to Poisson process with the frequency ν0,
yields ⟨z1v1⟩ = −z0v0 − v20ν
−1
0 − u2ν0
(
ν20 + ω2
)−1
,⟨
z21
⟩
= z20 + 2z0v0ν
−1
0 + 2v20ν
−2
0 + u2
(
ν20 + ω2
)−1
.
For the next collision, keeping the possibility of
frequency change in Poisson process, with indexes
change one comes to the equalities
⟨z2v2⟩ = −z1v1 − v21ν
−1
1 − u2ν1
(
ν21 + ω2
)−1
,⟨
z22
⟩
= z21 + 2z1v1ν
−1
1 + 2v21ν
−2
1 + u2
(
ν21 + ω2
)−1
.
For the averaged result of two successive collisions
the equalities
z2v2 − z0v0 = v20ν
−1
0 − v21ν
−1
1 + (ν1 − ν0)×
×u2
(
ν0ν1 − ω2
) [(
ν20 + ω2
) (
ν21 + ω2
)]−1
,
z22 − z20 = 2v21ν
−2
1 + (ν1 − ν0) ν
−1
1 ×
×
{
2z0v0ν
−1
0 + 2v20ν
−2
0 +
[
2ω2 + ν1 (ν1 − ν0)
]
×
×u2
[(
ν20 + ω2
) (
ν21 + ω2
)]−1
}
take place (averaging symbol in them is not written).
It is assumed that collision frequency is much
less than oscillation frequency and the full velocity
is much greater than the oscillation velocity. The de-
pendence of the collision frequency on the electron
energy may influence on the rate of increase of the
averaged squares of coordinate and velocity.
Let us consider two cases, in which from electron
energy it is independent either mean free path λ, or
collision frequency ν, respectively.
If λ = const, then for the velocity and coordinate
after the great number N of collisions, taking into
account the relationship ⟨νN ⟩≈
⟨
v2N
⟩1/2
λ−1, one gets⟨
v2N+1
⟩
−
⟨
v2N
⟩
= 2u2,
⟨
v2N
⟩
≈2u2N ,
⟨zN+2vN+2⟩ − ⟨zNvN ⟩≈
≈
(
⟨νN ⟩−1 ⟨
v2N
⟩
− ⟨νN+2⟩−1 ⟨
v2N+2
⟩)
/2,
⟨zNvN ⟩≈ − ⟨νN ⟩−1 ⟨
v2N
⟩
/2,
⟨
z2N+2
⟩
−
⟨
z2N
⟩
≈2λ2,⟨
z2N
⟩
≈λ2N≈
(
2u2
)−1
λ2
⟨
v2N
⟩
. (7)
But if ν = const then the relevant relationships
have the following form:⟨
v2N
⟩
≈2u2N ,⟨
z2N+2
⟩
−
⟨
z2N
⟩
≈2ν−2
⟨
v2N
⟩
,⟨
z2N
⟩
≈(u/ν)2N2≈
(
4u2ν2
)−1 ⟨
v2N
⟩2
. (8)
4. MOTION IN THE FIELD WITH
CORRELATED PHASE JUMPS
In this case, it is assumed that the phase at instant
of jump is changed so, that field strength projection
on the electron motion direction at any time is the
same, as one in the considered process with elastic
collisions in the sinusoidal field, and difference is only
in absence of change of motion direction at instant
of phase jump. In such case, at once after time
instant t1 the velocity is changed according to the
time dependence v(t) = v1 − ucos [ω (t− t1) + φ1],
and the equalities (6) and
v1 − ucosφ1 = v0 + ucos (ωt1 + φ0) are kept, to
ensure the continuity of velocity at the instant
of jump and the correspondence of the val-
ues of acceleration projection on the electron
motion direction to ones in elastic collisions.
