Fast-slow mode coupling instability for coasting beams in the presence of detuning impedance

A destabilizing effect of the detuning impedance has been recently observed in simulations of the CERN Proton Synchrotron (PS) at the injection energy: while without the detuning impedance the instability is faster in the vertical plane as expected (due to the elliptical shape of the vacuum chamber), with detuning impedance the instability appears to be faster in the horizontal plane. In order to understand the detuning impedance destabilizing effect, we study the collective behavior for the simpler case of a coasting beam with PS-like parameters and a simplified impedance model. The analysis, carried out from both numerical and theoretical points of view, highlights a new destabilizing mechanism related to the coupling of slow and fast waves.


I. INTRODUCTION
The CERN Proton Synchrotron (PS) provides high intensity proton bunches within the injector chain to the Large Hadron Collider (LHC). The machine and beam parameters have been recently pushed in order to meet the challenging goals of the LHC injectors upgrade (LIU) project [1,2] started in 2010. For the future operation at higher intensities and beam brightness required by the LIU project, it is of fundamental importance to detect and cure the observed machine instabilities along the LHC beam production cycles. Headtail horizontal instabilities have been observed, measured and systematically characterized versus chromaticity at the injection kinetic energy of 1.4 GeV with a bunched beam in [3]. The analysis of the instability based on Sacherer's theory [4] was found to be in good agreement with measurements concerning the headtail mode number [3]. One of the possible explanations for the horizontal plane to be more unstable than the vertical one (which is not expected for a flat vacuum chamber, as, in first approximation, the one of the PS) was a different chromaticity between the two planes. The mode pattern and behavior with chromaticity was also studied with a macroparticle code [5]: HEADTAIL [6] simulations were performed scaling the PS resistive wall wakefield to reduce the simulation time and without accounting for the effect of the direct space charge and detuning impedance. The simulation results confirmed the measurements [3]. Recently, the instability was simulated with the PyHEADTAIL code [7] by using the PS impedance model developed in [8,9] and it revealed a destabilising effect of the detuning impedance for some chromaticities in the horizontal plane, otherwise more stable than the vertical plane in the presence of only driving impedance [10].
In order to understand the instability mechanism and to investigate why the horizontal plane is destabilised by the detuning impedance, a series of studies was recently performed at CERN. The effect of the detuning impedance was investigated with a two-particle model [11], with systematic bunched beam PyHEADTAIL simulations [12], with the eDELPHI Vlasov solver [13], and with the circulant matrix formalism [14,15].
In this context, we analyze the effect of the detuning impedance on a coasting beam with PS parameters and a simple impedance model of two resistive parallel plates. On one hand, this approach offers the advantage of a simpler collective effects treatment to investigate if a destabilising effect of the detuning impedance can already be observed in the case of a coasting-beam (with the simplified impedance model) and for zero chromaticity. On the other hand, it does not aim to directly explain the PS instability, even if it shows that indeed a destabilizing effect is present also in this case.
The theoretical background for the study of coasting beam collective effects is described in Sec. II. Simulations accounting for the driving impedance only will be compared with the classical theory developed in [16][17][18]. In Sec. III the theory is extended in order to include the detuning impedance (not accounted for in the classical treatment [16][17][18]). These modifications are essential ingredients to be able to reproduce simulations performed in presence of both driving and detuning impedances. In Sec. IV the theory is further extended to account possible couplings between fast and slow coasting beam betatron waves: the fast-slow mode coupling instability is described and benchmarked with PyHEADTAIL simulations. Section V summarizes the results described in the previous sections.

