Sacherer, F J

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( ( SI/Note llR/72-l 25.5.1972 TRANSVERSE RESISTIVE-WALL INSTABILITIES OF THE BUNCHED BEAM IN THE SPS (prepared for the CERN Laboratory II 1972 Spring study) l. Conclusions 2. Review Frank J. Sacherer 3. Formula for single-bunch modes 4. Formula for coupled-bunch modes 5. Criterion for coupled motion 6. Numbers for the SPS 6.1 Normal acceleration in 4620 buckets 6.2 Injection and debunching of 20 PS bunches References Figures ( ( - 1 -1. CONCLUSIONS i) Single-·bunch modes are harmless. ii) Coupled-bunch modes are serious, with growth times of the order of 1.6 ms for 10 13 particles at injection and increasing linear-ly with Y to 32 ms at 200 GeV. iii) Sextupoles or the natural machine chromaticity have little or no eff<>ct on bunched beam instabilities. iv) Octupole Q -spreads of order of :: 5 x 10-3 across the beam are z sufficient to cure the instability. 2. REVIEW The original calculations for bunched beam instabilities were made by l) Courant and Sessler , who assumed that the bunch would move as a rigid unit. This l<as extended by Lee, Mills and Morton2) to include breathing motion and higher "throbbing beam" modes. Both calculations neglect synchrotron motion and require that the transverse motion be Further progress was made by Pellegrini 3) the and same all along the bunch length. 4) Sands who included synehro-tron motion, the effect of machine chromaticity, and also the higher head-tail modes in which different parts of the bunch oscillate with different phases. Examples of these modes are the standing-wave patterns )=0 (rigid-dipole mode) that show the variation of transverse dipole moment along the bunch. The same is true for quadrupole and higher transverse rnultipole modes. If machine chron1aticity or sextupole terms are included, the patterns acquire an ad-- 2 -ditional traveling-wave component. As far as transverse oscillations are concerned, these modes form a complete set, which may be driven unstable by beam-equipment interactions such as res is tive-vrall ~ cavities, pick·-up electrodes, etc. If many bunches are present, all with the same frequency, the bunch phases will be locked together in patterns with an in.tegral number of wave-lengths around the machine circumference. On the other hand, a sufficient spread in bunch frequencies prevents this phase-lock, and single-bunch modes result with a consequent reduction in growth rate. 3. FORMULA FOR SINGLE-BUNCH MODES We consider only the dipole modes since they generally have the fastest growth rates, The resistance in a smooth, round vacuum chamber causes a growth rate with where 1 = - Im /';;w (1) F9,' and F9, are form factors that depend on the type of mode- shown in Figs. 1, 2 and 3 G(21T,Q) is the bunch function of Courant and Sessler 1 c9, = 1 for 9, = 0 and falls approximately as 9, + 1 for the hi£;he:r modes E; = .!:. aQ machine chromaticity X = Q loS. Q <lP R n 1 1 n = --2 YT yz ( ( ( ( - 3 -Nb particles/bunch r = e2/m c 2 = 1.53 x l0- 18 m 0 0 6w "' 8 = skin depth at rev. freq. ~ 2.4 mm b vacuum chamber half-height~ 2.6 em If X -+ 0, the lowest head-t ai 1 mode (£ = 0) approaches the rigid-bunch mode of Courant and Sessler, and (1) becomes (2) This agrees with the C + S result except for i) an error of I21T' in their eq. (4.4) that MortonS) has pointed out previously ii) they neglect synchrotron motion so that for them X Q ~ Equation (1) also includes the head-tail results of Pellegrini and Sands except that more realistic modes are