Оn synchrotron radiation polarization from runaway electrons in toroidal magnetic fields

The formation of synchrotron radiation polarization of runaway electrons is studied. The radiated power and directions of polarization in the synchrotron radiation spot in the dependence of pitch angles of emitting electrons are estimated. The polarization measurements can give additional diagnost...

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Дата:	2018
Автор:	Sobolev, Ya.M.
Формат:	Стаття
Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2018
Назва видання:	Вопросы атомной науки и техники
Теми:	Термоядерный синтез (коллективные процессы)
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/147343
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Цитувати:	Оn synchrotron radiation polarization from runaway electrons in toroidal magnetic fields / Ya.M. Sobolev // Вопросы атомной науки и техники. — 2018. — № 4. — С. 97-101. — Бібліогр.: 14 назв. — англ.

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spelling	irk-123456789-1473432019-02-15T01:24:25Z Оn synchrotron radiation polarization from runaway electrons in toroidal magnetic fields Sobolev, Ya.M. Термоядерный синтез (коллективные процессы) The formation of synchrotron radiation polarization of runaway electrons is studied. The radiated power and directions of polarization in the synchrotron radiation spot in the dependence of pitch angles of emitting electrons are estimated. The polarization measurements can give additional diagnostics for the radiating relativistic electrons. Розглянуто умови формування поляризації синхротронного випромінювання утікаючих електронів. Надані оцінки потужності випромінювання і напрями поляризації в плямі синхротронного випромінювання утікаючих електронів у залежності від пітч-вуглів випромінюючих електронів. Вимірювання поляризації синхротронного випромінювання може слугувати додатковим засобом для діагностики релятивістських електронів. Рассмотрены условия формирования поляризации синхротронного излучения убегающих электронов. Оценена излучаемая мощность и направления поляризации в пятне синхротронного излучения убегающих электронов в зависимости от питч-углов излучающих электронов. Измерение поляризации синхротронного излучения может быть дополнительным средством для диагностики излучающих релятивистских электронов. 2018 Article Оn synchrotron radiation polarization from runaway electrons in toroidal magnetic fields / Ya.M. Sobolev // Вопросы атомной науки и техники. — 2018. — № 4. — С. 97-101. — Бібліогр.: 14 назв. — англ. 1562-6016 PACS: 41.60.Ap, 52.55.Fa, 97.60.Gb http://dspace.nbuv.gov.ua/handle/123456789/147343 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Термоядерный синтез (коллективные процессы) Термоядерный синтез (коллективные процессы)
spellingShingle	Термоядерный синтез (коллективные процессы) Термоядерный синтез (коллективные процессы) Sobolev, Ya.M. Оn synchrotron radiation polarization from runaway electrons in toroidal magnetic fields Вопросы атомной науки и техники
description	The formation of synchrotron radiation polarization of runaway electrons is studied. The radiated power and directions of polarization in the synchrotron radiation spot in the dependence of pitch angles of emitting electrons are estimated. The polarization measurements can give additional diagnostics for the radiating relativistic electrons.
format	Article
author	Sobolev, Ya.M.
author_facet	Sobolev, Ya.M.
author_sort	Sobolev, Ya.M.
