LONGITUDINAL MOTION OF PARTICLES IN THE ELECTRON SYNCHROTRON AT FIRST FEW REVOLUTIONS

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1 367 expect incidentally that with an increasing distribution, due for example to injection errors, the bunch at the beginning of the cycle may be unstable, which should result in an increase of coherent oscillations at low frequencies (ω ~ 1). This method using a self-consistent kinetic equation is applicable in more general cases than those considered above. In particular it may be useful in analytic investigations of certain features of injection in accelerators with phase lock. REFERENCES (1) W. Schnell: Int. Conf. on High Energy Accelerators. CERN, 1959, p (2) Yu. S. Ivanov and A. A. Kuzmin: Pribory i Tehnika Eksperimenta, 4, 106, (1962). (3) E. L. Burstein et al.: Pribory i Tehnika Eksperimenta, 4, 102, (1962). (4) H. Hereward: Intern. Conf. on High Energy Accelerators, Brookhaven, 1961, p (5) E. A. Zhilkov and A. N. Lebedev: Atomnaya Energia, 18, p. 22, (1965). (6) E. A. Zhilkov: Pribory i Tehnika Eksperimenta, 1, 17 (1965). (7) G. Newton, L. Gould and J. Kaiser Analytical Design of Linear Feedback Controls (N. Y. - London, 1958). LONGITUDINAL MOTION OF PARTICLES IN THE ELECTRON SYNCHROTRON AT FIRST FEW REVOLUTIONS S. A. Kheifets Institute of Physics, Erevan (USSR) The accelerating system of a circular accelerator, must be found, taking into account producing the stability regions of longitudinal the electric field produced by the particles themselves. motion, brings about a rather effective An attempt to solve this problem was made grouping of the accelerated particles. The frequency by K. Robinson (5) who built a system of differential of motion of bunches coincides with equations describing the longitudinal motion that of the accelerating field which in the electron of particles, taking into account the induced synchrotron is practically equal to the proper voltage, as well as by Passov (6) who developed frequency of the accelerating gaps (resonant an analogue model of the motion. cavities). This leads to a very effective generation Nevertheless, the equations due to Robinson of electromagnetic oscillations in the resonant are obtained only neglecting terms of the order cavities. of magnitude Δω/ω, where ω is frequency shift of the high frequency generator. According to The resulting electric field is phase shifted the results found in references (7, 8) these terms with respect to the field, produced by an external may be of considerable significance. generator, and depends upon the current The present study treats the above problem as of the particles being accelerated. This phenomenon a system of equations in finite differences. Such has been treated in a great number of approach seems to be closer to the physical picture studies (1, 2, 3, 4). of the phenomena taking place in the accelerator However, all these investigations either do not and at the same time it suits better the take into account the reverse effect of the induced treatment by means of a computer, since an analytical voltage on longitudinal motion of particles, solution to the problem appears un attainable. or induced voltage is calculated at a given motion Besides, the equations obtained are much of particles. Strictly speaking, the problem more accurate since they remain terms of the order should be solved in a self-consistent manner in of Δω/ω. the sense that the motion of particles in the synchrotron To simplify the problem it should be assumed

2 368 Session VII that on the orbit there is one resonant cavity with negligible lenght of accelerating gap. When a particle of the energy Ε passes the resonant cavity at the moment t, its energy changes as E i+1, = Ei + ev(t i ), where V(t i ) is the voltage on the accelerating gap at the moment t. Analogous relationship nominally holds for an equilibrium particle E i+1, s = E i,s + ev i, s On the rest part of the orbit the particle energy remains constant and the time of the next passage is determined by the relationship t i+1,s = t i,s + T o, t i+1 = t i + T(E i+1 - E i+1,s ), where T o is the period of revolution of the equilibrium particle, determined by the frequency ω o Fig. 1 - A blok-diagram of the program for calculation of longitudinal motion of particle at first few revolutions.

3 369 of the external generator: T 0 = 2πq/ω 0 q is the wave numer of the high frequency. Expanding the funtcion T(E i+1 - E i,1,s), into a series and introducing the notation E i = (E i E i,s )/E i,s, ε i =Ε i,s /E inj ; φ i = ω 0 t i, α = ( InL/ In E) s an initial system of equations for longitudinal motion of the particle is obtained. v(φi)-vi,s E i+n = E i + εi ] ] εi εi + I φ i +1 = φ i +2πqα E i πq E i + 1 = E i + V i,s φ i + 1,s = φ i,s + 2πq Here ν = ev/e inj The last factor in the first equation is related Fig. 2 - Losses of particle during first few revolutions. Fig. 3 - Full line-voltage in the cavity including induced voltage; broken line-voltage in the cavity without induced voltage. 24

