Prior, C R

Rees, G H

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Particle Accelerators, 1995, Vol. 48, pp. 251-257Reprints available directly from the publisherPhotocopying permitted by license only©1995 (Overseas Publishers Association)Amsterdam BY: Published under license byGordon and Breach Science Publishers SAPrinted in MalaysiaIMAGE EFFECTS ON CROSSING ANINTEGER RESONANCEG.H. REES and C.R. PRIORRutherford Appleton Laboratory, Chilton, Didcot, UK(Received 17 March 1994; infinalform 6 September 1994)The ISIS proton synchrotron requires improved horizontal and vertical orbit corrections when the space chargetune depressions are sufficient to move some single particle betatron tUBes, but not the coherent tunes, across aninteger resonance. Image effects are studied in an attempt to explain the observations.KEY WORDS: Collective effects resonances1 INTRODUCTIONAt ISIS, the tunes are split by half an integer around the integer 4, and the maximumincoherent space charge tune depressions are "'-'0.4 in both planes. The performance atthese space charge levels is very sensitive with respect to the horizontal and vertical closedorbit corrections, and in particular, to the fourth harmonic corrections..The transverse space charge forces are directed towards the centre of charge of a beam sothe closed orbit excitation terms should not directly affect a single particle's motion aroundits beam centre. Thus, if the equations of betatron motion are written with respect to thecentre of charge, and if the effect of image forces is neglected, the equations should showno dipole error excitation and there should be no effect on crossing an integer resonance.The beam centre, itself, is affected, but not significantly until the coherent tune approachesthe integer.Image effects change this picture. Coherent image effects occur when a beam is not at thecentre of its vacuum enclosure, and the resulting forces are not, in general, uniform alongan axis of displacement. Incoherent image effects occur for both centred and off-centredbeams, with the forces varying across the beam cross-section. The magnitude of the imageforces, and their effect when crossing an integer resonance, are the subject of this paper.2 BETATRON EQUATIONSThe following nomenclature is introduced, and an approximation then given for the singleparticle betatron equation:251252 G.H. REES and C.R. PRIORXj is the transverse betatron motion of the j-th particle,Xc is the transverse betatron displacement of the beam centre,xj' is the second derivative of X with respect to fJ(= s/R),s is the path length measured along the synchrotron orbit,R is the mean radius of the machine,Q is the unperturbed betatron tune in the (x, s) plane,8Qj is the direct transverse space charge tune depression of the j -th particle,F (fJ) are the closed orbit excitation terms,Fcj (fJ) is the coherent image excitation term for the j-th particle, andFij (fJ) is the incoherent image excitation term for this particle.xl' + Q2Xj = 28QjQ(xj - xc) + Fij(fJ)(xj - xc) + F(fJ) + Fcj(fJ)xc (1)Similar equations may be written for all beam particles, and an average taken:X~ + Q2xc = 0 + F(fJ) + (Fcj«())}xc (2)Subtracting (2) from (1), setting Fc«()) = (Fcj (fJ)), and including the Fij «()) effect in 8Qj,gives:(Xj - xc)" + Q(Q - 28 Qj )(Xj - xc) = [Fcj (fJ) - Fc«())]xc (3)If the coherent image terms are neglected, the right hand side of equation (3) becomeszero, and there is no effect on (Xj. - xc) when crossing an integer resonance, as discussed.If the terms are included, their magnitude is of interest. It may be shown that Fcj «()) hasthe form of a constant plus terms in y2, y4 ... , where y is the transverse particle motionorthogonal to x. Improving the closed orbits reduces the excitation terms in (3), and is theprobable reason why ISIS requires improved closed orbits when operating at maximumintensity.3 APPROXIMATIONS FOR COHERENT AND INCOHERENT IMAGE TERMSA first approximation is due to