title	Оn synchrotron radiation polarization from runaway electrons in toroidal magnetic fields
title_short	Оn synchrotron radiation polarization from runaway electrons in toroidal magnetic fields
title_full	Оn synchrotron radiation polarization from runaway electrons in toroidal magnetic fields
title_fullStr	Оn synchrotron radiation polarization from runaway electrons in toroidal magnetic fields
title_full_unstemmed	Оn synchrotron radiation polarization from runaway electrons in toroidal magnetic fields
title_sort	оn synchrotron radiation polarization from runaway electrons in toroidal magnetic fields
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2018
topic_facet	Термоядерный синтез (коллективные процессы)
url	http://dspace.nbuv.gov.ua/handle/123456789/147343
citation_txt	Оn synchrotron radiation polarization from runaway electrons in toroidal magnetic fields / Ya.M. Sobolev // Вопросы атомной науки и техники. — 2018. — № 4. — С. 97-101. — Бібліогр.: 14 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT sobolevyam onsynchrotronradiationpolarizationfromrunawayelectronsintoroidalmagneticfields
first_indexed	2025-07-11T01:54:47Z
last_indexed	2025-07-11T01:54:47Z
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fulltext	ISSN 1562-6016. ВАНТ. 2018. №4(116) 97 ON SYNCHROTRON RADIATION POLARIZATION FROM RUNAWAY ELECTRONS IN TOROIDAL MAGNETIC FIELDS Ya.M. Sobolev Radio Astronomy Institute NASU, Kharkov, Ukraine E-mail: sobolev@rian.kharkov.ua The formation of synchrotron radiation polarization of runaway electrons is studied. The radiated power and directions of polarization in the synchrotron radiation spot in the dependence of pitch angles of emitting electrons are estimated. The polarization measurements can give additional diagnostics for the radiating relativistic electrons. PACS: 41.60.Ap, 52.55.Fa, 97.60.Gb INTRODUCTION Synchrotron radiation provides information about the runaway electrons in tokamaks. Polarization meas- urements will give extra features. In [1 - 3], the synchrotron radiation from runaway electrons in tokamaks was observed. The synchrotron spectra of runaway electrons in tokamak were analyzed in [4, 5]. In [6, 7] the polarization pattern of the synchro- tron radiation of runaway electrons in tokamak has been calculated. Analysis of experimental data has shown that the de- scription of the synchrotron radiation of runaway elec- trons in tokamaks is the case when you need to take into account the curvature of magnetic field lines [8 - 10]. In this paper, we will continue to study the polariza- tion of synchrotron radiation in a toroidal magnetic field configuration taking more attention to forming the po- larization pattern. The topology of toroidal magnetic fields, the elec- tron energies take values that are typical for medium- size tokamaks. The nested circle tori for magnetic fields and drift trajectories are considered. The compact formulas that describe the coordinates of the radiation points are given. The synchrotron radia- tion formulae for an electron moving along a circle are used to describe the radiation of ultrarelativistic elec- trons from a small arc of its trajectory. The mechanism of the appearance of linear polarization in the synchro- tron radiation spot is elucidated. The paper is organized as follows. The formulas for coordinates of particles emitting to the detector are giv- en in Section 2. The contribution to radiation from two different regions on the torus is described. The merging of these two areas into one is described. In Section 3, the formulae for calculating the direc- tions of polarization and power of synchrotron radiation from electrons with drift trajectories on the torus are considered. In Section 4, the polarization and power in synchro- tron radiation spot is calculated for the range of pitch angles of the runaway electrons. 