The diffusion effects in relativistic electron beam in an undulator

We consider diffusion processes in momentum space of a relativistic electron beam moving in a spatially periodic magnetic field of an undulator. Basing on the dynamics of individual particles motion under the action of the pair interaction forces the longitudinal diffusion coefficient has been deriv...

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Published in:	Вопросы атомной науки и техники
Date:	2017
Main Author:	Ognivenko, V.V.
Format:	Article
Language:	English
Published:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2017
Subjects:	Динамика пучков
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Cite this:	The diffusion effects in relativistic electron beam in an undulator / V.V. Ognivenko // Вопросы атомной науки и техники. — 2017. — № 6. — С. 85-87. — Бібліогр.: 9 назв. — англ.

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spelling	Ognivenko, V.V. 2018-06-16T07:10:00Z 2018-06-16T07:10:00Z 2017 The diffusion effects in relativistic electron beam in an undulator / V.V. Ognivenko // Вопросы атомной науки и техники. — 2017. — № 6. — С. 85-87. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 41.60.-m, 41.60.Cr, 52.25.Gj https://nasplib.isofts.kiev.ua/handle/123456789/136200 We consider diffusion processes in momentum space of a relativistic electron beam moving in a spatially periodic magnetic field of an undulator. Basing on the dynamics of individual particles motion under the action of the pair interaction forces the longitudinal diffusion coefficient has been derived. The conditions for the high-gain self-amplification of spontaneous radiation in ultrashort-wavelength FELs have been discussed. Розглянуто процеси дифузії в просторі імпульсів релятивістського електронного пучка, що рухається в просторово періодичному магнітному полі ондулятора. Ґрунтуючись на динаміці руху окремих частинок під дією сил парної взаємодії, отримано поздовжній коефіцієнт дифузії. Обговорюються умови реалізації інтенсивного самочинного посилення спонтанного випромінювання в ультракороткохвильових ЛВЕ. Рассмотрены процессы диффузии в пространстве импульсов релятивистского электронного пучка, движущегося в пространственно периодическом магнитном поле ондулятора. Основываясь на динамике движения отдельных частиц под действием сил парного взаимодействия, получен продольный коэффициент диффузии. Обсуждаются условия реализации интенсивного самопроизвольного усиления спонтанного излучения в ультракоротковолновых ЛСЭ. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Динамика пучков The diffusion effects in relativistic electron beam in an undulator Ефекти дифузії в релятивістському електронному пучку в ондуляторі Эффекты диффузии в релятивистском электронном пучке в ондуляторе Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	The diffusion effects in relativistic electron beam in an undulator
spellingShingle	The diffusion effects in relativistic electron beam in an undulator Ognivenko, V.V. Динамика пучков
title_short	The diffusion effects in relativistic electron beam in an undulator
title_full	The diffusion effects in relativistic electron beam in an undulator
title_fullStr	The diffusion effects in relativistic electron beam in an undulator
title_full_unstemmed	The diffusion effects in relativistic electron beam in an undulator
title_sort	diffusion effects in relativistic electron beam in an undulator
author	Ognivenko, V.V.
author_facet	Ognivenko, V.V.
topic	Динамика пучков
topic_facet	Динамика пучков
publishDate	2017
language	English
container_title	Вопросы атомной науки и техники
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
format	Article
title_alt	Ефекти дифузії в релятивістському електронному пучку в ондуляторі Эффекты диффузии в релятивистском электронном пучке в ондуляторе
description	We consider diffusion processes in momentum space of a relativistic electron beam moving in a spatially periodic magnetic field of an undulator. Basing on the dynamics of individual particles motion under the action of the pair interaction forces the longitudinal diffusion coefficient has been derived. The conditions for the high-gain self-amplification of spontaneous radiation in ultrashort-wavelength FELs have been discussed. Розглянуто процеси дифузії в просторі імпульсів релятивістського електронного пучка, що рухається в просторово періодичному магнітному полі ондулятора. Ґрунтуючись на динаміці руху окремих частинок під дією сил парної взаємодії, отримано поздовжній коефіцієнт дифузії. Обговорюються умови реалізації інтенсивного самочинного посилення спонтанного випромінювання в ультракороткохвильових ЛВЕ. Рассмотрены процессы диффузии в пространстве импульсов релятивистского электронного пучка, движущегося в пространственно периодическом магнитном поле ондулятора. Основываясь на динамике движения отдельных частиц под действием сил парного взаимодействия, получен продольный коэффициент диффузии. Обсуждаются условия реализации интенсивного самопроизвольного усиления спонтанного излучения в ультракоротковолновых ЛСЭ.
