The simulation problem of self-consistent dynamics of electron beam in small-size betatrons has been considered. The description of numerical model developed on the basis of macroparticle method is presented. The results of process modelling of electron injection and capture on the acceleration mode in betatrons with axially-symmetrical and asymmetrical magnetic field are shown. Optimal input injection parameters by beam current and energy (20...40 keV and 0,1...1,0 А) providing the maximum number of the capture electrons are defined. The techniques of increasing capture efficiency due to using variations of external magnetic field and additional energy selection of circuital decelerating EMF of the captured electrons are numerically studied. It allows an increase in capture coefficient from 4 to 7,4 % and capture at the acceleration up to 7,4·1010 electrons

Gries, Thomas

Kyosev, Yordan

Peeva, Ketty

Reinbach, Ingo

RWTH Publications

Max-product fuzzy relation equation as inference engine for prediction of textile yarn properties

Grigoriev, V. P.

Ofitserov, V. V.

Semeshov, V. А.

Electronic archive of Tomsk Polytechnic University

IntroductionWide application of smallsize betatrons requires essential increase in radiation intensity. Therefore, one ofthe most important problems in betatron production isenhancement of efficiency of electron capture at the acceleration mode. Coulomb interaction of beam electrons is known to play a decisive role in capture mechanism [1, 2]. The existing approach [3, 4] to researchingthis capture mechanism in classical betatrons based onthe analytical methods describes well the physics of thephenomenon for the case of axialsymmetrical magnetic fields. It makes possible to obtain qualitative dependences of capture parameters at linear approximation ofrelatively small parameter ΔR/Ro supposing that thewidth of vacuum chamber ΔR is significantly smallerthan the radius of equilibrium orbit Ro.The given assumption is not applicable to smallsizebetatron where beam cross size can be compared withthe radius of equilibrium orbit. Therefore, it is necessary to develop methods of obtaining quantitative captureparameter characteristics reflecting not only the fieldinfluence of beam spatial charge, but also the system geometry as well as complex profile of the external magnetic field. This problem can be solved on the basis ofnumerical simulation using modern macroparticlemethod [5]. The advantage of the given approach consists in obtaining detailed information in beam internalstructure along with quantitative estimation of captureparameters. Below a threedimensional model of electron capture on the acceleration mode in the cylindricalcoordinate system {r, z, θ} is presented. In the model acomplete selfconsistent dynamic system of electronbeam describing physical processes for general case ofaxialasymmetrical magnetic fields which influence sufficiently the capture efficiency at multiturn injection isused. It gives the most adequate description of beamnonlinear dynamics, which is particularly important forsmallsize betatrons.Model descriptionNumerical model has been developed on the basis ofelectron beam description by macroparticles [6]. It isdesigned for threedimensional modelling region coinciding with the region of betatron accelerating chamber:D={–L≤z≤+L, R1≤r≤R2, 0≤θ≤2π}. Here R1, R2 are radiiof the internal and external chamber wall, L is the axialchamber halfwidth. Macroparticles have the form ofcoaxial sectors (fig. 1), containing the charge Q=Ne·eand the mass M=Ne·me, where Ne is the number of electrons in macroparticle; e, me are the charge and the massof electron. They are enlarged particles with specificcharge equal to that of elementary electron and are characterized by the coordinates {r,z,θ} and the velocities{Vr, Vz, Vθ}. The particle size is determined by the size ofcomputational grid. The external field in general case isdescribed by the axial and radial components of magnetic field induction Bz*(r,z,θ), Br*(r,z,θ) and by azimuthcomponent of electrical field density Eθ*(r,z,θ). Thus,the model gives the threedimensional description ofphysical processes both in dynamics and by the field.Fig. 1. Macroparticle modelMacroparticle dynamics is described by the nonrelativistic equation of electron motion [7]:(1)whereInitial conditions are defined by injector parametersand are set in the form:r=Rин, z=Zин, θ=θин, Vr=0, Vz=0, Vθ=Vин.