$A fractal accelerator on the basis of a corrugated plasma waveguide with superconducting walls$

A fractal accelerator on the basis of a corrugated plasma waveguide with superconducting walls

The use of a two-stage mechanism for the acceleration of charged particles is proposed. The principle of acceleration is based on the use of the fractal properties of the wave spectrum of a corrugated plasma waveguide with superconducting walls. The first stage provides for the excitation of a corru...

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Datum:	2019
Hauptverfasser:	Tkachenko, I.V., Tkachenko, V.I.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2019
Schriftenreihe:	Вопросы атомной науки и техники
Schlagworte:	Advanced methods of acceleration
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/195167
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spelling	irk-123456789-1951672023-12-03T16:37:14Z A fractal accelerator on the basis of a corrugated plasma waveguide with superconducting walls Tkachenko, I.V. Tkachenko, V.I. Advanced methods of acceleration The use of a two-stage mechanism for the acceleration of charged particles is proposed. The principle of acceleration is based on the use of the fractal properties of the wave spectrum of a corrugated plasma waveguide with superconducting walls. The first stage provides for the excitation of a corrugated plasma waveguide by short electron bunches in the direction of motion (the length of the bunch is significantly less than the period of the corrugation). On the second stage the test charged particles accelerates in an infinite number of harmonics of the electric field exited by an electron bunches. Calculations show that with the implementation of such an acceleration mechanism, the average speed of a non-relativistic test particle can increase several times, and its acceleration length can be up to one meter. Запропоновано використання двоступеневого механізму прискорення заряджених частинок. Принцип прискорення заснований на використанні фрактальних властивостей хвильового спектра гофрованого плазмового хвилеводу з надпровідними стінками. Перший ступінь забезпечує збудження гофрованого плазмового хвилеводу короткими в напрямку руху електронними згустками (довжина згустку значно менше періоду гофра). Другий ступінь здійснює прискорення пробних заряджених частинок у збудженому електронними згустками нескінченному за кількістю гармонік електричному полі. Розрахунки показують, що при реалізації такого механізму прискорення середня швидкість нерелятивістської пробної частинки може збільшуватися в кілька разів, а довжина її прискорення може становити відстань до одного метра. Предложено использование двухступенчатого механизма ускорения заряженных частиц. Принцип ускорения основан на использовании фрактальных свойств волнового спектра гофрированного плазменного волновода со сверхпроводящими стенками. Первая ступень обеспечивает возбуждение гофрированного плазменного волновода короткими в направлении движения электронными сгустками (длина сгустка значительно меньше периода гофра). Вторая ступень осуществляет ускорение пробных заряженных частиц в возбужденном электронными сгустками бесконечном по количеству гармоник электрическом поле. Расчеты показывают, что при реализации такого механизма ускорения средняя скорость нерелятивистской пробной частицы может увеличиваться в несколько раз, а длина ее ускорения может составлять расстояние до одного метра. 2019 Article A fractal accelerator on the basis of a corrugated plasma waveguide with superconducting walls / I.V. Tkachenko, V.I. Tkachenko // Problems of atomic science and technology. — 2019. — № 4. — С. 59-64. — Бібліогр.: 15 назв. — англ. 1562-6016 PACS: 52.40.Fd; 52.40.Mj http://dspace.nbuv.gov.ua/handle/123456789/195167 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	Advanced methods of acceleration Advanced methods of acceleration
spellingShingle	Advanced methods of acceleration Advanced methods of acceleration Tkachenko, I.V. Tkachenko, V.I. A fractal accelerator on the basis of a corrugated plasma waveguide with superconducting walls Вопросы атомной науки и техники
description	The use of a two-stage mechanism for the acceleration of charged particles is proposed. The principle of acceleration is based on the use of the fractal properties of the wave spectrum of a corrugated plasma waveguide with superconducting walls. The first stage provides for the excitation of a corrugated plasma waveguide by short electron bunches in the direction of motion (the length of the bunch is significantly less than the period of the corrugation). On the second stage the test charged particles accelerates in an infinite number of harmonics of the electric field exited by an electron bunches. Calculations show that with the implementation of such an acceleration mechanism, the average speed of a non-relativistic test particle can increase several times, and its acceleration length can be up to one meter.
