Diffusion in momentum of relativistic electrons with a thermal spread passing through an undulator

Diffusion in momentum of relativistic electrons with a thermal spread passing through an undulator

The longitudinal momentum diffusion of electrons moving in a spatially periodic magnetic field of an undulator is investigated, taking into account their initial energy spread. Expressions for the coefficient are obtained and the dependences of the diffusion coefficient are determined both on the di...

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Datum:2021
1. Verfasser: Ognivenko, V.V.
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Veröffentlicht: Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2021
Schriftenreihe:Вопросы атомной науки и техники
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Parametric radiation
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spelling irk-123456789-1952662023-12-03T18:02:06Z Diffusion in momentum of relativistic electrons with a thermal spread passing through an undulator Ognivenko, V.V. Parametric radiation The longitudinal momentum diffusion of electrons moving in a spatially periodic magnetic field of an undulator is investigated, taking into account their initial energy spread. Expressions for the coefficient are obtained and the dependences of the diffusion coefficient are determined both on the distance traveled by the electrons in the undulator and on the value of the initial energy spread of the electrons. The possibility of decreasing the wavelength in X-ray free electron lasers is discussed. Досліджено дифузію за поздовжнього імпульсу електронів, що рухаються в просторово-періодичне магнітне поле ондулятора, з урахуванням їх початкового енергетичного розкиду. Отримано вираз для коефіцієнта дифузії й визначені залежності його як від відстані, вздовж ондулятора, так і від величини початкового енергетичного розкиду електронів. Обговорюється можливість зменшення довжини хвилі в рентгенівських лазерах на вільних електронах. Исследована диффузия по продольному импульсу электронов, движущихся в пространственнопериодическое магнитное поле ондулятора, с учетом их начального энергетического разброса. Получены выражения для коэффициента диффузии и определены зависимости его как от расстояния, пройденного электронами в ондуляторе, так и от величины начального энергетического разброса электронов. Обсуждается возможность уменьшения длины волны в рентгеновских лазерах на свободных электронах. 2021 Article Diffusion in momentum of relativistic electrons with a thermal spread passing through an undulator / V.V. Ognivenko // Problems of Atomic Science and Technology. — 2021. — № 4. — С. 102-105. — Бібліогр.: 16 назв. — англ. 1562-6016 PACS: 41.60.-m, 41.60.Cr, 52.25.Gj DOI: https://doi.org/10.46813/2021-134-102 http://dspace.nbuv.gov.ua/handle/123456789/195266 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 Parametric radiation
Parametric radiation
spellingShingle Parametric radiation
Parametric radiation
Ognivenko, V.V.
Diffusion in momentum of relativistic electrons with a thermal spread passing through an undulator
Вопросы атомной науки и техники
description The longitudinal momentum diffusion of electrons moving in a spatially periodic magnetic field of an undulator is investigated, taking into account their initial energy spread. Expressions for the coefficient are obtained and the dependences of the diffusion coefficient are determined both on the distance traveled by the electrons in the undulator and on the value of the initial energy spread of the electrons. The possibility of decreasing the wavelength in X-ray free electron lasers is discussed.
format Article
author Ognivenko, V.V.
author_facet Ognivenko, V.V.
author_sort Ognivenko, V.V.
title Diffusion in momentum of relativistic electrons with a thermal spread passing through an undulator
title_short Diffusion in momentum of relativistic electrons with a thermal spread passing through an undulator
title_full Diffusion in momentum of relativistic electrons with a thermal spread passing through an undulator
title_fullStr Diffusion in momentum of relativistic electrons with a thermal spread passing through an undulator
title_full_unstemmed Diffusion in momentum of relativistic electrons with a thermal spread passing through an undulator
title_sort diffusion in momentum of relativistic electrons with a thermal spread passing through an undulator
publisher Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate 2021
topic_facet Parametric radiation
url http://dspace.nbuv.gov.ua/handle/123456789/195266
citation_txt Diffusion in momentum of relativistic electrons with a thermal spread passing through an undulator / V.V. Ognivenko // Problems of Atomic Science and Technology. — 2021. — № 4. — С. 102-105. — Бібліогр.: 16 назв. — англ.
