ii. 12. 13.

A. N. Vargin and M. M. Pasynkova, Teplofiz. Vys. Temp., 12, No. 3, 503 (1972). C. H. Popenoe and J. B. Shumaker, J. Res. Natl. Bur. Stand., Sect. A, 69, 495 (1965). T. L. Eddy and C. J. Cremers, J. Quant. Spectrosc. Radiat. Transfer, 17-~ No. 3, 287

14.

R. L. Giannaris and F. R. Incropera, J. Quant. Spectrosc. Radiat. Transfer, ii, No. 3, 291 (1971). W. H. Venable and J. B. Shumaker, J. Quant. Spectrosc. Radiat. Transfer, 9, No. 9, 1215 (1969). J. B. Shumaker, J. Quant. Spectrosc. Radiat. Transfer, 14, No. i, 19 (1974). F. R. Incropera and E. S. Murrer, J. Quant. Spectrosc. Radiat. Transfer, 12, No. i0, 1369 (1972). G. A. Kobzev and V. M. Sergeev, Opt. Spektrosk., 33, 1019 (1972). J. A. R. Samson, in: Advances in Atomic and Molecular Physics, Vol. 2, Academic Press, New York (1966), p. 198. L. M. Biberman, V. S. Vorob'ev, and I. T. Yakubov~ Kinetics of a Nonequilibrium LowTemperature Plasma [in Russian], Nauka, Moscow (1982). G. A. Koval'skaya and V. G. Sevast'yanenko, in: Physical Kinetics [in Russian], ITPM, Novosibirsk (1974). V. S. Vorob'ev, Teplofiz. Vys. Temp., 4, No. 4, 494 (1966). D. R. Bates, A. E. Kingston, and R. McWhirter, Proc. R. Soc. London, Ser. A, 267, 297 (1962); 270, 155 (1962). L. M. Biberman, V. S. Vorob'ev, andA. N. Lagar'kov, Opt. Spektrosko, 19, No. 3 (1965).

(1977). 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

TWO-CONDUCTOR STRIPS WITH DRIFT TUBES EXCITED BY AN ELECTRON BEAM IN AN ION ACCELERATOR N. M. Gavrilov, E. V. Gromov, A. V. Nesterovich, I. B. Nikitin, and A. V. Shal'nov

UDC 621.384.6

Excitation by a heavy-current electron beam in an ion accelerator is studied for a two-conductor strip with drift tubes whose supports on one bar are displaced at half the distance between the supports of another bar. The relations obtained on the basis of harmonic analysis and the equations from TWT theory indicate the potential of constructing such a system.

High-frequency excitation by an electron beam directly in an ion accelerator has always been of great interest in relation to the production of heavy-current electron beams. Such a problem can be solved without the use of high-frequency generators or channels, which simplifies the operation of the accelerator, decreases its size, and facilitates an increase in the acceleration rate of the ions [i]. The electron beam can used to focus ions for axial alignment, where the focusing of a more rapid electron beam is simpler than that of an ion beam [2]. One can obtain such a system without using high-efficiency acceleration devices which function by the interaction of high-frequency fields from two flows of particles with different velocities. We will study the excitation of a resonator by an electron beam in an acceleration system made of a two-conductor strip with drift tubes, where the ions are accelerated in the gaps between the tubes [3]. A schematic representation is given in Fig. la. When a high-frequency field is excited along the longitudinal rods, the neighboring drift tubes are excited with opposite phases so that the harmonic of the high-frequency potential with a period equal to twice the spacing period of the tubes, which runs in the direction of the motion of the ions, accelerates them. Each section of the bar between the supports is a

1986.

Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 24-29, June Original article submitted November 20, 1984.

0038-5697/86/2906-0445512.50

9 1986 Plenum Publishing Corporation

445

a

Y,

o ''~'~../.---x~-~ y~....4"SI~S/--k'y~-I -

1:%~

b

c

F i g . 1. General form of a two-conductor strip with drift tubes (a): the instantaneous distribution of the high-frequency potential a l o n g t h e s y s t e m ( b ) ; and t h e c o r r e s p o n d i n g i n t e n s i t y distribution of the electric f i e l d on t h e a x i s ( c ) .

