Single Bunch Stability to Monopole Excitation

We study single bunch stability with respect to monopole longitudinal oscillations in electron storage rings. Our analysis is different from the standard approach based on the linearized Vlasov equation. Rather, we reduce the full nonlinear Fokker-Planck equation to a Schr\"odinger-like equation which is subsequently analyzed by perturbation theory. We show that the Haissinski solution [Nuovo Cimento Soc. Ital. Fis. 18B, 72 (1973)] may become unstable with respect to monopole oscillations and derive a stability criterion in terms of the ring impedance. We then discuss this criterion and apply it to a broadband resonator impedance model.


Abstract
We study single bunch stability with respect to monopole longitudinal oscillations in electron storage rings.Our analysis is di erent from the standard approach based on the linearized Vlasov equation.Rather, we reduce the full nonlinear Fokker-Planck equation to a Schr odinger-like equation which is subsequently analyzed by perturbation theory.We show that the Haissinski solution 3 may become unstable with respect to monopole oscillations and derive a stability criterion in terms of the ring impedance.We then discuss this criterion and apply it to a broad band resonator impedance model.

Introduction
Single bunch longitudinal instability is one of the factors limiting the performance of electron storage rings.Theoretical analysis of this instability is usually based on the Fokker-Planck equation for the particle distribution function.This equation includes the e ects of both dynamic Hamiltonian and stochastic forces.The Hamiltonian part describes the synchrotron motion while radiation terms account for the much slower e ects of the synchrotron radiation and de ne the rms beam size at low intensity.Steadystate solution of the Fokker-Planck equation was rst obtained in 1973 by J. Haissinski 3 and since then it was con rmed in numerous experiments held below the instability threshold.Unfortunately, apart from a few limiting cases, nding other possible solutions of the Fokker-Planck equation that could account for the instability turned out to bequite di cult.This is why much of the analysis to explain the instability was done utilizing the linearized Vlasov equation technique, where the Fokker-Plank equation was linearized with respect to the Haissinski solution.In this approach the Haissinski solution is also used to introduce the action-angle variables that make the Haissinski Hamiltonian independent of angle, which results in great simpli cation of further analysis.The linearized Vlasov equation technique naturally leads to the concept of azimuthal phase space modes, that are basically the components of the perturbation to the Haissinski solution with certain azimuthal symmetry.The rst three of such modes are sketched in Fig. 1.Note, that in this gure and throughout the rest of the paper we assume the simplest phase space topology, where action-angle variables can be de ned uniformly across the whole plane.In other words, we neglect the possibility of several potential well minima.

Monopole
As seen from Fig. 1 the monopole mode is quite special because, in contrast to other modes, its physical space projection does not change signicantly on the time scale of a synchrotron period.This argues that radiation rather than Hamiltonian forces de ne the dynamics of this mode.Also, by de nition of action-angle variables, the unperturbed Haissinski solution has monopole azimuthal structure.These two features of the monopole mode explain why it is omitted from the standard linearized Vlasov analysis.Indeed, in that approach the sole e ect of radiation terms in the Fokker-Plank equation is that they de ne the Haissinski solution which subsequently cancels them out, so that only Hamiltonian terms remain in the linearized Vlasov equation.The possibility, that a perturbation is monopole, but with radial structure di erent from the Haissinski solution, is neglected.
In this paper we are exploring the possibility that an instability can be associated with the monopole mode.Rather than extending the linearized Vlasov technique we nd it more convenient to transform the Fokker-Plank equation to a Schr odinger-like equation and then analyze the latter using the Haissinski solution as a basis.Advantages of this approach are that it is tractable and it conveniently allows us to use some well known facts about Schr odinger equation solutions.
The only essential approximation that we make in this paper is that we assume that the monopole mode can be considered separately form other azimuthal modes.This is, by no means, general.On the contrary, it is known that, for example, some collective instabilities result from azimuthal mode coupling, hence, concentrating on one mode in that case would be inappropriate.However, at lower intensity, when incoherent frequency shifts are small compared to the synchrotron frequency, the cross talk between di erent azimuthal modes is negligible.Whether a monopole mode or any other single azimuthal mode can become unstable at this low i n tensity is, in our opinion, a quantitative question that depends on the exact measure of the storage ring impedance.In fact, there are computer simulations for model impedances 2 showing that radial modes that belong to one azimuthal mode become unstable before any signi cant azimuthal mode coupling occurs.Of course, if a monopole mode does become unstable by itself, its independence from the other modes applies only to the initial stage of instability.The nonlinear stage may berather complex and it falls beyond the scope of this paper.
Another argument to justify an independent consideration of the monopole mode refers to separate time-scales in this problem.Speci cally, i f w e assume that the monopole mode instability develops slowly compared to the lamentation time, then the whole beam e ectively maintains its initial monopole structure.Quantitatively, this amounts to a requirement that the incoherent frequency spread characteristic of the Haissinski equilibrium be much higher than the radiation damping rate.Such a condition is not unusual for many electron storage rings.

