Betatron oscillation resonances in an electron synchrotron

NUCLEAR

INSTRUMENTS

AND METHODS

5 2 (i967) 325-330; © N O R T H - H O L L A N D

PUBLISHING

CO.

B E T A T R O N O S C I L L A T I O N R E S O N A N C E S IN A N E L E C T R O N S Y N C H R O T R O N S. L. ARTJEMJEVA, O. F. KULIKOV, YU. N. METALNIKOV, E. M. MOROZ, V. A. P E T U K H O V and K. N. SHORIN

Lebedev Physical Institute of the USSR Academy of Sciences, Moscow, U.S.S.R. Received 31 January 1967 The conditions under which certain betatron resonances of the 2 na to the 5 th order are excited, and their influence on accelerator operation, have been studied in an electron racetrack. The precise measuring methods ( ~ 0 . 1 % ) of parameters n and 2 which

determine the betatron oscillation frequencies and resonance relations, have been elaborated. High speed filming of optical synchrotron radiation has been broadly used in experimental investigations.

1,Introduction Fro the theory and application of cyclic accelerators and storage systems, the problem of betatron oscillation resonances is a matter of considerable interest at least from two points of view. On the one hand, resonance antidamping of betatron oscillations may result in particle losses, beam intensity limitations and instabilities in operating installations. On the other hand, some resonances may be effectively used for ejection, injection and targeting of the beam; there are cases where application of a resonance is the only solution to the problem. For all such applications (or the occurences) of resonances of undoubted interest are the methods of increasing measurement accuracy of oscillation fiequencies, methods of reliably identifying resonances, investigating their effect on the operation of the installations, investigation of the conditions of their excitation and action. In the present paper are described the methods and results (mainly experimental), of such investigations performed on an electron racetrack having a maximum energy of 680 MeV.

0,?5 t/,

~70

2. Betatron oscillation frequencies and resonance conditions

0,60

In an ideal racetrack (i.e., in a weak-focusing accelerator where the magnetic field intensity is assumed to be independent of the azimuth in the magnet sectors and equal to zero in the straight sections), the numbers Qz and Qr of axial and radial betatron oscillations for one revolution of a particle are determined by the following well-known formulas1):

Fig. 1. Grate of resonance lines. Initial values of n and 2 are indicated by dashed lines. Resonances reliably identified are represented by heavy lines.

various resonances of the type KzQz + 1,:,Q, = K ,

Qz,r = Nl~/(2n); cosft = c o s ~ - ½ ~ s i n ~ ; Oz = 2nn÷/N; Or = 27t(1-n)~/N.

(1)

Here N is the number of magnet sectors, n is the magnetic field index in the sectors, and 2 = Nl/(2nR) the ratio of the total length of the straight sections to the total length of all circular sections of the orbit. Eq. (1) enables us to plot a grid of resonance curves n(2) for

(2)

where Kz, Kr, K are positive or negative integers, and IKz I + IKrl is the order of a resonance. Some resonance lines significant for operation of accelerators or storage systems are shown in fig. 1. Axial, radial, sum and socalled ideal (x = pN, p = integer) difference resonances are shown up to the eighth order; other difference resonances up to the fourth order inclusive.

325

326

s.L. ARTJEMJEVA et al.

Expanding the trigonometric functions in series, one can obtain from eq. (1) the following expressions

1,0

2 = ( Q =2

00

[-

Q 2, - 1 ) 2

{

1 - ( ~"1n 2 / N 2 )(Q=2 + Q ,2- 1 ) " 2

2

2

2

•[ I - 2 Q ~ / ( Q ~ + Q . ) + 2 { Q z / ( Q = + Q , ) }

2

2 } ] /

{Q~I(1 +,t)} [1-(½~/N ~) {2QzlO +2)}~] ,., 1 - {O21(i + 2)} [1 -(k~21u 2) {2Q,/(1 + 2)}2],

,

(3)

0

~9

zo

3o

9

s,0

y e~

q8

,, ~ -

(4)

which relates the ideal racetrack parameters 2 and n to the betatron oscillation frequencies Q~ and Qr. If the values of Q~ and Q, are known, a real accelerator, in which the magnetic field penetrates the straight sections, may be reduced to an ideal one by means ofeqs. (3)and (4). For the 680 MeV pulsed synchrotron, from an experiment described below, we have obtained Qz = 0.9056 and Qr = 0.5532. Substitution of these values in eqs. (3) and (4) yields 2 = 0.124 and n = 0.728. For comparison with the experimental results, it is of interest to calculate the values of 2 and n on the basis of the measured magnetic field distribution in the straight sections of the accelerator. Since the magnetic field penetrates the straight sections, the parameter 2 and the method of calculating this, enabling us to reduce a real accelerator to an ideal racetrack, are not strictly defined. Using the results of 2), one can obtain the following approximate formula for 2, which takes into account the presence of the field in the straight sections: 2 = max I N / e ( 1 - 2 n ) / ( L - N I ) ] .

