Features of new cyclotron resonances, as well as conditions for resonant acceleration of charged particles in a vacuum without a magnetic field

It is shown that the known conditions for cyclotron resonances are strictly valid only under autoresonance conditions or in the nonrelativistic case. In other cases, it is necessary to use the conditions established in the present work. The main features of charged particle dynamics under new resona...

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Veröffentlicht in:	Problems of Atomic Science and Technology
Datum:	2023
Hauptverfasser:	Buts, V.A., Zagorodny, A.G.
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Sprache:	English
Veröffentlicht:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2023
Schlagworte:	Relativistic and nonrelativistic plasma electronics
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spelling	Buts, V.A. Zagorodny, A.G. 2023-12-11T11:45:54Z 2023-12-11T11:45:54Z 2023 Features of new cyclotron resonances, as well as conditions for resonant acceleration of charged particles in a vacuum without a magnetic field / V.A. Buts, A.G. Zagorodny // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 3-7. — Бібліогр.: 13 назв. — англ. 1562-6016 PACS: 05; 41.75.Jv; 52.45.Ac; 76.40.+b DOI: https://doi.org/10.46813/2023-146-003 https://nasplib.isofts.kiev.ua/handle/123456789/196163 It is shown that the known conditions for cyclotron resonances are strictly valid only under autoresonance conditions or in the nonrelativistic case. In other cases, it is necessary to use the conditions established in the present work. The main features of charged particle dynamics under new resonant conditions are presented. Conditions are found for infinite acceleration of electrons by a transverse electromagnetic wave in a vacuum without a magnetic field. Показано, що відомі умови для циклотронних резонансів строго справедливі лише в умовах авторезонансу або в нерелятивістському випадку. В інших випадках необхідно використовувати умови, виписані в роботі. Наведено результати дослідження основних особливостей динаміки заряджених частинок за нових резонансних умов. Знайдено умови необмеженого прискорення електронів поперечною електромагнітною хвилею у вакуумі без магнітного поля. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Problems of Atomic Science and Technology Relativistic and nonrelativistic plasma electronics Features of new cyclotron resonances, as well as conditions for resonant acceleration of charged particles in a vacuum without a magnetic field Особливості нових циклотронних резонансів, а також умови резонансного прискорення заряджених частинок у вакуумі без магнітного поля Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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title	Features of new cyclotron resonances, as well as conditions for resonant acceleration of charged particles in a vacuum without a magnetic field
spellingShingle	Features of new cyclotron resonances, as well as conditions for resonant acceleration of charged particles in a vacuum without a magnetic field Buts, V.A. Zagorodny, A.G. Relativistic and nonrelativistic plasma electronics
title_short	Features of new cyclotron resonances, as well as conditions for resonant acceleration of charged particles in a vacuum without a magnetic field
title_full	Features of new cyclotron resonances, as well as conditions for resonant acceleration of charged particles in a vacuum without a magnetic field
title_fullStr	Features of new cyclotron resonances, as well as conditions for resonant acceleration of charged particles in a vacuum without a magnetic field
title_full_unstemmed	Features of new cyclotron resonances, as well as conditions for resonant acceleration of charged particles in a vacuum without a magnetic field
title_sort	features of new cyclotron resonances, as well as conditions for resonant acceleration of charged particles in a vacuum without a magnetic field
author	Buts, V.A. Zagorodny, A.G.
author_facet	Buts, V.A. Zagorodny, A.G.
topic	Relativistic and nonrelativistic plasma electronics
topic_facet	Relativistic and nonrelativistic plasma electronics
publishDate	2023
language	English
container_title	Problems of Atomic Science and Technology
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
format	Article
title_alt	Особливості нових циклотронних резонансів, а також умови резонансного прискорення заряджених частинок у вакуумі без магнітного поля
description	It is shown that the known conditions for cyclotron resonances are strictly valid only under autoresonance conditions or in the nonrelativistic case. In other cases, it is necessary to use the conditions established in the present work. The main features of charged particle dynamics under new resonant conditions are presented. Conditions are found for infinite acceleration of electrons by a transverse electromagnetic wave in a vacuum without a magnetic field. Показано, що відомі умови для циклотронних резонансів строго справедливі лише в умовах авторезонансу або в нерелятивістському випадку. В інших випадках необхідно використовувати умови, виписані в роботі. Наведено результати дослідження основних особливостей динаміки заряджених частинок за нових резонансних умов. Знайдено умови необмеженого прискорення електронів поперечною електромагнітною хвилею у вакуумі без магнітного поля.
