SYNCHRO-BETATRON RESONANCES. TOSHIO SUZUKI KEK, National Laboratory for High Energy Physics Tsukuba,IbarWO,305,Japan

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1 Particle Accelerators, 1990, Vol. 27, pp Reprints available directly from the publisher Photocopying permitted by license only 1990 Gordon and Breach, Science Publishers, Inc. Printed in the United States of America SYNCHRO-BETATRON RESONANCES TOSHIO SUZUKI KEK, National Laboratory for High Energy Physics Tsukuba,IbarWO,305,Japan Abstract Theories of synchro-betatron resonances are reviewed. Emphasis is on single beam effects with space charges and wake fields, with which I am more or less concerned. Other important topics such as two-beam effects and spin motion are omitted. The basic driving mechanism of modulation and sidebands is explained in some detail. INTRODUCTION A particle motion in accelerators is described in terms of six phase space coordinates (four transverse and two longitudinal) and spin. Thus a full description including longitudinal or synchrotron motion is important because the motion is usually not decoupled into each space; often the transverse and longitudinal motions are coupled. In this case, synchrotron motion causes modulations of the parameters or forces and sideband appears as a result. This is the essence of the theories of synchro-betatron couplings or resonances. History of synchro-betatron resonances goes back to the discovery of synchrotron sidebands1 in the betatron motion in accelerators with chromaticity and the theory of Orlov 2 of synchro-betatron resonances caused by chromaticity. The latter theory is extended much later 3 to include modulation of betatron oscillations by transverse space charge effect. The dispersion effect was found experimentally in NINA4 as the first observed effect. This effect was analyzed by Piwinski and Wrulich 5 and others. Then DORIS storage ring suffered from beam-beam effects caused by crossing angle and this effect was analyzed fully by Piwinski6. SPEAR storage ring suffered from the effects by spurious dispersion and longitudinal wake fields7, but to my knowledge, this effect has not been analyzed fully by an analytical method. The intensity-dependent synchro-betatron resonances were observed at PETRA8 and TRISTAN9 and they are considered to limit the stored current at injection in TRISTAN. Theoretical mechanism associated with the closed orbit distortion and transverse wake fields was proposed by Sundelin 10 and this effect was further studied by Suzuki 11 near [403]1157

2 158/[404] T. SUZUKI resonance by using a perturbation theory. A fonnalism was proposed by Hagel and Suzuki12 to study the wake field effects not only at resonance, but also in the nonresonant regions. Another wake field effect is the so-called coherent synchro-betatron resonances. It was found in computer simulations13 on transverse mode coupling instability. It was ascribed to a localized nature of impedance. This effect was further analyzed by a twoparticle model14, Vlasov equation15,16,17 and further computer simulations. This effect has not been experimentally identified. These are the topics that I want to review in this paper. Another important topic is the coupling of spin motion with synchrotron motion l8, but I omit this. Also the beambeam effect 6 is omitted. Also, since there are several good review articles l9, I want to stress more elementary aspects; modulation and sidebands. Finally, I should mention that though sinusoidal synchrotron oscillations are assumed in my previous works11,12,20 and in the works of some others, this was criticized recently by Baartman21. Nonlinear synchrotron oscillations are important for higher sidebands and for large synchrotron oscillation amplitudes. In this respect, the work ofpiwinski and Wrulich 5 treats nonlinear synchrotron oscillations correctly. MODULAnON AND SIDEBAND We assume a linear betatron oscillation with chromaticity ~ and synchrotron amplitude a. v~ = v~o + ~acosvse (1) Then the transverse coordinate is given by y = A cos (vr 0 9+ ~a t-' Vs sinvs9) (2) Then from well-known formulae 22, which are important in modulation theory, cos (asine) = Jo(a) + 2 L J2k(a) cos2ke, k=1 (3) sin (asine) = 2 I, J2k+l(a) cos(2k+l)e, k=l (4) we get y A{Jo(b) cos v~o}

