NONLINEAR DIAGNOSTICS USING AC DIPOLES*

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1 t 1999 Particle Accelerator Conference, New York, NY, March 29-April 2, 1999 NONLINEAR DIAGNOSTICS USING AC DIPOLES* BNL S. Peggs, BNL, Upton, NY There are three goals in the accurate nonlinear diagnosis of a storage ring. First, the beam must be moved to amplitudes many times the natural beam size. Second, strong and long lasting signals must be generated. Third, the measurement technique should be non-destructive. Conventionally, a single turn kick moves the beam to large amplitudes, and turn-by-turn data are recorded from multiple beam position monitors (BPMs) [1-6]. Unfortunately, tune spread across the beam causes the center of charge beam signal to decohere on a time scale often less than 100 turns. Filamentation also permanently destroys the beam emittanee (in a hadron ring). ~us, the strong single turn kick technique successfully achieves only one out of the three goals. AC dipole techniques can sehieve rdl three. Adiabatically excited AC dipoles slowly move the beam out to large amplitudes. The coherent signals then recorded last arbitrarily long. The beam maintains its original emittance if the AC dipoles are also turned off adiabatically, ready for further use. The AGS already uses an RF dipole to accelerate polarized proton beams through depolarizing resonances with minimal polarization loss [7]. Similar AC dipoles will be installed in the horizontal and vertical planes of both rings in RHIC [8]. The RHIC AC dipoles will also be used as spin flippers, and to measure linear optical functions [9]. 1 LINEAR MOTION Horizontal motion is deseribed using complex phasors z =z +ix =aei4 (1) so that the unperturbed one turn motion is just Zt+l = Rzt (2) where R = exp(i 27rQx ). Here Qx is the betatron tune, and the normalized coordinates z and z both have the dimensions of length. An AC dipole just after the reference point gives a real normalized angular kick on turn t of Azt = Az = ~ (3) where QD is the drive tune and *Ois the initial phase. he AC dipole strength is J = BL is the integrated field amplitude, (Bp) is the rigidhy, is the Ttviss fimction at the dipole. If z = Z. just before the first dipole kick, then the net displacement phasor on turn T is zt = RTZO+(RTAZO+RT laz1...+ RIAZT-l) (4) Figure 1: Adiabatic excitation of an AC dipole, in the rotating frame. The circles represent single particle motion. The exact general solution for linear motion is [10] %T = A ze 2 QXT + 6_ ei2uqdt 6+ e-i2 QDT (5) where 2 = Z. J_ + d+ is a constant given by the initial conditions, and the complex AC dipole strengths are J exp( i[@- *o]) &=z (6) sin(nq_) 6+=7 d exp(i[nq+ +.]) sin(mq+ ) whereq- = QD Qx andq+= QD+Qx. The oscillating closed orbit is defined as that orbit which exactly repeats itself after one modulation period. The solution on turn T is obtained by putting 2 = O,so that Zco = & ei2rqdt _ ~+ e-i27rqdt (7) generally following a tilted ellipse, not a circle, in normalized phase space. The semi-minor and semi-major axes are 116-I Id+ll and Id-I+ Id+( long. In practice the aspect ratio of the ellipse is close to 1 when the AC dipole is driven at a tune close to the fractional betatron tune (Q_ w O). When Iii_1>> lb+i the approximate motion is zt ~ ~ e~2~qxt + &_ ei2zqdt (8) Motion in the rotating frame, which rotates with the AC dipole drive at 2TQD per turn, isdenoted by over-bars. Assuming the previous approximation to be accurate, ~T = 6- + ;e-i2tq-t (9) That is, a test particle slowly circulates the vector c$_at a radius of constant length I; 1,as illustrated in Fig. 1. When a bunch is considered, a distribution of 2 values must be used. A smoothly distributed beam has (2) = O and (22) = 2&U where () represents a bunch average, and CUis the unnormalized root mean square emittance. The center of charge motion in the non-rotating frame is just Worksuppoti by the DoE. (ZT) = d_ ei2fiqdt (10)