issn	1562-6016
url	https://nasplib.isofts.kiev.ua/handle/123456789/196163
citation_txt	Features of new cyclotron resonances, as well as conditions for resonant acceleration of charged particles in a vacuum without a magnetic field / V.A. Buts, A.G. Zagorodny // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 3-7. — Бібліогр.: 13 назв. — англ.
work_keys_str_mv	AT butsva featuresofnewcyclotronresonancesaswellasconditionsforresonantaccelerationofchargedparticlesinavacuumwithoutamagneticfield AT zagorodnyag featuresofnewcyclotronresonancesaswellasconditionsforresonantaccelerationofchargedparticlesinavacuumwithoutamagneticfield AT butsva osoblivostínovihciklotronnihrezonansívatakožumovirezonansnogopriskorennâzarâdženihčastinokuvakuumíbezmagnítnogopolâ AT zagorodnyag osoblivostínovihciklotronnihrezonansívatakožumovirezonansnogopriskorennâzarâdženihčastinokuvakuumíbezmagnítnogopolâ
first_indexed	2025-11-26T00:41:52Z
last_indexed	2025-11-26T00:41:52Z
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fulltext	ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 3 RELATIVISTIC AND NONRELATIVISTIC PLASMA ELECTRONICS https://doi.org/10.46813/2023-146-003 FEATURES OF NEW CYCLOTRON RESONANCES, AS WELL AS CONDITIONS FOR RESONANT ACCELERATION OF CHARGED PARTICLES IN A VACUUM WITHOUT A MAGNETIC FIELD V.A. Buts 1,2 , A.G. Zagorodny 3 1 National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine; 2 Institute of Radio Astronomy of NAS of Ukraine, Kharkiv, Ukraine; 3 Bogolyubov Institute for Theoretical Physics, NAS of Ukraine, Kyiv, Ukraine E-mails: vbuts1225@gmail.com; azagorodny@bitp.kiev.ua It is shown that the known conditions for cyclotron resonances are strictly valid only under autoresonance condi- tions or in the nonrelativistic case. In other cases, it is necessary to use the conditions established in the present work. The main features of charged particle dynamics under new resonant conditions are presented. Conditions are found for infinite acceleration of electrons by a transverse electromagnetic wave in a vacuum without a magnetic field. PACS: 05; 41.75.Jv; 52.45.Ac; 76.40.+b INTRODUCTION In plasma physics and plasma electronics, two types of fundamental interaction processes play important role. These are the wave-particle and of wave-wave interactions. Below we will explore the wave-particle process in which resonances are of great importance. First of all, this concerns the Cherenkov resonance and cyclotron resonances. These resonances are the most widely used. Obviously, it is very important for applica- tions to increase the intensity of fields interacting with particles. The main parameter that characterizes the level of interacting fields is the wave strength parameter /eE mc  (nonlinear parameter). Usually, it is as- sumed that this parameter is small ( 1  ). Indeed, this parameter would be of the order of unit in the ten- centimeter range, if the intensity of electromagnetic fields is about 10 5 V/cm. For laser radiation, this intensi- ty should exceed 10 10 V/cm. So, to describe the interac- tion with the field of such it is necessary to take into account nonlinear effects. Note that the generally ac- cepted conditions for cyclotron resonances contain only the strength of the external magnetic field. In [1], the nonlinear particle dynamics was taken into account. The conditions for cyclotron resonances were formulated, which explicitly contain the parameter of the wave strength. The papers [1, 2] describe a large number of new features of particle dynamics in high-amplitude fields. The purpose of this work is to generalize the obtained results. In this case, the main attention is paid to identifying the conditions for the possibility of reduc- ing the field strength to have particle acceleration. Phys- ical considerations indicate that this can be achieved in the region of parameters that corresponds to autoreso- nance. In [1, 2] a wave with only one polarization was considered. In this paper, another polarization for cyclo- tron resonances is considered, and the results for wave- particle interaction for two polarizations will be pre- sented. The work consists of Introduction, three sections and Conclusions. In the Section 1, we formulate the state- ment of the problem and basic equations. In the Sec- tion 2, new variables are