DYNAMIC SPACE CHARGE CALCULATIONS FOR HIGH INTENSITY BEAMS IN RINGS*

Transcription

1 DYNAMIC SPACE CHARGE CALCULATIONS FOR HIGH INTENSITY BEAMS IN RINGS* J. A. Holmes, J. D. Galambos, D. Jeon, D. K. Olsen, J. W. Cobb, SNS Buldng, Oak Rdge Natonal Laborator, Oak Rdge, TN M. Blaskewcz, A. U. Lucco, and J. Beebe-Wang, Brookhaven Natonal Laborator, Upton, NY 11973, USA Abstract Space-charge-nduced emttance growth and halo generaton could lead to unacceptabl hgh beam loss n hgh ntenst rngs, such as the SNS [1]. In such accelerators, uncontrolled losses to the walls as small as one part n 10 4 would lead to actvaton, makng mantenance dffcult. For ths reason t s essental to understand the effects of space charge on beam dnamcs, and halo generaton n partcular, n hgh ntenst rngs. We have undertaken the stud of space charge dnamcs n hgh ntenst rngs usng a partcle trackng approach, wth self-consstent evaluaton of the space charge forces through a partcle-n-cell model. Because of the strngent loss requrements, t s necessar to thoroughl guarantee the relablt of these calculatons to hgh precson through comparson wth eperments and through convergence studes. In ths paper we present the results of convergence studes n the parameters of the model, namel, the number of macropartcles, the resoluton n the adopted FFT algorthm, the smoothng parameter, and the tme step sze. Although present calculatons have been etended to more than 10 5 macropartcles on ndvdual UNIX workstatons, t wll be necessar to ncrease another one to two orders of magntude to obtan the necessar precson. To accomplsh ths, we have constructed and are now usng a LINUX parallel computer from low cost components. 1 INTRODUCTION The operatng requrements of some new rngs, such as the Spallaton Neutron Source (SNS) accumulator rng, nclude hgh ntenst beams, low uncontrolled losses, and a consderable beam flght path n the rng. In these crcumstances, space-charge-nduced halo generaton s a potental loss mechansm, and requres stud. Hgh ntenst rngs are characterzed b the separaton of longtudnal and transverse ales. In SNS, for eample, the longtudnal bunch length s on the order of 100m, compared wth transverse beam dmensons of a few cm; and the longtudnal tune s about 10-3, compared wth transverse tunes of about 5.8. For ths reason t s possble, wth good appromaton, to separate the longtudnal and transverse dnamcs n hgh ntenst rngs and, for the stud of space charge effects, to consder the transverse dnamcs. Ths smplfcaton allows the derpton n terms of four, rather than s, dmensonal phase space, whch necesstates a smaller soluton space to obtan a gven numercal precson than s requred n s dmensons. In order to stud transverse beam dnamcs n hgh ntenst rngs, and space charge effects n partcular, we have adopted a partcle-trackng approach []. The ntegraton heme s chosen to be second order smplectc, wth a matr representaton of all lnear focusng elements, ncludng dsperson, and the ncluson of all nonlnear effects as kcks. Our treatment of space charge uses a partcle-n-cell (PIC) model [] wth fast Fourer transforms (FFTs) to evaluate the forces. Ths s carred out n a modfed verson of the partcle trackng and njecton code, ACCSIM [3], and n a new code, SAMBA, whch we are now developng. In PIC calculatons, numercal convergence and accurac must be carefull assured. We present n ths paper the results of convergence studes n the parameters of the model. Secton presents the equatons and numercal soluton of the model; Secton 3 presents the convergence results; and Secton 4 contans dusson and conclusons. EQUATIONS AND NUMERICAL SOLUTION The essence of our transverse partcle trackng model s the followng par of dnamc equatons for the macropartcle coordnates and : + K ( s) = F + K ( s) = F + F + F nl nl 1 δp + ρ ( s) p0 (1) where s s the azmuthal coordnate, K, ( s) are the lnear magnet focusng strengths, F, ( s) are the space charge forces, F nl, ( s) are the eternal nonlnear forces, 1 δp and s the dsperson term. The lnear focusng ρ( s) p0 forces, eternal nonlnear forces, and bendng radus are all dependent on the lattce; whle the momentum

