Transverse mode coupling instability of the bunch with oscillating wake field and space charge
aa r X i v : . [ phy s i c s . acc - ph ] J a n Transverse mode coupling instability of the bunchwith oscillating wake field and space charge
V. Balbekov
Fermi National Accelerator LaboratoryP.O. Box 500, Batavia, Illinois 60510 ∗ (Dated: November 11, 2018)Transverse mode coupling instability of a single bunch caused by oscillating wake field is consideredin the paper. The instability threshold is found at different frequencies of the wake with spacecharge tune shift taken into account. The wake phase advance in the bunch length from 0 upto 4 π is investigated. It is shown that the space charge can push the instability threshold up ordown dependent on the phase advance. Transition region is investigated thoroughly, and simpleasymptotic formulas for the threshold are represented. PACS numbers: 29.27.Bd
I. INTRODUCTION
The transverse mode coupling instability (TMCI) ofa bunch with both wake field (WF) and space charge(SC) was first considered in paper [1]. It has been shownthere that, at a moderate ratio of the SC tune shift to thesynchrotron tune (∆
Q/Q s ), the SC pushes up thresholdof the instability caused by a negative wake. Hereafterthe result has been confirmed in papers [2]-[4].However, more confusing picture appears at the largervalue of this ratio like a hundred or over. It has beensuggested in Ref. [2] that the threshold growth ceasesbehind this border coming to 0 at ∆ Q/Q s → ∞ . Bycontrast, it was asserted in Ref. [3] that negative wakecannot excite the TMCI in this limiting case.An explanation of the collision has been proposed inRef. [5] where it has been noted that approximate meth-ods of solution were applied in all mentioned articles.Typically, an expansion of the bunch coherent displace-ment in terms of some series of basic functions was usedthere, with subsequent truncation of the series. It turnsout that, with this approach, the TMCI threshold canstrongly and not monotonously depend from the actualnumber of used basic functions.The problem has been clarified in recent publication [6]where the method of solution has been developed whichdoes not use the expansion technique at all, and thereforeis applicable at arbitrary SC tune shift. In particular,it has been shown that the TMCI threshold of negativewakes is asymptotically proportional to ∆ Q/Q s in con-trast with the case of positive wake whose threshold is ∝ Q s / ∆ Q . The statement has been extended in the pa-per on any monotonous WF, e.g. on the resistive wallimpedance.In the presented paper, this method is applied for in-vestigation of oscillating (resonant) wake fields. It isshown that the SC tune shift can suppress or intensifythe instability, dependent on the wake phase advance ∗ Electronic address: [email protected] from the bunch head to its tail. It is shown as well that,at some conditions, the dependence of the TMCI rate onthe wake amplitude is a non-monotonic function.Some of these problems were considered earlier usingother models and techniques of the calculations [5], [7].All the results are in rather good consent to be convincedof applicability of the used methods. Their detailed com-parison is carried out at the end of Sec. V.
II. PHYSICAL MODEL
