Microwave instability of coasting ion beam with space charge in small isochronous ring

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1 SLAC-PUB-56 Mirowave intability of oating ion beam with pae harge in mall iohronou ring Yingjie Li* Department of Phyi, Mihigan State Univerity, Eat Laning, MI 4884, USA Lanfa Wang SLAC National Aelerator Laboratory, Menlo Park, CA 945, USA (Dated June, 3) A novel oupling of the tranvere betatron motion to the longitudinal mirowave intability i tudied. Beide the radial oherent dipole mode pae harge field, imulation and theoretial tudie in thi paper how that the longitudinal oherent dipole mode pae harge field due to deformation of beam hape alo play an important role in the iohronou regime, it indue betatron oillation frequenie in temporal evolution of growth rate petra of longitudinal harge denitie, radial offet and energy deviation of loal entroid, it make the intantaneou intability growth rate dependent on both the amplitude and phae of modulation of the above parameter. Thi paper alo introdue a D diperion relation inorporating the Landau damping effet due to finite energy pread and emittane, imulation reult how it i more uitable for aurate predition of the mirowave intability growth rate than the onventional D growth rate formula. Nonlinear beam dynami i alo tudied by o-rotational motion of two-maro-partile model in EäB field, energy pread meaurement and imulation. PACS number: 9.7.Bd, 9..dg I. INTRODUCTION In reent year, high power iohronou ylotron have been onidered for appliation in ientifi reearh, medial therapy, et. A bottlenek that limit the operation of a high power iohronou ylotron i the beam intability indued by the pae harge fore. Gordon [] explained the phyial origin of vortex motion and deformation of beam hape in ylotron. In the lat deade, additional extenive tudie on the pae harge effet in iohronou regime have been done through numerial imulation, experiment and analytial model [-9]. Pozdeyev and Bi propoed their own model and theorie to explain the mehanim of mirowave intability of a oating beam with pae harge in a irular aelerator operating in the iohronou regime [5-7], repetively. The two model only take into aount the radial oherent dipole mode pae harge field due to beam entroid wiggle and longitudinal monopole mode pae harge field originating from line harge denity modulation, none of them diue the effet of longitudinal oherent dipole mode pae harge field on evolution of beam intability due to beam hape deformation. The perturbation wavenumber k in Pozdeyev model and growth rate formula [5, 6] i defined in the inuoidal radial entroid offet funtion of a beam with uniform harge denity, while in Bi model and growth rate formula [7], the perturbation wavenumber k i defined in the line denity modulation funtion, atually the growth rate petra of the line harge denitie and radial entroid offet are different, the relation and interation between the evolution of two petra of different meaning in phyi are not diued in the two model. The miing term of unperturbed line denity in the alulation of radial oherent pae harge field in [7] alo make the growth rate formula not ompatible with the aling law with repet to beam intenity oberved in imulation and experiment [3, 5, 6]. Both model ue D onventional growth rate formula derived exluively for a mono-energeti beam, thi may lead to the overetimation of intability growth rate, epeially for the perturbation of hort wavelength, beaue the Landau damping effet aued by finite energy pread and emittane are all negleted. Simulation tudie in thi paper how that the intantaneou mirowave intability growth rate in iohronou regime are dependent on both the modulation amplitude and phae of line harge denitie, radial offet and mean energy deviation of loal entroid, the temporal evolution of growth rate petra of the above parameter are uually haraterized by a betatron oillation uperimpoed on exponential growth. Thee phenomena annot be explained by the two model with onventional D mirowave Work upported in part by US Department of Energy under ontrat DE-AC-76SF55.

2 intability growth rate formula whih an only predit a pure exponential growth. A theoretial diuion in thi paper explain that the above novel beam behavior primarily originate from the longitudinal oherent dipole mode pae harge field of the deformed beam, the interation and orrelation of temporal evolution of harmoni petra between line harge denitie, radial offet and energy deviation of loal entroid are alo revealed by a et of longitudinal and radial equation of motion taking into aount both the radial and longitudinal oherent dipole mode pae harge field. To predit the mirowave intability growth rate more aurately, thi paper introdue and modifie a D diperion relation [] whih inorporate the Landau damping effet ontributed from finite energy pread and emittane. It an explain the uppreion of mirowave intability growth rate for hort perturbation wavelength and predit the fatet-growing wavelength. A imple example verifiation of Cerfon theory of drift veloity in the Eä B field [9] uing two-maro partile model i alo provided in thi paper, it imply explain the harateriti binary luter merging phenomenon in iohronou ring; the imulation and experimental tudie of energy pread evolution of a long oating bunh how the energy pread of luter hange lowly at large turn number, it may reult from nonlinear advetion of the beam in the Eä B veloity field. Thi paper i organized a follow. Se. II give a brief introdution to Small Iohronou Ring (SIR) and imulation ode ued. Simulation and theoretial tudie of the temporal evolution of beam parameter affeted by both longitudinal and radial oherent pae harge field are provided in Se. III and IV, repetively. The limit of onventional D growth rate formula i pointed out and a modified D diperion relation with Landau damping effet for a Gauian beam model i introdued in Se. V. Se. VI diue the nonlinear beam dynami by two-maro-partile model, energy pread meaurement and imulation. II. SMALL ISOCHRONOUS RING AND CYCO To imulate and tudy beam dynami of high power iohronou ylotron, a low energy, low beam Small Iohronou Ring (SIR) wa ontruted between -4 at the National Superonduting Cylotron Lab (NSCL) at Mihigan State Univerity (MSU) a a thei projet for two graduate tudent and wa in operation until [3-4]. The Small Iohronou Ring i deigned to tudy pae harge effet of H + ion beam with a typial kineti energy of kev, beam urrent of 5-5 ua, bunh length of 5 m 5.5 m, the radial and vertial tune are.4 and., repetively, it bare lip fator i ä -4. The Small Iohronou Ring onit of a multi-up Hydrogen ion oure, injetion line and torage ring. The ion oure an produe three peie of Hydrogen ion and an analyzing dipole magnet i ued to elet the H + ion whih are uually ued in the experiment. The H + ion beam with deired beam length an be produed by a hopper and it Courant-Snyder parameter may be mathed to the torage ring by an eletrotati quadrupole triplet. The torage ring ha a irumferene of 6.58 meter, it mainly onit of four idential flat field bending magnet with edge fouing, and the 6 o pole fae rotation angle of eah magnet provide both vertial fouing and iohronim. After injetion to the torage ring by a pair of fat puled eletrotati infletor, the bunh an oat in the ring up to turn. There i an extration box loated in the drift line between the nd and 3 rd bending magnet, a pair of fat puled eletrotati defletor in the extration box an kik the beam either up to a phophor reen above the median ring plane, or down to the fat Faraday up below the median ring plane. The phophor reen and fat Faraday up are ued to monitor the tranvere and longitudinal beam profile, repetively. We an alo perform energy pread meaurement if the fat Faraday up aembly i replaed by an energy analyzer aembly. CYCO [3] i a 3D Partile-In-Cell (PIC) imulation ode that i developed by Pozdeyev to tudy the beam dynami with pae harge in iohronou regime. It an numerially olve the omplete and elfonitent ytem of ix equation of motion of harged partile in a realiti 3D field map inluding pae harge field. Beaue of large apet ratio between the vauum hamber width and height of the torage ring, the ode only inlude the image harge effet in the vertial diretion, the retangular vauum hamber i implified a a pair of infinitely large ideally onduting plate parallel to the median ring plane. III. SIMULATION STUDIES ON EVOLUTIONS OF BEAM PARAMETERS IN ISOCHRONOUS RING

