Oscillations and coherent radiation of harmonics in radiation spectrum of system of electrons moving in spiral in medium
Using the Lorentz self-interaction method completing by Dirac hypothesis it is investigated the spectral distribution of the radiation power for the system of electrons moving along a spiral in transparent isotropic medium. The overlapping between neighbour harmonics as well as oscillations in the s...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Zitieren: | Oscillations and coherent radiation of harmonics in radiation spectrum of system of electrons moving in spiral in medium / A.V. Konstantinovich, I.A. Konstantinovich // Вопросы атомной науки и техники. — 2011. — № 5. — С. 67-74. — Бібліогр.: 40 назв. — англ. |
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| author | Konstantinovich, A.V. Konstantinovich, I.A. |
| author_facet | Konstantinovich, A.V. Konstantinovich, I.A. |
| citation_txt | Oscillations and coherent radiation of harmonics in radiation spectrum of system of electrons moving in spiral in medium / A.V. Konstantinovich, I.A. Konstantinovich // Вопросы атомной науки и техники. — 2011. — № 5. — С. 67-74. — Бібліогр.: 40 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | Using the Lorentz self-interaction method completing by Dirac hypothesis it is investigated the spectral distribution of the radiation power for the system of electrons moving along a spiral in transparent isotropic medium. The overlapping between neighbour harmonics as well as oscillations in the spectral distribution of one, two, three, and four electrons radiation power are studied for the case when the transversal component of electron velocity is bigger than the light phase velocity in medium but still less than the light velocity in vacuum. The effect of coherence in the spectrum of synchrotron-Cherenkov radiation for the system of two, three and four electrons is analyzed.
Використовуючи метод сили самодії Лоренца, доповнений гіпотезою Дірака, досліджено спектральний розподіл потужності випромінювання системи електронів, що рухаются вздовж гвинтової лінії у прозорому ізотропному середовищі. Перекриття гармонік та осциляції у спектральному розподілі потужності випромінювання одного, двох, трьох і чотирьох електронів досліджено для випадку, коли поперечна компонента швидкості електрона більша від фазовоі швидкості світла у середовищі, але менша від швидкості світла у вакуумі. Досліджено ефект когерентності у спектрі синхротронно-черенковського випромінювання системи двох, трьох та чотирьох електронів.
Используя метод силы самодействия Лоренца, дополненный гипотезой Дирака, исследовано спектральное распределение мощности излучения системы электронов, движущихся вдоль винтовой линии в прозрачной изотропной среде. Перекрытие гармоник и осцилляции в спектральном распределении мощности излучения одного, двух, трех и четырех электронов исследовано для случая, когда поперечная компонента скорости электрона больше фазовой скорости света в среде, но меньше скорости света в вакууме. Исследован эффект когерентности в спектре синхротронно-черенковского излучения системы двух, трех и четырех электронов.
|
| first_indexed | 2025-12-07T16:31:30Z |
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| fulltext |
ELECTRODYNAMICS
OSCILLATIONS AND COHERENT RADIATION OF
HARMONICS IN RADIATION SPECTRUM OF SYSTEM OF
ELECTRONS MOVING IN SPIRAL IN MEDIUM
A.V. Konstantinovich1∗, I.A. Konstantinovich1,2
1Chernivtsi National University, 58012, Chernivtsi, Ukraine
2Institute of Thermoelectrics, NAN and MON of Ukraine, 58027, Chernivtsi, Ukraine
(Received February 24, 2011)
Using the Lorentz self-interaction method completing by Dirac hypothesis it is investigated the spectral distribution
of the radiation power for the system of electrons moving along a spiral in transparent isotropic medium. The
overlapping between neighbour harmonics as well as oscillations in the spectral distribution of one, two, three, and
four electrons radiation power are studied for the case when the transversal component of electron velocity is bigger
than the light phase velocity in medium but still less than the light velocity in vacuum. The effect of coherence in
the spectrum of synchrotron-Cherenkov radiation for the system of two, three and four electrons is analyzed.
