Synchrotron radiation and its polarization from runaway electrons in toroidal magnetic fields

The formulas to describe synchrotron radiation and its polarization in toroidal magnetic field configurations are presented. The radiated power and direction of polarization of the synchrotron radiation spot of runaway electrons for medium-sized tokamaks are estimated. Two polarization models are pr...

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Published in:	Вопросы атомной науки и техники
Date:	2015
Main Author:	Sobolev, Ya.M.
Format:	Article
Language:	English
Published:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2015
Subjects:	Термоядерный синтез (коллективные процессы)
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Cite this:	Synchrotron radiation and its polarization from runaway electrons in toroidal magnetic fields / Ya.M. Sobolev // Вопросы атомной науки и техники. — 2015. — № 4. — С. 135-139. — Бібліогр.: 12 назв. — англ.

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spelling	Sobolev, Ya.M. 2017-01-17T18:25:26Z 2017-01-17T18:25:26Z 2015 Synchrotron radiation and its polarization from runaway electrons in toroidal magnetic fields / Ya.M. Sobolev // Вопросы атомной науки и техники. — 2015. — № 4. — С. 135-139. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 41.60.Ap, 52.55.Fa, 97.60.Gb https://nasplib.isofts.kiev.ua/handle/123456789/112143 The formulas to describe synchrotron radiation and its polarization in toroidal magnetic field configurations are presented. The radiated power and direction of polarization of the synchrotron radiation spot of runaway electrons for medium-sized tokamaks are estimated. Two polarization models are proposed. It is shown that the polarization measurements give additional diagnostics for the radiating relativistic electrons. Пропонуються формули, які описують синхротронне випромінювання (СВ) та його поляризацію в тороїдальній конфігурації магнітного поля. Дані оцінки потужності випромінювання і напрям поляризації у плямі синхротронного випромінювання утікаючих електронів для токамаків середніх розмірів. Розглянуто дві поляризаційні моделі. Показано, що вимірювання поляризації СВ може слугувати додатковим засобом для діагностики релятивістських електронів. Предлагаются формулы для описания синхротронного излучения (СИ) и его поляризации в тороидальных конфигурациях магнитного поля. Оценены излучаемая мощность и направления поляризации в пятне синхротронного излучения убегающих электронов с параметрами, характерными для токамаков средних размеров. Рассматриваются две модели формирования поляризации. Показано, что измерение поляризации СИ может быть дополнительным средством для диагностики излучающих релятивистских электронов. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Термоядерный синтез (коллективные процессы) Synchrotron radiation and its polarization from runaway electrons in toroidal magnetic fields Синхротронне випромінювання та його поляризація від утікаючих електронів у тороїдальному магнітному полі Синхротронное излучение и его поляризация от убегающих электронов в тороидальном магнитном поле Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Synchrotron radiation and its polarization from runaway electrons in toroidal magnetic fields
spellingShingle	Synchrotron radiation and its polarization from runaway electrons in toroidal magnetic fields Sobolev, Ya.M. Термоядерный синтез (коллективные процессы)
title_short	Synchrotron radiation and its polarization from runaway electrons in toroidal magnetic fields
title_full	Synchrotron radiation and its polarization from runaway electrons in toroidal magnetic fields
title_fullStr	Synchrotron radiation and its polarization from runaway electrons in toroidal magnetic fields
title_full_unstemmed	Synchrotron radiation and its polarization from runaway electrons in toroidal magnetic fields
title_sort	synchrotron radiation and its polarization from runaway electrons in toroidal magnetic fields
author	Sobolev, Ya.M.
author_facet	Sobolev, Ya.M.