3 In order to aquire detailed information and propertie of mirowave intability evolution in iohronou ring aurately and omprehenively, in thi etion, we will preent the imulation method and reult of exploring the beam parameter pae in iohronou regime baed on Fat Fourier Tranformation (FFT) tehnique. A. Effet of modulation trength and phae on intantaneou intability growth rate Firt, a default mono-energeti and traight H + bunh of beam intenity I = ua, kineti Energy E k = 9.9 kev, bunh length τ b = 3 n ( 4 m), radial and vertial emittane ε x, = ε y, = 5 p mm*mrad wa reated by CYCO. The initial bunh ha a uniform ditribution in both 4D tranvere phae pae (x, x, y, y ) and longitudinal harge denity. Then variou beam parameter modulation are reated a below: () Line harge denity modulation funtion: S = Λˆ / Λ ^ Λ ( k ) = Λ o( kz ) () with modulation trength =,.5,. and wavelength l = π/k =. m, Λ i the unperturbed line harge denity; () Radial entroid offet modulation funtion : (a) x = a o(k z); (b) x = - a in(k z) where a =,.5 mm,. mm,. mm,.5 mm,. mm and l = π/k = l =. m. (3) Coherent loal kineti energy deviation funtion: (a) E = ; (b) E /E k = x /R. where E k = 9.9 kev, R i average ring radiu. The eond relation between E and x atifie iohronou ondition of low energy SIR beam. The ode CYCO wa run for a beam of initial ditribution with ome ombination of the above modulation parameter, the intantaneou intability growth rate of line harge denity perturbation magnitude of wavelength λ = m at turn are alulated by performing FFT analyi for the entral part of beam profile between turn and turn a hown in Figure. τ - ( - ) x S =, x =, E = S =.5, x = a o(k z), E = S =., x = a o(k z), E = S =.5, x = -a in(k z), E = S =., x = -a in(k z), E = S =., x = -a in(k z), E = *x /R*E k a (mm) FIG.. (Color) Effet of modulation trength and phae of variou beam parameter on intantaneou beam intability growth rate. Simulation reult in Figure indiate that both amplitude and phae of line harge denity modulation trength, radial entroid offet, and oherent loal energy deviation may affet the intantaneou intability growth rate. B. Betatron oillation in long term intability growth urve The imulation tudy of long-term linear tage of mirowave intability wa arried out by lowering the beam intenity in Se. A to I = ua and keeping other beam parameter unhanged. Figure how the evolution of top view of beam profile. The beam move from left to right. We an oberve the beam hape i deformed due to pae harge fore. 3

4 FIG.. (Color) Evolution of top view of beam profile. FFT analyi wa performed for petral evolution of line harge denitie, radial offet and energy deviation of entroid with repet to longitudinal oordinate z, the analyi reult of ome hoen perturbation wavelength are hown in Figure 3 - Figure 5, repetively. It i learly hown that there are many oillation uperimpoed on the exponential growth urve in the figure. Beaue the radial betatron tune of SIR beam i.4, the betatron oillation period i about /.4 7 turn, it i eay to judge that thee oillation are indued by the oupling of betatron oillation. It i the firt time to learly oberve thi oupling to our knowledge. Atually, an indiation of imilar oillation an be found in the Fig. 9 of [6], where there i a fat intability growth due to high beam intenity. The beam behavior in Se. A and B annot be explained by the exiting model and theorie, in fat, they are all aued by the longitudinal dipole mode pae harge field due to beam deformation whih will be explained in Se. IV. Normalized line harge denity λ=6.67 m λ=5. m λ=4. m λ=.857 m λ=. m λ=. m Turn # FIG. 3. (Color) Evolution of harmoni amplitude of normalized line harge denity. 4