Laslett, l who treats the beam as a line charge between twoinfinite parallel plane conductors. The resultant field is Ec for a coherent displacement,xc, and Ei for incoherent motion, Xj, with Ec and Ei determined from an infinite series ofimages:Ec = [1l'2 /4h2]AXcEi = [1l'2/ 12h2]A(xj - xc)IMAGE EFFECTS AT INTEGER RESONANCE 253where h is the half spacing of the parallel plates, and 'A is the line charge density of thebeam.These may be compared with the direct incoherent space charge force, E, for a beam ofelliptical cross-section of semi-axes, a and b, and of uniform transverse density (the factor1/41l'80 for MKS units is omitted from the expressions):E = [4/b(a + b)]'A(xj - xc)Ec/ E = [b(a + b)/ h2][1l'2 /16]xc/(xj - xc)Ei/E = [b(a + b)/ h2][1l'2 /48]Coherent and incoherent image coefficients, ~1, and 81, respectively, are then defined:~1 = [1l'2/ 16] = 0.617£1 = [1l'2/48] = 0.206(4)(5)A better approximation is to take a beam of finite size between the parallel plates, andthough actual beams have elliptical cross-sections, a rectangular beam ofuniform transversedensity is of interest as it results in analytical solutions for the image terms. The followingcentral image coefficients, for such a beam, of half width, a, and half height, b, are obtainedby an extension to the previous method of summing an infinite series of images:~1 = [1l'h/4a] [coth (1l'a/2h) - cosech (1l'a/2h)]£1 = [1l'h/4a][(2h/1l'a) - cosech (1l'a/2h)](6)(7)The formulae are identical to those published elsewhere for the case of a zero heightbeam. Further image coefficients may be obtained for the y-extremities of the beam:~e = [1l'h/8a] [coth(1l'a/ h) - cosech(1l'a/ h)]£e = [1l'h/8a][(h/1l'a) - cosech(1l'aj h)](8)(9)In the limit, as (a) tends to zero, it may be shown that equations (6) and (8) both reduceto equation (4), and similarly, equations (7}and (9) reduce to equation (5). Taking a valueof (hja)2 = 2, from ISIS, gives the following coefficients:~1 = 0.560; ~e = 0.447£1 = 0.179; £e 0.128These indicate a significant variation of the coherent and incoherent, x-direction, imageforces along the y-axis of the beam.254 G.H. REES and C.R. PRIORFurthermore, in the y-direction, there are additional image forces, and it may be shownthat these orthogonal forces contribute some incoherent, non-linear self focussing. The fieldalong the major (or minor) axis of the beam may be written, for constants nand m:El-y = -x~(n(Yj - Yc) + m(Yj - Yc)2 + ... ) (10)Neglecting the higher order forces, a transverse incoherent image term may be defined, ofthe form:(11)4 COMPUTATIONSThe interiors of the ISIS chambers do not have the previously assumed parallel plategeometry. They are square or rectangular and the ISIS beam cross sections are circularor elliptical. Detailed computations are then required to find the space charge fields.Results have been obtained by C. Prior2 for a circular beam of uniform transverse densityand radius, a, in a square cross section chamber, of half side length, d, with (dja)2 = 2.The computer program uses a Poisson solver based on finite element techniques, with themesh as shown in Figure 1.6.25(":70.0-6.25cm -+--~----'--~--T----'---_....L.-_--"--------'''---1-6.25cmO.GFIGURE 1: Computer mesh for horizontally displaced beam in ISIS.IMAGE EFFECTS AT INTEGER RESONANCEUsing image coefficients to represent the results, these are:~1 = 0.595; ~e 0.30681 < 0.075; 8e = 0255(12)The coherent image coefficients are not very different from those derived in Section 3,except for a greater variation along the y-axis of the beam. On the other hand, the incoherentimage terms, 81 and 8e , for the circular beam, are reduced and are significant for £1 only,for large x amplitude particles; £..L becomes the important incoherent term.These representative image terms for ISIS may be used to estimate the scale of the effectsfor some incoherent crossings of the resonances, Qx = 4 and 2Qx = 8.5 IMAGE EFFECTS ON CROSSING AN INTEGER RESONANCEFor the case