3 SYNCHRO-BETATRON RESONANCES [405]/159 + L J2k(b) {COS(VpO + 2kvs) + COS(VpO- 2kvs)e} k=1 + L J2k+l(b) {cos(vpo + (2k+l)vs)9 - cos(vpo - (2k+l)vs)9), (5) k=o where b ~a Vs (6) Thus, we observe the appearance of sidebands at vpo ± mvs (m; integer). These sidebands or satellites were observed actually in accelerators1. More than thirty years ago, the observation of satellites was in itself a surprise. We add a comment here that the amplitude A contains a B-function and B function is also modulated by synchrotron oscillations so that Eq. (5) should be modified accordingly. A reflection on Eqs. (3) and (4) shows that it is just a Fouriers series expansion of a periodic function; the coefficients being given by Bessel functions. Then any nonlinear function f(acos9), which denotes, e.g., a wake function that depends on the longitudinal positions, can be expanded into a Fourier series. In this case, the coefficients will not be given by analytic functions and must be evaluated numerically in general. Further a function f(p(9)) of nonlinear synchrotron oscillation p(9) can also be expanded into Fourier series since p(9) is periodic anyway. Thus, in general, f(p(9)) = L am cosme + L bmsinm9, m m 1 r2n am = 1t J O f(p(9)) cosm9d9, (7) (8) bm = kfo1t f(p(s)) sinmsds, (9) where the coefficients are evaluated generally by a numerical method. The sidebands appear also in this general case. In connection with nonlinear synchrotron oscillations, Baartman 21 analyzed its role in synchro-betatron coupling theory and found that sinusoidal approximation 20 for synchrotron motion breaks down particularly for higher order satellites and for large synchrotron amplitudes. Since sinusoidal approximation for synchrotron motion is often used, we should reanalyze some of the previous theories.

4 160/[406] T. SUZUKI CHROMATICITY AND TRANSVERSE SPACE CHARGE EFFECT It is expected that sidebands caused by chromaticity affect the resonance conditions. This was studied by Orlov 2 a long time ago. However, this effect is usually considered to be small and moreover the chro,maticity is corrected in many machines. The larger effect in proton accelerators and storage rings is the tune modulation caused by transverse space charge effect. Though the mechanism is quite similar, this was studied by Bruck3. I plan to use this theory to estimate the beam loss and emittance growth of high intensity proton accelerators and storage rings such as the TRIUMF KAON factory23. According to Bruck, we assume a parabolic bunch and sinusoidal synchrotron oscillations. Then, the tune shift!i.v is modulated as where!i.v " is the maximum tune shift, ct>max is the maximum phase (bunch length) andct>m is the synchrotron amplitude of the particle we study. The essential point is that we study a single particle motion under collective space charge force and with betatron resonances. We use a first order perturbation theory. (10) Then, in storage rings, the amplitude of a particle satisfying the resonance condition with sidebands begins to grow. However, no overall emittance growth is expected. In accelerators where some relevant parameters change in time, some particles sweep (cross) the resonance and some emittance growth is expected. DISPERSION EFFECT When there is dispersion in RF cavity sections, the sudden change of equilibrium orbit occurs by acceleration. The position and angle of the particle does not change by acceleration in first order, and this means that the betatron oscillation amplitude is changed. By simple inspection, we observe that the betatron oscillation is excited when the betatron tune between the successive cavities is near integer and that it is damped when the tune is near half-integer. This effect was first observed in NINA4, where the betatron tune was about 5.25 and the RF periodicity was five. In subsequent accelerators and storage rings, particularly for electrons, either the dispersion in cavity sections is made zero or special care is taken for RF periodicity. Even if the design dispersion is made zero, there are spurious horizontal and vertical dispersions. These associated with decelaration by longitudinal wake fields caused synchro-betatron resonances in SPEAR This will be further discussed in the next section.