2 δp devaton s determned b the longtudnal p 0 dnamcs. The space charge forces are evaluated selfconsstentl: F F = P = P N = 1 N = 1 ( ) ( ) + ( ) + ( ) + ε + ε where the summaton s over the number of Z rpλ macropartcles N, P =, Z s the beam charge 3 γ β AN number, A s the mass number, r p s the classcal radus of the beam partcle, γ and β are the relatvstc knematc parameters, λ s the longtudnal beam partcle denst, and ε s a numercal smoothng parameter. The numercal model solves the dnamc equatons usng an eplct, second order, smplectc ntegraton heme: Dvde the lattce nto N az lnear elements (drfts, bends, quadrupoles); Transport macropartcles through these elements usng a transport matr approach, ncludng dsperson; Treat nonlnear forces as kcks, appled at the ends of lnear elements wth strengths approprate to a second order smplectc heme; Eternal nonlnear forces can be appled ndependentl to each macropartcle; Space charge forces nvolve nteracton of all beam macropartcles, and requre specal treatment. The space charge forces are evaluated as nonlnear kcks usng a PIC model and FFTs: At each azmuth s, select a regular rectangular (, ) mesh of N FFT N FFT ponts, centered on the beam. Because of the perodct of the FFT, the mesh must be at least twce the etent of the beam n each drecton; Obtan the partcle denst on the mesh b blnear dstrbuton of the macropartcle charges to adjacent mesh ponts; Obtan the FFT of space charge forces at mesh ponts as the convoluton of the FFT of the partcle denst that of the force due to a unt charge. The nverse FFT of ths quantt gves the space charge forces on the mesh ponts. () Obtan the space charge force on each macropartcle as a blnear nterpolaton of the forces at the mesh ponts to the locaton of the macropartcle. The advantage of usng ths FFT procedure s one of speed, wth the number of operatons alng as N for partcle dstrbuton and force nterpolaton and as N FFT ln( N FFT ) for the transforms. A lmtaton of the FFT s that t s not conducve to the ncluson of wall effects. The stud of space charge usng ths numercal model nvolves a number of parameters: N the number of macropartcles; N FFT the spatal resoluton (grd parameter for the FFT algorthm); ε the smoothng parameter; and N the azmuthal ntegraton step sze. az We now stud the convergence propertes of an eample calculaton wth respect these parameters. 3 CONVERGENCE OF THE MODEL FOR AN EXAMPLE CASE We now stud the effect of the parameters of the numercal heme on the convergence of the model for an eample case, namel halo generaton b the parametrc resonance [4] n a doublet lattce. We consder a doublet lattce wth fourfold smmetr smlar to the SNS FODO lattce, length 0.668m, havng lnear focusng onl (48 quadrupoles, 3 sector bends, and 80 drfts) and bare tunes of ν =5.85 and ν =5.70. We consder a coastng beam wth a K-V dstrbuton, energ and energ spread E 0 =1GeV, Ema =9.4MeV, =4.7MeV, and emttances E rms ε, = 100π mm-mrad, and number of partcles 14 N = The ntal beam s rms msmatched, resultng n envelope ollatons of about 10% around the matched values. We follow the subsequent evoluton for 150 turns. The evoluton s observed to be as follows: Because the beam s msmatched envelope ollatons occur, as shown n Fg. 1 whch plots the beam averages of < ( < > ) > and < ( < > ) > ( < ( ) > and ) < ( > ). Because the tune s less than the tune, the focusng s slghtl weaker n than n, and through couplng the ollaton energ s transferred from the drecton nto the drecton. As the beam relaes some partcles cross the separatr of the parametrc resonance, drven b the ollatons n < ( ) >, becomng halo partcles and leadng to a growth n the rms emttance (Fg. ). Ths halo

