Radiation friction of relativistic charged particles moving in a periodic field

Radiation friction of relativistic charged particles moving in a periodic field

The motion of relativistic charged particles beam in an external periodic field is considered, taking into account the influence of incoherent fields produced by particles on this motion. On the basis of the dynamics of individual particles motion under the action of the pair interaction forces each...

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Published in:Вопросы атомной науки и техники
Date:2022
Main Author: Ognivenko, V.V.
Format: Article
Language:English
Published: Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2022
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Beam dynamics
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/195399
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Cite this:Radiation friction of relativistic charged particles moving in a periodic field / V.V. Ognivenko // Problems of Atomic Science and Technology. — 2022. — № 3. — С. 91-93. — Бібліогр.: 7 назв. — англ.

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spelling Ognivenko, V.V.
2023-12-05T09:47:34Z
2023-12-05T09:47:34Z
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Radiation friction of relativistic charged particles moving in a periodic field / V.V. Ognivenko // Problems of Atomic Science and Technology. — 2022. — № 3. — С. 91-93. — Бібліогр.: 7 назв. — англ.
1562-6016
PACS: 41.60.−m, 52.25.Gj
https://nasplib.isofts.kiev.ua/handle/123456789/195399
The motion of relativistic charged particles beam in an external periodic field is considered, taking into account the influence of incoherent fields produced by particles on this motion. On the basis of the dynamics of individual particles motion under the action of the pair interaction forces each of them we derived the coefficient of friction. The expression for the friction force, which describes the average change in the momentum of charged particles per unit time, in the case of motion of an initially monoenergetic particle beam, is obtained.
Розглянуто рух пучка релятивістських заряджених частинок у зовнішньому періодичному полі, з урахуванням впливу на цей рух некогерентних полів, створюваних частинками. На основі динаміки руху окремих частинок під дією сил парної взаємодії кожної з них отриманий коефіцієнт тертя. Отримано вираз для сили тертя, що описує середню зміну імпульсу заряджених частинок за одиницю часу у випадку руху початково моноенергетичного пучку частинок. Установлений взаємозв’язок між средньоквадратичним розкидом по імпульсах і силою гальмування частинок.
Рассмотрено движение пучка релятивистских заряженных частиц во внешнем периодическом поле, с учетом влияния на это движение некогерентных полей, создаваемых частицами. На основе динамики движения отдельных частиц под действием сил парного взаимодействия каждой из них получен коэффициент трения. Получено выражение для силы трения, описывающее изменение среднего значения импульса заряженных частиц за единицу времени в случае движения первоначально моноэнергетического потока частиц. Установлена взаимосвязь между среднеквадратическим разбросом по импульсам и силой торможения частиц.
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
Вопросы атомной науки и техники
Beam dynamics
Radiation friction of relativistic charged particles moving in a periodic field
Радіаційне тертя релятивістських заряджених частинок, що рухаються в періодичному полі
Радиационное трение релятивистских заряженных частиц, движущихся в периодическом поле
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Radiation friction of relativistic charged particles moving in a periodic field
spellingShingle Radiation friction of relativistic charged particles moving in a periodic field
Ognivenko, V.V.
Beam dynamics
title_short Radiation friction of relativistic charged particles moving in a periodic field
title_full Radiation friction of relativistic charged particles moving in a periodic field
title_fullStr Radiation friction of relativistic charged particles moving in a periodic field
title_full_unstemmed Radiation friction of relativistic charged particles moving in a periodic field
title_sort radiation friction of relativistic charged particles moving in a periodic field
author Ognivenko, V.V.
author_facet Ognivenko, V.V.