Notation and basic equations
For a relativistic bunch, = E=mc 2  1, longitudinal dynamics is conveniently described in dimensionless variables x = z= 0 ; p=, = 0 ; ~ =! s 0 t; 1 where z is position of a particle with respect to the bunch centroid z 0 in the head of a bunch, is the relative energy spread E = E ,! s 0 is the synchrotron frequency.The subscript "0" refers to zero-current equilibrium quantities, related by !0s 0 =c = j j 0 , where is the momentum compaction.
The Fokker-Planck equation for the distribution function x; p; ~ in these variables can bewritten e.g. 1 as @ @~ +f H; g p;x = d !s0 @ @p @ @p +p where N is number of particles in a bunch, r 0 is the classical electron radius, and C is the ring circumference.We have also de ned a dimensionless function Sx 0 Z x 0 dx 0 W 0 x 0 ; 5 in terms of the wake eld Wz for two particles separated by distance z.Causality w as assumed in the form Wz = 0 for z 0. Haissinski particle density H in these variables depends only on J, and arbitrary distribution function J; ; ~ = H J + J; ; ~ can be expanded in azimuthal harmonics J; ; ~ = H J + : 11 In the J; v ariables the form of the left-hand-side of the Fokker-Planck equation is unchanged, appearing as in Eq. 2. The right-hand-side can be obtained using the invariance of the Poisson brackets 4 .Namely, for any F Fx; p, @Fx; p @p =f x; Fg x;p = fx; Fg ;J @ @J @x @ F ! , @ @ @x @J F ! : 12 Hence, for the zero-th Fourier harmonic 0 J; ~ h J; ; ~ i , where h:::i de ne phase averaging, we have @ 0 @~ +f H; 0 g J; = @ @J @x @ F! ; 13 where F @ =@p + p .On the other hand, for any Hamiltonian H = p 2 =2 + U x; ~ the canonical momentum can be found as p = fx; Hg ;J .
Therefore, if we neglect non-zero azimuthal modes by assuming H = HJ; ~ , = 0 J; ~ , we get @ @ = @ @J " D @x @ 2 Eh @ @J +! J ; i : 14 where !J; @H @J =! H J + Z dJ 0 0 J 0 ; @S 0 ; 0 @J : 15 As discussed earlier, the monopole mode should not change much on the time scale of a synchrotron period.This is why time was renormalized above to the radiation damping constant 16 For the x derivative in Eq. 14 we can write 17 where integration is performed over one synchrotron period and the last equality assumes small deviation from the Haissinski solution.Finally, i n troducing the di usion coe cient as DJ J ! H J ; 18 we can rewrite Eq. 14 as @ @ = @ @J DJ h @ @J +! J ; where Jy in the right-hand-side of Eqs.21-22 is given implicitly by Eq. 20.Now the Fokker-Planck equation Eq. 19 takes the form @f @ = @ 2 f @y 2 ,U S y; f+ 1 2 _ y; f; 23 where U s y; 0 y; =2 2 , 00 y; =2 24 and dot and prime denote partial derivatives with respect to and y respectively.This equation is strongly nonlinear since, according to Eq. 15, is also related to f by y; = Z dy 0 S 0;0 J; J 0 e ,y 0 ; =2 fy 0 ; , e , H y 0 = 2 f H y 0 ; 26 Note, that f H y is the steady-state solution of Eq. 23 and it corresponds to the Haissinski solution, as can bechecked by direct substitution.However, it is not obvious how to proceed to other, time-dependent solutions of Eq. 23.On the other hand, if the _ term is neglected, the analysis of Eq. 23 can bemade by analogy with more familiar quantum mechanical problems.Indeed, without that term Eq. 23 can be thought o f a s a S c hr odinger equation for a particle in the potential well U S y; .y Whether the _ term y Of course, the similarity is quite formal.The problem is purely classical, so no ãppear anywhere.Also, there is no imaginary constant in front of the time derivative.
is negligible for a general case remains unclear.However, since this term is zero for the Haissinski solution, one can safely neglect it for solutions that are close to f H .This includes, for example, the important case of the early time behavior of a system initialized with the Haissinski distribution at = 0.
In the next two sections we will follow this approach.Namely, we will rst analyze Eq. 23 with the last term neglected, and then account for it by perturbation theory.