PACS: 41.60.-m, 41.60.Ap, 41.60.Bq, 41.60.Cr, 03.50.-z, 03.50.De
1. INTRODUCTION
The properties of synchrotron radiation of charged
particles moving in a circle in vacuum in framework
of classical electrodynamics were studied in papers
[1 - 3]. The particularities of radiation spectrum of
charged particles moving in magnetic field in vacuum
were examined by Ternov in report [4] and analyzed
in studies [5 - 8]. The properties of electromagnetic
radiation of the system of non-interacting electrons
moving in a spiral in constant magnetic field in vac-
uum were reported in papers [9 - 17].
The radiation spectrum of one electron moving in
a medium in magnetic field was under investigation
in papers [18 - 24]. The oscillations in synchrotron-
Cherenkov radiation spectrum of one electron were
obtained at its motion in a circle [18] and in a spiral
[24]. The hopping change of the function of spec-
tral distribution of radiation power of an electron is
studied in [12].
The coherence effects in the structure of radia-
tion spectrum of a system of non-interacting electrons
moving one by one along a spiral in a transparent
medium were considered in papers [25 - 31]. If the
dimension of a system of electrons is smaller compar-
ing to the radiation wavelength, both for a quantum-
mechanical system [32] and for a system of electrons
[33 - 36] the super-radiant regime is possible.
The aim of this paper is the investigation of the
oscillations and overlapping between neighbour har-
monics of the synchrotron-Cherenkov radiation spec-
trum of two, three, and four electrons moving in a
spiral in magnetic field in transparent medium for
the case when the transversal component of elec-
trons velocity is bigger than the light phase velocity
in medium but still less than the light velocity in
vacuum and the parallel component of electrons ve-
locity is much smaller than the light phase velocity in
medium. The spectral distributions of electrons radi-
ation power are calculated by the instrumentality of
high accuracy numerical methods and studied within
the analytical methods. The coherent radiation of
harmonics in the spectrum of synchrotron-Cherenkov
radiation for two, three and four electrons in the case
when the distance between electrons (phase shifts be-
tween electrons) is much smaller than the radiation
wavelength are studied. These studies present a great
interest for the investigation of radiation spectrum
structure of bunches of charged particles moving in
magnetic field.
2. TIME AVERAGED RADIATION
POWER OF SYSTEM OF ELECTRONS
MOVING ALONG A SPIRAL IN
TRANSPARENT MEDIUM
According to [12, 29] the time averaged radiation
power P
rad
of charged particles moving in medium
is determined by the relationship:
P
rad
= lim
T→∞
1
2T
T∫
−T
dt
∫
τ
d~r×
×
(
~j (~r, t)
∂ ~ADir (~r, t)
∂t
− ρ (~r, t)
∂ΦDir (~r, t)
∂t
)
. (1)
Here ~j (~r, t) is the current density and ρ (~r, t) is the
charge density. The integration is over some volume τ .
∗Corresponding author E-mail address: aconst@hotbox.ru
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2011, N5.
Series: Nuclear Physics Investigations (56), p.67-74.
67
According to the hypothesis of Dirac [3, 12, 29, 37,
38], the scalar ΦDir (~r, t) and vector ~ADir (~r, t) po-
tentials are defined as a half-difference of the retarded
and advanced potentials.
ΦDir =
1
2
(
Φret − Φadv
)
, ~ADir =
1
2
(
~Aret − ~Aadv
)
.
(2)
Then, the sources functions of N charged point par-
ticles are defined as [12, 29]
~j (~r, t) =
N∑
l=1
~Vl (t) ρl (~r, t) , ρ (~r, t) =
N∑
l=1
ρl (~r, t) ,
ρl (~r, t) = eδ (~r − ~rl(t)) , (3)
where ~rl (t) and ~Vl (t) are the motion law and velocity
of the lth particle, respectively.
The law of motion and the velocity of the lth elec-
tron in magnetic field are given by the expressions
~rl (t) = r0 cos {ω0(t + ∆tl)}~i+r0 sin {ω0(t + ∆tl)}~j+
+V‖(t + ∆tl)~k, ~Vl (t) =
d~rl (t)
dt
. (4)
Here r0 = V⊥ω−1
0 , ω0 = c2eBextẼ−1, Ẽ =
c
√
p2 + m2
0c
2, the magnetic induction vector ~Bext ‖
0Z, V⊥ and V‖ are the components of the velocity,
the ~p and Ẽ are the momentum and energy of the
electron, e and m0 are its charge and rest mass.
