Transverse stability with nonlinear space charge

Transverse stability with nonlinear space charge is studied within the context of coasting beams. For bare tune spreads originating from chromaticity or frequency slip, the space charge tune spread has a fairly small effect and the incoherent space charge force is well modeled by a transverse capacitance. For tune spreads due to octupoles or fringe fields, beams are more stable when the bare tune increases with betatron amplitude.


I. INTRODUCTION
Early work on transverse coasting beam instability usually treated the direct space charge force as a source of transverse capacitance [1,2]. Somewhat later, attempts to incorporate the (betatron) amplitude dependent nature of the space charge tune shift were made [3,4]. These works generally started with the dispersion relation derived for coasting beams in the presence of a dipolar transverse field [1,2,[5][6][7]. The dispersion relation was then phenomenologically modified to incorporate the nonlinearity in the space charge force.
The present work begins with equations of motion which include nonlinear, interparticle forces [8]. Only one dimension is considered. This simplifies both the notation and the complexities of coupling between the transverse degrees of freedom. Since the intended application is synchrotrons, it is assumed that the magnitude of the tune shift is small compared to the average tune. This allows the equations to be heterodyned. Finally, first-order perturbation theory on the Vlasov equation is used to obtain the dispersion relation. Throughout, the momentum and energy conserving properties of the interparticle forces are retained.

II. THE MODEL
Consider a one-dimensional model with coordinate x. For N interacting particles the equation of motion for particle j is *Email address: blaskiewicz@bnl.gov In Eq. (1), particle j has bare tune Q j , and the average bare tune is Q 0 . The space charge force is characterized by the antisymmetric, nonlinear function f͑x͒. For Q j ϵ Q 0 one can average Eq. (1) over j: The double sum over the space charge force vanishes due to the antisymmetry of f͑x͒. Assuming x~exp͑2iQu͒ and jW 1 iaj ø Q 0 , Q 6͑Q 0 2 W͒ 1 ia. The wall induced tune shift and growth rate are 2W and a, respectively. With no bare tune spread, space charge has no effect on the coherent frequency [3,4,9]. The problem at hand is to determine the behavior of the system when bare tune spread is present. Define the functions A j ͑u͒ and B j ͑u͒ via Since the tune shift is small compared to the tune, the functions A j and B j vary slowly compared to the sinusoids. In Eq. (1) make the substitutions where the 9 denotes differentiation with respect to u. De- where and similarly for B. To proceed we multiply Eq. (4) by cosx and average over the fast variable, x. This gives By Newton's third law f͑2x͒ 2f͑x͒.
which defines the function G͑x 2 ͒. After this substitution one obtains Multiplying Eq. (4) by sinx and averaging gives If A and B in Eqs. (6) and (7) are treated as functions of u alone then these equations can be derived from the Hamiltonian where and the potential U͑x 2 ͒ satisfies dU͑x 2 ͒͞dx 2xG͑x 2 ͒. If a 0, substituting W ! W͞2 allows the equations of motion to be derived from this Hamiltonian when the coordinate dependence of A and B is included. Only the presence of a precludes a full Hamiltonian treatment. This is reasonable given the nonsymplectic motion of a damped oscillator. Now go to the continuum limit and introduce the distribution function F͑A, B, D, u͒ where F͑A, B, D, u͒ 3 dAdBdD gives the fraction of particles in phase space volume dAdBdD so the distribution is normalized to one, Consider a particle with phase space coordinates A, B, D. This particle's coordinates evolve according to the continuum versions of Eqs. (6) and (7) 044202-2 044202-2 where the equation for D has been added for completeness. If A is taken as a position variable with B as its conjugate momentum, and D is a position variable with conjugate momentum P D , these equations of motion can be obtained from the Hamiltonian where and similarly for B. The equations of motion are The Hamiltonian given by (12) is not the continuum limit of Eq. (8), but it generates the equations of motion (9)- (11). With Hamiltonian (12) the phase space distribution evolves according to the Vlasov equation, Since P D is absent from the Hamiltonian, the tune distribution is constant in time (u), As a final check of the physical equivalence of the Vlasov equation and Eqs. (6) and (7), consider the evolution of the Klimontovich distribution [10] F͑A, B, D, u͒ where the parameters A k ͑u͒, B k ͑u͒, and D k ͑u͒ can depend on u, but not on A, B, or D. With its sum of delta functions, this is the distribution for N identical particles. Substituting this distribution into the Vlasov equation (13) with Hamiltonian (12) gives Eqs. (6) and (7) and dD k ͑u͒͞du 0.