topic	Термоядерный синтез (коллективные процессы)
topic_facet	Термоядерный синтез (коллективные процессы)
publishDate	2015
language	English
container_title	Вопросы атомной науки и техники
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
format	Article
title_alt	Синхротронне випромінювання та його поляризація від утікаючих електронів у тороїдальному магнітному полі Синхротронное излучение и его поляризация от убегающих электронов в тороидальном магнитном поле
description	The formulas to describe synchrotron radiation and its polarization in toroidal magnetic field configurations are presented. The radiated power and direction of polarization of the synchrotron radiation spot of runaway electrons for medium-sized tokamaks are estimated. Two polarization models are proposed. It is shown that the polarization measurements give additional diagnostics for the radiating relativistic electrons. Пропонуються формули, які описують синхротронне випромінювання (СВ) та його поляризацію в тороїдальній конфігурації магнітного поля. Дані оцінки потужності випромінювання і напрям поляризації у плямі синхротронного випромінювання утікаючих електронів для токамаків середніх розмірів. Розглянуто дві поляризаційні моделі. Показано, що вимірювання поляризації СВ може слугувати додатковим засобом для діагностики релятивістських електронів. Предлагаются формулы для описания синхротронного излучения (СИ) и его поляризации в тороидальных конфигурациях магнитного поля. Оценены излучаемая мощность и направления поляризации в пятне синхротронного излучения убегающих электронов с параметрами, характерными для токамаков средних размеров. Рассматриваются две модели формирования поляризации. Показано, что измерение поляризации СИ может быть дополнительным средством для диагностики излучающих релятивистских электронов.
issn	1562-6016
url	https://nasplib.isofts.kiev.ua/handle/123456789/112143
citation_txt	Synchrotron radiation and its polarization from runaway electrons in toroidal magnetic fields / Ya.M. Sobolev // Вопросы атомной науки и техники. — 2015. — № 4. — С. 135-139. — Бібліогр.: 12 назв. — англ.
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first_indexed	2025-11-24T02:19:43Z
last_indexed	2025-11-24T02:19:43Z
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fulltext	ISSN 1562-6016. ВАНТ. 2015. №4(98) 135 SYNCHROTRON RADIATION AND ITS POLARIZATION FROM RUNAWAY ELECTRONS IN TOROIDAL MAGNETIC FIELDS Ya.M. Sobolev Institute of Radio Astronomy NASU, Kharkov, Ukraine E-mail: sobolev@rian.kharkov.ua The formulas to describe synchrotron radiation and its polarization in toroidal magnetic field configurations are presented. The radiated power and direction of polarization of the synchrotron radiation spot of runaway electrons for medium-sized tokamaks are estimated. Two polarization models are proposed. It is shown that the polarization measurements 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] the formulas that take into account curvature of magnetic field lines have been derived. In [2 - 4] the radi- ation formulae of a relativistic charge moving along bent spiral path have been generalized. The new expres- sions for frequency spectra and polarization components of synchrotron radiation were obtained. The synchrotron radiation spectrum of runaway electrons in tokamak was recieved in [5]. In [6] the polarization pattern of the synchrotron ra- diation of runaway electrons in tokamak has been calcu- lated. The synchrotron radiation is the powerful tool to di- agnose the parameters of relativistic runaway electrons. This diagnostic provides a direct image of the runaway beam inside the plasma. The radiation was measured in infra-red and visual wavelength ranges [7 - 10]. Analysis of experimental data has shown that the description of the synchrotron radiation of runaway electrons in tokamaks is the case when you need to take into account the curvature of magnetic field lines. In this paper, we will continue to study the polariza- tion of synchrotron radiation in a