5 Centroid offet (mm) λ=6.67 m λ=5. m λ=4. m λ=.857 m λ=. m λ=. m Turn # FIG. 4. (Color) Evolution of harmoni amplitude of radial entroid offet. Energy deviation (ev) λ=6.67 m λ=5. m λ=4. m λ=.857 m λ=. m λ=. m Turn # FIG. 5 (Color) Evolution of harmoni amplitude of energy deviation. IV. THEORETICAL STUDIES OF EVOLUTIONS OF BEAM PARAMETERS IN ISOCHRONOUS RING In iohronou ring, the longitudinal line harge denity gradient of a oating bunh may indue energy deviation and the aoiated entroid offet. For impliity, we aume both entroid offet x (z, and line harge denity Λ (z, onit of only a ingle harmoni omponent negleting the nonlinear oupling between the hoen omponent with other omponent. For general purpoe, we alo aume there i no orrelation between wavenumber k of x (z, and k of Λ(z,. Uing the relation of phae Φ = k - ωt + φ = k (z + v - ωt + φ = kz - ωt + kv t + φ, where φ i initial phae at t =, =, v = β i the veloity of on-momentum partile, then the loal beam entroid x (z,, line harge denity Λ (z, and beam intenity I (z, an be expreed a: x z, = aˆ e ( i( kz ω t+φ ) () 5

6 Λ( ˆ ( kz ω t+φ) z, = Λ + Λ( z, = Λ + Λe i (3) I( z, = I + I ( z, = I + Ie ˆ i( kz ω t+φ) (4) where φ ( = k v t + φ,, φ( = kv t + φ, I ˆ = Λ ˆ β, and amplitude â, Λˆ, Î are all real number. ω and ω are perturbation frequenie of line harge denitie and radial entroid offet, repetively. A. Longitudinal dipole mode oherent pae harge field and impedane The entroid wiggle produe not only the radial dipole mode oherent pae harge field E x a pointed out by [6], but alo it longitudinal ounterpart () E a hown in Figure 6. FIG. 6. (Color) Spae harge omponent of loal entroid with oordinate (x, z) at time t. The total longitudinal pae harge field on a loal beam entroid an be approximated a: () E ( z, = E ( z, + E () ( z, (5) where the firt and eond term in Eq. (5) are longitudinal pae harge field of monopole and dipole mode generated by line harge denity modulation and entroid wiggle, repetively. If we adopt Pozdeyev irular beam model of radiu r in free pae [6], the firt term of Eq. (5) an be alulated by etting r = and r w = in Eq. (7) of [7] a: E ˆ i( kz Λe z, = i πε r ωt + φ ) () ( k [ kr K ( kr )] (6) The orreponding longitudinal monopole mode pae harge impedane in low energy and hort wavelength limit i [6]: ZR Z ( k) = Z, ( k) = i [ kr K( kr )] k r β (7) Where Z = 377 Ω i the impedane in free pae, r i beam radiu, k i harge denity perturbation wave number, R i average ring radiu, β i relativiti peed fator. Aording to Eq. (4) of [6], the radial pae harge field on a partile at oordinate (x, z) an be etimated in SI unit ytem a: 6

7 E Λ x, z, = [ x x ( z, k r K( k r )] πε r (8) x ( Where Λ i the unperturbed part of line harge denity, x (z, i the time-dependent loal radial entroid offet. Aording to Panofky-Wenzel theorem, the longitudinal dipole mode oherent pae harge field on thi partile an be alulated a: () E ( x, z, Ex ( x, z, Λ x ( z, E ( x, z, = x = x = kk( kr ) x x z πε r z (9) Let x = x in Eq. (9), with Eq. (), the longitudinal pae harge field of dipole mode beome: () Λaˆ i( k z ωt + φ ) E ( z, = i k K( kr ) e πε r The dipole mode pae harge wake potential over ring irumferene C [7, 8] i: () V = E ( z, C = Z I x ( z,, () () From Eq. () and Eq. (), the longitudinal dipole mode pae harge impedane an be alulated a: Z R Z (, = i k K kr ) βr () Figure 7 how the longitudinal monopole and dipole mode pae harge impedane of a irular H + beam with emittane of 5 π mm*mrad and kineti energy of 9.9 kev in SIR. x 7 x (Ω) Z,SC (Ω m - ) Z,SC Monopole mode Dipole mode λ (m) FIG. 7. (Color) Longitudinal monopole and dipole mode pae harge impedane. B. Radial and longitudinal equation of motion Aording to Eq. (5), (6) () and [7], the radial and longitudinal equation of motion of loal beam entroid are: 7

8 " voh ( k ) x + x R δ = R (3) ˆ i ' eλe δ = i πε r ( k ωt+ φ ) z'= ηδ [ kr K( kr )] eλ aˆ i k K ( kr ) e kβ E πε rβ E i( k ω t+ φ, ) (4) (5) Where δ i the frational momentum deviation of loal beam entroid, n oh i the oherent radial betatron tune, η i lip fator, E i total energy. The prime tand for differentiation with repet to. If δ ( = ) =, then δ an be alulated by integration a: δ ˆ i( k ω t+ φ ) eλe eλ a i( k ω t+ φ, ) (, = [ kr K( kr )] k K( kr ) e πε 4 r k β E πεr β E ˆ (6) In the firt order approximation, if entroid offet amplitude i mall, the eond term on RHS of Eq. (6) due to longitudinal dipole pae harge field an be negleted, Eq. (3) beome: x " v + ( k R oh ) x eλˆ e = πε r i( k ωt+ φ ) [ kr K( kr )] k β ER (7) We an ee Eq. (7) deribe a fored harmoni oillation whih reult in k k, ω ω and Im(ω) Im(ω ), then in firt order approximation, the growth rate petra of line harge denity modulation and loal entroid offet are approximately the ame. If the eond term on RHS of Eq. (6) i omparable to the firt term thu annot be negleted, then Eq. (7) will beome a fored nonlinear eond order differential equation and will not be diued in thi paper due to it omplexity. C. Betatron oillation in evolution of beam parameter If the longitudinal dipole mode oherent pae harge field i taken into aount, then we may define a time-dependent equivalent longitudinal monopole pae harge field longitudinal monopole pae harge impedane [ E [ Z ( k, ] [ E ( z, ] (), eq by Eq. (8) -Eq. () a below: () ( z, ] eq + eq and equivalent () () = E ( z, E ( z, (8) ( ) ˆ i( kz ω t+φ) ( z, C = Z, ( k) Ie E (9) E ˆ () () i( kz ω t+φ ) ( z, C = Z, ( k) Ix ( z, = Z, ( k) Ia e ( ) ˆ i( kz ω t+φ) ( z, ] eqc = [ Z, ( k, ] eqie [ E () Eq. (8) - Eq. () give: 8