series Вопросы атомной науки и техники
work_keys_str_mv AT ognivenkovv diffusioninmomentumofrelativisticelectronswithathermalspreadpassingthroughanundulator
first_indexed 2025-07-16T23:09:20Z
last_indexed 2025-07-16T23:09:20Z
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fulltext ISSN 1562-6016. ВАНТ. 2021. № 4(134) 102 https://doi.org/10.46813/2021-134-102 DIFFUSION IN MOMENTUM OF RELATIVISTIC ELECTRONS WITH A THERMAL SPREAD PASSING THROUGH AN UNDULATOR V.V. Ognivenko National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine E-mail: ognivenko@kipt.kharkov.ua The longitudinal momentum diffusion of electrons moving in a spatially periodic magnetic field of an undulator is investigated, taking into account their initial energy spread. Expressions for the coefficient are obtained and the dependences of the diffusion coefficient are determined both on the distance traveled by the electrons in the undula- tor and on the value of the initial energy spread of the electrons. The possibility of decreasing the wavelength in X-ray free electron lasers is discussed. PACS: 41.60.-m, 41.60.Cr, 52.25.Gj INTRODUCTION The change in the mean square value of the momen- tum in the flow of electrons interacting by means of the electromagnetic fields they produce at the stage of spon- taneous emission was studied in [1]. In this work, the radiative relaxation of the electrons at the pre-Brownian stage of the evolution of the system [2], when the elec- trons have the same initial energy, is considered. Since electrons move with different velocities in real flows, it is necessary to establish a quantitative criterion that determines the possibility of neglecting the differ- ence in the initial velocities of electrons when describ- ing the radiation relaxation of the beam. In addition, one should take into account that the difference in the ve- locities of electrons makes it possible to study the diffu- sion of electrons in momentum space at the kinetic stage of the evolution of the system [3], when the motion of the electrons in the process of radiation relaxation be- comes completely random. The description of electron momentum diffusion in such flows is also of consider- able interest in connection with the development of X- ray free electron lasers (FEL) based on self- amplification of spontaneous emission by a monoener- getic ultrarelativistic electron beam moving in an undu- lator [4 - 10]. 1. DIFFUSION COEFFICIENT The expression for the diffusion coefficient obtained on the basis of the dynamics of individual particles mo- tion can be written in the form [1, 2]               ,v)(, , 2 00001 01 2 0 sszss s z s s z t t zz dqtqfqtXF qtXFdtp dt dD q     (1) where zzz ppp  , p is the momentum of the elec- tron,   s s z xtx ;,F is the pair interaction force,           sss qtxttxqtX 0 00 0 ,;,,  ,  pr,x , p=mv, m is the mass of the electron, 2122 )v1(  c , с is the velocity of light,       000 , sssx pr are the equilibrium trajectory and momentum of sth electron in an undula- tor,  sssss tyq 00000 ,,,x p are the initial coordinates and momentum of the sth electron at the time st0 when it intersects the z=0 plane, sssss dtdydddq 00000 xp ;  sqf 0 is the single particle momentum distribution of electrons, q is the region of integration over the initial coordinates and momenta of emitting electrons. We will assume that the relativistic electrons beam moves in the positive direction of the z axis in a static periodic magnetic field of the undulator  )sin()cos(0 zkzkH uyuxu eeH  , where uuk 2 , 0H and u are the amplitude and period of the magnetic field, zyx eee ,, are the unit vec- tors of the Cartesian coordinate system x, y, z. In the approximation of a small value of the undula- tor parameter, the expression for the diffusion coeffi- cient has the form [11]           )(K 0 2 001 222 0 szssz t t u z pwdpdtkeD   , (2) where           q ssszsb ss ss dtdydxn qtRqtR qtGqtG 000 011*0* 0110 v ;,;, ;,;, rr rr , (3)   *0* 0 22 * 2 0 2 * 2 0* 0 0 cos sin 2 ;, RkR R R R RkR R qtG s z zs zs zs s zsz zss                      r  *0 2 RRk zszuzs   , uzszss kk 2 0  . To obtain the explicit expressions for the diffusion coefficient, we use simplifying assumptions. Let us con- sider the spread in momenta of the electrons moving near the beam axis x0=y0=0. We will take into account the