1. TOROIDAL MAGNETIC FIELDS AND ELECTRON TRAJECTORIES Cartesian coordinates are taken as shown in Fig. 1. The torus is described with coordinates ( )ϕϑ,,r : coor- dinates ( )ϑ,r with centre at major radius R and the toroidal angle ϕ , ϕϑ eee ,,r are the corresponding orts, [ ]ϑϕ eee ,r= . Magnetic surfaces and drift surfaces have a toroidal topology. The major radius of nested magnet- ic surfaces is 0RR = , the major radius of drift surfaces is δ+= 0RR . So, there are two coordinate systems ( )ϕϑ,,r and ( )ϕϑ ,, ffr corresponding to drift and magnetic torus, respectively. Fig. 1. Coordinate systems. Cartesian coordinates ( )zyx ,, of the emitting electron; coordinates of the torus ( )ϕϑ,,r ; cylindrical coordinates ( )z,,ϕρ , ϑρ cosrR −= ; cartesian coordinates of the detection point ( )ZYD ,,− ; Larmor’s circumference and trihedron orts of the drift trajectory DDD bντ ,, (bottom right corner) Suppose, the toroidal magnetic field ϕB and plasma current I are clockwise, radiating electrons are moving counterclockwise. Magnetic fields take the form [11, 12] ( ) ( )         + − − = ff f f ff ff Rrq r Rr Br ϕϑϑ ϑ eeB 00 0 cos1 , , (1) here ffr ϑ, are the radius and poloidal angle of magnet- ic surface, ( )frq the safety factor. Suppose that the safety factor of the magnetic field lines ( )frq is equal that of the particle drift orbits ( )rqD when equal radii of magnetic field line surface fr and drift surface r are considered [1]. The guiding center is moving along drift trajectory with speed \|\|v . The electron pitch angle is (–D,Y,Z) -D x y (x,y,z) J j n v R z bD νD Θ rB ISSN 1562-6016. ВАНТ. 2018. №4(116) 98 \|\|vv⊥≈α 1<< , with respect to the drift trajectory, where BBrv ω=⊥ , ( )mceBB γω = , ( )ϑ,rBB = the magnetic field at the point with drift coordinates ϑ,r , 1>>γ is the Lorentz factor. The electron velocity vector is given by [7] ( )DDD vv bντv Θ−Θ−+= ⊥ cossin\|\| , (2) here DDD bντ ,, are the tangent, normal, and binormal to the drift path, the angle Θ is measured from the normal Dν to the direction of vector Db− , Bω=Θ , Fig. 1. We note that the position of the electron on the Lar- mor circle is described by an angle Θ , which is meas- ured from the direction of the normal Dν to the drift trajectory. 1.1. RADIATION POINTS The detector is located at the point OP with Carte- sian coordinates ( )ZYD ,,− , Fig. 1. The line of sight is directed from the detection point OP to the radiation point eP , denote unit vector in the direction eO PP as n . The high energetic electrons emit their synchrotron ra- diation practically along their velocity vector (2), Fig. 1. The coordinates of radiation point are founded out after equating components zy nn −− , to the directional cosines of velocity vector, i. e., equations 0=+ vnv are solved. For any ϑ,r the third coordinate of radiation point takes a value ϕπϕ ∆+= 2/ , where ϕ∆ is the first- order correction with respect to 1<<Rr , 1<<α . The electron position in Larmor circle, angle Θ , is given by [6, 7] ( ) ( ) D Z Rrq r D α ϑϑ ϑα −−=Θ 0 0 cos cos cos , (3) where ( )22 0cos RqDD D+=ϑ . From equation 0=+ yy vnv follows1 ( ) D YR Rq r D − +Θ+ − =∆ sin cos sin 0 0 α ϑ ϑϑϕ , (4) here ( ) ( ) ( ) 122 22 0 11 1 + −− += nD ar arqnrq models the safe factor of drift path, 7...1=n , a is the small tokamak radius [11]. Larmor radius Br is not taken into account because of its smallness. Formulas (3) and (4) can be illustrated by visual ge- ometric constructions. The formulae (3), (4) are consistent with eqs. (9), (10) in [12], only you need to pay attention to the fact that our angle Θ is measured from the normal to the drift trajectory. As, the unit vector Dν has almost constant orientation along the unit vector y−e , this simplifies calculations. Further, we assume 0=Z , 0=− RY . 1In [7, eq. 5] should be ( ) ϑsinRqr D instead of ( ) .cosϑRqr D The range of angles ϑ depends on pitch-angles α . As can see from eq. (3), there are constraints on ac- ceptable angles ϑ when ( )0cosϑα DDRqr< , Fig. 2. The angle 0ϑ specifies the direction of the line AA ′ to the vertical axis ( Z -axis), Fig. 2. Using (3), we obtain 2π=Θ at the point ( )2, 0 πϑ +rA . At this point, the drift velocity vector is directed to the detector (the projection to the ( )zx, plane, more correctly). Suppose, the radiation point is in the line OC then the point C has coordinates ( )00 ,ϑr . There are not radi- ation points for 0rr > along this direction. From eq. (3) we obtain ( ) 000 cosϑα Rrqr D= . The larger pitch-angles α give larger values of the radius 0r , (the dash-dotted curve in Fig. 2) Fig. 2. Scheme of the synchrotron radiation spot. Z=0, R=Y If 0=α the radiation spot is converted into line AA ′ , which has an angle 20 πϑϑ +=A , where DqRA == ϑϑ ctgtg 0 , [12]. There are two solutions of