topic Beam dynamics
topic_facet Beam dynamics
publishDate 2022
language English
container_title Вопросы атомной науки и техники
publisher Національний науковий центр «Харківський фізико-технічний інститут» НАН України
format Article
title_alt Радіаційне тертя релятивістських заряджених частинок, що рухаються в періодичному полі
Радиационное трение релятивистских заряженных частиц, движущихся в периодическом поле
description The motion of relativistic charged particles beam in an external periodic field is considered, taking into account the influence of incoherent fields produced by particles on this motion. On the basis of the dynamics of individual particles motion under the action of the pair interaction forces each of them we derived the coefficient of friction. The expression for the friction force, which describes the average change in the momentum of charged particles per unit time, in the case of motion of an initially monoenergetic particle beam, is obtained. Розглянуто рух пучка релятивістських заряджених частинок у зовнішньому періодичному полі, з урахуванням впливу на цей рух некогерентних полів, створюваних частинками. На основі динаміки руху окремих частинок під дією сил парної взаємодії кожної з них отриманий коефіцієнт тертя. Отримано вираз для сили тертя, що описує середню зміну імпульсу заряджених частинок за одиницю часу у випадку руху початково моноенергетичного пучку частинок. Установлений взаємозв’язок між средньоквадратичним розкидом по імпульсах і силою гальмування частинок. Рассмотрено движение пучка релятивистских заряженных частиц во внешнем периодическом поле, с учетом влияния на это движение некогерентных полей, создаваемых частицами. На основе динамики движения отдельных частиц под действием сил парного взаимодействия каждой из них получен коэффициент трения. Получено выражение для силы трения, описывающее изменение среднего значения импульса заряженных частиц за единицу времени в случае движения первоначально моноэнергетического потока частиц. Установлена взаимосвязь между среднеквадратическим разбросом по импульсам и силой торможения частиц.
issn 1562-6016
url https://nasplib.isofts.kiev.ua/handle/123456789/195399
citation_txt Radiation friction of relativistic charged particles moving in a periodic field / V.V. Ognivenko // Problems of Atomic Science and Technology. — 2022. — № 3. — С. 91-93. — Бібліогр.: 7 назв. — англ.
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AT ognivenkovv radiacionnoetrenierelâtivistskihzarâžennyhčasticdvižuŝihsâvperiodičeskompole
first_indexed 2025-11-25T22:45:22Z
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fulltext ISSN 1562-6016. ВАНТ. 2022. №3(139) 91 https://doi.org/10.46813/2022-139-091 RADIATION FRICTION OF RELATIVISTIC CHARGED PARTICLES MOVING IN A PERIODIC FIELD V.V. Ognivenko National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine E-mail: ognivenko@kipt.kharkov.ua The motion of relativistic charged particles beam in an external periodic field is considered, taking into account the influence of incoherent fields produced by particles on this motion. On the basis of the dynamics of individual particles motion under the action of the pair interaction forces each of them we derived the coefficient of friction. The expression for the friction force, which describes the average change in the momentum of charged particles per unit time, in the case of motion of an initially monoenergetic particle beam, is obtained. PACS: 41.60.−m, 52.25.Gj INTRODUCTION As is known, charged particles moving in external periodic fields emit electromagnetic radiation. For parti- cles with very high energy, the main part of the intensity of the radiation is concentrated in the region of high frequencies [1]. The effect of this radiation on the mo- tion of charged particles leads to a change in the rms value of the momentum of the particles. These effects are significant in the case when the system of charged particles is homogeneous in space, and the self- consistent fields are negligible. The flow of relativistic electrons moving in an external periodic field (undulator), which is a source of intense ultrashort- wavelength electromagnetic radiation, satisfies just such conditions. The change in the rms value of the longitu- dinal momentum of electrons moving in an external static and periodic in space magnetic field is considered in [2, 3]. These studies are of particular importance in connection with the work aimed at creating X-ray free electron lasers (FEL) operating in the mode of self- amplified of spontaneous emission (see, for example, [4]). The question naturally arises of how the incoherent field of spontaneous electromagnetic radiation affects the change in the mean value of the electron momen- tum. This work is devoted to the study of this issue. 