Radiation power spectral distribution of two electrons moving in magnetic fields

Integral expressions for spectral distributions of the radiation power for systems of non-interacting point charged particles moving on arbitrary trajectory in electromagnetic fields in isotropic transparent media and in vacuum are investigated using the Lorentz self-interaction method. Special atte...

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Дата:	2005
Автори:	Konstantinovich, A.V., Melnychuk, S.V., Konstantinovich, I.A.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 2005
Назва видання:	Semiconductor Physics Quantum Electronics & Optoelectronics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/121648
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Radiation power spectral distribution of two electrons moving in magnetic fields / A.V. Konstantinovich, S.V. Melnychuk, I.A. Konstantinovich // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 2. — С. 70-74. — Бібліогр.: 13 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1216482017-06-16T03:03:12Z Radiation power spectral distribution of two electrons moving in magnetic fields Konstantinovich, A.V. Melnychuk, S.V. Konstantinovich, I.A. Integral expressions for spectral distributions of the radiation power for systems of non-interacting point charged particles moving on arbitrary trajectory in electromagnetic fields in isotropic transparent media and in vacuum are investigated using the Lorentz self-interaction method. Special attention is given to the research of the fine structure of the synchrotron radiation spectral distribution of two electrons spiraling in vacuum in a relativistic case. The spectra of synchrotron, Cherenkov and synchrotron-Cherenkov radiations for a single electron are analyzed. 2005 Article Radiation power spectral distribution of two electrons moving in magnetic fields / A.V. Konstantinovich, S.V. Melnychuk, I.A. Konstantinovich // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 2. — С. 70-74. — Бібліогр.: 13 назв. — англ. 1560-8034 PACS: 41.60.-m, 41.60.Ap, 41.60.Bq, 41.60.Cr, 41.20.-q, 41.20Bt, 03.50.-z, 03.50.Dе http://dspace.nbuv.gov.ua/handle/123456789/121648 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Integral expressions for spectral distributions of the radiation power for systems of non-interacting point charged particles moving on arbitrary trajectory in electromagnetic fields in isotropic transparent media and in vacuum are investigated using the Lorentz self-interaction method. Special attention is given to the research of the fine structure of the synchrotron radiation spectral distribution of two electrons spiraling in vacuum in a relativistic case. The spectra of synchrotron, Cherenkov and synchrotron-Cherenkov radiations for a single electron are analyzed.
format	Article
author	Konstantinovich, A.V. Melnychuk, S.V. Konstantinovich, I.A.
spellingShingle	Konstantinovich, A.V. Melnychuk, S.V. Konstantinovich, I.A. Radiation power spectral distribution of two electrons moving in magnetic fields Semiconductor Physics Quantum Electronics & Optoelectronics
author_facet	Konstantinovich, A.V. Melnychuk, S.V. Konstantinovich, I.A.
author_sort	Konstantinovich, A.V.
title	Radiation power spectral distribution of two electrons moving in magnetic fields
title_short	Radiation power spectral distribution of two electrons moving in magnetic fields
title_full	Radiation power spectral distribution of two electrons moving in magnetic fields
title_fullStr	Radiation power spectral distribution of two electrons moving in magnetic fields
title_full_unstemmed	Radiation power spectral distribution of two electrons moving in magnetic fields
title_sort	radiation power spectral distribution of two electrons moving in magnetic fields
publisher	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
publishDate	2005
url	http://dspace.nbuv.gov.ua/handle/123456789/121648
citation_txt	Radiation power spectral distribution of two electrons moving in magnetic fields / A.V. Konstantinovich, S.V. Melnychuk, I.A. Konstantinovich // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 2. — С. 70-74. — Бібліогр.: 13 назв. — англ.