format	Article
author	Tkachenko, I.V. Tkachenko, V.I.
author_facet	Tkachenko, I.V. Tkachenko, V.I.
author_sort	Tkachenko, I.V.
title	A fractal accelerator on the basis of a corrugated plasma waveguide with superconducting walls
title_short	A fractal accelerator on the basis of a corrugated plasma waveguide with superconducting walls
title_full	A fractal accelerator on the basis of a corrugated plasma waveguide with superconducting walls
title_fullStr	A fractal accelerator on the basis of a corrugated plasma waveguide with superconducting walls
title_full_unstemmed	A fractal accelerator on the basis of a corrugated plasma waveguide with superconducting walls
title_sort	fractal accelerator on the basis of a corrugated plasma waveguide with superconducting walls
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2019
topic_facet	Advanced methods of acceleration
url	http://dspace.nbuv.gov.ua/handle/123456789/195167
citation_txt	A fractal accelerator on the basis of a corrugated plasma waveguide with superconducting walls / I.V. Tkachenko, V.I. Tkachenko // Problems of atomic science and technology. — 2019. — № 4. — С. 59-64. — Бібліогр.: 15 назв. — англ.
series	Вопросы атомной науки и техники
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first_indexed	2025-07-16T23:00:35Z
last_indexed	2025-07-16T23:00:35Z
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fulltext	ISSN 1562-6016. ВАНТ. 2019. №4(122) 59 A FRACTAL ACCELERATOR ON THE BASIS OF A CORRUGATED PLASMA WAVEGUIDE WITH SUPERCONDUCTING WALLS I.V. Tkachenko1, V.I. Tkachenko1,2 1National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine; 2V.N. Karazin Kharkiv National University, Kharkiv, Ukraine E-mail: tkachenko@kipt.kharkov.ua The use of a two-stage mechanism for the acceleration of charged particles is proposed. The principle of acceler- ation is based on the use of the fractal properties of the wave spectrum of a corrugated plasma waveguide with su- perconducting walls. The first stage provides for the excitation of a corrugated plasma waveguide by short electron bunches in the direction of motion (the length of the bunch is significantly less than the period of the corrugation). On the second stage the test charged particles accelerates in an infinite number of harmonics of the electric field exited by an electron bunches. Calculations show that with the implementation of such an acceleration mechanism, the average speed of a non-relativistic test particle can increase several times, and its acceleration length can be up to one meter. PACS: 52.40.Fd; 52.40.Mj INTRODUCTION The term wakefield acceleration indicates that Che- renkov radiation [1], propagating behind the particle in a medium or a slowing structure, is used to accelerate a low-current bunch to higher energies [2]. Currently, this method of acceleration is the subject of intensive research (see, for example, [3, 4]) and in this direction quite encouraging results have been ob- tained. However, the investigations of the wakefield accel- eration in periodic plasma waveguides (for example, corrugated sinusoidally) remain an open question con- cerning accounting of the full set of radial and longitu- dinal electric fields harmonics. Typically, one or more resonant [2] eigenmodes are taken into account from an infinite number them, and the rest is neglected because of assumptions about their weak interaction with the accelerated particles. In fact, unaccounted waves, despite their small con- tribution to the acceleration of charged particles, can greatly change the dispersion of the plasma waveguide [5, 6]. Therefore, the study of taking into account of the full spectrum of a superconducting corrugated wave- guide with plasma filling to accelerate charged particles is, as before, a very topical issue. The task is to indicate the plasma and corrugated waveguide parameters, which would ensure the acceler- ation of charged particles over the shortest interaction length. To solve this problem, it is necessary, first, to study the dispersion properties of the waveguide