introduced and conditions for new cyclotron resonances are formulated in more gen- eral case than in [1]. It is shown that the main features of resonances are the same for the fields of different polarizations. It is shown that the known conditions of cyclotron resonances are strictly valid only for autores- onances and for the case of nonrelativistic motion of particles ( 1  ). In all other cases, the new conditions should be used. The Section 3 describes new variables that made it possible to find the conditions for unlimited acceleration of charged particles (electrons) by a laser field in a vacuum without an external magnetic field. In Conclusions, the most important results of the work are formulated. 1. STATEMENT OF THE PROBLEM AND BASIC EQUATIONS Consider a charged particle that moves in the sta- tionary homogeneous external magnetic field and in the field of a plane electromagnetic wave that in the general case has the components Re( exp( ))E i t i E α kr ,  Re exp( ) c i t i         H kE kr , (1) where 0EE α ,  , ,x y zi  α is the wave polariza- tion vector. We choose a coordinate system in which the wave vector of the wave has only two components xk and zk . It is also convenient to use the dimensionless dependent and independent variables / mcp p , t  , c  r r ,   kr. The equations of motion in terms of these variables are given by      1 Re Re , 1 , i iHd e e d d d d d                               p kp k ε ph ε p r p kp v (2) , mailto:vbuts1225@gmail.com mailto:azagorodny@bitp.kiev.ua 4 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) where 0/ Hh H , 0 /H eH mc  , 0ε α , 0 0( / )eE mc  ,   kr , k is the unit vector in the direction of the wave vector, 2 1 2(1 )  p is the particle’s dimensionless energy (measured in the units of 2mc ), p is the momentum of the particle, 0 H is the external magnetic field directed along the z-axis. Multiplying the first equation of the set (2) by p, we obtain a useful equation that describes the change in particle energy, i.e.  Re ie  v . (3) Equations (2) and (3) have integrals given by        0 0 Re Re - =const. i H i H i e i e              0 0 p ε rh k p k ε r h (4) Here the subscript “0” denotes the values of the ini- tial variables. 2. CYCLOTRON RESONANCES The set of vector equations (2), (3), even taking into account the integrals (4), can be fully analyzed only by numerical methods. However, many important features of charged particle dynamics can be discovered using new variables cos , sin , , sin , cos . x y z H H p p p p p p p p x y                      (5) An analysis of the general case leads to the need to analyze very cumbersome and complex systems of equations. In order to simplify expressions obtained for analysis, below we consider the dynamics of particle motion in the field of a polarized wave. Many features of particle dynamics in the field E polarization were described in [1, 2]. Below we will consider the dynam- ics of particles in the field H polarized wave  , ,x z yE E H . We will assume that the wave vector has the following components ( , )x zk k . In view of the above, the complete system of equations that describe the dynamics of particles in new variables can be re- duced to the following     1 cos x x H y x x x z z dp p d k p p                                 kp , 0 y H x dp p h d      ; (6)    1 cosz z z x x z z dp k p p d                         kp ; x x p x v    ; y y p y v    ; z z p z v    ; 1 .    kp To find the conditions for the resonant interaction of waves with particles (the conditions for cyclotron reso- nances), we use the following expansion of functions into series in terms of Bessel functions [3]: exp( sin ) ( )exp( )n n i J in         . (7) Taking this expansion into account, the system of equations (6) can be rewritten     1 1 cos cos cos cos x z z x x x z z k p k p p k p p                                    ; (8)     1 cosz z x x z z z x x z z k p k v k v p p                     1 (1 )cos .. .. cos sin H H p v v                 ;  .. ..H         , where cos ( )cos( )n n J n        ;       1 1 cos 1 .. .. sin cos cos x z z x x x z z k p k p p k p p                                   . The cyclotron resonance conditions will be as the stationarity conditions for the phase of one term from the sum on the right side of system (8):  1n z xk z k n