INSTITUTE OF PHYSICS PUBLISHING PLASMA SOURCES SCIENCE AND TECHNOLOGY

Plasma Sources Sci. Technol. 14 (2005) 98–108 doi:10.1088/0963-0252/14/1/013

Theoretical modelling of an x-ray sourcebased on stochastic acceleration ofcharged particles at electron cyclotronresonanceO V Manuilenko and J K Lee

Department of Electronic and Electrical Engineering,Pohang University of Science and Technology, Pohang 790-784, Korea

E-mail: jkl@postech.ac.kr

Received 28 January 2004Published 27 January 2005Online at stacks.iop.org/PSST/14/98

AbstractThe motion of charged particles in the field of H-waves of a cylindrical anda rectangular waveguide in the presence of a uniform magnetic field isinvestigated analytically and numerically. We obtain the equations ofcharged particle motion in the H-wave field under cyclotron resonanceconditions, evaluate the width of nonlinear resonances, and establish thecriterion for dynamical chaos emergence. The criterion is used to analysethe possibility of stochastic acceleration of charged particles by the field ofH-waves of circular and rectangular waveguides in the external magneticfield for x-ray production. The stochastic acceleration of particles in twoand three standing waves of a circular resonator and twocounter-propagating waves of a rectangular waveguide are investigated. It isshown that, over the course of several hundreds of periods of the microwavefield, the kinetic energy of the electrons reaches values of around 1.0 MeVand the average kinetic energy reaches values of the order of 0.3 MeV.

1. Introduction

Sources of x-ray radiation (XRS), in the main, are based onprocesses of two types: the first is the radiation of an ion (atom)under electron transition to a lower energy level (from whichan electron was removed previously) and the second is theradiation of accelerated charged particles. The processes ofthe first type underlie the schemes of x-ray lasers (collisionaland recombinational pumping). The processes of the secondtype underlie the beam XRS, synchrotron XRS, x-ray FEL,XRS based on electron channelling in crystal, etc.

Plasma x-ray sources are an important class of bothtypes of XRS. A plasma with the necessary parameters iscreated with the help of lasers, beams, Z-pinch, etc. One ofthe main problems in the development of plasma XRS (bothcoherent and non-coherent) is the fast energy transfer froman external pumping source to plasma electrons, i.e. theproblem of fast and effective heating (acceleration) of plasmaelectrons.

In a typical situation, waves are absorbed in a spatialresonant range, where the particles acquire or give up energydepending on the wave–particle phase and then they ‘forget’the phase as a result of collisions, so that in the subsequentpassage through the resonant zone this phase can be taken tobe a random one. In this case a particle acquires energy onaverage. This method cannot be applied if the plasma has tobe heated in a very short time. Heating can be accomplished bydirect conversion of electromagnetic wave energy into thermalenergy of the plasma particles, where ‘intermediate’ stagesof the heating process, namely collisions, are eliminated. Thiscan be done, in particular, due to the development of stochasticmotion of plasma particles in regular electromagnetic fields(stochastic heating) [1]. Another method of heating plasmaswithout collisions is by using stochastic electromagnetic fields.

When the conditions for stochastic instability of particlemotion are satisfied, one can introduce the effective collisionfrequency νeff ∼ 1/Tc, where Tc ∼ 1/(δω · ln K) is

0963-0252/05/010098+11$30.00 © 2005 IOP Publishing Ltd Printed in the UK 98

X-ray source based on stochastic acceleration of electrons at ECR

a decorrelation time, K = �ω/δω is the ratio of thenonlinear resonance width�ω to the distance between adjacentresonances δω (overlapping parameter). It should be notedthat Tc is comparable to the period of the microwave (MW)field when K � 1. Therefore, the heating due to thedevelopment of stochastic instability is sufficiently fast andthe heating rate is controllable externally. This energytransfer mechanism from field to particles can be usedfor heating different types of plasma. The possibility ofstochastic heating of plasma in open traps was investigatedin [1–7]. The collisionless heating of electrons in differentdischarges was studied in [8]. The efficiency of stochasticheating of a solid-state plasma with laser radiation wasdemonstrated in [9]. Recently, stochastic acceleration wasused to explain high-energy electrons arising in laser–matterinteraction experiments [10].