structure, which takes into account absolutely all eigenmodes, and, secondly, to study their interaction with accelerated charged particles. When considering the latter issue, the problem of excitation of such waveguides, for example, by charged particles, is to be solved and then the excited wave spec- trum utilization for the acceleration of the test charged particles may be proposed. According to the method of excitation of a longitu- dinal electric field in a plasma, the proposed method of acceleration can be referred to as the bunch method [2]. Therefore, we restrict ourselves to considering just such an acceleration scheme: we describe the dispersion properties of corrugated plasma waveguides, investigate the possibility of their excitation by electron bunches, and finally consider the possibility of using such sys- tems to accelerate test charged particles. 1. FEATURES OF THE DISPERSION PROPERTIES OF CORRUGATED PLASMA WAVEGUIDES The study of the dispersion properties of waveguides with an ideally conducting metal wall, corrugated ac- cording to the law 0( ) (1 cos( ))H z a k za= + and filled with plasma (here 00 2 kL π= is a corrugation period, b a a = is a corrugation parameter; b is a corrugation depth, 2a is a transverse size of the waveguide) showed the formation of a dense wave spectrum [7 - 9]. This means that in the phase space at any deliberate- ly selected point with coordinates ( )0 0,qω , where: 0ωω = , 03 qk = both the frequency and the longitudi- nal wave number, there exists an infinite number of waves. In this case, a charged particle that moves in such medium with a velocity 0V , will resonantly interacts with an infinite number of longitudinal waves, that sat- isfy the requirement of phase matching 0 0 0V qω= , and are characterized by both longitudinal 0q and trans- verse 2 lk a a π π ⊥ = + wave numbers (case of flat geome- try, 0; 1; 2;...,l K= ± ± , K − integer). Although the excited waves are polarized due to the discrete nature of the wave spectrum (their phase veloci- ty is equal to фV , and the group wavelength is zero), the presence of the transverse component of the wave vector gives them the character of Cherenkov waves [1]. Moreover, the velocity of the particle must satisfy the condition: 0 0 ф q V V k⊥ > . mailto:tkachenko@kipt.kharkov.ua ISSN 1562-6016. ВАНТ. 2019. №4(122) 60 The resonant nature of particles interaction with the waveguide eigenwaves requires that the parameter m∆ introduced in [10 - 13] should be considered zero. In addition, depending on which of the forbidden bands of the corrugated plasma waveguide the frequency 0ω falls, it is necessary to take into account the correspond- ing decrement 0( )d ω , which is proportional to the depth of the corrugation a (in the opacity band, the amplitude of wave A decays according to the law dteAA −= 0 ). In this case, the reactive bunch instabilities [10 - 12] may change to dissipative [11, 14], which is non-threshold, and is characterized by smaller values of increments. Thus, to study the acceleration of charged particles in corrugated plasma waveguides with superconducting walls, it is necessary to study the problem of the reso- nant interaction of charged particles with a plasma in the presence of a dense wave spectrum. 2. ANALYSIS OF THE EXCITED WAVE SPECTRUM OF A CORRUGATED PLASMA WAVEGUIDE Lets consider a two-step scheme for the acceleration of charged particles in an infinite corrugated plasma waveguide. The first step is the excitation of a corrugated plasma waveguide with a short electron bunch in the direction of the particles longitudinal movement (the length of the bunch is significantly less than the period of the corru- gation). The second step is the acceleration of test charged particles in an electric field excited by an electron bunch of an infinite number of harmonics. Calculations of the dispersion properties of a flat plasma filled corrugated waveguide, carried out in [7 - 9], allow us to imagine the strength of the longitudinal electric field .zE in the form: ( ) 3( , , ) ik z