In this case the time averaged radiation power of
the point electrons can be obtained after substitution
of (2) to (4) into (1). Then, it is found [12, 29] that
P
rad
=
∞∫
0
dω W (ω) , (5)
W (ω) =
2e2
π
∞∫
0
dxωµ (ω)
µ0
4π
sin
(
n(ω)
c
ωη(x)
)
η(x)
×
×SN (ω) cos (ωx)
{
V 2
⊥ cos (ω0x) + V 2
‖ −
c2
n2 (ω)
}
,
(6)
where η (x) =
√
V 2
‖ x2 + 4
V 2
⊥
ω2
0
sin2
(ω0
2
x
)
,
µa (ω) = µ (ω)µ0 is the absolute magnetic permeabil-
ity, n (ω) is the refraction index, ω is cyclic frequency,
c is velocity of light in vacuum.
In the case of electrons moving one by one along
a spiral the coherence factor takes the form [12, 29]:
SN (ω) =
N∑
l,j=1
cos {ω (∆tl −∆tj)} . (7)
The coherence factor SN (ω) determines a redis-
tribution of radiation power of electrons in spectral
distribution of this radiation.
3. SPECTRAL-ANGULAR AND
SPECTRAL DISTRIBUTION OF THE
RADIATION POWER OF THE SYSTEM
OF ELECTRONS MOVING ALONG A
SPIRAL IN TRANSPARENT MEDIUM
After some transformations of relationships (5)
and (6) the contributions of separate harmonics to the
electrons radiation power can be expressed as [29]:
P
rad
=
e2
c
∞∑
m=−∞
∞∫
0
dωµ (ω)
µ0
4π
n (ω)ω2
π∫
0
sin θdθ×
×SN (ω) δ
{
ω
(
1− n (ω)
c
V‖ cos θ
)
−mω0
}
×
×
{
V 2
⊥
[
m2
q2
J2
m (q) + J
′2
m (q)
]
+
+
(
V 2
‖ −
c2
n2 (ω)
)
J2
m (q)
}
, (8)
where q = V⊥
n (ω)
c
ω
ω0
sin θ, ω0 =
eBext
m0
√
1− V 2
c2
,
Jm (q) and J
′
m (q) are the Bessel function with integer
index and its derivative, respectively, θ is the angle
formed by wave vector ~k and axis 0Z [29].
The relationship (8) un the case of one electron
was also obtained using retarded potentials within
the method of enclosing surfaces [20].
Each harmonic in relationship (8) is a set of the
frequencies, which are the solutions of the equations
ω
(
1− n (ω)
c
V‖ cos θ
)
−mω0 = 0. (9)
After integrating in (8) over θ variable we have
obtained the spectral distribution of the system of
electrons radiation power on harmonics [29]
P
rad
=
e2
V‖
∞∑
m=−∞
∞∫
0
dωµ (ω)
µ0
4π
η
(
u2(m)
)
ω×
×SN (ω)
{
V 2
⊥
[
m2
q2
1u2 (m)
J2
m (q1u (m))+
+J
′2
m (q1u (m))
]
+
(
V 2
‖ −
c2
n2 (ω)
)
J2
m (q1u (m))
}
,
(10)
where
η
(
u2(m)
)
=
{
1, u2(m) > 0
0, u2(m) < 0
, q1 = V⊥
n (ω)
c
ω
ω0
.
u2(m) = 1− c2 (ω −mω0)
2
n2 (ω) V 2
‖ ω2
. (11)
The band boundaries in the radiation spectrum
are determined by the function η
(
u2(m)
)
.
The coherence factor S1 (ω) of a single electron is
defined as
SN (ω) = S1 (ω) = 1. (12)
In the case of two electrons the coherence factor
S2 (ω) is defined as [16]
S2 (ω) = 2 + 2 cos (ω∆t12) . (13)
68
Here ∆t12 = ∆t2 −∆t1 is the time shift between the
first and second electrons moving along a spiral.
