Resistive-wall instability evaluation along the ramp in the SOLEIL II booster

The ultra-low emittance specification of the SOLEIL II storage ring requires a challenging lattice design of the booster that will inject the beam into it. The dimension of the vacuum chamber in the new booster must be reduced compared to that in the present machine. The resistive-wall (RW) instability is then expected to become more important than in the current booster. However, the Amplitude-Dependent Tune Shift (ADTS) is also expected to be stronger due to the strong sextupole magnets necessary for chromatic error correction in the new lattice. It could then be an important effect to fight against this instability. Therefore, evaluating this instability is important to ensure the machine’s feasibility. This work studies the beam dynamics along the ramp in the RW instability regime using the code mbtrack2. The turn-by-turn tracking allows us to see the evolution of the beam thoroughly and understand how RW, synchrotron radiation, and ADTS impact the beam stability.


Introduction
SOLEIL is a 3 rd generation light source (GLS) that has been providing photon beams to external users since 2008.An upgrade project to a 4 th GLS called SOLEIL II had passed the CDR phase and is currently in the TDR phase [1,2,3,4] .A very low emittance booster is needed to be able to inject the beam into the new storage ring, such a very low emittance booster can only be achieved by using a multi-bend achromat (MBA) lattice [5,6] with a much-reduced vacuum chamber aperture of 32x22 mm 2 compared to the current machine of averagely 50x35 mm 2 .This raises concern regarding the transverse resistive-wall (RW) instability since its growth rate is inversely proportional to the vacuum chamber radius to the third power.
The RW instability refers to a multi-bunch transverse instability originated from the RW wakefields that can usually last for several turns inside the machine.The instability manifests as an exponential growth of the beam center of mass (CM) betatron oscillation.The formula for a uniform beam filling at zero chromaticity as Eqs.(31,32) in [7] can help us estimate that, for the SOLEIL II booster whose basic parameters can be found in Table 1, the threshold current to trigger the instability is only 0.6 µA (equivalent to charge 0.3 pC) and its growth time τ RW can be as short as 1.8 ms at the injection (150 MeV).This is relatively fast compared to the beam ramp-up time in the booster of 167 ms.At the same energy, the synchrotron damping time τ rad is 31 s for the vertical plane, making the radiation damping too weak to defeat the RW instability.
Although, at the extraction where the beam energy E 0 is high, the synchrotron damping should be strong enough to restrain the instability since τ rad ∝ E −3 0 [7] but τ RW ∝ E 0 [8], it was not certain how the beam would behave during the ramp-up process where both mechanisms compete with each other and other lattice parameters are ever-changing.In addition, strong sextupole magnets, which are required to correct the large natural chromaticity of a lowemittance ring, can have a significant impact due to the Amplitude-Dependent Tune Shift (ADTS) they can induce on the beam.This work's purpose is to simulate the beam dynamics using turn-by-turn tracking to investigate the stability of the beam during the ramp-up.

Booster ramp model
The lattice parameters during the ramp in the booster continuously change accordingly to the beam energy.The energy gain per turn (∆E) turn can be determined by the dipole magnet field ramp rate Ḃ (in T/turn) [10]: where ρ is the dipole bending radius, e is the particle charge, and L is the machine circumference.
The dipole field as a function of time can be written as where B max = 1.29 T is the maximum field, f rep is the repetition rate of the booster.In this study, it is assumed that the minimum field B min is zero for simplicity.But, it is possible in reality to choose B min > 0 to reduce the effect of Eddy's current.In fact, there will be another dipole family of which ρ = 9.44 m and B max = 0.97 T, but it also provides the same (∆E) turn curve due to the same ρ Ḃ as the first family.Any dipole can then be used to calculate the energy gain per turn.Another necessary parameter is the RF voltage V RF variation.The model used in this study can be expressed by the same equation as the dipole field, Eq. ( 2), by replacing B max by the maximum voltage V max = 3 MV, while the minimum voltage V min = 0. Note that V RF at the injection (25 ms after the start of the ramp) is 160 kV.The shape of both the dipole field and the RF voltage is, therefore, a sinusoid that passes through zero.After the variation of E 0 and V RF have been determined, the variation of other lattice parameters can be obtained via proportionality as follows [8]: Energy loss per turn : Radiation damping time : Natural energy spread : Natural H emittance : where superscript i denotes the values at the injection used as a reference point.Moreover, at any given turn, the RF cavity must be able to provide energy equal to (∆E) turn + U 0 , the synchronous phase ϕ s then needs to be varied as Here, the cosine RF convention is used.