of ISIS, the relevant motion is in the horizontal plane, now designated as the(x, s) plane, and so is given by the incoherent betatron equation for (Xj - xc):The terms on the RHS are considered separately, and are given here very approximatelyin terms of 8Q, the average over the appropriate range of s, of the peak incoherent tuneshift:[Fcj(B) - Fc(B)]F..L «())0.5Q 8Q[~1 - ~e]2.5Q 8Q[Yc/d]2(14)(15)5.1 Effect ofHorizontal Closed Orbit Errors on Horizontal Motion in ISISThe excitation term, [Fcj «()) - Fc«())]xc, is finite for some particles due to the variationwith y of the coherent image forces. It is a maximum for those particles with maximumhorizontal and minimum vertical betatron amplitudes, or vice-versa, and is, approximately,for ISIS:0.5Q 8Q[~1 - ~e]xc ~ (4.39)(0.2)(0.289)xc ~ 0.25xcThe maximum value for the incoherent tune shift is 8Qj rv 0.4, sufficient for a particle tocross the resonance, Qx = 4. ISIS has n (= 2) such crossings, of random excitation phaseas the closed orbit changes continually. Assuming a linear rate of change for the tune per256 G.H. REES and C.R. PRIORturn, 8Qj/N = (0.4/1000), the maximum additional motion for a peak fourth harmonicclosed orbit deviation, Xc, of 1 mm, is given by the integer resonance crossing formula: 35.2 Effect ofVertical Closed Orbit Errors on Horizontal Motion in ISISThe effect is largest for those particles with maximum horizontal amplitudes (see later).The excitation term for ISIS is approximately:A fourth harmonic component ofthe vertical closed orbit, yc, becomes an eighth harmoniccomponent of y;, and hence leads to some excitation of the second order resonance,2 Qx = 8. The full bandwidth of the resonance for ISIS may be written:42rr!:le = f (fJx/rr) (4.4(Ycld)2) exp 2i [J-tx - (Qx - 4)] dO / Ro/).e ~ 8.8(yc/d)2/ Q/).e ~ 10-3 for (yc/d)2 = [1.414/45]2/2Here fJx and /-Lx are the lattice horizontal fJ and phase shift parameters, respectively, andthe factor 2 in the denominator is the scaling for the eighth harmonic. Finally, the growthin the horizontal betatron amplitude of some particles, when crossing the resonance, maybe related to the bandwidth, /).e, by:4If the same values for N, nand 8Qj are introduced as in 5.1, it may be seen that a 4%growth in the maximum betatron amplitude of some beam particles is expected at ISIS fora 1.414 mm peak fourth harmonic vertical closed orbit deviation.5.3 Observations at ISISISIS operates with the natural value of chromaticity, so there are momentum dependent tuneshifts in addition to the space charge tune shifts. The lowest tune values are reached by someparticles of maximum positive momentum increase, and these may experience the largesthorizontal betatron growth on resonance crossing. There is some experimental confirmationof this for, at the highest intensities achieved, particle beam loss is not completely containedat the beam loss collector system, but is also found on some outer machine radii at otherazimuths. Loss is reduced by fine tuning of the closed orbits over the acceleration cycle.6 CONCLUSIONIMAGE EFFECTS AT INTEGER RESONANCE 257Image effects are significant on ISIS at space charge levels where there are some incoherenttune crossings of the Qx = 4 and 2Q~ = 8 resonances. Improved closed orbits in both thehorizontal and vertical plane are required to reduce the effects.REFERENCES1. LJ. Laslett, On Intensity Limitations Imposed by Transverse Space Charge Effects in Circular ParticleAccelerators, Proceedings of1963 Summer Study on Storage Rings, BNL - Report 7534, pp. 324-367.2. C.R. Prior, Computations at RAL, 1993, using a 2-D transverse space charge code. Code is reported in the1982 HIP Workshop Proceedings, GSI-82-8, p. 376.3. J. Le Duff, Integer Resonance Cro~sing, 1979 HIP Proceedings, LBL-I0301, p. 310.4. G. Guignard, Selection of formulae concerning Proton Storage Rings, CERN 77-10, p. 69.

Image effects on crossing an integer resonance

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