eq. (3), i. e. Θ± . Substituting Θ=Θ1 and Θ−=Θ2 into eq. (4), we obtain azimuthal coordinates 1ϕ and 2ϕ of two radiation points. The angular difference between these points is ϕϕϕ ∆=− 221 , ( ) ( ) ( )0 2 2 0 2 0 cos122 ϑϑαϕ −            −=∆ r r rq rq D D . The maximum value of the angle ϕ∆2 is reached on the line AA ′ whenever 2π±=Θ , then αϕ =∆ . Along the dashed line in Fig. 2, as follows from (4) for ( ) constrqD = , we obtain ( )201122 rr−=∆ αϕ , where ( )001 ,rrr −∈ is the minimum distance between the dashed line and the point O . ( )2112 beambeam rrrl −≈ is the dashed line length, beamr is the radius of electron beam. z y r1 r0 R A′ B A B′ D′ O C ϑ0 D ϑ0 Θ=0 Θ=π/2 ISSN 1562-6016. ВАНТ. 2018. №4(116) 99 There is one radiation point at lines BB ′ and CC ′ , 0=∆ϕ for these lines of sight. The line BB ′ is de- scribed by ( ) ( ) ( )000cos rqrrqr =−ϑϑ as it follows from eq. (3). The distance 10 rrr −=∆ from the boundary line BB ′ to the point, for which the angular distance be- tween the radiation points decreases to γ1 , γϕ 1~∆ , can be estimated as ( )205.0 αγrr =∆ . The ultrarelativistic electron radiates into a narrow cone along the instantaneous velocity vector, the angu- lar width of the cone is of the order γ1 . From the vec- tor product, [ ] γ1~/, zvvn− , we can estimate that cor- rections to the rhs of (4) are of the order γ1 . The electron passes the distance BSC vl ωπ /2\|\|≈ in one revolution around magnetic field line. This distance corresponds to an angle ( )Rv BSC ωπα \|\|2≈ . Then, the ratio of the angle between the radiation points to the angle SCα is \|\|sin2 vRq BSCSC παωαϕ Θ=∆= , or πΘ= sinaSC qq , where aq is given by (7). 2. SYNCHROTRON RADIATION FORMULAE The synchrotron radiation of an ultrarelativistic elec- tron moving along a circle is formed on a small segment of circle with an angular opening γ2 . For an ensemble of electrons we integrate the spectral angular distribu- tion over the solid angle assuming that the magnetic field is not strong anisotropic in angles γ1~ [13, 1]. The spectral power density of synchrotron radiation emitted by an relativistic electron moving along a circu- lar path with a curvature radius curvR at wavelength λ is expressed by [13, 14] ( ) ( ) ( )         ±= ∫ ∞ yKdxxKce d dP y i 3/23/523 2 3 2 γλ π λ λ , (5) where λλcy = , ( )334 γπλ curvc R= is the characteris- tic wavelength, πσ ,=i denote the cases of π and σ - polarization, “plus” corresponds to σ -polarization, “minus” corresponds to π -polarization, [ ]kee ,πσ = is the unit vector of σ -polarization, k is the unit wave vector, νK is the Macdonald function. Linearly polar- ized radiation is described by eq. (5) [13]. As shown in [8 - 10], the curvature radius of the tra- jectory, which is formed by the motion of an electron with guiding center on a circle of radius R ( DR for the trajectory (2)), is equal to 2cos21 aa D curv qq RR +Θ− = , (6) where Θcos is given by eq. (3), \|\|vRq curvBa αω= . At each point of radiation, there are electrons with ( )π2,0∈Θ , but radiation is detected from the electron with a definiteΘ . This angle defines the direction of acceleration of the radiating electron. In this case, the radius of curvature is greater on the side of the small circle closer to the center of the large circle of torus. The parameter aq is defined as the quotient of the acceleration of the particle motion in the small Larmor circle to the centrifugal acceleration due to movement in the larger circumference of radius equal to the curvature radius of drift trajectory. Also, it is the ratio of Larmor’s speed BBrω to the speed of centrifugal drift. Available coordinates ( )ϕϑ,,r of the guiding center of electron are given by (3), (4). At the point, the curva- ture radius ( )ϕϑ,,rRD of the drift trajectory is calculat- ed. Further, the value ( )ϑ,rB of magnetic field is found at this point. The parameter aq can be written as ( ) ( ) γ αϑϑ ,,2 rRrB mc eq Dffa = , (7) where ϑδ cos11 20 r q RR D D         +−++= is the curvature radius of drift trajectory in a first-order approximation. To calculate the magnetic field at the point      = 2 ,, πϑrPe the displacement δ in equatorial plane of the drift torus with respect to magnetic torus is taken into account, then ϑδδ cos222 rrrf −+= and f f r r δϑϑ − = coscos . ( ( )eBmcq f 2γδ = see in [1, 4]). Therefore, the