const         , (9)        0 1 1 1 1/ cos 1 .. .. sin H n z z x x H k v n k v k v n                            It is convenient to rewrite the resulting expression in the form of three terms: 0 1 2 0n      , (10) where 0 1 H z zk v n            ;  1 1 1/ cosxk v      ;  1 1 1/ cosxk v      ;  2 .. .. sinx H k v n         . The first term in expression (10) describes the usual condition for cyclotron resonance. The last term de- scribes the role of the electric intensity of the external wave ( 2   ). Note that the second term in 2 can be omitted in almost all cases of interest to us. The middle term ( 1 ), as will be seen below, determines the charac- teristic features of the particle dynamics on the steps. In addition, the presence of this term significantly changes the cyclotron resonance conditions even when the wave strength parameter can be neglected ( 0  ). The pres- ence of this term indicates the fact that the commonly used widely used cyclotron resonance conditions (the first term in (10) 0 0  ) are strictly valid only under the conditions of cyclotron autoresonance ( 0xk  ) or in nonrelativistic case ( 1  ). In all other cases, this term ; . ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 5 can significantly change the resonance conditions. In particular, this term can lead to a limitation of the ener- gy transferred from the wave to the particles. 2.1. NUMERICAL INVESTIGATION The analytical results described above point to a sig- nificant difference between the dynamics of particles at cyclotron resonances and the usual dynamics of parti- cles at these resonances. However, these results were obtained using a fairly large number of large and small parameters. Therefore, it is of considerable interest to find out how these results look when numerically solv- ing the original equations not transformed to new varia- bles. Such decisions have been made. Fig. 1. The dependence of the longitudinal momentum of the particle on time. Options: 0.5, 0.05, 0.99x z H     ; 0.8, 0.6z xk k  ; (0) (0) (0) 0.01x y zp p p   Some of the most representative results of these nu- merical studies are presented below in Figs. 1-4. When selecting pictures for demonstration, preference was given to those values of the parameters at which the new features of particle dynamics were most clearly mani- fested. In addition, such parameters were chosen at which the wave strength parameter was, if possible, smaller. Fig. 1 shows the dependence of the longitudinal momentum of the particle on time for a sufficiently small value of the wave strength parameter ( 0.5  ). Despite such a small parameter, one can see the most noticeable feature characteristic of particle dynamics at new cyclotron resonances. This feature consists in the appearance of steps in the time dependence of momenta and energy. Fig. 2. The same parameters as in Fig. 1 Fig. 2 shows the plot of the dependence of the longi- tudinal impulse on time, as well as the form of the func- tion 1 . One can see the dependence of the characteris- tics of the steps, as well as the dependence of the mo- ments of jumps on the characteristics of the function 1 . This function first appeared in [2]. The presence of this function leads to a significant limitation of the use of familiar conditions for the implementation of cyclotron resonances. It can be seen from this figure that jumps between steps occur in the region of the maximum value of this function. In addition, the time width of the steps is also determined by the period of the function 1 . An interesting feature is manifested in the transverse dy- namics of particles at resonances. This feature is shown in Fig. 3. Fig. 3. Time dependence of the transverse coordinates of the particles. Options: εx = 0.9, εz = 0.09, ωH = 0.99; 0.8, 0.6z xk k  ; (0) (0) (0) 0.01x y zp p p   It can be seen that the maxima of the function ( )y  goes to zero. In addition, it turns out that at the same points vanishes and ( )x  . This feature of the transverse particle trajectory leads to the fact that a point appears on the transverse particle trajectory through which all particle trajectories pass. This feature of the trajectory is shown in Fig. 4. Fig. 4. Trajectory of particles in the transverse plane. Options: εx = 0.9, εz = 0.09, ωH = 0.99; (0) (0) (0) 0.01x y zp p p   It is