The analogous expression for the coherence fac-
tor was obtained by Bolotovskii [39]. The coherence
factor of three electrons takes the form [16]
S3 (ω) = 3 + 2 cos (ω∆t12) + 2 cos (ω∆t23)+
+2 cos {ω (∆t12 + ∆t23)} . (14)
Here ∆t23 is the time shift between the second and
third electrons. The coherence factor of four electrons
is de-fined as [16]
S4 (ω) = 4 + 2 cos (ω∆t12) + 2 cos (ω∆t23)+
+2 cos (ω∆t34) + 2 cos {ω (∆t12 + ∆t23)}+
+2 cos {ω (∆t23 + ∆t34)}+
+2 cos {ω (∆t12 + ∆t23 + ∆t34)} , (15)
where ∆t34 is the time shift between the third and
fourth electrons.
4. PECULIARITIES OF THE SPECTRAL
DISTRIBUTION OF RADIATION
SPECTRUM OF ONE, TWO, THREE, AND
FOUR ELECTRONS MOVING ALONG A
SPIRAL IN TRANSPARENT MEDIUM
Our high accuracy numerical method of calcula-
tions of the radiation spectra was carried out on the
basis of relationships (5) to (7) and (12) to (15). The
spectral distribution of synchrotron-Cherenkov radi-
ation power was obtained for Bext = 10−4T, µ = 1,
µ0 = 4π×10−7H/m, n = 2.0, V⊥med = 0.2×109m/s,
V‖med = 0.15 × 108m/s, ω0j = 0.1307 × 108rad/s,
r0j = 15.3m (j=1,2,. . . ,21) and V⊥med = 0.2 ×
109m/s, V‖med = 0.3 × 108m/s, ω022 = 0.1298 ×
108rad/s, r022 = 15.4m, c = 0.2997925 × 109m/s.
Fig.1. Oscillations in synchrotron-Cherenkov radi-
ation spectrum at low harmonics for Bext = 10−4T,
n = 2, V⊥med = 0.2×109m/s, V‖med = 0.15×108m/s.
ω0j = 0.1307×108rad/s, r0j = 15.3m (j=1,2,. . . ,21).
Curve 1. One electron with radiation power
P int
med1 = 0.315× 10−19W
The radiation power for one electron P int
med1 =
0.315 × 10−19W in interval 0 to 40ω01 is deter-
mined after integration of relationships (5) taking
into account (6), (7) when SN (ω) is substituted by
S1 (ω) = 1 (curve 1 in Figs.1 and 2).
It is interesting to compare the radiation power
spectral distributions for one electron (see curve 1 in
Fig.1) to that of two, three, and four electrons (curves
2 to 4 in Fig.2, respectively).
Fig.2. Oscillations in synchrotron-Cherenkov
radiation spectrum at low harmonics. Curve 2.
Two electrons at ∆t
(2)
12 = 0.001π/ω02 with
radiation power P int
med2 = 0.1257 × 10−18W,
P int
med2/P int
med1 = 3.99. Curve 3. Three elec-
trons at ∆t
(3)
12 = ∆t
(3)
23 = 0.001π/ω03 with
P int
med3 = 0.2820 × 10−18W, P int
med3/P int
med1 = 8.95.
Curve 4. Four electrons at time shifts
∆t
(4)
12 = ∆t
(4)
23 = ∆t
(4)
34 = 0.001π/ω04 with
radiation power P int
med4 = 0.4999 × 10−18W,
P int
med4/P int
med1 = 15.84
For the time shift ∆t
(2)
12 = 0.001π/ω02 (curve 2
in Figs. 2 and 3) the coherence factor S2 (ω) ≈ 4 at
low harmonics and two electrons radiate as a charged
particle with the charge 2e and the rest mass 2m0,
i.e. by a factor of four more than a single electron
(P int
med2 = 0.1257× 10−18W, P int
med2/P int
med1 = 3.99).
Fig.3. Synchrotron-Cherenkov radiation spec-
trum at low harmonics. Curve 5. Two elec-
trons at ∆t
(5)
12 = 0.1π/ω05 with radiation power
P int
med5 = 0.6564× 10−19W, P int
med5/P int
med1 = 2.08
For the time shifts ∆t
(3)
12 = ∆t
(3)
23 = 0.001π/ω03
the coherence factor S3(ω) ≈ 9 and at low har-
monics three electrons radiate as a charged particle
with the charge 3e and the rest mass 3m0(curve 3
in Figs. 2 and 4), i.e. by a factor of nine more
69
than a single electron (P int
med3 = 0.2820 × 10−18W,
P int
med3/P int
med1 = 8.95).
