In a recent paper, T. Raubenheimer and F. Zimmermann described a new, fast transverse instability caused by the interaction of a train of bunches with the residual gas. Ions produced by transversely offset bunches in the head of a train induce oscillations of the tail of the train. The ions may be cleared out by a gap after one revolution, but the memory remains in the train. Amplitude of oscillations keeps growing exponentially as exp s{top}s{sub c}, until the amplitude of a bunch centroid is on the order of the transverse rms {sigma} of a bunch. The rise time s{sub c}, of the oscillations of a bunch centroid for the PEP-II HER was found to be a fraction of a millisecond, even taking into account the spread of ion frequencies.Computer simulations confirm the exponential growth. However, the results of the simulations show that the exponential regime holds only for a short period of time and then changes to a much slower growth. Initial growth is rapid; it would be difficult to observe it directly in experiments. From a practical point of view, the important questions are, what is the amplitude at which a transition to slow growth takes place and, secondly, what is the growth rate after that transition compared to the rate, which could be handled with a reasonable feedback system. The exponential regime is limited by nonlinearity of the beam-ion interaction. As a result, exponential growth at large amplitudes is replaced by a linear dependence of the amplitude on time. The transition from exponential growth to a linear regime depends on the initial conditions: exponential growth is noticeable only for very small initial amplitudes. An estimate of the growth rate at large amplitude is obtained and compared with computer simulations

Heifets, S.A.

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SLAC-PUB4959 January 1996 Saturation of the Ion Induced Transverse Blow-up Instability* S.A. HEIFETS Stanford Linear Accelerator Center Stanford University, Stanford, California 94309 Presented at the International Workshop on Collective Effecs and Impedance for B Factories Tsukuba, Japan, June 12-17, 1995 * Work supported by Department of Energy contract DE-AC03-76SF00515. Introduction In a recent paper['], T. Raubenheimer and F. Zimmermann described a new, fast transverse instability caused by the interaction of a train of bunches with the residual gas. Ions produced by transversely offset bunches in the head of a train induce oscillations of the tail of the train. The ions may be cleared out by a gap after one revolution, but the memory remains in the train. Amplitude of oscillations keeps growing exponentially as elcp s/sc until the amplitude of a bunch centroid is on the order of the transverse rms u of a bunch. The rise time sc of the oscillations of a bunch centroid for the PEP-I1 HER was found to be a fraction of a millisecond, even taking into account the spread of ion frequenciesc2'. Computer simulations[11[2'confirm the exponential growth. However, the results of the simulations show that the exponential regime holds only for a short period of time and then changes to a much slower growth. Initial growth is rapid; it would be difficult to observe it directly in experiments. From a practical point of view, the important questions are, what is the amplitude at which a transition to slow growth takes place and, secondly, what is the growth rate after that transition compared to the rate, which could be handled with a reasonable feedback system. The exponential regime is limited by nonlinearity of the bem-ion interaction. As a result, exponential growth at large amplitudes is replaced by a linear depen- dence of the amplitude on time. The transition from exponential growth to a linear regime depends on the initial conditions: exponential growth is noticeable only for very small initial amplitudes. An estimate of the growth rate at large amplitude is obtained and compared with computer simulations. For completeness, the basic formulas of the original paper are re-derived. Basic equations First, we reproduce the basic equations of the paper [I]. Consider a train of bunches n = 1, .., nb. The head of the n-th bunch is located at the position z ( s )  = s - zn of the ring at the moment s = ct where z, = (n - 1)sb is the distance of the bunch from the head of the train, 2 The vertical motion of electrons of the n-th bunch is described by the equation where Wb is a betatron frequency (in cm-l), and the potential U of the beam-ion interaction is The ion density n;(F, s) is given by the initial density nf; ns(?,s) = dF‘6[x - Xk(?, s)]S[y - Yh(F‘, s)]S[z - z’]np(.‘). (3) k Here, &(?, s) and Yk(yl, z ,  s) are trajectories of ions generated by the k-th bunch at the location z of the ring. The initial value y’ for Yk, Yk(y’, z, sk )  = yl, is equal to  the offset y’ = Yb(sk,zk) of the k-th bunch at the location z of the ring at the moment Sk < s, sk = z + 4 .  