In a microtron, the path length change from pass to pass is a fixed multiple of the RF wavelength, and the accelerating system can be reasonably well approximated as a single cavity. Under such circumstances it is possible to derive an analytical formula for the multipass beam breakup threshold current. The threshold current determined by numerical simulations agrees very well with the formula for a machine with a small number of passes. The analytic formula can serve as a useful guide in examining optics designs to improve the BBU threshold

Merminga, L.

Yung, B.C.

UNT Digital Library

JLAB-ACP-97-02 BEAM BREAKUP IN A MICROTRON* El VbED @o/qF-476533-- Df/Ele/LfOl - '''Y Byung C. Yunn and Lia Menninga 3uw 17 Thomas Jefferson National Accelerator Facility 12000 Jefferson Avenue, Newport News, VA 23606, USA I Abstract In a microtron, the path length change from pass to pass is a ked multiple of the RF wavelength, and the accelerat- ing system can be reasonably well approximated as a sin- gle cavity. Under such circumstances it is possible to de- rive an analytical formula for the multipass beam breakup threshold current. The threshold current determined by numerical simulations agrees very well with the formula for a machine with a small number of passes. The an- alytic formula can serve as a useful guide in examining optics designs to improve the BBU threshold. The bunch then recirculates and enters the cavitv for the 1 INTRODUCTION Multipass beam breakup (BBU) is an important performance-limiting factor in microtrons. Extensive computer simulations are generally unavoidable to deter- mine BBU thresholds since the tens of recirculations in- volved make analysis very difficult. When the betatron phase advance through the accelerating cavities is small, a good approximation is to replace the whole accelerat- ing system by a single cavity. Furthermore, the total kick received passing through the cavity can be treated as if re- ceived by a point kick located at the middle of the cavity in describing beam motion. The resulting simplification allows an analytic treatment of the multipass BBU phe- nomenon without losing too much information. We have developed an analytic formula for the BBU threshold current in this single cavity approximation. A benchmarking of the formula using a 25-pass racetrack microtron as an example has shown its potential as a d e  sign tool. Tie-consuming computer simulations could be minimized at the early stage of design. 2 A MODEL OF BBU We start by defining OUT model of BBU in a micromn un- der the assumption that a single cavity approximation can be made. For the convenience of readers, we will closely follow the notations of references [l] and [2], which rep- resent previous study of an analytical model of multipass BBU in recirculating linacs. Consider a bunch(say, the Nth one) entering a cavity at its pth pass through the linac with its motion represented by a two component column matrix, Up (N) of 8 and pE . While traversing the cavity, the bunch will get a momen- tum kick due to transverse wakes excited in the cavity by preceding bunches in addition to gaining nominal energy. *This work was supported by the U.S. b e n t  of mew, under contract NO. DE-ACO5-84ER40150. (P + 11th time. A dipole mode of muency  L with a quality factor Q is assumed to be generated at the cavity for the following discussions. The equation of transverse bunch motion described in this physical picture is where q=1 M=l and x sinw((N - M)T + T, - T ~ ) ,  T P i q  is the transfer matrix from the cavity at the qth pass to the cavity at the pth pass. I is the average current and G is a 2 x 2 matrix with all elements equal to zero except G21 = 1. n, is the number of passes and RL is the trans- verse shunt impedance per unit length of the higher order mode w excited in the cavity, which is assumed to have an effective length of L. We also note that Rl is related to other commonly used transverse shunt impedance Z" as RL = 9. 7, is the summation over recirculation times up to the pth pass, and 71 = 0. In general, rP has to be an integer multiple of the RF period, rrj, if recir- culations are all in the same direction. However, it is not necessary to be an integer multiple of the bunch spacing T,  when the beam is subharmonically bunched. 3 THRESHOLDCURRENT As noted in [2], wemust deal with an (np- l>dimensional eigenvalue problem for the threshold current for multipass BBU in the most general circumstances even for a single cavity case. Deferring an analysis of general subharmonic bunching schemes until a later time, we restrict our at- tention to the case when every bucket of the accelerating mode of the cavity (i.e. T = T r j )  is filled. The model simplifies significantly in this situation and the eigenvalue problem reduces to ktf ~STRlBllTION OF THIS DOCUMENT IS UNLIMITED ASTE DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. 