Cavity loss factors can be easily computed for ultrarelativistic beams using
time-domain codes like MAFIA or ABCI. However, for non-ultrarelativistic beams
the problem is more complicated because of difficulties with its numerical
formulation in the time domain. We calculate the loss factors of a
non-ultrarelativistic bunch and compare results with the relativistic case.Comment: 3 pages, 3 figures; presented at European Particle Accelerator
  Conference, Stockholm, Sweden (June 22-26, 1998

Kurennoy, Sergey S.

English

arXiv

Cavity loss factors can be easily computed for ultrarelativistic beams using time-domain codes like MAFIA or ABCI. However, for non-ultrarelativistic beams the problem is more complicated because of difficulties with its numerical formulation in the time domain. The authors calculate the loss factors of a non-relativistic bunch and compare results with the relativistic case

Kurennoy, S. S.

UNT Digital Library

. 4 Title: CAVITY LOSS FACTORS FOR NON-ULTRARELATIVISTIC BEAMS Author(s): Sergey S. Kurennoy LANSCE-1 Submitted To: Presented at European Particle Acceferator Conference Stockholm, Sweden June 2246,1998 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 responsibility for the accuracy, completeness, or use- fulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any spe- cific commercial product, process, or service by trade name, trademark, manufac- turer, 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. DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. CAVITY LOSS FACTORS FOR NON-ULTRARELATIVISTIC BEAMS Sergey S. Kurennoy LANSCE-1, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Abstract Cavity loss factors can be easily computed for ultrarela- tivistic beams using time-domain codes like MAFIA or ABCI. However, for non-ultrarelativistic beams the prob- lem is more complicated because of difficulties with its nu- merical formulation in the time domain. We calculate the loss factors of a non-ultrarelativistic bunch and compae results with the relativistic case. 1 INTRODUCTION It is common to believe that loss factors of a bunch mov- ing along an accelerator structure at velocity t~ = Pc with ,f? < 1 are lower than those for the same bunch in the ul- trarelativistic case, p + 1. The loss factors are then com- puted numerically for the ultrarelativistic bunch, which is a relatively straightforward task, and considered as upper estimates for the case in question, p < 1. We study ,&dependence of loss factors in an attempt to develop a method to obtain answers for p < 1 case from the results for p = 1. It is demonstrated that the above as- sumption on the upper estimate might be incorrect in some cases, depending on the bunch length and properties of the structure (cavity + pipe) under consideration. 2 BEAM COUPLING IMPEDANCE AND LOSS FACTORS OF A CAVITY In the frequency domain and in the ”closed-cavity” approx- imation (which means very narrow beam pipes) the beam coupling impedance calculation can be reduce_d to an inter- nal eigenvalue boundary problem. Let H ,  be a com- plete set of eigenfunctions (EFs) for the boundary prob- lem in a closed cavity with perfect walls. The longitudinal impedance is then given by (e.g., [l]) where I,(P,w) = sL dzexp(- iwz /~c)E, , (O,z )  is the overlap integral, and W, is the energy stored in the s-th mode. Here E,,(O, 2) is the longitudinal component of the s-th mode electric field taken on the chamber axis. There is a resonant enhancement of the s-th term in the series (1) for Z ( p , w )  as w + ws. Let us introduce a fi- nite, but small absorption into the cavity walls by adding an imaginary part to the eigenvalue: w, 4 wi  - i w t  = w t ( l  - i / 2Qs) .  Here the Q-value of the s-th mode is Qs = wLWs/Ps >> 1, where P, is the averaged power dissipated in the cavity walls ( plus, in a real structure, due to radiation into beam pipes). For w E wL the s-th term in Eq. (1) dominates: The quantity R,(P) is the shunt impedance of the s-th cav- ity mode, and, unlike the &-factor, it depends on p. The beam loss factor is 00 k = Jd dw Re Z(p, w)lX(w)I2 , (3) where X(w) = ds exp [iws/(,&)]X(s) is a harmonic of bunch spectrum. For a Gaussian bunch with m s  length 21, the line density is X(s) = exp ( - s2 /2Z2) / ( f iZ )  and X(w) = e x p { - [ ~ Z / ( P c ) ] ~ / 2 ) } .  