issn	1562-6016
url	https://nasplib.isofts.kiev.ua/handle/123456789/136200
citation_txt	The diffusion effects in relativistic electron beam in an undulator / V.V. Ognivenko // Вопросы атомной науки и техники. — 2017. — № 6. — С. 85-87. — Бібліогр.: 9 назв. — англ.
work_keys_str_mv	AT ognivenkovv thediffusioneffectsinrelativisticelectronbeaminanundulator AT ognivenkovv efektidifuzíívrelâtivístsʹkomuelektronnomupučkuvondulâtorí AT ognivenkovv éffektydiffuziivrelâtivistskomélektronnompučkevondulâtore AT ognivenkovv diffusioneffectsinrelativisticelectronbeaminanundulator
first_indexed	2025-11-24T04:31:51Z
last_indexed	2025-11-24T04:31:51Z
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fulltext	ISSN 1562-6016. ВАНТ. 2017. №6(112) 85 THE DIFFUSION EFFECTS IN RELATIVISTIC ELECTRON BEAM IN AN UNDULATOR V.V. Ognivenko National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine E-mail: ognivenko@kipt.kharkov.ua We consider diffusion processes in momentum space of a relativistic electron beam moving in a spatially period- ic magnetic field of an undulator. Basing on the dynamics of individual particles motion under the action of the pair interaction forces the longitudinal diffusion coefficient has been derived. The conditions for the high-gain self- amplification of spontaneous radiation in ultrashort-wavelength FELs have been discussed. PACS: 41.60.-m, 41.60.Cr, 52.25.Gj INTRODUCTION As it is known, relativistic electron beams, moving in a spatially periodic static magnet field (undulator) are the sources of intense narrowband electromagnetic radiation. The wavelength of this radiation is proportional to the period of an external magnetic field and inversely propor- tional to the square of energy of electron. Such mecha- nism of interaction between ultra-relativistic electrons and external periodic magnetic field has been used to obtain the electromagnetic radiation in nanometer range of wavelengths by now [1 - 3]. At a spontaneous incoherent radiation of electromag- netic waves by relativistic electrons, moving in an undula- tor, there is a change of the average momentum of elec- trons, as a result of braking by the force of radiation fric- tion. Moreover, influence of incoherent electromagnetic field of spontaneous radiation of individual electrons leads to the increase in root-mean-square spread in mo- menta in a relativistic electronic beam, moving in an un- dulator [4, 5]. The study of motion dynamics of electrons at the stage of spontaneous incoherent radiation is of in- terest regarding the researches directed on creation of sources of coherent electromagnetic radiation in X-ray range of wavelengths by means of relativistic electron beam passing through an undulator. The interaction of initially monoenergetic electron beam with an undulator field has been considered in [5] and the expression describing the change of a root-mean- square longitudinal momentum of electrons, in the case when the spread in energy of electrons at the entrance of the undulator can be neglected, has been found. The mo- tion of the beam of electrons, having at the entrance of the undulator some initial spread in longitudinal momentum, is considered in the given work. In the limit case of small value of the undulator parameter the expression for the diffusion coefficient in momentum space is obtained, which can describe both the initial stage of prebrownian motion of electrons in the electromagnetic field of undula- tor radiation, when approach of a monoenergetic electron beam is valid, and in the case of kinetic stage of particles diffusion. 1. PROBLEM STATEMENT Let's consider a beam of relativistic electrons, moving in the spatially periodic static magnet field of helical un- dulator     zkzkH uyuxu sincos0 eeH  , (1) uuk  2 , Н0 and u are the amplitude and period of magnetic field, yx ee , are the unit vectors along axes x and y the Cartesian system of coordinates. Moving in an undulator, electrons radiate. The electric field produced by individual electron (s-th) in undulator can be found from formulas for the field of a charge, moving with acceleration [6].          32322 11 sss sssss sss sss s Rc e R e βn vβnn βn βn E        , (2)    ssss xt EnrH ;, , (3) where sss R Rn ,    tt sss rrRR  , cvβ  , dtdvv  ,   2121   , c is the speed of light in vacu- um, е is the electron charge, the prime denotes the values taken in retarding time t, defined by the equation:   ctRtt s  . Considering the motion of a test particle in an undula- tor the equation describing its motion is possible to be written down in the form      s si s z zi xttxF dt d ;, p ,    tm t dt d i ii   pr , (4)                zssszss s z xtx c xtxextxF ;, 1 ;,E;, Hv , (5) where Fz (s) (xi,t;xs) is the longitudinal component of pair interaction force of two electrons, m is the mass of elec- tron, xs(t){rs(t), ps(t)} set of the Cartesian coordinates and momentum of s-th electron. 