Radiation spectrum of an electron moving in a spiralin medium
The action of the Doppler effect on the particularities of radiation power spectral distribution of an electron moving in a spiral in a medium has been investigated near the Cherenkov threshold. The fine structure of the electromagnetic radiation spectrum at the Cherenkov threshold and the synchrotr...
Saved in:
| Published in: | Condensed Matter Physics |
|---|---|
| Date: | 2007 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут фізики конденсованих систем НАН України
2007
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/117911 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Radiation spectrum of an electron moving in a spiralin medium / A.V. Konstantinovich, I.A. Konstantinovich // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 5-9. — Бібліогр.: 15 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-117911 |
|---|---|
| record_format |
dspace |
| spelling |
Konstantinovich, A.V. Konstantinovich, I.A. 2017-05-27T11:49:11Z 2017-05-27T11:49:11Z 2007 Radiation spectrum of an electron moving in a spiralin medium / A.V. Konstantinovich, I.A. Konstantinovich // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 5-9. — Бібліогр.: 15 назв. — англ. 1607-324X DOI:10.5488/CMP.10.1.5 PACS: 41.60.-m, 41.60.Ap, 41.60.Bq, 03.50.-z, 03.50.De https://nasplib.isofts.kiev.ua/handle/123456789/117911 The action of the Doppler effect on the particularities of radiation power spectral distribution of an electron moving in a spiral in a medium has been investigated near the Cherenkov threshold. The fine structure of the electromagnetic radiation spectrum at the Cherenkov threshold and the synchrotron-Cherenkov radiation spectrum for an electron moving along a spiral with non-relativistic longitudinal component of the velocity (i.e., the component parallel to the magnetic induction vector) in a transparent isotropic medium have been obtained and studied. Дослiджено вплив ефекту Доплера на особливостi спектрального розподiлу потужностi випромiнювання електрона, що рухається вздовжгвинтової лiнiї у середовищi, поблизу черенковського бар’єру. Отримана i дослiджена тонка структура електромагнiтного спектра на черенковському бар’єрi та спектра синхротронно-черенковського випромiнювання електрона, що рухається вздовж гвинтової лiнiї у прозорому середовищi, з нерелятивiстською поздовжньою компонентою швидкостi (компонентою, що паралельна вектору магнiтної iндукцiї). en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Radiation spectrum of an electron moving in a spiralin medium Спектр випромiнювання електрона, що рухається вздовж гвинтової лiнiї у середовищi Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Radiation spectrum of an electron moving in a spiralin medium |
| spellingShingle |
Radiation spectrum of an electron moving in a spiralin medium Konstantinovich, A.V. Konstantinovich, I.A. |
| title_short |
Radiation spectrum of an electron moving in a spiralin medium |
| title_full |
Radiation spectrum of an electron moving in a spiralin medium |
| title_fullStr |
Radiation spectrum of an electron moving in a spiralin medium |
| title_full_unstemmed |
Radiation spectrum of an electron moving in a spiralin medium |
| title_sort |
radiation spectrum of an electron moving in a spiralin medium |
| author |
Konstantinovich, A.V. Konstantinovich, I.A. |
| author_facet |
Konstantinovich, A.V. Konstantinovich, I.A. |
| publishDate |
2007 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Спектр випромiнювання електрона, що рухається вздовж гвинтової лiнiї у середовищi |
| description |
The action of the Doppler effect on the particularities of radiation power spectral distribution of an electron moving in a spiral in a medium has been investigated near the Cherenkov threshold. The fine structure of the electromagnetic radiation spectrum at the Cherenkov threshold and the synchrotron-Cherenkov radiation spectrum for an electron moving along a spiral with non-relativistic longitudinal component of the velocity (i.e., the component parallel to the magnetic induction vector) in a transparent isotropic medium have been obtained and studied.