toroidal magnetic field configuration, using formulas from [3, 6]. The topology of toroidal magnetic fields and drift trajectories, the electron energies take values that are typical for medium-size tokamaks. The paper is organized as follows. The formulas proposed to describe the synchrotron radiation are given in Section 1. Terms of using these formulas instead of classical formulas of synchrotron radiation are written out. In Section 2, the spectrum and polarization properties of radiation emitted by relativistic electrons are derived. The formulas for calculating the direction of polarization are given. The contribution to the radiation in the defined direction from one or two emitting points is considered. In Section 3, the polarization in synchrotron radia- tion spot is calculated and discussed. The influence on the polarization pattern of unequal shift of drift trajectories is shown. 1. RADIATION FORMULAE IN CURVED MAGNETIC FIELDS The spectral power density of the synchrotron radia- tion emitted by relativistic electrons moving within magnetic field lines with curvature radius BR and mag- netic field B is expressed by [3, 4] ),( C C C ai i qyfP d dP λλ = , (1)            −+ + 2 +      += ∫ ∫ − + ∞ − \|1\| 1 22 C 2 \|q1\| CC C C C 2 /1arcsin)(1 )( 16 39),( a a a q y q y a a i y iai q xyqxdxF xdxFyqyf π π π here 22 \|\| 4 Ω= βγ c e P 2 C 3 2 is the total power emitted by a charged particle moving with velocity \|\|V along a circu- lar orbit of radius BR , λλCy =C , ( )Ω≈ 3γπλ 34C the characteristic radiation wavelength, BRV\|\|=Ω , ( )Da Rrq 2 B 2 B Ω=ω , ( )mceBB γω = , γ,, mee = are the electron charge, mass, and Lorentz-factor, Br is the Larmor radius, i    denote the cases of π- and σ- polarization, ( )( )3/13/53/23/5 2/1/ KKdxdKKF −=+=π , ( )( )3/13/53/23/5 32/1/ KKdxdKKF +=−=σ , νK is the Macdonald function. Formulas (1) contain the parameter aq that deter- mines the radiation mode. 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 on the larger circumference of radius equal to the radius of curvature of the magnetic field line. Also, it is the ratio of Larmor’s speed BBrω to the speed of centrifugal drift. This parameter indicates the extent of changing the curvature radius along the parti- cle trajectory. Formula (1) has classical synchrotron or curvature radiation limits when 1>>aq or 1<<aq , respectively [3, 4]. ISSN 1562-6016. ВАНТ. 2015. №4(98) 136 The parameter aq can be written as                        = 05.0 MeV40 cm200T2 5.1q α E RB a , where energies E are measured in MeV, magnetic field B is measured in T. Equations (1) are valid for arbitrary values of the pa- rameter aq . Formula (1) should be used whenever the parameter value of the order of unity, 1~aq . In other cases, the classical formulas of synchrotron radiation can be applied. Taking sum of components σπ FFF += in eq. (1) we obtain the formula of total spectral power. Then, the formula obtained and eq. (15) in [5] corresponds to each other (more details see in [4]). In Fig. 1, the spectra of total emitted power ( σπ FFF += in eq. (1)) for different particle energies E and pitch angles α are shown. The vertical lines show the range of wavelengths of visible light and infra- red wavelengths. The magnitude of magnetic field is G102 4⋅=B . The radiated power increases in magnetic field G103 4⋅=B . Fig. 1. Spectra according eq. (1): magnetic field G102 4⋅=B , curvature radius cm180=BR . Radiation bands: visible light 0.38…0.75 µm (left) and infrared wavelength 3…8 µm (right) 2. 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. Suppose, the toroidal magnetic field ϕB and plasma current I are clockwise, radiating electrons are moving counterclockwise. Magnetic fields take the form [11] ( ) ( )       + − − = ϕϑϑ ϑ eeB 00 0 cos1 , Rrq r Rr Br , (2) here ffrr ϑϑ == , are the radius and poloidal angle of magnetic surface, ( )rq the safety factor. The subscript f is omitted in eq. (2). 