2 3 PHASE SPACE PAINTING Since the beam is injected into the ring at a dispersion-ffee region [5], beam phase%pace painting in the transverse direction is conveniently de-coupled tkom the longitudinal beam manipulation [11]. Furthermore, painting in the horizontal and vertical direction can be adjusted independently. The izjection system is designed to accommodate both x-y correlated and x-y anti-correlated painting schemes, illustrated in Fig. 1. inidim md m,* y ~ injecticmbegin,! foil *L x / injectionutd! I injectionbegin (a) Fig. 1 Basic painting scenarios. (a) x-~correlated painting, (b) x-y anti-correlated painting. Table 3 Comparison of x-y correlated and anti-correlated phase space painting scenarios painting scenarios correlated I anti-correlated FoiI hitting rate 1:2 Aperture inj. 1 :1.5 Susceptibility to coupling yes no Capability to KV painting no yes In both painting examples, shown in Fig. 3 and 4, the time constants of the bumps were 0.3 ms or longer in consideration of edcty current effect in tbe vacuum chamber. Since painting is adopted in both directions without steering of the injecting beam the minimization of foil hitting and the design of downstream beam line and beam dump for [the un-stripped beam become straightforward. -%!%743.%~ no Y (Iml) 1 The most easily achievable bumps are the ones moving the closed orbit monotonically as m exponential function of time with a reasonably long time constant. With a x-y correlated bump setting, as shown in Fig. 2, phase spaces in both dimensions are painted from small to large emittances. Ideally, the resulting rectangular transverse profile, Fig. 3, can easily meet the target requirements. :~ =~ I 91/ I -4%%+4 I.i I I 6oT21mo + (*) %o-tzo -aoo l \ i *I \ 1 Fig. 3 Phase space (distribution at the end of injection o %4 -sm & e -* g -,.5 -> 2 -r 2.5 o.o~ time(ins) Fig. 2 Orbit bufi setting for a x-y correlat&i painting. However, such a beam profile is susceptible to transverse coupling due to magnet misalignment and space charge forces, which in turn results in an effective doubling of the maximum emittauce in both directions. On the other haml with the x-y anti-correlated scheme the total transverse emittance is approximately constant during the injection. The resulting oval beam profiie achieved by x-y anticorrelated painting, fig. 4, is immune to the transverse coupling. Table 3 compares the two painting schemes. achieved by a x-y correlated phase space painting. ma. -Wk5b+m :! m., -z.4@hhaj%-dd Fig. 4 Phase space distribution at the end of injection achieved by ax-y anti-correlated phase space painting.

3 f 4 SPACE CHARGE EFFECT We compare space charge effects between the beams in the current design of FODO nominal tune lattice and in an alternative design of FODO split tune lattice [12] developed recently as part of design studies in BNL. I@ macro particles, with transverse phase space distribution similar to the one shown in Fig. 4 were tracked in the two lattices for 100 terns by SIMPSONS 2D code. Space charge effect in beams with peak current O-1OOA corresponding to the proton accumulation of 0-2MW, was investigated. The rms emittance growth and hme shift due to space charge as functions of peak current are shown in Fig. 5 and Fig. 6 respectively. The lattice functions and space charge effects in 2MW beams in the two lattices are compared in Table 4. With the split tune lattice, the vertical beam envelope variation (~~~ti) is significantly reduced. Correspondingly, the vertical emittance and beam Ml/halo generation is also dramatically reduced. 5 PSEUDO END TO END SIMULATION Driven by searchiig for suitable painting schemes, injections with various bump settings were simulated in the accumulator ring. Then particle distribution obtain from the ring simulations were tracked to the target through the RTBT line with PARMILA. PARMILA was modified to include scattering effect of a 4mm thick inconel window, wkdchis about 2 meter from the target. Then the current distributions are checked against the beam requirement at the target. (Table 1). It was in a 2-D plan of time constant and bump strength, there is a nice-sized region in which all the bump settings give satisfactory distributions. Fig. 7 shows one example of satisfactory distribution at the target that was achieved by the bump settings shown in Fig. 2. Complete end-toerid simulations, from the ion source to the target though linac, H13BT,accumulator ring and RTBT is in progress. I.2, I Peaak C&ren? (A) - m 00 Fig. 5 RMS emittance growth as function of peak current. 0,= 00 Avxfrom.ilnulatio,mm.w@ xx Avy from Sirn.lelio.,matchd p A.xfrom sim.ltion. fit 00 A.,frotnaimu Ialion, WI i s = g+,, , - ~ ~w.. - Fig. 6 c A. fr.rrl analytical rnd.1 x.. ~ Maximum space charge tune shifts. Max. IDzl,(m) P=@ Max. B~~ (at locm), (kg) Max. tune shif4 Av, (m y) SC ~ growth Afl/@,(@) 0.10/ I 0.06 Emittance gxowth, AE/&,(w y) -0 I /.002 Tail/halo (lo- /turn) 1.1/ / 0.1 Fig. 7 Current density distribution at the target in units of 10A/m. one quadrant of the beam footprint is shown. 5 ACKNOtiEDGEMENT The authors would like to thank 1?.Jones and S. Machida for developing ACCSJM and SIMPSONS and making them available. They would also like to thank A. Luccio and N. Maiitsky for helping to set-up the codes in BNL. 6 REFERENCES [1] W.T. Wenget uj,pmt. of PAC 97,p.972, (1997). [2] D.Rapariaefal, theseproceedings. [3] C.J. Gardneret al, Proc. of PAC 97,p.962, (1997). [4] D.Rapariaet al, Proc. of PAC 97,p.162, (1997). [5] L.N. Blurnberget al, Proc. of PAC 97,p.159, (1997). [6] SNS Collaboration@ Spallation Neutron Source Design Manual, p , June [7] F. W. Jones, T_ker s Guide to ACCSIM, TRIUMF Design Note TIWDN-90-17, June 1990 (and later additions). [8] S. Machid~ The Simpsons User s Manual, Dallas, [9] G. Boicort et al, Parmila User s Manual. LA-UR [10] N. Malitsky et al, these proceedings. [11] J. Beebe-Wang, Study of Long@dinal Injection/stacking in the SNS Accumulator Ring, these proceedings. [12] C.J. Gardner et al, these proceedings.

4 DISCLAIMER This repofi 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, make any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, 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.

5 DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.

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