III. SOLUTIONS
To solve to the Vlasov equation, start by introducing the action angle variables J and c defined implicitly by The coordinate change is canonical so dAdB dJdc.
The threshold for instability is determined using first order perturbation theory. Let with r andF 0 each normalized to unity. The perturbation is parametrized as and where n 0 2W 1 ia is the tune shift for D ϵ 0 and g 1 g 1 ͓F 1 ͔ does not vary with J or D: Applying first order perturbation theory gives With the simple c dependence the solution is straightforward givinĝ Both g 1 and U 1 ͑J͒ depend on F 1 which in turn depends on n.
First consider the case where r͑D͒ is a Lorentzian, The right-hand side of Eq. (19) depends on F 1 only throughF 1 . Multiplying both sides by the denominator on the right and p J and integrating with respect to J yields In fact, X ϵ 0, and is a statement of Newton's third law.
To show this, consider the first term in brackets in the integrand. Integrate this by parts, which yields Insert the integral expressions for U 0 and U 1 . In the double integral interchange J and J 1 in the second term in brackets. The result is and the integral is zero since U sinc has period 2p. Therefore X ϵ 0 and one concludes that the coherent frequency with a Lorentzian distribution in D is given by Notice that the nonlinearity in the space charge force as well as the shape of the unperturbed distribution are irrelevant in Eq. (21). Only r͑D͒ was constrained, and with this constraint the space charge force has no effect on the coherent frequency. Next consider the case whereF 0 ͑J͒ is a step function which is constant for J , J 0 and vanishes for where the equality of the two expressions is a special case of the argument used to obtain X ϵ 0. With this definition Eq. (18) becomes 044202-4 044202-4 Integrating Eq. (23) over D yields a slightly modified dispersion integral from the result with no space charge. Suppose the space charge free dispersion relation is n 0 D͑n͒. When space charge is included the dispersion relation is n 0 1 Q e sc D͑n 1 Q e sc ͒. Generally, Q e sc is positive. The peak of the threshold curve given by Im͑n͒ 0 shifts toward negative values of Re͑n 0 ͒. Both this result and the previous one using the Lorentzian distribution support the usual prescription of treating the space charge force as a source of transverse capacitance. The final problem is to understand the effects of space charge tune spread when the distributions r͑D͒ and F 0 ͑J͒ are more realistic.
Solving the problem with more realistic distributions of r͑D͒ and F 0 ͑J͒ appears to require a model of the pairwise space charge force. A simple model, in the raw position variable, which includes nonlinearity is given by For real space charge both k and s are positive and the coefficients are chosen to simplify later expressions. After performing the averaging procedures the space charge potentials are given by The parameter g 1 is defined in Eq. (16) and Multiplying Eq. (18) by p J and integrating over J and D gives g 1 , while multiplying by J 3͞2 and integrating yields g 2 . The dependence on F 1 is thus reduced to two numbers. Define the general dispersion integral The dispersion relation is equivalent to the two simultaneous equations, and g 1 ∑ ͑n 0 1 k͒I 2 ͑n͒ 2 2k Taking the ratio of the right-hand and left-hand sides of Eqs. (29) and (30) results in the final dispersion integral, Given F 0 ͑J, D͒, k, s, and n, Eq. (31) gives n 0 . When combined with Eq. (29) or (30) one obtains g 2 ͞g 1 which via Eq. (26) gives U 1 ͑J͒͞g 1 and finally via Eq. (18) the eigenfunctionF 1 ͑J, D͒͞g 1 . The constant g 1 cancels and the problem is, in principle, solved. Of course, one is generally interested in the problem of finding n given n 0 or, less ambitiously, whether a given value of n 0 , which is proportional to the transverse impedance, corresponds to an unstable system. This will be addressed later using threshold curves. The next problem is to find a suitable expression for Eq. (28). In principle one could do the double dispersion integral numerically for arbitrary r and F 0 but, given the cubic approximation to the space charge force, any reasonable functions would be equally realistic. Toward this end, expressions which yield a final result in terms of simple functions will be obtained.
