Russian Physics Journal, Vol. 46, No. 3, 2003

A CYCLIC INDUCTION ACCELERATOR V. A. Moskalev and G. I. Sergeev

UDC 621.384.634

The main feature of the examined cyclic induction accelerator is the separation of control and accelerating electromagnetic fluxes. The control magnetic field is formed by analogy with the magnetic field of a weakly focusing synchrotron, and the accelerating vortex electric field is generated by electromagnetic cores – inductors. Such a design of the cyclic induction accelerator allows the active steel volume and the power of a supply unit to be reduced significantly, and the separation of control and accelerating magnetic fluxes allows the energy lost by particles by synchrotron emission to be compensated using a relatively simple method. Recent investigations have demonstrated that in this accelerator, electrons can be accelerated to energies exceeding 300 MeV. INTRODUCTION

Betatrons are inexpensive and reliable electron accelerators simple in operation. Their main advantage over accelerators of other types is the absence of high-frequency accelerating systems. The induction acceleration method used in betatrons is low-impedance one and suits for acceleration of high currents better than the high-frequency resonant method. A disadvantage of betatrons is the necessity of creating a magnetic field in the region encompassing orbits of particles being accelerated, and this field is doubled, on average, compared to the field in the equilibrium orbit. This leads to a fast increase in the accelerator mass as the energy of particles increases. Therefore, betatrons are used to accelerate electrons to low energies, when the dimensions of the facility are not so large. Betatrons between 3 and 30 MeV are most widespread now, though there are some betatrons for 50 MeV [1]. The largest betatron for 300 MeV was built in the USA in 1950 [2]. Its emission was not reported. The electron charge being accelerated per one turn is typically about several nanocoulombs, whereas in highcurrent betatrons it may reach 1 µC. The best results were obtained for a 70-MeV ironless betatron operating in the regime of single pulse emission. For an injected energy of 2 MeV, a 1.5-µC charge was accelerated. The possibility of induction acceleration of currents of several kiloamperes was investigated in the Naval Research Laboratory (Washington, USA) using a modified MBA betatron (acceleration of a 1-kA current to an energy of 20 MeV was reported in [3, 4]). To confine such currents in orbit of a betatron, technologies developed previously for physical facilities intended for plasma confinement in a magnetic field (toroidal or stellar one) were used. However, no results confirmed by extraction of an accelerated electron beam from the betatron or by its discharge on a target were obtained, and at present these investigations are stopped for a time. To build higher-energy betatrons, two problems must be solved. First, the emitter mass grows fast as the energy increases (in proportion to the energy cubed). Second, as predicted by D. D. Ivanenko and I. Ya. Pomeranchuk [5], the conditions of proper functioning of betatrons are violated, because the electrons loose a portion of their energy by synchrotron emission. In connection with the foregoing, in the present work we consider a cyclic induction accelerator as an alternative to classical betatrons.

Scientific-Research Institute of Introscopy at Tomsk Polytechnic University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 48–54, March, 2003. Original article submitted July 11, 2002. 270

1064-8887/03/4603-0270$25.00 2003 Plenum Publishing Corporation

1 2 3 2

Fig. 1

Fig. 2

Fig. 1. Cyclic induction electron accelerator. Fig. 2. Cross-sectional view of the magnetic block that combines beam focusing and bending of the trajectory of particles being accelerated.