Schr odinger equation analysis
After neglecting the _ term Eq. 23 reads @f @ = @ 2 f @y 2 ,Uy; f: 27 First, we solve a linear problem for which !J = ! 0is a constant.In this case y = 2 p ! 0 J and the Schr odinger potential is simply U 0 S y = y 2 16 , 1 2 , 1 4y 2 ; 28 which makes Eq. 27 a solvable eigenvalue problem.It is easy to check that the solution is 0 m ye , 0 m ; 29 where 0 m = m, m = 0 ; 1 ; 2 :::; 30 0 m y = y=2 1=2 e ,y 2 =8 L m y 2 =4; 31 and L m denotes the Laguerre polynomial of order m.As expected, the linear problem does not have a n y unstable solutions.Any initial distribution exponentially approaches the Haissinski solution 0 0 y on the time-scale de ned by radiation damping.
As an example, we consider a broad-band resonator impedance model with shunt impedance R s , resonance frequency !R , and quality factor Q. 39 and Z 0 is the impedance of free space.The frequency !H J and the e ective potential V S y for the parameters Q = 1 and = 3 are shown in Fig. 2 for two values of intensity I = 1 and I = ,1.Negative intensity corresponds to negative momentum compaction and the importance of this case will be discussed later.For these parameters the e ective potential indeed has a minimum where V S y 0. On the rst sight, we could expect a mode trapped near the bottom of the potential well with the eigenvalue negative at large current.This, however, is not true since 0 = f H is the solution of Eq. 33 with 0 = 0 .Because f H y does not have zeros, this solution has the lowest eigenvalue and the rest of m are positive.Therefore, in this approximation, the Haissinski solution is stable.How much this conclusion depends on the assumption that the _ term in Eq. 23 is negligible can beanalyzed by perturbation technique.

Perturbation theory
Eq. 23 and the condition of self-consistency Eq. 25 is a strongly nonlinear system of equations.We want to analyze it with perturbation theory, D n e H y=2 n : 44 We assume that eigenfunctions n satisfy the equation The solution of this equation is given by the roots of the determinant for the matrix M m;n + 2 n;k k + k : 51 Positive roots 0 would mean instability to monopole excitation of a bunch.
Since the matrix M is in nite it is unclear how to nd its determinant i n the general case.However, it is easy to see, that o -diagonal terms of n;k are small, while the diagonal terms quickly converge to zero.This is why w e expect that a good approximation for the roots can be found by truncating the matrix M. If we truncate matrix M to the lowest nontrivial rank then zero determinant occurs for = , 1 1 + 2 1 ; 1 53 Because 1 0, this root is positive when 54 and this may be viewed as the criterion for the onset of the monopole instability.
It is easy to see, however, that usually 1;1 0. Indeed, let's introduce impedance Z so that Wz = Z d 2 Ze ,iz=c : 55 Z d e i 0 xJ; =c : 57 Now 1;1 depends on jF 1 j 2 which is positive and even function of .Hence, 1;1 is given by the odd part of the impedance, I m Z which is negative for inductive impedance and positive for capacitive impedance.As a result, for the most common case of inductive impedance and positive momentum compaction, 1;1 0 and the Haissinski solution is stable.!J w as assumed constant and 1 was taken from the solution to the linear problem Eq. 31.
However, the situation is not that simple for negative momentum compaction or in the case of capacitive impedance.Each of these conditions have been proposed by various authors to get shorter bunches and also as a remedy against longitudinal instabilities.For illustration, we continue our example of the broad band resonator model for Q = 1 .Using Eqs.56-57 we numerically compute the quantity 2 1;1 as function of at intensity I = ,1.The result together with the threshold value given by Eq. 54 is plotted in Fig. 3.It shows that a bunch is monopole unstable at this intensity provided its zero current length exceeds about one twelveth of the resonator wavelength.Note, that according to Fig. 2 this intensity is not high at all since, for example, for = 3 , it only leads to about 5 increase in the incoherent frequency spread.