From the equalities v1 = v0 + 2ucos (ωt1 + φ0),
and (4), with taking of products and squares
of their different parts and averaging by φ0,
one comes to the equalities
⟨
v21
⟩
φ
= v20 + 2u2,
⟨z1v1⟩φ = (z0 + v0t1) v0 + u2ω−1sin (ωt1),⟨
z21
⟩
φ
= (z0 + v0t1)
2
+ (u/ω)2 [1− cos (ωt1)].
The next averaging, over t1, according to Pois-
son process with frequency ν0, gives the equal-
ities ⟨z1v1⟩ = z0v0 + v20ν
−1
0 + u2ν0
(
ν20 + ω2
)−1
,⟨
z21
⟩
= z20 + 2z0v0ν
−1
0 + 2v20ν
−2
0 + u2
(
ν20 + ω2
)−1
.
For the velocity and coordinate after the great
number N of phase jumps one comes to the
relationships
⟨
v2N+1
⟩
−
⟨
v2N
⟩
= 2u2,
⟨
v2N
⟩
≈2u2N ,
⟨zN+1vN+1⟩ − ⟨zNvN ⟩≈⟨νN ⟩−1⟨
v2N
⟩
,⟨
z2N+1
⟩
−
⟨
z2N
⟩
≈2⟨νN ⟩−1⟨zNvN ⟩.
If the frequency of phase jumps is
changed so, that ⟨νN ⟩−1⟨
v2N
⟩1/2≈λ = const,
then one gets ⟨zN+1vN+1⟩ − ⟨zNvN ⟩≈λu(2N)1/2,
⟨zNvN ⟩≈λu(2N)3/2/3,
⟨
z2N+1
⟩
−
⟨
z2N
⟩
≈4λ2N/3,⟨
z2N
⟩
≈2λ2N2/3≈
(
6u4
)−1
λ2
⟨
v2N
⟩2
. (9)
But if the frequency of phase jumps is constant,
ν = const, then the corresponding relationships have
the following form: ⟨zNvN ⟩≈u2ν−1N2,⟨
z2N
⟩
≈2(u/ν)2N3/3≈
(
12u4ν2
)−1 ⟨
v2N
⟩3
. (10)
5. CONCLUSIONS
The relationships (5), (7–10), for the different meth-
ods of electron energy increase, give the square of
the characteristic dimension of the device, in which
a given value of square of velocity may be achieved.
The relationships (7), (8) deal with collision heating,
141
in the different assumptions, and the relationships
(5), (9), (10) deal with electron motion without col-
lisions in the field with phase jumps. In the case (5),
the values of phase before and after jump are indepen-
dent, but in the cases (9) and (10) they are connected
in such a way as ones in elastic collisions. Comparing
of the relationships (5) and (10) with (8) shows that
the similarity between electron energy increase due
to the phase jumps and due to the elastic collisions is
not full. If the frequency of phase jumps is close to the
frequency of elastic collisions then in all cases the en-
ergy gained by electron is proportional to the number
of phase jumps or elastic collisions. But the square of
dimension of the motion space necessary to achieve
a given energy in the case of the acceleration in the
field with phase jumps is proportional to the third
power of the energy, whereas in the case of the colli-
sion heating it is proportional to the second power of
the energy, so, the collision heating requires much less
device dimensions than the collisionless heating. The
main cause is change of the electron motion direction
in collisions, in contrast to the case of motion in the
field with phase jumps, where only the acceleration
value is changed at instant of the phase jump, and
an electron continues to move in the same direction
as before the phase jump.
As for comparing of the acceleration in the field
with phase jumps and the acceleration in the con-
stant field, it is obvious that any decrease of the field
strength from the maximum attainable value (and so
much the greater, the change of the strength direc-
tion) leads to decrease of the energy passed from field
to electron during the same displacement, to decrease
of acceleration rate, and to increase of necessary de-
vice dimensions.
It may be paid attention to the fact that when
electron energy increases due to elastic collisions the
relationship between the square of characteristic di-
mension of the space of electron motion and the
square of energy gained by electron is the same as it
is when electron is accelerated in the constant field.