Following [6], we represent the bunch transverse co-herent displacement in the rest frame as the part of thefunction¯ X ( z, t ) = ¯ Y ( z ) exp (cid:2) − iQ β ( z/R + Ω t ) − i Ω νt (cid:3) (1)where z is the longitudinal coordinate, t is time, R isthe machine radius, Ω is the revolution frequency, Q β and ν are the central betatron tune and the additionto it due to the wake field. The machine chromaticityin not included in the expression as a factor of a smallimportance for the TMCI threshold [5].In framework of the model, it is convenient to use an-other longitudinal coordinate ϑ ∝ z which is adjusted tothe bunch located on the interval 0 ≤ ϑ ≤ π . Then therearranged function ¯ Y satisfies the equation Q s ¯ Y ′′ ( ϑ ) + ν ˆ ν ¯ Y ( ϑ ) = 2ˆ νπ Z πϑ q ( ϑ ′ − ϑ ) ¯ Y ( ϑ ′ ) dϑ ′ (2)with the boundary conditions¯ Y ′ (0) = ¯ Y ′ ( π ) = 0 (3)where Q s is the synchrotron tune, and ˆ ν = ν +∆ Q with∆ Q as the space charge tune shift [6]. The function q ( ϑ )is proportional to the usual transverse wake function atcorresponding distance W ( z ϑ ): q ( ϑ ) = r RN b W πβ γQ β (4)where r = e /mc is the particle electromagnetic radius, N b is the bunch population, β and γ are the normalizedvelocity and energy of the particles.The oscillating wake of the form q ( ϑ ) = q exp( − αθ ) cos κϑ (5)will be considered in this paper. Then Eq. (2) obtainsthe form ¯ Y ′′ ( ϑ ) + P ¯ Y ( ϑ ) =2 Q π exp( αθ ) Z πϑ exp( − αθ ′ ) cos κ ( ϑ ′ − ϑ ) ¯ Y ( θ ′ ) dϑ ′ (6)with the notations P = ν ˆ νQ s , Q = q ˆ νQ s (7)It is easy to see that the integral-differential Eq. (6) isreducible to the net differential equation of 4 th order:¯ Y IV − α ¯ Y ′′′ + ( P + κ ) ¯ Y ′′ + (cid:18) Q π − α P (cid:19) ¯ Y ′ + κ P ¯ Y = 0 (8)with boundary conditions given by Eq. (3) and more¯ Y ′′ ( π ) = −P ¯ Y ( π ) , ¯ Y ′′′ ( π ) = − Q π ¯ Y ( π ) . (9)Similar in appearance equations of third order were con-sidered in Ref. [6]. Like them, it is possible to solveEq. (8) step-by-step starting from the point ϑ = π withinitial conditions ¯ Y ( π ) = 1 , ¯ Y ′ ( π ) = 0 , (10a)¯ Y ′′ ( π ) = −P , ¯ Y ′′′ ( π ) = − Q π (10b)and going down to the point ϑ = 0. Some trial valueof P has to be used each time with chosen parameters κ, α and Q ; however, only those of them resulting in therelation Y ′ (0) = 0 should be taken as the valid solutionssatisfying all the required conditions. With any Q , thereis an infinite discrete sequence of the eigennumbers P n satisfying these conditions. III. EIGENNUMBERS OF THE EQUATION
It is the general requirement for applicability ofthe single bunch approximation that the dampingcoefficient is large enough to neglect the bunch-to-bunchor the turn-by-turn interaction. However, it does notexclude that the damping is negligible within the bunch:exp( − πα ) ≃
1. Therefore the case α = 0 is consideredfor the beginning. -30 -20 -10 0 10 20 30 40 50 Q -100102030 P κ = 0.5 FIG. 1: Several lowest real eigennumbers of Eq. (6) andEq. (8) at α = 0 , κ = 0 .
5. The picture insubstantially differsfrom the case of rectangular wake κ = 0 [6]. In particular,numbers of the coalescence lines are: n = (0,1); (2,3); (4,5);etc. All the lines diverge at Q → −∞ . -30 -20 -10 0 10 20 30 40 50 Q -100102030 P κ = 1 FIG. 2: The same as in Fig. 1 at κ = 1. The main differenceis other combinations of the coalesced modes. In particular,the modes n = 0 and n = 5 join smoothly at Q ≃
Several eigennumbers of Eq. (8) are represented inFigs. 1-3 where different frequencies of the wake areconsidered (the used value of κ is designated right inthe graph). Six lowest real eigennumbers P n are plot-ted against Q in each the case. As one would expect, P n = n at Q = 0 independently on κ . The index n willbe used further to mark the lines.The overall behavior of the plots considerably dependson the frequency. The upper graph ( κ = 0 .
5) is similarto the case of rectangular wake κ = 0, both qualitativeand quantitative [6]. The next case κ = 1 has a differentappearance because its lowest line n = 0 outspreads farto the right where it connects with the upper line n = 5.Figure 3 with κ = 1 . n = 0 and n = 1 starting together inthe point Q ≃ − -30 -20 -10 0 10 20 30 40 50 Q -100102030 P κ = 1.5 FIG. 3: The same as Fig. 1 at κ = 1 .