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 [7]. The guiding center is moving along drift trajectory with speed \|\|v . The electron pitch angle is \|\|vv⊥≈α 1<< , where BBrv ω=⊥ , ( )mceBB γω = , ( )ϑ,rBB = the magnetic field at the point with drift coordinates ϑ,r . The electron velocity vector is given by ( )DDD vv bντv Θ−Θ−+= ⊥ cossin\|\| , (3) here DDD bντ ,, are the tangent, normal, and binormal to the drift path, the angle Θ is measured from normal Dν to the direction of vector Db− , Bω=Θ (Fig. 2). Fig. 2. Coordinate systems on torus. Cartesian coordi- nates ( )zyx ,, ; orts on a small circle of the torus ϕϑ eee ,,r (bottom left corner); Larmor’s circumference and trihedron orts of the drift trajectory DDD bντ ,, (bottom right corner) 2.1. RADIATION POINTS The detector is located at the point OP with Carte- sian coordinates ( )0,, YD− (see Fig. 2). The line of sight is directed from the detection point OP to the radia- tion point eP , denote unit vector in the direction eO PP as n . The high energetic electrons emit their synchrotron radiation practically along their velocity vector (3). 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. ISSN 1562-6016. ВАНТ. 2015. №4(98) 137 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] ( ) ( )0 0 cos cos cos ϑϑ ϑα −=Θ Rrq r D , (4) where ( )22 0cos RqDD D+=ϑ . From equation 0=+ yy vnv follows Θ−+ +− −=∆ sincoscos αϑϑϕ Rq r D rRY D , (5) ( ) ( ) ( ) 122 22 0 11 1 + −− += nD ar arqnrq models the safe fac- tor of drift path, 1...7n = , a is the small tokamak ra- dius [11]. Larmor radius Br is not taken into account because of its smallness. (see also, eqs. (14), (15) in [10]) The range of angles ϑ depends on pitch-angles α . As can see from eq. (4), there are constraints on ac- ceptable angles ϑ when ( )0cosϑα DDRqr< . The shift δ of the drift trajectory is given by [7, 10] ( ) ( ) ( )eBmcrqr 2γδ = . (6) Using this formula, one can model dependence of δ on the radius r . 2.2. PARAMETERS FOR SYNCHROTRON RADIATION FORMULAS Available coordinates ( )ϕϑ,,r of the guiding center of electron are given by (4), (5). At the point, the curva- ture radius ( )ϕϑ,,rRD of the drift trajectory is calculat- ed. Then, the radius BR is replaced by ( )ϑ,rRD in for- mula (1). 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 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 . Therefore, the magnetic field val- ue is expressed by       − += 0 0 cos1 R rBB δϑ . (8) In Fig. 3, the total radiated power is shown. Note that the maximum (white color) and minimum (black color) values differ by 16 times. An inclination angle of the radiation spot is given by (4) and is 02/ ϑπ + (measured in the poloidal plane in such way as angle ϑ ). If the drift trajectory is wound in opposite direction ( ϑe− direction), one obtains a mirror image relative to axis 0y. In this case, the inclination angle is 02/ ϑπ − . Fig. 3. Total radiated power at the wavelength λ=0.5 µm; R0=186 cm; δ=10 cm; r=10…30 cm; D=186 cm; B0=2·104 G; α=0.12; E=40 MeV; q0=1; n=2; a=45 cm We can say that the figure ‘explains’ the experimen- tally observed synchrotron spot [8] Fig. 12; [9] Fig. 6. 2.3. POLARIZATION VECTOR As known, the direction of the larger axis of polari- zation ellipse for synchrotron emission of relativistic electrons moving on a circular orbit coincides with the direction of particle’s acceleration [12]. Let χ be an angle between the axis z0 and the electron 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 . The acceleration is found as the vector sum of the acceleration due to movement along drift trajecto- ry and the acceleration owing to Larmor rotation. Then ( ) 2 1 coscos12cos21 sinsincoscos1 sin aa D a a qbqb q qbbq +Θ−−+       Θ+−Θ+− = ϑϑ ϑϑ χ , (9) and ( ) ( ) 2 1 coscos12cos21 sincos1sin cos aa a D a qbqb q q bq +Θ−−+ Θ−−Θ = ϑϑ ϑ χ ,(10) where ( ) ( )ϑ,1 rRrq rb DD = , ( ) ( )     −⋅= rq rqbb D D 1 1 . To describe the polarization properties we use the Stokes parameters VUQI ,,, . According ([12] eq. (5.28)) λλ πσ d dP d dPI +∝ , ISSN 1562-6016. ВАНТ. 2015. №4(98) 138 χ λλ πσ ~2cos      −∝ d dP d dPQ , χ λλ πσ ~2sin      −∝ d dP d dPU , 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 is measured clockwise from the selected direction. The homogeneity of the electron distribution function implies that 0=V . 