electromagnetic field of the emitting electrons mov- ing only behind the considered (test) electron. We as- sume that the distance between the considered electrons is much greater than the thermal dispersal of these elec- trons during the process under consideration, i.e. r >>. Nevertheless, in this case, thermal dispersal of the electrons at a distance greater than the wavelength of undulator radiation   2 0 2 21 su K  is possible. Assuming also that the beam radius is greater than the wavelength of undulator radiation in the transverse ISSN 1562-6016. ВАНТ. 2021. № 4(134) 103 direction:   2br , where zszsu  , using Eq. (2), we find        ,,cossin )( v4 K 2 0 1max 24 0 4 00 0 12 244           zrkd pwdpdz nke D uzs szssz z z bu z (4) where  cos1 zs . Suppose that the function w (pz0s) is a Maxwellian                 2 2 0 0 2 exp 2 1 th zmsz th sz p pp p pw . (5) We will assume that the initial thermal spread in the beam, as well as an increase in the energy spread of the electrons, due to the radiation interaction, satisfy the conditions pth  pzm m. Substituting (5) into (4) and integrating over the longitudinal momentum using the value of the integral [12], we obtain the following expression for the diffusion coefficient [11]                 2 0 11max 4 0 12 4244 ,sin v4 K zzzrd dz nke pD z z bmu zz (6) where    x z xx c          cosexp 2 2 ,        z zm u p p k 1 , uth z c kp pz 2  ,             sin , cos min 2 1 max zs b zszs rz r . Pre-Brownian stage. In the case czz  from ex- pression (6), taking into account (1), we obtain              rrr r z bum z zzzzBz zzz znkep dz d ,2 ,1615 v K 22 2244 2 , (7) where   x x x arctgx x xB 32 15 2 1ln2 416 3511 3 5 2                        . Kinetic stage. In the case czz  , the asymptotic expression for the diffusion coefficient (6) at rzz  takes the form     , 25 8 v K 16 5, 223 2 2 2 22225 2 2                                          th zmzcrp pp r cru z bm zz p pp z zze z zB zzknezpD th zmz (8) where   x arctgxxB       5 6 15 22 2 12 2 ,    x tx dtexex 0 22 21 . For very large deviations of the momentum from the equilibrium value (x is large) and very small deviations of the momentum (x is small), we have                           1... 15 4 3 2121 1.... 4 15 2 31 2 1 42 2 422 xxxx x xxx x (9) Fig. 1 shows the graph of the function  x . Fig. 1. Function  entering the formula for the diffusion coefficient Expression (8) at rzz  takes the form   2 2 22 2 24425 v16 K5 th zmz p pp cru z bm z ezzkneD      . (10) Thus, at large distances, the diffusion coefficient is independent of z. Such a dependence of the diffusion coefficient on the coordinate or on time corresponds to the kinetic stage of the evolution of a system consisting of a large number of particles (see, for example, [3, 13- 15]). 2. MONOENERGETIC BEAM For self amplification of spontaneous ultrashort wave- length electromagnetic radiation, the spread in the mo- mentum of relativistic electrons should be sufficiently small uuzmth Lpp  (see, for example, [5, 16]). This inequality can be rewritten as cu zL 2 , where DuuL 1 is the distance at which the intensity of electromagnetic radiation reaches its maximum value,     1 0 3132 1 4  AbbuD IIrK , bbzb nreI 2 0v , emcI A 3 =17 kА. Therefore, at the stage of sponta- neous emission (z<Lu), the initial energy spread of elec- trons can be neglected, assuming that the electrons are monoenergetic:    0zzszs pppw  . In this case, us- ing equations (2) and (3), the expression for the mean square spread of the longitudinal momentum can be written in the form   V kc nep u b z 2 44 2 K  , (11)            yx z xdxgydydV , 0 23 0 1 00 11 1max , (12) where   x xx x xg cos2sin21 2        ; ISSN 1562-6016. ВАНТ. 2021. № 4(134) 104  yx mmax at  1*0  yy ; 1max x at   11*  yy ;     2 0 0 1 y yy z zr m    ;              22 1 22 1 1* ,0max r ry ; zku , 11 zku ,  bubzzr rkr 200 . Formula (12) can be transformed into a simpler form                     422 8 0 1,2, 8 15 2 2 x xxudxxudxV r , (13) where      xgxxu 22,  ,  r ,min2 . Fig. 2 plots the normalized momentum spread     2222 cmp z as a function of coordinate z, calculated by formulas (11), (13) for the values Н0=1.4 kG, u=4 cm, rb=50 μm, Ib=1 kA (a), 4 kA (b), (nb=2.6 1015 cm-3, 1.32·1016 cm-3, respectively). The curves correspond to the radiation wavelength  and the energy of