Дослiджено вплив ефекту Доплера на особливостi спектрального розподiлу потужностi випромiнювання електрона, що рухається вздовжгвинтової лiнiї у середовищi, поблизу черенковського бар’єру. Отримана i дослiджена тонка структура електромагнiтного спектра на черенковському бар’єрi та спектра синхротронно-черенковського випромiнювання електрона, що рухається вздовж гвинтової лiнiї у прозорому середовищi, з нерелятивiстською поздовжньою компонентою швидкостi (компонентою, що паралельна вектору магнiтної iндукцiї).
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/117911 |
| citation_txt |
Radiation spectrum of an electron moving in a spiralin medium / A.V. Konstantinovich, I.A. Konstantinovich // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 5-9. — Бібліогр.: 15 назв. — англ. |
| work_keys_str_mv |
AT konstantinovichav radiationspectrumofanelectronmovinginaspiralinmedium AT konstantinovichia radiationspectrumofanelectronmovinginaspiralinmedium AT konstantinovichav spektrviprominûvannâelektronaŝoruhaêtʹsâvzdovžgvintovoíliniíuseredoviŝi AT konstantinovichia spektrviprominûvannâelektronaŝoruhaêtʹsâvzdovžgvintovoíliniíuseredoviŝi |
| first_indexed |
2025-11-25T22:43:37Z |
| last_indexed |
2025-11-25T22:43:37Z |
| _version_ |
1850569969569366016 |
| fulltext |
Condensed Matter Physics 2007, Vol. 10, No 1(49), pp. 5–9
Radiation spectrum of an electron moving in a spiral
in medium
A.V.Konstantinovich∗, I.A.Konstantinovich
Fedkovych Chernivtsi National University, 2, Kotsyubinsky Str., Chernivtsi, 58012, Ukraine
Received September 4, 2006, in final form December 27, 2006
The action of the Doppler effect on the particularities of radiation power spectral distribution of an electron
moving in a spiral in a medium has been investigated near the Cherenkov threshold. The fine structure of
the electromagnetic radiation spectrum at the Cherenkov threshold and the synchrotron-Cherenkov radiation
spectrum for an electron moving along a spiral with non-relativistic longitudinal component of the velocity
(i.e., the component parallel to the magnetic induction vector) in a transparent isotropic medium have been
obtained and studied.
Key words: synchrotron radiation, synchrotron-Cherenkov radiation
PACS: 41.60.-m, 41.60.Ap, 41.60.Bq, 03.50.-z, 03.50.De
1. Introduction
In case of the charged particles moving in magnetic field, three kinds of radiation are possible
in a medium [1–5]: synchrotron, Cherenkov, and synchrotron-Cherenkov. In papers [5–11] the au-
thors studied the particularities of the synchrotron-Cherenkov radiation spectrum in a transparent
medium with the relativistic longitudinal component of the velocity (the component parallel to the
magnetic induction vector).
The aim of this paper is to investigate the spectral distribution of the radiation power for an
electron moving in magnetic field using the improved Lorentz’s self-interaction method [4,5]. Using
the exact integral relationships for the spectral distribution of the radiation power of an electron
moving along a spiral in a transparent isotropic medium, the fine structure of the electromagnetic
radiation spectrum at the Cherenkov threshold and synchrotron-Cherenkov radiation spectrum was
investigated by means of analytical and numerical methods. The action of the Doppler effect on the
particularities of radiation spectrum of a single electron at its motion in a spiral in a transparent
medium with non-relativistic longitudinal component of the velocity (i.e., the component parallel
to the magnetic induction vector) is investigated.
2. Time-averaged radiation power of charged particles
The time-averaged radiation power P
rad
of charged particles moving in a transparent isotropic
medium is expressed in [4,5] as
P
rad
= lim
T→∞
1
2T
T∫
−T
dt
⎧⎨
⎩
∫
τ
(
�j (�r, t)
1
c
∂ �ADir (�r, t)
∂t
− ρ (�r, t)
∂ϕDir (�r, t)
∂t
)
d�r
⎫⎬
⎭ . (1)
Here �j (�r, t) is the current density and ρ (�r, t) is the charge density. The integration is over the
volume τ . According to the hypothesis of Dirac [12–15], the scalar ϕDir (�r, t) and vector �ADir (�r, t)
potentials are defined as a half-difference of the retarded and advanced potentials.