magnetic field value is expressed by       − += 0 0 cos1 R rBB δϑ . (8) 2.1. POLARIZATION VECTOR To describe the polarization properties we use the Stokes parameters VUQI ,,, . According [13, eq. (3.13)] the Stokes parameters for the radiation from an individ- ual electron are given by λλ πσ d dP d dPIe += , (9) χ λλ πσ 2cos      −= d dP d dPQe , (10) χ λλ πσ 2sin      −= d dP d dPUe , 0=eV , (11) where ( )πχ ,0∈ is the angle between some arbitrary fixed direction, axis z0 in our case, and the major axis of the ellipse of polarization. The angle χ is measured clockwise from the selected direction if the coordinate axes is taken as in Fig. 2. λddPi is given by eq. (5). As consequence of the integration the synchrotron radiation formulae over the solid angle, the parameter V , which indicates the presence of elliptically polarized radiation, is equal zero, 0=V [13]. For the ensemble of particles, the Stokes parameters are equal to the sum of the parameters of the individual particles. ISSN 1562-6016. ВАНТ. 2018. №4(116) 100 Then the degree of polarization is defined as [13] ( ) I UQ 22 + =Π λ , (12) and the angle χ Q U =χ2tg . (13) The rule according [13] is “from two values of the angle πχ ≤≤0 we choose that which lies in the first quadrant if 0>U and in the second if 0<U .” As known, the direction of the larger axis of polari- zation ellipse (σ -component) for synchrotron emission of relativistic electrons moving on a circular orbit coin- cides with the direction of particle’s acceleration [14]. Let χ be an angle between the axis z0 and the elec- tron acceleration a . Taking into account the smallness of angle between the line of sight and axis x0 , we find that a ay=χsin , a az=χcos . Then 2cos21 cos1sin aa a qq q +Θ− Θ+− =χ , (14) and 2cos21 sincos aa a qq q +Θ− Θ =χ . (15) Here, corrections of the first order of smallness, as in [7], are not given, It follows from (3) that each ϑ corresponds to two values of Θ . Substituting them into (4) we get two val- ues of the angle ϕ. It turns out that electrons radiate from two different places. We add up Stokes parameters for these two emitting regions, 21 QQQ += , 21 UUU += . Expressions (14), (15) are substituted for χsin , χcos in trigonometric formulas for doubled angles in (10), (11). Using Eqs. (3)-(15) we will calculate the distribu- tions of the total intensity I , degree of polarization Π and polarization directions χ in the synchrotron radia- tion spot from a homogenous electron beam with radius ],[ maxmin rrr∈ . Fig. 3. Total radiated power at pitch angles α = 0.08…0.15 rad: λ = 5 µm, R0 = 186 cm, δ = 10 cm, r = 5…24 cm, D = 186 cm, B0 = 2⋅104 G, \|γ = 80, q0 = 1, n = 2, a = 45 cm 3. DISCUSSION We take a small arc of the electron path and use in- tegrated over solid angle the classical synchrotron radia- tion formulae (5). The distribution of total power I in the area of syn- chrotron spot is shown in Fig. 3 if electrons uniformly distributed in a limited range of angles The radiation point coordinates are plotted in (y, z)-plane. The color (gray colormap) shows the radiation power (in arbitrary units) at a given wavelength λ (in this case, the wave- length λ = 5 µm), the range of pitch angles is from 0.08 to 0.15 rad. One can see the broadening of the spot with increasing pitch angles α. The increased brightness is observed inside the spot where the emissions of parti- cles with different α are summed. The radiation will be strongly influenced by the dis- tribution of electrons over the pitch angle. The presence of particles with different pitch angles and inhomogene- ity of magnetic field directions will cause the radiating hollow cone with an opening angle α to become filled, or partially filled. Then, we can integrate eqs. (5) over angles Θ to obtain formulae from [8 - 10]. 