series	Semiconductor Physics Quantum Electronics & Optoelectronics
work_keys_str_mv	AT konstantinovichav radiationpowerspectraldistributionoftwoelectronsmovinginmagneticfields AT melnychuksv radiationpowerspectraldistributionoftwoelectronsmovinginmagneticfields AT konstantinovichia radiationpowerspectraldistributionoftwoelectronsmovinginmagneticfields
first_indexed	2025-07-08T20:16:44Z
last_indexed	2025-07-08T20:16:44Z
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fulltext	Semiconductor Physics, Quantum Electronics & Optoelectronics. 2005. V. 8, N 2. P. 70-74. © 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 70 PACS: 41.60.-m, 41.60.Ap, 41.60.Bq, 41.60.Cr, 41.20.-q, 41.20Bt, 03.50.-z, 03.50.Dе Radiation power spectral distribution for two electrons moving in magnetic fields A.V. Konstantinovich, S.V. Melnychuk, I.A. Konstantinovich Chernivtsi National University, 2, Kotsyubynsky str., 58012 Chernivtsi, Ukraine Abstract. Integral expressions for spectral distributions of the radiation power for systems of non-interacting point charged particles moving on arbitrary trajectory in electromagnetic fields in isotropic transparent media and in vacuum are investigated using the Lorentz self-interaction method. Special attention is given to the research of the fine structure of the synchrotron radiation spectral distribution of two electrons spiraling in vacuum in a relativistic case. The spectra of synchrotron, Cherenkov and synchrotron- Cherenkov radiations for a single electron are analyzed. Keywords: Cherenkov radiation, synchrotron radiation, synchrotron-Cherenkov radiation, Lorentz self-interaction. Manuscript received: 10.02.05; accepted for publication 18.05.05. 1. Introduction Investigations of the radiation spectra of charged particles moving in magnetic fields in transparent isotropic media and vacuum are important from the viewpoint of their applications in electronics, astrophysics, plasma physics, physics of storage rings, etc. [1–3]. When charged particles move in magnetic field, three kinds of radiation are possible in a medium [4-5]: synchrotron, Cherenkov, and synchrotron- Cherenkov ones whereas in vacuum only synchrotron radiation takes place. A question calling for further investigations is the coherence of synchrotron radiation [6-7]. Investigations of the fine structure of synchrotron, Cherenkov, and synchrotron-Cherenkov radiation spectra in vacuum and transparent media for the low-frequency spectral range are of great interest, too [4-7]. Using the exact integral relationships for the spectral distribution of radiation power of two electrons spiraling one after another in vacuum, the fine structure of the synchrotron radiation spectrum in relativistic case was investigated by analytical and numerical methods. The Doppler effect influence on peculiarities of the radiation spectrum of a single electron spiraling in transparent media and vacuum is investigated. 2. Instantaneous and time-averaged radiation powers of charged particles The instantaneous radiation power of charged particles ( )tP rad in an isotropic transparent medium and in vacuum is expressed in [8, 9] as ( ) =tP rad ( ) ( ) ( ) ( ) rd t trtr t trA c trj DirDir r r r rr rr∫ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −= τ ∂ ∂ϕρ ∂ ∂ ,,,1, . (1) Here ( )trj ,r r is the current density and ( )tr ,rρ is the charge density. The integration is over some volume τ . According to the hypothesis of Dirac [8-10], the scalar ( )trDir ,rϕ and vector ( )trADir ,r r potentials are defined as a half-difference of the retarded and advanced potentials: ( )advretDir ϕϕϕ −= 2 1 , ( )advretDir AAA rrr −= 2 1 . (2) After substituting (2) into (1) we obtain the relationship for instantaneous radiation power of charged particles moving in isotropic transparent media as a function of spectral distribution ( ) ( )ωω , 0 tWdtPrad ∫ ∞ = , (3) ( ) ( )∫ ∫∫ ∞ ∞− ∞ ∞− ∞ ∞− ×′′= ωωμ π ω tdrdrd c tW rr 2 1, ( ) ( )×′− ′− ⎥⎦ ⎤ ⎢⎣ ⎡ ′− × tt rr rr c n ω ωω cos sin rr rr (4) ( ) ( ) ( ) ( ) ( ) ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ′′−′′× trtr n ctrjtrj ,,,, 2 2 rrrrrr ρρ ω , Semiconductor Physics, Quantum Electronics & Optoelectronics. 2005. V. 8, N 2. P. 70-74. © 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 71 where ( )ωμ is the magnetic permeability, ( )ωn is the refraction index, ω is the cyclic frequency, and c is the velocity of light in vacuum. The time-averaged radiation power of charged particles is defined by the expression ( )dttP T P T T rad T rad ∫ − ∞→ = 2 1lim . (5) It can be obtained after substitution of the instantaneous radiation power expressed by relation- ships (3) and (4) into (5). 3. Systems of non-interacting point charged particles According to [4, 5], the source functions of N charged point particles are defined as ( ) ( ) ( )∑ = = N l ll trtVtrj 1 ,, rrrr ρ , ( ) ( )∑ = = N l l trtr 1 ,, rr ρρ , ( ) ( )( )trretr ll rrr −= δρ , , (6) where ( )trl r and )(tVl r are the motion law and the velocity of the thl particle, respectively. Let us consider a system of point non-interacting charged particles ( eql = , 00 mm l = ) moving one by one along an arbitrary defined trajectory. Then the motion law and the velocity of the thj particle of this system are determined by the relationships ( ) ( )jpj ttrtr Δ+= rr , ( ) ( )jj ttVtV Δ+= rr . (7) Substituting the relationships (6) and (7) into (3) and (5) we obtain the expression for the averaged of charged particles system in transparent media (magnetic permeability ( )ωμ and dielectric permittivity ( )ωε are real): ( ) ( )×′= ∫∫∫ ∞∞ ∞−− ∞→ ωωωωμ π N T T T rad Sdtddt Tc eP 0 2 2 2 1lim ( ) ( ) ( ) ( ) ( ) ( )×′− ′− ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ′− × tt trtr trtr c n pp pp ω ωω cos sin rr rr (8) ( ) ( ) ( )⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ −′× ω2 2 n c tVtV rr , ( ) ( ){ }∑ = Δ−Δ= N jl jlN ttS 1, cos ωω . (9) The coherence factor ( )ωNS determines a redistribution of the charged particles radiation power between harmonics. We study N electrons spiraling one by one in magnetic fields in transparent media. The law of motion and the velocity of the j-th electron are given by the expressions ( ) ( ){ } +Δ+= ittrtr jj rr 00 cos ω ( ){ } ( )jj ttVjttr Δ++Δ++ \|\|00 sin r ω , ( ) ( ) dt trd tV j j r r = . (10) Here 1 00 − ⊥= ωVr , 1 0 ~−= EceBextω , 22 0 2~ cmpcE += , the magnetic induction vector extB s \|\|0Z, ⊥V and \|\|V are the components of the velocity, pr and E~ are the momentum and energy of the electron, e and 0m are its charge and rest mass. We obtain the time-averaged radiation power of N electrons on substitution of the expressions (10) into (8). Then ( )∫ ∞ = 0 ωω dWP rad , (11) ( ) ( ) ( ) ( ) ( ) ( ) ×⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = ∫ ∞ x x c n Sdx c eW N η ωηω ωωωμ π ω sin 2 0 2 2 ( ) ( )⎥⎥⎦ ⎤ ⎢ ⎢ ⎣ ⎡ −+× ⊥ ω ωω 2 2 2 \|\|0 2 coscos n cVxVx , (12) where ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛+= ⊥ xVxVx 2 sin4 02 2 0 2 22 \|\| ω ω η . The coherence factor ( )ωNS of N electrons is defined as (9). 