3 RMS Ampltudes Squared [mm^] Envelope Ollatons "^" "^" Fgure 1. Average values of < ( ) > and < ( ) > plotted once each turn and showng envelope ollatons. RMS Emttances (*4) [mm-mrad] RMS Emttances "e_" "e_" Fgure. Transverse rms emttances number. prme [mrad] Beam Cross Secton n Y-Yprme ε and ε versus turn "_p" [mm] Fgure 3. Fnal beam dstrbuton n showng beam core and halo partcles. phase space generatng process removes energ from the envelope ollatons, whch then dmnsh. As ths drvng force s removed, the nstablt saturates (Fg. ) wth the beam n a new stead state wth halo. Fgure 3 plots the fnal beam cross secton n the phase plane. That the observed evoluton s a result of phscs, and not a numercal nstablt, was tested b carrng out the same calculaton wth the onl dfference beng the method for dstrbutng the charge to the FFT grd. In ths latter method, each partcle s regarded as a representatve of a populaton of macropartcles, all havng the same acton values ( J, J ) but dstrbuted unforml over betatron ollaton phases ( φ, φ ). In ths phase averaged model envelope ollatons are suppressed so that the parametrc resonance s not drven, but no changes are made to numercal methods or parameters. The result of the phase averaged calculaton s emttance conservaton and no halo generaton, as epected for a stable numercal heme. Ths ndcates that the observed nstablt s a result of the dnamcs. We also performed convergence studes n the four numercal parameters N, N FFT, ε, and N az. Fgures 1-3 are taken from the base case havng N =7680 macropartcles, N N = 3 3 FFT grd cells, FFT FFT ε =4.4mm (> cell sze at all azmuths), and N az =488 steps/turn. Convergence studes ncluded the followng ranges of parameters: N =7680, 3070, 1880, N N = 3 3, 64 64, FFT FFT ε =6.5, 4.4, 3.15mm N az = 488, 880 In the macropartcle varaton, the man effects of ncreasng the number of macropartcles are a dela n the onset of the nstablt and a faster rse of the nstablt followng onset. The ntal and fnal confguratons are bascall unchanged as are the man features of the evoluton. The same effect s observed when the number of FFT grd ponts s ncreased. In both of these cases, the passage to hgher resoluton decreases the numercal vost, thus sharpenng the observed evoluton. Choce of the smoothng parameter s perhaps one of the more subjectve aspects of PIC calculatons b our chosen method. Too small a value obures the collectve dnamcs n nose due to bnar nteractons of macropartcles wth nearb FFT nodes. Too large a value smooths awa the dnamcs. As a gudng constrant, we mantaned the values of the smoothng parameter to be larger than the FFT cell sze n all cases. As the value of the smoothng parameter was vared, t was observed that ncreasng the smoothng parameter results n a decreased saturaton level of the nstablt, both n terms of rms emttances and fracton of beam partcles n the halo.

4 Fnall, ncreasng the number of azmuthal ntegraton ponts resulted n no basc changes n the results. In order to llustrate the effects of the numercal parameter varatons, we now compare the beam evoluton for the base case and a bg case havng the followng parameter values: N = macropartcles, N FFT N FFT = FFT grd cells, ε = 4.4mm, and N az = 880 steps/turn. In both cases the man features of the beam evoluton are the same. In the bg case the onset of nstablt s delaed, the growth of the nstablt s faster, and the beam halo at saturaton contans a slghtl hgher fracton of the beam. Fgure 4 shows the beam averages of < ( ) > and < ( ) > for the two cases plotted versus turn number. The most notceable dfference s n the decrease of the vertcal fluctuatons drvng the nstablt. In the bg case ths decrease s both delaed and sharper relatve to the base case. Fgure 5 plots the evoluton of the rms emttances for the two cases, and t s agan seen that the rse n the emttance for the bg