The bare tune distribution is approximated as Given a set e 1 , e 2 , · · · e M the C k 's are chosen to keep the integral over D equal to unity and to cancel the tails of the distribution. The latter is done by demanding With r͑D͒ given by Eq. (32) the integral with respect to D in Eq. (28) is easily done and results in where I m k ͑n͒ is the integral over J for each value of m. There are many possible choices forF 0 ͑J͒. For definiteness takê This representation ofF 0 ͑J͒ is smooth enough to give bounded values of I m k ͑n͒ even if Im͑n 1 e m ͒ 0 [11]. The space charge tune shift as a function of action is given by Define q 2kJ 0 ͞s 2 and let y J͞J 0 . The required dispersion integrals are where the real and imaginary parts of n are explicit. The integral in Eq. (37) is along the real axis and only results with n I 1 e m $ 0 are meaningful. The integrals are elementary but tedious. An efficient representation involves the function and I m k J k21 0 G k ͑z͒͞q. With these definitions [12], The remaining integrals are given by G 2 ͑z͒ 1 1 zG 1 ͑z͒ and G 3 ͑z͒ 1͞2 1 zG 2 ͑z͒. The real and imaginary parts of these functions for y 0 1 are shown in Figs. 1 and 2, respectively. The functions are bounded and continuous. Notice that Im͑G 1 ͒ 6pu͑1 2 u͒ for 0 , u , 1, and that the discontinuity in its derivative at u 0 would be present for anyF 0 ͑J͒ that had a nonzero derivative as J ! 0 1 . For instance, a thermal distribution would yield a G 1 with a discontinuous derivative at u 0 [11]. Therefore, the present choice ofF 0 ͑J͒ 3͑1 2 J͞J 0 ͒ 2 ͞J 0 has no unrealistic singularities and should fairly represent real beams. given by Eq. (28) with no space charge. For a given curve, giving n a positive imaginary part always results in a value of n 0 which lies above the curve so those values of n 0 which lie on and below the curve result in no exponential growth. In fact, solutions corresponding to values of n 0 below the curve are damped [13].
Adding space charge to the picture requires choosing both k and q. To get the maximum effect in one dimension choose q͞k so that particles at opposite edges of the beam exert no force on each other. The difference in their positions is x 2 p 2J 0 , and setting Eq. (24) to zero gives q͞k 3͞16. Real beams have a two-dimensional cross section. Take an unperturbed distribution F 0 ͑J x , J y ͒͑ J 0 2 J x 2 J y ͒. This yields a round beam with a onedimensional projection F 0 ͑J x ͒~͑J 0 2 J x ͒ 2 and similarly for y. The x component of the space charge force is where R is the beam radius. Neglect the third term in brackets yielding a force which vanishes at the edge of the beam, and a soft upper limit for realistic distributions. Using first-order perturbation theory the x tune shift is given by where Q 0 sc is the x tune depression in the center of the beam. The probability distribution of Q sc is With Eq. (40) the distribution of Q sc may be obtained in closed form. Plots of this expression as well as the distribution for a 1D case with the same average and standard deviation are shown in Fig. 5. The effective nonlinearity is q͞k 3͞5, just more than 3 times the onedimensional case. Figure 6 shows the threshold curves for the ten-term Gaussian distribution with k 0, 5, 10 and q͞k 3͞16. Adding space charge shifts the curves to the left, which is expected since space charge is capacitive. These curves are shifted by k 2 q, which is slightly less than the space charge tune shift averaged over F 0 , which is k 2 3q͞4. For fixed q, the width of the dispersion curve is independent of k. Figure 7 shows similar curves for q͞k 3͞5.  The center of the curve is shifted by half the peak value of the space charge tune shift in the 2D beam. The shape of the distribution changes noticeably as well, but for jRe͑n 0 ͒j & q the threshold value of the transverse resistance is reduced.