i tnE r z t a f r ez n nn ω∞ − = ⋅∑ = −∞ , (1) where an − harmonic amplitudes; ( )f rn − membrane functions that describe the transverse distribution of the electric field strength and all are equal to 1 at 0r → ; 3 3 0nk k nk= + ; 3k − longitudinal wave number; ω − wave frequency. Let us consider the excitation of a field in the form (1) and determine the possibility of setting the coeffi- cients na necessary for acceleration. The method of incomplete numerical simulation can show that a short electron bunch which parameters correspond to the pa- rameters of a very narrow forbidden band in a corrugated plasma waveguide (decrement 0( )d ω is small compared to the bunch instability increment) excites wave (1), but with special conditions for the coefficients na : a suffi- ciently large number of these coefficients coincide with each other (Fig. 1). This statement can be justified by a series of simple transformations of the expression for the field (1). As a result of such transformations, it is easy to obtain the following total δ-functional dependence of the electric field intensity on time: 0 0(0, , ) ( )n n i tE z t E e t tz ω δ =−∞ ∞−= −∑ , (2) where 0E is the amplitude, generally weakly dependent on time. In expression (2), due to the smallness of the contri- bution, the balance of the sum, which is determined by the coefficients of expression (1) remaining from the first sum, not equal to each other, is omitted. In Fig. 1 shows the temporal dynamics of the electric field strength at a particular point in space, obtained by the method of incomplete numerical simulation, similar to the calculations carried out in [2] and the sources cited there. It can be seen that the periodic amplitude pulsations are of sufficient magnitude to approximate them by a sum of δ-functional dependencies. Fig. 1. The temporal dynamics of the electric field strength at a single point in space ζ with the number of modes M = 100, the absorption index d = 0.01 Thus, when a short electron bunch moves in a corru- gated plasma waveguide, a spectrum (1) is excited, in which K harmonics can be equal to each other. From the mathematical point of view, this possibility arises be- cause in the infinite determinant obtained in the study of the dispersion properties of a flat corrugated plasma waveguide, the term K coincides. In the case when K >> 1 (1 K − the multiplicity points of the intersection of the dispersion curves [8 - 10]), the infinite sum of harmonics can be represented as an infinite sum δ- functions. In expression (2), the amplitude 0E is the amplitude of K harmonics, nt − the time that determines the passage of the n-th period of the waveguide by the bunch. 3. ANALYSIS OF THE CORRUGATED PLASMA WAVEGUIDE EXCITED WAVE SPECTRUM Now, after the accelerating field (2) is determined, let’s consider the motion of a test positively charged particle with the mass i em m= Λ ⋅ (where Λ >> 1) in this field. The change in time of its coordinates ζ near the axis of the waveguide is described by the equation: 2 1 3 0 0 02 ( / ) ( / )d i k k n ie d n ζ ζ ω τ τ e− ∞ + − Ω= Λ ∑ = −∞ , (3) where 0 0, ,t zkτ ω ζ= = 0 0 0 2 0 2 , e ek E m π ω e = 0 iω ωΩ = − ∆ , 0ω − is the frequency at which the intersection of an infinite number of dispersion curves of a corrugated ISSN 1562-6016. ВАНТ. 2019. №4(122) 61 plasma waveguide is observed, ω∆ − the width of the forbidden band, which corresponds to the frequency 0ω , 3k − is the longitudinal wave number. We assume that the decrement is small, and in the calculations should be assumed to zero. Equation (3) in a view as the above can be reduced to the form: 3 2 1 0 0 02 ( ) ( ( ) 2 ) k i i i d ke n nd ωζ τ τ ζ ω δ ζ τ π τ e− ∆ − + ∞ =Λ −∑ =−∞ . (4) Let’s find a solution to this equation. First of all, we note that in the time intervals 1k kτ τ τ +< < , where kτ is determined from the equation ( ) 2k kζ τ π= , the test particle moves with a constant velocity, the dimensionless value of which we denote as 0k kV w V= , where kw is a test particle’s velocity in the interval 1k kτ τ τ +< < , 