For the time shifts ∆t
(4)
12 = ∆t
(4)
23 = ∆t
(4)
34 =
0.001π/ω04 the coherence factor S4(ω) ≈ 16 and four
electrons radiate as a charged particle with the charge
4e and the rest mass 4m0 (curve 4 in Figs. 2 and 5),
i.e. by a factor of sixteen more than a single electron
(P int
med4 = 0.4999× 10−18W, P int
med4/P int
med1 = 15.84).
Fig.4. Synchrotron-Cherenkov radiation spectrum
at low harmonics. Curve 6. Three electrons at
∆t
(6)
12 = ∆t
(6)
23 = 0.1π/ω06 with radiation power
P int
med6 = 0.9944× 10−19W, P int
med6/P int
med1 = 3.16
Fig.5. Synchrotron-Cherenkov radiation spectrum
at low harmonics. Curve 7. Four electrons at
∆t
(7)
12 = ∆t
(7)
23 = ∆t
(7)
34 = 0.1π/ω07 with radiation
power P int
med7 = 0.1331 × 10−18W, P int
med7/P int
med4 =
4.23
In the frequency range of 0−40ω0j for smaller time
shifts we have obtained the coherent radiation with
radiation power P
rad
proportional to N2 (curves 2,
3, and 4 in Fig. 2) for such the electron system so
far as the dimension of this system is smaller in com-
parison to the radiation wavelength [38], see also [32
- 36].
For the component of velocity V⊥med = 0.2 ×
109m/s, V‖med = 0.15 × 108m/s the spectral distri-
bution of synchrotron-Cherenkov radiation power of
one, two, three, and four electrons moving along spi-
ral at the first harmonics has a form of discrete bands
(see Figs.1-2).
Fig.6. Oscillations in synchrotron-Cherenkov
radiation spectrum at low and middle harmon-
ics. Curve 8. One electron with radiation power
P int
med8 = 0.1950× 10−18W
Fig.7. Oscillations in synchrotron-Cherenkov radia-
tion spectrum at low and middle harmonics. Curve 9.
Two electrons at ∆t
(9)
12 = 0.001π/ω09 with P int
med9 =
0.7702 × 10−18W, P int
med9/P int
med8 = 3.95. Curve 10.
Three electrons at time shifts ∆t
(10)
12 = ∆t
(10)
23 =
0.001π/ω010 with radiation power P int
med10 = 0.1698×
10−17W, P int
med10/P int
med8 = 8.71. Curve 11. Four elec-
trons at ∆t
(11)
12 = ∆t
(11)
23 = ∆t
(11)
34 = 0.001π/ω011
with P int
med11 = 0.2933 × 10−17W, P int
med11/P int
med8 =
15.04
Fig.8. Synchrotron-Cherenkov radiation spectrum
at low and middle harmonics. Curve 12. Two elec-
trons at ∆t
(12)
12 = 0.1π/ω012 with radiation power
P int
med12 = 0.3916× 10−18W, P int
med12/P int
med8 = 2.02
70
Fig.9. Synchrotron-Cherenkov radiation spectrum
at low and middle harmonics. Curve 13. Three elec-
trons at ∆t
(13)
12 = ∆t
(13)
23 = 0.1π/ω013 with radiation
power P int
med13 = 0.5878 × 10−18W, P int
med13/P int
med8 =
3.01
Fig.10. Synchrotron-Cherenkov radiation spectrum
at low and middle harmonics. Curve 14. Four elec-
trons at ∆t
(14)
12 = ∆t
(14)
23 = ∆t
(14)
34 = 0.1π/ω014
with radiation power P int
med14 = 0.7837 × 10−18W,
P int
med14/P int
med8 = 4.02
Fig.11. Oscillations in synchrotron-Cherenkov ra-
diation spectrum at low, middle, and high harmon-
ics. Curve 15. One electron with radiation power
P int
med15 = 0.7764× 10−18W
Fig.12. Oscillations in synchrotron-Cherenkov ra-
diation spectrum at low, middle, and high har-
monics. Curve 16. Two electrons at time shift
∆t
(16)
12 = 0.001π/ω016 with radiation power P int
med16 =
0.2956× 10−17W, P int
med16/P int
med15 = 3.81. Curve 17.