Electron cyclotron resonance (ECR) is widely usedin many applications, such as plasma heating in fusiondevices, plasma production (ECR plasma sources), ion sources(multiple and single charge ions), ECR plasma deposition andetching, etc [11–15]. Recently, the ECR technique beganto be used for x-ray production [16–30]. The advantagesof the ECR x-ray source over the other sources are (a) it ischeap, (b) it does not require high dc potentials, (c) it canbe made as a portable device with tunable x-ray length. TheECR x-ray source, however, needs a MW oscillator (usuallya magnetron), a set of magnets, and a vacuum pumpingsystem. The maximum electron energy in the ECR x-raysource (minimal length of the x-rays due to collision with thetarget) is determined by the MW field strength and frequency,external magnetic field strength, and the type of working cavitymode [16–30].

Experiments on the heating of plasmas by MW sourcesbased on the ECR have proved the efficiency of this methodfor generating plasmas with densities up to 1012–1014 cm−3.If the plasma is confined in a trap by a mirror magneticfield, hot electrons are generated in the central plane ofthe trap [31–42]. These experiments were done for athermonuclear plasma. It was recently demonstrated that theappearance of hot electrons is possible in compact deviceswith intermediate and weak magnetic fields (<1 kG) and lowMW power [16–22, 25–29]. Such devices can be used forthe generation of x-rays, and multiple charge ion production[13, 14].

A compact ‘table top’ x-ray source is proposed in [16–18].It is based on a MW discharge in a magnetic mirror trapat low gas pressure under ECR conditions. This device isdriven by a 2.45 GHz MW source of power up to 700 W.The magnetic mirror is formed by two circular permanentmagnets (5000 G at the surface), located at a distance ofless than 10 cm from each other. The mirror ratio is 2.4,the resonance zones (magnetic field strength is 875 G) beinglocated a few centimetres from the mirror axis. The MWs areinjected at the mid-plane across the mirrors magnetic field [34].X-rays with energy up to 100 keV were observed from a smalltantalum target, placed in the mid-plane, a few centimetresfrom the axis in the resonance zone, which is hit by energeticelectrons. The x-ray production is maximal at an Ar pressureof 3 × 10−5 Torr. Experiments have shown that the energeticelectrons are located on a ring that contains the mirror axis

and are very close to the ring where acceleration by ECR takesplace [17, 18].

Ikegami et al [34] found a resonant increase of the x-rayemission when the second and third harmonics of the electron-cyclotron frequency associated with the magnetic field in themid-plane of the magnetic mirror reached the frequency of theheating MW. Strong x-ray emission when a small tungstenplate was placed at the location of the axially symmetricresonant surface, where the magnetic field corresponds to theECR and its second, third and fourth harmonics, was reportedby Golovanivsky et al [19]. Resonant increase of the x-rayemission in a MW discharge at a half-gyrofrequency wasstudied by Vodopyanov et al [20].

The maximal energy that electrons gain under ECR islimited due to relativistic detuning of the resonance. In thehomogeneous electric and magnetic fields the maximal energyis determined by the electric field strength. Therefore, iflarge values of electron energy are needed high MW power isrequired. In the case of the ECR produced in the magnetic fieldslowly growing in time, the relativistic change of the electronmass can be compensated for the change in the magneticfield in the time [22–24]. An x-ray source based on thisphenomenon was proposed in [22].

An inexpensive x-ray source for the energy range100–150 keV, based on an electron-cyclotron accelerator wasproposed, designed and tested in [25–29]. The electron-cyclotron-based accelerator uses an electric field (electric fieldstrength in the cavity approximately 10 kV cm−1) from aninexpensive MW oven magnetron (2.45 GHz), a magneticfield (up to 1065 G) which can be produced by permanentmagnets, and a MW cavity instead of traditional cyclotrondees. The electron source is a filament that can be heatedto increase its electron emissivity. Electrons circling thecentral filament gain energy from the MW field, increasingthe radius until they strike the target, which is a thinmolybdenum plate.

It has been shown recently that the stochasticacceleration of low-density plasma electrons by the regularresonator/waveguide modes in an external magnetic field issufficiently simple in realization, highly effective and fast [30].This mechanism can be utilized for x-ray production. Inthis paper, we aim to carry out theoretical and numericalinvestigations of stochastic acceleration of electrons in acylindrical resonator and in a rectangular waveguide to obtainconditions for x-ray production. In section 2, we determinethe conditions for the transition to chaotic motion and analysethe chaotic dynamics of charged particles in a circularwaveguide and resonator. In section 3, we investigate thedynamics of charged particles in a rectangular waveguide.In section 4, we summarize our conclusions.