(5)

Here le is the effective length of the straight section; L = c / f o is the closed orbit length (fo is the accelerating voltage frequency, c the particle velocity); u is the ratio of the average streight section field to the average sector magnetic field of the accelerator. Fig. 2 illustrates this calculation of 2 on the basis of magnetic measurement data. Curve 1 represents the magnetic field variation along the 680 MeV synchrotron orbit 5000 Oe. Along the abscissa is plotted the distance from the middle of the straight section, and along the ordinate the magnetic field strength in relative units. Curves 2 to 5 illustrate successive stages of the calculations. Curve 6 represents the 2 values calculated under various assumptions relative to the effective straight section length 1,. One can see that for le close to the geometrical distance between the edges of adjacent sectors l = 67 cm, the value of 2 is slightly less than the value obtained experimentally. Taking into account that in all the assumed cases the calculated 2 is less than the experimental one, it is reasonable to choose the maximum 2 = 0.120 as the effective value of 2.

q6

qs

Lo,zo

o,,, 0,t5 q2 03

o

~o

50

do

&to _

e,i-)

Fig. 2. Some characteristics of a synchroton magnetic field; 1 - H(y);

2 - str - the result of averaging of H(y) along the assumed length le; 3 - s the result of averaging of H(y) over the assumed length of magnet sector; 4 -K = < H ( y ) > s t r / < H ( y ) > s ; 5 - Z1(0= Nl/(L- Nl); 6 - 22(l) = (1 - 2x)21(l). The difference between the values of ;t obtained experimentally and those calculated by means of eq. (5) is not large. This difference may result in an error of less than 0.2% for betatron oscillation frequencies. At the same time, the above comparison indicates that it is necessarry to include the factor (1-2~) in calculating 2. If this correction is not introduced, the errors in the betatron frequencies assume intolerably large values of

3-5%. The effective field index n of a weak-focusing accelerator also may be determined from magnetic measurement data by means of the equation n = - ( R / H ) [(dH/dR)l + 2u(dH/dR)2] +

+(N/n){f-(fin/R)}, (6) which follows from 2). Here ( d H / d R ) l and (dH/dR)2 are the magnetic field derivatives in the sector and in the straight section of the accelerator, respectively; 26 is the difference between 2 g I N and the sector azimuthal angle. One can see

BETATRON OSCILLATION RESONANCES IN AN ELECTRON SYNCHROTRON

327

from eq. (6), that the major part of n isn 1 = - ( R / H ) . • ( d H / d R ) l , i.e., the field index in the sector. A variation in n is equal to the variation in nl because the other terms in eq. (6) are small and are not dependent on n. The parameter 2 changes insignificantly in this case. This fact was used for orientation on the grid of resonances (fig. 1) in the experiments described below. In addition, to identify the type of the resonances crossed by the beam, some distinguishing features of the cross-sectional distribution of the particles were used. In particular, a cross-sectional view of an electron beam crossing sum resonances has the shape of a geometrical figure with hyperbolic boundaries. The angle c¢between the asymptotes depends on the resonance type and initial conditions3), in accordance with formula:

tglct=az/ax=(Kz/Kr)~az,o/ax,o

= {(Kz/Kr)(I - n ) / n } I,

(7) Here a~ and ax are the axial and radial oscillation amplitudes, respectively, which have increased due to the sum resonance, and az, o and ax, 0 are the corresponding amplitudes before the resonance was crossed. The last part of eq. (7) is applicable if before the resonance the main cause of the oscillation antidamping is gas scattering.

3. Experimental technique and apparatus For experimental investigation of the resonances, conditions (2) were created by a short-term variation of the field index n~ by providing an additional pulse of current I n in a gradient winding of the accelerator4); the amplitude and the shape of the pulse may be varied. Beam losses due to a resonance at some values of I, were observed by means of a circulating beam intensity monitor or, in some cases, by a 7-radiation monitor. The beam intensity signal~C~ together with the current signal I n was fed to a double-beam oscilloscope (fig. 3a,c). Variation of the betatron amplitude distribution due to the resonances was investigated by observing optical radiation of the beam. Quantitative data were obtained both from the oscillograms and from highspeed filming of the electron optical radiation. Two modes of operation of the camera were used in filming the beam cross-section. The first mode is the usual frame filming, when a framing prism is used and the image follows the film within the limits of one frame (fig. 3d). The second is the so-called photorecorder mode of filming, where the framing prism is removed (fig. 3b). The former gives a full picture of the beam cross-section, but may lead to some difficulties when investigating time-varying processes. The latter ensures continuous observation, but only for one component of