1. BASIC EQUATIONS We consider an external field that is periodic along the z-axis of the Cartesian coordinate system and uni- form in the transverse direction (perpendicular to the z- axis). Let the flow of relativistic charged particles move along the z axis in this periodic field. We will assume that the forces of pair interaction between particles are known. The equations of longitudinal motion of an in- dividual charged particle can be written in the form:         s ss s zzz xtXttxFp dt d 0,F, , (1)   m p z dt d z s , (2) where zp and z are the momentum and the longitudinal coordinate of a particle,  s zF is the force of pair interac- tion between particles,       ssss xtxttxxtX 00 ,;,,  ,  pr,x is the set of coordinates and momentum of a particle,  000 ,prx are the initial values of coordi- nates and momentum of a particle,  is the relativistic factor. We will consider the motion of particles in the time t , when the trajectory of the particles does not change significantly Then, on the right-hand side of equation (1), expanding the expression for the force in a series in small parameters   tzzz ss 0 and  0zzz ppp  , we obtain the following equation           txtx zzz ttxFttxFp dt d 0 ,,0           ξ ξ , (3) where  zpz,ξ ,  zpz  ,ξ . The change in the momentum and coordinate, ac- cording to (1) and (2) is determined by the equations     t t zz txFtdp 0 ,0 , (4)          t t zs txFtt td m z 0 , 1 0 3 . (5) Substituting (4) and (5) into (3), we obtain the fol- lowing equation describing the change with time in the average value of the momentum of the particle    zzz AttxFp dt d  ,0 , (6) where             0 0 0 t z z z t ˆA dt F x t ,t L x,t t F x t ,t     , (7)   3 z t L̂ x,t z pm       , the angle brackets denote en- semble averaging. The first term in equation (6) describes the average force acting on the test particle, or the deceleration force of an individual charge by the field of its own radiation. The second term in equation (6) is the radiation friction force. Averaging in the right-hand side of Eq. (7) is per- formed using the distribution function of dynamic states of the particles in the phase space of the coordinates and momentum of the particles at initial time 0t (in the co- ordinate 0z ) [5, 6]. We consider the influence of 92 ISSN 1562-6016. ВАНТ. 2022. №3(139) average forces on the motion of particles insignificant if these forces are not equal to zero. Averaging in the integrand of Eq. (7) in the same way as it was done in [2, 6], neglecting the particle cor- relation at the initial moment of time, and taking into account Eq. (1) for the microscopic force, we obtain             0 0 0 0 0 t s z z s s t s z s s s s A dt F X t ,x L̂ x,t t F X t,x f x dx ,             (8) where  sxf 0 is the one-particle distribution function. To calculate Az we note that the force of pair interac- tion in the integrand (8) is determined by the motion of the charged particles along equilibrium trajectories in an external periodic field. 2. FLOW OF PARTICLES IN A PERIODIC FIELD Let charged particles move in a static and periodic in space magnetic field     zkzkH uyux sincos0 eeH  , (9) where 0H , u are the amplitude and period of the magnetic field, uuk  2 . We consider the interaction of charged particles via the electromagnetic field produced by them. The trans- verse equilibrium velocity of charged particles in the magnetic field (9) has the form         tzktzkt suysuxs sinvcosv0   eev . The expression for the force of pair interaction be- tween particles under the assumption of a small value of the amplitude of the external periodic field ( 12 2 00  mceH ) can be written in the form [3]       s uzss z qtrG trR kKe tx 0 * 22 ;, , ,F    , (10) where   , cos sin 2 ;, ** 0 22 * 2 0 * 2 0* 0 0 RkR R R R RkR R qtrG os z zs zs zs s zsz zss                           *0 2 RRk zszuzs  , 22 0 2 0* zsz RRR   ,    20 2 00 ss yyxxR  ,  tzzR sz 0 , uzszss kk 2 0  , 0 v   cK , ukmc He K 2 0  . Let's consider the initial (pre-Brownian) stage of charged particles diffusion in momentum space, assum- ing that particles are monoenergetic at the entrance to the external periodic field. Furthermore, we are interest- ed in small changes in the energy of charged particles due to their radiation friction 0 . Substituting the expression for the force (10) into right-hand side of Eq. (8), after integration over the momentum, we obtain the following equation for the radiation friction force                   t ss ssss bs t t uz xtRxtR xtXGxtXG ndq tdkKeA 0*0* 00 0 244 ,, ,, 0 , (11) where sssszs dtdydxdq 00000 v ,  is the range of inte- gration over the initial coordinates of the emitting charges. We assume that the beam is a solid cylindrical of ra- dius rb with a uniform average density nb. On the right-hand side of equation (11), it is conven- ient to pass from the integration variables x0s, y0s, t0s to new variables r', , , using the formulas:  sincosxx 00 rzss ,  sinsinyy 00 rzss ,    cos1 rzzs ,  20 , 0 . The limits of integration over the initial coordinates in equation (11) can be written in the form (cf. [2]): at rzz         cos ,, 2 1 111max zmzm z zrzr , (12) at rzz                  2, sin 0,, , 1* 1*11 1max z r zzr zr zm b , (13) where    zzarctgz r2*  , bzmzmr rz  . Substituting the expression (10) into equation (11) and integrating over the initial coordinates using bound- ary conditions (12), we obtain the following expression for the radiation friction force at z<zr z buz z nkreA    2 24 0 2 K 32 15 , (14) where 2 2 0 mc e r  . For z>zr, calculating the integrals on the right-hand side of Eq. (11), taking into account the limits of inte- gration (13), we obtain          r bbuz z z zBrnkreA 24 0 2 K , (15) where   2 5 1 35 1 3 16 4 2 1 15 ln . 