0V is the initial velocity of the test particle. At time kτ , the velocity abruptly changes from kV to 1kV + . The relationship between these veloci- ties can be obtained from equation (4) by performing its integration within 1, ( )k k k kτ e τ τ e e τ τ+− < < + << − and then directing the small parameter e to zero. As a result, one can get a recurrent relation that determines the test particle’s velocity at each of the acceleration intervals: 1 0 3 0 0 2 2exp( ) cos( 2 ).1 kk kV V kk k V V k Vk k k ω π ππ ω e−Λ ∆ = + − ⋅ −+ In an infinite corrugated plasma waveguide, the pa- rameter 3 0k k can take any value less than one, i.e. it is determined by its fractional part (if the parameter 3 0k k is greater than one, then the integer part will be sub- tracted due to the periodicity of the function, the argu- ment of which it is). Therefore, in further calculations, we will assume this parameter to be less than one. Fig. 2 shows the graphs of the growth rate of a test positively charged particle’s velocity as a function of distance, measured by the number of periods of a corru- gated plasma waveguide k. In the calculations, the fol- lowing parameters of the accelerating system were cho- sen: 1 0e−Λ = 0.15; 0ω ω∆ = 0.02. Fig. 2. Dependence of the positively charged test parti- cle’s velocity at the distance 0L L k= versus of the parameter value: 1 − χ =0.1; 2 − χ =0.3; 3 − χ =0.5; 4 − χ =0.7; 5 − χ =0.9; 6 − χ =0.0 (numbering of curves is located under the corresponding curves) It can be seen from the figure that acceleration is more efficient near one parameter value 0.3χ = . For other values of the parameter χ , either a lower acceler- ation rate or a decrease in the initial velocity is ob- served, i.e. there is a deceleration. Thus, the above example confirms the possibility of the two-step mechanism for the acceleration of charged particles in a plasma waveguides with a superconduct- ing corrugated walls. In this case, the non-relativistic dimensionless velocity of the test particle may increase several times. 4. THE INVESTIGATION OF THE TWO-STEP METHOD OF CHARGED PARTICLES ACCELERATION For a more detailed analysis of the proposed two- step acceleration of charged particles, we use the meth- od of incomplete numerical simulation, which has re- cently been used to study the generation of electromag- netic waves in plasma waveguides [10 - 14]. In our case, we will investigate the motion of parti- cles of two types in the plasma in the presence of a large number of waves. In the longitudinal direction, we as- sume the plasma is unlimited, and we exclude trans- verse boundary from consideration due to a strong ex- ternal magnetic field, which allows the use of a one- dimensional description of the motion of charged parti- cles. For this, it is necessary that the condition 2 1/2 0 0(4 )e e e e eeH m c e n mω π= >> Ω = to be fulfilled where en0 is equilibrium plasma density; em is the electron mass; 0H is external magnetic field intensity; c is the speed of light. First, we define the type of electromagnetic field that will be excited, and then accelerate the particles. The wave packet in plasma is represented as the sum of a large number of waves that move with different phase velocities: ( ) 1 ( , ) ( )sin M m m m m m E z t E t k z t tω φ = = − +  ∑ . (5) Here M >> 1 is a large number of modes that propa- gate in the plasma; mmmm kE ϕω ,,, are an ampli- tude of electric field, frequency, wave number and phase of the m-th mode. We consider the electric field intensity in the form that takes into account only resonant harmonics. To do this, we assume that 0ωω ∆= mm , 0kmkm ∆= ( 1...m M= ), where 0ω∆ , 0k∆ are the conditional size of the perturbation waveform partitioning; 0k∆ is cho- sen in such a way that the relation 000 ω∆=∆ Vk is fulfilled. Due to such transformations, we obtain the expression for the wave packet: ( ) ( )0 0 1 ( , ) ( )sin M m m m E z t E t m k z V t tφ = = ∆ − +  ∑ . (6) The system of equations that describes the time vari- ation of the amplitudes and phases of the electric