case s delaed but sharper n comparson wth the base case. The dfference n saturaton value of the two emttances s ver small, wth that of the bg case beng slghtl larger. RMS Ampltudes Squared [mm^] Envelope Ollatons for Base Case and Bg Case "Base_" "Bg_" "Base_" "Bg_" Fgure 4. Average values of < ( ) > and < ( ) > plotted once each turn and showng envelope ollatons. Plots are for the base case and the bg case. 4 DISCUSSION AND CONCLUSIONS Convergence studes have been carred out for the calculaton of halo generaton va the parametrc resonance drven b rms msmatch. In these studes the number of macropartcles per FFT grd cell vared from a mnmum of 7.5 to a mamum of 30, although the effectve number n the occuped cells s at least four RMS Y Emttance (*4) [mm-mrad] RMS Y Emttance for Base Case and Bg Case "Base_Case" "Bg_Case" Fgure 5. Transverse rms emttances for the base case and the bg case. ε versus turn number tmes hgher snce the FFT grd s twce the etent of the beam n both drectons. The number of ntegraton ponts per quadrupole par s 0 n the case of 488 ntegraton steps and 37 for 880 steps. It was found that the base case of 7680 macropartcles resulted n a reasonable derpton of the overall dnamcs of the beam. Increased resoluton revealed a dela n the onset of the nstablt together wth a faster rse tme. However, n the case studed here more than 1% of the beam macropartcles mgrated to the halo and the vertcal rms emttance growth was more than 3%. In hgh ntenst rngs, such as SNS, the lmt on uncontrolled losses s 10-4, or two orders of magntude below the observed losses here. To accuratel calculate the dnamcs of cases wth such small losses, hgh resoluton wll be a requrement. In order to perform hgh resoluton calculatons n a routne fashon, we wll requre greater computng resources than have been used here. The present calculatons were performed on a varet of IBM RS-6000 and DEC Alpha workstatons. The fastest of these machnes was a 400MHz Alpha, and the tmngs on ths machne were: appromatel turns/mnute, or 10 hours for 150 turns, for the base case ; and about 3 turns/hour, or 400 hours for 150 turns, for the bg case. In order to perform hgh resoluton calculatons routnel, we have taken two steps: we have obtaned accounts at NERSC, and we have assembled our own parallel computer, the SNS Wonderland Cluster. Ths cluster conssts of fve 533 MHz DEC Alpha computers (one gatewa node and four compute nodes), each wth 4MB of 9ns SRAM Cache, 56MB memor, 4.5GB SCSI Hard Drve, and 10/100 Ethernet. The gatewa node contans an addtonal 18GB storage, and the cluster s connected b a 10/100 Ethernet swtch. The cluster s runnng the RedHat Lnu operatng sstem and supports PVM and MPI message passng nterfaces. Because partcle trackng calculatons are CPU and cache bound, the 533MHz processor speed and 4MB cache are well chosen.

5 Also, the cluster s easl upgradeable, so that more compute power can be added when desred. 5 ACKNOWLEDGEMENT * Research on the Spallaton Neutron Source s sponsored b the Dvson of Materals Scence, U.S. Department of Energ, under contract number DE-AC05-96OR464 wth Lockheed Martn Energ Research Corporaton for Oak Rdge Natonal Laborator. REFERENCES [1] Natonal Spallaton Neutron Source Conceptual Desgn Report, Volumes 1 and, NSNS/CDR-/V1,, (Ma, 1997); On the World Wde Web at [] R. W. Hockne and J. W. Eastwood, Computer Smulaton Usng Partcles, Adam Hlger, IOP Publshng Ltd. (New York: 1988); C. K. Brdsall and A. B. Langdon, Plasma Phscs va Computer Smulaton, McGraw-Hll Book Compan (New York: 1985). [3] Jones, F., Users Gude to ACCSIM, TRIUMF Desgn Note TRI-DN-90-17, (1990). [4] J. S. O Connell, T. P. Wangler, R. S. Mlls, and K. R. Crandall, Proc Part. Accel. Conf., Washngton, DC (1993) 3657; J. M. Lagnel, Nucl. Inst. Meth. A345 (1994) 46; J. M. Lagnel, Nucl. Inst. Meth. A345 (1994) 405; R. L. Gluckstern, Phs. Rev. Lett. 73 (1994) 147.