However, it is obvious that collision heating is suit-
able for electron energy increase only up to energy
values, at which the cross-sections of non-elastic pro-
cesses are relatively small.
References
1. V.I. Karas’, Ja.B. Fainberg, A.F.Alisov, et al. In-
teraction with plasma and gases of microwave
radiation with stochastically jumping phase //
Plasma Physics Reports. 2005, v. 31, p. 748-760.
2. V.I. Karas’, A.F.Alisov, O.V.Bolotov, et al. Op-
tical radiation special features from plasma of
low-pressure discharge initiated by microwave ra-
diation with stochastic jumping phase // PAST.
Series: Plasma Electronics and New Methods of
Acceleration. 2013, N4(86), p. 183-188.
ÎÖÅÍÊÀ ÐÀÇÌÅÐÎÂ ÎÁËÀÑÒÈ ÂÇÀÈÌÎÄÅÉÑÒÂÈß, ÍÅÎÁÕÎÄÈÌÎÉ ÄËß
ÓÑÊÎÐÅÍÈß ×ÀÑÒÈÖ Â ÑÒÎÕÀÑÒÈ×ÅÑÊÎÌ ÏÎËÅ
Â.Îñòðîóøêî
Ðàññìîòðåíî óñêîðåíèå ýëåêòðîíà â ïîëå ñ çàâèñèìîñòüþ îò âðåìåíè, îòëè÷àþùåéñÿ îò ñèíóñîèäàëüíîé
ââåäåíèåì ñêà÷êîâ ôàçû. Óñòàíîâëåíà ñâÿçü ìåæäó ýíåðãèåé ýëåêòðîíîâ è õàðàêòåðíûì ðàçìåðîì
óñòðîéñòâà, â êîòîðîì òàêàÿ ýíåðãèÿ ìîæåò áûòü äîñòèãíóòà, ïðè ðàññìîòðåííîì ñïîñîáå óñêîðåíèÿ è
ïðè ñòîëêíîâèòåëüíîì íàãðåâå. Ïîêàçàíî, ÷òî óñêîðåíèå â ïîëå ñî ñêà÷êàìè ôàçû òðåáóåò çíà÷èòåëüíî
áîëüøåãî ðàçìåðà îáëàñòè äâèæåíèÿ ýëåêòðîíîâ, ÷åì íàãðåâ äî òàêîé æå ýíåðãèè â ñèíóñîèäàëüíîì
ïîëå ïðè íàëè÷èè óïðóãèõ ñòîëêíîâåíèé.
ÎÖIÍÊÀ ÐÎÇÌIÐI ÎÁËÀÑÒI ÂÇÀ�ÌÎÄI�, ÍÅÎÁÕIÄÍÎ� ÄËß ÏÐÈÑÊÎÐÅÍÍß
×ÀÑÒÈÍÎÊ Ó ÑÒÎÕÀÑÒÈ×ÍÎÌÓ ÏÎËI
Â.Îñòðîóøêî
Ðîçãëÿíóòî ïðèñêîðåííÿ åëåêòðîíà ó ïîëi iç çàëåæíiñòþ âiä ÷àñó, ÿêà âiäðiçíÿ¹òüñÿ âiä ñèíóñî¨äàëü-
íî¨ ââåäåííÿì ñòðèáêiâ ôàçè. Âñòàíîâëåíî çâ'ÿçîê ìiæ åíåðãi¹þ åëåêòðîíiâ òà õàðàêòåðíèì ðîçìiðîì
ïðèñòðîþ, ó ÿêîìó òàêà åíåðãiÿ ìîæå áóòè äîñÿãíóòà, ïðè ðîçãëÿíóòîìó ñïîñîái ïðèñêîðåííÿ òà ïðè
çiòêíåííîìó íàãðiâàííi. Ïîêàçàíî, ùî ïðèñêîðåííÿ ó ïîëi çi ñòðèáêàìè ôàçè ïîòðåáó¹ çíà÷íî áiëüøî-
ãî ðîçìiðó îáëàñòi ðóõó åëåêòðîíiâ, íiæ íàãðiâàííÿ äî òàêî¨ ñàìå åíåðãi¨ ó ñèíóñî¨äàëüíîìó ïîëi çà
íàÿâíîñòi ïðóæíèõ çiòêíåíü.