2. Stochastic acceleration of charged particles byregular electromagnetic fields in a circularwaveguide

Let us consider the motion of a charged particle in the H-wavesof a circular waveguide (radius a) placed in a uniform magneticfield H0, which is directed along one of its axes. In cylindrical

99

O V Manuilenko and J K Lee

coordinates (r, ϕ, z) the Hmn wave has components

Er = E0m

k⊥rJm(k⊥r) sin(mϕ) sin(kzz − ωt),

Eϕ = E0J′m(k⊥r) cos(mϕ) sin(kzz − ωt),

Ez = 0, Hr = −kz

kEϕ, Hϕ = kz

kEr,

Hz = E0k⊥k

Jm(k⊥r) cos(mϕ) cos(kzz − ωt),

(1)

where Jm is the Bessel function of order m, J ′m is the derivative

of the Bessel function, k⊥ = νmn/a, νmn is the root of J ′m, the

index n is the root number, kz =√

k2 − k2⊥, k = ω/c and ω is

the wave frequency. In the dimensionless variables ωt → t ,�rω/c → �r , �p/Mc → �p, k⊥c/ω → k⊥, kzc/ω → kz, whereM is the particle mass, the equations of motion for a chargedparticle have the form

dpx

dt=

(1 − kzpz

γ

)εx +

py

γ(ωH + hz) − νpx,

dpy

dt=

(1 − kzpz

γ

)εy − px

γ(ωH + hz) − νpy,

dpz

dt= kz

γ

(pxεx + pyεy

) − νpz,

dγ

dt= pxεx + pyεy

γ− ν

(γ 2 − 1

)γ

,

d�rdt

= �pγ

,

(2)

where �ε = e �E/Mcω, �h = e �H/Mcω, ωH = eH0/Mcω,γ =

√1 + �p2 is the particle energy and ν is the

dissipation.Writing equations (2) in the guiding centre variables

px = p⊥ cos �, py = p⊥ sin �, x = xc − p⊥ωH

sin �,

y = yc +p⊥ωH

cos �, x = r cos ϕ, y = r sin ϕ,

xc = rc cos ϕc, yc = rc sin ϕc,

(3)where the index c marks the coordinates of the particles guidingcentre, we obtain the following set of equations:

dpz

dt= ε0kzp⊥

2γIm

∞∑s=−∞

isJ ′s (k⊥ρ)Jm−s(k⊥rc)e

i(m−s)ϕc−i�s

− ε0kzp⊥2γ

Im∞∑

s=−∞isJ ′

s (k⊥ρ)Jm+s(k⊥rc)

× ei(m+s)ϕc+i�s − νpz,

dp⊥dt

= ε0

2

(1 − kzpz

γ

)Im

∞∑s=−∞

isJ ′s (k⊥ρ)Jm−s(k⊥rc)

× ei(m−s)ϕc−i�s − ε0

2

(1 − kzpz

γ

)

× Im∞∑

s=−∞isJ ′

s (k⊥ρ)Jm+s(k⊥rc)ei(m+s)ϕc+i�s − νp⊥,

dγ

dt= ε0p⊥

2γIm

∞∑s=−∞

isJ ′s (k⊥ρ)Jm−s(k⊥rc)e

i(m−s)ϕc−i�s

− ε0p⊥2γ

Im∞∑

s=−∞isJ ′

s (k⊥ρ)Jm+s(k⊥rc)

× ei(m+s)ϕc+i�s − ν(γ 2 − 1)

γ,

d�n

dt= kzpz

γ− 1 +

nωH

γ− nε0

2p⊥

(1 − kzpz

γ

)

× Re∞∑

s=−∞is

s

k⊥ρJs(k⊥ρ)Jm+s(k⊥rc)e

i(m+s)ϕc+i�s

− nε0

2p⊥

(1 − kzpz

γ

)Re

∞∑s=−∞

iss

k⊥ρJs(k⊥ρ)

× Jm−s(k⊥rc)ei(m−s)ϕc−i�s +

nε0k⊥2γ

× Re∞∑

s=−∞isJs(k⊥ρ)Jm+s(k⊥rc)e

i(m+s)ϕc+i�s

+nε0k⊥

2γRe

∞∑s=−∞

isJs(k⊥ρ)Jm−s(k⊥rc)

× ei(m−s)ϕc−i�s ,

drc

dt= ε0

2ωH

(1 − kzpz

γ

)Re

∞∑s=−∞

is+1Js(k⊥ρ)