* 2; 1 1 ( / ) ; .E E E V c V Vθ θ θ γΣ = + = − =?22*202*202* *20{( ) } ,{( ) },1 2{ } ( ),z rr zr zd r e r B E rd t md z e r B Ed t md e z B r B E rd t m r rθθ θγθγθ θγΣ= + += − += − + −? ???? ??Mathematics and mechanics. Physics61UDC 537.333SIMULATION OF INJECTION AND CAPTURE OF BEAM ELECTRONS IN SMALLSIZE BETATRONS BY THE METHOD OF MACROPARTICLESV.P. Grigoriev, V.V. Ofitserov, V.А. SemeshovTomsk Polytechnic UniversityEmail: grig@am.tpu.ruThe simulation problem of selfconsistent dynamics of electron beam in smallsize betatrons has been considered. The description of numerical model developed on the basis of macroparticle method is presented. The results of process modelling of electron injection andcapture on the acceleration mode in betatrons with axiallysymmetrical and asymmetrical magnetic field are shown. Optimal input injection parameters by beam current and energy (20...40 keV and 0,1...1,0 А) providing the maximum number of the capture electrons aredefined. The techniques of increasing capture efficiency due to using variations of external magnetic field and additional energy selection of circuital decelerating EMF of the captured electrons are numerically studied. It allows an increase in capture coefficient from 4 to7,4 % and capture at the acceleration up to 7,4.1010 electrons.Properly Coulomb beam field is described by thePoisson equation for scalar potential:(2)where ϕ=ϕ(r,z,θ,t), ρ=ρ(r,z,θ,t).Boundary conditions meet the requirements of idealconductivity of the chamber wall and azimuth periodicity:Through the potential the components of beamelectric field density are calculated:(3)The equation of defining charge density closes thesystem:(4)where M is the mesh point; Ω is the volume of grid cell;n(Ω) is the number of macroparticles in Ω.Modeling selfconsistent beam dynamics is performedby means of numerical solution of equations set (1–4). Atevery step of macroparticle distribution in calculation areathe grid density of the beam charge is calculated ρ(r,z,θ,t).Then the Poisson equation is solved (2) for the beam potential ϕ(r,z,θ,t), and components of selffield are calculated by the formulas (3). The equations of motion (1) areintegrated by the RungeKutt method of the second accuracy order. In the relation (4) the volume produced by gridcell (fig. 1) is recognized as Ω. To solve the Poisson equation (2) the iteration dissipative difference scheme is used[8], which permits us to smooth calculative highfrequency «noises», inherent to macroparticles models, in the truevalue of the basic physical processes. The influence of theself magnetic field on the electron motion in the region ofthe parameters considered can be neglected.Investigation of electron injection and capture at accelerationAt numerical modelling the injection of electron beam having kinetic energy Win=20...40 keV is considered.Geometrical parameters of the equipment: L=20 mm,R1=20 mm, R2=80 mm, R0=45 mm. Particle injection issimulated by the injector arranged in the median planeon the radius Rin=60 mm and having axial widthΔZin=10 mm. The components of the external magnetic field are set analytically. Simulation is performed forthe following external magnetic field configurations:а) Axialsymmetrical field. The components of themagnetic field are presented in the form of:(5)where Bz0 =Bz*(R0) is the value of the field Bz* on the equilibrium orbit r=R0, is the coefficient ofmagnetic field decay.b) Azimuth variation field. For the foursector magnetic field we have:(6)where δВz is the depth of the field variation.The results of simulation for four variants of calculation at the current of injected beam I=0,5 A and magnetic field configurations (5, 6), table, are presented infig. 2–5. In figures the configuration images of the beamin the planes {r, z} and {r, θ} – (а), (b) are shown, thephase image of the beam in {r, Vr} in (в) and electron distribution histograms by energy in the chamber is in (d).Table. The maim parameters of numerical calculation andthe number of captured electrons NcapFig. 