ELECTROMAGNETIC SYSTEM OF A CYCLIC INDUCTION ACCELERATOR

A cyclic induction accelerator described by M. S. Khvastunov [6] is free from the above-indicated disadvantage. In this accelerator, the system of the guide field of a weakly focusing synchrotron was used as a magnetic system. The accelerator was built of separate alternating blocks of control and accelerating electromagnets. Figure 1 shows the design of an accelerator with six periodic blocks. The control magnetic field determining an electron orbit (1) is created by electromagnets (2), and the electric field accelerating the particles is generated by ferromagnetic cores – inductors (3) placed in rectilinear sections of the trajectory of particles. The accelerator is analogous to the fixed-field betatron described in [7], but differs from it by much less radial dimensions of the vacuum chamber, inductors, and magnets due to the use of the confining field rising with time. The magnetic system of the accelerator is compact; therefore, it makes no sense to use a large number N of periodic blocks. For small N, strong focusing has no significant advantages over weak focusing. For this reason, weak focusing was chosen, because it could be easily obtained. In addition, weakly focusing magnetic systems widely used in betatrons are preferable to strongly focusing ones, because their relatively high optical dispersion allows the Landau decay mechanism to be used to control the collective beam instabilities. The electromagnetic system of the cyclic induction accelerator comprises control electromagnets and accelerating cores – inductors. Figure 2 shows the cross-sectional view of the control block of electromagnets. The electromagnet comprises the magnetic circuit 1 and the magnetizing coils 2. The magnetic circuit is made up of electrical-sheet steel laminations stacked together and has the working gap 3 in which the accelerating chamber is placed. If magnetic circuits of control magnets are put close together in the radial direction as shown in Fig. 1, we obtain the magnetic circuit of the typical six-post betatron with a cylinder cut inside and a working gap forming an interpolar space. The cylinder radius Rc is equal to the radius of central inserts. Therefore, the curvature radius of particles R0 and the main parameters of the interpolar space of the cyclic induction accelerator can be calculated using procedures described in [8, 9] that demonstrated high efficiency in calculations of electromagnets of betatrons. In so doing, the coefficient of magnetic flux scattering by the poles must be corrected, because in the cyclic induction accelerator, the magnetic flux is scattered by both external and internal end surfaces of the poles of the magnetic circuit. The electromagnet design shown in Fig. 2 is typical of electromagnets of proton synchrotrons. It allows the magnet height and mass to be decreased. For further decrease of the mass and overall dimensions of the magnet in high-energy

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accelerators, magnetizing coils are made water-cooled, for example, as in the high-current betatron for 50 MeV described in [10]. The current in the window in this case will be about 107 A/m2. The control magnets are primarily intended for beam focusing and bending of the trajectory of particles being accelerated. At the same time, they generate a time-varying magnetic flux Φc inside the circle of radius R0, which also contributes to the electron acceleration. The main contribution to the accelerating EMF comes from the accelerating cores – inductors – generating the magnetic flux Φa. The inductor cores made up of electrical-sheet steel are wrapped on a toroidal mandrel. The internal core radii are determined by the dimensions of the vacuum chamber and by insulation distances. Magnetization can be easily performed in steel of the cores. The operation of betatrons with magnetization and the schematics of power supply units were considered in detail in [11, 12]. With allowance for the magnetization, induction in steel changes from –1.75 to +1.75 T, which corresponds to the limiting hysteresis loop. For inductors without air gaps, the magnetomotive force is minimum, and magnetizing coils have small number of turns. To decrease inter-turn scattering fluxes, magnet coils are made up of sheet copper. Our investigations demonstrated that despite a certain increase in radial dimensions (with simultaneous decrease in vertical dimensions), the electromagnet mass significantly decreases. The separation of accelerating and control electromagnetic fields allows the energy lost by electrons by synchrotron emission to be compensated for electrons accelerated up to energies exceeding 300 MeV by generation of these fields at different reduced frequencies. ESTIMATION OF THE PARAMETERS OF THE ELECTROMAGNET OF A CYCLIC INDUCTION ACCELERATOR

The electromagnetic system of a cyclic induction accelerator differs significantly from analogous systems of classical betatrons. In the cyclic induction accelerator, the orbit perimeter exceeds the orbit length 2πR0 in a classical betatron by the length of rectilinear sections 6lr. In this case, the induction EMF is insufficient for acceleration of electrons to the preset energy because of the longer orbit. Second, in calculations of the magnetic flux scattered by the pole we must take into account that the magnetic flux, unlike the classical configuration of the betatron field, is scattered from both external and internal surfaces of the pole (here there is no flux completed through the yoke) and also from the sides of segments of the control magnet. Therefore, to develop a procedure for calculating the electromagnetic system of the cyclic induction accelerator, it is worth using procedures developed previously for calculating analogous systems of betatrons and synchrotrons and of betatrons with fixed magnetic fields. As already indicated above, the well-known procedures described in [8, 9] can be used to calculate the parameters of the interpolar space of the cyclic induction accelerator. To estimate the charge being accelerated, the effect of the space charge of the injected beam should be taken into account. All of the accelerators with high charges, for example, highcurrent and modified betatrons and adhesion facilities [13], have increased dimensions of the interpolar space. The magnetic field decay factor n is typically set in the limits from 0.5 to 0.75 to avoid dangerous resonances. We should also take into account that after injection of a high charge, the operating point of the accelerator is shifted thereby changing the magnetic field decay factor by dn [14, 15]: dn =

QeR0 µ0 c 2  1 − β2  4πа 2 E  β2

  . 