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Images are produced fipm the best available original document. - iIlegible . . where 200 H(L2) sinwr 1 + P(a) - 2H(a)coswr' 160 h ( d )  = and H ( Q )  = e-%e-i*r. This still is a complicated nonlinear equation for d which requires a numerical approach to solution in gen- eral. Note that the imaginary part of d changes sign from plus to minus as we cross threshold current. Solv- ing Fq(2) to first order in current, we find the following expression for the BBU threshold current: 100 5o 3 1 8 0  3160 3200 3220 3 2 4 0 ( ~ ~ ~ )  0 2c2 I& = - ewLRlM' (3) Figure 2: I&nA) vs. HOM frequency (MHz): Compar- ison of simulation results with I& predicted with Eq.(4) where M is This is our central result. The important parameters determining the BBU threshold current are clearly spelled out. 4 APPLICATIONS We have selected a 25-pass microtron essentially identical to the NIST RTM (racetrack microtron)[3] as our exam- ple to test the validity of Eq.(3) and have performed ex- tensive simulations of the microtron BBU problem using the computer code TDBBU[4] developed at Jefferson Lab originally for the 5-pass nuclear physics CEBAF acceler- ator. Figure 1 summarizes results of a TDBBU threshold current scan of the microtron when limited to 1 to 3 re- circulations. The parameters for simulation and analytic study are: Injection Energy = 5 MeV 2000 1500 1 0 0 0  500 0 .: . . I  . .  . . I  for a 2-pass system. Energy Gain per pass = 1.404 MeV 7 - l  = 2380 MfIZ (%)L = 250 51 72 = 387 and each subsequent path length is increased by one RF wavelength per pass. The HOM frequency is allowed to vary between 3150 MHz and 3250 MHz and Q of the mode is set to 1oooO. Let us start with a 2-pass system. Since np = 2, we get (4) 2c2 I& = - ewLR1c; 'es  sinwn This is a threshold formula studied in detail by Sereno[S]. Clearly, the '12' element of transfer matrix and phase ad- vance are the two most important parameters in designing a system like this as far as the BBU threshold is con- cerned. Regions of high threshold correspond to regions where sinwg term is positive. I& is infinity at least in the first order approximation we make. As one can see from Figure 1, it is finite even though very high. We have compared I& given by Eq.(4) with the TDBBU scan (dot-dashed line of Figure 1) in Figure 2. The minimal threshold current of this 2-pass machine is 21 mA. For a 3-pass system, I& is obtained from Eq.(3) with M replaced by . . .  3160 3180 0200 ' 'H ~ )  old current scan should be auasi-wriodic with a wriod of Noting that T, = 777 we find that the 38th, 39th and 77th subharmonics of the fundamental accelerating fre- quency appear in Eq.(5). As aresult, the pattern of thresh- 200 1 6 0  100 50 0 Figure 3: I&(mA) vs. HOM fresuency (MHz): Compar- ison of simulation results with I& predicted with Eq.(5) for a 3-pass system. adding with the T21' term being 180 deg apart in phase. We also notice that the graph is truly periodic with the pe- riod of 2380 MHz, if the small perturbation in amplitudes due to exponential terms is neglected. We have compared analytic I& with the TDBBU scan(dotted line of Figure 1) in Figure 3. The agreement is excellent and the lowest threshold current we found is 9 mA in this %pass opera- tion. The expression for the I& of a 4-pass system is already too long to fit on one line. It involves 6 interpass transfer matrices T2s1, T31', T3*2, pt', pi2, and pi3 with associ- ated sinusoidal terms resulting in a chcterist ic 3 min- ima threshold current pattern. This arises as a result of su- perposing terms which contain roughly 3 frequencies, the 39th, 78th and 117th subharmonics of 2380 MHz. Com- parison of the TDBBU scan(solid line of Figure 1) with the analytically obtained I& exhibits a good agreement as shown in Figure 4. The lowest threshold current is 6.4 mA, occurring at the HOM frequency of 3210 MHz.  = 200 1 6 0  1 0 0  so r. Figure 5: I&(mA) vs. HOM frequency (MHz): Compar- ison of simulation results with I& predicted with Eq.(3) for a 25-pass microtron. 5 CONCLUSION The agreement between the approximate expression Eq.(3) and simulation is quite satisfactory even for a =-pass machine which involves superposition and interference of 300 distinct fresuencies and recirculation matrices. Our analytic formula is able to pinpoint regions of HOM fre- quency where one should look to find lower threshold cur- rent. Furthermore, it provides a qualitative understanding of the characteristic pattern of valleys and peaks in the threshold cwent scan of a typical multipass machine. The analytic formula for BBU threshold current pre- sented here in Eq.(3) makes various parameter interrela- tions clear and should serve as a useful guide in designing microtrons. 6 ACKNOWLEDGEMENTS We are grateful to J. J. Bisognano and S. Comeliussen for reading the paper. 7 REFERENCES [l] J. J. Bisognano and R. L. Gluckstem, Proc. 1987 Particle [2] B. C. Yunn, Pm. 1991 Particle Accelerator Con., 1785 [3] S. Penner et&, E E E  TmnsactionmNuclearScience, 2269 Accelerator Cons, 1078 (1987). (1991). .141 G. A. KrafFt etul., "Calculating Beam Breakup in Supercon- [51 N. Figure 4: &(mA) vs. HOM fresuency (MHz): Compar- ison of simulation results with analytic prediction of G for a 4-pass system. ducting Linear Accelerators", CEBAF Publication (1991). ph.D. Dissertation univ. of (1994). 

Beam breakup in a microtron

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Beam breakup in a microtron

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