9 In the ae of k k /, [ Z ( k, ] I Im( ω ω) t i[(k ) z Re( ω ω) t+ (φ φ)] [ Z, (, )] =, ( ) +, ( )ˆ k SC k t eq Z k Z k a e e () Iˆ, eq will depend on longitudinal oordinate z, the beam dynami i nonlinear, thi ompliated ae will not be diued in thi paper. In the ae of k = k/, [ Z ( k, ], eq will be independent of longitudinal oordinate z, the dynami i linear; Uually, Re(ω ) ω β, Re (ω), where ω β i the angular betatron frequeny. Then Eq. () an be implified a: [ Z, ( k, ] eq iω t B ( t ) t β = Z ( k)[ + Ae e ] (3), where ˆ A, (4) Z, ( k ) a I i( φ, φ ) = e Z ˆ, ( k ) I B( = Im[ ω ( ω ( ]. (5) Note A i a time-independent omplex number, while B ( i real and funtion of time t. Aording to Eq. (3), the amplitude of perturbed line harge denity an be expreed a: t ˆ Im[ ω ( t )] t ˆ τ ( t ) Λ ( = Λe = Λe (6) When energy pread, emittane and loal entroid offet are mall, modulation wavelength of line harge denity are greater than beam diameter, the intability growth rate an be etimated by the onventional D mirowave intability formula [6]: ηeikrz( k) τ ( k) = ω i πβ E (7) Where ω i the angular revolution frequeny of on-momentum partile, e i unit harge, I i unperturbed beam intenity. Z (k) i the longitudinal monopole mode pae harge impedane whih i given by Eq. (7) If Z, ( k) of Eq. (7) i replaed by [ Z, ( k, ] eq of Eq. (3), the time-dependent growth rate beome:. τ iω t ( The amplitude of line harge denity perturbation in Eq. (6) beome: B ( t ) t β k, = τ ( k ) Re{[ + Ae e ] } (8) t Re{[ B ( t ) t iω t + Ae e β ] } τ ( k ) Λ ) ˆ ( t = Λe (9) 9

10 If aˆ =, predited by Eq. (7). A =, ( k, = τ ( k ) τ, Λ ( = Λˆ e t τ ( k ), it i a purely exponential growth a B( t If aˆ A and, Ae <<, beaue B( = Im[ ω ( ω ( ] Im( ω ) = τ ( k ), Eq. (9) an be approximated a: t τ ˆ τ φ Λ( Λe [ + A e o( ω β t + φ, t )] (3) Eq. (3) i an exponential growth funtion modulated by betatron frequeny ω β, together with Eq. (7) and Eq. (6), the betatron frequenie in evolution of line harge denitie, entroid offet and energy deviation in Figure 3 - Figure 5 an be undertood. From Eq. (3) and Eq. (4), we an ee the intantaneou mirowave intability growth rate depend on the urrent modulation trength I ˆ / I, the entroid offet amplitude â, the phae angle φ, andφ thi harateriti property of beam profile, evolution ha already been oberved in Figure of Se. III. Explanation of the dependene of intantaneou intability growth rate on oherent energy deviation E /E k i not eay, it require a more ompliated formula than Eq. (7) whih i only valid for D mono-energeti beam, thi will not be diued in thi paper due to it omplexity. A. Limit of D growth rate formula V. D DISPERSION RELATION The tranformation of longitudinal oordinate z i [, ]: ' z = z + R x + R x + (3) 5 5 R56δ where R 5 (), R 5 (), and R 56 () are tranfer matrix element, x and x are radial oordinate and veloity lope, repetively; z i longitudinal oordinate with repet to bunh enter, δ = p/p i frational momentum deviation. For a beam in torage ring, R 56 () at = C i related to ompation fator α by [3]: R56 ( C) D( ) D( ) = + = d =< > C γ C R( ) R( ) α (3) where D() i loal diperion funtion, R() i loal urvature of orbit, C i the ring irumferene. We an ee z i determined by R 5, R 5, and R 56 together. In the formalim of [6] in whih a onventional D growth rate formula Eq. (7) i adopted, only the ontribution of ompation fator α or R 56 to derivation of oherent lip fator η i onidered. For a beam with mall momentum pread δ (old beam) and finite emittane, the term R 5 x +R 5 x may uppre the intability growth rate of hort wavelength thu hould not be negleted. B. D diperion relation For a hot beam, Landau damping effet due to emittane and energy pread are important, and a multidimenional diperion relation inluding both longitudinal and tranvere dynami i needed [,4,5]. For a oating ultra-relativiti eletron beam model whoe initial equilibrium ditribution in radial and longitudinal phae pae are Gauian funtion, the D diperion relation for CSR intability in torage ring in CGS unit wa derived in Appendix B of [] a: L

11 ir ( k / ) [ o( / R)] ( k / ) enb R σ x ν ν σ ν µ ν δ = kz( k) de [ in ] e v γ (33) v R Where r e = e /m i laial eletron radiu in CGS unit, γ i relativiti energy fator, ν i radial betatron tune, R i average ring radiu in mooth approximation, Z(k) i CSR impedane with unit m -, σ x i the RMS beam radiu, σ δ i the RMS unorrelated energy pread of ultra-relativiti eletron. Eq. (33) i an integral equation whih determine relation between k and µ. For a fixed k, µ an be olved numerially. If µ i real and µ>, the CSR intability grow at a rate of τ - = µβ. Eq. (33) an be tailored to tudy mirowave intability of non-relativiti H + beam of SIR indued by pae harge after ome modifiation. Eq. () and Eq. (4) of [] hould be modified a: dz x x δ = + d R ( ) R ( ) γ (34) R D ( ' ) D ( ' ) 56 ( ) d ' d ' = R ( ' ) R ( ' ) + γ (35) Eq. (B4) of [] hould be modified a: R (, ') = ( ') ( )( ν ν γ 56 ') (36) R R ν( ') R ν( ') ' ) = [( ') in( )] ( )( ') + in[ ] ν ν R ν γ ν R ( 56 3 (37) The impedane and laial partile radiu hould be modified a: 4πε m e Z = Z CSR ( k, ) Z, SC ( k ) C β m + H (38) r e e = m e CGS e 4πε m + H SI (39) Where m e- and m H + are the ret mae of eletron and H + ion, repetively. Finally, the D diperion relation in SI unit ytem for SIR beam beome: ie = βγm e Λ µ R ν oh kz, SC ( k) de [ in ] H C + ν oh γ v oh R ( kσ x / ν in ) [ o( ν in / R)] [ kσ δ (/ ν in / γ ) ] (4) Where Λ i the unperturbed line harge denity, i the peed of light, Z i the longitudinal pae harge impedane in Ω, ν and ν in are oherent and inoherent radial betatron tune modified by pae harge effet, repetively., SC