2 32 B x arctgx x x x x                        . Let's note that in the limiting case z>>zr   216 35 3 5  xB . CONCLUSIONS Thus, an expression for the force of radiation friction of relativistic charged particles passing in an external periodic field is obtained in this work. The pre- Brownian stage of charged particles motion is investi- gated, when the difference in the initial velocities of particles at the entrance to the undulator can be neglect- ISSN 1562-6016. ВАНТ. 2022. №3(139) 93 ed. It should also be noted that expression for the fric- tion force in the case of the pre-Brownian motion of Coulomb interacting charged particles was obtained in [7]. As follows from (14, 15), the radiation friction force is proportional to the beam density. This force increases as the beam of particles moves in a periodic field. At distances z>zr this force increases in proportion to the distance in the first degree. At distances z<zr, the radiation friction force increases in proportion to the square of the distance passed by the beam. This depend- ence of the force on the distance is due to an increase in the number of particles in the field of which the test particle is located, as the beam passes through an exter- nal periodic field. From formulae (14, 15) and the expression for the diffusion coefficient zD , obtained in [2], the relation follows z z z p D A  , which relates the force of radiation friction and the dif- fusion coefficient. As follows from formulae (14, 15) for ultrarelativistic electron beams, which are used to obtain ultrashort-wavelength radiation in the FELs, the radia- tion friction force must be taken into account along with the radiation damping force of an individual charge. In addition, explicit expressions for the radiation friction force make it possible to kinetically describe the relaxa- tion of relativistic flows of charged particles in external periodic fields. REFERENCES 1. L.D. Landau, E.M. Lifshitz. The classical theory of fields. Pergamon Press, Oxford, 1968. 2. V.V. Ognivenko. Momentum spread in a relativistic electron beam in an undulator // J. Exp. Theor. Phys. 2012, v. 115, № 5, p. 938-946. 3. V.V. Ognivenko // J. Exp. Theor. Phys. 2021, v. 132, № 5, p. 766-775. 4. Ye.N. Ragozin, I.I. Sobel'man. Lazernyye istoch- niki v myagkoy rentgenovskoy oblasti spectra // UFN. 2005, № 12, p. 1339-1341. 5. N.N. Bogolyubov. Problems of Dynamical Theory in Statictical Physics. M.: “Gostekhteorizdat”, 1946; Interscience, New York, 1962. 6. V.V. Ognivenko. Dynamical derivation of momen- tum diffusion coefficients at collisions of relativistic charged particles // J. Exp. Theor. Phys. 2016, v. 122, № 1, p. 203-208. 7. V.V. Ognivenko. The pair interaction forces and the friction and diffusion coefficients of particles in momentum space // Problems of Atomic Science and Technology. Series “Plasma Physics”. 2017, № 1, p. 195-198. Article received 11.01.2022 РАДІАЦІЙНЕ ТЕРТЯ РЕЛЯТИВІСТСЬКИХ ЗАРЯДЖЕНИХ ЧАСТИНОК, ЩО РУХАЮТЬСЯ В ПЕРІОДИЧНОМУ ПОЛІ В.В. Огнівенко Розглянуто рух пучка релятивістських заряджених частинок у зовнішньому періодичному полі, з урахуванням впливу на цей рух некогерентних полів, створюваних частинками. На основі динаміки руху окремих частинок під дією сил парної взаємодії кожної з них отриманий коефіцієнт тертя. Отримано вираз для сили тертя, що описує середню зміну імпульсу заряджених частинок за одиницю часу у випадку руху початково моноенергетичного пучку частинок. Установлений взаємозв'язок між средньоквадратичним розкидом по імпульсах і силою гальмування частинок. РАДИАЦИОННОЕ ТРЕНИЕ РЕЛЯТИВИСТСКИХ ЗАРЯЖЕННЫХ ЧАСТИЦ, ДВИЖУЩИХСЯ В ПЕРИОДИЧЕСКОМ ПОЛЕ В.В. Огнивенко Рассмотрено движение пучка релятивистских заряженных частиц во внешнем периодическом поле, с учетом влияния на это движение некогерентных полей, создаваемых частицами. На основе динамики дви- жения отдельных частиц под действием сил парного взаимодействия каждой из них получен коэффициент трения. Получено выражение для силы трения, описывающее изменение среднего значения импульса заря- женных частиц за единицу времени в случае движения первоначально моноэнергетического потока частиц. Установлена взаимосвязь между среднеквадратическим разбросом по импульсам и силой торможения час- тиц.

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