II. TRANSVERSE COASTING BEAM INSTABILITY SIMULATIONS WITH DRIVING IMPEDANCE
In this section we introduce the theory and the simulations of coasting beam transverse instabilities following the approach of [17]. This will be the starting point for the following developments presented in Sec. III to correctly account for the additional effect of the detuning impedance.
Let us consider a coasting proton beam of charge qN with q the elementary charge and N the number of charges in the beam. The line density distribution ρðsÞ is supposed to be constant and equal to ρðsÞ ¼ qN=C where C is the accelerator circumference. We consider a transverse perturbation dðs; tÞ as a function of space s and time t with the form of where R ¼ C=2π is the average radius of the machine, n an arbitrary integer number and Ω is the angular frequency of the perturbation. Equation (1) describes a wave of n periods within the accelerator circumference, oscillating with the collective frequency Ω.
The dipole moment of the coasting beam is given by Given a beam circulating with velocity v ¼ βc with β the relativistic factor and c the speed of light, a particle in s 0 at t ¼ 0 will move according to Dðs 0 þ vt; tÞ.
Let us now consider a particle performing betatron oscillations, for example, in the vertical plane y, and perturbed by the wakefields produced by the perturbation. The equation of motion can be written as [17] yðs; tÞ þ ω 2 β yðs; tÞ ¼ where ω β ¼ Q y ω 0 is the vertical betatron angular frequency with Q y the machine vertical tune and ω 0 the revolution angular frequency, γ is the relativistic Lorentz factor, m 0 the rest mass of the particle, hF y iðs; tÞ is the average force acting on the particle motion, and _ y means derivation of y with respect to t.
In the frame of wakefields, the force induced by a source particle on a following test particle at distance z > 0, can be expressed as whereȳ and y respectively indicate the displacement of the source and test charges. The Taylor expansion around the center of the beam vacuum chamber gives where we neglected the constant term for simplicity. We define W driv y ¼ ∂W y =∂ȳj y¼0 the driving wake function and W det y ¼ ∂W y =∂yj¯y ¼0 the detuning wake function [19] in units of V/(C m). We introduce as well the driving and detuning impedances as the Fourier transform of their corresponding wake functions: Considering, for the moment, the absence of detuning wake, and the perturbation defined in Eq. (2), the total force acting on a charge at a fixed point in the accelerator can be rewritten as Using the impedance definition of Eq. (6) we have hF driv y iðs; tÞ ¼ − qv C Dðs; tÞjZ driv y ðΩÞ: ð8Þ This force acts on a charge at a fixed location into an accelerator. A particle that passes in s ¼ s 0 at t ¼ 0 will be affected by a force at the time t equal to hF driv y iðs; tÞ (3) can then be written as yðs; tÞ þ ω 2 β yðs; tÞ ¼ − qv γm 0 C Dðs 0 þ vt; tÞjZ driv y ðΩÞ: ð9Þ Assuming yðs; tÞ ¼ Ae jððΩ−nω 0 Þt−ns 0 =RÞÞ as trial particular solution of the differential equation, we can find the constant A substituting the expression into Eq. (9) and getting To be self-consistent with the initial assumption of coherent steady-state oscillations of Eq. (2), yðs; tÞ is also equal to From Eqs. (11) and (10), we get the dispersion relation Let us now solve for Ω. We rewrite Eq. (12) as from which, defining Ω ¼ nω 0 − ω β þ ΔΩ n , and assuming ΔΩ n ≪ nω 0 − ω β , we have The rise time, in number of turns, associated to each line is defined as τ n ¼ −1=ImðΔΩ driv n Þ=T 0 and we define the normalized frequency shift as ReðΔΩ driv n Þ=ω 0 . Having chosen n as an integer number, the solutions with n − Q y > 0 will sample ReðZ driv y Þ > 0 and the waves are therefore unstable (so called slow waves) with positive rise time, while solutions with n − Q y < 0 will sample ReðZ driv y Þ < 0 and the waves are stable (so called fast waves) with negative rise time.
The PyHEADTAIL code was originally conceived to simulate the collective effects of bunched beams only [6]. We have introduced some modifications to adapt the code to simulate coasting beams: the longitudinal charge density covers the full machine circumference, a simple drift replaces the rf focusing, the wake interaction is computed with the inverse Fourier transform of the product between the impedance and the charge transverse dipole distribution.
In order to benchmark the new PyHEADTAIL code with the complex mode shift of Eq. (14) we assumed a uniform longitudinal charge distribution with neither momentum spread nor chromaticity.
In the frame of this study, we consider the resistive wall impedance of a flat beam pipe with PS-like parameters: two parallel plates of stainless steel (electrical resistivity of 7.2 × 10 −7 Ωm) with 27.5 mm half gap, infinitely thick and 628.32 m long. The impedance is computed with the ImpedanceWake2D code [20,21]. Figure 1 shows the computed driving and detuning impedances: as expected the vertical detuning impedance is half the driving one in the same plane, while the horizontal detuning impedance is opposite in sign to the driving one in the same plane; the driving horizontal is half the vertical one [22]. More accurate models for the PS transverse impedance are available [9,[23][24][25] but beyond the scope of this work. It is important to notice the inductive bypass effect toward low frequencies [26,27], at which the effect of the image charges dominates the imaginary part of the impedances.
For the simulation, we assume equal working points in the horizontal and vertical planes Q x;y ¼ 6.4 but no coupling between the two planes. The beam is approximated by 50000 macroparticles, sliced with 128 bins and tracked for 2000 turns with variable intensity ranging from 1 × 10 13 to 1 × 10 14 charges. Figure 2 shows the amplitude spectrum of the vertical plane as a function of the number of turns: since the coasting beam spectrum samples the impedance in the classical thick wall regime, i.e., a monotonically decreasing real part versus frequency, the lowest frequency slow wave corresponding to n − Q y ¼ 0.6 is the most unstable. Figure 3 shows the rise time of the instability (at the top) and the normalized frequency shift of the most unstable mode (at the bottom), compared to Eq. (14). The agreement is excellent and it can be seen that, as expected, for all the intensities the vertical plane is more unstable than the horizontal one. It is important to notice that the resolution on the normalized frequency shift has been improved over the classical FFT algorithm by using a harmonic analysis method based on iterative frequency interpolation [28].

