Multi-particle simulation of the fourth-order particle resonance

As discussed in the previous section, self-consistency was not included for the analytical investigation by using particle-core model, so that beam emittance was assumed to be constant. However, in the real beam propagation, particle resonance and parametric instability are accompanied by emittance growth and halo formations. The first experimental observation of the 4σ=360^◦fourth-order particle resonance was reported in Ref. [28]. Figure 6 of Ref. [28] shows that there is no sign of emittance growth when zero-current phase advance is below 90^◦. Numerical simulation studies for finding the lower boundary of the fourth-order particle resonance stop band were shown in Fig. 5 of Ref. [53]. The emittance growth

from the fourth-order particle resonance takes place forσ⊥0>90^◦independent of the beam current.

Figure 4.3: Numerical simulation results of the change of RMS emittance, phase advances, and (x−x^′) phase space plots for 4 different cases along the solenoid focusing linacs. (a)σ⊥0is set to be constant, (b) σ⊥ to be constant, (c)σ⊥0 changes above 90^◦, and (d) σ⊥0 increases above 90^◦. In (a) and (d), the envelope instability occurs following the fourth-order particle resonance. In (b) and (C), only the fourth-order particle resonance is manifested along the linac.

It has been established that the 4σ =360^◦ fourth-order particle resonance is predominantly man- ifested over the envelope instability when σ⊥ is kept constant along the linac [31]. Depend on the accelerator design structures that set the conditions ofσ⊥0andσ⊥along the linac, the excitation of the fourth-order particle resonance and the excitation of the envelope instability can be observed or not.

Figure 4.3 shows the numerical simulation results of the change of RMS emittance and phase advances for 4 different cases along the solenoid focusing linacs.

Figure 4.3 (a) : σ⊥0=100^◦ is constant along 200 lattice periods. Along the 50 periods, the 4σ = 360^◦fourth-order particle resonance is manifested and beam draws four-fold structures on the (x−x^′) phase space. The resonance particles induce the beam mismatch and accordingly excite the envelope instability asσ⊥ approaches 90^◦. The maximum emittance growth rate accompanied by the envelope instability is 2.08 over 200 lattice periods.

Figure 4.3 (b) : σ⊥ ∼70^◦ is set to be constant along 200 lattice periods and σ⊥0 decreases from 108^◦to 98^◦. The 4σ=360^◦fourth-order particle resonance is maintained for over 200 periods and the envelope instability is not excited after all. Emittance growth from the fourth-order resonance is not dominantly shown than the case of Fig. 4.3 (a).

Figure 4.3 (c) : σ⊥0changes above 90^◦andσ⊥increases above 90^◦along 200 periods. In this case, the envelope instability is not excited whileσ⊥0decreases significantly and gets closer to 90^◦.

Figure 4.3 (d) : σ⊥0increases constantly. The 4σ =360^◦ fourth-order particle resonance is mani- fested over short periods and the envelope instability is excited asσ⊥increases approaching to 90^◦.

As shown in Fig. 4.3, the 4σ =360^◦ fourth-order particle resonance is predominantly manifested

over the envelope instability and the envelope instability occurs following the fourth-order resonance accompanied by large emittance growth after long periods. Also, the stop band of the fourth-order particle resonanceisσ⊥0>90^◦andσ⊥<90^◦, whereGletter is omitted in general convention.

When fourth-order particle resonance is manifested under the conditions ofσ⊥0>90^◦andσ⊥<90^◦, the resonant particles trapped within four separate islands oscillate and merge together, incurring beam emittance growth. If the emittance grows sufficiently to generate beam mismatch, it rapidly excites the envelope instability after all. However, the question remains that within the stop band of the fourth- order particle resonance,does the emittance growth always excite the envelope instability after the fourth-order particle resonance is manifested? The answer is no. It is clear that the fourth-order resonance stop band in the tune depression space is wider than the envelope instability stop band, which is represented by growth factor as discussed in Sec. 3.1.

Figure 4.4: Plots of RMS emittance growth, phase advance, and mismatch factor with cross sections of particle distributions on (x−x^′) phase space for initially well-matched Gaussian beams over 200 solenoid focusing lattice. (a)σ⊥0=89^◦ andσ⊥=74^◦, there is no fourth-order resonance. (b)σ⊥0=100^◦and σ⊥=72^◦, the fourth-order particle resonance is predominantly manifested and the envelope instability is excited after 100 lattice periods accompanied by beam mismatch and large emittance growth. (c) σ⊥0=100^◦andσ⊥=68^◦, the fourth-order particle resonance is remained for over 200 lattice periods without excitation of envelope instability.