× (J ′m−s(k⊥rc)e

i(m−s)ϕc−i�s − J ′m+s(k⊥rc)

× ei(m+s)ϕc+i�s ) − ε0

2ωH

k⊥p⊥2γ

Re∞∑

s=−∞is+1

× (Js+1(k⊥ρ)Jm−s−1(k⊥rc) − Js−1(k⊥ρ)

× Jm−s+1(k⊥rc))ei(m−s)ϕc−i�s − ε0

2ωH

k⊥p⊥2γ

× Re∞∑

s=−∞is+1(Js+1(k⊥ρ)Jm+s+1(k⊥rc) − Js−1(k⊥ρ)

× Jm+s−1(k⊥rc))ei(m+s)ϕc+i�s − νp⊥

ωHsin(� − ϕc),

rc

dϕc

dt= ε0k⊥ρ

4γRe

∞∑s=−∞

is(Js−1(k⊥ρ)Jm−s+1(k⊥rc)

+ Js+1(k⊥ρ)Jm−s−1(k⊥rc))ei(m−s)ϕc−i�s

− ε0k⊥ρ

4γRe

∞∑s=−∞

is(Js+1(k⊥ρ)Jm+s+1(k⊥rc)

+ Js−1(k⊥ρ)Jm+s−1(k⊥rc))ei(m+s)ϕc+i�s

+ε0

2ωH

(1 − kzpz

γ

)Re

∞∑s=−∞

ism + s

k⊥rc

Js(k⊥ρ)

× Jm+s(k⊥rc)ei(m+s)ϕc+i�s − ε0

2ωH

(1 − kzpz

γ

)

× Re∞∑

s=−∞is

m − s

k⊥rc

Js(k⊥ρ)Jm−s(k⊥rc)

× ei(m−s)ϕc−i�s +νp⊥ωH

cos(� − ϕc),(4)

where �s = kzz− s�− t , ρ = p⊥/ωH. The addition theoremfor cylindrical functions [43] was used in (4).

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X-ray source based on stochastic acceleration of electrons at ECR

Neglecting the dissipation in (4), it is possible to determinethe integral:

pz − kzγ = const. (5)

Taking into account that efficient wave–particle interac-tion takes place under the resonance conditions

kzpz0,s + sωH

γ0,s

= 1, s = 0, ±1, ±2, . . . , (6)

by averaging equations (4), one can obtain the equations forparticle motion in the isolated resonance (6):

dγ̃

dt= ε0p⊥0,s

2γ0,s

Jm+s(k⊥rc,s)J′s (k⊥ρ0,s)Im(isei�̃s ),

d�̃s

dt= γ̃

γ0,s

(k2z − 1),

(7)

where γ̃s is the deviation of the particle energy from γ0,s ,|γ̃s | � γ0,s , �̃s = �s +(m + s)ϕc. In deriving equations (7) wehave neglected the dissipation terms, assumed that n = 1, andthat the particle is located near the waveguide axis. In derivingthe equation for the phase �̃s we have neglected terms of theorder ε0, as the perturbation of γ is of the order of

√ε0.

However, it should be noted that it is valid only in the casewhen p⊥ is far from zero.

It is easy to find the nonlinear resonance half-widthfrom (7):

�γs =√

2|ε0p⊥0,sJm+s(k⊥rc,s)J ′s (k⊥ρ0,s)|

k2⊥

. (8)

The distance between adjacent cyclotron resonances s ands + 1 under interaction with the travelling Hmn wave (1) can befound from the integral (5) and resonance conditions (6):

γ0,s+1 − γ0,s = ωH

k2⊥

, (9)

and the criterion for dynamical chaos emergence and stochasticacceleration of electrons can be written as

�γs + �γs+1 � ωH

k2⊥

. (10)

The main features of the motion of the dynamicalsystem (4) can be demonstrated clearly on the plane {pz, γ }(figure 1). The motion of the particle is the phase oscillationnear the resonance centre (6) along the integral (5). If thecondition (10) is not satisfied, the particle dynamics is regular,and it is described by the average system (7). In the oppositecase, when criterion (10) is satisfied, the particle motionis chaotic. Particles gain energy on average, moving inconformity with the integral (5).