Three electrons at time shifts ∆t
(17)
12 = ∆t
(17)
23 =
0.001π/ω017 with radiation power P int
med17 = 0.6126×
10−17W, P int
med17/P int
med15 = 7.89. Curve 18. Four
electrons at time shifts ∆t
(18)
12 = ∆t
(18)
23 = ∆t
(18)
34 =
0.001π/ω018 with radiation power P int
med18 = 0.9721×
10−17W, P int
med18/P int
med15 = 12.52
Fig.13. Synchrotron-Cherenkov radiation spectrum
at low, middle, and high harmonics. Curve 19. Two
electrons at ∆t
(19)
12 = 0.1π/ω019 with radiation power
P int
med19 = 0.1520× 10−17W, P int
med19/P int
med15 = 1.96
For the velocities c > V⊥med > c/n ( V⊥med =
0.2×109 m/s, V‖med = 0.15×108m/s) we have found
the oscillations in the radiation spectrum of one elec-
tron (see curve 1 in Figs.1, 2, curve 8 in Figs.6, 7,
and curve 15 in Figs.11, 12) as well as in that of two,
three, and four electrons moving one by one along the
spiral with a smaller selected time shifts 0.001π/ω0j
(see curves 2 to 4 in Fig.2, curves 9 to 11 in Fig.7,
and curves 16 to 18 in Fig.12).
The oscillating character of the spectral distribu-
tion of the synchrotron-Cherenkov radiation of elec-
trons moving in magnetic field in the medium at
c > V⊥med > c/n is defined by a properties of the
Bessel functions [40] (Figs.1-16).
If V‖med → 0, then the spiral transforms into
a circle and the radiation spectrum takes a discrete
character. At moving in a circle at V⊥med > c/n , the
spectral distribution of the synchrotron-Cherenkov
radiation of the system of electrons has also an os-
cillating character, too [21].
71
Fig.14. Synchrotron-Cherenkov radiation spectrum
at low, middle, and high harmonics. Curve 20.
Three electrons at ∆t
(20)
12 = ∆t
(20)
23 = 0.1π/ω020
with radiation power P int
med20 = 0.2264 × 10−17W,
P int
med20/P int
med15 = 2.92
Fig.15. Synchrotron-Cherenkov radiation spectrum
at low, middle, and high harmonics. Curve 21. Four
electrons at ∆t
(21)
12 = ∆t
(21)
23 = ∆t
(21)
34 = 0.1π/ω021
with radiation power P int
med21 = 0.3008 × 10−17W,
P int
med21/P int
med15 = 3.87
For the time shift ∆t
(19)
12 = 0.1π/ω019 between two
electrons we have found that there is no radiation at
frequencies 10(2i−1)ω019 (i=1,2,. . . ,10) (curve 19 in
Fig.13) and at the frequencies 20iω019 (i=1,2,. . . ,10)
the coherence factor takes the maximum value equal
to four.
In the case of time shifts ∆t
(20)
12 = ∆t
(20)
23 =
0.1π/ω020 between three electrons we have found that
there is no radiation at frequencies 20(3i − 2)ω020/3
and 20(3i−1)ω020/3, (i=1,2,. . . ,10) (curve 20 in Fig.
14) and at the frequencies 20iω020 (i=1,2,. . . ,10) the
coherence factor takes the maximum value equal to
nine.
In the case of time shifts ∆t
(21)
12 = ∆t
(21)
23 =
∆t
(21)
34 = 0.1π/ω021 between four electrons we have
found that there is no radiation at frequencies 5(4i−
3)ω021, 5(4i−2)ω021, and 5(4i−1)ω021, (i=1,2,. . . ,10)
(curve 21 in Fig. 15) and at the frequencies 20iω021
(i=1,2,. . . ,10) the coherence factor takes the maxi-
mum value equal to sixteen.
For time shifts 0.1π/ω0j (Figs.3-5, Figs.8-10, and
Figs.13-15) the coherence factor takes the maximum
value SN (ω) = N2 at frequencies 20iω0j (i=1, 2,. . . ).