Fig. 3. Example of recording resonance processes. a. Oscillograms of the beam intensity signal from pick-up electrodes and of the current pulse in the gradient winding. 1 - The beam losses caused by resonance Qz+ 2Qr = 2; 2 - The beam losses by Qz + 4Qr = 3; 3 - The beam losses by 2Qr = 1. 4 - The beam losses caused by the high order sum coupling

resonance. b. Result of photorecording of the beam radial size dynamics. c. Oscillograms of beam losses and of current in the gradient windings. 5 - Beam losses at resonance 3Qr = 2. 6 - Beam losses at 2Q~- Qr = 1. d. A series of frames showing the dynamics of bunch crosssection. the oscillations. Frame filming is better when searching for resonances; photorecording yields better quantita-

tive results.

328

s.L. ARTJEMJEVA et al.

The time relationship of the effects recorded on the oscillograms and on the film was established by means of time marks transmitted to the oscilloscope and for the camera by one and the same generator. The time marks also enabled us to e~tablish the relationship between the events under investigation and various stages of the acceleration cycle. To identify the resonances more reliably, the frequencies Qr, z were independently measured for various values of the current I, in the gradient winding. For this purpose, the betatron oscillations were additionally excited by a transverse high-frequency field, the field frequency f being chosen equal to one of the expected values of the betatron frequenciesf~,= = Q~,~'fo, or to the difference fo - f r , z (wherefo is the accelerating field frequency). The exciting rf voltage was produced by a standard signal generator. Through a wideband amplifier this voltage was fed to a pair of horizontal plates located inside the vacuum chamber (for fz measurement), or to a pair of fiat coils placed in the gaps between the chamber and pole pieces of the accelerator magnet (for f~ measurement). The input voltage of the amplifier was modulated by a rectangular pulse of controllable length. During each measurement the freq u e n c y f o f the excitation field was kept constant, and the frequencies Qr,z varied by varying the gradient winding current. The characteristic signal of oscillation antidamping was detected by the camera at the instants when f = f o - J r , ~ or f =f~,z. We could calibrate the current pulse I. in terms of oscillation frequencies f,,z or Q,,z by measuring them at various frequencies f The oscillation frequencies near the resonances under investigation were measured in the same manner.

Signals shown in fig. 4a,b,c were obtained for the case of radial oscillation antidamping with the camera operating in the photorecording mode. One can see that variation of the frequency causes a shift of the antidamping signal in time. In most cases the oscillation antidamping by the transverse excitation field did not result in noticeable beam losses; antidamping could be detected by filming only. An analogous method was used to measure the betatron frequencies Qr,z under normal operation of the accelerator (without exciting the gradient winding) and to determine the corresponding values of n and 2, viz. (no, 2o), the measurement technique is illustrated in fig. 3. Oscillograms 3a,c show current pulses I, (for both modes of operation) and the corresponding signals JV£ from the beam intensity monitor. The indexes 1 to 6 indicate the beam losses appearing when crossing various resonances; 3b,d show the corresponding dynamics of the beam cross-section (fig. 3b was obtained in the photorecording mode, 3d by frame filming). 4. Experimental results

By means of the experimental method described above and the relations (3) and (4), we were able to determine with a high degree of accuracy the combination of values Qz, Qr, n, 2 in the usual mode of operation of the accelerator as well as under given resonance conditions. The initial values o f n and ), are marked by the intersection of the dashed lines on fig. 1. The resonances reliably identified in the experiments are represented by heavy lines. Table 1 summarizes some quantitative results obtained from the investigation of the indicated resonances

(a)

(b)

(c) Fig. 4. The radial oscillations antidamping caused by an rf field, obtained with the camera operating in the photorecording mode. Frequency of rf fieldf = 10.020 Mc (a); f = 9.928 Me (b); f = 9.550 Mc (c). The accelerating field frequencyfo = 20.534 Me.

BETATRON

OSCILLATION

RESONANCES

IN

AN

ELECTRON

329

SYNCHROTRON

TABL~ 1

Type of the resonance

Qz+2Qr=2

Qz+4Qr=3

2Qr=l

3Qr= 2

2ez-Q r=l

0.606

0.611

C r o s s - s e c tion

o f t h e beam

i

Qz

0.9120

Q9256

E

Qp

0.5443

0.5181

0.488 < Or <0.503

n x

/1

0.737

0.762

> 0.774

50*

-- 34*

'7"

kt.I

8 S 0

L)