Ions are generated uniformly along the ring with the rate dN; p 300K’ dz torr T _-   o.o6(-)(-)Nb cm-’, (4) where p is the residual gas pressure and the ionization cross-section is assumed to  be 2 Mbarn. Initial distribution of ions is defined by the density of the parent bunch (5) dN. dz .P(q = Lp(x)p(y - Yb(z + zk, zk ) )  7 where p(z) ,  p(y) are Gaussian distributions with rms ux, try, respectively. We assume a flat beam ox >> uy, and take ux,y constant around the ring, and for all bunches. Therefore, the ion frequencies for ions with atomic number A, are also constant. The frequency spread in a nonlinear regime is produced auto- matically by nonlinearity of the motion. For a flat beam we take xk(?, s) = z cos gX, +x = wX(s - S k ) ,  and Sk = z + zk. Then Here $(x) has rms u% cos2 6, N 4 1 2 .  The equation for a bunch centroid is obtained by averaging Eq. (1) with the density pn(?, s) = Nbp(Z)p(y-yb(S, z,))p(z-(s-z,)) of the n-th bunch. The following relation simplifies the result Averaging Eq. (7) gives (9) Here yg(s - 2% + .zk,zk) is the offset of the k-th bunch at the moment Sk = z + Zk when it generates ions at the same location z = s - zn, where the n-th bunch is situated at the moment s. Define the position Pm(s, zo) of the group of ions generated by the rn-th bunch at the location z0 Pm(% 20) Ym(Yb(z0 f zm, zm), ZO, s) (10) with the initial condition Ym(zo+zm, 20) = yb(.zg+z,, zm) at s = zoSz,. Equation 3 4 a 4- m 2 3  " ba V V n 00 4 V - x I h N - h N" d 0 42 . n N 4 3 + I 4 4 V <$ I 4 v s -F: I 0 I 4 V a 3 d N ." h - n E 0 3 h N" s ,. 4 v Y tt V P i fr 4 I 4 v h I I I w I V 4 V s I 3 h I - 4 4 v a 3 c3 0h E I1 h N" n f : 3" II h N .. E + ,. 9 v $.f 4 ;h "a v 3 + N v s h a3 ri -a V 3 II ,. 4 v - E  x h t- 4 v n m 4V h 2 V h 2 V - hN" I 4 a 3 h V 00 lo) E- "D 0 - h s I h e N 4 v h -rv t? s A A - aJ P 3 CJ s I 4 4 ,. v h I I s - rv A A b" 8 J -3 4 d aJ a I I1 A 2 Icn" V mlb" I II A II V I The solution was found in the original paper'llin power series Indeed, substitution of the term ~ ( ~ ) ( - s ,  2) in the right hand side (RHS) of Eq. (17) gives the next term y(")(s, z )  provided the small terms of the order of (wyz/2m)-l, can be neglected. These conditions, compared to the original paper"], include the additional factor m. From the expansion in Eq. (19), it is easy to see that signifi- cant m N and (bJbs/2m)-1 -8zz are large, and neglected terms are small provided Under these conditions, Eq. (18) gives the result"] The solution grows exponentially as ezcp[wyz s / s e f f ] .  This expression depends only on two parameters: number of bunches in the train nb, and characteristic time sc - yb - e(n/nb) s/sc = (w~sbnb)2 a0 SC +f f For larger s >> seff, the solution can be found in series giving oscillatory dependence on time (24) This solution describes oscillations with the amplitude modulated in time s with frequency 2wb / K z . 7 The PEP-I1 HER parameters are: E = 9 GeV, pressure 5 nTorr, ox = 0.1 cm, uy = 0.02 cm, bunch spacing s b  = 1.24 m, number of bunches nb = 1650, Nb = 8.3 x IO1' (at the average beam current I b  = 3 A), and Wb = 0.07 m-' (for Vb = 23.5). That gives dN,/ds = 24.9 ern-', wy = 0.96 x Zcm-', n = 2.1 x cmP3, seff = 1.2 x 10' cm for A = 28. In this case zmaX is larger than the ring circumference. Clearly, the regime shown in Eq. (24) can not be achieved, and the exponential growth of Eq. (22) can be stopped only by nonlinear effects. The  nonlinear regime Exponential growth, as it was noticed in the original paper''', tends to saturate at large amplitudes. Let us consider this regime in more details. Eq. (17) shows that Y k  grows faster than a bunch centroid Yb. Ions can be cleared out by a gap in the bunch train, but memory is retained in the beam, and the amplitude of ions is determined by the amplitude of bunch centroids. Motion becomes substan- tially nonlinear when the amplitude growth, Eq. (21), changes the ion frequency W ' ( U / ~ ~ ) ~  1 where the nonlinearity of ion oscillations w' = ~ w , / ~ ( u / u , ) ~  on the order of w' N 0 . 1 ~ ~ .  This provides criterion for the time when transition to the nonlinear regime for the last nb-th bunch takes place: - - 3u SC S e f f  a0 ' E (Wysbnb) 2: Clearly, the exponential regime can be expected for s >> s, only if the initial amplitude a0 is small. In the substantially nonlinear regime when amplitudes of ions and electrons are large, IYb - Fk I 2 oy, the equations of motion are 8 (27) The analysis of these equations is complicated. To achieve a qualitative result, we proceed as follows. Take yb(s, z,) = a(zm) cos @(s, z,) where, by analogy with the linear case, @(s, Z m )  = Wbs - (wb + wy)zm + ps- (28) Note, that Hence, yb(s, s - zo) = a(s - 20) cos Q(s, zo). The RHS of Eq. (27) is - 7T - Zoyw&n[Y,(s, zo) - a(s - zo) cos *(s, zo)].  