(1)

Here Q is the injected charge, e is the electron charge, R0 is the equilibrium orbit radius, µ0 is the magnetic permeability, c is the velocity of light, a is the effective beam radius, E is the energy of electrons, and β is the relative velocity of electrons. After termination of injection, dn can be greater than necessary for the displacement of the operating point in any resonance. Since the effect of the space charge decreases as the energy of particles increases, the operating point returns to the preset value intersecting the dangerous region. This results in a resonant buildup of betatron beam oscillations and the loss of a considerable portion of the charge being accelerated.

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In [16, 17] it was pointed out that the effect of resonances is especially pronounced for betatrons with n(R) = const. In betatrons in which n monotically increases with the radius R, linear and nonlinear resonances through which the beam passes during acceleration affect only insignificantly the beam stability, since conditions of resonance are not satisfied in the entire working region. The accelerating field in the cyclic induction accelerator is generated on the axes of inductors and in the gaps between the control magnets. The accelerating EMF averaged over the orbit perimeter is given by the equation ε = ω ( Ф a + Ф c ) cosω t = ( εa + ε c ) cosω t ,

(2)

where the subscript “a” refers to the inductors, the subscript “c” refers to the magnets, and ω is the angular frequency of the exciting field. This EMF is distributed over the orbit perimeter Π: П = 2 π R0 + 6lr ,

(3)

where R0 is the curvature radius of electrons in the control magnets, and lr is the length of the rectilinear gap between the control magnets. The results of calculations of the electromagnetic system of the cyclic induction accelerator demonstrated that the total length of free gaps may be as great as 55–65% of the orbit perimeter. The gap length can be estimated from the expression lr = ( 0.24 − 0.25 ) 2πR0 .

(4)

The energy to which the electrons are accelerated can be found by multiplying the accelerating EMF ε given by Eq. (2) into the number of turns of electron around its orbit and by integrating the expression obtained in the limits from 0 to π/2. As a result, we obtain the expression E=

βc . π ( Φ a + Φ c ) sin ω t

(5)

It can be used to estimate the magnetic flux of the inductors required for the stable electron motion in the orbit with the perimeter Π: Φa =

WП − Φc , β с sin ω t

(6)

and also the magnetic flux producing the accelerating EMF: Φc =

2 π B0 R0n 2− n R0 − Rс2 − n σ 0 . 2−n

(

)

(7)

Here B0 is the maximum induction created by the rotating magnets in the orbit of radius R0 , and σ 0 = 1 + 0.56 δ0 R0 is the coefficient that considers the scattered flux.

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ε , kV 75 50

εc

ε ca εa εc

25

∆Es.e 0

π/6

π/3

π/2

ωt

Fig. 3. Time dependences.

Compensation for energy losses by synchrotron emission

The EMF components εa and εc , induced by the magnetic field of the betatron, follow synchronously cosine laws and are equal to zero at the end of the acceleration cycle at ωt = π/2 (Fig. 3), whereas the energy ∆Es.e, lost by electromagnetic emission per turn of electron around its orbit, increases with the electron energy E in proportion to E4 and reaches its maximum at the end of the accelerating cycle: ∆Es.e = 88.5 ⋅ 103 E 4 R0 .

(8)

Here E is the energy of electrons (GeV) and R0 is the orbit radius (in m).

As ∆Es.e approaches the value ( εa + εc ) cos ωt , the electron energy increment decreases compared to that required

for electron motion in the equilibrium orbit in a rising magnetic field, and the radius of the trajectory decreases. The electron will continuously spiraling into smaller radius and finally will leave the acceleration region. To compensate for the energy lost by electrons by synchrotron emission, the accelerating EMF εa must be increased by the energy losses ∆Es.e . Curve εca in Fig. 3 shows the sum εca = εa + ∆Es.e that follows a cosine law at the new frequency ωa . In this case, formula (2) for the accelerating EMF averaged over the orbit is written in the form ε c = εac cos(ωa t ) + εc cos(ωc t ) ,

(9)

where ωa is the angular frequency of the accelerating field, and ωc is the angular frequency of the control field. The frequency of the cosine curve εca cos(ωa t ) differs from that of the cosine curve εc cos(ωc t ) ; the ratio of the frequencies ωa and ωc is ωa ωc = α c (π 2) ,

(10)

where α c is the compensation angle at which the condition εca = ∆Es.e is satisfied for the cosine curve εca cos(ωc t ) . Since εca

α= 0

274

= εa

α= 0

at the beginning of the acceleration cycle ( α = 0) , the angle α c can be found from the formula

(

α c = arccos ∆Es.e

εa α=0 ) .