5. The main differenceis that real parts of the modes n = 0 and n = 1 are coalecsedat Q < −
1, and diverge at
Q → ∞ . IV. THE BUNCH EIGENTUNES
Obtained functions P n ( Q ) have to be imaged into theplane ( q , ˆ ν ) with help of the transformationsˆ ν ± n = ∆ Q ± r ∆ Q P n Q , q = Q Q ˆ ν ± n (11)which follow from Eq. (7). Any point of the family isthe real eigentune of the bunch with given SC tune shift.They form several lines representing tunes of differenthead-tail modes which depend on the wake frequency.As it has been shown above, P n = n at Q = 0.Corresponding solutions are [2]:ˆ ν ± n = ∆ Q ± r ∆ Q n Q , at q = 0 (12)At ∆ Q = 0, it results in ˆ ν m = mQ s where m = ± n .Such oscillations are known as multipoles. Any eigen-mode of the bunch is a combination of several multipoles.Nevertheless, the multipole numbers m can be used forthe classification of the lines in the plots. For example,a line index m = − ν = − s at q = 0 and ∆ Q = 0. An im-portant role in this consideration play coalescing curves.We will use the symbol m = ( m , m ) to mark the coa-lescence of the lines of kinds m and m .Using Figs. 1-3 and Eq. (11), one can get the bunchtunes at selected wake frequency and different ∆ Q . Someresults are plotted in Figs.4-6 where the solid or thedashed lines are used repeating appearance of the originblocks of Figs. 1-3. Different colors are used in the result-ing figures for different SC tune shifts: ∆ Q = 0 (blue),or ∆ Q/Q s = 1 . Q/Q s = 3 (red). Theextreme points of the lines characterized by the relation dq = 0 highlight beginning of the instability regions.The case κ = 0 . κ = 0 which has -8 -6 -4 -2 0 2 4 6 8 q /Q s0 -3-2-1012345 ( ν + ∆ Q ) / Q s ∆ Q/Q s0 = 0 ∆ Q/Q s0 = 0 ∆ Q/Q s0 =1.5 ∆ Q/Q s0 =1.5 ∆ Q/Q s0 = 3 ∆ Q/Q s0 = 3 FIG. 4: The bunch eigentunes with different ∆ Q at κ = 0 . -8 -6 -4 -2 0 2 4 6 8 q /Q s0 -3-2-1012345 ( ν + ∆ Q ) / Q s FIG. 5: The bunch eigentunes with different ∆ Q at κ = 1.The same legends as in Fig. 4 are used. been considered in Ref. [6] with help of the square wakemodel. In particular, the blue lines form the chart whichis well known as the TMCI tunes without space charge∆ Q = 0 (see e.g. [8]). At κ = 0 .
5, the instability appearsbecause of coalescence of the modes m = 0 and m = ± | q | /Q s = 0.81 (0.57 at κ = 0 orwith the square wake model).All the lines move to the left-up at increasing ∆ Q .It means formally that the TMCI threshold increases inmodulus tending to −∞ at q <
0, and decreases tendingto 0 at q >
0. The coinciding results have been obtainedwith the square wake model [6]. However, only the case q < κ ≤ .
5. It is necessary to takeinto account that the mode m = ( − , −
3) depends lesson ∆ Q and moves slower to the left than the mode m =(0 , − Q/Q s < ∼ −
6. It has been stated in Ref. [6] at κ = 0, and remainsin force at the moderate value of κ .Similar picture takes place at κ = 1 (Fig. 5). The -8 -6 -4 -2 0 2 4 6 8 q /Q s0 -3-2-1012345 ( ν + ∆ Q ) / Q s FIG. 6: The bunch eigentunes with different ∆ Q at κ = 1 . main difference is that the instability appears because ofa coalescence of the modes m = − m = − q /Q s = − .
56 at ∆ Q = 0 (thedashed blue lines). The mode m = 0 could participateonly by the coalescence with the mode m = − κ = 1 . Q = 0, the instability arises by acoalescence of the modes m = − m = − q /Q s = − .
13 (dashed blue lines). How-ever, the lower modes m = 0 and m = 1 come in to playat ∆ Q = 0 resulting in the instability with the threshold q /Q s = − .
66 at ∆
Q/Q s = 1 . q /Q s = − . Q/Q s = 3. The appearance of positive multipolesas well as the lowering of the TMCI threshold at increas-ing ∆ Q is the characteristic of a positive wake [6]. It isan expected effect in the case because the wake is mostlypositive in the bunch at q < . π .Evolution of the spectral lines m = 0 and m = 1at κ = 1 . Q is represented in moredetails in Fig. 7. The blue lines, showing their initial po-sition at ∆ Q = 0, move toward each other to meet at∆ Q/Q s ≃ . Q/Q s = 0 .
35. It is seen that very nar-row region of instability − . < q /Q s < − Q obtaining the location − . < q /Q s < − . Q/Q s = 0 . − . < q /Q s < − . Q/Q s = 1 . m = ( − , −
2) is essentially loweras it follows from Fig. 6. -12 -10 -8 -6 -4 -2 0 q /Q s0 -3-2-1012 ( ν + ∆ Q ) / Q s ∆ Q/Q s0 = 0 ∆ Q/Q s0 = 0.35 ∆ Q/Q s0 = 0.7 ∆ Q/Q s0 = 1.5 FIG. 7: The lowest bunch mode m = (0 ,
1) as a function ofthe wake amplitude at κ = 1 . Q/Q s . At∆ Q/Q s > .