used here that result in some-what larger growth rates, A report covering the derivation of (1) should ·be available soon. 4. FORMULA FOR COUPLED-BUNCH MODES For M identical bunches Eq. (1) still applies if G(2TI,Q) is replaced by For large M (say M > 5), Hubner and Zotter 6) show that this approaches l + i Sign (n - Q) M ~ ~In- Q\ where n is an integer. In the limit of large M, the near-field term in (l) approaches zero because X-+ 0, and 6w l + i Sign (n - Q) ~ M6w w ·..;r;;- Q\ (3) - 4 -which is identical to the coasting-beam are sketched in Fig. 4 for Q = 28 t 'l.ll • -[:i' MAW w • at • 3(> • 7) result of LNS , These frequencies The frequency diagram for 4,600 bunches differs from Fig. 4 only near the origin, and the differences are too small to be seen. 5. CRITERION FOR COUPLED MOTION The most serious mode, n = 29, is also the most resistant to perturbation$ while the modes with n much different from 29 are closely spaced and thus more easily modified. A criterion for the strength of perturbation required to destroy a given mode can be derived from perturbation theory. It is convenient to derive this criterion in the continuum limit M -~ oo~ but the. result is also valid for the discrete case. In the continuum limit, the bunch-coupling matrix ( ( - 5 -Eq. (3.20) of Courant and Sessler becomes 2n A'l'(e) = I K(e'- e) '!'(e') de', 0 where ~(8) is the amplitude of oscillation of the bunch at 8, and 8 is measured from a reference bunch. The eigenfunctions and eigenvalues of (4) are in8 -· e 2n . 8 J e~n K(8) de 0 (4) (5) U + iV evaluated at (n - Q)w . 0 For the resistive-wall interaction, A = 6w of Eq. (3). n n We now perturb Eq. (4) by allowing the bunches to oscillate with different frequencies, 2n A'l'(e) = w(e)'¥(8) + I K(e' - e) 'l'(e') de' ' 0 (6) where w(8) is the frequency of the bunch at 8, but measured with respect to the average frequency so that 2n I w(e) cte = o • (7) 0 The new eigenvalues are f d • 8) oun from perturbat1on theory . Define the matrix elements 2n (e) w(8) '!' (e) de= l I ei(n- m)e w(e) d8 , n 2n 0 (8) where because of (7) w = 0 for all m. Then to first order, there is no mm change in A , n - 6 -To second order, • (9) This is valid provided the shift A - A 0 is much less than the spac1ng n n between level n and the rest. For the resistive-wall interaction, mode n = 29 (assumin'g Q::::: 28 i) has the largest spacing and is therefore the most difficult to destroy. The condition that this mode remain intact is with n 29 • (10) Because > l ' * and because w = w , condition (10) becomes rnn nm with n 29 (ll) From (8) we see that L: j w j 2 1s the sum of Fourier components of w(8). nm That is, if then Thus (10) becomes l 21T w( 8) - Z k = 2rr J w2 (e) 0 d8 ilce ~e ' with E~ = • l 2rr 2rr J w(e) 0 (12) ( ( ( ( - 7 -or w < ILI!c I rms for coupled motion, where w is the rms spread of bunch frequencies and 1\"A is the spacing rms between the mode in question and the nE~xt nearest mode. This critE~rion is completely general and applies to longitudinal as well as transverse motion, to coupling caused by cavities, pick-up electrodes, as well as (13) res is ti ve-wall interactions. It is analogous to the rule-of-thumb cri-terion for Landau damping 1n coasting beams, namely the spread in particle frequencies should exceed the frequency shift caused by the coherent mo-tion. The frequency spread may result from rf quadrupoles, or naturally from population differences between bunches via the coherent Laslett Q-shift 1\w • In the latter case (13) becomes c ( LIN) 1\w < 6/c N rms c for coupled motion. . - 9) As po1nted out by D. Mohl , 1\w c should not include the usual DC magnetic field terms since these shift the frequency of each bunch the same amount, independent of population differences. In this caselO) TT Nro§c QB[3 2 ,,3b2 N = total number of particles B = bunching factor < 1 b vacuum chamber half-heip,ht /C • f. . 