For space charge dominated machines, reducing the effective space charge tune shift by half roughly doubles the intensity threshold estimates for transverse instabilities. An independent estimate of this effect can be obtained as follows. Consider a beam composed of rigid cylinders instead of points. The force per unit length of cylinder 1 on cylinder 2 is given by where n 1 ͑x͒ and n 2 ͑x 0 ͒ are the two-dimensional charge distributions of the generator and recipient of the Coulomb force, respectively. Assume n 1 ͑x͒ ͑3͞p͒ 3 ͑1 2 jxj 2 ͞R 2 ͒ 2 with a total charge of 1. The horizontal electric field generated by n 1 is proportional to the space charge force given by Eq. (39). Make the same approximation as before, neglecting the term proportional to xjxj 4 . Next, one calculates F 1,2 for n 2 ͑x͒ d͑x 2 x 2 ͒͞x 2 and n 2 ͑x͒ n 1 ͑x 2 x 2 ͒͞x 2 where x 2 is the offset . For jx 2 j ø R the total force on the extended distribution is exactly half the total force on the offset delta function. This coincides with the shift observed using the dispersion integrals and is easily extendable to other distributions. For a Kapchinskij-Vladimirskij distribution which has constant charge density within the beam, the force on the appropriate extended distribution is equal to the force on the delta distribution. This follows from Eq. (23) and is found in [1,2,5,6]. For a Gaussian cross section the force on the extended distribution is exactly half the force on a delta distribution with the same dipole moment. This last result is valid even if the beam is elliptical.
Finally, the effect of amplitude dependent tune spread due to octupoles or other nonlinear elements is addressed. These nonlinearities are with respect to the center of the beam pipe and introduce tune spread which can damp dipole modes. For one transverse dimension, nonlinearities add a term to the Hamiltonian U oct ͑J͒ rJ 2 ͞2J 0 , where r is proportional to the strength of the octupoles. Factorizing F 0 F 0 ͑J͒r͑D͒͞2p the calculation proceeds as before and Eq. (14) becomes The net effect is to make the substitution dU 0 ͞dJ ! dU 0 ͞dJ 1 rJ͞J 0 in the dispersion integrals for I k ͑n͒ in Eq. (28). Equations (31)   for either sign of r. However, for r , 0, space charge nonlinearity dramatically reduces the area of the stable region and r . 0 is the best choice. The octupoles should cause the tune to increase with betatron amplitude, as found previously [4]. In two dimensions, nonlinear elements add a term dH r xx J 2 x 1 r yy J 2 y 1 r xy J x J y to the Hamiltonian. Making all three coefficients positive requires three families of octupoles, and the benefits to stability need to be weighed against any reductions in dynamic aperture. This is especially important when space charge tune shifts *0.1 are expected. Another possibility is to exploit the effect of quadrupole end fields. Equation (17) in [14] shows that fringe field effects in quadrupoles almost always lead to a betatron tune increase with amplitude. While that reference suggested lengthening magnets to reduce nonlinear effects, introducing tune spread by using short, strong quadrupoles may be an attractive alternative to strong octupoles.

IV. CONCLUSIONS
A momentum and energy conserving model of the direct space charge force has been used to predict the effects of this force on transverse stability. It was found that the nonlinearity in the space charge force can modify dispersion diagrams even though it causes no Landau damping on its own. Even using a soft upper limit for the amount of space charge tune spread, it was found that space charge reduces stability for a reasonable machine impedance. When using traditional dispersion diagrams with smooth beams and tune spreads created by momentum spread, using half of the peak space charge tune shift in the dispersion relations gives fairly good estimates. Earlier work, showing that nonlinear elements should cause the betatron tune to increase in amplitude, were confirmed.

ACKNOWLEDGMENTS
The author thanks Deepak Raparia for careful reading of the manuscript. This work was performed under the auspices of the United States Department of Energy.