field (6) can be obtained by substituting the results of inte- grating the equations of plasma particles motion into the ISSN 1562-6016. ВАНТ. 2019. №4(122) 62 Poisson equation using the harmonic oscillation method [10 - 12]. Considering the above, we can write the equations (in dimensionless variables) which describe the interac- tion between the wave spectrum of the corrugated plas- ma waveguide and charged particles (electrons and positive charged test particles): 1 1 sin( ) N m i m m i d m d d N ξ φ τ e e = = ⋅ + −∑ , (7) 1 1 cos( ) N m i m im d m d N φ ξ φ τ e = = ⋅ + ⋅ ∑ , (8) where 1/3 0 0k V tτ β= , 2/3 0 4 m m ob E k en β π e = , 1ob oen nβ = << − is the ratio of the equilibrium density of the bunch and plasma, respectively. Charged particles are modeled using N layers moving with velocity 0V (1 i N≤ ≤ , i − is the layer number), the coordinate of which is given in the form ( )0 0i ik z V tξ = ∆ − . The system of equations (7), (8) must be supple- mented by equations that describe the motion of charged particles of a bunch (in dimensionless variables): 12 1 2 1 1 1 sin( ), 1 , sin( ), 1 . M m i m mi M m i m m m i N d d m N i N ξ φ ξ τ ξ φ e e = − = − + ≤ ≤=  Λ ⋅ + + ≤ ≤  ∑ ∑ (9) Equations (9) describe the motion of electrons (up- per expression on the right side of (9)) and positively charged test particles with mass Λ and quantity 2N (lower expression on the right side of (9)). The total number of charged particles is N: 1 2N N N+ = . From the equations (7)-(9) in the absence of absorp- tion can be obtained the integral I which was helped us to control the accuracy of the numerical calculations. This integral has the following form: 1 2 1 1 1 1 11 2 2 2 2 1 cos . M M e m m N N i i N N Nm mI V V N N d di i t N d N d e e ξ ξ τ τ = = = = + = + −Λ ⋅ = +∑ ∑ Λ + ⋅ − =∑ ∑ (10) System (7) - (9) satisfies the condition of the particle coordinate's periodicity − the equations do not change when replaced 2i i kζ ζ π→ + . In addition, there is the following symmetry of the amplitudes and phases of the field: ,m m m mφ φe e− −= − = − . The system of equations (7) - (9) was investigated numerically using the method of bunches particles mod- eling by large particles [1, 10 - 12] with the following values of the counting parameters: the number of modes M = 50…100; the initial coordinates of the electron lay- ers iζ are distributed on the (0…2π) interval in the form of two large particles (the number of layers is N = 2); the initial distribution of the amplitudes of the wave disturbances was set to be uniform with the initial val- ues of the amplitudes 210me −= and phases 310mφ −= for all values of m. 5. NUMERICAL SIMULATION RESULTS Let us discuss the results of numerical simulation of the interaction of particles of two types with a plasma corrugated waveguide's dense wave spectrum. The choice of one decrement value for two types of particles in equation (7) indicates that we have chosen such sys- tem parameters for which the interaction occurs in the same forbidden band. The ratio of the number of test particles in the bunch, which is accelerated, and electrons in the bunch, which generates oscillations in the waveguide, is deter- mined by the parameter 1 2N Nθ = [15]. First, as a test calculation, consider the acceleration of such test parti- cles as positrons. Let the number of particles in bunches be: 1θ = . In Fig. 3 the time variations of the average velocity of the test particle V , electron bunch eV , and integral of the system of equations in the absence of absorption (d = 0) I are shown. In the calculations it was assumed that the initial velocities of the particles are the same. Fig. 3. Dependence of the average velocities of the test particle V and the electron bunch eV on the dimensionless time τ : 1 – d = 0,0; 2 – d = 0.2 It can be seen from the figure that there is a syn- chronous change in the average velocities of the test particle and the electron bunch. In the absence of ab- sorption, the integral remains almost unchanged, which indicates the reliability of the obtained numerical