35, the mode becomes unstable at sufficientlylarge q but obtains the stability again at the further increaseof the wake. However, the bunch can remain unstable becauseof excitation of the mode m = ( − , − κ - q / Q s ∆ Q/Q s0 = 0 ∆ Q/Q s0 =1.5 ∆ Q/Q s0 = 3 FIG. 8: The TMCI threshold against the wake frequency atrelatively low space charge tune shift. The shift stabilizes thebunch at κ < κ > . V. THE INSTABILITY THRESHOLD
The developed method allows to find the TMCI thresh-old with any set of parameters. Some results are repre-sented in this section.Dependence of the threshold on the wake frequency isshown in Fig. 8 at ∆
Q/Q s = 0, 1.5, and 3. One cansee the oscillations of the threshold and the slow growthof its average value with frequency at κ > .
5, whicheffects have to be expected. However, the most impor-tant conclusion is that the SC stabilizes the TMCI at∆
Q/Q s < Q/Q s > .
5. Thetransition takes place rather rapidly at ∆ Q ≃ . ∆ Q /Q s0 -( q / Q s ) t h r e s h κ = 0 κ = 0.25 κ = 0.5 κ = 0.75 κ = 1 κ = 1.5 κ = 1.75 κ = 2 κ = 2.25 κ = 2.5 FIG. 9: The TMCI threshold against the SC tune shift atdifferent wake frequencies. The parts with frequencies κ < κ > . Dependence of the threshold on the SC tune shift isshown in Fig. 9 over a wider range of the parameters.The most remarkable effect is that the lines representingdifferent wake frequencies are distinctly broken down intotwo groups: with upper threshold at ∆
Q/Q s < Q/Q s > .
5. They have differentasymptotic behavior: q ∼ ∆ Q in the upper group and q ∼ Q s / ∆ Q in the lower one. Such a behavior can beobtained with help of Eq. (11) at the assumption |P| ≪ (∆ Q/ Q ) . (13)With sign ’–’ before the root, it givesˆ ν ≃ − P Q s ∆ Q , q ≃ − QP ∆ Q (14)According these equations, the value of q varies whenthe point ( Q , P ) moves along one of the curves like shownin Fig. 4 or Fig. 5. The TMCI threshold obtained bythe last Eq. (14) appears in the point where the condition dq = 0 is fulfilled that is d P / P = d Q / Q . It is the pointof tangency of the curve with the straight line P = k tg Q .Therefore the asymptotic expression for the threshold is: q = k th ∆ Q where the constant depends on the wake frequency κ asit is represented in Table IAnother possibility follows from Eq. (11) and Eq. (13)with sign ’+’ before the root:ˆ ν ≃ ∆ Q, q ≃ Q s Q ∆ Q According these equations the value of q varies when thepoint ( Q , P ) moves along one of the curves like shown inFig. 6. The relation dq = 0, that is d Q = 0, should beapplied additionally to find the instability threshold. It follows from Fig. 6 that the corresponding value of the Q ex ≃ − . κ = 1 .
5. Other values are represented inTable II. The results are in a conflict with the output ofRef. [2] where it is predicated that the space charge hasalmost no affects on the TMCI threshold at κ = 2. TABLE I: Dependence of the asymptotic coefficients on thewake frequency: q = k tg ( κ )∆ Qκ k tg -1.34 -1.43 -1.65 -1.17 -0.92TABLE II: Dependence of the asymptotic coefficients on thewake frequency: q = Q s Q ex ( κ ) / ∆ Qκ Q ex -1.1 -1.1 -1.3 -2.4 -3.4 The data submitted in Fig. 9 at κ ≤ Q ≫ Q s .The coincidence of the plots like shown in Fig. 8 con-firms that the TMCI threshold is not very sensitive toused models of the bunch and of the potential well. Thethresholds presented in Fig. 9 at κ > . VI. TRANSITIONAL WAKE FREQUENCY.
The broad blank area between the top and bottomparts of Fig. 9 is assigned for the wake of the frequency1 < κ < .
