Möhl, D

Schönauer, Horst Otto

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LANDAU DAMPING BY NON-LINEAR SPACE-CHARGE FORCES AND OCTUPOLES D. Möhl and H. Schönauer, CERN, Geneva, Switzerland Summary and Conclusions Increasing the PSB and CPS intensities to 1013 ppp will require more powerful octupoles in order to over-come the transverse instabilities that are to be expect-ed. Unfortunately, theory fails to give quantitative predictions at low energies, where direct space-charge forces dominate over image forces. To be able to speci-fy the future needs more precisely, an extension of the theory would be desirable, particularly since experimen-tal evidence is such that it is hard to extrapolate to higher intensities. The theory presented here solves the dispersion relation, including the influence of external and space-charge induced non-linearities in both transverse dimen-sions as well as a spread in the LNS coefficient U. Results are preliminary in the sense that modulation of the space-charge forces due to synchrotron motion is not included. They show that the combined effect of exter-nal and space-charge non-linearities in the two trans-verse planes can considerably enhance Landau damping in low-energy machines (up to 10 GeV, say). The sign of the octupole current is important especially for a flat beam, and should be chosen in such a way that V increas-es with amplitude in the direction where the beam is wide. Stability diagrams for several typical conditions are presented and applied to the PSB. 1· Introduction Octupole lenses have been installed both in the CPS and its Booster years ago, and they have successfully cured transverse instabilities. It is planned to use more powerful lenses in the CPS, and possibly in the Booster too, in order to tame future intensities. It is puzzling, however, to observe that the currents needed to avoid instabilities are considerably lower than predicted by theory1,2. In addition, lower thresh-olds for radial rather than for vertical resistive wall instability were observed frequently in the PSB. These observations suggest that the influence of space-charge non-linearities and the spreads due to motion in the second transverse direction might be important. An attempt to include these effects3 already showed the importance of the variations due to the betatron am-plitudes in the second transverse plane. This approach, however, still predicted too high octupole strengths and failed to give the correct behaviour in the limit of vanishing external octupole force. In this limit in-coherent space-charge forces should have no influence on dipole motion, a fact which can be deduced from the single particle-equation4 and is fully confirmed by computer simulation5. To make more reliable extrapolation it was felt necessary to extend the theory in order to remove these deficiencies. The main difference of the present work in comparison with Ref. 3 is that we include space-charge non-linearity in both the driving term (the U-term of Ref. 1) and in the single-particle frequency. A general model would have to include: i) frequency spreads due to external non-linearities in both transverse (x and y) directions; ii) spreads due to x- and y-non-linearities of space-charge forces; iii) spreads due to the energy distribution of the beam; iv) spreads due to longitudinal variation of transverse space-charge forces; and probably other effects. In the present paper we content ourselves with considering the effects (i) and (ii). This is a valid approximation at least for the coasting beam experiments which were performed in the Booster and which exhibited the "anomalities" discussed above. We shall find that the combination of external and space-charge non-linearities can considerably en-hance the stability conditions, and that the spreads from both transverse directions are important. 2. Dispersion Relation L t us discuss dipole oscillations in one trans-verse (x) direction. Neglecting energy spread, the dis-persion relation of Laslett, Neil and Sessler (LNS) can be written as (U + V + iV) ∞ -½ h'(a)a2 da ·= 1. (1) ∫ ω - (n - νx)Ω 0 Here the tune νx = νx(a) depends on a, the amplitude of the incoherent betatron oscillation; h(a) is the corresponding distribution function, ∫0∞ h(a)a da = 1. The coefficient U + V + iV can be expressed in terms of the revolution frequency (Ω) and the incoherent ( νic) and coherent ( νc) Laslett tune shifts (generalized to include wall resistivity and equipment interaction): U + V + iV = Ω(Δνc - Δ ν i c ) . (2) 1 Finally ω = (n - ν)Ω is the mode frequency (coasting beam mode n) to be obtained by solving Eq. (1); Ιm(ω) is