It follows from (10) that it is difficult to obtain thechaotic motion of a particle due to nonlinear cyclotronresonances overlapping under interaction with one travellingHmn wave (1). Indeed, if a particle is in a cyclotron resonancewith number s and the conditions for the optimal resonanceinteraction are satisfied (m + s = 0, and rc,s ≈ 0), thenthe width of the nearest resonance with number s ± 1 is

0 0

Figure 1. Resonances (6) and integral (5) in the {pz, γ } plane.

|J1(k⊥rc,s±1)|−1/2 times smaller than the width of resonance s;i.e. the widths of the neighbouring resonances decrease rapidlywith distance from the s resonance. Numerical simulation fora H11 travelling wave and a = 8 cm, ω = 1.76 × 1010 rad s−1,H0 = 1 kG shows that, for low energy (γ = 1 + 10−4)

particles, stochastic acceleration does not occur up to the waveamplitudes ε0 = 1.

In order to ensure chaotic particle motion an additionalwave can be added. In this case the lines corresponding tothe additional wave–particle resonances arise in the {pz, γ }plane. These additional wave–particle resonances intersect theresonance lines corresponding to the primary wave–particleresonances. At the points of intersection of resonances,the integrals of motion are completely destroyed, and theparticle motion is chaotic [44]. This outline of stochasticacceleration of particles can be fulfilled in the waveguide withtwo propagating waves or in the resonator with two standingwaves. The results of the numerical analysis of electronmotion in the field of one (H11, the amplitude of the wave isε0 = 0.05) and two standing waves with the same amplitudes(H11 and H21, the amplitudes of these waves are ε0 = 0.05for one case and ε0 = 0.1 for other case) are presented infigures 2–9. Equations (2) were integrated numerically for1000 electrons. At the initial moment the electrons wererandomly distributed over the coordinates and directions ofthe transverse momentum inside a cylinder of radius a/8 andlength 2π min−1{kz1, kz2}. The initial energy of all the particleswas γ = 1 + 10−4, pz = 0, the other parameters were:a = 8 cm, ω1 = 1.76 × 1010 rad s−1, H0 = 1 kG, ν = 0.The lines in the {pz, γ } plane in figures 2 and 6 show theresonances (6) for H11 (s = −1) and H21 (s = −2). Thedotted lines in figures 2 and 6 show the integrals (5). Thepoints show the final positions of electrons. The curve infigures 2 and 6 give the minimal particle energy in the {pz, γ }plane, namely, γmin = √

1 + p2z . It is seen from figure 2 that

electrons gain energy up to γ = 1.875 in the field of onestanding wave H11 with amplitude ε0 = 0.05. The particleswith energies near γ ≈ 1.5 can resonantly interact with the

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O V Manuilenko and J K Lee

Figure 2. Electron motion in the field of one standing wave(H11, ε0 = 0.05) and uniform magnetic field in the {pz, γ } plane.The final positions of electrons are presented. The lines show theresonances (6) for H11 (s = −1) and H21 (s = −2). The dotted linesshow the integrals (5).

1×104 2×104 3×104

Figure 3. Time evolution of the average energy of electrons movingin the field of one standing wave (H11, ε0 = 0.05) and a uniformmagnetic field.

second H21 wave. The time evolution of the average particleenergy is presented in figure 3. As shown in figure 3, theaverage energy 〈γ 〉 ≈ 1.2, which corresponds to a kineticenergy of 0.1 MeV. Figures 4 and 5 show the space distributionfunction for accelerated electrons at t = 3.5 × 104 and thetime evolution of the average particle radius. It is seen fromfigure 5 that the average radius 〈r〉 reaches saturation. Figure 6shows the final positions of electrons in the {pz, γ } planeon interaction with two standing waves H11 and H21 withamplitudes ε0 = 0.05. It is seen from figure 6 that electronsfrom the initial stage diffuse towards greater energies, up toγ = 3.0 (the kinetic energy is approximately 1 MeV). The timeevolution of the average particle energy is presented in figure 7.

Figure 4. Electron radius distribution function in the field of onestanding wave (H11, ε0 = 0.05) and a uniform magnetic field att = 3.5 × 104.

1×104 2×104 3×104

Figure 5. Time evolution of the average radius of electrons movingin the field of one standing wave (H11, ε0 = 0.05) and a uniformmagnetic field.

The average energy value reached is 〈γ 〉 ≈ 1.45 (the averagekinetic energy is approximately 0.23 MeV) for ε0 = 0.05and 〈γ 〉 ≈ 1.7 (the average kinetic energy is approximately0.36 MeV) for ε0 = 0.1. It is seen from figure 7 that theaverage particle energy reaches saturation for approximately4000 wave periods (ε0 = 0.05) and 500 wave periods (ε0 =0.1). The space distributions for accelerated particle and thetime evolution of the average particle radius are presented infigures 8 and 9.