4. Fine structure of the radiation spectra of two electrons moving along a spiral in vacuum Peculiarities of the radiation spectra of two electrons moving one by one in a spiral in vacuum can be invest- tigated combining analytical and numerical methods. The time-averaged radiation power of two electrons we can obtain from expressions (11) and (12). Then ( )∫ ∞ = 0 ωω dWP rad , (13) ( ) ( ) ( ) ( ) ×⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = ∫ ∞ x x cSdx c eW η ωη ωω π ω 1sin 2 2 0 2 2 ( )[ ]22 \|\|0 2 coscos cVxVx −+× ⊥ ωω , (14) where ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛+= ⊥ xVxVx 2 sin4 02 2 0 2 22 \|\| ω ω η . The coherence factor ( )ω2S of two electrons is defined as ( ) ( )tS Δ+= ωω cos222 . (15) Semiconductor Physics, Quantum Electronics & Optoelectronics. 2005. V. 8, N 2. P. 70-74. © 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 72 Here 12 ttt Δ−Δ=Δ is the time shift of the electrons moving along a spiral. The analogous expression for the coherence factor was investigated by Bolotovskii [11]. From relationships (13) and (14) on some transformations the contributions of separate harmonics to the averaged radiation power can be written as ( ){ }[ ]×Δ+= ∫∑∫ ∞ = ∞ tdd c eP m rad ωθθωω π cos12sin 0 2 1 0 3 2 × ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ −⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −× 0\|\| cos11 ωθωδ mV c (16) ( ) ( ) ( ) ( ) ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ −+ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ′+× ⊥ qJсVqJqJ q mV mmm 222 \|\| 22 2 2 2 , where θ ω ω sin 0c Vq ⊥= , ( )qJ m and ( )qJ m′ are the Bessel function with integer index and its derivative, respectively. Each harmonic is a set of the frequencies, which are the solution of the equation 0cos11 0\|\| =−⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ωθω mV c . (17) The limits of the thm harmonic are determined by the frequencies c V m m \|\| 0min 1+ = ω ω , c V m m \|\| 0max 1− = ωω , (18) and the total radiation power emitted by a separate electron is determined according to [12] as 2 2 2 22 03 2 1 3 2 − ⊥ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −= c VV c ePtot vac ω , (19) where 2 2 0 0 1 c V cm eBext −=ω . Our numerical calculations of the radiation power spectral distribution were performed at 1ext =B Gs. For the velocities components vac⊥ V = 0.4713c and vac\|\| V = 0.3333c the radiation power spectral distri- butions of two electrons in dependence of their location along a spiral are shown in Fig. 1 (curves 1–3) and magnitudes of radiation power are presented in Table 1. It is interesting to compare the radiation power spectral distribution for two electrons with that of a single electron (curve 0 in Fig. 1a). The radiation power of a single electron in vacuum 15tot vac0 10529.0 −⋅=P erg/s calculated according to the relationship (19) is in good agreement with the power 15int vac0 10526.0 −⋅=P erg/s xetermined on integration of the relationships (13) and (14). For the time difference 011 /001.0 ωπ=Δt (curve 1 in Fig. 1a) the coherence factor ( ) 42 =ωS , and two electrons radiate as a charged particle with the charge e2 and the rest mass 02m , i.e., by a factor of four more intensively than the single electron. For the time difference 022 /ωπ=Δt (curve 2 in Fig. 1b), we have found the peaks of the spectral distribution function appoximatelly at the frequencies 022 ωi , i = 1, 2, 3, 4 whereas the radiation was absent at the frequencies ( ) 0212 ω−i , i = 1, 2, 3, 4. Fig. 1. Spectral distribution of radiation power for a single electron (curve 0) and two electrons spiraling one by one at: a) 011 /001.0 ωπ=Δt (curve 1); b) 022 /ωπ=Δt (2); c) =Δ 3t 03/2 ωπ= (3). a b c Semiconductor Physics, Quantum Electronics & Optoelectronics. 