the growth rate, V being the collective betatron frequen­cy. To include space-charge non-linearities as well as y-variation of νx we can use the derivation of Eq. (1) given by Hereward6. We start from the single-particle equation i + Ω2 [ν2(xi,yi) - 2ν0Δνic(xi,Υi)]xi = 2Ω2ν0 [Δνc - Δνic(xi,yi)] , (3) w h e r e i s the dipole motion of the beam and Xi the mo­tion of the test particle (coherent and incoherent). The corresponding dispersion relation is ∫0∞ (U + V + iV)(a,b) {-½ h'(a)a2)b g(b) dadb = 1 . (1a) ω -[n - ν(a,b)]Ω Here b is the incoherent y-amplitude, g(b) the corres­ponding distribution, ∫0∞ g(b)bdb = 1. The tune ν(a,b) = ν0(a,b) - Δνic(a,b) includes the incoherent tune shift, and so does U + V + iV through Eq. (2). The terms v(a,b) and (U + V + iV)(a,b) are obtained by averaging over the incoherent betatron motion. To evaluate Eq. (la) we make some further approxi­mations: We assume that only the incoherent tune shift is non-linear, whereas the image term Δν is the same for all particles. This is a valid approximation at low energy, where Δνc << Δνi and/or for thin beams. The case where both Δν = Δνc(a,b) and Δνic = Δνic(a,b) but where external non-linearities are negligible (νi = const) was discussed in Ref. 4. We shall further expand ν(a,b) and only retain terms up to the octupole moment: 380 νic(a,b) = νic(0) + νic a2 + νic b 2 2 b 2 = Δ - Δa a2 - b 2, (4) and ν0(a,b) = ν0(0) + ν0 a2 + ν0 b 2 a2 2 = ν0 + νa a2 + ν b b 2 . The symbols Δ, Δ , Δ (all positive) and νa, νb are hereby defined and will be used throughout the rest of the paper. Quantities such as ν, etc., will be denoted as "external spread" and "space-charge spread", respectively; and are typical amplitudes to be defined below; "a"corresponds to the plane of the instability. 3. Results Here we shall present the solution of the disper­sion relation (la) for two different distributions and for several typical conditions. We assume νa = 0, νb = 0 in Section 3.1j » in Section 3.2 and Figs. 1 and 2; 2â = in Section 3.3 and Figs. 3 and 4; and 3 « in Section 3.3 and Fig. 5. 3.1 Mo external non-linearities (i.e. νa = 0, νb = 0) In this case incoherent space-charge has no effect, as was found already in Ref. 4. In fact it is easily verified from the single-particle equation (3) that xi = is a solution provided that νi2 = ν02 - 2ν0Δνc is the same for all particles. Under this condition Δνic simply drops out from Eq. (3). 3.2 ν-spread due to betatron amplitudes in the plane of the instability only (i.e. >> ) 3.2.1 "Semicircular" distribution It is instructive to start with the distribution which -- without space charge forces -- leads to a cir­cular range of stability and gives the rule-of-thumb criterion 4 |U + v + iv| ≤ δνFWHH (5) Ω for the stabilizing ν-spread. The derivative of this function which enters into Eq. (la) is a half circle: -a2 h'(a2) da2 = 2 √1 - ( a2 - 1)2 da2, (6) 2 π2 0 ≤ a2 ≤ 2 2 Including now space-charge, the solution of the dis­persion relation gives the stability boundary V2 + (U + V ) 2 = Ω2 (νa Ω2 ( δ ν F W H H ) 2 . (7) (1+  4 16 Here we have introduced a parameter qa = νa/ a, the ra­tio of external spread to space-charge spread. U in­volves an average of Δνic U = Δνc - (Δ - Δaâ2)· (8) Ω For 1/qa = 0 we recover the rule of thumb (5). With space-charge spread, the stability circle is distorted into an ellipse with real half-axis: = (Ω/4) δνFWHH (1+2/qa).(( The stable range is thus increased or decreased depending on the sign of qa, i.e. depending on the polarity of the octupoles. Typically the space-charge non-linearity is such that 2/qa = ± |U|/(Ωνa2) (see Appendix). Hence for V << |U| -- the case of interest in the PSB and in the CPS below 10 GeV -- the stability condition reduces to 4 M ≤ δ ν F W H H δνFWHH {0.672, (9) Ω 1 + ½ and the stabilizing octupole ν-spread is about 0.67 or twice the value obtained from conventional theory (1/qa = 0). Normalization: In the following we shall use a suitable normalization to reduce the number of parameters. We normalize frequencies (U,V) and spreads to the space-charge spread S = ΩΔ in the plane of the instability. We denote u = U + V |beam centre, ν = V · (10) S S This normalization is convenient because the range over which u can reasonably vary is small (values between-3 and-5 depending on the emittance ratio and the plane considered, are typical for the PSB), see Appendix. Figure 1 displays stability boundaries2 for the distribution (6) in this normalization. Note that the normalization (10) shifts the ellipse (7) towards more negative u-values because u refers to the beam centre. The reduction of the stable area for the "wrong" octupole polarity (qa < 0) can be seen from a comparison between Fig. 1a and Fig. lb for given |qa|, and is obvious­ly due to cancellation of external and space-charge spread. 