the appearance of such a common point for all trajectories that leads to jumps of the particle from one step to another. Moreover, this process (these jumps) is random. The characteristics of such randomness were first described in [1]. Similar regimes with dynamic 6 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) chaos in systems with one degree of freedom are de- scribed in [4 - 6]. Note that this randomness is similar to throwing a die with an unlimited number of faces. This feature of the particle dynamics is clearly manifested at sufficiently significant particle energies ( 20  ). At smaller ones, it is not so noticeable. It is also useful to add that such dynamics (as shown in Fig. 1, for exam- ple) is characterized by intermittency. In our case, this means that the particle dynamics on the steps is regular. Randomness occurs only at moments of jumps. Moreover, the magnitude of the jump is significantly larger than the changes in the magnitudes of the impuls- es at the steps themselves. The main feature of regimes with intermittency is the appearance of higher moments, which turn out to be larger than the lower moments [7]. Some features and details of this regime are described in [8]. 3. RESONANCES AND ACCELERATION OF PARTICLES IN VACUUM WITHOUT EX- TERNAL MAGNETIC FIELD Accelerating charged particles in a vacuum is an at- tractive option. This is especially true for laser accelera- tion schemes. There are many attempts to find such acceleration schemes. There is a large number of works that describe various scenarios for such acceleration. One of the last works in this direction is the work [9] (see also the literature cited therein). Below we will show that taking into account the strength of an elec- tromagnetic wave that interacts with particles, as well as the presence in this wave of the transverse component of the wave vector, allows us to formulate the resonant conditions for the interaction of waves with particles, as well as to carry out unlimited acceleration of charged particles in vacuum by transverse electromagnetic waves without an external magnetic field. The initial system of equations is written above (see system of equations (2)). In this system, you need to put 0H  . Formulas (5) in this case must be replaced by other formulas. As the latter, we accept the following trans- formation formulas: \|\|cos , sin , , sin , cos . x y zp p p p p p p p x y                    (11) To simplify the form of the formulas below, we will analytically present only the expressions for the case when the wave has only the following components  , ,y z xE H H . The system of equations (2) for new variables in this case can be rewritten:   1 2 2 sin 1 cos sinx z zk p k p p                      ,     2 sin cos 2 1 sin sin x y x x z z p k p k p k p                      (12) cos sinx p v v            . Using formula (7) and the considerations that were used above to obtain cyclotron resonant conditions, we can leave only one resonant term in the equation for the new angular variable on the right side:   1 2 1 ( )sin( ) 2 z z n nk p J p                . (13) Here n z xn k z k n           , /xk p  . At 0n  the resonant conditions will be condition   ; 0n const       . (14) These conditions can be conveniently rewritten 0 sin     , (15) where  0 1 z z x xk v k v    , 0 1 ( ) 2 z zk p J p            . Equation (15) is the Adler equation [10]. This equa- tion has been studied in synchronization theory and is widely used (see, for example [11 - 13]). 3.1. NUMERICAL RESULTS The particle dynamics is very sensitive to even small changes in parameters. So, a small change in the con- figuration of an electromagnetic wave can significantly change this dynamics. A typical example is shown in Figs. 5 and 6. Fig. 5. The dependence of the longitudinal momentum of the particle on time. Options: (0) 10, (0) 0.4, (0) 0.1z x yp p p   ; 0.5, 0, 0y x z     ; 0.995, 0.099z xk k  , max 104zp  Let us pay attention to the appearance of flat sec- tions in the dependence of momentum on time (see Fig. 5). This feature is characteristic of solutions of the Adler equation. Fig. 6. Time dependence of the longitudinal momentum. Options: (0) 10, (0) 0.1, (0) 0.1z x yp p p   ; 0.5, 0, 0.025x y z     ; 0.999, 0.051z xk k  , max 1922zp  , ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 