2. DIFFUSION COEFFICIENT Distribution of the electrons in the beam at the en- trance of the undulator is random, therefore the total elec- tromagnetic field of radiation by individual electrons at the initial stage is incoherent. Assuming that at initial instant of time the motion of the electrons is uncorrelated, and there are many electrons in the beam, the diffusion coefficient in a longitudinal momentum can be taken from the equations of test electron motion [5, 7]                         ,v,;, ,;, 2 00010 001 0 0 0012 sszsssiz q ssizziz dqtqfqtxtxF qtxttxFdp dt d D       (6) where q0s=(p0s, x0s, y0s, t0s), dq0s=dp0sdx0sdy0sdt0s, =t-t0i, f1 is the single-particle distribution function,       000 , iiix pr . As we consider time intervals  small in ISSN 1562-6016. ВАНТ. 2017. №6(112) 86 comparison with the time of the significant change of the electrons trajectory, in the right-hand side of Eq. (6), in the pair interaction force, we have replaced coordinates and the momenta with unperturbed trajectory   t0 r and momenta   t0 p of electron in the undulator. Let's assume that the electron beam is cylindrical with radius br and constant average density of electrons bn for brr  , and at the initial time (at z=0) distribution function in momenta takes the form:                    2 2 0 2 exp 2 th zz th b p pp p n f pp . The equilibrium velocity and trajectory of electron in the field (1) are:          tzkrtzkrttt suusysuusxossss cossin00 0 eevrr  ,       tzktzkt susysusxss sinvcosv0 0   eevv , where 00v   zu u k cK r , 0 v   cK , ukmc He K 2 0  ,    szs tttz 00v  . In the case of small undulator parameter К<<1, con- sidering only the second term of force (5), from Eqs. (2), (3) and (4) we get the expression for pair interaction force between electrons:        ssisiissusizs zGkeqttF  ,K;, 22 0 0 r , (7)                               0 2 * 2 0*0 cos sin 1 , RkR x RkR x Rk yxG s s s s s s , where     2, Rxkyx ssu  ,     2122 * , yxyxR  ,  szsisi ttzz 0v  , sisi   0rr , usss kk 2 000  . Let's substitute expression for force (7) in the equation (6). Assuming that change in momentum occurs at a dis- tance greater than a period of the undulator, in expression (7) we retain the terms inversely proportional to the first degrees R . We will also consider that the basic contribu- tion to the integral (6) will give terms containing a differ- ence of phases at time t and t-. Then Eq. (6) can be re- duced to the form:     , 0 tKdD zz , (8)           wr pp r dVfdkKetK sssu cos , 000 2 0 22 , (9) where     sissi zz   0rrr ,   si vvw ,     tt , sssss dtdydxdV 00000 v . The limits of integration in the Eq. (9) are defined by the time of radiation propagation from electron-radiator (s-th) to considered test electron and the transverse di- mensions of the beam: 2 0 iiis zRz  ,   bsoss ryxr  212 0 2 . (10) In the right-hand side of Eq. (9) at the integration over initial coordinates it is expedient to transform to the new variables  ,,r :  cossinrxx soios ,  sinsinryy soios ,  cosrzsi . Let's find the diffusion coefficient for electrons mov- ing near to the beam axis. Thus the range of the integra- tion on r and  according to the Eq. (10) is:   2cos ii zr  , bs rr  sin . (11) In Eq. (9) we will consider the forces exerted on the test electron by the electrons, moving behind it zs<zi (0<</2). Integrating in Eq.(9) on r and  at z>z, and substituting the obtained expression in Eq.(8), we find:      22 0 0 222 cosv zz xa zibbuz dxbxenrkKeD  , (12) where 02 z uth p kp a  , 0 0 z zzi u p pp kb   , brz 0*  . Using this formula, it is possible to find the diffusion coefficient in momentum space for the various times at the certain initial energy spread of electrons. 3. DISCUSSION From the expression (12), connecting the correlation function and diffusion coefficient, it follows that the cor- relation function can be written in the form:          zbbu benrkKettK c 00 222 vcos, 2 , (13) where  002 zuthzc vkpp . From (13) we see that correlation function oscillates on  with decreasing amplitude for large values of t. For  in formula (13) the correlation function tends to zero. Such dependence of correlation function on time describes chaotization of particles motion. Characteristic time of particles motion chaotization is c, which is equal to the displacement time in the longitudinal direction, as a result of thermal motion, at the distance equal to the half of the wavelength of undulator radiations thc v5.0   , 2 02 u . For >>c the motion of particles becomes chaotic. The expression (12) describes