Assuming all Qs >> 1 and integrating formally Eq. (1) for the Re Z(p, w ) ,  one can express the loss factor as a series where the loss factors of individual modes k, in the last equation are written for the Gaussian bunch. In principle, Eq. (4) give us the dependence of the loss factor on p. However, the answer was obtained in the “closed-cavity’’ approximation. Moreover, it is practical only when the number of strong resonances is reasonably small, since their the P-dependence varies from one reso- nance to another: where y = l / d m .  It is obvious from Eq. ( 5 )  that for long bunches loss factors will decrease rapidly with p de- crease, as exp ( -@-2). Indeed, the lowest resonance fre- quencies are w: M c /d ,  where d is a typical transverse size of the cavity. The exponent argument - ( l / ~ ? ) ~  will have a large negative value for E 2 d, and the exponential de- crease for small p will dominate the impedance ratio. The impedance ratio dependence on ,B is more complicated, and we consider below a few typical examples. 3 EXAMPLES 3. I Cylindrical Pill-Box For a cylindrical cavity in the limit of a vanishing radius of beam pipes, b + 0, one can obtained explicit expressions of the mode frequencies and impedances, e.g., [l]. Let the cavity length be L and its radius be d. The mode index s = (m, n, p )  means that there are m radial variations and p longitudinal ones of the mode E-field. The resonance frequency is W m n p  = d p L n  + (.rrpd/L)2c/d, where pmn is the n-th zero of the first-kind Bessel function Jm(x) .  The longitudinal shunt impedance is -2 x [ (s)’ - (?)’I {E:} (3). (6) The upper line in {. . .} corresponds to even p and the lower one to odd p ,  and 6 is the skin-depth. The ratio of loss factors Eq. ( 5 )  for the lowest E-mode, Eolo, is then Obviously, it is almost independent of p when the bunch is short, I << d, and the cavity is short compared to its ra- dius, L << d. For longer cavities, however, the ratio oscil- lates and might exceed 1. This strong resonance behavior is clearly seen in Fig. 1 for large L / d ,  while for small L / d  the IC-ratio slowly decreases with p decrease. For some par- ticular parameter values, IC010(,8) can be many times larger than IColo(1). A picture for a longer bunch is similar except the resonances at small P s  are damped heavily. / 2 0 u -  0. Ud v 4  Figure 1: Ratio of loss factors (7) for a short bunch, Z/d = 0.05, versus P and resonator length L / d .  3.2 APT I-cell Cavity As a more realistic example, we consider an APT super- conducting (SC) 1-cell cavity with a power coupler [2].  Of course, such a cavity with wide beam pipes to damp higher order modes can not be described completely by the for- malism of Sect. 2, except for the modes below the pipe cutoff. Direct time-domain computations with the codes MAFIA [3] and ABCI [4] show the existence of only 2 longitudinal modes below the cutoff for the p = 0.64 cav- ity, and only 1 for /3 = 0.82, in both cases including the fundamental mode at fo = 700 MHz. The loss factor con- tributions from these lowest resonance modes for a Gaus- sian bunch with the length E = 3.5 mm for = 0.64, and I = 4.5 mm for p = 0.82, are about 113 of the total loss factor. We use MAHA results for the field of the lowest mode to calculate the overlap integral and study the loss factor dependence on p. The on-axis longitudinal field of the fundamental mode is fitted very well by a simple formula E,(z) = E,(O) exp [ - ( ~ / a ) ~ ] ,  where a = 0.079 m for P = 0.64 and a = 0.10 m for p = 0.82, see [5] for detail. The ratio of the shunt impedances in Eq. ( 5 )  is then easy to get analytically where w = 2 ~ f o .  The resulting dependence shows a smooth decrease at lower ps. The loss factor for the lowest mode for p = 0.64 is 0.614 times that with P 1, and for ,L? = 0.82 is 0.768 times the corresponding = 1 result. 