Fourth order resonance of a high intensity linear accelerator

For a high intensity beam, the 4\nu=1 resonance of a linear accelerator is manifested through the octupolar term of space charge potential when the depressed phase advance \sigma is close to and below 90 but no resonance effect is observed when \sigma just above 90 . To verify that this is a resonance, a frequency analysis is performed and a study of resonance crossing from above and from below the resonance is conducted. It is observed that this fourth order resonance is dominating over the better known envelope instability and practically replacing it. Simulation study shows a clear emittance growth by this resonance and its stopband. One of the authors [DJ] proposed an experiment to measure the stopband of this resonance to GSI using the UNILAC.


I. INTRODUCTION
Recently many high intensity linear accelerators (linacs) have been designed or constructed like the SNS (USA) [1], and J-PARC (Japan) [2], or people are trying to increase the intensity of existing linacs such as the UNILAC of GSI (Germany) [3]. For the high intensity linacs, it is the utmost goal to minimize the beam loss of halo particles by avoiding or minimizing contributions of various halo formation mechanisms. One such mechanism is the envelope instability [4]. So far, the high intensity linac design such as the SNS linac has avoided the zero current phase advance o ¼ 90 [1] because of the envelope instability. Until 1998, mismatch was the primarily studied mechanism of halo formation. In late 1998, Jeon found a case of halo formation induced by the 2 x À 2 y ¼ 0 resonance from the space charge potential in the ring [5]. Further studies of halo formation and/or emittance growth by space charge and resonances were reported in [6] and space charge coupling resonance studies of the linac were reported in [7]. Besides these, halo formation by nonround beam was reported in [8] and halo formation by rf cavity in [9]. A space charge driven fourth order mode instability of a ring lattice was reported [10]. Various resonance conditions outside of a Kapchinskij-ladimirskij (KV) beam were reported in [11] and various KV beam mode instabilities in [4], where the KV beam itself does not have any nonlinear potential within the beam.
In this paper, we report a discovery of the 4 ¼ 360 resonance (the equivalent of the 4 ¼ 1 resonance of a circular accelerator) in high intensity linear accelerators that was never reported before and discuss a proposed experiment. For a high intensity linear accelerator, the tune can be defined as =360 .
Numerical simulation of a linac is performed with a fairly well matched beam with 50 000 to 100 000 macroparticles using the PARMILA code with a 3D space charge routine [12]. A space charge grid with (N x ¼ 8, N y ¼ 8, N z ¼ 20) is used for the simulation. A 10 emA 40 Ar þ10 beam with initial ¼ 0:103 and initial normalized rms emittance " x ¼ " y ¼ 0:115 ½mm mrad, " z ¼ 0:119 ½mm mrad is used for most of the simulations unless specified otherwise. The beam is accelerated to ¼ 0:24. The space charge phase advance depression is about À20 . The coupling between the transverse and longitudinal planes is minimal because the depressed longitudinal phase advance is about 10 well separated from the depressed transverse phase advance. A linac with an focusing-focusing-defocusing-defocusing (FFDD) transverse focusing lattice is used for the simulations. This resonance is observed independent of the choice of the lattice such as FD (focusing quad and defocusing quad) or FFDD.
The study shows that the 4 ¼ 360 (or 4 ¼ 1) resonance occurs when the depressed phase advance is close to and below 90 for a high intensity linac just like a ring through the space charge octupole potential for a variety of beams including Gaussian, waterbag, etc. However, no resonance effect is observed when is slightly above 90 , as shown in Fig. 1. When is slightly above 90 , we observe no fourfold structure unlike the case with below 90 and the emittance growth is very small, which is due to the tiny initial mismatch. As shown in Fig. 1, the emittance growth increases as approaches the resonance ¼ 90 .
One characteristic of this resonance is the behavior difference when we cross the resonance from below and from above the resonance. This is due to the stable fixed points of the resonance. When we cross the resonance from above, the four stable fixed points emerge from the origin and move away, scooping particles from the core (see the bottom plot of Fig. 2). On the other hand, when we cross the resonance from below, stable fixed points move in from far toward the origin. So the particles cannot be captured by the stable fixed points, and they move around the fixed points (the top plot of Fig. 2). Figure 2 shows this behavior difference well, resulting in the beam distribution difference when we cross the 4 ¼ 360 resonance from below (top plot where varies from 70 to 95 ) and from above (bottom plot where varies from 100 to 75 ).
Depending on the direction to cross the 4 ¼ 360 resonance, the emittance growth also differs for the same reason. Figure 3 shows the plot of the emittance growth vs a parameter S ðÁ=360 Þ 2 =ðd=dn=360 Þ [13], where Á is the tune spread (proportional to the stop band width of the resonance) and d=dn is the phase advance change per cell. Emittance growth for two different groups is shown when we cross the 4 ¼ 360 resonance from above the resonance (downward crossing) and from below (upward crossing). We observe that the emittance growth is mostly linear for both cases except for S > 2. A large value of S means slow resonance crossing or wide resonance stop band.
Another characteristic of the resonance is the existence of a resonant frequency component. Because of the fixed points of the resonance, some particles have the same transverse oscillation frequency as the driving frequency of the 4 ¼ 360 resonance. A Fourier analysis is performed on the 2nd moment hX 2 i (or hY 2 i) of the beam distribution along the linac when the depressed phase advance of the linac lattice is 75 , 85 , and 92 , and a clear 4 ¼ 360 resonance peak is observed at the oscillation tune of individual particles (number of transverse oscillation particles made within one cell) with a value of 0.25 ( ¼ 90 =360 ) for the cases with < 90 , as shown in Fig. 4. We observe no resonance peak at the tune of 0.25 when ¼ 92 just above the resonance as is expected for > 90 . For each value, we keep the depressed phase advance pretty constant throughout the lattice.  1. (Color) Plots of the rms emittance growth vs the depressed phase advance for two different tune depressions, that is, two different beam current values. There is no resonance effect or emittance growth for > 90 . An emittance growth of more than 400% is observed for ¼ 87 .