5 SYNCHRO-BETATRON RESONANCES [407]1161 A more or less complete theory was presented by Piwinski and Wrulich 5. They showed that not only betatron oscillations are affected by synchrotron oscillations through RF acceleration, but also synchrotron oscillations are affected by betatron motion through the path-lengthening effect of betatron oscillations. These two give a symplectic description of the phenomenon. They presented a theory which agrees well with their simulations. Corsten and Hagedoom 24 derived a Hamiltonian of synchro-betatron couplings from more basic Hamiltonian of the Lorentz force. Suzuki 20 developed an analytical theory using sinusoidal synchrotron oscillations that can be used for hand calculation. Though his theory agrees well with the simulations of Piwinski and Wrulich for smaller synchrotron oscillation amplitudes and smaller satellite orders, the agreement is not generally good in other cases. This point was critically analyzed by Baartman 21 who stressed the importance of nonlinear synchrotron oscillations. WAKE FIELD EFFECTS Intensity dependent synchro-betatron resonances were observed in SPEAR 11 7, PETRA8 and TRISTAN9. They are considered now to limit the stored current at injection in TRISTAN. A mechanism was proposed by Sundelin lo that explains the beam growth driven by the combination of transverse wake field and closed orbit distortions. The work was extended further by Suzuki ll, who showed that a single bunch effect is dominant and a multi-tum effect is rather small as observed in PETRA8. He gave some estimate of the beam size in TRISTAN at several sideband resonances. A simple analytical theory which is valid also in the non-resonant regions was developed by Hagel and Suzuki l2. We will review the wake field problem based on their work. The equations of betatron and synchrotron oscillations with wake fields are (11) dvi ~E de = ah(e)i (12) d(llli)i ---ae- = LeVI o(e- (1) {sinvi - sinvs} I - 21tRe 2 I, 0(8 - SI)G('Vi - 'Vj), l,j~i (13)

6 162/[408] T. SUZUKI where Yi : R: 9 : K(9) : 5p(9) : 91 : a: h: E: ~: e: VI: 'I's: G('I'): transverse coordinate ofi-th particle average radius azimuthal position linear transverse focusing force periodic ~-function azimuthal position ofrf classical electron radius Lorentz factor transverse wake function RF phase ofi-th particle momentum compaction factor hannonic number energy energy deviation from synchronous value elementary change RF voltage synchronous phase angle longitudinal monopole wake function. In these equations, we neglected path lengthening by betatron oscillations and longitudinal dipole wake functions. We separate the transverse coordinate as Yi = YBi + ll(~)i + Yeo, (14) where ybi is the coordinate of betatron oscillations, 11 is the dispersion, (~P/P)i is the momentum spread of the i-th particle and Yco is the closed orbit distortion. Then we obtain the term including dispersion and derivative of the momentum spread, which shows the dispersion effect, the term including dispersion and transverse wake field, which is zero for stationary longitudinal phase space distribution, the term including closed orbit distortions and transverse wake field, which shows Sundelin effect, and the term including the betatron coordinates YBj of the other particles and the transverse wake field, which shows so-called coherent synchro-betatron resonances to be described in the next section. They made the following approximations. 1) linear betatron oscillations

