On the transverse instability of an intensive electron beam in a betatron

NUCLEAR

INSTRUMENTS

AND

METHODS

([970) 69-73;

84

©

NORTH-HOLLAND

PUBLISHING

CO.

ON THE TRANSVERSE INSTABILITY OF AN INTENSIVE ELECTRON BEAM IN A BETATRON V. A. M O S K A L E V

Tomsk Polytechnical Institute, Tomsk, USSR Received 30 J u n e 1969 and in revised form 3 February 1970 Experimental data on transversal instability of intensive electron b e a m in high c u r r e n t b e t a t r o n s are described.

The bremsstrahlung intensity in those cases is practically invariable. The current of the accelerated electrons surviving to the end of the accelerating cycle, which was measured by the method described in ref. 2, is about 3 x l012 electrons in the pulse under any number of the " spill "--pulses and in their absence as well. The estimation of the total number of the electrons wasted away on the chamber walls during acceleration showed that 30 to 90 per cent of the total number of electrons involved in acceleration were lost. Thus, at the beginning of the cycle about 1013 electrons in a pulse are captured to the orbit. It should be noted that the technique used for measuring the accelerated electron charge inevitably gives results obviously less than the value measured, for it is assumed that at about 1 MeV energy all the electrons arriving at the target are completely absorbed and do not knock out secondary electrons from the target material. As it is shown in refs. 3 and 4, over the energy range from some dozens of keV to a few MeV, the greater part of the primary electrons are reflected from the target surface.

A transverse instability of a beam resulting in a loss of a significant portion of the accelerated electrons has been experimentally observed in a 25 MeV betatron of the T o m s k Polytechnical Institutel). The electron beam circulating on the equilibrium orbit periodically obtains exceedingly large amplitudes of axial oscillations and a part of the electrons wastes away on the accelerating chamber's walls. As a result a " t r a c e " of electrons in the form of dark ring-shaped paths over and under the equilibrium orbit remains on the conducting lining of the chamber, the orbit having 8 - 1 0 m m radial width of the " t r a c e " and sharp borders. Fig. 1 shows one of the pulse oscillograms of such on "electron spill" onto the accelerating chamber's walls, the oscillogram having been obtained by the technique described in ref. 2. The variation of the conditions of the electron injection into the accelerating chamber may essentially change the picture represented in fig. 1. The number of the "spill"--pulses may vary from one to 8-9 and also an oscillogram containing no pulses may be obtained. Electron

energy cn ~.

oJ

to

•--

C~l

(~

~r

t

I

r

"tO

tO m

t

I

t

t 5

t6

t7

to

l,l,ooj,,o1,0

t 1 t2 205

(MeV)

t 3

t4

180

2 5 0 I Ia s e c

I i

Fig. I. Oscillogram o f electron beam instability.

69

70

V. A. MOSKALEV

The reflection factor fl grows with an increase o f the atomic number Z of the target material and for Z = 74 (tungsten) fl = 0.78. By introducing the correction factor K = (1 - / 3 ) - ~ the agreement between the actual values and the results of measuring the charge captured in acceleration by our method becomes much better. To begin with, it was assumed that the electron loss during the acceleration was due to the resonance growth of the betatron oscillations, because the value of the factor N of the magnetic field decrease in one o f the betatrons was equal to 0.5. However, in other betatrons with N = 0.6 just the same beam instability in the vertical direction has been observed. Recent theoretical papers 5'6) shed light on the nature of the phenomenon. Ref. 5 considers the transverse electromagnetic inner interactions in an intensive azimuthally homogeneous beam of particles, taking into account the effect o f resistance of the vacuum c h a m b e r walls. The beam whose particles have the same velocity v is shown to be unstable with respect to the development of transverse waves with phase velocity of about (1 - v ) v (for the betatron), where v is the n u m b e r o f transverse free betatronic oscillations per turn. The rate o f the instability growth is proportional to N/a ½, where N is the n u m b e r of particles in a beam, and a is the conductivity o f the chamber coating material. The final formula for determination o f the time constant of the instability growth is n't o C_.2..~

ZO=

"YV z

h3

e2 ~ - - ~ - ~

[87ro'(l--v~)COo] ~-,

where mc 2

v z h 3-

±

K = ---;- ~r - - , [8 rctr(l - o:)] ~ 7T¢e" 1 0.775" 16" 5 3 2.8 x 10- iS ~ n" 9 × 1020 [8" 3 . 1 4 - 6.3 × 1025 (0.225)] ½ = 3.23 x 103 . Substituting in (2) the values y, fl and 090 corresponding to different moments of the acceleration cycle t,(e.g., tl, t2, etc. oscillograms) we obtain for z o r0 =