3.2.2 Parabolic distribution Figure 2 gives similar results for a parabolic distribution: h(a) = 1 [1 -a2 ], 0 ≤ a2 ≤ 2â2. (11) 2â2 2â2 Again for negative qa the stable area is largely reduced. One particularity of Fig. 2 is that all curves intersect the point v = 0, u = -2. It appears that this effect is due to the sharp cut-off of this distribution. 3.3 ν-spread due to betatron amplitudes in both transverse directions (parabolic distribution in each plane) Now we have to introduce another two parameters: Ρ = νb , r = ; (12) ν p denotes the ratio of the two external spreads and r the ratio of the space-charge spreads; b is the equiva­lent of â for the second transverse direction. Note that r is determined by the emittance ratio and varies from 0 for â » to about 2 for a << ; r = 1 corres-ponds to = 2 . Figures 4 and 5 refer to parabolic distributions h(a) g(b) in Eq. (la) and demonstrate the influence of the second transverse plane. In Figs. 3 and 4, r = 1, i.e. the same spreads Δ and Δ have been assumed. The case ρ = 1 as taken in Fig. 3 assumes that two sets of octupoles (in F- and D-sections say) are used such that νa and νb have the same sign. In this case we recover the reduction of the stable area for negative qa as for the one-dimensional case (Figs. 1 and 2). In Fig. 4, νa and νb have been taken of opposite sign, which is typical for simple lens arran­gements. 381 In this case, the reduction of stable areas is less noticeable, because the tendency of cancellation for the one octupole polarity is to some extent smoothed out: cancellation in one plane goes together with addi­tion in the other plane. Finally Fig. 5 corresponds to a beam which is wide in the direction perpendicular to the plane of insta­bility ( >> ), as is usually the case for vertical instability of a multiturn beam. Hence we take r = 2 and assume in addition "simple octupoles" with large negative p. One finds again that one octupole polarity is fa­vourable although the "right" sign is now the opposite to the preferred one in the case of a beam which is wide in the other direction (p = 0, r = 0). We conclu­de that the octupole moment in the direction where the beam is wide should be chosen with care. Application to the PSB The coasting beam and bunched beam instabilities observed in the PSB7,8 occur sometimes horizontally, sometimes vertically, in an apparently irregular way. In order to explain this feature, we apply the results of Section 3.3 to the PSB. We compute the thresholds for both planes as a function of the emittance ratio ΕΗ/ΕV, assuming the product EHEV and hence the area of the beam cross-sections to be a constant. We include image forces and averaging over the strongly varying beam dimensions within a machine period. The arrange­ment of the octupoles is as described in Ref. 9. Tresh¬ olds are expressed by the octupole currents required to stabilize the beam. Figure 6 shows the result for N = 2.5 1012 p/p and    V = 5200 (π mrad mm) 2 at 50 MeV. One observes that positive octupole current ( ν/ r2 > 0) is more favou­rable and that for this polarity vertical stability requires stronger octupole currents for EH/EV ≤ 1.7 whereas horizontal stability is more critical for EH/Ev ≥ 1.7, The intersection point EH/EV = 1.7 falls into the region of emittance ratios actually observed in many machine experiments. This might explain why slight dif­ferences in beam parameters can favour the one or the other direction. The predicted octupole currents are still higher than measured values. This is probably owing to the neglect of synchrotron motion in our model for the case Fig. 1 : Stability diagram for a beam with amplitude distribution (6). The LNS-coefficients u and v are nor­malized to the space-charge ν-spread (for this distri­bution and vanishing image forces, u is around -3). The beam is stable if the point (u,v) is on the left-hand side of the curve. Fig. 2 : Same physical conditions as in Fig. 1, but parabolic amplitude distribution (11). For this distri­bution u is typically between -3.5 and -5. qa is a measure of the octupole ν-spread due to betatron oscillations in the plane of the instability. Parts (a) and (b) of Figs 1-5 refer to the two polari­ties of the octupoles, respectively; p is the ratio of the octupole ν-spreads due to incoherent motion in the two transverse directions; r is the corresponding ratio for space-charge ν-spreads.In Figs 1 and 2, p and r are 0 because the beam is wide in the direction of the instability ( >> ). 