2005. V. 8, N 2. P. 70-74. © 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 73 For the time difference 033 /2 ωπ=Δt (curve 3 in Fig. 1c), we have found the peaks of the spectral distribution function appoximatelly at the frequencies 03ωi , i = 1, 2, …, 8, and the radiation was absent at the frequencies 032 1 ω⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + i , i = 1, 2, …, 8. The dependence of the radiation power magnitude for two electrons moving one by one in dependence of their location in a spiral is presented in Fig. 2. With increasing Δt, the radiation power of the system of two charges tends to double radiation power of a single charge. 5. Spectral distribution of synchrotron-Cherenkov radiation power in low-frequency range Let us consider the Doppler effect influence on syn- chrotron-Cherenkov radiation in transparent media. The expressions for the synchrotron-Cherenkov radiation power in such a medium can be obtained starting from (13). Then for the single electron spiraling we have found [3, 5] ( )∫ ∞ = 0 rad ωω dWP , (20) ( ) ( ) ( ) ( ) ( ) ×⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = ∫ ∞ x x c n dx c eW η ηωω ωωμ π ω sin 2 0 2 2 ( ) ( ) ( )⎥⎥⎦ ⎤ ⎢ ⎢ ⎣ ⎡ −+× ⊥ ω ωω 2 2 2 \|\|0 2 coscos n cVxVx , (21) where ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ += ⊥ x V xVx 2 sin4 02 2 0 2 22 \|\| ω ω η . For the case of low-frequency spectral range, i.e., at const=ε and 1=μ , the power of the Cherenkov radiation at rectilinear motion in a medium (n is the constant) is determined as [5]: ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −= 22 2 2 max2 2 tot ch 1 2 nV cV c eP ω . (22) For the refraction index n = 2 at the velocities V⊥med = 0.15. 1010 cm/s = 0.05003c and =med\|\|V 11101493.0 ⋅= cm/s = c4981.0 as well as =⊥medV 9109.0 ⋅= cm/s c03002.0= and 11 med\|\| 101498.0 ⋅=V cm/s c4997.0= (curves 5 and 6 in Fig. 3, respectively) the conditions for the existence of synchrotron-Cherenkov radiation are fulfilled. The spectral distribution for these two cases is shown in Fig. 3. The upper boundary of the first harmonic band in curve 5 is located at the frequency 5max 1ω = 265ω05 and for curve 6 it is at the frequency 6max 1ω = 1852ω06. The values of the synch- rotron-Cherenkov radiation power are listed in Table 2. The power of the Cherenkov radiation at rectilinear motion 11tot ch7 106979.0 −⋅=P erg/s (relation (22)) is in good agreement to the synchrotron-Cherenkov radiation Fig. 2. Radiation power of two electrons moving one by one vs their location in a spiral for vac⊥ V = 0.4713c and vac\|\| V = = 0.3333c. Fig. 3. Spectral distribution of synchrotron-Cherenkov radia- tion power at n = 2. Curve 5 – med⊥V = 0.05003c and \|med\|V = = 0.4981c, curve 6 – med⊥V = 0.03002c and \|med\|V = 0.4997c, curve 7 – med⊥V = 0.00033c and \|med\|V = 0.5006c. Table 1. Radiation power for two electrons moving one by one in a spiral in relativistic case (Bext = 1 Gs, ω0j = 14.36⋅106 rad/s, r0j = 984 cm, j = 1, 2, 3, c = 2.997925⋅1010 cm/s). Curve j jtΔ vacV⊥ vacV\|\| ,int vacjP 1510− erg/s 1 01 001.0 ω π c⋅47.0 c⋅33.0 2.113 2 02/ωπ c⋅47.0 c⋅33.0 0.7742 3 03/2 ωπ c⋅47.0 c⋅33.0 1.046 Semiconductor Physics, Quantum Electronics & Optoelectronics. 2005. V. 8, N 2. P. 70-74. © 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 74 power 11int 7med 106992.0 −⋅=P erg/s calculated according to the relationships (20) and (21) at the motion of the charged particle having a small transverse velocity com- ponent (the absolute values of the velocities are the same). The synchrotron-Cherenkov radiation is the unified process with interesting properties [2-7, 13]. The analytical and numerical calculations showed that the Doppler effect influence on the peculiarities of the radiation power spectral distribution of the electrons were essential near the Cherenkov threshold. 