12 Due to large ratio between vauum hamber width, height and beam radiu r, we an ue Pozdeyev model [6] of uniform irular beam with entroid wiggle in free pae to alulate the radial pae harge field and modified tune. By mooth approximation and the equation of oherent and inoherent radial motion [6], the pae-harge-modified oherent and inoherent betatron tune an be derived eaily a: α oh α inoh ν oh, ν inoh (4) Where α oh = ε m eρ H +ω [ kr K ( kr )], α in = e ρ ε m ω + H (4) ε = F m - i permittivity of free pae. ω i angular revolution frequeny of on-momentum partile, ρ i volume harge denity, K (x) i the modified Beel funtion of the eond kind. Thu if we neglet the tranvere dynami ontributed from R 5 x and R 5 x, the D lip fator i: D eρ η SC = α [ kr K( kr )] oh = (43) voh γ ε m +ω H whih i eentially the ame a Eq. () derived in [6]. Uing Eq. (3)(37) (4-43), the D lip fator an be etimated a: η D SC = R56 ( C) 3 D = in( πν 3 oh) ηsc C ν oh γ πν oh (44) For a mono-energeti beam, if we neglet the exponential damping term and the eond term in the quare braket of integrand in Eq. (4), and ue the equalitie: de µ = µ, ν oh γ = η D SC (45) the mirowave intability growth rate τ - (k) = µβ predited by Eq. (4) an be redued to Eq. (7). We an ee that the term R 5 x and R 5 x an affet the mirowave intability in iohronou ring in two way: (a). Eq. (44) how it an enhane the intability by inreaing 5% of D lip fator derived in [6] (b). It provide Landau damping by the exponential uppreion fator in Eq. (4).

13 Suppreion fator σ E = ev σ E =5 ev σ E = ev σ E =5 ev σ E =5 ev σ E = ev λ (m) FIG. 8. (Color) Exponential uppreion fator for a beam with urrent of ua, emittane of 5 π mm mrad and variou RMS energy pread. Figure. 8 how the alulated exponential uppreion fator in Eq. (4) for a beam with urrent of ua, emittane of 5 π mm mrad, mean kineti energy of 9.9 kev and variou unorrelated RMS energy pread σ E. We an ee the Landau damping effet are more effetive for maller perturbation wavelength. C. Benhmarking of D diperion relation Note that the D diperion relation Eq. (4) i derived for a Gauian beam model without oherent radial entroid offet and energy deviation, it i only valid for predition of long term mirowave intability growth rate in iohronou ring, not the intantaneou one. We an fit the urve of line harge denity evolution in Figure 3 to get imulated intability growth rate. If we replae t by t = N t T, where N t i the turn number, T i the revolution period of H + ion, Eq. (3) may be expreed a a general fitting funtion below: T Nt τ ( ) ˆ QNt Λ Nt Λe + Pe o( WT N t + Φ) (46) where Λˆ, P, Q, W, Φ, τ are fit oeffiient, /τ i jut the long term intability growth rate. Amplitude (arb. uni λ = (m) τ - = 5.5e+3 (- ) W =.498e+6 ( - ) (a) Data Fitting Turn # Amplitude (arb. uni 5 5 λ = (m) τ - = 7.67e+3 (- ) W =.55e+6 ( - ) (b) 5 Data Fitting Turn # 3

14 Amplitude (arb. uni λ =.86 (m) τ - = 7.e+3 (- ) W =.5e+6 ( - ) () 5 Data Fitting Turn # Amplitude (arb. uni λ = 5 (m) τ - = 5.74e+3 (- ) W =.54e+6 ( - ) (d) Data Fitting Turn # FIG. 9. (Color) Curve fitting reult for growth rate of line harge denitie (a). λ =. m; (b). λ =. m; (). λ =.86 m; (d). λ = 5. m; The urve fitting reult for growth rate of line harge denitie of ome hoen modulation wavelength are how in Figure 9. For beam energy of 9.9 kev, the nominal angular betatron frequeny i ω β =.499 ä 6 radian/eond. We an ee the oillation in the urve are betatron oillation. Figure 9 alo how that for λ >. m, the linear effet of the entroid offet modulation wavelength λ =λ dominate over the nonlinear effet of λ =λ on line denity evolution; while for mall wavelength λ=. m, the nonlinear effet of the entroid offet modulation with wavelength λ =λ on line denity evolution are more notieable. Growth rate ( - ) x Calulated by D formula Calulated by D diperion relation Simulation 4 6 λ (m) FIG.. (Color) Comparion of intability growth rate between theory and imulation. Figure how the omparion of mirowave intability growth rate between theoretial value predited by D formula Eq. (7) with D lip fator expreed in Eq. (43), D diperion relation Eq. (4), and imulation reult by urve fitting for Figure 9. The D formalim overetimate the growth rate of hort wavelength and annot predit the fatet-growing wavelength ine Eq. (7) neglet the Landau damping effet, it alo underetimate the growth rate of large wavelength beaue the lip fator i underetimated a explained in Eq. (44). The D diperion relation ha a better performane than the onventional D formula, epeially on the predition of the growth rate of mall wavelength and the fatet-growing wavelength, beaue it inlude the Landau damping effet. The differene of long term growth rate between theoretial predition and imulation may originate from the emittane and energy pread growth of the oating beam in imulation. 4