382 Fig. 3 : Parabolic amplitude distributions (11) in both transverse directions: r=1 means ; p -1 is typical for one set of octupoles and not too flat a beam Fig. 4 : As for Fig. 3, but p 1 requires two sets of octupoles. Fig. 5 : As for Fig. 3, but for a beam wide in the di-rection perpendicular to that of the instability ( > > ) . Fig. 6: Octupole currents required to stabilize the PSB beam versus emittance ratio, calculated for para-bolic amplitude distributions (11) in both transverse planes. Nominal parameters at 50 MeV: EHEV = 130×40 (π mrad mm) 2, N = 2.5 1012 p/p. 383 of bunched beams and because of neutralization in the coasting beam case. To conclude, let us make a corapa¬ rison with conventional theory (external spread in one transverse direction, no spread in ν i c ) : the latter requires stabilizing octupole currents of 400 A (hori­zontal) and 520 A (vertical instability), for the nomi­nal EH/EV = 130/40, all other parameters as for Fig. 6. This is to be compared with the values of 180 A and 70 A taken from Fig. 6. References 1) L.J. Laslett, V.K. Neil and A.M. Sessler, Rev. Sci. Instrum. 36, 436 (1965). 2) . Hübner, A.G. Ruggiero and V.G. Vaccaro, Proc. 7th Int. Conf. on High-Energy Accelerators, Yerevan, 1969, p. 343. K. Hübner and V.G. Vaccaro, CERN/ISR-TH/70-44. 3) K.H. Reich, H. Schönauer, CERN/SI/Int.BR/72-2. 4) G. Merle and D. Möhl, CERN/ISR/300/GS/69-66. 5) G. Besnier, Thesis, University of Rennes, France. 6) H.G. Hereward, CERN/MPS/DL 69-11. 7) The PSB Staff (reported by K.H. Reich), The CERN PS Booster, design expectations.., this conference. 8) J. Gareyte and F. Sacherer, Head-tail instabilities in the CERN PS and Booster, this conference. 9) C. Bovet, K.H. Reich, CERN/SI/Int.DL/69-3. Appendix Calculation of incoherent ν-shift and V-spreads In order to assure self-consistency with the para­bolic distribution that we mainly use, we should solve the potential problem for the corresponding charge dis­tribution. This seems to be difficult for the kind of factorized amplitude distribution h(a)g(b) assumed above, which represents a beam of rectangular cross-section. Hence we restrict ourselves to the computation of ν-shifts and ν-spreads for a beam of elliptical cross-section and parabolic density p(x.y) = λ [1- (x - ) 2 - y2 ] , (A1) â 2â2 2 2 where λ is the linear density and â/2, √2 are the beam radii, corresponding to the ellipse p(x,y) = 0, which contains all particles. This choice provides at least approximate self-consistency: projected densities ob­viously do agree with those obtained from the factorized amplitude distribution. Neglecting image forces, the force in the x-direc¬ tion is given by eFx = e2λ 1 {(x - ) -1 2â+ (x - ) 3 -πE0 â(â + 6 â2(â + ) -1 2 ( x - )}· (A2) 2 ( â + ) Performing an averaging process over incoherent betatron motion (e.g. by the method of harmonic balance), x - = a cos (νxΩt), y = b cos (νyΩt), we take <(x - 3 a3 cos νxΩSt = 3 a2(x - ) ( 3) 4 4 <y2(x - )> = 1 b 2 a cos νxΩt = 1 b 2 ( x - ) , 2 2 and obtain for νic(a,b): Δνic. (a,b) = NrpR 1 [1 - 2â + a2 -ε0πβ2γ3 â(â + ) 8(â + ) â2 b b2 ], (A4) 4(â + ) 2 where the r.h.s. should be averaged over the circumfe-rence. We identify the quantities introduced in Eq.(A4): Δ = NrpR 1 εnπβ2γ3 â(â + ) Δ = 1 + â ( 5) â + â2 8 Δb Δ 1 4 b · 2 â + This gives a rough estimate of the order of u Lof Eq.(10) J to be expected. For our parabolic distribution, u re-presents the normalized ν-shift in the beam centre: u -Δ - Δνc -Δ = 8 · Δa Δ a2 1+ This would imply values of u between-4 and-8, which is valid only for a machine of small wiggle in the β-func¬ tions. Computed values for the PSB, assuming nominal emit¬ tances (EH = 130π, EV = 40π mrad at 50 MeV and inclu­ding image contributions, give values of u = -3.8 (horizontal) and u = -4.8 (vertical). Note that r is approximately given by r = 2/(2â + ) and can only take values of 0 < r < 2 (Eqs.(12), (A5)). For the semicircular distribution (6), a numerical estimate gives u = -Δ/Δ = - 3. This value of u is used to derive the stability criterion (9). From (8) and (10) we obtain for V<<|u| 1 = = U/Ω + Δ = U -u , qa νaâ2 νaâ2 Ωνaâ2 qa and 2 = ± | U | 2 - ± f U |, (A6) qa Ωνaâ2 |u + 1| Ωνaa2 f = 2 1. |u + 1| Hence |u| < = Ω νaâ2| + 2 |, 2 qa 2 |U| |1 ± f ≤ νaâ2 ~ 1 δν F W H H, Ω 2 2 or 4 |U| ≤ δ ν F W H H 1 δνFWHH {0.672· Ω |1 + f/2| Acknowledgements We are grateful to K.H. Reich for the encouragement to carry out the work presented here, and to Mrs. M. Lelaizant for carrying out numerical computations. 384 

Landau damping by non-linear space-charge forces and octupoles

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