the change of root- mean-square value of the momentum of electrons also at times <<c. The expression for diffusion coefficient in this case becomes:     z nr kKepD zi bb uziz v 0 222   . In this limiting case the change in time of root-mean- square value of the longitudinal momentum is described by the formula      u bz Rzz r NFp    2 9 0 21 2 , (14) which coincides with the corresponding formula of [5], where     3 0 2 0 2 0 2 032 zRz HrF  , 4 0 3 8 zubnN  . For <<c the motion of particles occurs under the influ- ence of pair interaction forces of particles, the change in time of which is negligible. Therefore, the r.m.s. value of momentum is proportional to the time. For >> c the motion of electrons is random. The ex- pression for diffusion coefficient becomes:             2 2 00 23 044 2 exp 2v th zzi th z zi bb uz p pp p pnr kKeD  . (15) ISSN 1562-6016. ВАНТ. 2017. №6(112) 87 The r.m.s. value of the momentum increases propor- tionally to the square root of time. Such dependence of momenta spread in time describes the completely chaotic motion of particles. The distance in the undulator at which the particles motion chaotization occurs can be written as czcz 0v . Then, for the electron beam with some initial energy spread, we find:  uthzc kppz 02 . Thus, for the momentum spread at the entrance of the undulator thp so that 10 zuth pzkp the r.m.s. deviation of the longitudinal momentum from equilibrium value increases proportionally to the distance traversed by elec- trons in the undulator (14). In this case the monoenergetic beam approximation [5] is applicable. In the opposite limit of large energies spread 10 zuth pzkp the kinetic stage of the radiative relaxation of an electron beam in the undulator occurs. At this stage the r.m.s. spread increases proportionally to the square root of time (15). As we see from (14), at the initial energy spread for which the mode of self amplification of spontaneous un- dulator radiations occurs [8, 9], the energy spread in the beam can increase as a result of the radiative relaxation [5]. At small momentum spread 10 zuth pzkp the anal- ysis of radiation formation in the mode of self amplifica- tion of spontaneous emission needs to be carried out while taking into account the effect of the radiative relax- ation of the beam. REFERENCES 1. V. Ayvazyan et al. First operation of a free-electron laser generating GW power radiation at 32 nm wave- length // Eur. Phys. J. D. 2006, v. 37, p. 297-303. 2. P. Emma et al. First Lasing and Operation of an Ång- strom-Wavelength Free-Electron Laser // Nature Pho- tonics. 2010, v. 4, p. 641-647. 3. S.T. Ishikawa et al. A compact X-ray free-electron laser emitting in the sub-ångström region // Nature Photonics. 2012, v. 6, p. 540-544. 4. V.V. Ognivenko. Radiative relaxation of relativistic electron beam in helical undulator // Problems of Atomic Science and Technology. Series “Plasma Elec- tronics”. 2006, № 5, p. 7-9. 5. V.V. Ognivenko. Momentum spread in a relativistic electron beam in an undulator // J. Exp. Theor. Phys. 2012, v. 115, № 5, p. 938-946. 6. L.D. Landau, E.M. Lifshic. Theorya polya. M: “Nau- ka”. 1967, 460 p. (in Russian). 7. V.V. Ognivenko. Dynamical derivation of momentum diffusion coefficients at collisions of relativistic charged particles // J. Exp. Theor. Phys. 2016, v. 122, № 1, p. 203-208. 8. K.-J. Kim. An analysis of self-amplified spontaneous emission // Nucl. Instr. and Meth. 1986, v. A 250, № 1-2, p. 396-403. 9. С. Pellegrini. Free electron lasers: development and applications // Particle Accelerators. 1990, v. 33, p. 159-170. Article received 09.10.2017 ЭФФЕКТЫ ДИФФУЗИИ В РЕЛЯТИВИСТСКОМ ЭЛЕКТРОННОМ ПУЧКЕ В ОНДУЛЯТОРЕ В.В. Огнивенко Рассмотрены процессы диффузии в пространстве импульсов релятивистского электронного пучка, дви- жущегося в пространственно периодическом магнитном поле ондулятора. Основываясь на динамике движе- ния отдельных частиц под действием сил парного взаимодействия, получен продольный коэффициент диф- фузии. Обсуждаются условия реализации интенсивного самопроизвольного усиления спонтанного излуче- ния в ультракоротковолновых ЛСЭ. ЕФЕКТИ ДИФУЗІЇ В РЕЛЯТИВІСТСЬКОМУ ЕЛЕКТРОННОМУ ПУЧКУ В ОНДУЛЯТОРІ В.В. Огнівенко Розглянуто процеси дифузії в просторі імпульсів релятивістського електронного пучка, що рухається в просторово періодичному магнітному полі ондулятора. Ґрунтуючись на динаміці руху окремих частинок під дією сил парної взаємодії, отримано поздовжній коефіцієнт дифузії. Обговорюються умови реалізації інтен- сивного самочинного посилення спонтанного випромінювання в ультракороткохвильових ЛВЕ.

The diffusion effects in relativistic electron beam in an undulator

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