∗E-mail: aconst@ukr.net
c© A.V.Konstantinovich, I.A.Konstantinovich 5
A.V.Konstantinovich, I.A.Konstantinovich
After substituting the scalar ϕDir (�r, t) and vector �ADir (�r, t) potentials into (1) we obtain the
relationship for the average radiation power of charged particles moving in a transparent isotropic
medium:
P
rad
=
∞∫
0
dω W (ω) , (2)
where
W (ω) = lim
T→∞
1
2T
T∫
−T
dtW (t, ω) , (3)
W (t, ω) =
1
πc2
∞∫
−∞
d�r
∞∫
−∞
d�r′
∞∫
−∞
dt′ωμ(ω)
sin
(
n(ω)
c ω|�r − �r′|
)
|�r − �r′|
× cos {ω (t − t′)}
{
�j (�r, t)�j
(
�r′, t′
)
− c2
n2 (ω)
ρ (�r, t) ρ
(
�r′, t′
)}
. (4)
Here W (ω) is a function of the average spectral distribution of radiation power, W (t, ω) is a
function of the instantaneous spectral distribution of radiation power, μ (ω) is the magnetic per-
meability, n (ω) is the refraction index, ω is the cyclic frequency, and c is the velocity of light in
vacuum.
3. Radiation spectrum of an electron moving along a spiral with a low lon-
gitudinal component of the velocity
The motion law and the velocity of an electron moving in a spiral in a transparent medium are
given by the expressions
�r (t) = r0 cos(ω0t)�i + r0 sin(ω0t)�j + V‖t�k, �V (t) =
d�r (t)
dt
. (5)
Here r0 = V⊥ω−1
0 , ω0 = ceBextẼ−1, Ẽ = c
√
p2 + m2
0c
2, the magnetic induction vector �Bext ‖ 0Z,
V⊥ and V‖ are the components of the velocity, �p and Ẽ are the momentum and energy of the
electron, e and m0 are its charge and rest mass. We obtain the time-averaged radiation power of
an electron after substituting (5) into (2) to (4). Then (see [4,5])
P
rad
=
∞∫
0
dω W (ω) , (6)
W (ω) =
2e2
πc2
∞∫
0
dxωμ (ω)
sin
(
n(ω)
c ωη(x)
)
η(x)
cos (ωx)
{
V 2
⊥ cos (ω0x) + V 2
‖ − c2
n2 (ω)
}
, (7)
where η (x) =
√
V 2
‖ x2 + 4
V 2
⊥
ω2
0
sin2
(ω0
2
x
)
.
After some transformations from relationships (6) and (7) for the longitudinal component of
the velocity V‖ < c/n (ω) the contributions of separate harmonics to the averaged radiation power
of the electron can be expressed as
P
rad
=
e2
c3
∞∑
m=1
∞∫
0
dωμ (ω) n (ω) ω2
π∫
0
sin θdθδ
{
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)
6
Radiation spectrum of an electron moving in a spiral in medium
where q = V⊥
n (ω)
c
ω
ω0
sin θ, Jm (q) and J
′
m (q) are the Bessel function with an integer index and
its derivative, respectively.
For the velocities V‖ < c/n (ω) each harmonics is a set of the frequencies, which are determined
from the solution of the equation
ω
(
1 − n (ω)
c
V‖ cos θ
)
− mω0 = 0. (9)
4. Spectral distribution of radiation power of an electron moving along a
spiral near the Cherenkov threshold
Let us consider some partial case when ε = const and μ = 1, i.e. the low-frequency spectral
range is under investigation. The spectral distribution of radiation power W (ω) in transparent
isotropic medium at the velocity absolute value Vmed = c/n = 0.2306 ·1011 cm/s at Bext = 1Gs,
μ = 1, n = 1.3 is presented in figures 1 and 2. These calculations were carried out by using the
relationship (7).