7 It is easy to see from Eq. (15) that the capture of par- ticles into unlimited acceleration can occur at sufficient- ly small  . Really:    2 2 2 0 1 1 1 1/z z x x z xk v k v v k         , p   , 1/  . (16) CONCLUSIONS Let us formulate the most important results of the work. 1. As follows from the new expressions for cyclo- tron resonances (10), the known conditions for the oc- currence of cyclotron resonances ( 0 0  ) can be strict- ly fulfilled only for autoresonance ( 0xk  ) or in the nonrelativistic case ( 1  ). In all other cases, formula (10) should be used. 2. Formula (10) was obtained for an H-polarized wave. However, the resonance structure does not de- pend on polarization. For the E-wave, only the third term ( 2 ) in formula (10) will change slightly. 3. The most important result of the work is the demonstration of the fact that taking into account the electric strength of the wave, as well as taking into ac- count its transverse component of the wave vector, made it possible to discover the resonant condition for unlimited acceleration of charged particles by the field of transverse electromagnetic waves in vacuum without an external magnetic field. It turned out that rather mod- erate field strengths can be used to capture particles in unlimited acceleration ( 1/  ). REFERENCES 1. V.A. Buts, A.G. Zagorodny. New cyclotron reso- nances and features of charged-particle dynamics in the presence of an intense electromagnetic wave // Phys. Plasmas (28). 2021, 022311; https://doi.org/10.1063/5.0037808. 2. V.A. Buts, A.G. Zagorodny. New resonances in wave-particle interactions // Phys. Plasmas. 2023, v. 30; https://doi.org/10.1063/5.0143202. 3. G. Korn and T. Korn. Handbook of Mathematics. M.: “Science”, 1968. 4. D. Dixon, F.W. Cummings, and P.E. Kaus. Continu- ous chaotic dynamics in two dimensions // Physica. 1993, D 65, p. 109-116. 5. J. Alvarez-Ramirez, J. Delgado-Fernandez, and G. Espinosa-Parades. The origin of a continuous two-dimentional ‘chaotic’ dynamics // Int. J. Bifur- cation Chaos. 2005, v. 15(9), p. 3023-3029. 6. V.A. Buts. Singular solutions and dynamic chaos // Problems of Atomic Science and Technology. Series “Plasma Physics”. 2015, № 4, p. 232-236. 7. Ya.B. Zeldovich, S.A. Molchanov, A.A. Ruzmaikin, D.D. Sokolov. Intermittence in a random environ- ment // UFN. 1987, v. 152, № 1, p.3-32. 8. V.A. Buts, V.V. Kuzmin. The role of higher mo- ments on the distribution of particles in the momen- tum space at cyclotron resonances // Problems of Atomic Science and Technology. Series “Plasma Electronics and New Methods of Acceleration”. 2023, № 4, p. 16-20. 9. N.N. Rozanov, N.V. Vysotina. Direct acceleration of charge in vacuum by pulses radiation with linear po- larization // ZhETF. 2020, v. 157, issue 1, p. 63-66. 10. Robert Adler. A Study of Locking Phenomena in Oscillators // Proc. I.R.E. Waves and Electrons. 1946, v. 34(6), 351 p. 11. A. Pikovsky, M. Rosenblum, and J. Kurths. Syn- chronization. Fundamental Nonlinear Phenomenon. M.: “Tekhnosfera”, 2003 12. V.M. Kuklin, D.N. Litvinov, S.M. Sevidov, A.E. Sporov. Simulation of synchronization of non- linear oscillators by the external field // East Eur. J. Phys. 2017, v. 4, № 1, p. 75-84. 13. V.A. Buts, A.G. Zagorodny. On effective accelera- tion of charged particles in vacuum // Problems of Atomic Science and Technology. 2021, № 4, p. 39- 42; https://doi.org/10.46813/2021-134-039. Article received 15.06.2023 ОСОБЛИВОСТІ НОВИХ ЦИКЛОТРОННИХ РЕЗОНАНСІВ, А ТАКОЖ УМОВИ РЕЗОНАНСНОГО ПРИСКОРЕННЯ ЗАРЯДЖЕНИХ ЧАСТИНОК У ВАКУУМІ БЕЗ МАГНІТНОГО ПОЛЯ В.О. Буц, А.Г. Загородній Показано, що відомі умови для циклотронних резонансів строго справедливі лише в умовах авторезонан- су або в нерелятивістському випадку. В інших випадках необхідно використовувати умови, виписані в робо- ті. Наведено результати дослідження основних особливостей динаміки заряджених частинок за нових резо- нансних умов. Знайдено умови необмеженого прискорення електронів поперечною електромагнітною хви- лею у вакуумі без магнітного поля. https://doi.org/10.1063/5.0143202 https://doi.org/10.46813/2021-134-039

Features of new cyclotron resonances, as well as conditions for resonant acceleration of charged particles in a vacuum without a magnetic field

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