electrons 0 2  mcEb : 2 (5.76) (black), 1.5 (6.65) (red), 1.0 (8.15) (green), 0.5 Å (11.53 GeV) (blue). Fig. 2. Dependence of the normalized longitudinal momentum spread  2 on the z coordinate in meters; Ib=1 kA (a), 4 kA (b). The numbers on the curves represent the radiation wavelength  in angstroms The dependences on the z coordinate were calcu- lated up to the values of Lu. The white circles on the curves indicate the values relz of the momentum spread in coordinates in which the displacement of the elec- trons relative to the equilibrium trajectory, due to the radiation spread in the momentum, reaches half the wavelength of the radiation:     0 2 0 2122 zzzzzrel ppz rel . The black circles on the curves correspond to the displacement of the electrons by the wavelength  of the radiation. It can be seen from the Fig. 2 that the spread in mo- menta increases with an increase in the electron energy. The distance zrel also increases with an increase in the electron energy in proportional to 21 0 at z>zr. Since the characteristic length of the undulator Lu is proportional to the energy of the electrons, the relative distance urel Lz decreases with increasing the energy of the electrons. CONCLUSIONS The analysis shows that at the initial stage of the beam motion, when the distance traveled by it in the undulator is small compared to the distance zc, the rms spread <(Δpz)2>1/2 linearly depends on the coordinate z, for z>zr. The evolution of electrons in momentum space at z<<zc corresponds to the pre-Brownian motion of electrons. The distance zc is equal in order of magnitude to the distance, during which time τc (τc=zс/vz) the thermal dispersal of two electrons relative to each other reaches half the wavelength of the radiation: 22v cth . The time τc does not depend on either the beam density or the force of the pair interac- tion of electrons and is the duration of the "synchro- nous" interaction of two electrons. During this time, the phase of the force acting on the test electron changes to . At distances much larger than zc, the force of pair in- teraction changes rapidly. As a result, at large distances (z>>zr) the diffusion coefficient does not depend on the coordinate, and the rms spread in the longitudinal mo- mentum is proportional to z1/2. In this case, the kinetic stage of electron diffusion in momentum space is real- ized, and their motion becomes completely random. As follows from the above calculations, the radiative interaction of electrons can lead to an increase in the energy spread in the beams, which are used to obtain coherent electromagnetic radiation of the nanometer and a shorter wavelength range. For example, for an electron beam and an undulator with the parameters given above (see Fig. 2), an in- crease in the energy spread as a result of radiation re- laxation can prevent the production of coherent radia- tion with a wavelength less than 1.5 Å at Ib =1 kA and 1.0 Å at Ib = 4 kA, respectively. REFERENCES 1. V.V. Ognivenko. Momentum spread in a relativistic electron beam in an undulator // J. Exp. Theor. 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Article received 03.06.2021 ДИФФУЗИЯ ПО ИМПУЛЬСУ РЕЛЯТИВИСТСКИХ ЭЛЕКТРОНОВ С ТЕПЛОВЫМ РАЗБРОСОМ, ПРОХОДЯЩИХ В ОНДУЛЯТОРЕ В.В. Огнивенко Исследована диффузия по продольному импульсу электронов, движущихся в пространственно- периодическое магнитное поле ондулятора, с учетом их начального энергетического разброса. Получены выражения для коэффициента диффузии и определены зависимости его как от расстояния, пройденного электронами в ондуляторе, так и от величины начального энергетического разброса электронов. Обсуждает- ся возможность уменьшения длины волны в рентгеновских лазерах на свободных электронах. ДИФУЗІЯ ПО ІМПУЛЬСУ РЕЛЯТИВІСТСЬКИХ ЕЛЕКТРОНІВ З ТЕПЛОВИМ РОЗКИДОМ, ЩО ПРОХОДЯТЬ В ОНДУЛЯТОРІ В.В. Огнівенко Досліджено дифузію за поздовжнього імпульсу електронів, що рухаються в просторово-періодичне маг- нітне поле ондулятора, з урахуванням їх початкового енергетичного розкиду. Отримано вираз для коефіціє- нта дифузії й визначені залежності його як від відстані, вздовж ондулятора, так і від величини початкового енергетичного розкиду електронів. Обговорюється можливість зменшення довжини хвилі в рентгенівських лазерах на вільних електронах.

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