170180190200210220 -20 -15 -10 -5 0 5 10 15 20 y (cm) z (c m ) 170180190200210220 -20 -15 -10 -5 0 5 10 15 20 25 y, cm z, c m Fig. 4. Polarization of the synchrotron spot (poloidal projection) at the wavelength λ = 5 µm (top), the normals to the trajectory (2) that correspond Θ± (bottom). Parameters: B0 = 2⋅104 G, \|γ = 80, α = 0.12 rad, r = 6…24 cm, R0 = 186 cm, δ = 10 cm, D = 186 cm, q0 = 1, n = 2, a = 45 cm The directions of polarization in the synchrotron ra- diation spot are shown in Fig. 4. The length of dashes is proportional to the degree of polarization. In the upper part of the figure, a case when the contribution to the polarization comes from two emission points has been considered. The degree of polarization varies from 0 to 72%. This range of values depends on α and γ. Polariza- tion directions are orthogonal. There are regions with small and zero polarization. ISSN 1562-6016. ВАНТ. 2018. №4(116) 101 To understand how such polarization pattern is formed, the particle accelerations (directions of normal to the trajectory (2)) are shown in the lower part of Fig. 4. Two arrows at each point correspond to two val- ues of the angle Θ± in eq. (3). The contribution to radi- ation comes from two waves of equal intensity linearly polarized along these directions. The radiation of two waves of equal intensity linearly polarized along these directions is added at each point. The Stokes parameter 0=U in this case. The degree of polarization of the total wave vanishes if the initial waves have orthogonal directions of polarizations. For the range of pitch angles, these regions of zero polarization expand. It should be noted the change of the polarization di- rection on 90°. The polarization from ‘horizontal’ paral- lel to the y axis becomes ‘vertical’ parallel to the Z axis. An important result is the presence of regions of ze- ro polarization. Their presence allows us to speak of contributions to radiation from two different regions, which in turn will speak of the stability of the magnetic field directions. Also, one can talk about the directions of the mag- netic field having a polarization pattern. New relationships between the parameters of the emitting electrons and the magnetic field obtained from measurements of polarization can give additional data for the diagnostics of electron beams. They will complement the ratios obtained from measurements of the total intensity of the synchrotron radiation of runaway electrons. REFERENCES 1. I. Entrop, R. Jaspers, N.J. Lopes Cardozo, K.H. Finken. Runaway snakes in TEXTOR-94 // Plasma Phys. Control. Fusion. 1999, v. 41, p. 377-398. 2. Y. Shi, J. Fu, J. Li, Y. Yang, et al. Observation of runaway electron beams by visible color camera in the Experimental Advanced Superconducting To- kamak // Rev. Sci. Instrum. 2010, v. 81, p. 033506. 3. A.C. England, Z.Y. Chen, D.C. Seo, et al. Runaway Electron Suppression by ECRH and RMP in KSTAR // Plasma Sci. Technol. 2013, v. 15, p. 119-122. 4. R.J. Zhou, I.M. Pankratov, L.Q. Hu, et al. 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Sokolov, I.M. Ternov. Radiation from relativ- istic electrons. New York, AIoP, 1986, 324 p. Article received 26.06.2018 О ПОЛЯРИЗАЦИИ СИНХРОТРОННОГО ИЗЛУЧЕНИЯ УБЕГАЮЩИХ ЭЛЕКТРОНОВ В ТОРОИДАЛЬНОМ МАГНИТНОМ ПОЛЕ Я.М. Соболев Рассмотрены условия формирования поляризации синхротронного излучения убегающих электронов. Оце- нена излучаемая мощность и направления поляризации в пятне синхротронного излучения убегающих элек- тронов в зависимости от питч-углов излучающих электронов. Измерение поляризации синхротронного излу- чения может быть дополнительным средством для диагностики излучающих релятивистских электронов. ПРО ПОЛЯРИЗАЦІЮ СИНХРОТРОННОГО ВИПРОМІНЮВАННЯ УТІКАЮЧИХ ЕЛЕКТРОНІВ У ТОРОІДАЛЬНОМУ МАГНІТНОМУ ПОЛІ Я.М. Соболєв Розглянуто умови формування поляризації синхротронного випромінювання утікаючих електронів. На- дані оцінки потужності випромінювання і напрями поляризації в плямі синхротронного випромінювання утікаючих електронів у залежності від пітч-вуглів випромінюючих електронів. Вимірювання поляризації синхротронного випромінювання може слугувати додатковим засобом для діагностики релятивістських еле- ктронів. http://vant.kipt.kharkov.ua/ARTICLE/VANT_2013_4/article_2013_4_108.pdf http://vant.kipt.kharkov.ua/ARTICLE/VANT_2013_4/article_2013_4_108.pdf Introduction 1. Toroidal magnetic fields and electron trajectories 1.1. Radiation points 2. SYNCHROTRON RADIATION FORMULAE 2.1. Polarization vector 3. discussion references О ПОЛЯРИЗАЦИИ СИНХРОТРОННОго ИЗЛУЧЕНИЯ УБЕГАЮЩИХ ЭЛЕКТРОНОВ В ТОРОИДАЛЬНОМ МАГНИТНОМ ПОЛЕ ПРО поляризаціЮ синхротроннОГО випромінювання утікаючих електронів У тороідальному магнітному полі

Оn synchrotron radiation polarization from runaway electrons in toroidal magnetic fields

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