Space charge forces, representing the weak-strong case, are simulated by kicks from a line charge having a 2D-Gaussian transverse charge distribution. A series of particles having initial coordinates consistent with the dimensions of the injected beam are tracked sequentially, and tunes are obtained from analysis of the coordinates, x, x{prime}, y, and y{prime}, at the end of each turn. Stability is tested using 30K turn tracking runs during which the momentum error {delta} varies as {delta} = {delta}sin(2{pi}Q{sub s}t)

Dell, G.F.

Peggs, S.

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1995 P a r t i c l e  A c c e l e r a t o r  Conference and I n t e r n a t i o n a l  Conference on  High-Energy A c c e l e r a t o r s  May 1-5, 1995 Dallas, Texas BNL-6 1 I75 Simulation of the Space Charge Effect in RHIC * G.F. Dell and S. Peggs Brookhaven National Laboratory Upton, New York 11973, USA Abstract Space charge forces, representing the weak-strong case, are simulated by kicks from a line charge having a 2D-Gaussian transverse charge distribution. A series of particles having initial coordinates consistent with the dimensions of the injected beam are tracked sequentially, and tunes are ob- tained from analysis of the coordinates, x, x‘, y, and y‘, at the end of each turn. Stability is tested using 30K turn tracking runs during which the momentum error 6 varies as 6 = 6sin(27rQSt). I. INTRODUCTION RHIC will have 57 bunches of lo9 Au ions with a spac- ing of 67.3 m. There is a 9.8 m region on either side of the interaction point in which the colliding beams will be collinear. They will then be separated by the DX dipoles. With this geometry there are no parasitic beam-beam col- lisions. Hence the tune shift from beam-beam forces is limited to head on beam-beam collisions at the interaction points and is small. The tune shift from the space charge is expected to be - 0.021. As RHIC will operate between the 5th and 6th integer resonances at nominal tunes of u, = 28.190 and u, = 29.180, the anticipated space charge tune shift may drive particles on the edge of the beam to the 5th inte- ger resonances at 28.200 and 29.200. The present study was made to explore possible limitations to the dynamic aperture caused by space charge forces. The beam has transverse dimensions much smaller than the bunch length and is treated as a line charge hav- ing a transverse charge distribution represented by a 2- dimensional Gaussian. Test particles at an arbitrary dis- placement from the axis of the beLm experience kicks at- tributed to the space charge force. The tunes of small am- plitude betatron motion are depressed by as much as 0.021, and the arc quadrupoles are adjusted to restore the tunes to the nominal values of 28.190,29.180. However, particles with large amplitudes experience little space charge force and have tunes that can approach or cross the 5th integer resonances at u, = 28.200 and uy = 29.200. Information on tune and amplitude growth of the motion is obtained by tracking on a grid with initial displacements defined by the action Jt = ( n ~ ) ~ / 2 P  = J, + Jy with 0.5 5 n 5 7. 11. THEORY The electric field from a 2D-Gaussian charge distribution at a point (x,y) has been expressed in terms of the complex ‘Work performed under the auspices of the U.S. Department of Energy error function by Bassetti and Erskine [l] The above expressions are valid when u, > by, where: s,, = 2(u,2 - a;), (3 ) by a x  a x  CY z ~ = x - + + y - ,  and z = z + iy, (4) ( 5 )  is the complex error function. Okamoto and Talman [2] have used Pade approximations for rapid evaluation of the complex error function. This for- malism has been incorporated in Teapot [3] to evaluate the beam-beam force. The calculation of the kick is divided into three cases according to the profile of the beam cross section: 1). when a, and uy are within one part in lo4 of being equal, .a radial Gaussian dependence, (1 - e--r2/202) is used, 2). when u, > by, Eqn.1 - Eqn.6 are used, and 3). when by > u,, x and y are interchanged in Eqn.1 - Eqn.6. Elements of class “beambeam”, one of the Teapot elements with “extended allowed parameters”, are used to gener- ate kicks from the charge distribution. These elements are intended for calculation of the beam-beam force and are usually placed at the interaction points. In the present study we use 144 of these elements to simulate the space charge force and have distributed them smoothly at  the SF and SD sextupoles in all six arcs. The different optical functions at SF, where u, > uy, and SD, where by > ux, require two families, and the number of particles/bunch as well as the local values of ux and by must be specified for each. Keil [4] considers the beam-beam force and points out the electric and magnetic components differ by a factor of P 2 -  The magnitude of the kick given to the test particle depends upon whether the test particle is travelling with, 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, 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. DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. or against, the beam. For the head on beam-beam effect the test particle travels against the beam, and the electric and magnetic fields add as (1 + P2).  For space charge the test particle travels with the beam, and the electric and magnetic fields subtract as (1 - a2) and contribute a l/y2 factor to the space charge force. We make use of the transverse dependence of the two dimensional charge distribution given by the beam-beam formalism in Teapot. Compensation for the difference be- tween tune shifts from beam-beam and space charge forces can be made by adjusting the number of particles per bunch until the observed tune shifts equal the Laslett tune shift, Eqn.7. However, for the present study where kicks are present at all SF and SD sextupoles, the observed tune shifts are sufficiently close to the Laslett tune shift that no change of & is necessary. (7) Where N b  = lo9 is the number of gold ions per bunch, rp = 1.510-l8 m is the classical proton radius, Ro = 610.18 m is the average radius of RHIC, y = 12.89 at in- jection, trms = u2/P = 1.29710-7m, UL = 0.3961 m, is the rms bunchlength, and Z=79, A=I97 are the charge and mas  number. For fully stripped gold, Av = -0.021. 