HIGH-EFFICIENCY TRAPPING BY AMPLITUDE MODULATION OF R.F. VOLTAGE

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1 134 Session III Conclusion in some accelerators. In fact, the entire particles can be stacked in the FFAG and brought Further study of the specific accelerator (the ZGS in the example chosen) is required to determine the feasibility of using the FFAG accel resonant extraction. If the problem of injecting out using radio-frequency phase displacement and erator as an injector. However, a part from the many turns ( ) into the final accelerator were beam instabilities for which remedies have been solved, the entire beam could be transferred in indicated, the FFAG is technically sound and 0,05 seconds. Furthermore, because of the high should be considered as a possible injector. Beam energy spread in the stacked beam for this mode, stacking provides a degree of flexibility not available none of the beam instabilities would be present. DISCUSSION WIDERÖE: How great is the mean extracted current of this machine? SNOWDON: protons/second. REFERENCES (1) F. T. Cole, G. Parzen, E. M. Rowe S. C. Snowdon, K, R. MacKenzie and Β. Τ. Wright: Nuclear 25, 189 (1964) (2) F. E. Mills et al.: Proc. Int. Conf. on High Energy Accelerators, Dubna, 1963 (Atomizdat, Moscow, 1964), p. 723 (3) Proposal for a High Intensity Accelerator, MURA Report (1963) (4) G. Parzen and P. Morton: Rev. Sci. Instr (1963); G. Parzen: Effects of Radial Str Sector FFAG Accelerator, Brookhaven National Laboratory, AADD-22 (5) C. D. Curtis: Injection into FFAG Accelerators. Conf. on Proton Linear Accelerators, Yale U Mills et al.: Rev. Sci. Instr. 35, 1451 (1964) (6) R. W. Fast and R. S. Christian: Contoured Poles for DC Accelerator Magnets, Int. Symp. on M Stanford (1965) (7) J.W. Hicks: An Adjustable Current Shunt for Trimming the Ampere Turns in a Magnet Operating tant DC Source, Int. Symp. on Magnet Technology, Stanford (1965). (8) S. C. Snowdon: Study of a 500 MeV High Intensity Injector, MURA Report 700 (1964, unpublished); E. M. Row One the Suitability of the 500 MeV FFAG Synchrotron as an Injector for the AGS and ZGS, MURA Report, 701 (1964, unpublished). Insufficiency ofr.f. system described in MURA-700, kindly pointed out by M. Plotkin, has been remedied here. (9) L. J. Laslett: On Intensity Limitations Imposed by Transverse Space-Charge Effects in Circular Parti tors. Proc Summer Study on Storage Rings, Accelerators, and Experimentation at Super-High Energies, Brookhaven, BNL-7534, p (10) L. J. Laslett, V. K. Neil and A. M. Sessler: Rev. Sci. Instr (1965). (11) V. K. Neil and A. M. Sessler: Rev. Sci. Instr. 36, 429 (1965). (12) M. Q. Barton: A Summary of the Cosmotron Experiments on Coherent Vertical Instability. BNL unpublished). (13) B. Gittelman: Bull. Am. Phys. Soc. 9, 715 (1964). HIGH-EFFICIENCY TRAPPING BY AMPLITUDE MODULATION OF R.F. VOLTAGE G. I. Batskikh Radiotechnical Institute, Accademy of Sciences (USSR) (Presented by A. A. Vassiliev) 1. INTRODUCTION The problem of increasing efficiency of particle trapping is one of the actual problems of charged particle accelerators technique. Different methods of increasing efficiency of particle trapping are used at present. On the Brookhaven AGS, for example, the problem is

2 Session III 135 Fig. 1 - Curves of the particle distribution in the stationary bucket. Fig. 3 - Mutual location of separatrices for accelerating (φ s = 30 a)nd bunching (φ s = O) of particles. Fig. 2 - Curve of the function F(ψ) = cos ψ - ψ sinφ s for φ s = 30. solved by reducing equilibrium phase at the beginning of acceleration cycle by means of reducing the speed of magnetic field growth at nominalr.f. tension (1). European Organisation for Nuclear Research (CERN) makes use of preliminary bunching of particles (2). For this purpose the r.f. tension with frequency somewhat different from that of beam rotation is fed to accelerating elements during injection. Effected by the voltage the beam is compressed in phase and some time later the beam is assumed Z-shaped form in the phase plane. After this, the phase and the frequency of ther.f. tension are changed by means of phase lock in such a way that «to cover» by separatrix the compressed beam. The paper deals with the method of bunching under the effect of ther.f. tension with equilibrium phase equal to zero and the frequency equal to (or multiple with) beam rotation frequency. Fig. 4 - Dependence of particle trapping efficiency (η) on bunching time (τ). Here, as in the previous case, the beam is compressed in phase but the trapping is made by increasing the r.f. tension without changing its frequency and phase. This method of bunching is discussed as applied to 1 GeV cybernetic accelerator developed in the Radiotechnical Institute of the USSR Academy of Sciences (3). The peculiarity of accelerator is constant magneticfieldduring injection; however, the method may be employed for other accelerators as well. 2. PARTICLE MOTION INTO STATIONARY BUCKET Assuming that accelerator parameters change rather slowly we may have the following form of the phase equation: ψ + ω c 2 (sin ψ sinφ s ) = 0 [1]