Figure 1. Spectral distribution of radiation
power of an electron moving in a spiral in a
medium at the velocity absolute value Vmed =
c/n = 0.2306 · 1011 cm/s at low harmonics at
Bext = 1Gs, μ = 1, n = 1.3, V⊥med = 0.2301 ·
1011 cm/s, V‖med = 0.15 · 1010 cm/s, r01 =
2048 cm, ω01 = 0.1124 · 108 rad/s, c = 0.2998 ·
1011 cm/s, P int
med1 = P
rad
1 = 0.2629·10−13 erg/s.
Figure 2. Spectral distribution of radiation
power of an electron moving in a spiral in a
medium at the velocity absolute value Vmed =
c/n = 0.2306 · 1011 cm/s at high harmonics at
Bext = 1Gs, μ = 1, n = 1.3, V⊥med =
0.2301 · 1011 cm/s, V‖med = 0.15 · 1010 cm/s,
r02 = 2048 cm, ω02 = 0.1124 · 108 rad/s, c =
0.2998·1011 cm/s, P int
med2 = 0.2085·10−12 erg/s.
For the case Vmed = c/n = 0.2306 · 1011 cm/s at the Cherenkov’s threshold the overlapping
between the harmonics was studied. As one can see in figure 1, the overlapping between the
harmonic shifts to the higher harmonics with a decreasing longitudinal component of the velocity.
At higher harmonics the maxima in the spectral distribution of the radiation are caused mainly
by the overlapping between the neighbouring mth and (m + 1)th harmonics (see figures 1 and 2)
as well as by some contributions of other harmonics.
The spectral distribution of the synchrotron-Cherenkov radiation power in transparent medium
above the Cherenkov threshold (a velocity absolute value Vmed > c/n) at Bext = 1Gs, μ = 1,
n = 1.3, is presented in figures 3 and 4. The high-accuracy numerical calculations according to
relationships (6) and (7) have shown that the synchrotron-Cherenkov radiation of the electron
moving in a spiral in a medium (μ = 1, n = 1.3) is an unified process [3,5] and made it possible to
obtain the magnitude of radiation power P int
medj = P
rad
j .
7
A.V.Konstantinovich, I.A.Konstantinovich
Figure 3. Spectral distribution of synchrotron-
Cherenkov radiation power at the velocity ab-
solute value Vmed > c/n at low harmonics at
Bext = 1Gs, μ = 1, n = 1.3, V⊥med =
0.24 · 1011 cm/s, V‖med = 0.15 · 1010 cm/s,
r03 = 2285 cm, ω03 = 0.1050 · 108 rad/s,
P int
med3 = 0.3058 · 10−13 erg/s.
Figure 4. Spectral distribution of synchrotron-
Cherenkov radiation power at the velocity ab-
solute value Vmed > c/n at high harmonics at
Bext = 1Gs, μ = 1, n = 1.3, V⊥med =
0.24 · 1011 cm/s, V‖med = 0.15 · 1010 cm/s,
r04 = 2285 cm, ω04 = 0.1050 · 108 rad/s,
P int
med4 = 0.3579 · 10−12 erg/s.
At higher harmonics, the maxima in the spectral distribution of the synchrotron-Cherenkov
radiation are caused mainly by the overlapping between the mth and (m + 1)th harmonics (see
figures 3 and 4) as well as by some contributions of other harmonics.
As one can see in figures 1 to 4 all the calculated spectral dependencies of the radiation power
spectral distribution are of a non-monotonous character near the Cherenkov threshold.