re- sults. Let’s now consider the acceleration of a heavy (Λ>> 1) positively charged particle. In Fig. 4 shows the temporal dynamics of the aver- age velocities of particles which move in a corrugated plasma waveguide. In the numerical simulation, the following parameter values were specified: M = 50; N = 2; d = 0.2; 10Λ = at the same initial particle velocities. It follows from the figure that at the beginning of the interaction, the average velocity of the accelerating par- ticles increases more slowly than the main bunch loses. In the absence of absorption, the integral of the system changes by a relatively small value at 2.5τ ≤ , i.e. at such time intervals when acceleration of test particles is observed. ISSN 1562-6016. ВАНТ. 2019. №4(122) 63 Fig. 4. Dependence of average velocities of a heavy test particle V and electron bunch eV on dimensionless time τ : 1 − d = 0; 2 − d = 0.2 The difference between the energies of these bunch- es is concentrated in the internal energy of the plasma and in the energy of the oscillations. The above examples of test particles acceleration are made for cases when the initial velocities of particles of two types are comparable. This explains the small value of the change in the average velocity of the test particle. If the initial velocity of the test particle is less than the initial velocity of the electron bunch, then the average momentum of the test particle may increase twice. This is indicated by numerical calculations, the result of one of which is shown in Fig. 5. Fig. 5. Dependence of average velocities of a heavy test particle V and an electron bunch eV on dimensionless time τ : 1 − d = 0; 2 − d = 0.02 Calculations carried out for different values of the initial velocities of the test particle and the accelerated bunch, show that the greatest increase in the average speed of the test particle is observed at the initial speed of the test particle 0 1.1d d τ ς τ = = − . The results of such a calculation are shown in Fig. 5. The same in- crease in the average velocity was established in a test calculation for positrons. For large values of the initial velocity 0 1.1d d τ ς τ = > − the acceleration of the test particle is not observed. Smaller values of the initial velocity lead to a small change in the average velocity of the test particle. Accounting for the absorption of oscillations in the forbidden band (see Figs. 4, 5, curve 2) increases the acceleration rate of the test particle. Let’s now estimate the length at which the accelera- tion of the test particles is observed, and compare with the result of the analysis of the wave spectrum corrugat- ed plasma waveguide. Based on the results of calculations shown in Fig. 1, one can estimate the acceleration length of the test par- ticle. It is determined by the ratio 1 1 0 0100 2 100L L kπ −≈ ⋅ = ⋅ ⋅ . On the other hand, the ac- celeration distance of the test particle, obtained as a re- sult of numerical calculations, gives the dimensionless acceleration time 9...10τ ≈ , which, in terms of the dis- tance passed by the main bunch, gives the acceleration length 1 32 10 −⋅= kL . A comparison of these values makes it possible to state that the acceleration distances are of the same order already at the plasma density 05.03/1 ≈β . CONCLUSIONS It is shown that in a corrugated superconducting plasma-filled waveguide, it is possible to use a two-step mechanism for the acceleration of charged particles. The first step is the excitation of a corrugated plasma waveguide with a short electron bunch in the direction of the longitudinal movement (the length of the bunch is significantly less than the period of the corrugation). The second step is the acceleration of test charged parti- cles in an electric field excited by an electron bunch of an infinite number of harmonics. Calculations show that with the implementation of such an acceleration mecha- nism, the average velocity of a test non-relativistic par- ticle may increase several times. A system of equations which describes a two-step method of the charged particles acceleration is obtained. Test calculations show that the proposed accelera- tion mechanism ensures complete transfer of the elec- tron bunch energy to the energy of a test particle of the same mass, but with a charge of opposite sign. Numerical calculations have shown that the acceler- ation length of the test particle can be up to one meter (under 0 02 /k Lπ= cm-1) with a plasma density satisfy- ing the condition 1/3 0.05β ≈ . REFERENCES 1. B.M. Bolotovskij. Teoriya effekta Vavilova- Cherenkova // UFN. 1957, t. 42, v. 3, p. 201-246 (in Russian). 