(11)

After substitution of Eq. (11) into Eq. (10), we obtain

(

ωa ωc = ( 2 π ) arccos ∆Es.e

εa α=0 ) .

(12)

For accelerators with rectilinear gaps, the average radius of the equilibrium orbit Rav is introduced: Rav = П 2π = R0 + 3lr π .

(13)

In this case, the maximum induction of the control field averaged over the accelerator ring will be Bav =

πR0 B0 . πR0 + 3lr

(14)

The criterion of compensation for the energy lost by electrons by synchrotron emission and of stable accelerator operation is the allowable deviation of the instantaneous particle orbits from the equilibrium orbit. The instantaneous orbit Rav is determined by the expression Rav =

Ek − Es.e , 300 Bav sin(ωc t )

(15)

where Ek is the energy of accelerated particles at a fixed time, and Es.e is the total energy lost by the electron by synchrotron emission by the given moment of time. The energy Ek can be found by multiplying the accelerating EMF εc given by Eq. (9) into the number of turns of electron around its orbit and by integrating the expression obtained in the limits from 0 to π/2. As a result, we obtain Ek =

  εc c  εa sin ( ωc t )   ,  sin  ωa t + П  ωa ωc   

(16)

where c is the velocity of light. The expression for Es.e can be derived if we replace E in Eq. (8) by E sin ( ωc t ) , multiply this expression into the number of turns of electron around its orbit, and integrate the final equation. We obtain Es.e =

88.5cE 4 2πR0 ω

 3ωc t 3sin ( 2ωc t ) sin 3 ( ωc t ) cos ( ωc t )  − −  . 16 4  8 

(17)

The current orbit radius Rav can be found by substituting Eqs. (16) and (17) into Eq. (15).

An accelerator for 150 MeV

By the query of Hiroshima University, we estimated the parameters of a cyclic induction accelerator for 150 MeV intended for generation of parametric x rays. We used the following input parameters: – air gap length along the equilibrium radius of the electron orbit, cm 15

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– decay factor of the magnetic field in the control magnets 0.46 – induction in steel of the control magnets, T 1.6 – induction in steel of the accelerating cores (with allowance for the magnetization), T 3 The draft yielded promising results. For an orbit perimeter of 7.57 m (an average orbit radius of 1.2 m), the mass of active materials of the emitter did not exceed 23 t. The main mass of active steel equal to 16 t was concentrated in six inductors, and the total mass of control magnets was 4 t. The emitter height, equal to 1.35 m, was determined by the inductor diameter, and the average emitter radius was 1.75 m. Lifting gears with capacities up to 3 t could be used for mounting of the accelerator. Because the curve ε c (Fig. 3) actually differs from a cosine curve, the instantaneous orbit during the acceleration cycle deviates from the equilibrium one. The maximum deviation of the orbit is at the angle ωt equal to 50°. It must not exceed 0.5 cm. By this time, the electrons have been accelerated up to 80% of the maximum energy, that is, the maximum deviations are observed for the focused and shaped beam. For a radius of the accelerating chamber of 15 cm, such a deviation will not lead to the particle escape from the beam. For a comparison, we point out that the mass of a classical betatron for 150 MeV with the current density in the window ∆ = 10 ⋅106 A/m2 (water cooling) exceeds 300 t. At present, we are investigating the feasibility of replacing the ferromagnetic inductors by ironless accelerating modules [18] to decrease considerably the accelerator mass. CONCLUSIONS

Thus, the cyclic induction accelerator is promising for generating of electron beams with energies up to several hundreds of megaelectronvolts. The separation of control and accelerating electromagnetic fluxes being the distinctive feature of this accelerator allows the emitter mass to be significantly decreased and the energy lost by the accelerated beam by synchrotron emission to be compensated. To accelerate a current of several thousand amperes up to 500 MeV, the energy of the electron beam per pulse must be ~1 kJ. REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

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