To reduce the amplitude of waves necessary for stochasticacceleration of particles, it is possible to excite one more mode,whose frequency differs from that of the first two modes.

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X-ray source based on stochastic acceleration of electrons at ECR

Figure 6. Electron diffusion in the field of two standing waves H11,H21 with amplitudes ε0 = 0.05 and a uniform magnetic field. Thefinal positions of electrons are presented. The lines show theresonances (6) for H11 (s = −1) and H21 (s = −2). The dotted linesshow the integrals (5).

Figure 7. Time evolution of the average energy of electrons movingin the field of two standing waves H11, H21 with the sameamplitudes (ε0 = 0.05 for one case and ε0 = 0.1 for other case) anda uniform magnetic field.

The results of the numerical analysis of electron motion inthe field of one (H11 with frequency ω1 = 1.76×1010 rad s−1)

and three standing waves (H11, H21 with frequency ω1 =1.76 × 1010 rad s−1, and H11 with ω2 = 1.35 × 1010 rad s−1)are presented in figures 10–15. The amplitudes of the waveswere equal ε0 = 0.02. The other parameters and initialconditions were chosen to be the same as in the previouscase. Figure 10 shows the final positions of electrons in the{pz, γ } plane on interaction with one H11 standing wave. Theparticles gain energies up to γ = 1.25 and can interact withthe H11 wave with frequency ω1 = 1.35 × 1010 rad s−1. The

Figure 8. Space distribution of electrons in the field of two standingwaves H11, H21 with amplitude ε0 = 0.05 at t = 3.5 × 104.

⟨r⟩

1×104 2×104 3×104

Figure 9. Time evolution of the average radius of electrons movingin the field of two standing waves (H11 and H21 with amplitudesε0 = 0.05) and a uniform magnetic field.

time evolution of the average energy of electrons moving inthe field of the H11 wave is presented in figure 11. It isseen from figure 11 that the saturation level is sufficientlysmall. Figure 12 shows the final positions of electrons inthe {pz, γ } plane on interaction with three waves. It is seenfrom figure 12 that electrons from the initial stage diffusetowards greater energies, up to γ = 2.5 (the kinetic energyis approximately 0.8 MeV). The time evolution of the averageparticle energy is presented in figure 13. The average energyvalue reached is 〈γ 〉 ≈ 1.41 (the average kinetic energy isapproximately 0.21 MeV) for ε0 = 0.02 and 〈γ 〉 ≈ 1.56(the average kinetic energy is approximately 0.29 MeV) forε0 = 0.05. It is seen from figure 13 that the average particle

103

O V Manuilenko and J K Lee

Figure 10. Electron motion in the field of one standing wave(H11, ε0 = 0.02) and a uniform magnetic field in the {pz, γ } plane.The final positions of electrons are presented. The lines show theresonances (6) for H11 (s = −1 with different frequenciesω1 = 1.76 × 1010 rad s−1 and ω2 = 1.35 × 1010 rad s−1) and H21

(s = −2, ω1 = 1.76 × 1010 rad s−1). The dotted lines show theintegrals (5).

1×104 2×104 3×104

Figure 11. Time evolution of the average energy of electronsmoving in the field of one standing wave (H11, ε0 = 0.02,ω1 = 1.76 × 1010 rad s−1) and a uniform magnetic field.

energy reaches a level of saturation for approximately 1600wave periods (ε0 = 0.02) and 500 wave periods (ε0 = 0.05).Figures 14 and 15 show the radial distribution of electrons att = 3.5 × 104 and the time evolution of the average particleradius.

Dissipation is always present in a real situation andhas various physical causes, such as particle collisions andemission processes [45]. The results of the numerical analysisof electron motion in the field of two and three standing waveswith dissipation ν = 8 × 10−4 are presented in figures 16–19.The other parameters and initial conditions were chosen to bethe same as in figures 6 and 8. It is easy to see that dissipation

Figure 12. Electron diffusion in the field of three standing waves(H11, H21 with frequencies ω1 = 1.76 × 1010 rad s−1 and H11 withfrequency ω2 = 1.35 × 1010 rad s−1) in the {pz, γ } plane. Theamplitude of the waves is ε0 = 0.02. The final positions of electronsare presented. The lines show the resonances (6) for H11

(s = −1 with frequencies ω1 = 1.76 × 1010 rad s−1 and ω2 =1.35 × 1010 rad s−1) and H21 (s = −2, ω1 = 1.76 × 1010 rad s−1).