KKi

,

(3)

Ni

where K i = ( ~ i / f l i ) X / ¢ ' O o i corresponds to the m o m e n t t 1 of the oscillogram. Hence, the number of particles N~ circulating on the orbit by the m o m e n t tg is determined as N~ =

KKi

(4)

"cOl The time constant z equals the time during which the amplitude of the beam vertical oscillations (i.e. the beam vertical size) increases by the factor e. Hence, the vertical size A of the beam increases according to the exponential

(1)

A = Aoe t/~°.

(5)

Hence where mo and e are the mass and the charge of the particle respectively, c is the velocity o f light, y is the particles energy in terms o f rest energy, fl is a relative particle velocity, h is the vertical size o f the accelerating chamber and 090 is the particle revolution frequency. F o r our case, the values in the equation are as follows: h = 16.6 cm; vz = x/n = 0.775, where n = 0 . 6 is the field index; 090 = c/(2rcRo), where R o = 24 cm is the radius o f the equilibrium orbit; a = 6.3x ×10~6sec-1 is the conductivity o f the chamber conductive material (tin). The n u m b e r o f particles circulating on the betatron orbit at the predetermined m o m e n t of the acceleration cycle can be calculated as follows. After substituting certain values eq. (I) for our betatron becomes

Zo = K ~

\/090,

(2)

zo =

t In A - In Ao

,

(6)

where A is the m a x i m u m beam vertical size equal to the c h a m b e r height h. In our case h = 16.5 cm on the radius o f the equilibrium orbit Ro = 24 cm, t is the time required for the amplitude to increase from Ao to A. It follows f r o m the experiment that after the electron "spilling" onto the c h a m b e r walls, the beam experiences compression during the first part t t o f the time interval tl between the neighbouring "spills" and it expands during the second part t~_ a o f this interval. The beam compression after the "spill" of a part of the total charge m a y be due to the following factors. 1. The electrostatic repulsion of the extreme electrons of the beam by the charge, accumulated on the chamber coating after previous spillings o f electrons. If the mean value of the resistance o f the chamber

INSTABILITY

OF

AN

INTENSIVE

conductive coating for the earthed current is assumed to be R x = 200 f~, and if the capacitance of the coating relatively to the earth is determined as the capacitance o f two cylindrical condensers in series one o f which has percelain as dielectric and the second has an air one (the thickness of the chamber percelain wall being ~ 1 cm, e = 5 . 5 + 6 . 5 and the gap between the c h a m b e r wall and the pole being ~ 1 cm), it is possible to make an approximate estimation of the time constant R kCk o f the chamber coating. F o r the 25 MeV betatron this constant appears to be equal to 2 × 10- 8 sec, while the time o f the electron revolution on the orbit changes from 6.2× 1 0 - 9 s e c during injection (Ui = 350kV) up to 5 × 10 - 9 sec at the relativistic velocities. Thus, the time o f charge leakage from the coating is greater than the time of the electron revolution by more than factor ten o f magnitude. 2. Electron loss on the chamber walls causes not only the decrease of the particles number in the beam but also the reduction o f medium energy of the transverse motion of the remaining electrons, i.e. it results in the beam cooling. The cross section o f the beam decreases because of cooling. The variation o f both the electron number A N and the temperature parameter Aa during the time A t has been considered by Seidl in ref. 7 and he has shown that the beam cross section sharply decreases just after the electron loss and than this compression goes on much slower. 3. Because of the partial loss o f the beam charge and interaction o f the circulating electrons with the electrostatic field of the coating, the "spill", the coherency of the beam oscillations may be disturbed which results in a decrease o f the amplitude o f the electron vertical oscillations due to the effect o f the focusing forces by the betatron magnetic field. Let us assume now that interval ti is divided into two parts which are proportional to the ratio ti_ lit v Then time t of eq. (6) may be determined from the oscillogram as t = t~

(7) 1

To determine the initial size Ao~, i.e. the size up to which the beanl will be compressed during time t, let us use the eqw~ti~n of the charge density obtained by Cherdantsev: i~ 7 Q 2 ( e n - - 1 ) - - F = 0,

with /?=In

---P, P0

BEAM

(8)

IN

71

A BETATRON

,Q2= 4rce2 (1 - f l 2 ) p o t co, m

F=

f/2.