1T2 Sl = 1mage coef 1c1ent ~ T6 (lS) For large y, other population dependent frequency shifts become important, including the resistive-wall term (1), the effect of cavities, dielectric or oxide coatings on the vacuum chamber, plus neutralizing electrons and 1.ons. - 8 -6. NUMBERS FOR THE SPS 6.1 Normal acceleration ln 4620 buckets Unless stated othen .. dse wt~ take N = 1013 particles y 10 b 2.6 em L 15 em B 0.1 ( Q 28.75 YT = 28 F, = - 1. 33 w 2.74 X 105 rad/s. 0 Then and the betatron phase shift X is sketched >n Fig. 5. 8 X q.!:.S. R ll 6 lf / \ -:a. 0 to -':1. -'l -& -If Fig. 5 ( ( - 9 -If we ignore the non-adiabatic region near transition, then X·( 2 rad and the 9, = 0 mode dominates, ~w [G + 215 F 0 (x)J , w where the maximum value of Im F (X) is 0 0.1 before transition (X positive) 0.6 after transition (X negative). If the bunches were decoupled, G = G(28.75) N- 1 + 1, so near-fields dominate and with 1 T 0.096(1 + 215 X 0.6) T = 80 ms. 12.5 This is the maximum possible growth rate and occurs just after transition. We conclude that single bunch modes are not se"ious. The more likely case is coup led motion Hi th G = 1 + i ..}2 4620 ·~ = (1 + i) 6500 for the mode Hith 29 wavelengths around the machine circumference, Now near-fields are negligible and 6w = - 0.096(1 + i)6500 = - 624(1 + i) with T l. 6 ms. The criterion for coupled motion 1s 6ll ll>W + ... I ( -!, ) < I M I c .~ rms where we take - 10 -l:!.w = 475. c 883 Thus to decouple the bunches requires a relative population spread of (n:) > 1. 85 rms or a full spread at half-height exceeding 370%. One expects (nN) to be N rms at most 0.1, corresponding to a 20% spread at half-height. Therefore, even large contributions couple the bunches. to D.w from electrons, ions, etc. are unlikely to de-c This remains true at higher energies. Sextupole terms or changes in the machine chromaticity change the beta-tron phase shift X• As X increases, the instability is shifted to the higher head-tail modes which have slower growth rates. However, to achieve a signifi-cant reduction in growth rate would require an order of magnitude increase in chromaticity. Because of the short bunch length, it is very difficult to have a large betatron phase shift between head and tail. We conclude that ( changes in chromaticity have a negligible effect. ( Octupoles cure the instability if they produce enough frequency spread within a bunch to prevent its coherent motion, that is provided I:!.Qoct > total frequency shift due to coherent motion lu+iVI/w. 0 . .. 11) Thts has been computed by D. Mohl who finds I:!.Q > 0.01 oct across the beam vertically. At N = 1012 we need 0. 001. ( ( - ll -6.2 In·jection and debunching of 20 PS bunches Sinee there are no synchrotron forces,x = Q!: R Equation (1) becomes [ (2ifR' J l1W = - 22.0 C~ G F~ (X) +VL F*(X) For decoupled motion, G(28. 75) ~- 1 + i, and the frequency shifts llw58 for single-bunch modes are given in Table I. Table I - Single-bunch modes I ~-Condition L (m) X (rad) 2- ct ~ F (x) F ( x) - llwsB 2- 2, just injected :o 0.0765 0 1 48 l.O o.eo + 0.013i l/10 de bunched y .l o.ee 0 1 14 0.95 0. B5 + 0.11j 1_ de bunched 173 4.4 1 0.5 6.3 0.56 0.90 + 0.16i :2 de bunched 345 [3.8 4 0.2 4.5 0.20 0.55 + 0.17i The mode number .