/1

0.736

0.761

46*

31"

and also the corresponding beam cross-sectional distributions obtained by filming. All these data and photographs relate to experiments at electron energies from 120 to 150 MeV. The resonances were identified by comparison of the beam cross-section frames with the data obtained by radio frequency antidamping at various exciting field frequencies and at various phases of the current pulse. For some resonances it was possible to measure Qr,= close to the resonance line and check the corresponding formulas (2). Under the conditions of the experiment, the influence of resonance on acceleration can be seen from the oscillograms in fig. 3. A decrease in the beam intensity due to a resonance is indicated by an arrow and by the corresponding resonance condition. Among the dangerous resonances the closest to the operating point is a sum coupling resonance of the third order Qz+2Q, = 2. In accordance with a) when crossing this resonance, the betatron oscillation amplitudes may increase without limit and result in a noticeable beam loss. In thiscase, motion of the particles along the beam cross-section is described by hyperbolas, the angle between their asymptotes being determined from relation (7). With respect to this resonance, table 1 shows close agreement between theory and experiment. It should be noted that it is this resonance that for a long time had been an unknown cause of instability and limitations of the accelerator intensity. As a consequence of this investigation, we introduced a correcting current into the gradient winding to eliminate the undesirable influence of this resonance.

0.777

0.606

0,612

The sum coupling resonance of the fifth order, also is located near the operating point of the accelerator, should not according to 3) result in an unlimited increase of the oscillation amplitudes. Nevertheless, the experiment shows that this resonance results in a noticable intensity drop. Identification of this resonance is absolutely reliable: the resonance frequencies obtained by the measurement, Q,=0.5181 and Q==0.9256, yield Q = + 4 Q , = 2 . 9 9 8 ; the characteristic cross-section of the beam (table 1) gives an angle ct between the asymptotes, in close agreement with eq. (7). Filming of the beam clearly shows that the radial oscillation amplitudes increase when this resonance is crossed. The resonance 2Q, = 1 (when particles execute one radial oscillation every two revolutions) is characterized by a distinct doubling of the beam, the distance between the two beam nuclei approaching 16 cm. Beam cross-section pictures show that a sum coupling resonance acts almost simultaneously with this resonance, the shift of these resonances in time (and in n value) being small (see the characterstic hyperbolic crosssection, table 1). The order of this sum resonance is probably higher then those indicated in table 1. The beam losses seen on the oscillogram and indicated by the index "4" (fig. 3a) are caused precisely by this resonance. Such a noticeable effect of a high-order resonance is apparently due to the simultaneous action of the strong resonance Qr = ½The third order nonlinear resonance 3 Q, --- 2 (table 1). was identified by measuring frequencies Qr.

Q=+4Q,---3, which

330

s.L. ARTJEMJEVA et al.

The picture of the cross-sectional distribution of the particles in the beam clearly illustrates the action of this resonance. From fig. 1, one can see that the resonance lines 3Q, = 2 and 2 Q z - Q , = 1 intersect near the line 20 . This circumstance explains some features of their action, in particular the action of the difference resonance. Under conditions of the experiment, the beam crosses in succession the difference resonance 2Q~ - Q, = 1, the resonance Q~ = ~ and then the first resonance again. A detailed analysis of the beam cross-section pictures and the oscillograms shows that the first transition through the coupling resonance does not result in intensity losses, but antidamping by the resonance Q, = ~ produces beam losses during the second crossing of the resonance 2 Q z - Q, = 1. The combined action of these two resonances on the beam is illustrated by the oscillogram of fig. 3c.

5. Conclusion Some other resonances were observed and experimentally investigated in the synchrotron. In the above work, we have limited ourselves to investigating some effects which we believe are of practical importance in the operation of accelerators and storage rings. The reso-

nances Qr = ½ and Q, = $ are interesting as methods of oscillation antidamping for beam-ejection purposes, and sum resonances may be a cause of instabilities. Under the specific conditions of various installations, difference coupling resonances may be dangerous as well. The described technique makes it possible to investigate the resonances with great precision. The authors would like to take this opportunity to thank V. E. Pisarev, M. G. Sedov, A. N. Lebedev for their interest in this work and their valuable advice; also, A. S. Jarov and O. E. Shirokov for their assistance in solving a number of experimental problems.

References 1) M. S. Rabinovich, Trudy FIAN 10 (1958) 23. 2) E. M. Moroz, Trudy FIAN 13 (1960) 130. 3) A. A. Kolomenskij and A. N. Lebedev, Teorijazyclicheskikh uskoritelej (Gos. izdat, phys.-mat, lit., Moskva, 1962). Transl. Theory of cyclic accelerators (North-Holland Publishing Company, Amsterdam, 1966). 4) O. F. Kulikov, E. M. Moroz, V. A. Petuchkov, K. N. Shofin and A. S. Jarov, Nucl. Instr. and Meth. 40 (1966) 173. 5) F. T. Cole, R. O. Haxby, L. W. Jones, C. H. Pruett and K. M. Terwilliger, Rev. Sci. Instr. 28 (1957) 403.