4 370 Session VII to the damping of phase oscillations. It is evident that,if ν (φ i ) = ν 0 cos φ 0, then the first two equations are equivalent to ordinary equations of synchrotron motion. The frequency of small oscillations is given by the formula 2πqdv 0 sin φ s Ω 2 = ; v s cosφ s = Τ 2 0ε V0 Wen η particles move along the orbit, the system of equations is of the form E v(φi) vis i+1 = (E i + εi ) εi ) εi+n [1] φ i+h = φ i + 2πqα E i+n + 2πq [2] ε i+n = ε i + ν i,s [3] φ i+n,s = φ i,s + 2πq [4] The function v(φ i ) included in equation [1] taes into account both the voltage, produced by the external generator and the voltage, generated by the particles themselves It is easy to show that v i and u i may be written in the form of the following recurrent relationships; Δ V i = e -η (φ i -φ i-1 ) {(Δ V i-1 + δ i-1,0 Δ v 0 ) cos ξ ( Δ U i-1 sin ξ (φ i φ i-1 )} Δu i, = e -η (φ i -φ i-1 ) {( v i-1 + δ i-1,0 Δν 0 )sinξ(φ i -φ i-1 ) + + Δu i-1 cosξ(φ i φ i-1 )} [8] Such form is more convenient for computer calculations since there is no need in remembering the results of all the previous passages. Now let us consider alteration of the resonant cavity resistance due to the load by the bunch. An effective decrement of the electromagnetic field damping in a loaded resonant cavity may be found in the form Σ ν(φ i) δ i, 0 η = η β Σ φ i+1 'i ν 2 (φ) d φ [9] ν (φ i ) = v 0 cos φ i Δv i (φ i ) [5] where v 0 = ev 0 /E inj is the amplitude of voltage, produced by the external generator. The second term addend in this expression is the sum of voltages resulting from all the previous passages of all the particles through the resonant cavity. The amplitude and the phase of each addend in this sum were calculated in reference (9). Each term of sum from the initial moment oscillates with the proper frequency of the resonant cavity w cav and damps whit the decrement η = 1/2 Q, where Q is the goodness of the resonant cavity: Δν i = Δν 0 Σ δ po e -η(φi-φp) cosξ(φ i φ p ) [6] and the function ξ = ω cav /ω 0 = 1 + Δω/ω 0 1 Ε p < E perm δp0 = { 0 E p E perm accounts for the loss of particles. This condition means that the lost one is considered to be the particle whose energy deviation from the equilibrium energy exceeds the permissible value of Єperm. If we introduce an auxiliary function i = 1 Δ u i = Δ v i Σ δp0 δ e-η (φ i -φ p ) sin ξ (φ i φ p ) [7] p=1 β = 2πRNe 2 /E inj Τ 0 [10] Here η 0 is the decrement of the damping of a non-loaded resonant cavity, R is its shunt resistance, Ν is the number of particles in each bunch. The coefficient β is related to the amplitude of voltage induced by each bunch at one passage: β = Δv 0/η 0 Equations [1], [4], [5], [8] and [9] form a complet system of equations of the problem, permitting to analyse (at least numerically) the longitudinal motion for any initial distributions of particles with respect to energies ε 0 and phasesφ 0 (ε 0 =1). Fig. 1 is a block diagram of the program to calculate the longitudinal motion of particles at first few revolutions. Fig. 2 shows the curves of particle losses taking place of first few revolutions, when particles were injected uniformly throughout the phases. Tre rapid drop of the current at first few revolutions doesn't differ from the curve of losses, calculated without considering the klystron effect. The particle losses, occuring after the current is set up, when its value is fairly, large, take place due to interaction between particles and the accelerating system.

5 371 Fig. 3 shows the curves of voltage, found in this case, effecting the particles asa wel as the curve v 0 cos φ (t), given for comparison The small steps on the curves are due to the finite number of particles approximating the continuous distribution. Finally, Fig. 4 shows the phase trajectories of particles. Along the ordinate axis in this case there is plotted the value of, while along the abscissa axis we have Φ = φ-φ s. In conclusion the author deems it his pleasant duty to express gratitude to Yu. F. Orlov and A. I. Baryshev for the interest shown by them towards this study as well as to the workers of the Institute Calculating Centre, K. G. Chaish-vily in particular, for their assistance in carring out the calculations. Fig. 4 - Phase trajectories of particles. REFERENCES (1) S. A. Kheifets and A. I. Baryshev; J.T.Ph. 31, 605 (1961). (2) V. K. Neil: UCRL (3) V. K. Neil and, A. H. Sessler: UCRL (4) R. B. Neal: Stanford M. L. Report 338, (1957). (5) K. W. Robinson: CEAL-1010, (1964). (6) C. Passow: DESY 64/4, (1964). (7) J. J. Henry; J. Appl. Phys. 8, 1338, (1960). (8) S. A. Kheifets and A. I. Baryshev: J.T.Ph., 33, 3, 1963). (9) A. I. Baryshev and, S. A. Kheifets: Radiotechnica i Electronica, 7, 3, 483, (1962). COMPUTER RESULTS ON NON-LINEAR PARTICLE-BEAM INTERACTIONS (FOR 10' REVOLUTIONS) IN THE ORSAY STORAGE RING R. A. Beck * and G. Gendreau * Laboratoire de l'accélérateur Linéaire, Orsay (France) Some numerical calculations of this type have been made previously many authors (1), (2), (3), (4) to explain the small luminosity obtained in the Stanford electron-electron storage rings, but for a small number of revolutions (some thousands) compared to the damping time and with restrictions on the distribution inside the beam. Our purpose isfirstto investigate the behaviour of a single positron colliding with an actual * On leave from Commissariat à I'Energie Atomique (France) in the frame of a collaboration. electron beam during 10 5 turns i.e. 1/2.7 the damping time) and then to extrapolate the results to the case of a weak positron beam. Our UNIVAC program calculates: 1) the field created by a 3-dimensional gaus-sian distributed electron beam having a coherent motion (amplitude Z 0 ); 2) the transverse kick (ΔΧ ' n, ΔΖ ' n) given to a positron by such a beam; 3) amplitude and phase of transverse oscillations (neglecting damping) in the interaction re