Paper presented at the Workshop on Space Charge Physics in High Intensity Hadron Rings, sponsored by Brookhaven National Laboratory, May 4-7,1998

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1 Paper presented at the Workshop on Space Charge Physics in High ntensity Hadron Rings, sponsored by Brookhaven Nationa Laboratory, May 4-7,998 Noninear Sef Consistent High Resoution Beam Hao Agorithm in Homomorphic and Weaky Chaotic Systems J. H. Wheaton, R J. Raridon, D. K. Osen, J. D. Gaambos, J. A. Homes Oak Ridge Nationa Laboratory Engineering Technoogy, Physics, and Computationa Physics and Engineering Divisions Prepared by the Oak Ridge Nationa Laboratory Oak Ridge, Tennessee Managed by Lockheed Martin Energy Research Corp. for the U.S. Department of Energy Under contract DE-AC05-960R22464 RECEVED The submitted manuscript has been authored by a contractor of the U.S. Government under contract No. DE-AC05-960R Accordingy, the US.Government retains a nonexcusive, royaty-free icense to pubish or reproduce the pubished form of this contribution, or aow others to do so, for US.Government purposes.

2 DSCLAMER 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 empoyees, makes any warranty, express or impied, or assumes any ega iabiity or responsibiity for the accuracy, competeness, or usefuness of any information, apparatus, product, or process discosed, or represents that its use woud not infringe privatey owned rights. Reference herein to any spccific commercia product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessariy constitute or impy its endorsement, mommendation, or favoring by the United States Government or any agency thenof. The views and opinions of authors expressed herein do not necessariy state or refect those of the United States Government or any agency thereof.

3 DECLAMER Portions of this document may be iegibe eectronic image products. mages are produced from the best avaiabe origina document.

4 ABSTRACT A technique is described which enabes high resoution of hao in beam dynamic studies by direct simuation. The method consists in first soving the beam dynamics probem using coarse initia data. The regions of the initia data, which resut in beam hao, or extremums in phase space, are identified. The dynamics are resoved by continuing the cacuation using initia data points sighty offset from those that resut in hao formation, thus fiing in the hao structure. The soution is repeated with appropriate scaing of such things as charge per orbit etc. This process may be continued indefinitey. The method can aso shed some ight on the hao generation in weaky chaotic systems. The scheme is essentiay different from the Af method in that no assumption is made about f,. As an exampe, a bihrcation in a non-trivia space charge dominated homomorphic probem is resoved sef-consistenty using minor computationa resources, rather than having to perform the cacuation for 250 triion effective partices. NTRODUCTON Recent acceerator systems require a very ow beam-wa interception (as ow as part in 0'). Modeing the beam dynamics incuding significant statistica sampe in the hao woud require considering on the order of a biion trajectories. n certain cases, which are not chaotic or weaky chaotic, a technique may be used which requires substantiay fewer resources than the brute force approach. The idea is easiy understandabe by reference to a specific exampe as shown in Fig.. Here is modeed an ion extraction from a pasma. The foowing Vasov-Poison equations [ )] ( ) V cp r, t = f ( r, v, ) dv- exp - cp (r,t at v.vf oi io r,v,t vxb-vq.v,f r,v,t = O

5 provide an adequate description. Pasma eectron density is described by a Botzman distribution. Eectrostatic equipotentias are shown in this 2-D exampe by dashed ines. Soid ines show ion trajectories (ony on the top haf of the figure). The extraction sheath, found sefconsistenty, is shown by the high fied region in the midde of the figure. (Pasma eectrons are abundant on the eft side of this sheath and absent on the right side.) One eectrode is shown in the midde of the figure on the top and bottom. The beam goes of the right side of the figure. The occupation in phase space of the exiting beam is shown in Fig. 2. Of particuar interest are the beam aberrations shown in Fig. 2 due to the noninear transverse forces in the proximity of the eectrode (shown in Fig. ). The scheme is basicay to take those trajectories which are in the outer reaches of phase space, trace them back to their starting position (on the eft side of Fig. ) and then create, say ten trajectories, in the phase space neighborhood of the ten most aberrated trajectories. This is a very simpe thing to do in the exampe iustrated in Fig.. First, the 0 most aberrationa trajectories are repaced by 00 trajectories with a space charge weight of one-tenth the origina. The sef-consistent soution is shown in Fig. 3 (corresponding to Fig. ) and Fig. 4 (corresponding to Fig. 3). This is denoted as the second major iteration. A comparison of Fig. 3 with Fig. shows exacty what trajectories were seected. A comparison of Fig. 4 with Fig. 2 shows the corresponding region in phase space that was refined. Continuation of the process is done by again seecting the ten most aberrated trajectories in Figs. 3 and 4, repacing each of them with ten trajectories with a weighting of one-tenth of the repaced trajectory. This is the third major iteration. The resut after 3 major iterations is shown in Figs. 5 and 6. As can be seen in Tabe, after 3 major iterations, ony 0,306 cumuative trajectories were cacuated, but if the resoution were uniform over a of the phase space 252 triion trajectories woud have

6 been cacuated - a saving of a factor of 25 biion in computation time. f a of these other 252 triion trajectories were actuay computed, the resut woud simpy fi in the ine in the centra part of the phase space diagram - Fig. 6. Whie this factor may not be easiy reaizabe in more compex probems with nontrivia occupation of 6D-phase space, arge factors of resource savings in sef-consistent hao cacuations are sti possibe. For non-chaotic, competey deterministic systems, there is no imit to the savings possibe. For weaky chaotic systems, there wi be a imit to the resoution obtainabe, but information wi be found on these chaotic regions of the phase space. For highy chaotic systems, the method wi not be reiabe.

7 Major teration Actua Apparent 26 x 2 = = 342 E = =432 = , = 522 z = , = 62 E = 270 2,520, = 702 = ,200, = 792 z = ,000, =882 E = ,520,000, = 972 z: = ,200,000, ~ = ,000,000, =52 = ,520,000,000, = T 24290=332 E = T 2 z: c 2 3 E r = 8974 (M (B)

8 Figure Legends Figure. on orbits (soid ines starting at the eft from inside the pasma) are acceerated by externay appied eectric fieds (the equipotentias indicated by dashed ines). The sefconsistent pasma sheath is shown by the high fied region (cose spacing of equipotentias) shown. An eectrode (constant equipotentia) is shown at the midde top and bottom. The ion beam, which exists on the right, has a hao produced by aberrations caused by noninear fieds near the eectrode. Figure 2. Phase space pot of the beam existing on the right hand side, Fig., the transverse distance is the horizonta axis and the transverse veocity is the vertica axis. The points with arge transverse speeds represent the aberrations (hao) of the beam, in this case. Figure 3. Same as Fig. except the most aberrated 0 orbits are repaced by severa more (00) for enhanced resoution. Figure 4. Same as Fig. 2 but now refers to the exiting beam of Fig. 3, Figure 5. Same as Fig. 3 but the process has been continued more times giving much greater resoution near the region of aberrations. Figure 6. resoved. Same as Fig. 4, but now referring to Fig. 5. Now the aberrations are fuy

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International Journal of Mass Spectrometry

International Journal of Mass Spectrometry Internationa Journa of Mass Spectrometry 280 (2009) 179 183 Contents ists avaiabe at ScienceDirect Internationa Journa of Mass Spectrometry journa homepage: www.esevier.com/ocate/ijms Stark mixing by ion-rydberg