£ is choosen for the fastest growing mode. For coupled motion, l + i 20 G = 7z7f= 4 28.3(l+i), fl23 + 35.7i 242 + 55.5i 51.7 + 22i 5.5 + 7.7i and the frequency shifts llwCB for the fastest growing coupled-bunch mode are given in Table II. T (msec) 28 113 45 130 - 12 -Table II - Coupled-bunch nmdes -- ---------~--Condition -,\wOE r~ junt injected 1470 + 631i 1 msec:) ~ • 0 -·------1/10 de bunched 854 + 625i 1 .6 -· . J_ de bunched 231 + 184i " 2 / .4 ( deb.unched 31.4 + 23.9i 4 2 --The criterion for coupled motion is and it appears from Table III that the motion is coupled. Table III - Criterion for coupled motion Condition B - !:!.we 1/:!.wc + l:!.wsBI i l:!.wCB l ~6NJ K rm:3 ( just injected 0.01 4750 5673 1600 0.28 1/10 deblmched 0.1 475 717 1060 1.48 1 de bunched 0.5 95 147 294 2 .. 0 " de bunched 1.0 4'7.5 53 39.4 0.7~ . ( ( - l3 -The instability is cured by either sextupole or octupole terms. The spread in frequency due to the natural machine chromaticity is or !:Jw w -3 1. 33 X 1. 3 X 10 = 1.7 X 10- 3 -3 5 1.7 X 10 X 28 X 2.74 X 10 4 1. 3 X 10 . This spr<,ad is larger than the frequency shift li,tuc + !:JwCB I due to the coherent motion. We conclude that there should be no instability during injection and debunching, unless the natural chromaticity is reduced. The same conclusion is reached in Ref. 11 where a more conservative Landau damping criterion is employed. Acknowledgement I thank Dieter M(ihl for reading the manuscript and suggesting several improvements. Dis tri bun~ on Members of \.Jerking Group Conveners of other working groups HST C. BcJvet, K.H. Reich, E.J.N. Wilson, H.O. W~ster :GEFEJ1ENCES 1.. I<J.Do Cou.rant and A .• M. :..>essJ.e:r.·:~ Transverse coherent resistive instabilitj_es of azirauthal1y hw1.ched. beams in ·particle 1 '· "' Q • •• · ~- 1 5··~9 (' qo·6) aece era\iOrS? 11.ey., uC"l., .i.USlT. 2.}_~ .. 1 ~t-.:.. ~. " 2 .. M .. J. Lee, :I!' •. E~ MiJJs and l)oiJ~ Morto:n 9 r.rhrobb1ng bea.Jn transverse resistive instabilities in ci:rc1.:;·.lar accelerators and storage rings, SLAC report 76 (1967). 3. C. Pellegrini, On a new installili ty i.n ele ctron--pos;. tron storage rings (the head .... tail ei'±'ect), Nuovo Cimento 64A~ 477 (1969). M. Sands, The bead-· tail effect, an :i.nstabili ty mechanism in storage rings, SLAC-TlJ-69/B, and he&d-·tail effect IJ : from a resj_stive-wall walce, SIAC·-~'lJ-69/10 (1969). 5. P.L. Morton, An investigation of space-charge effects for the booster injector to the CPS, CERN Interna.l Report SI/Int. DL/68-3 (1968). 6. K. HUbner and B. Zotter, Transverse instabilities of the b=ched beam in the ISR, CERN Interr1al Report; CER.N/ISR-TH/71-113 (1971). L.J. Iaslett~ V.K. Neil and A.M. Sessler, Rev.Sci. Instr. 426 (1965). _2_6, 13. See, for e:xample, I. Schiff, Quantum l'liechenicE!, 2nd Ed., (McGraw-Hill Book Company, Inc., New York, 1955), p. 151. g. D. Mohl, Transverse coherent instabilj ties in multi-bunch beame; a ren:ark concerning the criterion for the abcoence of coherent bunch mod.8s, Lawrence Eerkcoley· Laboratory Report LBL-570 (19'71). 10. L.J. Laslett and L. Ilesegoi;ti, The space-charge hmi t imposed by coherent oscillation of a bunched beam, in I'rooeedings of the VI th International Conference on High Energy Accelerators, CEAL-2000, p. 150 (1967). ll. D. 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Transverse resistive-wall instabilities of the bunched beam in the SPS

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