1×104 2×104 3×104

Figure 13. Time evolution of the average energy of electronsmoving in the field of three standing waves with the sameamplitudes (ε0 = 0.02 for the lower curve and ε0 = 0.05 for theupper curve) and a uniform magnetic field. The other parameters arethe same as in figure 8.

restricts the stochastic acceleration of particles. However, theprocess of energy gain is sufficiently fast.

3. Stochastic acceleration of charged particles byregular electromagnetic fields in a rectangularwaveguide

Let us consider now the motion of a charged particle inthe H-waves of a rectangular waveguide (dimensions {a, b})placed in a constant magnetic field H0, which is directed along

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X-ray source based on stochastic acceleration of electrons at ECR

Figure 14. Space distribution of electrons in the field of threestanding waves with amplitudes ε0 = 0.02 at t = 3.5 × 104.

⟨r⟩

1×10 2×10 3×10

Figure 15. Time evolution of the average radius of electronsmoving in the field of three standing waves with amplitudesε0 = 0.02 and a uniform magnetic field.

one of its axes. The Hmn wave has components

Ex = E0ky

k⊥cos(kxx) sin(kyy) sin(kzz − ωt),

Ey = −E0kx

k⊥sin(kxx) cos(kyy) sin(kzz − ωt),

Ez = 0, Hx = −kz

kEy, Hy = kz

kEx,

Hz = E0k⊥k

cos(kxx) cos(kyy) cos(kzz − ωt),

(11)

where kx = mπ/a, ky = nπ/b, kz =√

k2 − k2⊥, k⊥ =√

k2x + k2

y . Equations (2) in the guiding centre variables (3)

Figure 16. Electron diffusion in the field of two standing wavesH11, H21 with amplitudes ε0 = 0.05 and a uniform magnetic field.The dissipation ν = 8 × 10−4. The final positions of electrons arepresented. The lines show the resonances (6) for H11 (s = −1) andH21 (s = −2).

1×104 2×104 3×104

Figure 17. Time evolution of the average energy of electronsmoving in the field of two standing waves and a uniform magneticfield. The parameters are the same as in figure 10.

take the form

dpz

dt= −ε0kzp⊥

γ

∞∑n=−∞

J ′n(k⊥ρ)Gn sin(�n − �n)−νpz,

dp⊥dt

= −ε0

(1 − kzpz

γ

) ∞∑n=−∞

J ′n(k⊥ρ)Gn

× sin(�n − �n) − νp⊥,

dγ

dt= −ε0p⊥

γ

∞∑n=−∞

J ′n(k⊥ρ)Gn

× sin(�n − �n) − ν(γ 2 − 1)

γ,

105

O V Manuilenko and J K Lee

Figure 18. Electron motion in the field of three standing waves(H11, H21 with frequencies ω1 = 1.76 × 1010 rad s−1 and H11 withfrequency ω2 = 1.35 × 1010 rad s−1) in the {pz, γ } plane. Theamplitude of the waves is ε0 = 0.02. The dissipation ν = 8 × 10−4.The final positions of electrons are presented. The lines show theresonances (6) for H11 (s = −1 with frequenciesω1 = 1.76 × 1010 rad s−1 and ω2 = 1.35 × 1010 rad s−1) and H21

(s = −2, ω1 = 1.76 × 1010 rad s−1).

1×104 2×104 3×104

Figure 19. Time evolution of the average energy of electronsmoving in the field of three standing waves and a uniform magneticfield. The parameters are the same as in figure 12.

d�s

dt= kzpz

γ− 1 +

sωH

γ+

sε0k⊥γ

∞∑n=−∞

Jn(k⊥ρ)Gn

× cos(�n − �n) − sε0

p⊥

(1 − kzpz

γ

)

×∞∑

n=−∞

n

k⊥ρJn(k⊥ρ)Gn cos(�n − �n),

(12)

whereGn = √c2n + s2

n , tg(�n) = sn/cn, cn = cos(kyyc) cos(nϕ)

×cos(kxxc − nπ/2), sn = sin(kyyc) sin(nϕ) sin(kxxc − nπ/2),

Figure 20. Electron diffusion in the field of two waves H10, H11

with amplitude ε0 = 0.0125 and a uniform magnetic field. The finalpositions of electrons are presented. The lines show theresonances (6) for H10 (s = −1) and H11 (s = −2).