Magnitude R o is calculated by the formula m0c Po =

2

(2_1)~,

(9)

4 7reR ~

where R0 is the equilibrium orbit radius. Value F is derived from the equation d_.__FF_ F = - 02 " ( e " - 1). dr/ Assuming that the beam changes only its vertical size, with the width being unchanged in the radial direction, the solution o f the density equation (8) for the case of a plane beam becomes ~/ _ dr/ _ f2 [ e 2 ( " - " ° ) - 1 - 2 dt

e 2 " - " ° - e~)]~ = ¢b(r/),

(10) where r/o = r/(0) for t = 0, i.e. at the beam has the m a x i m u m value For t = 0 and r/o = r/the right-side to 0. Function t(r/) is derived from dt

1

1

dr/

~

¢p(r/)

the m o m e n t when A = h and i/ = 0. part in (10) reduces eq. (10)

Ol)

Hence

dt-

dr/

~o(r/)

(12)

So, after the integration over the range from r/o to r/,

no ¢p(r/)

t~ t ti + ti

ELECTRON

r e 2 t r t - "°)-- 1 --2(eZ~- ~°-- e~)] ~' ,J~o ~'~ L

(13) In a general case this integral is not calculable analytically, therefore the solution can be found by a numerical integration. F r o m the plot t = f(r/) drawn up for every interval ti+ 1 - t ; the beam cross section may be found as well as its vertical size .40 to which the beam contracts during time t. The accepted division o f the interval t i into two unequal parts has proved to correspond to the beam contraction after every "spill" to about equal vertical size close to .40 --- 10 cm for the oscillogram under consideration. In this case

72

v. A. MOSKALEV TABLE

the time t0~ necessary to reach A0 = 10 grows from interval to interval in a c c o r d a n c e with the g r o w t h o f the accelerated electron energy. Substituting A0 to (6) we find value r0~ for the c o r r e s p o n d i n g m o m e n t o f the accleration cycle. Then placing z01 in (4) we o b t a i n the n u m b e r o f particles N i which are circulating on the o r b i t in a time interval between ith a n d ( i + 1)th

Section of oscillogram

Duration (/~sec)

Electron energy at t = t~ (MeV)

Number of particles N~ x 1012

t1-0 t2 - tj t3 - t2

205 65 100

1.96 2.47 3.26

8.0 4.4 3.4

t4-- t3

110 130 180 250

4.52 5.13 6.61 8.35

2.7 3.2 2.35 2.03

ts-t4

30

t5

t6 -

25

I

t7

-

-

t6

20 ~15

~ ~ £

9 8 7 6

×

5

Z

electron spill-pulse. The results o f calculations for the oscillogram illustrated in fig. 1 have been listed in table 1. P o i n t 5-4 for the oscillogram c o n c e r n e d falls out o f the general trend o f the charge d i s p l a c e m e n t curve shown in fig. 2. A s it is seen f r o m the graph, the electron loss due to the b e a m transverse instability vanishes at the energy o f ~ 10 M e V a n d by that time a b o u t 2 × 10 a2 electrons r e m a i n on the orbit. T h e m e a s u r e m e n t o f the electron charge at the end o f the acceleration cycle has given the value o f a b o u t 3 x 1012 particles, which is in a g o o d agreement with the results o f the calculations. M a k i n g use o f these calculations it is possible to give a qualitative picture o f the b e a m b e h a v i o u r in a high-current b e t a t r o n d u r i n g the acceleration process. This is represented in fig. 3. [t shows that with the increase o f energy E a n d o f the n u m b e r o f particles N, the time t which is

4

1OO

200

I.L sec

5O0

1000

Fig. 2. Electron beam losses during accelerating process.

4962 2 . 4 6 7

3.26

,,

4.52

¥

5.13 "

6.61 "v

.8 . 3 4 5 250

E (MeW

~ se

×\\\~\\\,x\\\\\\\~,,

A ~~\\\\\\ \k:\k~XXX\\\\\\\\\\\\\\\x ~ ~ J ~\\\\\~\\\\\'&\\\~\\\\~\\\\~%:'b,\\\\\\\\k~S~\\\\\\\\\\\\\~:~~\ i'

tll

tt2 '

'4 Beam

t

°

t13-

.....4

t5

/\,,

v/

!~

t7

t

exis

,.. ],3,,> J ....

I

2.03

Fig. 3. Possible picture of electron beam behaviour.