6. Conclusions In the radiation spectrum of charged particles the Doppler effect establishes the limits between the bands of separate harmonics. The synchrotron-Cherenkov radiation is the unified process with interesting properties. The influence of the Doppler effect on the peculiarities of the spectral distribution of the electron radiation power in a medium is significant only nearby the Cherenkov threshold. References 1. I.M Ternov, Synchrotron radiation // Usp. Fiz. Nauk, 165 (4), p.429-456 (1995) (in Russian). 2. A.V. Konstantinovich, S.V. Melnychuk, I.M. Raren- ko, I.A. Konstantinovich, V.P. Zharkoi, Radiation spectrum of the system of charged particles moving in nonabsorbable isotropic medium // J. Physical Studies, 4 (1), p.48-56 (2000) (in Ukrainian). 3. A.V. Konstantinovich, S.V. Melnychuk, I.A. Kons- tantinovich, Fine structure of radiation spectrum of charged particles moving in magnetic fields in non absorbable isotropic media and in vacuum // Roma- nian Journal of Physics, 48 (5-6), p. 717-725 (2003). 4. J. Schwinger J., Tsai Wu-yang, T. Erber, Classical and quantum theory of synergic synchrotron- Cherenkov radiation // Ann. of Phys., 96 (2), p.303– 332 (1976). 5. I.A. Konstantinovich, S.V. Melnychuk, A.V. Kons- tantinovich., Classical radiation theory of charged particles. II.Medium influence on the radiation spectrum of charged particles moving in electromagnetic fields // Bulletin of Chernivtsi National University. Physics and Electronics, 132, p. 24-39 (2002) (in Ukrainian). 6. A.V. Konstantinovich, S.V. Melnychuk, I.A. Kons- tantinovich, Radiation spectra of charged particles moving in a spiral in magnetic fields // Proceedings of the Romanian Academy. A, 4(3), p. 175-182 (2003). 7. A.V. Konstantinovich, S.V. Melnychuk, I.A. Kon- stantinovich, Radiation power spectral distribution of electrons moving in a spiral in magnetic fields // Journal of Optoelectronics and Advanced Materials, 5(5), p. 1423-1431 (2003). 8. A.V. Konstantinovich, S.V. Melnychuk, I.A. Kons- tantinovich, Classical radiation theory of charged particles. I.Retarded and advanced potentials and electromagnetic fields intensites and the Lorentz’s self-interaction method // Bulletin of Chernivtsi National University. Physics and Electronics, 102, p. 5-13 (2001) (in Ukrainian). 9. J. Schwinger, On the classical radiation of accelerated electrons // Phys. Rev., 75(12), p. 1912- 1925 (1949). 10. P.A.M. Dirac, Classical theory of radiating elect- rons// Proc Roy. Soc. A., 167 (1), p.148–169 (1938). 11. B.M. Bolotovskii, The theory of the Vavilov- Cherenkov effect // Usp. Fiz. Nauk., 62(3), p. 201- 246 (1957) (in Russian). 12. A.A. Sokolov, V.Ch. Zhukovskii, M.M. Kolesniko- va, N.S. Nikitina, O.E. Shishanin, On the synchrot- ron radiation of the electron moving in a spiral // Izv. Vuzov. Fizika, No 2, p.108–116 (1969.). 13. V.N. Tsytovich, On the radiation of the rapid electrons in the magnetic field in the presence of medium // Bulletin of Moscow State University, 11, p.27-36 (1951) (in Russian). Table 2. Synchrotron-Cherenkov radiation power of the electron at Bext = 1 Gs, (n = 2, Bext = 1 Gs, ω0j = 15.23⋅106 rad/s). Curve j med⊥V med\|\|V jr0 , cm 11int med 10, − jP erg/s 5 c05003.0 c4981.0 98.5 0.4688 6 c03002.0 c4997.0 59.1 0.5266 7 c00033.0 c5006.0 0.6 0.6992

Radiation power spectral distribution of two electrons moving in magnetic fields

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