2.4. ONE OR TWO EMISSION POINTS ? It follows from (4) that each ϑ corresponds to two values of Θ . Substituting them into (5) we get two val- ues of the angleϕ . It turns out that electrons radiate from two different places. We add up Stokes parameters for these two emitting points, 21 QQQ += , 21 UUU += . Expressions (9), (10) are substituted for χ~sin , χ~cos in trigonometric formulas for doubled angles. Then the degree of polarization is defined as [12] ( ) I UQ 22 + =λπ , (11) and angle χ~ Q U =χ~2tg . (12) By definition, the angle χ~ describes the direction in the picture plane in which the intensity of the polarized components has maximum. Using Eqs. (3)-(12) 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∈ . If ( )rδδ = , ( )00 δ+= RY is taken. 3. DISCUSSION The monoenergetic distribution function and a given pitch angle are supposed [7 - 10]. The distribution is also homogenous in space. The formula (1) of synchrotron radiation in curved magnetic fields is valid for arbitrary values of parameter aq . The distribution of total intensity in the area of syn- chrotron spot is shown in Fig. 3. The radiation point coordinates are plotted in ( )zy, -plane. The color (gray colormap) shows the radiation power (in arbitrary units) at a given wavelength λ (in this case, the wavelength mμ5.0=λ ). The parameters for calculations are taken as in the middle-sized tokamaks [7 - 10]. The power emitted at a given spot point depends strongly on the values of α and γ . In Fig. 3 we see that the radiation is stronger in the area of larger mag- netic fields (closer to axis z ) than in the area with smaller magnetic field (further from axis z ). This dependence may be the cause of the observed heterogeneity in the synchrotron radiation spots ([8] Fig. 12; [9] Fig. 6). ` Fig. 4. Polarization of the synchrotron spot (poloidal projection) at the wavelength λ=5 µm. Parameters: B0=2⋅104 G, E=50 MeV, α=0.12, r=10…30 cm, R0=180 cm, δ=10 cm, D=186 cm, q0=1, n=2, a=45 cm Fig. 5. Polarization of the synchrotron spot with drift trajectories shift R0+δ(r) at the wavelength λ=5 µm the values of δ(r) are given by (6): q0=1, n=2, a=45 cm. Other parameters as in Fig. 4 At different values of the given (experimental) pa- rameters, the parameter aq often takes value near unity that evidenced in favor of using Eqs. (1). 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. Two models have been taken to calculate the polarization. In the up- per part of the figure, a case when the contribution to ISSN 1562-6016. ВАНТ. 2015. №4(98) 139 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. Po- larization directions are orthogonal. The lower part of the figure shows the case when the radiation comes from a single point (as it discussed in subsection 3.4). The absolute value of the degree of polarization is almost unchanged. The degree of polari- zation is about 70%. The directions of polarization be- have differently than in the upper part of the figure. In the case of small pitch angles α the polarization directions are determined by accelerations of guiding center, i. e. by the normal vector to the drift trajectory (along axis y). For large α the polarization direction is generated by centrifugal accelerations within Larmor circle. This gives the direction along axis z. In Fig. 5, as in [10], the displacement (6) of drift tra- jectories is taken into account. As in Fig. 4, the upper part of the figure shows the case with two radiating points from the given direction, but the lower part shows the case with contributions from a single point. Upper parts in Figs. 4, 5 show polarization direc- tions as well as areas of zero polarization. It should be noted the change of polarization direction on 90°. Assume that a similar cause may be responsible for changing the direction of polarization in the emission of pulsars. Thus the synchrotron radiation is polarized and the distribution pattern of polarization directions in the spot and the degree of polarization can give additional data for diagnostics of electron beams. REFERENCES 1. K.S. Cheng, J.L. Zhang. General radiation formulae for a relativistic charged particle moving in curved magnetic field lined: The synchro-curvature radia- tion mechanism // Astrophys. J. 1996, v. 463, № 1, p. 271-283. 