5. However, the transition is not a smoothprocess in the case. Actually, at constant SC tune shiftand increasing wake, the instability can arise, disappear,appear again in other form, etc. This statement is illus-trated below on the example of the wake with κ = 1 . Q ≃ − , P ≃ − q , ˆ ν ) which are shown onFig. 11. Like Fig. 7, instability of the mode m = (0 , − Q/Q s = 1, the instability -30 -20 -10 0 10 20 30 40 Q -10-50510152025 P FIG. 10: Several lowest real eigennumbers of Eq. (6) andEq. (8) at α = 0 , κ = 1 .
25. In comparison with Figs. 2 and3, there is an additional curl of the lower line -25 -20 -15 -10 -5 0 5 q /Q s0 -10123456 ( ν + ∆ Q ) / Q s ∆ Q/Q s0 = 4 ∆ Q/Q s0 = 5 ∆ Q/Q s0 = 6 FIG. 11: The bunch eigentunes at the wake parameters α = 0 , κ = 1 .
25. The eigentunes at ∆
Q/Q s = 4 areshown by dotted blue lines the lower of them representingthe mode m = ( − , − Q/Q s = 5 and 6 (green and red ovals). arises at q /Q s ≃ −
3. However, it disappears alreadyat q /Q s ≃ − . Q reaching approximatelythe area − < q /Q s < − . Q/Q s = 4 Thiscase is shown by the blue lines in the top-right cornerand in the bottom-left one of Fig. 11. It should benoted that the mode m = ( − , −
3) has the threshold q /Q s = − . Q/Q s = 4 which tune is shownby the blue dotted line in the bottom-right corner ofFig. 11. Therefore, with such SC tune sift, the broadpicture is: ∆ Q/Q s0 -40-30-20-100 ( q / Q s ) t h r e s h STABILITYINSTABILITYSTABILITYINSTABILITY
FIG. 12: Thresholds of different bunch modes at κ = 1 . m = (0 , − m = ( − , − The bunch is stable at | q /Q s | < . m = (0 , −
1) becomes unstable at0 . < | q /Q s | < . m = (0 , −
1) and m = ( − , − . < | q /Q s | < m = ( − , −
3) is unstable at | q /Q s | > m = (0 , − Q/Q s = 5 or 6, correspondingly. As aresult, following picture is played e.g. at ∆ Q/Q s = 6(red lines):The bunch is stable at | q /Q s | < . m = (0 , −
1) becomes unstable at0 . < | q /Q s | < . . < | q /Q s | < . m = ( − , −
3) becomes unstable at6 . < | q /Q s | < . m = (0 , −
1) and m = ( − , − . < | q /Q s | < . m = ( − , −
3) remains unstable at | q /Q s | > . m = (0 , −
1) may be unstable in the regions between thered lines, or between the red and the violet lines but notbetween the violet lines. The legends in the picture con-cern namely this mode. Another mode m = ( − , −
3) isunstable everywhere below the green line. The asymp-totic of this threshold is q = − . Q which expressionis quite coordinated with Table II.Thus, with the oscillating wake, the TMCI thresholdcan be a nonmonotonic function of both the SC tune shiftand the WF amlitude. The result is similar to that foundin Ref. [1] in framework of the expansion technique. VII. CONCLUSIONS
The TMCI threshold of the oscillating wake withoutspace charge increases at increase of the wake frequency.The space charge effect depends on the wake phase ad-vance on the bunch length φ .At φ < π , the threshold is about proportional to thespace charge tune shift ∆ Q .At φ > . π , the threshold is about ∝ Q s / ∆ Q with Q s as the synchronron tune.At φ ≃ . π , the TMCI threshold can be a non- monotonic function of both the space charge tune shiftand the wake amplitude. The variability occurs becausedifferent multipole numbers are responsible for the insta-bility at different conditions.The phase advance more of 4 π is not considered in thepaper. VIII. ACKNOWLEDGMENT
Fermi National Accelerator Laboratory is operatedby Fermi Research Alliance, LLC under Contract No.DEAC02-07CH11395 with the United States Departmentof Energy. [1] M. Blaskiewicz, Fast head-tail instability with spacecharge, Phys. Rev. ST Accel. Beams 1, 044201 (1998).[2] A. Burov, Head-tail modes for strong space charge, Phys.Rev. ST Accel. Beams 12, 044202 and 12, 109901 (2009).[3] V. Balbekov, Transverse instability of a bunched beamwith space charge and wakefield, Phys. Rev. ST Accel.Beams 14, 094401 (2011).[4] M. Blaskiewicz, Comparing new models of transverse in-stability with simulations, in