III. ENVELOPE INSTABILITY
High intensity linac design has avoided 90 phase advance because of the well-known envelope instability. Our study indicates that the 4 ¼ 360 (or 4 ¼ 1) resonance is dominating over the envelope instability and practically replacing it. We did not observe the envelope instability during the simulation for the phase advance around 90 as shown in Fig. 5. Figure 5 shows the rms beam size as the beam crosses the 4 ¼ 360 (or 4 ¼ 1) resonance from above, where varies linearly from 99 to 75 . Here the beam size increase from about the 100th gap on is due to the resonance and the envelope instability is not observed. The characteristic of the envelope instability is the erratic change in the beam envelope (or rms beam size) and the coherent increase/decrease of the entire beam, not just part of the beam. Instead we see a smooth development of the fourfold structure due to the resonance as shown in Fig. 2, resulting in the change in the rms beam size shown in Fig. 5.
Considering this, it should be stated that the high intensity linac design should avoid ¼ 90 phase advance because of the 4 ¼ 360 (or 4 ¼ 1) resonance rather than the better known envelope instability. The effect of the envelope instability can actually be minimized-in theory-by nearly perfect envelope matching, whereas the 4 ¼ 360 resonance is independent of the rms matching. The solution is to stay outside of 90 À ÁQ 90 , where ÁQ is the stop band width of the fourth order resonance which depends on space charge tune depression. beam size is shown as the beam crosses the 4 ¼ 360 (or 4 ¼ 1) resonance from above ( varies linearly from 99 to 75 ). As we monitor the change in the phase space throughout the crossing, we only observe a smooth development of the fourfold structure due to the resonance as in Fig. 2 and do not see any coherent change of the whole beam. The envelope instability is not observed, which is also indicated by the lack of the erratic change in the beam size. The beam size increase is due to the 4 ¼ 360 resonance as we observe the evolution of the phase space beam distribution. One cell consists of four gaps due to FFDD transverse focusing.  3. (Color) Plot of emittance growth when we cross the 4 ¼ 360 resonance from above (downward crossing) and from below (upward crossing). The emittance growth is proportional to S up to around S ¼ 2. Here S ¼ ðÁ=360 Þ 2 =ðd=dn=360 Þ, where Á is the tune depression (proportional to the stop band width of the resonance) and d=dn is the phase advance change per cell.

IV. PROPOSAL TO MEASURE THE RESONANCE STOP BAND USING GSI UNILAC
One of the authors (Jeon) made a proposal to GSI colleagues to measure the stop band of this fourth order resonance in April 2007, and since then significant simulation efforts were undertaken to optimize the conditions to measure the stop band. Simulations were performed to predict the emittance growth at the end of the UNILAC Tank 1 as a function of the zero current phase advance. It should be noted zero current phase advance was used as an independent variable because x and y are different for the same zero current phase advance because the initial x and y normalized rms emittances of the real beam are different, " x ¼ 0:151 and " y ¼ 0:214 mm mrad. For instance, x ¼ 90 corresponds to x;o ¼ 109:4 and y ¼ 90 to y;o ¼ 105:6 . Simulations were done for the real UNILAC machine lattice and the same realistic 7.1 emA 40 Ar þ10 beam with 199 K macroparticles used for the previous benchmarking effort [14]. This input beam was obtained by reconstructing from the emittance measurement conducted before the UNILAC Tank 1. Figure 6 shows the simulation prediction of the average of the experimental transverse rms emittances ð" x þ " y Þ=2 at the end of UNILAC DTL Tank 1 vs the zero current phase advance. We observe a clear emittance growth due to the resonance.
The proposed experiment was conducted successfully in December 2008, confirming this simulation prediction to be pretty accurate compared with the experiment result. The experimental result is submitted in a separate paper [15]. Because the UNILAC Tank 1 is relatively short (about 16 cells), maximum emittance growth takes place around o ¼ 100 . Figure 7 shows the simulated beam distribution in x phase space at the exit of UNILAC Tank 1 for zero current phase advance o ¼ 100 illustrating well the fourfold structure of the 4 ¼ 360 resonance. The resonance stop band can be deduced from further simulation after it is confirmed that the simulation agrees well with the experiment.

V. CONCLUSIONS
The most important result presented in this paper is the discovery of a fourth order resonance in high intensity linear accelerators that was never reported before. The 4 ¼ 360 resonance of a linear accelerator (equivalent of the 4 ¼ 1 resonance of a circular accelerator) is manifested through the space charge potential for a high intensity beam when the depressed phase advance is close to and below 90 but no resonance effect is observed just above 90 . The frequency analysis study and the study of resonance crossing from above and from below the resonance confirm that the observed phenomenon is the 4 ¼ 360 (or 4 ¼ 1) resonance. It is observed that this fourth order resonance is dominating over the better known envelope instability and practically replacing it. It needs to be rephrased that the high intensity linac design should avoid 90 phase advance because of the 4 ¼ 360 (or 4 ¼ 1) resonance rather than the better known envelope instability.

ACKNOWLEDGMENTS
This work is a result of the collaboration between GSI-FAIR and SNS. The authors would like to express their gratitude to Professor I. Hofmann for his advice and comments. One of the authors (D. J.) is grateful for the hospital-