cos(ϕ) = kx/k⊥ and �n = kzz − n� − t + nπ/2. Afteraveraging over the resonant phase �̃s = �s−�s (see, also (7)),we obtain equations describing the particle motion at the res-onance conditions (6):

dγ̃

dt= −ε0p⊥0,s

γ0,s

GsJ′s (k⊥ρ0,s) sin(�̃s),

d�̃s

dt= γ̃

γ0,s

(k2z − 1),

(13)

and the half-width of the nonlinear resonance:

�γs = 2

√|ε0p⊥0,sGsJ ′

s (k⊥ρ0,s)|k2⊥

. (14)

The distance between adjacent resonances is defined by (9),and the criterion for the dynamical chaos arising has theform (10).

The half-widths of resonances (6) in the circular(8) and rectangular (14) waveguides are similar to thatfor the case of particle motion in the field of a planeelectromagnetic wave, propagating perpendicularly to aconstant magnetic field, with a vector of the electric fielddirected perpendicularly to a constant magnetic field aswell [9]. The difference is in the coefficients

√Jm+s(k⊥rc)

and√

Gs(kxxc, kyyc, s), i.e. the half-width of the resonancedepends on the particle position with respect to the axis of thewaveguide.

The results of the numerical analysis of electron motionin the field of two counter-propagating waves H10 and H11

are presented in figure 20. The equation of motion (2)was integrated numerically for 1000 electrons. At theinitial moment, the electrons were randomly distributed overthe coordinates and directions of the transverse momentuminside a cylinder of radius a/8 and length 2π min−1{kz1, kz2}.The amplitudes of these waves were ε0 = 0.0125. The initialenergy of all the particles was γ = 1 + 10−4, pz = 0, the other

106

X-ray source based on stochastic acceleration of electrons at ECR

parameters were a = b = 16 cm, ω = 1.76 × 1010 rad s−1,H0 = 1 kG and ν = 0. It is seen from figure 20 that electronsgain energies up to γ = 3 and the average energy reaches valueγ = 1.73. The numerical analysis has shown that dissipationrestricts the stochastic acceleration of particles in the case of acylindrical resonator as well.

4. Conclusions

The main results of this paper can be summarized as follows.The possibility of efficient low energy electron accelerationin the fields of regular electromagnetic waves/modes ofcircular and rectangular waveguides/cavities due to theonset of stochastic instability of charged particle motion isdemonstrated. The energy of particles increases from almostzero (γ ≈ 1) to a relativistic value of γ ∝ 2.5–3.0, at areasonable amplitude of the MW field (E0 ∝ 10 kV cm−1) for atime of the order of several hundreds of MW field periods. Theamplitude of a MW field, necessary for stochastic acceleration,can be decreased with the help of excitation of additionalwaves/modes of the waveguide/resonator.

The equations of motion of a charged particle in thefield of a H-wave of cylindrical and rectangular waveguideswere written in the form of an expansion in terms ofcyclotron resonances. This allowed us to determine theconditions for efficient resonant wave–particle interaction.We have obtained expressions for the widths of nonlinearcyclotron resonances and conditions for the onset of dynamicalchaos. The results of numerical simulation indicate stochasticmotion in the systems under study and the possibility ofstochastic acceleration of charged particles up to relativisticenergies.

The investigated x-ray source is close to the electroncyclotron accelerator based x-ray source proposed in [25](see also [26–29]). Electrons are accelerated by thesingle H111 wave in the cavity with an externally applieddc magnetic field. The relativistic detuning limits theparticles’ energy [26]. To obtain a high electron energya high field strength is required. In our case, due toexcitation of additional waves/modes in the waveguide/cavity,high electron energies can be obtained at lower fieldstrengths. The experiments [25, 28] were done withoutaxial particle containment. The axial particle containmentwas not investigated in the present work. The numericalinvestigation of stochastic acceleration of electrons by regularelectromagnetic waves/modes of circular and rectangularwaveguides/cavities in mirror magnetic fields will constituteour future work.

Mode selection is usually needed for particle accelerationin a circular waveguide by the chosen wave. Additional wavesfacilitate the onset of dynamical chaos. Therefore, modeselection is not really necessary in our case.

Acknowledgment

This work is done as part of the Tera-level Nano DevicesProject, 21c Frontier R&D Program of Korea Ministry ofScience and Technology.

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