3 136 Session III where ψ is the phase angle taken from zero crossing of accelerating voltage, = d 2 ψ/dt 2, ω c is the angular frequency of the small phase oscillations. Equation [1] is the non-linear differential equation. After multiplying it by 2 = 2 dψ/dt and integrating we have = 2ω c {COS ψ COS ψο + + (ψ - ψ 0 ) sinφ s + /2} ½ 0 [2] where and ψ o are the starting conditions for injected particles. For simplicity it is assumed that particles are not energy-spread ( = 0). Then, for non-accelerating r.f. voltage (φ s = 0) espression [2] will have the form: = 2ωΦ {cos ψ cos ψ ο } ½ [3] Further ω c will be marked by indices Φ and y corresponding to particle buching (φ s = 0) and to particle acceleration (φ s 0). Equation [3] describes the particle motion trajectories in the phase plane with different starting conditions in case of φ s = 0. We are interested in the distribution of injected particles within stationary bucket. For this purpose, we determine the speed of the particle motion in the phase plane. Now, we put down equation [3] in the following form: dψ dt = [4] 2 ωc Φ {sin 2 ψ ο /2 - sin 2 ψ/2} ½ and after changing variables we have sin ψ/2 = sinα sin ψ ο /2 [5] 2 cos α sin ψ ο /2 dα dψ = [6] {1-sin 2 α sin 2 ψ ο /2} ½ Substituting [5] and [6] into [4] we have dα dt = [7] ωc Φ {1 - sin 2 α sin 2 ψ ο /2} ½ After integrating equation [7] may have the form: 1 t = ψο ωc Φ Κ (α, ψ ο/2) [8] where Κ (α, ψ ο /2) is the elliptic integral of the first kind. From relation [5]it is evident that α = π/2 at ψ = ψ o and α = 0 at ψ = 0. The change of ψ from ψ ο to 0 corresponds to particle travel of a quarter of a revolution in phase plane during rotation around the centre of a stationary bucket (a quarter of the phase oscillation period). Consequently, using expression [8] an angular frequency of particle rotation on the phase plane may be expressed in the following form: πω c Φ ω = 2K (π/2, ψο /2) [9] where Κ (π/2, ψ ο /2) is the complete elliptic integral of the first kind. Making use of complete elliptic integral tables and knowing the trajectory of the particle motion in phase plane [3] we determine the curves of the particle distribution in the stationary bucket for different instants by means of formula [9] (Fig. 1). Here τ = t/τ cφ, where Τ cφ is the small phase oscillation period in the case of bunching. It is evident that at τ = ¼ and τ = ⅓ particle distribution curves are developed Z-shape. The particles are concentrated in phases, but at the same time the height of Fig. 5 - Dependence of particle trapping efficiency on the value of m. Fig. 6 - Mutual location of an accelerating and stationary buckets an da beam for φ s = 30, m =1,7 and τ = ⅓.