Resistive-wall wakefield
For a circular thick-wall beam pipe of radius r with the electrical conductivity σ, the exact wakefield as a function of the position in time relative to the source particle τ can be written as Eq. ( 25) in [11] 1 .This equation can be approximated by an asymptotic formula when τ ≫ τ 0 as Eq. ( 26) in the same article, where τ 0 is the characteristic length in time depending on r and σ.
To calculate the wakefields of an elliptical beam pipe using the same equations, the equivalent radius r eq needs to be computed such that the wake of the circular pipe of radius r eq is equal to that of the elliptical pipe.Supposing the elliptical pipe has the vertical half-aperture size of b, the equivalent radius can be given by r eq = b/(Y D y ) 1/3 , where Y D y is the Yokoya's factor [12,13,14] for the dipolar vertical impedance.For the aperture size of the SOLEIL II booster beam pipe, it is found that Y D y = 0.8651 and r eq = 11.54 mm.In this study, the exact wakefield formula is used to calculate the short-range wakefields for the within-bunch interaction.The wake is applied to each particle according to its longitudinal position.Whereas the approximate one is used to calculate the long-range wakes for the interbunch interaction.Each bunch is considered as a point charge represented by its CM.The length of the long-range wake was found to give a converged result from 10 turns.Note that the bunch spacing is 2.84 ns which is much larger than τ 0 = 0.27 ps (determined by the pipe conductivity and r eq previously found), making the use of the approximate formula safe in this case.

Amplitude-Dependent Tune Shift
Several sextupole magnets are needed for chromaticity correction.Their strength will be increased along with the beam energy during the ramp to keep the chromaticity constant.This means the normalized force will also be constant.The tune shift ∆ν in the horizontal plane x and the vertical plane y can be given in a general form as where u and v can be either x or y, J is the Courant-Snyder invariant which is equivalent to the rms emittance in case of a multi-particle system, and C is the ADTS coefficient specific for each combination.Note that Eq. ( 8 The code mbtrack2 [15] was used to track the particles turn by turn along the ramp.The beam energy is increased each turn discreetly following Eq.(1).The particle tracking is done by the one-turn maps while taking into account both the short-range and the long-range RW wakefields.The lattice parameters are varied in each turn depending on the energy ratio as described in Eqs.(3)(4)(5)(6)(7).Several simulation results are shown in Fig. 1.
In case (a.1),where ADTS is absent and chromaticity ξ is zero, the beam exhibits a large inflation at the middle of the ramp before damping down at the end.This is caused by the competition between the RW instability growth rate τ −1 RW and the synchrotron radiation damping rate τ −1 rad .The former is inversely proportional to the beam energy (τ −1 RW ∝ E −1 0 ), whereas the latter is proportional to the beam energy cubed (τ −1 rad ∝ E 3 0 ).The turning point of the curve is where the radiation damping can totally compensate for the instability growth.The beam is then damped when the damping rate is larger than the growth rate.
It can be seen that chromaticity is important in fighting against the instability in the absence of ADTS.Only when the chromaticity is larger than or equal to 2 will the instability be totally suppressed and the emittance damp down normally.This result agrees with the well-known RW instability theory that the threshold current tends to increase at high chromaticities.Simulations done by using the Vlasov solver rwmbi [16] in case of MAX IV [11] and in case of the SOLEIL storage ring [17] also confirm this statement.The same conclusion was also drawn in case of SPring-8 [18].Although, these studies were done at a fixed beam energy, it clearly shows the weakening of the instability.
However, chromaticity loses its significance when the ADTS is included in the simulation, as shown in case (a.2) where no instability is observed regardless of the chromaticity.Here, the same ADTS coefficients are applied in every case since they are not significantly different among these ξ values.The impact of ADTS on the instability can be understood via the incoherent beam spectrum in Fig. 1 (b.2).At the injection, the beam has a large vertical emittance (about 0.7 µm rad, approximately one thousand times larger than at the extraction), the incoherent tune shift is then also large as described in Eq. ( 8) resulting in a wide tune spread, which is known as a necessary condition for the Landau damping effect [19].In other words, the instability is efficiently suppressed due to strong ADTS at the beginning of the ramp.Compared to case (b.1), even if the beam is injected with the identical emittance, but due to the lack of ADTS, the tune spread remains small, and therefore no Landau damping is present to fight against the instability.
The beam coherent spectrum in (c.1) confirms the instability that builds up in the absence of ADTS.As the chromaticity is zero, this spectrum shows purely the feature of coupled-bunch instability [20] caused by long-range RW wakefields.The overall spectrum amplitude is high and a dominant frequency, corresponding to the coupled-bunch mode n = 103, can be observed.On the contrary, in the presence of ADTS shown in (c.2), the spectrum amplitude at the same point on the ramp as (c.1) is low with no single frequency standing out, suggesting the beam is stable.A similar stabilizing effect of ADTS has also been shown in the case of single-bunch instabilities in the HEPS booster [21].

Conclusion
The RW instability could result in a severe beam blow-up in the SOLEIL II booster when the chromaticity is lower than 1 and the ADTS effect is neglected.However, it has been shown that the ADTS from the nominal sextupole strength can totally suppress the instability regardless of the chromaticity.A large injected emittance could also be of interest since the ADTS effect would be exploited when the RW instability is strong.

Figure 1 .
Figure 1.(a) The beam vertical emittance along the ramp (ξ denotes chromaticity).(b) The incoherent beam spectrum at various points during the ramp in case ξ = 0 (f 0 = 1.916MHz is the revolution frequency).(c) The coherent beam spectrum at 104 ms also when ξ = 0.Each has two sub-cases where the effect of ADTS was absent (1) and where it was present (2).

Table 1 .
Basic Parameters of the SOLEIL II Booster

Table 2 .
Table 2 shows the parameter values at the injection and the extraction retrieved from Eqs. (3-7).The injected beam parameters are also shown in the same table.Lattice Parameters and Beam Parameters