7 SYNCHRO-BETATRON RESONANCES [409]/163 2) path lengthening due to betatron oscillations is neglected. 3) longitudinal dipole wake field is neglected. 4) Though not essential~ sinusoidal synchrotron oscillations are assumed. Then the problem is reduced to that of forced linear oscillations instead of nonlinear coupled motions. They also assumed stationary Gaussian distribution as the longitudinal phase space distribution and a broadband resonator impedance for the wake fields. Then analytical solutions with some numerical Fourier transforms are obtained. This work still gives only the basic formalism and more extensive calculations are necessary to compare with experiments and simulations. COHERENT SYNCHRO-BETATRON RESONANCES This was found in a computer simulation13 of transverse mode coupling instability. It is driven by a localized impedance. A theory based on the Vlasov equation was presented by Ruggiero15 for the case of one cavity in a ring. He revealed some general mechanisms and showed that it is due to the coupling of mode m = 0 with a higher azimuthal mode (say, m =4) whereas transverse mode coupling instability is driven by the coupling of m =0 and m =- 1 modes 25. Chin16 studied the effect of symmetrically placed cavities and Suzuki1? studied general arrangements of cavities. He also studied the case where many cavity gaps are replaced by a cavity offinite length. In this study~ it is found that the coherent synchro-betatron resonances are satellite resonances of a halfinteger resonance. This conclusion is consistent with the calculation based on the twoparticle model 14. The prediction of Suzuki is that the finite length of the cavity and the particular arrangement of cavities suppress the coherent synchro-betatron resonances at LEP except for m = 1 satellite. The Fokker-Planck treatment was given by Ruggier0 26. These resonances have not yet been actually identified in actual machines. CONCLUSIONS Various mechanisms and theories of synchro-betatron resonances for a single beam are reviewed with emphasis on space charge and wake field effects. Other topics, not treated here, are two beam effects and spin motion. In all these, driving mechanism is the same: force depends on momentum or on the longitudinal position in the bunch, and synchrotron oscillations cause modulation of the force producing sidebands. In many works, sinusoidal synchrotron oscillations are assumed producing Bessel sidebands. Baartman 21 criticized this and stressed the importance of nonlinear synchrotron

8 164/[410] T. SUZUKI oscillations. In this respect, some of the previous works should be reanalyzed. For the wake field effect and space charge effects, some more calculations or simulations are necessary to quantitatively explain the observed phenomena. REFERENCES 1. C. L. Hammer, R. W. Pidd and K. M. Terwilliger, Rev. Sci. Instrum, 26 (1955). 2. Yu. F. Orlov, Sov. Phys.-JETP, 5, 45 (1957). 3. H. Bruck, Part. Accel., 11,37 (1980); Part. AcceI., 11, 181 (1981). 4. M. C. Crowley-Milling and I. I. Rabinowitz, IEEE Trans. NucI. Sci., NS-18, 1052 (1971). 5. A. Piwinski and A. Wrulich, DESY 76/07 (1976). 6. A. Piwinski, DESY 77/18 (1977). 7. SPEAR GROUP, IEEE Trans. NucI. Sci., NS-24, 1863 (1977). 8. D. Degele et ai., DESY 80/10 (1980). 9. K. Nakajima et ai., These proceedings. 10. R. M. Sundelin, IEEE Trans. NucI. Sci., NS-26, 3604 (1979). 11. T. Suzuki, Nucl. Instrum. Methods, A241, 89 (1985). 12. J. Hagel and T. Suzuki, CERNILEP-TH/88-57, (1988). 13. D. Brandt and B. Zotter, Proc. 12th Int. Conf. on High-Energy Accelerators, Fermilab, 1983 (Fermilab, Batavia, 1984) p B. Zotter, IEEE Trans. NucI. Sci., NS-32, 2191 (1985). 15. F. Ruggiero, Part. Accel., 20, 45 (1986). 16. Y. H. Chin, CERN SPS/85-33 (DI-MST) (1985). 17. T. Suzuki, CERNILEP-TH/87-55 (1987). 18. A. W. Chao, AlP Conf. Frac. No.87 (AlP, 1982), p A. Piwinski, Proc. 11th Int. Conf. on High Energy Accelerators, CERN, 1980 (Birkhauser, Basel, 1980) p T. Suzuki, Part. AcceI., l8., 115 (1985). 21. R. Baatman, TRIUMF Int. Rep, TRI-DN-89-K40 (1989). 22. M. A. Abramowitz and I. A. Stegun, Handbook of mathematical functions (Dover, New York, 1965). 23. T. Suzuki, TRIUMF Int. Rep. TRI-DN (1986) and TRI-DN-88-K2 (1988). 24. C. J. A. Corsten and H. L. Hagedoom, Nucl. Instrum. Methods, 212,37 (1983). 25. K. Satoh and Y. Chin, Nucl. Instrum. Methods, 207, 389 (1983). 26. F. Ruggiero, IEEE Trans. NncI. Sci., NS-32, 2344 (1985).