5. Conclusions
For a non-relativistic longitudinal component of the electron velocity, the overlapping between
the harmonics begins at higher harmonics number with a decreasing longitudinal component of
the velocity.
At higher harmonics, the maxima in the spectral distribution of the synchrotron-Cherenkov
radiation are caused mainly by the overlapping between the mth and (m + 1)th harmonics.
The spectral dependence of the radiation power spectral distribution is of a non-monotonous
character near the Cherenkov threshold.
8
Radiation spectrum of an electron moving in a spiral in medium
References
1. Tsytovich V.N. Bulletin of Moskow State University, 1951, No. 11, 27 (in Russian).
2. Konstantinovich A.V., Nitsovich V.M. Izv. Vuzov. Fizika, 1973, No. 2, 59 (in Russian).
3. Schwinger J., Wu-Yang Tsai, Erber T. Ann. Phys., 1976, 96, 303.
4. Konstantinovich A.V., Melnychuk S.V., Rarenko I.M., Konstantinovich I.A., Zharkoy V.P. J. Physical
Studies, 2000, 4, 48 (in Ukrainian).
5. Konstantinovich A.V., Melnychuk S.V., Konstantinovich I.A. Journal of Optoelectronics and Advanced
Matterials, 2003, 5, 1423.
6. Konstantinovich A.V., Melnychuk S.V., Konstantinovich I.A. Romanian Journal of Physics, 2003, 48,
No. 5–6, 717.
7. Konstantinovich A.V., Melnychuk S.V., Konstantinovich I.A. Proceedings of the Romanian
Academy A., 2003, 4, No. 3, 175.
8. Konstantinovich A.V., Konstantinovich I.A. Romanian Journal of Optoelectronics, 2004, 124,
No. 3, 13.
9. Konstantinovich A.V., Melnychuk S.V., Konstantinovich I.A. Romanian Journal of Physics, 2005, 50,
No. 3–4, 347.
10. Konstantinovich A.V., Konstantinovich I.A. Physics and Chemistry of Solid State, 2005, 6, No. 4, 535
(in Ukrainian).
11. Konstantinovich A.V., Melnychuk S.V., Konstantinovich I.A. Journal of Materials Science: Materials
in Electronics, 2006, 17, No. 4, 315.
12. Dirac P.A.M. Proc. Roy. Soc. A, 1938, 167, 148.
13. Sokolov A.A. Bulletin of Moskow State University, 1947, No. 2, 33 (in Russian).
14. Schwinger J. Phys. Rev. 1949, 75, 1912.
15. Konstantinovich A.V., Melnychuk S.V., Konstantinovich I.A. Bulletin of Chernivtsi National Univer-
sity, Physics and Electronics, 2001, No. 102, 5 (in Ukrainian).
Спектр випромiнювання електрона, що рухається вздовж
гвинтової лiнiї у середовищi
А.В.Константинович, I.А.Константинович
Чернiвецький нацiональний унiверситет iм. Юрiя Федьковича,
вул. Коцюбинського 2, 58012, Чернiвцi
Отримано 4 вересня 2006 р., в остаточному виглядi – 27 грудня 2006 р.
Дослiджено вплив ефекту Доплера на особливостi спектрального розподiлу потужностi випромiню-
вання електрона, що рухається вздовж гвинтової лiнiї у середовищi, поблизу черенковського бар’є-
ру. Отримана i дослiджена тонка структура електромагнiтного спектра на черенковському бар’єрi та
спектра синхротронно-черенковського випромiнювання електрона, що рухається вздовж гвинтової
лiнiї у прозорому середовищi, з нерелятивiстською поздовжньою компонентою швидкостi (компо-
нентою, що паралельна вектору магнiтної iндукцiї).
Ключовi слова: синхротронне випромiнювання, синхротронно-черенковське випромiнювання
PACS: 41.60.-m, 41.60.Ap, 41.60.Bq, 03.50.-z, 03.50.De
9
10
|