Fig.16. Oscillations in synchrotron-Cherenkov ra-
diation spectrum at low, middle, and high harmonics
for Bext = 10−4T, n = 2, V⊥med = 0.2 × 109m/s,
V‖med = 0.3 × 108m/s, ω022 = 0.1298 × 108rad/s,
r0 = 15.4m. Curve 22. One electron with radiation
power P int
med22 = 0.7827× 10−18W
For high harmonics at V⊥med = 0.2 × 109m/s,
V‖med = 0.3×108m/s the overlapping between neigh-
bour harmonics does not lead, in fact, to periodical
changes of the spectral distribution for synchrotron-
Cherenkov radiation power. Only the oscillations of
this function are observed (curve 22 in Fig.16). The
obtained results are in good agreement to those ob-
tained in [24].
5. CONCLUSIONS
The near-periodical variations of the spectral
distribution function at c > V⊥med > c/n
of synchrotron-Cherenkov radiation are preferably
caused by the overlapping between the mth and
(m+1)th harmonics at some contribution of the other
ones. At increasing parallel component of the velocity
V‖med the near-periodical variations of the spectral
distribution of the synchrotron-Cherenkov radiation
power considerably decrease.
At small time shifts 0.001π/ω0j between electrons
the system of two, three, and four electrons in the fre-
quency range of 0 − 40ω0j there arises the coherent
synchrotron-Cherenkov radiation with coherent fac-
tor SN (ω) = N2 so far as the dimension of this system
is smaller in comparison to the radiation wavelength.
For the velocities c > V⊥med > c/n ( V⊥med =
0.2×109 m/s, V‖med = 0.15×108m/s) there arise the
oscillations in the radiation spectrum of two, three,
and four electrons moving one by one along the spiral
with a smaller selected time shifts 0.001π/ω0j .
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ОСЦИЛЛЯЦИИ И КОГЕРЕНТНОЕ ИЗЛУЧЕНИЕ ГАРМОНИК В СПЕКТРЕ
ИЗЛУЧЕНИЯ СИСТЕМЫ ЭЛЕКТРОНОВ, ДВИЖУЩИХСЯ ВДОЛЬ ВИНТОВОЙ
ЛИНИИ В СРЕДЕ
А.В. Константинович, И.А. Константинович
Используя метод силы самодействия Лоренца, дополненный гипотезой Дирака, исследовано спектраль-
ное распределение мощности излучения системы электронов, движущихся вдоль винтовой линии в
прозрачной изотропной среде. Перекрытие гармоник и осцилляции в спектральном распределении
мощности излучения одного, двух, трех и четырех электронов исследовано для случая, когда попе-
речная компонента скорости электрона больше фазовой скорости света в среде, но меньше скорости
света в вакууме. Исследован эффект когерентности в спектре синхротронно-черенковского излучения
системы двух, трех и четырех электронов.
ОСЦИЛЯЦIЇ ТА КОГЕРЕНТНЕ ВИПРОМIНЮВАННЯ ГАРМОНIК У СПЕКТРI
ВИПРОМIНЮВАННЯ СИСТЕМИ ЕЛЕКТРОНIВ, ЩО РУХАЮТЬСЯ ВЗДОВЖ
ГВИНТОВОЇ ЛIНIЇ У СЕРЕДОВИЩI
А.В. Константинович, I.А. Константинович
Використовуючи метод сили самодiї Лоренца, доповнений гiпотезою Дiрака, дослiджено спектральний
розподiл потужностi випромiнювання системи електронiв, що рухаются вздовж гвинтової лiнiї у прозо-
рому iзотропному середовищi. Перекриття гармонiк та осциляцiї у спектральному розподiлi потужностi
випромiнювання одного, двох, трьох i чотирьох електронiв дослiджено для випадку, коли поперечна
компонента швидкостi електрона бiльша вiд фазовоi швидкостi свiтла у середовищi, але менша вiд
швидкостi свiтла у вакуумi. Дослiджено ефект когерентностi у спектрi синхротронно-черенковського
випромiнювання системи двох, трьох та чотирьох електронiв.