2. Moment of beam injection, t=5 nsInjection simulation is carried out up to the momentt=50 ns, when the charge limit is accumulated(≈1011 el.), then up to t=150 ns the capture process is investigated at the acceleration of this accumulated beam.In modeling the beam acceleration by circuital field isnot taken into consideration due to small growth of beam particle energy making up hundredth parts of thepercent per revolution. Hence, the external field is considered to be constant in time. This is a rather justifiedassumption as during the time of setting the given processes (t<200 ns), corresponding to 50–100 rev., thegrowth of beam particle energy is negligibly small (more than 0,5 %). In the course of the calculations thecontrol of complete energy conservation of interactingparticle system the model parameters provide the energy conservation with the error of less than 1 %.       Variant ofcalculationField variation δВz, %Energy extraction ΔW, eV/revNcap .10–100 ns 30 ns 50 ns 150 ns1 0 0 0 5,5 7,1 4,22 50 0 0 8,4 8,1 5,13 0 200 0 7,3 9,4 6,34 50 200 0 9,1 10,8 7,4var * var var[1 cos(4 )], ,Z Z Z r zn zB B B B Brδ θ= + = −**zzBrn rB∂= − ⋅ ∂* 0 * *0 , ,nz z r znR nzB B B B rr= = −( )( ) ,QnMρ Ω= Ω1, , .r zE E Er z rθϕ ϕ ϕθ∂ ∂ ∂= − = − = −∂ ∂ ∂0, ( , ,0) ( , , 2 ).sr z r zϕ ϕ ϕ π= =2 22 2 201 1 1 ,rr r r z rϕ ϕ ϕ ρθ ε∂ ∂ ∂ ∂⎛ ⎞ + + = −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠Bulletin of the Томsк Pоlytеchnic University. 2007. V. 310. № 162bc dFig. 3. Moment of beam injection, t=50 ns Fig. 4. Beam capture at t=100 nsDuring the investigation the capture mechanism atacceleration of the injected beam in betatron is revealed.It is sufficiently influenced by the fields of spatial beamcharge. The beam capture process at acceleration is anonstationary one with entering stable quasistationarystate characterised by the absence of particle loss andkeeping the beam structure in coordinate and phasespaces (fig. 2–5). In betatron chamber a monochromatic beam is introduced (fig. 2, d). In the process of radial oscillation redistribution of energy among the beamparticles owing to the influence of self Coulomb fieldstakes place (fig. 3, d). A part of beam particles gains energy and settles on the injector and chamber walls, butanother part of the beam loses energy and is captured atacceleration (fig. 4, d – 5, d). Energy losses of capturedelectrons amount up to 15 % of injection energy, thisproviding the radial removal from the injector for them(fig. 5, а).Fig. 5. Beam capture at t=150 nsAt injection energy Win=40 keV in the volume ofchamber the charge corresponding to Nacc≈7·1010 el. isaccumulated, captured at acceleration Ncap≈4·1010 el.The results of modeling according to the size of captured charge coincide with experimental data in the rangeof the order [2, 9]. It follows from the calculation results(Table) that increase in capture efficiency is possible bymeans of additional beam energy selection of circuitaldecelerating EMF and variations of the external magnetic field. A more sufficient increase in capture efficiency is achieved by selection of beam energy. The energy selection of 200 eV/rev allows the capture at theelectron acceleration Ncap≈6·1010 el, in this case the capture coefficient increases by the factor of 1,5. Combination of field variation and decelerating EMF providesthe maximum capture parameters: Ncap≈7,4·1010 el.,Kcap≈7,4 %. The capture coefficient is calculated by theformula [4]: Kcap=Ncap/Nмах, where Nмах is the maximumnumber of electrons captured in the betatron chamberby the external magnetic field. For the betatron parameters presented above the value of Nмах≈1·1012 electrons.Conclusion1. The numerical model of electron beam selfconsistent dynamics in smallsize betatrons describingphysical processes both in axialsymmetrical and inasymmetrical magnetic fields has been developed.2. Modelling the processes of injection and capture atacceleration in smallsize betatron reveals the optimal values of the current of injected beam0,1...1,0 A at the energy 20...40 keV. At the injectioncurrent values less than 0,1 A one can neglect the influence of self fields on the beam dynamics, whilethe current 1 A is a limit, above which formation of«virtual» cathode in injector takes place.3. In the course of the numerical experiments it is stated that self Coulomb fields conditioned by radialnonhomogeneity of the charge density play sufficiMathematics and mechanics. Physics63bc dbc dbc dent part in beam capture. The values of charge limitaccumulated at the injection (7·1010 el.) and captured at the acceleration (4·1010 el.) are obtained.4. Possible increase in capture efficiency by means ofvariation of external magnetic field and beam energy selection of circuital decelerating EMF is investigated. Thus, at the electron energy 40 keV and theenergy selection value 200 eV/rev, capture efficiencyincreases by the factor of 1,5, which permits thecapture at the acceleration 6·1010 el. The maximumcapture parameters are obtained at combination ofvariation of the external magnetic field and decelerating EMF: Ncap≈7,4·1010 el., Kcap≈7,4 %.Bulletin of the Томsк Pоlytеchnic University. 