Discussion
We have investigated single bunch stability with respect to longitudinal monopole oscillations.These oscillations may become unstable as a result of an imbalance between radiation excitation and damping.Since this phenomenon falls beyond the coverage of the linearized Vlasov approach, we chose to employ a di erent technique that has not been used for instability analysis.This technique involves the transformation of the phase-averaged Fokker-Planck equation to a Schr odinger equation with an additional term arising from the self-consistent potential.This Schr odinger equation is analyzed similar to quantum mechanics and the e ects of the additional term are found by perturbation analysis.
Utilizing this technique we have obtained a simple criterion, Eq. 54 for the onset of monopole instability.According to this criterion, monopole mode instability does not appear in the most common case of storage ring operation with positive momentum compaction when the impedance is largely inductive.However, for the case of negative momentum compaction 0, as we have illustrated in Fig. 3, bunches may become monopole unstable at modest intensity.We expect a similar behavior for the somewhat rare case of predominantly capacitive impedance and 0. As discussed in the introduction, the essential assumption we make in our analysis is that the monopole mode can be considered separately from the rest of azimuthal modes.This assumes the absence of azimuthal mode coupling which implies some limitations on the growth rate of the instability.
It also assumes that the other azimuthal modes are stable by themselves.It is interesting, that since the monopole instability criterion Eq. 54 e ectively includes only the imaginary part of impedance, the second assumption is rather relaxed.Indeed, as follows from the linearized Vlasov analysis, the azimuthal modes other than monopole, become unstable due to the asymmetry in the Haissinski potential that comes from the real part of impedance We omit a somewhat exotic possibility o f m ultiple minima in the Haissinski potential.Therefore, monopole mode instability can exist when the remaining azimuthal modes are stable.
It is conceivable, that the monopole instability could be one of the factors that prevent high current operation of storage rings with negative momentum compaction.Many attempts of such operation have been tried to shorten a bunch and to avoid various instabilities e.g. 7 , 8 , often the so-called microwave instabilitye.g.6 .Unfortunately, since, usually only the static bunch shape and or the energy spread measurements are reported, it is hard to infer what particular instability was the limitation.However, in some cases, it appears that there is something other than the microwave instability, because the threshold reduction predicted for this instability e.g. 9 is not observed.It would be nice to nd a concrete evidence of monopole instability in either future experiments or in the log books from the past ones.Such evidence might include, for example, growth of the longitudinal beam size, in the absence of synchrotron sidebands to the revolution harmonics of a beam position monitor signal.
Finally, w e hope that the technique described in this paper can be applied to other problems in accelerator physics that lead to the one dimensional Fokker-Planck equation.This include, for example, microwave instability, beam-beam interaction in collider rings, and even halo formation in rings and linacs.It would beespecially interesting if, for any of these problems, along with a steady state solution with = 0 , there exists an exponentially growing solution of the Schr odinger equation that has a negative eigenvalue 0. This could qualitatively explain the relaxation oscillation behavior seen in many numerical and real life experiments.

Figure 1 :
Figure 1: Example contour plots of the lowest three azimuthal modes

2 2
::g denotes the Poisson brackets, Hx; p; ~ is the self-consistent Hamiltonian Hx; p; ~ p 0 x 0 ; p 0 ; ~ S x 0 , x : 3 and is normalized as R dpdx x; p; ~ = 1.We have neglected the nonlinearities of RF potential well and de ned the parameter as e im 0 ,ik S xJ 0 ; 0 , x J ; in our derivation we allow arbitrary time dependence of HJ; .4Transformation to a Schr odinger-like equationThe Fokker-Planck equation Eq. 19 has a standard one-dimensional form that permits transformation to a Schr odinger-like equation 5 .Let us intro-

Figure 2 : 2 Z
Figure 2: Incoherent frequency !H J and corresponding e ective potential V S y right shown for broad band Q = 1 resonator impedance with = 3 and two values of intensity I = 1 and I = ,1 dashed 00n y , U H y n y = , n n y :45The above expansion includes the eigenfunction 0 y = p Z H e , H y=2 with 0 = 0. Note, that by the same argument as above, the remaining eigenvalues are positive.The linearized Fokker-Planck equation Eq. 43 and the condition of selfconsistency Eq. 41 lead to the following system _ y 0 e , H y 0 =2 S 0;0 Jy; J y 0 : 48 Due to orthogonality, the normalization condition Eq. 42 gives just C 0 = D 0 : 49 Since 0 = 0 this is automatically satis ed by Eq. 46 with the initial condition C 0 0 = D 0 0.Looking for exponentially varying solutions C n a n e ; D n b n e we transform the system Eqs.46-47 to the matrix equation b

Figure 3 :
Figure 3: Illustration for the monopole instability criterion Eq. 54 for broad band Q=1 resonator impedance for negative momentum compaction and intensity I = ,1.Numerical calculation was done using Eqs.56-57 where The function Sz in this case is e.g. 6