2. V.A. Balakirev, N.I. Karbushev, A.O. Ostrovskij, Yu.V. Tkach. Teoriya cherenkovskih usilitelej i gen- eratorov na relyativistskih puchkah. Kiev. “Naukova dumka”, 1993, 208 p. (in Russian). 3. V.A. Balakirev, V.I. Karas’, I.M. Onishenko, et al. Doslidzhennya zbudzhennya kilvaternih poliv relyativistskimi elektronnimi zgustkami // UFZh. 1998, v. 43, iss. 9, p. 1160-1166 (in Ukrainian). 4. I.N. Onishenko. 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O vozniknovenii turbulentnosti pri vzaimodejstvii «monoenergeticheskogo» puchka s plazmoj // ZhETF. 1972, v. 62, iss. 3(9), p. 874 (in Russian). 11. A.A. Ivanov. Fizika silnoneravnovesnoj plazmy. M.: “Atomizdat”, 1977, 347 p. (in Russian). 12. A.N. Kondratenko, V.M. Kuklin. Osnovy plazmen- noj elektroniki. M.: “Energoatomizdat”, 1988, 320 p. (in Russian) 13. V.I. Tkachenko, P.V. Turbin. Dinamika volnovogo spektra v puchkovo-plazmennyh sistemah // El- ektromagnitnye volny i elektronnye sistemy. 2000, v. 5, № 1, p. 24-29 (in Russian). 14. D.A. Andronov, V.I. Tkachenko. Dinamika volnovyh processov v dissipativnyh puchkovo- plazmennyh sistemah // Visnik HNU im. V.N. Karazina, № 529, “Yadra, chastinki, polya. 2001, v. 3/15/, p. 47-49 (in Ukrainian). 15. I.V. Tkachenko, V.I. Tkachenko. Uskorenie zarya- zhennyh chastic v gofrirovannyh plazmennyh volnovodah s idealno provodyashimi stenkami // Tez. dokladov XIV konferenciya po fizike vysokih energij, yadernoj fizike i uskoritelyam. Kharkiv, NSC KIPT, 2016, p. 82 (in Russian). Article received 03.06.2019 ФРАКТАЛЬНЫЙ УСКОРИТЕЛЬ НА ОСНОВЕ ГОФРИРОВАННОГО ПЛАЗМЕННОГО ВОЛНОВОДА СО СВЕРХПРОВОДЯЩИМИ СТЕНКАМИ И.В. Ткаченко, В.И. Ткаченко Предложено использование двухступенчатого механизма ускорения заряженных частиц. Принцип уско- рения основан на использовании фрактальных свойств волнового спектра гофрированного плазменного волновода со сверхпроводящими стенками. Первая ступень обеспечивает возбуждение гофрированного плазменного волновода короткими в направлении движения электронными сгустками (длина сгустка значи- тельно меньше периода гофра). Вторая ступень осуществляет ускорение пробных заряженных частиц в воз- бужденном электронными сгустками бесконечном по количеству гармоник электрическом поле. Расчеты показывают, что при реализации такого механизма ускорения средняя скорость нерелятивистской пробной частицы может увеличиваться в несколько раз, а длина ее ускорения может составлять расстояние до одного метра. ФРАКТАЛЬНИЙ ПРИСКОРЮВАЧ НА ОСНОВІ ГОФРОВАНОГО ПЛАЗМОВОГО ХВИЛЕВОДУ З НАДПРОВІДНИМИ СТІНКАМИ І.В. Ткаченко, В.І. Ткаченко Запропоновано використання двоступеневого механізму прискорення заряджених частинок. Принцип прискорення заснований на використанні фрактальних властивостей хвильового спектра гофрованого плаз- мового хвилеводу з надпровідними стінками. Перший ступінь забезпечує збудження гофрованого плазмово- го хвилеводу короткими в напрямку руху електронними згустками (довжина згустку значно менше періоду гофра). Другий ступінь здійснює прискорення пробних заряджених частинок у збудженому електронними згустками нескінченному за кількістю гармонік електричному полі. Розрахунки показують, що при реаліза- ції такого механізму прискорення середня швидкість нерелятивістської пробної частинки може збільшува- тися в кілька разів, а довжина її прискорення може становити відстань до одного метра.

A fractal accelerator on the basis of a corrugated plasma waveguide with superconducting walls

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