15 VI. NONLINEAR BEAM DYNAMICS OF SIR BUNCH When a high intenity uniform long H + bunh of finite length i injeted to SIR, the nonlinear pae harge fore on beam head and tail i very trong, in addition, the bunh may break up into many mall luter only after everal turn of oating due to mirowave intability, the beam dynami are highly nonlinear in thee ae. In thi etion, we mainly diu the two apet of the nonlinear dynami: (a). o-rotational motion in E ä B field, (b). energy pread evolution. A. Co-rotational motion in E ä B field Aume at t =, there are two idential maro-partile plaed on the drift line between two ring bending magnet of SIR with initial ditane d, eah maro-partile ha pae harge Q and ma M whih atifie Q/M = e/m, where e and m are harge and ma of ingle H + ion, repetively. They have the ame initial relativiti veloity β =.33 (orrepond to kineti energy of.3 kev for H + ion), their initial moving diretion are from left to right and are all parallel to the deign orbit. If Q = 8. ä -4 oulomb, d =.5 m, the top view of the two miro-partile are hown in Figure. Thi kind of o-rotational motion an be explained by Cerfon theory of drift veloity in E ä B field. Radial oordinate x(m) Turn Turn5 Turn Turn Turn34 Turn Longitudinal oordinate z(m) FIG. (Color) Co-rotation of two-maro-partile model with d =.5 m. Aume SIR an be implified a an ideal irular torage ring of mean radiu R with a uniform effetive magneti field of trength B eff : mv B eff = (47) er where v i the veloity of on-momentum partile, the magneti field vetor B eff i perpendiular to and point into the paper. At time t the ditane between them i d (, the magnitude of pae harge field E. on eah maro-partile i: Q E. = 4 πε d ( (48) The pae harge field vetor E. i along the line onneting the two maro-partile and point outward. Then eah maro-partile will aquire a drift peed in their enter of ma frame: r r r E. B eff V drift = B eff (49) whih i perpendiular to both B eff and E., it magnitude i: 5

16 V drift E. Q = = (5) B 4πε B d ( eff eff If the trajetorie of two partile are approximated a loed ellipe (atually their trajetorie are ompliated and not loed), their mean ditane i etimated a <d> = (d min + d max ) /, then the angular frequeny of o-rotation an be etimated a: ω o rot. = V drift Q = 3 < d > πε B eff < d > (5) Co-rotation freq. (deg. * turn - ) Simulated freq. Theoretial freq Initial ditane d (m) FIG.. (Color) Co-rotation frequenie of two-maro-partile model with different d. By fitting lope of the urve of o-rotational angle v. turn number in Figure, we an get the imulated o-rotational frequeny. The imulated and theoretial o-rotation frequenie of the two-maro-partile model with Q = 8. ä -4 oulomb, d =. m,.5 m,. m, 3. m, 4. m, and 5. m are hown in Figure. We an ee the imulated and theoretial value math quite well whih verified Cerfon theory, in fat, thi two-maro-partile model may alo help u undertand the harateriti binary luter merging phenomenon in iohronou ring. B. Energy pread meaurement and imulation The evolution of beam energy pread of oating bunh indued by pae harge field play an important role in both linear and nonlinear regime of beam intability. In linear beam dynami, a large energy pread may help uppre the mirowave intability; in nonlinear beam dynami, the energy pread i one of important meaure of aymptoti bunh behavior. In thi etion, we diu the energy pread evolution by both imulation and experimental method. SIR Lab ha deigned a ompat eletrotati retarding field analyzer [9] whih an an aro the beam radially to meaure the energy pread of a bunh at variou hoen number of turn after injetion. The analyzer i roughly a ube with a mm (width) ä 4 mm (heigh entrane lit loated in the middle of the entrane plate. It mainly onit of a retangular houing tube, a retangular high voltage retarding tube, a fine retarding meh ( line per inh, 5% tranmiion rate), a eondary eletron uppreor, and a urrent olletor. The analyzer i intalled under the median ring plane in the extration box, it entrane plate i tilted at an angle with repet to vertial plane to align the analyzer axi parallel to that of defleted beam. A pair of high voltage puled eletrotati defletor i ued to kik the beam down to the energy analyzer when energy pread meaurement i performed. Uually, the energy pread i meaured at three radial poition: one i at the loation of the peak beam urrent, and the other two poition are loe to the beam ore edge on eah ide. 6

17 A H+ ion bunh of length 6 mm, peak urrent 8. ua, kineti energy.3 kev and emittane 3 π mm mrad i ued in the energy pread meaurement. A mono-energeti maro-partile bunh of the above initial parameter with uniform ditribution in both longitudinal line harge denity and 4D tranvere phae pae are alo ued in the imulation tudy by ode CYCO. In the analyi of imulation reult, the radial beam ize i ut into everal mm wide mall bin, the number of maro-partile, mean kineti energy and RMS energy pread in eah bin are alulated and ompared with experimental value. Figure 3 how the imulated and experimental radial lie beam denity. Figure 4 how the imulated top view and RMS lie energy pread at turn 4. Figure 5 how the imulated RMS lie energy pread up to turn 8. Figure 6 how the imulated top view and RMS lie energy pread at turn 3. Figure 7 how the omparion of RMS lie energy pread between imulation and experiment. Note that in thi paper the lie energy pread and lie denity denote all the lie are ut parallel to the longitudinal z diretion intead of the radial diretion that i onventionally ued in FEL. Conidering the harged long bunh i a haoti ytem, a mall differene in the initial beam ditribution may aue huge beam profile deviation at large turn number, we an ee that the imulated radial beam denity profile and RMS lie energy pread math the experimental value within an aeptable range. Radial denity (arb. uni Simulation Experiment Turn Turn Turn Turn 3 Turn 5 Turn Radial oordinate x (m) FIG. 3. (Color) Evolution of radial beam denity. 4 Turn 4 3 Turn 4 x (m) x (m) z (m) σ E (ev) FIG. 4. (Color) Simulated top view (lef and RMS lie energy pread (righ at turn 4. 7