Ni "1012

INSTABILITY

OF A N I N T E N S I V E

necessary to swing the beam from A 0 < h to A = h (where h is the vertical size of the chamber) is also increasing. After the charge value has lowered to 2 x 1012 particles, value Zo increases so much that t becomes greater than the time remaining to the end of the betatron operation cycle and the entire charge accelerating to the pre-calculated energy incidences onto the target. An increase of a captured charge does not result in any increase of the charge surviving to the end of the acceleration cycle, i.e. it does not result in an increase of the bremsstrahlung output. Thus, in our high-current betatrons we have obtained the maximal electron charge [(2-3)x 1012 particles per pulse] which might be accelerated in a betatron at preset parameters of a magnetic field. The charge being accelerated is limited by the origination of the beam transverse instability which is due to the interaction of the beam with the chamber conductive coating. The charge No captured in acceleration and circulating on the orbit at the beginning of the acceleration cycle (up to the energy of 2.1 MeV) can also be evaluated according to the data available from the oscillogram and the above mentioned calculations. The electron charge wasted away onto the chamber walls during the first "spill" pulse is equal to N 1 = (Nl_2--N2_3)

(S1/S2)

=

3.52x 10 12,

where N1_2 = 4.4× 1012 is the charge circulating on the orbit over the interval t 3 - t 2 and S 1 and S 2 are the areas of the first and the second "spill" pulses, respectively (taken from the oscillogram). Then, the charge N O circulating during the first 200 msec after the completion of the injection process is equal to N O = NI_2+N

= ( 4 . 4 + 3 . 5 2 ) × 1 0 1 2 ~ 8×1212.

Hence, it follows that any moderation of the beam instability during the acceleration process may result in an increase of the radiation output from a highcurrent betatron about several times as much, with the same basic parameters of the device. To suppress the instability in synchrotron a certain spread in particle energies or in betatron oscillation phases is created4). This greatly raises the radiation intensity. An analysis of the oscillograms allows one to assume one of the feasible ways of suppressing a beam

ELECTRON

B E A M IN A B E T A T R O N

73

instability in a high-current betatron. On all the oscillograms available (there are about 20 of them) it was always to_ 1 > tl_ 2, although over the interval t1_2 the charge N1_2 is less than No_ 1 corresponding to to_l, circulates on the orbit. As the highest charge circulates over the interval to_l it might be natural to expect that this interval should be the shortest one. The delay taking place in the development of the instability during this time interval appears to be due to the fact that at the initial period (and only during it) of the acceleration cycle there occur larger amplitudes of the betatron radial oscillations and a corresponding spread in energy of the particles. It seems likely that while high radial oscillations exist, vertical oscillations do not develop or develop very slowly. An intensive coherent oscillation of the beam in a vertical direction begins only after the beam contracts on the equilibrium orbit in the radial direction. If this effect really takes place then stimulation of the beam radial oscillations on the level of the maximum amplitudes during the entire acceleration cycle seems to be effective in elimination of the phenomenon of the beam transverse instability for that very number of the particles N which are captured in acceleration in the betatron concerned. Such an oscillation (swinging) is likely to give rise to spead in phases of the betatron oscillations, since the value N changes in accordance with the radius. The assumption that the radial swinging of the beam during the oscillation period exists, should be verified both theoretically and experimentally. This verification is intended to be carried out on a functioning high-current betatron. References 1) A. A. Vorobjev a n d V . A . Moskalev, Betatrons mit hoher Strahlungsintensitfit, Wiss. Z. Friedrich-Schiller-Univ. Jena, M a t h . - N a t u r w i s s . Reihe, no. 4 (1964) 501. o) V . A . Moskalev, Yu. M. Skvortsov, V . V . O k u l o v a n d V. G. Shestakov, O n m e a s u r e m e n t a n d recording of displaced current in a 25 MeV stereobetatrort (Electronic Uskoritely, Visshaya Shkola, Moscow, 1964) p. 204. ~) H. H. Seliger, Phys. Rev. 88, no. 2 (1952) 408. 4) W. Dressel, Phys. Rev. 144, no. 1 (1966) 332. 5) L. I. Laslett, V. K. Neil a n d A. H. Sessler, Rev. Sci. Instr. 36, no. 4 (1965) 436. 6) V. M. Balbekov and A. A. K o l o m e n s k y , O n coherent instability o f betatron oscillations in accelerators a n d storage rings, Soviet J. At. Energy (1965) 126. 7) p. A. Cherdantsev, Izv. Vysshikh Uchebn. Zavedenii, Fiz. 5 (1959) 45. 8) M. Seidl, Czech. J. Phys. 9 (1959) 721.