We assume where the first term corresponds to the initial condition Y,(zo + z,, zo) = yb(z0 + z,, z,) for ions generated by the m-th bunch at location zo of the ring, and the second term describes oscillations of these ions induced by kicks of the following bunches. The argument of the sign function oscillates with s with frequency wy. The main Fourier harmonics of the sign function oscillates with the same frequency, confirming the choice of the solution shown in Eq. (29). From Eq. (27) it follows that A,  N cy, but the phase Em depends on the relation between amplitudes of different bunches. Substitute now Y,(s,s - 8,) in Eq. (26) using The RHS of Eq. (26) takes the form n-1 - T - -KCTySb sign[a(z,) cos @(s, zn) - cos @(s, zk) - oy COS(@(S, zn) - &)I. (32) k 2 It oscillates with the frequency wb in the resonance with the betatron oscillations. 9 The solution of Eq. (26) grows linearly with time (33) fiuysb 2Wb yb(s, 2,) = a, cos Q(s, zn) - (-)sdn sin Q(s, zn) where coefficients dn depend again on the relation between the amplitudes of differ- ent terms in the sum of Eq. (32). The result, Eq. (33), can be written in the form Eq. (28) with the frequency shift p = fiuysb/(2wb), making analysis self-consistent. Note that the frequency shift p kills resonance and stops amplitude growth but it happens only at very large numbers of turns. Depending on the correlation of signs of different terms in Eq. (32), the am- plitude squared is proportional to n or to n2. Such uncertainty translates in the computer simulations to a non-monotonic growth of an amplitude for different bunches with the bunch number: the amplitudes of different bunches vary by sev- eral orders of magnitude and a bunch with a larger number does not necessarily have a larger amplitude. Assuming random signs of different terms in Eq. (32), we get the estimate for the time dependence of the amplitude of the n-th bunch when the amplitude is large As discussed below, this estimate qualitatively agrees with numerical simulations, although it does not predict rather random variation of amplitude with time for individual bunches. Because full scale modeling is time consuming, a simple code was used to model the instability based on Eqs. (17), (26), and (27). Equation (34) is compared below with the code to demonstrate that the estimate in Eq. (34) is consistent with Eqs. :17), (26) ,  (27). Equation (34) is also compared with Raubenheimer’s tracking simulations. A train of bunches was modeled in the code as a train of macroparticles, a single nacroparticle per bunch. Bunches generate a macroparticle representing ions at 10 all locations around the ring separated by a bunch spacing. Ions are generated by all bunches, which pass a given location and then are retained at this location for a specified number of turns, or until their amplitude is less than a certain aperture (3 x 103a, was used in the simulations). The interaction between ions and bunches is described as a kick to ions and bunches proportional to the difference of the positions of the microparticles if the difference is smaller than uy, inversely proportional to the difference if the difference is larger than 10uy N uz, and as a constant kick dependent only on the sign of the difference in other cases. Variation of the ion frequencies around the ring and variation of the betatron frequencies of the individual bunches were optional. The results of the simulations with this code are shown in Figs. 1-5. The train of ?Zb = 80 macroparticles moves in the 130 rf bucket long ring. Bunch spacing is Sb = 0.42 m, the ion frequency wy corresponds to wysb/c = 1.55, the interaction is defined by s e f f  = 2.35 x lo5 m, and characteristic time s,, defined by l/s, = (wy$bnb)2/s,ff, is s, = 15.3 m. The initial amplitude is not zero only for the first bunch y~(O)/u~ = Under this conditions, the linear theory predicts that the amplitude of the n- th  bunch grows to ( ~ , / a ~ ) ~  = 0.1 after s = s, = 2450(nb/n)' m. Fig. 1 shows the result of the simulations where all ions are cleared out after each turn and no frequency spread is included both for ions and bunches. The results qualitatively reproduce the main features of the exact They give an exponential growth in agreement with linear theory, and show that exponential growth slows down at large amplitudes. The behavior predicted by Eq. (34) for the same bunches is superimposed with the results of the code in the lower part of Fig. 1. The curves obtained without any fitting confirm that the estimate Eq. (34) derived from Eqs. (26) and (27) agrees reasonably well with the code describing the same equations. Fig. 2 is calculated with the same conditions but ions were retained in 10 turns rather than cleared out after the first turn. Results