111. ERRORS A .  Position Errors assigned to dipole position are a, = uy = 0.5 mm and to dipole orientation are ug = 1 mradian. De- termination of the center and orientation of the fields in quadrupoles using colloidal cells[5] justifies using a, = uy = 0.25 mm and ug = 0.5 mradian for the position and orientation errors of all quadrupoles. B. Magnetic Magnetic harmonic errors[6], a, and b, with 1 5 n 5 9, were obtained from magnetic measurements on 34 arc dipoles and 36 arc quadrupoles. Harmonic errors for special insertion magnets, DO (10 cm) and DX(l8cm) dipoles and the 13 cm triplet quadrupoles are based on a combination of magnet calculations and measurements of test magnets. IV. SIMULATION Tracking at constant momentum errors of 6 = 0 and &0.11% (f2.5up/p) is performed for test particles with ini- tial displacements defined by Jt = Jx + Jx = (n~)~/2; ;P for 0.5 5 n 5 7 with values of the initial x and y satisfy- ing J,/Jt = 0.96,0.75,0.50,0.25 and 0.04. Each particle is tracked for 512 turns, and the values of x, x’, y, and y’ relative to the closed orbit are written to a file for post processing from which the tunes at  each amplitude are ob- t ained . Stability is determined using 30K turn tracking runs dur- ing which the momentumerror 6 varies as d = 8sin(2sQSt). As a line charge is assumed, there is no longitudinal varia- tion of the charge density in a bunch. 0.16 . 1 7 ~  0.1 5 0.15 0.16 , . I  1 .  I 0.17 0.18 0.19 0.20 VI Figure 1. Dependence of tune on 6, J t ,  and aspect ratio x/y with all magnetic field and random positioning and orientation errors included. The nominal tunes are v, = 28.190, v,, = 29.180, and at injection, below transition, the chromaticities ex = <, = -3. Four different seeds are used to initiate the generation of random displacements, rotations, and magnet field er- rors. In each case two families of skew quadrupoles are used to globally decouple the betatron motion, and the arc quadrupoles are adjusted to obtain the desired tunes. V. RESULTS Tunes are obtained by averaging the phase advance per turn in each plane, and the results are plotted in tune space with one set of lines connecting particles with equal action and another set of lines connecting particles having the same ratio of x/y. Test particles were located wherever two or more lines intersect. “Leafprints” showing the de- pendence of v, and v,, on amplitude and aspect ratio x/y are shown in Figure 1 when only maghetic, positioning, and orientation errors are present. In general the tunes decrease as the amplitude increases. The addition of the space charge force depresses the tunes of particles at  small amplitudes. The “leafprint” in Figure 2 includes the effect of space charge. The arc quadrupoles have been adjusted to restore tunes at small amplitude to the nominal tunes. The leaves for S # 0 are distorted; the tunes at large am- plitude include shifts from magnetic field errors, while the tunes at small amplitudes also show a nonlinear chromatic dependence. This feature results from the test particle os- cillating around a closed orbit that is displaced by qx S from the center of the charge distribution. Figure 3 shows results of a finer scan intended to search for tune distortion near the resonances. The test is most sensitive for the 2u and 3u lines and shows no peculiarities. The stability of particles has also been tested by tracking all particles in the mesh for 30K turns. Figure 4 shows 0.21 0.20 ,&0.19 0.18 0.17 0.18 0.1 9 0.20 0.21 0.22 v x  Figure 2. ” Leafprint” when space charge forces are added to the configuration used in Figure 1. The nominal tunes have been shifted to vx = 28.210, vr = 29.200 to restore the tune at small amplitudes to v, = 28.190, vy = 29.180. survival plots for four particles representing different er- ror distributions. Solid lines connect data without space charge forces and dashed lines connect data with space charge forces included. The dynamic aperture is consis- tent with IOU and the plateau shows that all particles hav- ing initial amplitudes 5 10a survived the requested 30K turns. There is no significant decrease of the maximum dynamic aperture when space charge forces are included. As mentioned above, the regions most sensitive to pertur- bations from the 5th integer resonaces are 2 5 n 5 3 for v, = 28.200 and n 2 4 for the vy = 29.200 resonance. The plateau in Figure 4 shows that all particles having initial amplitudes in these ranges survived for the requested 30K turns and that, at the level of 30K turns, their stability is not limited by the 5th integer resonances. 0.17 ’ I I I 1 0.18 0.19 0.20 0.21 0.22 vz 1 0‘ 8-PW4-a ‘ O ~  1 o2 10‘ 0 5 0 A 0 ~ 20 0 Figure 4. Comparison of dynamic aperture with(dashed) and without(so1id) space charge at d = 0.0%. References [l] M. Bassetti and G.A. Erskine, “Closed expression for the electrical field of a two-dimensional Gaussian charge”, CERN-ISR-TH/80-06,(Unpublished). 121 Y. Okamoto and R. Talman, “Rational Approximation of the Complex Error Function and the electric field of a Two-dimensional Gaussian Charge Distribution”, Report CBN 80-13, Cornell University, Sept. 1980. [3] L. Schachinger and R. Talman, “Manual for the Pro- gram Teapot”, Cornel1 Univ., (November 20,1994). [4] E. Keil, CERN Accelerator School, Rhodes, Greece, Sept. 20 - Oct.1, 1993, Report CERN SL/94-78 (AP). [5] D. Trbojevic et al., “Alignment and Survey of the Ele- ments in RHIC”, these proceedings. [SI J. Wei et aI., “FieId QuaIity EvaIuation of the Super- conducting Magnets for the Relativistic Heavy Ion Col- lider” , these proceedings. Figure 3. “Leafprint” at  6 = 0.0% using a finer mesh to explore the 5th integer resonances. 

Simulation of the space charge effect in RHIC

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Simulation of the space charge effect in RHIC

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