142
|
| id | nasplib_isofts_kiev_ua-123456789-80498 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:29:24Z |
| publishDate | 2014 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Ostroushko, V. 2015-04-18T15:15:15Z 2015-04-18T15:15:15Z 2014 Estimation of dimensions of interaction space necessary for particle acceleration in stochastic field / V. Ostroushko // Вопросы атомной науки и техники. — 2014. — № 5. — С. 140-142. — Бібліогр.: 2 назв. — англ. 1562-6016 PACS: 29.17.+w, 52.80.Pi https://nasplib.isofts.kiev.ua/handle/123456789/80498 It is considered electron acceleration in the field having time dependence, which differs from sinusoidal one with phase jumps introduction. The connection is found between electron energy and characteristic dimension of device, in which such energy may be achieved with considered acceleration method or with collision heating. It is shown that the acceleration in the field with jumps of phase requires much greater dimension of space for electron motion than the heating to the same energy in sinusoidal field in presence of elastic collisions. Рассмотрено ускорение электрона в поле с зависимостью от времени, отличающейся от синусоидальной введением скачков фазы. Установлена связь между энергией электронов и характерным размером устройства, в котором такая энергия может быть достигнута, при рассмотренном способе ускорения и при столкновительном нагреве. Показано, что ускорение в поле со скачками фазы требует значительно большего размера области движения электронов, чем нагрев до такой же энергии в синусоидальном поле при наличии упругих столкновений. Розглянуто прискорення електрона у полi iз залежнiстю вiд часу, яка вiдрiзняється вiд синусоїдальної введенням стрибкiв фази. Встановлено зв'язок мiж енергiєю електронiв та характерним розмiром пристрою, у якому така енергiя може бути досягнута, при розглянутому способi прискорення та при зiткненному нагрiваннi. Показано, що прискорення у полi зi стрибками фази потребує значно бiльшого розмiру областi руху електронiв, нiж нагрiвання до такої саме енергiї у синусоїдальному полi за наявностi пружних зiткнень. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Теория и техника ускорения частиц Estimation of dimensions of interaction space necessary for particle acceleration in stochastic field Оценка размеров области взаимодействия, необходимой для ускорения частиц в стохастическом поле Оцiнка розмiрiв областi взаємодiї, необхiдної для прискорення частинок у стохастичному полi Article published earlier |
| spellingShingle | Estimation of dimensions of interaction space necessary for particle acceleration in stochastic field Ostroushko, V. Теория и техника ускорения частиц |
| title | Estimation of dimensions of interaction space necessary for particle acceleration in stochastic field |
| title_alt | Оценка размеров области взаимодействия, необходимой для ускорения частиц в стохастическом поле Оцiнка розмiрiв областi взаємодiї, необхiдної для прискорення частинок у стохастичному полi |
| title_full | Estimation of dimensions of interaction space necessary for particle acceleration in stochastic field |
| title_fullStr | Estimation of dimensions of interaction space necessary for particle acceleration in stochastic field |
| title_full_unstemmed | Estimation of dimensions of interaction space necessary for particle acceleration in stochastic field |
| title_short | Estimation of dimensions of interaction space necessary for particle acceleration in stochastic field |
| title_sort | estimation of dimensions of interaction space necessary for particle acceleration in stochastic field |
| topic | Теория и техника ускорения частиц |
| topic_facet | Теория и техника ускорения частиц |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80498 |
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