Pout IeVinj

o,2~ 7 0

Off

0,2

0,$

0,4

$,8 Pinp /IeVinj

Fig. 2. Characteristic dependences of the high-frequency power at the output of the system on the power at the input for different values of the parameter b in the accelerating section and for different values of the parameter L in the feedback link. half-wave oscillator with a iength of I e = %/2, where %e = c/f, f is the resonance frequency, and c is the speed of light. In this case, the supports of one bar on the casing of the resonator are positioned exactly between the supports of another bar, which allows one to obtain equal amplitudes for the intensities of this harmonic. In Fig. ib, one can find the instantaneous potential distributions VA and V B along bars A and B and the potential on the axis of the system vo for a fixed moment in ~ime which corresponds to the amplitude of the potential, where VA(z)----Fmlsin2~ ~- z , V B(z) ~ --Vmlcos 2~zl, and V m is the maximum potential of the bar. The distribution Vo(z) consists of a double-period even function which is a superposition of two functions with the periods %e and %i, where I i << %e (the indices "i" and "e" correspond to the parameters of the shorter and longer wavelength functions, respectively), and I i is the doubled period of the drift tubes. We will now consider a simple case, which does not affect the generality of the results when I e is a multiple of %i, i,e., the period Vo(z) is equal to %e" The first harmonic of the longer wavelength function, which possesses a much higher phase velocity, can be used for synchronized electron interactions. Its phase velocity is equal to c/2, since the length of its period is two times less than the wavelength along the single bar which is the central conductor of the coaxial strip (without accounting for reactive loads of the resonance profile). 446

We will determine the relation between the amplitudes of the harmonic's intensity IEe] and IEi] and the periods i e and Xi" In the general case, the even function can be represented as an infinite sum

2~vrt =

Vn cos

-7---

(1)

z,

Ae

n=o

In the simplest variant, Vo(z) can be approximated by the sum of two terms 2=

V,, (z) ~ < V (z) > -- V; cos -- /.i w h e r e
1

> = =[VA(z ) + VB(Z)]

is

the average

value

A V(a)

changes with a period of Xe, and Vi(z ) -

2

of the potential

VA (z)

-

(2)

l,% (z)

-

2

on t h e b a r ,

which

is the amplitude of the

shorter wavelength harmonic, which changes with the same period. Approximation (i) is more accurate if the real distribution Vo(z) is close to sinusoidal over each shorter wavelength period. The potentials on the restrainer of the tube and at its center are always equal for this approximation. The intensity for the longitudinal component of the longer wavelength and shorter wavelength harmonics change over space and time according to an harmonic rule

Ee,i(z, t)=Ee,i(z)e ]<~

dVe'i(Z) e/~~ dz

9

2~

V<,,i(z)

(4)

= V
where ~ = 2~f. Using the equations for the coefficients of a Fourier series, of the longer wavelength harmonic of the potential can be expressed as

~f

9 '-' Ve~

OVm (, ( zdz

< V(z]>c0s Ag

~ Xe

=

ke

9-\'] >. /

.

' "e

and t h e v a l u e o f t h e a m p l i t u d e longer wavelength period is

~

the amplitude

4Vm

9~ 7.e

3~

t. 0

of the

shorter

wavelength

harmonic

which

is

averaged

over

the

Xe t&=

ii

--

AV(Z)

'7

dz =

(

2)

g" t sm2~z'cos

9t e

(6)

0

Applying the appropriate transformations,

one obtains the following relation:

lEvi

I&I

__ 1,5 X,: .

(7)

J'~

Using this expression and those in [6], one can obtain an expression for the resistances of the couples Rcres

/

447

where Rsh

and

Q are

the shunt resistance and the Q-factor of the resonator.

It is evident from Eq. (7) that the amplitude of the longer wavelength harmonic is much less than that of the shorter wavelength harmonic, and, therefore, such an harmonic is ineffective for accelerating electrons. However, it can be used for the generation of a high-frequency field by a heavy-current beam when the excitation conditions are satisfied. In addition, it can be applied for raising the energy of a focused electron beam so that it does not fall into the acceleration mode of the shorter wavelength harmonic, which is related to losses in focusing properties. It is known that the excitation processes in resonance acceleration systems such as travelling or standing wave resonators (TWR, SWR) are identical and, for relativistic grouped beams, are the same as those described in [5]. The physical model for the process of establishing the steady-state mode (for any beam modulation depth, and any electron velocities) is similar to that for the process of selfexcitation in a travelling wave tube (TWT) when the reflection factor of the high-frequency power at the input and output end faces of the system is close to one. During the transition, the high-frequency power P which is increasing along the length is extracted from the beam and is incident at the input of the system. This process continues until the high-frequency power at the output is equal to the sum of the power at the input, the high-frequency power losses in the reverse wave, and the high-frequency power put into the resonator. The mathematically steady-state mode is characterized by the intersection of the dependence Poutput = f(Pinput), which can be derived through nonlinear analysis for the Poutput which gives the amplification mode of the TWT, with the straight line L = I0 log p. input losses in the feedback links (for a TWR) or the power put into the resonator using the coupling devices. We will use the information given in [6, p. 235] to illustrate this procedure, With the following parameters:

[| Rcres4vjin~ei/ei " C= L

[t1/3= 0 , 1

is

the

amplification

is the in~ection voltage of the electrons;

factor;

I e is

R ~ P~ --0,07; ~

the

electron

current;

b"(2~6e) ~"

[

e[e

Vinj e

]u~

is the plasma frequency of the ~lectrons; e and m e are the charge and the mass of an electron; go = 8.85-10 -~2 F/m is the dielectric constant; b e is the radius of the electron beam; be/X e = 0.16; a is the radius of the transmission channel; Ue is the dielectric constant;b = (Ue--Ve)/CVe; and ~e is the phase velocity of the longer wavelength harmonic. The dependence Poutput = f(Pinput) is shown in Fig. 2 for different electron injection rates. The damping factor for the straight harmonic is taken to be equal to zero. Lines are also given which correspond to different values for the damping of the inverse harmonic. It is evident that large electron injection rates result in high excitation powers, and the most typical values of the electronic efficiency ~ = P/leVin j e are in the range of 20-40%. The amplitude of the acceleration harmonic can be determined from (7)

(9) It is easy to see that the power of the high-frequency field P is proportional to IEil 2 and linearly grows with an increase in the power of the electron beam (the parameter ~ is practically independent of I e and Vin j e). As an example, we will give the following parameters for a system with an electron beam: X = 1.5 m, %i = 0.i ~e = 7.5 cm, Win j e ~ 200 keV, (Rsh/Q) i = 20 k~/m, and ~ = "1/3. For a current of I e = i00 A, the quantity IE~ is i0 KV/cm, which is typical for ion accelerators. Hence, the electron beam can pass without being stopped by the potential barrier of the shorter wavelength harmonic [2], where Vin j e > (IEiIXi)/~.

448

There exists a minimum length for the accelerator below which self-excitation by an unmodulated electron beam is impossible, since the power from this beam does not exceed the high-frequency power lost due to attenuation and grouping of the electrons. It is determined from the graph of Poutput = f(z),;-which is also based on a nonlinear analysis, and is on the order of several wavelengths in the above example. For excitation of a system with a short length, one must introduce power from an external generator Pext or employ preliminary modulation. The loaded Q factor of the system Q~ is determined by the equation

1

Q~

1

Q,,

1 Qext

1 ~

[5]

(lO)

Qe'

where Qo is the Q factor of the "cold" resonator; Qext is the external Q factor, which is determined by the losses in the system couplings and in the external links; and Qe is the electronic Q factor, which is a function of I e. If one accelerates large currents, the Q factor, which is on the order of Qe' can be much less than (Pext/l~Rshunte) Q0, and, therefore, the time for establishing the steady-state mode for TWR and SWR, which is ty = 2Ql/~ , can be much less than the duration of the transfer process in a resonator Without a beam, where, under the same conditions, it is two times less in the TWR than in the SWR. In our example, this time is on the order of microseconds. One can obtain high efficiency with such a system if the power of the electron beam is recovered by reducing the potential of the collector [4, p. 246] while connecting the cathode of the electron injector to the electron collector through an EMF source. LITERATURE CITED i.

2. 3. 4. 5. 6.

A. I. Dzergach and V. A. Krasnopol'skii, in: Proceedings from the Sixth All-Union Conference on the Acceleration of Charged Particles, Dubna, October 11-13, 1978, Joint Institute for Nuclear Research (OIYaI), Dubna, Vol. 2 (1978), p. 86. A. I. Akhiezer, G. Ya. Lyubarskii, and Ya. B. Fainberg, in: Theory and Design of Linear Accelerators [in Russian], Gosatomizdat, Moscow (1962), p. 131. V. K. Baev and V. P. Zubovskii, Inventor's Certificate No. 586780 (USSR); IPC: NO5N 7/06, Byull. OIPOTZ, No. i0, 297 (1982). M. B. Tseitlin and A. M. Kats, Travelling Wave Tubes [in Russian], Sov. Radio, Moscow (1964). A. N. Didenko and G. P. Fomenko, Radiotekh. Elektron., No. 6, 1017 (1971). J. Eo Rowe, Theory of Nonlinear Phenomena in Microwave Devices [Russian translation], Zo S. Chernov (ed.), Soy. Radio, Moscow (1969).

449