4 Session IIΙ 137 distribution curves grows which corresponds to the increase of the particle energy spread. Consequently, it is necessary in the time of trapping to have the accelerating bucket height more than that of particle distribution curve. It is necessary to make note of the fact that plotting the distribution curves (Fig. 1) we do not take into account non-constancy of speed of particle motion in the phase plane during single revolution. To check the precision of curve plotting by formula [9] we determined the time within which the particles with different starting conditions ψ ο reached the curve corresponding to τ = ⅓ (as we shall see later this curve is the most significant for us). Calculations show that error of curve plotting by formula [9] does not exceed 5% for particles with starting phases from 0 to 150 which is quite sufficient for practical calculations. 3. OPTIMUM CONDITIONS OF PARTICLE TRAPPING We put down equation [2] for the case of φ S 0 (particle acceleration) at = 0 in the following form: = 2ω cy {cos ψ ψ sin φ S cos ψ ο ψ ο sin φ S } ½ [10] Fig. 2 shows the curve of function F (ψ) = cos ψ --ψ and acceleration (m) it is expedient that r.f. tension during bunching be lower than that neces sin φ S (for the case of φ S = 30 ). It is known (4) that stable oscillations will be peculiar for the sary for particle acceleration with the selected particles for starting conditions of which the following expression is true: equilibrium phase (φ s ). It is not difficult to show that ψ 1 ψ 0 ψ 2 (See Fig. 2) ψ 2 is a minimum of the function F(ψ),it is equal to π φ s. Substituting the value ψ O = π φ S in expression [10] we receive separatrix equation for the acceleration case (φ S 0): = 2ω cy { cos ψ ψ sin φ s + + cosφ s (π φ s )sin φ s } ½ [11] A maximum of expression [11] (separatrix height) corresponds to value ψ = φ s,it is equal to max = 2ω cy {cosφ s (π/2 φ s )sin φ s } ½ For the case of φ s = 30 max = 1,17ω cy whereas for the case φ s = 0 (stationary bucket ) max = 2ω cφ, that is in both cases separatrix height is directly proportional to phase oscillation frequency. Fig. 3 illustrates mutual location of separatrices for accelerating (φ s 0) and bunching (φ s = 0) of particles at different phase oscillation frequency ratio in these cases (m ω cy /ω cφ ). It is evident that at an instant of trapping it is necessary to increase phase oscillation frequency which will result in growth of the bucket height and consequently the number of captured particles. By the curves of Figs. 1 and 3 graphical calculation were made of dependance of particle trapping efficiency η (percentage of the number of injected particles) on bunching time (τ) for different values of m (Fig. 4). It is evident that trapping efficiency curves for m > 1 have a maximum depending on bunching time. Values of maxima grow with increasing of m but it is unexpedient to choose value of m > 1.7 so far as further increasing of m has an influence on the trapping efficiency slightly (Fig. 5). At the same time the growth of m increases the time of bunching which is undesirable at all. Thus we may consider optimal the case when the stationary bucket height is equal to the accelerating bucket (m 1.7, see Fig. 6). Here, the time of bunching has to be equal to ⅓ of the phase oscillation period for the bunching case (see Fig. 4). Mutual location of buckets and the beam at an instant of particle trapping for this case is shown on Fig. 6. Theoretically the possible percentage of particle trapping in this case is equal to 80%. To provide for the necessary relation between phase oscillation frequencies during bunching U Φ = U y cos φ s where U Φ and U y are r.f. tension during bunching and acceleration of particles. For the case of φ s = 30 and m = 1.7 we have U φ = 0.3 U y. Consequently, the following conditions should be considered optimum ones for the accelerator operating with equilibrium phase of 30 : 1) A bunching r.f. tension is 0.3 of the accelerating one. 2) Time of bunching is ⅓ of small phase oscillation period in the case of bunching. In order to make use of such trapping it is necessary to increase the r.f. tension 3.3 times. At an instant of trapping the frequency and phase of r.f. tension remain unchanged. CONCLUSION The high efficiency particle trapping method which was discussed is proposed for use in 1 GeV cybernetic accelerator. Because of the fact that a Van de Graaff machine with r.f. ion m 2

5 138 Session III source is an injector of the 1 GeV accelerator we did not examine an influence of an energy spread. The given method of particle trapping has nearly the same efficiency as the method employed in the CERN-PS (2). The method of highefficiency trapping with the amplitude modulation has the possibility to avoid the transients in the beam phase-lock system which occur in the CERN-PS. As for the requirements tor.f. tension jump during particle trapping in 1 GeV cybernetic accelerator they are rather simple with the use of the method described. REFERENCES (1) M. Plotkin: private communication. (2) K. Johnsen: Int. Conf. on High Energy Accelerators, Brookhaven, 1961, p (3) A. L. Mintz, A. A. Vassiliev, G. I. Batskikh, N. L. Sosensky and A. I. Dzergatch: Int. Conf. lerators, Dubna, 1963, (Atomizdat, Moscow, 1964) p (4) M. S. Rabinovich: Ann. Phys. Inst., 10, 23 (1958). EFFECT OF NON-LINEARITY OF THE BEAM TRANSPORT SYSTEM ON THE INJECTION PROCESS OF THE PROTON SYNCHROTRON N. L. Sosensky Radiotechnical Institute, Academy of Sciences (USSR) (Presented by A. A. Vassiliev) Determination of relationship between amplitude increment for betatron oscillations of particles injected in proton synchrotron on the one hand, and magnetic field errors of quadrupole lenses on the other, have been the subject of our study. The ratio determining the particle loss at injection due to non-linearity of the beam transport system has been obtained. 1. Non-linearity of real quadrupole lenses used non-ideal system and C i for focusing force of i-lens in the beam transport system results in dismatching in the ideal system. of injected beam emittance with the proton Then the expressions for the components H ix synchrotron admittance and consequently in increasing of betatron oscillation amplitude. Be non-ideal system will look like (1): and H iy of the magneticfieldfor i-lens in the low we describe the approximate method for determining rather simple expressions for the number of particles receiving amplitude incre Ρ H ix = [C,u i +μ i F i (u i,v i } )] ments of betatron oscillations due to the non-linearity e of the beam transport system. 2. Let us consider the transport system consisting of a series of thin lenses spaced at some Ρ [1] H iy = [C i v i + μ i G i (u i,v i )] e intervals. Let us number the lenses one after another as they are arranged in the direction of particle motion as i = 1,2...n. Further we shall consider two types of ionooptic system: the one consisting of ideal (linear) lenses and the other consisting of lenses possessing a slight non-linearity. The following designations will be used: κ, y for the transverse coordinates of a particle in the injection plane of the accelerator ring when dealing with an ideal system, and u, v for the same coordinates when working with a non-ideal system. u i, V i will be used for particle coordinates in the plane of i-lens of in which e, p stand for the charge and the momentum of the particle,μ i «1 stands for small parameters (i = 1,2...n), F i, G i are functions specified by the type of non-linearity of i-lens.