3.3 APT Caviv, 5 cells For 5-cell APT SC cavities the lowest resonances are split into 5 modes which differ by phase advance per cell A@, and their frequencies are a few percent apart [2]. We use MAFIA results [6] for these modes to calculate their loss factors according to Eq. (4). The on-axis fields of two modes, with A@ = 0 (0-mode) and A@ = T (x-mode), which is the cavity accelerating mode, are shown in Fig. 2. Time-domain simulations with the code ABCI [4] give us the loss factor of a bunch at /3 = 1. The loss factor spectrum for the p = 0.64 cavity, integrated up to a given frequency, has two sharp steps: one near 700 MHz with the height 0.5 V/pC and the other near 1400 MHz with the height about 0.1 VlpC. They correspond to the two bands of the trapped monopole modes in the cavity, cf. Table 1. We calculate numerically overlapping integrals in Eq. (4) for a given p. The results for the loss factors of the lowest monopole modes are presented in Table 1. The totals for the TM010 and TM020 bands for p = 1 in Table 1 agree very well with the time-domain results. In fact, we are mostly concerned about only these two resonance bands, since the higher modes are above the cutoff, and they prop- agate out of the cavity into the beam pipes depositing most of their energy there. Our results for the design values of p are in agreement with those obtained in [2] .  Remarkably, I I 0.1 0.2 0.3 0.4 0.5 0.6 z, m Figure 2: Longitudinal on-axis electric field (arbitrary units) for 0-mode (dashed) and fundamental (n-) mode in a half of the 5-cell APT p = 0.82 cavity. the total loss factors for a given resonance band in Table 1 are lower for the design than at ,# = 1. The only ex- ception is the TMOZo band for the 0 = 0.82 cavity, but it includes some propagating modes, and its contribution is very small. The P-dependence of the loss factor for two TMolo modes mentioned above (0- and n-mode) is shown in Fig. 3. Obviously, the shunt impedance (and the loss factor) dependence on p is strongly influenced by the mode field pattern. 1.4 1.2 1 3 0.8 P 0.6 0.4 0.2 P Figure 3: Loss factor ratio vs p for 0-mode (dashed) and fundamental (n-) mode in the 5-cell APT p = 0.82 cavity. 4 SUMMARY The examples above compare loss factors for p < 1 with p + 1 results. More details can be found in [5].  Essen- tially, the frequency-domain approach has been applied in- stead of the time-domain one. It can be done only when we know the fields of all modes contributing significantly into the loss factor. Nevertheless, for many practical applica- tions, including SC cavities, the lowest mode contribution is a major concern, because propagating modes travel out of the cavity and deposit their energy away from the struc- ture cold parts. Table 1: Loss Factors (in V/pC) in APT 5-cell Cavities A@ 1 f ,MHz I k(P> 1 W) I W I W )  B = 0.64. TMnin-band , "-- 0 681.6 7.2 3.7 0.020 2 ~ 1 5  686.5 4.810-5 2.910-' 0.0016 3 ~ / 5  692.6 1.1 0.218 0.0005 4 ~ / 5  697.6 1.210-3 0.250 0.0049 n 699.5 0.184 9.210-3 19.92 Total 0.185 0.507 0.365 ,B = 0.64, TMozo-band 0 1396.8 6.5 5.4 lov4 1.187 2 ~ / 5  1410.7 1.2 9.0 0.0014 3n/5 1432.7 1.810-5 0.0173 0.0011 4 ~ / 5  1458.8 8.0 0.0578 1.4 Total 6.7 0.086 7.8 T 1481.0 3.510-7 0.0095 3.710-5 p = 0.82, TMolo-band 0.3 10- 6.9 10- 4.5 10- 697.2 1.3 0.220 5.9 699.9 0.285 0.240 1.188 Total 0.286 0.494 0.579 I I I I p = 0.82, TMozo-band 0 1  1357.7 I 4.210-5 1 0.810-6 I 52.4 1367.7 1.4 8.0 1 1384.5 1 1.610-: 1 1,410-: 1 ",":' 4 ~ / 5 *  1409.6 8.0 1.3 5.6 Total 1.6 10- 3.7 10- 4.32 1436.9 1.6 lo-' 2.2 Mode near the cutoff. **Propagating mode, above the cutoff. One interesting observation is that the Ioss factor of an individual mode at some p < 1 can be many times larger than for p = 1. Obviously, one should exercise caution in using p = 1 results as upper estimates for a ,B < 1 case. The author would like to thankFrank Krawczyk for fruit- ful discussions and for providing MAFIA results for the 5-cell cavities. Useful discussions with Robert Gluckstern and Thomas Wangler are gratefully acknowledged. 5 REFERENCES [ 11 S.S. Kurennoy, "Beam-Chamber Coupling Impedance. Cal- culation Methods", Phys. Part. Nucl. 24, 380 (1993); also CERN SL/91-31 (AP), Geneva, 1991. [2] EL. Krawczyk, in PAC97, Vancouver, BC (1997); also LA- [3] T. Weiland et al., Proc. 1986 Lin. ACC. Conf., SLAC Report [4] Y.H. Chin, Report LBL-35258, Berkeley, 1994. UR-97-1710, LOS Alamos, 1997. 303, p.282; MAFIA Release 4.00 (CST, Darmstadt, 1997). [SI S.S. Kurennoy, Report LA-CP-98-55, Los Alamos, 1998. [6] F.L. Krawczyk, private communication, November 1997. 

Cavity Loss Factors For Non-Ultrarelativistic Beams

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