18 Turn Turn Turn 4 Turn 6 Turn 8 8 σ E (ev) x (m) FIG. 5. (Color) Simulated RMS lie energy pread of turn -8. Turn 3 8 Turn x (m) x (m) z (m) σe (ev) FIG. 6. (Color) Simulated top view (lef and RMS lie energy pread (righ at turn 3. Simulation Experiment RMS energy pread (ev) 8 Turn Turn 3 Turn 5 Turn Radial oordinate (m) FIG. 7. (Color) Comparion of RMS lie energy pread. 8

19 Figure 3 - Figure 7 how that when an initially mono-energeti, traight long beam oat in the iohronou ring, the pae harge will indue the longitudinal denity modulation and energy pread, at maller turn number, the energy pread in beam head and tail i muh greater than that of beam ore around beam axi, a turn number inreae, the radial RMS lie energy pread ditribution tend to beome uniform and hange very lowly. At the ame time, the radial beam ize inreae, the beam entroid deviate from the deign orbit. The beam entroid wiggle may alo aue the differene in betatron oillation phae between beam luter (lie), if the beam i long enough, the radial entroid offet of different luter (lie) may be regarded a ditributed randomly and uniformly around the deign orbit, then the meaured RMS lie energy pread at different radial oordinate i jut the denity weighted mean RMS lie energy pread of beam ore of any individual luter (lie) thu i independent of the radial oordinate. Thi an be explained below in Figure 8. FIG. 8 (Color) Sketh of luter and energy analyzer. Figure 8 how the long bunh ha broken up into many idential mall luter (blue oval) whoe entroid ditribute randomly and uniformly around the deign orbit (the red dahed line). If we meaure the RMS lie energy pread at a radial poition a indiated by the olid blak line with arrow, the analyzer will ample lie of different luter. Let aume there are N luter in the whole bunh whih are tagged by ID#,, 3,.N, eah luter ha the ame total partile number and radial harge ditribution profile, if the lie ampled by the analyzer in eah luter ontain n i ( i =,, 3,.N ) harged partile and it RMS energy pread i σ i, the mean kineti energy of all lie are the ame a < E (x) > at large turn number, where x i the radial oordinate of blak olid line with repet to the red dahed line, then the RMS energy pread in the i th beam lie i: n i σ = [ E < E ( x) > ] (i=,, 3, N ) (5) i ni j = j The um of quare of Eq. (5) give: σ n + nσ +... nn σ N =< Ei > < E( x) > = σ E ( x) n + n +... n N (53) nσ + nσ +... nn σ N σ E ( x) = [ ] (54) n + n +... nn 9

20 The RHS of the Eq. (54) i the weighted mean RMS energy pread of the ampled beam lie of different luter ore at a fixed radial oordinate x, if the number of luter i large enough and the radial entroid offet of all the luter are randomly and uniformly ditributed around the deign orbit, it i atually equal to the weighed mean RMS lie energy pread of any given ingle luter ore thu i independent of the oordinate x. In real meaurement, the above ideal preondition are not atified ompletely, there are alway mall energy pread flutuation between different radial meaurement point. The equilibrium value of kineti energy deviation E eq (x) = E eq (x) - E k and radial oordinate x of an offmomentum partile atifie the relation: Ek Eeq ( x) = x (55) R Where E eq (x) i the equilibrium kineti energy and i equal to <E(x)> of beam lie entered at x at large turn number. For impliity, aume the radial beam denity ditribution i uniform, then the RMS energy pread of the equilibrium partile meaured by SIR energy analyzer with an entrane lit of width Γ = mm entered at x an be etimated a: Γ x + Ek Ek Γ σ E eq = [ [ ( x' x)] dx'] = 5. 7eV (56) Γ R 3R Γ x Thi value i proportional to lit width Γ and i independent of x, it i muh le than the aymptoti energy pread of about 5 ev at large turn number. It indiate that the number of partile of equilibrium energy only aount for a mall fration of total partile in a beam lie. The aturation of RMS lie energy pread of luter in SIR i an indiation of formation of nonlinear advetion of the beam in the E ä B veloity field. Aume an ideal dik-haped luter oat in an iohronou ring of effetive uniform magneti field B eff, the E ä B eff veloity field at any point of median plane inide the luter i along the tangential diretion in the ret frame of the luter, no partile tay at the beam head (tail) forever, then the energy pread within a given beam lie of mm width at any oordinate x will not build up with time ignifiantly. During the binary luter merging proe, the total harge and ize of new luter grow at the ame time, the mean harge denity doe not hange oniderably whih may reult in the aturation of mean RMS lie energy pread averaged along radial oordinate. VII. CONCLUSION In thi paper we diued the limit of two beam model on mirowave intability and explore novel beam dynami with pae harge in iohronou regime. By imulation tudie, we found the intantaneou mirowave intability growth rate depend on not only the amplitude of Fourier omponent of line harge denitie, oherent loal entroid offet and energy deviation, but alo their phae; the betatron oillation in the growth rate petra of above beam parameter are diovered, thee phenomena were explained by longitudinal dipole mode oherent pae harge field with the onept of timedependent equivalent longitudinal pae harge impedane for the ae of k = k/. Thi paper how that for a long oating bunh of pae harge in iohronou regime, both the radial and longitudinal oherent pae harge field hould be taken into aount in the beam dynami. The growth rate petra of line harge denitie, loal entroid offet and energy deviation may interat eah other and evolve in a elfonitent way. In addition, by introduing a modified D diperion relation, the Landau damping effet provided by finite emittane and energy pread are inluded in the beam dynami, thi i epeially important for the aurate predition of intability growth rate of hort perturbation wavelength and fatet-growing wavelength. Simulation reult for long term intability growth rate how that the D diperion relation ha a better performane than the onventional D intability growth rate formula. An aurate predition of intantaneou intability growth rate in iohronou regime require a more ompliated D diperion relation whih derived for a beam with initial modulation in both oherent radial offet and energy deviation of loal entroid, it will be diued in the future work.