2. Ya.M. Sobolev. Drift trajectory and synchrotron radia- tion of an ultrarelativistic electron moving in magnetic field with curved force lines // Problems of Atomic Sci- ence and Technology. Series “Plasma Electronics and New Acceleration Methods”. 2000, № 1, p. 27-30. http://vant.kipt.kharkov.ua/ARTICLE/VANT_2000_ 1/article_2000_1_27.pdf 3. Ya.M. Sobolev. Influence of magnetic line curvature on spectrum and polarization of synchrotron radia- tion of a charged particle // Radio Physics and Radio Astronomy. 2001, v. 6, № 4, p. 277-290. http://journal.rian.kharkov.ua/index.php/ra/article/vi ew/868/731 4. Ya.M. Sobolev. On the synchrotron radiation of ul- trarelativistic electrons moving along curved spiral trajectory // Problems of Atomic Science and Tech- nology. Series “Plasma Electronics and New Accel- eration Methods”. 2003, №4, p. 197-202. http://vant.kipt.kharkov.ua/ARTICLE/VANT_2003_ 4/article_2003_4_197.pdf 5. I.M. Pankratov. Towards analyses of runaway electrons synchrotron radiation spectra // Plasma Physics Reports. 1999, v. 25, № 2, p. 145-148. 6. Ya.M. Sobolev. 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Synchro- tron radiation spectra and synchrotron radiation spot shape of runaway electrons in Experimental Ad- vanced Superconducting Tokamak // Physics of Plasmas. 2014, v. 21, p. 063302. 11. J. Wesson. Tokamaks. Oxford: “Clarendon Press”. 2004, 762 p. 12. V.L. Ginzburg. Theoretical physics and astrophy- esics. M.: “Nauka”, 1987, 488 p. Article received 18.05.2015 СИНХРОТРОННОЕ ИЗЛУЧЕНИЕ И ЕГО ПОЛЯРИЗАЦИЯ ОТ УБЕГАЮЩИХ ЭЛЕКТРОНОВ В ТОРОИДАЛЬНОМ МАГНИТНОМ ПОЛЕ Я.М. Соболев Предлагаются формулы для описания синхротронного излучения (СИ) и его поляризации в тороидаль- ных конфигурациях магнитного поля. Оценены излучаемая мощность и направления поляризации в пятне синхротронного излучения убегающих электронов с параметрами, характерными для токамаков средних размеров. Рассматриваются две модели формирования поляризации. Показано, что измерение поляризации СИ может быть дополнительным средством для диагностики излучающих релятивистских электронов. СИНХРОТРОННЕ ВИПРОМІНЮВАННЯ ТА ЙОГО ПОЛЯРИЗАЦІЯ ВІД УТІКАЮЧИХ ЕЛЕКТРОНІВ У ТОРОЇДАЛЬНОМУ МАГНІТНОМУ ПОЛІ Я.М. Соболєв Пропонуються формули, які описують синхротронне випромінювання (СВ) та його поляризацію в торої- дальній конфігурації магнітного поля. Дані оцінки потужності випромінювання і напрям поляризації у плямі синхротронного випромінювання утікаючих електронів для токамаків середніх розмірів. Розглянуто дві по- ляризаційні моделі. Показано, що вимірювання поляризації СВ може слугувати додатковим засобом для діа- гностики релятивістських електронів. http://vant.kipt.kharkov.ua/ARTICLE/VANT_2000_1/article_2000_1_27.pdf http://vant.kipt.kharkov.ua/ARTICLE/VANT_2000_1/article_2000_1_27.pdf http://vant.kipt.kharkov.ua/ARTICLE/VANT_2003_4/article_2003_4_197.pdf http://vant.kipt.kharkov.ua/ARTICLE/VANT_2003_4/article_2003_4_197.pdf Introduction 1. RADIATION FORMULAE IN CURVED MAGNETIC FIELDs 2. Toroidal magnetic fields and electron trajectories 2.1. Radiation points 2.2. parameters for synchrotron radiation formulas 2.3. Polarization vector 2.4. one or two emission points ? 3. discussion references СИНХРОТРОННОЕ ИЗЛУЧЕНИЕ И ЕГО ПОЛЯРИЗАЦИЯ ОТ УБЕГАЮЩИХ ЭЛЕКТРОНОВ В ТОРОИДАЛЬНОМ МАГНИТНОМ ПОЛЕ синхротроннЕ випромінювання ТА ЙОГО поляризація ВіД утікаючих електронів У тороЇдальному магнітному полі

Synchrotron radiation and its polarization from runaway electrons in toroidal magnetic fields

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