74
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| id | nasplib_isofts_kiev_ua-123456789-111470 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:31:30Z |
| publishDate | 2011 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Konstantinovich, A.V. Konstantinovich, I.A. 2017-01-10T11:53:05Z 2017-01-10T11:53:05Z 2011 Oscillations and coherent radiation of harmonics in radiation spectrum of system of electrons moving in spiral in medium / A.V. Konstantinovich, I.A. Konstantinovich // Вопросы атомной науки и техники. — 2011. — № 5. — С. 67-74. — Бібліогр.: 40 назв. — англ. 1562-6016 PACS: 41.60.-m, 41.60.Ap, 41.60.Bq, 41.60.Cr, 03.50.-z, 03.50.De https://nasplib.isofts.kiev.ua/handle/123456789/111470 Using the Lorentz self-interaction method completing by Dirac hypothesis it is investigated the spectral distribution of the radiation power for the system of electrons moving along a spiral in transparent isotropic medium. The overlapping between neighbour harmonics as well as oscillations in the spectral distribution of one, two, three, and four electrons radiation power are studied for the case when the transversal component of electron velocity is bigger than the light phase velocity in medium but still less than the light velocity in vacuum. The effect of coherence in the spectrum of synchrotron-Cherenkov radiation for the system of two, three and four electrons is analyzed. Використовуючи метод сили самодії Лоренца, доповнений гіпотезою Дірака, досліджено спектральний розподіл потужності випромінювання системи електронів, що рухаются вздовж гвинтової лінії у прозорому ізотропному середовищі. Перекриття гармонік та осциляції у спектральному розподілі потужності випромінювання одного, двох, трьох і чотирьох електронів досліджено для випадку, коли поперечна компонента швидкості електрона більша від фазовоі швидкості світла у середовищі, але менша від швидкості світла у вакуумі. Досліджено ефект когерентності у спектрі синхротронно-черенковського випромінювання системи двох, трьох та чотирьох електронів. Используя метод силы самодействия Лоренца, дополненный гипотезой Дирака, исследовано спектральное распределение мощности излучения системы электронов, движущихся вдоль винтовой линии в прозрачной изотропной среде. Перекрытие гармоник и осцилляции в спектральном распределении мощности излучения одного, двух, трех и четырех электронов исследовано для случая, когда поперечная компонента скорости электрона больше фазовой скорости света в среде, но меньше скорости света в вакууме. Исследован эффект когерентности в спектре синхротронно-черенковского излучения системы двух, трех и четырех электронов. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Электродинамика Oscillations and coherent radiation of harmonics in radiation spectrum of system of electrons moving in spiral in medium Осциляцiї та когерентне випромiнювання гармонiк у спектрi випромiнювання системи електронiв, що рухаються вздовж гвинтової лiнiї у середовищi Осцилляции и когерентное излучение гармоник в спектре излучения системы электронов, движущихся вдоль винтовой линии в среде Article published earlier |
| spellingShingle | Oscillations and coherent radiation of harmonics in radiation spectrum of system of electrons moving in spiral in medium Konstantinovich, A.V. Konstantinovich, I.A. Электродинамика |
| title | Oscillations and coherent radiation of harmonics in radiation spectrum of system of electrons moving in spiral in medium |
| title_alt | Осциляцiї та когерентне випромiнювання гармонiк у спектрi випромiнювання системи електронiв, що рухаються вздовж гвинтової лiнiї у середовищi Осцилляции и когерентное излучение гармоник в спектре излучения системы электронов, движущихся вдоль винтовой линии в среде |
| title_full | Oscillations and coherent radiation of harmonics in radiation spectrum of system of electrons moving in spiral in medium |
| title_fullStr | Oscillations and coherent radiation of harmonics in radiation spectrum of system of electrons moving in spiral in medium |
| title_full_unstemmed | Oscillations and coherent radiation of harmonics in radiation spectrum of system of electrons moving in spiral in medium |
| title_short | Oscillations and coherent radiation of harmonics in radiation spectrum of system of electrons moving in spiral in medium |
| title_sort | oscillations and coherent radiation of harmonics in radiation spectrum of system of electrons moving in spiral in medium |
| topic | Электродинамика |
| topic_facet | Электродинамика |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/111470 |
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