are similar but behavior of different bunches is much less systematic than in the first case. Quadratic dependence on s is seen more clearly if results are plotted in the usual (not semi- logarithmic) scale, Fig. 2b. Note that the amplitude of the first bunch grows because the ion are not cleared out completelly by the gap. However, the amplitude of the first bunch remains smaller than that for the last bunch in the train by two orders of magnitude. The effect of the frequency spread of ions was studied by G. Stupakov"]. Fig. 3 shows the variation of the amplitude with time for the same parameters but ion frequency is varied periodically five times around the ring with 10% amplitude. The ions are cleared out in one turn. Fig. 4 is calculated under the same conditions but betatron frequency is in- creased from bunch to bunch monotonically, increasing totally by 5% along the bunch train. Fig. 5 compares Eq. (34) with the Raubenheimer tracking code. Parameters of the NLC damping ring were used: E = 2GeV, nb = 90, A = 28, Nb = 1.5 X lo1', pressure Torr, Sb = 0.42 m, Py = 8 m, a, = 42pm, uy = 7.7pm. That gives s, = 0.82 m, s e f f  = 1.3 x lo* m, and wysb = 1.4. Agreement is reasonable. Raubenheimer's numerical simulations for the ALS ring (wysb = 0.5) give se.f = 1.7 x lo4  m in good agreement with the linear theory for this machine. If this regime continued, the amplitude of the last bunch in the train of 150 bunches would grow to Yb/Uy = 5 x lo9 at s = 1500 m. The simulations show that this quantity is in the range of 0.1 to 10.0 for different bunches in the nonlinear regime after s = 1500 m. The estimate Eq. (34) for the last bunch is 0.3. Note that the range of the amplitudes for different bunches agrees with linear dependence on n in Eq. (34). For parameters of PEP-11, seff = 1.2 x lo7 m, wysb = 0.22, and n)  = 1658, equation (34) gives (yb/ay) = 7.5 x 1 0 - ' ( ~ / ~ )  n/nb, or Yb = u in 4.5 msec for the last nb = 1650 bunch. This rate is smaller than the PEP-I1 damping time but can be managed with the bunch-by-bunch feedback system. - 11 12 We can estimate the amplitude, at which the transition from the exponential regime of Eq. (22) to the linear regime of Eq. (34) takes place by calculating s at which both formulas give the same result: The break point amplitude for the n-th bunch is (35) It is very small in the tail of the train, indicating that tail of bunches is always described by Eq. (34). In the head of the train, the break point amplitude can be large but time Eq. (35) in this case is large too, and growth can be stopped by a feedback system. Conclusion Ion-induced fast transverse instability is constrained by nonlinear effects. Non- linear effects stop exponential growth of the amplitude at fractions of the rms bunch size. At large s the exponential growth described in the papers[11[21is replaced by the much slower linear growth of the amplitude. An estimate of the growth rate in this case is given by Eq. (34). The estimate is qualitatively consistent with numer- ical simulations. Instability at PEP-I1 can be controlled by the bunch-by-bunch feedback system with damping time of less than 4.5 ms. Acknowledgement I thank T. Raubenheimer for kindly providing me with the results of his sim- ulations. I also thank A. Chao, F. Zimmermann, and G. Stupakov for discussions. REFERENCES 1. T. Raubenheimer, F. Zimmermann. Interaction of a Charged Particle Beam with Residual Gas Ions or Electrons, SLAC, January, 1995 2. G. Stupakov, T. Raubenheimer, F. Zimmermann. Effect of Ion Decoherence on Fast Beam-Ion Instability, SLAC, January, 1995 DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsi- bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer- ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. 13 14 1 oo (v, 1 o-2 1 0 - ~  10-6 $ 0 15000 20000 8-95 s (m) 8004A1 1) Comparison of the results of simulations (above) with Eq. (34). Lines are for different bunch numbers. All ions are cleared out after each turn and no frequency spread is included both for ions and bunches. Below: results of Eq. (34) are added, no additional parameters are used. 1 oo lo-* CU, 6- 9 1 o-6 H I 0.8 % 2 0.4 6- Y 0 0 5000 10000 15000 8-95 s (m) 8004A2 2) The same as Fig. 1, but ions are retained for 10 turns. The same data are plotted in logarithmic (above) and linear (below) scales. 15 16 ' 0  02 0 b 0 a 0- Lo 0 0 7- 7 0 7- a I 0 7 0 0 0 In 7- 2 B 0 co 0 n s W 3 0 :  O B  o m  In 7- 0 0 O n  sg. v) 0 0 0 L(3 7- T- O ln c? o0 m 3P- 3cu d- (0 0 I I I 0 0 0 0 (u d- (D 0 I I I 0 0  0 0 7 7 7 7 v)  0 0 co4 n E W cn v) W 9 Portions of this document may be illegible in electronic imgge  product^. Imaeefi are pFoducced from the best available original dOCUXllent 

Saturation of the ion induced transverse blow-up instability

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