21 We ue a two-maro-partile model to tet the theory of drift veloity in E ä B field propoed by Cerfon. The good agreement between imulated and theoretially predited o-rotation frequenie provide a good example to validate Cerfon theory and aumption; thi model may alo help u undertand the approahing tage of the harateriti binary luter merging phenomenon in iohronou ring. The meaured and imulated RMS lie energy pread and radial denity profile of a long oating bunh reahe aeptable agreement, at maller turn number, due to finite bunh length, the energy pread of beam head and tail dominate over that of the entral part of bunh; while at large turn number, the randomly ditributed luter entroid offet tend to make radial energy pread ditribution of whole bunh uniform; the meaured energy pread i jut the denity weighted mean RMS lie energy pread of any ingle luter ore, it aturation behavior indiate the formation of nonlinear advetion of the partile due to the E ä B veloity field in eah luter, during the ueive binary luter merging proe of a long bunh, the mean RMS lie energy pread averaged over radial oordinate of merger remnant of parent luter pair with different ize reahe an aymptoti value. ACKNOWLEDGEMENTS We would like to thank the guidane of Prof. F. Marti and T. P. Wangler, we are alo grateful to Y.C. Wang, S. Y. Lee, K. Y. Ng, G. Stupakov, E. Pozdeyev, A. W. Chao, R. York, M. Sypher, V. Zelevinky, J. Baldwin, and J. A. Rodriguez for their fruitful diuion and uggetion. Thi work wa upported by NSF Grant # PHY 667. REFERENCE [] M. M. Gordon, Proeeding of the 5th International onferene on Cylotron (Oxford, United Kingdom, 969), p. 35. [] S. A. Bogaz, Proeeding of the IEEE Partile Aelerator Conferene (San Franio, CA, U.S.A, May 6-9, 99), p. 85. [3] E. Pozdeyev, Ph.D. thei, Mihigan State Univerity (3). [4] J. A. Rodriguez, Ph.D. thei, Mihigan State Univerity (4). [5] E. Pozdeyev, J. A. Rodriguez, F. Marti, R. C. York, Proeeding of Hadron Beam (Nahville, Tenneee, U.S.A, 8), WGA3, p. 57. [6] E. Pozdeyev, J. A. Rodriguez, F. Marti, R.C. York, Phy. Rev. ST Ael. Beam 5, 54 (9). [7] Y. Bi, T. Zhang, C. Tang, Y. Huang, J. Yang, J. Appl. Phy. 7, 6334 (). [8] C. Baumgarten, Phy. Rev. ST Ael. Beam 4, 4 (). [9] A. J. Cerfon, J. P. Freidberg, F. I. Parra, T. A. Antaya, Phy. Rev. ST Ael. Beam 6, 4 (3). [] S. Heifet, G. Stupakov, and S. Krinky, Phy. Rev. ST Ael. Beam 5, 644 (). [] M. Venturini, Phy. Rev. ST Ael. Beam, 344 (8). [] D. Ratner, A. Chao, Z. Huang, SLAC-PUB-339. [3] D.A. Edward, M. J. Sypher, An Introdution to the Phyi of High Energy Aelerator (John Wiley and Son, New York, 993). [4] A. Marinelli and J. B. Roenzweig, Phy. Rev. ST Ael. Beam 3, 73 (). [5] A. Marinelli, E. Heming, and J. B. Roenzweig, Phy. Plama 8, 35 (). [6] M. Reier, Theory and Deign of Charged Partile Beam, nd edition (Wiley-VCH, 8). [7] A. W. Chao, Phyi of Colletive Beam Intabilitie in High Energy Aelerator (Wiley Serie in Beam Phyi, 993). [8] K. Y. Ng, Phyi of Intenity Dependent Beam Intabilitie (World Sientifi Publihing Co. Pte. Ltd, 6). [9] Y. Li, G. Mahioane, F. Marti, T. P. Wangler, E. Pozdeyev, Proeeding of the IEEE Partile Aelerator Conferene (Vanouver, BC, Canada, 9), TU3PBC6, p. 734.

22 FIG.. (Color) Effet of modulation trength and phae of variou beam parameter on intantaneou beam intability growth rate. FIG.. (Color) Evolution of top view of beam profile. FIG. 3. (Color) Evolution of harmoni amplitude of normalized line harge denity. FIG. 4. (Color) Evolution of harmoni amplitude of radial entroid offet. FIG. 5 (Color) Evolution of harmoni amplitude of energy deviation. FIG. 6. (Color) Spae harge omponent of loal entroid with oordinate (x, z) at time t. FIG. 7. (Color) Longitudinal monopole and dipole mode pae harge impedane. FIG. 8. (Color) Exponential uppreion fator for a beam with urrent of ua, emittane of 5 π mm mrad and variou RMS energy pread. FIG. 9. (Color) Curve fitting reult for growth rate of line harge denitie (a). λ =. m; (b). λ =. m; (). λ =.86 m; (d). λ = 5. m. FIG.. (Color) Comparion of intability growth rate between theory and imulation. FIG. (Color) Co-rotation of two-maro-partile model with d =.5 m. FIG.. (Color) Co-rotation frequenie of two-maro-partile model with different d. FIG. 3. (Color) Evolution of radial beam denity. FIG. 4. (Color) Simulated top view (lef and RMS lie energy pread (righ at turn 4. FIG. 5. (Color) Simulated RMS lie energy pread of turn -8. FIG. 6. (Color) Simulated top view (lef and RMS lie energy pread (righ at turn 3. FIG. 7. (Color) Comparion of RMS lie energy pread. FIG. 8 (Color) Sketh of luter and energy analyzer.