2007. V. 310. № 164REFERENCES1. Vorobyev А.А., Kononov B.А., Evstigneev V.V. Betatron electronbeams. – Moscow: Atomizdat, 1974. – 152 p.2. Moskalev V.А. Betatrons. – Moscow: Energoizdat, 1981. – 282 p.3. Matveev А.N. On the mechanism of capture and limit current in betatrons // Journal of Experimental and Theoretical Physics. – 1958.– V. 35. – № 2. – P. 372–380.4. Kovrizhnykh L.М., Lebedev А.N. Consideration of electron common interaction in cyclic accelerators // Journal of Experimentaland Theoretical Physics. – 1958. – V. 34. – № 4. – P. 984–992.5. Antoshkin М.Yu., Grigoriev V.P., Koval Т.V., Sablin N.I. Numericalmodel for investigation of axialasymmetrical wave excitation in coaxial vircators // Radio Technology and Electronics. – 1995. – № 8.– P. 1300–1305.6. Hokney R., Eastwood J. Numerical simulation by the particlemethod. – Moscow: Mir, 1987. – 640 p.7. Bruk G. Cyclic accelerators of charged particles. – Moscow: Atomizdat, 1970. – 311 p.8. Ofitserov V.V. Numerical simulation of nonstationary processes incyclic induction accelerators // Computer Techniques. – 2003. –V. 8. – № 6. – P. 322–330.9. Kasyanov V.А., Shtein М.М., Chakhlov V.L. Formation of focusingpoint of smallsize deceleration radiation // Devices and Techniqueof Experiment. – 1998. – № 1. – P. 41–42.Arrived on 11.10.2006Resonance electron accelerator – synchrotron withweak focusing of electron beam presents a complexelectricalphysical device consisting of a number of large units and their supply systems. The basic element ofsynchrotron is an electric magnet producing progressivemagnet field necessary for motion of charged electronsalong the orbit of constant radius. It is the so called«control magnet field». Synchrotron magnetic conductor is made of sheet transformer steel and requires highaccuracy of electromagnet assembling and mountingmade up of four sectors including azimuth angle 90°each. In general this system is the most labourconsuming, technologically complex and expensive part ofthe accelerator. Thus, the steel weight of the synchrotron «Sirius» electromagnet per 1,5 GeV in Tomsk Polytechnic University is 120 tons [1, 2].Control magnet field in the suggested synchrotronconstruction is induced by singleturn winding enclosing an azimuth circle quarter [3].The winding presents two concentrically locatedstrips of conductive material. The strips are connectedwith each other at one end, but at the other one they areconnected with alternating power supply. Current flowsin both strips in the opposite directions.In the space between the strips magnet field providing the motion of accelerated particles along the orbit(control synchrotron field) is produced. This field is topossess the property of electron beam axial focusing.In accelerator with «steel» magnet system focusing isachieved by shaping magnet field in «barrellike» formby means of increasing the gap between the electromagnet poles as it moves away from the system centre.To shape magnet field in a necessary form inducedbetween the poles of singleturn winding these poles areto bend in the vertical direction so that the convex part ofthe strip would present outside from the system centre.One more labourconsuming element of synchrotron is a vacuum acceleration chamber. It is glued froma large number of ceramic curved sectors forming anarch of quarter circle length, the radius of which corresponds to that of path curve of the accelerated electronbeam. Four of such arched sections connected withUDC 621.364.634.3IRONFREE ELECTRON SYNCHROTRON WITH WEAK FOCUSINGV.А. MoskalevTomsk Polytechnic UniversityEmail: mva@tpu.ruA synchrotron construction the magnetic field of which is without steel core is suggested. Acceleration chamber is combined with magnetizing winding. The described version of accelerator is favorably different in small weight, simplified production and assembling technique.

Simulation of injection and capture of beam electrons in small-size betatrons by the method of macroparticles

Simulation of injection and capture of beam electrons in small-size betatrons by the method of macroparticles

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