III. TRANSVERSE COASTING BEAM INSTABILITY SIMULATIONS WITH BOTH DRIVING AND DETUNING IMPEDANCES
We now include the effect of the detuning impedance, first analytically following a treatment similar to the one discussed in Sec. II, then with macroparticles simulations.
From Eq. (5), the infinitesimal source charge ρðs; t 0 Þvdt 0 induces a detuning wakefield on the test particle yðs; tÞ given by dF det y ðs; tÞ ¼ − q C yðs; tÞW det y ðvt 0 − vtÞρðs; t 0 Þvdt 0 : Integrating over time we get the total force as hF det y iðs; tÞ ¼ − q C yðs; tÞ Differently from the driving force, which is produced only by the perturbation, the detuning force depends on the unperturbed longitudinal distribution ρðs; tÞ. When this is constant, the detuning force simplifies into hF det y iðs; tÞ ¼ − From the definition of impedance of Eq. (6) we have In turn the force can be expressed as hF det y iðs; tÞ ¼ − In presence of detuning impedance only, Eq. (3) is written as follows yðs; tÞ þ ω 2 β yðs; tÞ ¼ − Gathering the coefficients of yðs; tÞ, the effect of the detuning impedance is to add an additional tune shift to the bare machine working point. The new working point is defined as ω 0 β ¼ ω β − ΔΩ det n . Solving Eq. (20) for this new frequency and neglecting second order terms in ΔΩ det n , we have It is interesting to notice that in the specific case of the resistive wall impedance, the thick-wall regime has 1= ffiffiffi ffi ω p frequency dependence [29], i.e., with a singularity at DC, that disappears only if the inductive bypass effect is taken into account [26,27]. In other words, it is necessary to have the full/correct detuning impedance description up to DC (this is not the case for the driving impedance as the lowest needed frequency is the one of the lowest unstable mode).

IV. THE FAST-SLOW WAVES COUPLING
Let us now consider the effect of both driving and detuning impedances. Comparing respectively Eqs. (9), FIG. 3. PyHEADTAIL simulations (with dots) compared to theory [16] for driving impedance only (with full lines) for the horizontal (blue) and the vertical (red) planes. The rise time of the most unstable mode is shown at the top while the normalized frequency shift is shown at the bottom. (11) and (14) for the driving force, and Eqs. (20), (21) for the detuning force, we can writë yðs; tÞ þ ω 2 β yðs; tÞ ¼ 2ω β ðΔΩ driv n þ ΔΩ det n Þyðs; tÞ: In Sec. III we described the effect of the detuning impedance on transverse coasting beam instabilities and the way to correctly account for it from the analytical point of view. As the detuning impedance introduces a large frequency shift due to the sampling at DC, we investigate the possibility of fast-slow waves coupling, so far not investigated in the available literature.
Let us consider two nearby waves, namely a slow wave y 1 ðs; tÞ corresponding to n ¼ n 1 and a fast wave y 2 ðs; tÞ corresponding to n ¼ n 2 . Dropping the dependence on s and t, the coupled system can be written as where ΔΩ tot n 1 ;n 2 ¼ ΔΩ driv n 1 ;n 2 þ ΔΩ det n 1 ;n 2 . The nontrivial solution of the system represents the solution for the coupled waves. Looking for the solutions in the form of y 1;2 ¼ e jðΩ−n 1;2 ω 0 Þt we get the matrix system Looking for the nontrivial solutions of the system, for example using [30], we obtain the coupled complex frequency shift of the fast-slow waves. Figures 4 and 5 show, respectively for the horizontal and vertical plane, the normalized frequency shift and the rise time of the instability accounting for the coupling between adjacent fast and slow waves. As can be seen, the prediction from the coupled waves approach of (24) is approximated by the uncoupled one of Eq. (14) only for intensities below 1 × 10 13 charges. At 6 × 10 13 the horizontal plane shows coupling between fast and slow waves, with consequent shorter instability rise time. In the vertical plane, the fast and slow waves are repelling each other, and the difference in complex frequency shift is negligible. Figure 6 shows the comparison between Eq. (24) and PyHEADTAIL simulations performed with both driving and detuning impedances. The agreement is excellent and shows the importance of considering the coupling of fast and slow waves in order to correctly predict the rise time and frequency shift of the instability. This mechanism is conceptually analogous to the one of a transverse mode coupling instability (TMCI) occurring when bunched beam modes couple with each other [18]. Similarly, coupling occurs between fast and slow waves in coasting beams.

V. CONCLUSIONS
In this paper we described the effect of the driving and detuning impedance on transverse coasting beam instabilities.
The PyHEADTAIL code has been extended to include the study of coasting beam collective effects in presence of wakefields. The code was successfully benchmarked with the already developed theory of instabilities induced by the driving impedance.
The effect of the detuning impedance was studied both analytically and numerically: the impedance is sampled at DC for a perfect coasting beam, and can therefore induce significantly larger frequency shift compared to the driving impedance only. This result is presently being investigated following also the Vlasov's formalism and will allow future studies on the effect of a finite momentum spread and chromaticity.
This study revealed the possibility of a new instability mechanism: the coupling between adjacent fast and slow waves. The mechanism has been successfully reproduced with PyHEADTAIL numerical simulations.
This work supports, from a different point of view, the necessity of having a correct description of the inductive bypass impedance of a machine (i.e., a correct description of the impedance down to DC).