Figure 4.4 shows the change of RMS emittance growth, phase advance, and mismatch factor with cross sections of particle distributions on (x−x^′) phase space for initially well-matched Gaussian beams over 200 solenoid focusing lattice.σ⊥0is kept constant along the linac as the same case with Fig. 4.3 (a).

The numerical simulations are done by TraceWin PIC code [54]. Here, the mismatch factor is defined byM= [1+^∆+

√

∆(∆+4)

2 ]^1/2−1, where∆= (∆α)²−∆β∆γ, and∆α,∆β, and∆γare the differences of

Twiss parameters from the matched beam [35].

Figure 4.4 (a) : σ⊥0=89^◦ andσ⊥=74^◦, no fourth-order particle resonance and no envelope in- stability have manifested. Beam stays stable becauseσ⊥0 is lower than 90^◦, which is out of range of resonance stop bands.

Figure 4.4 (b) : Initial condition withσ⊥0=100^◦andσ⊥=72^◦. Between 10 to 50 lattice periods, the fourth-order particle resonance is manifested denoted by the four-fold structure on phase space of cell 20. The resonance particles make emittance grows and accordingly beam mismatch is generated.

The mismatch factor increases over 0.1 reaching the maximum value 0.43, which is enough to generate the envelope instability. After cell 50, the four-fold structures get blurred and the envelope instability occurs with an emittance growth up to 2.27 times of the initial value.

Figure 4.4 (c) : Initial condition withσ⊥0=100^◦andσ⊥=68^◦, where the beam current is higher than the case of Fig. 4.4 (b). Beam has a squared shape of four-fold structure and it persists over 200 periods. The emittance growth is only up to 1.03 times of the initial value, which is small compared to that of (b). In this case, the fourth-order particle resonance is manifested but do not have much effect on the beam mismatch factor to excite the envelope instability as beam propagates along long periods.

The 4σ =360^◦ fourth-order particle resonance is observed over short periods for initially well- matched Gaussian beam under the initial condition ofσ⊥0>90^◦andσ⊥<90^◦. However, the excitation of the envelope instability after the fourth-order resonance depends on the initial conditions of beam phase advances and the effects of the fourth-order particle resonance on the mismatch factor. In Sec. 5, we will discuss in detail about the interplay of the fourth-order particle resonance and the envelope instability forσ⊥0>90^◦of realistic Gaussian beams under solenoid and quadrupole focusing channels.

Also, the exact comparison between the parametric instability and incoherent particle resonance will be covered.

Chapter 5

Interplay of 4 σ = 360 ^◦ fourth-order

particle resonance and envelope instability

In Sec. 3.1, we discussed about the envelop instability of B and Q modes which occurs forσ⊥0>90^◦and within the range of tune depression, where the growth factor is different with 1. It is generated from the initial mismatch of perturbed beam core (envelope) in uniform density K-V beams. In Sec. 4, we verified the condition of the 4σ =360^◦ fourth-order particle resonance which occurs because of the realistic non-linear space-charge forces such as Gaussian density beams. We reported the theoretical analysis by using analytical calculation and particle-core model under the solenoid focusing fields. Furthermore, we have examined the numerical simulations to clarify the general stop band of the fourth-order particle resonance and observe the emittance growth from the fourth-order particle resonance and the excitation of the following envelope instability.

In this section, we verify the interplay of the 4σ =360^◦ fourth-order particle resonance and the envelope instability forσ⊥0>90^◦of realistic Gaussian beams under solenoid and quadrupole focusing channels. The two space-charge driven halo mechanisms set the operational limit in high-intensity ac- celerators that the zero-current phase advance should be less than 90^◦. However, the intrinsic principles of the fourth-order particle resonance and the envelope instability are evidently different. Also, the ex- citation position along the beam propagation periods and the emittance growth rates generated from the two mechanisms are different, which should be clearly noted for the exact understanding of the space- charge driven beam dynamics. The figures that we will introduce in this section have been published in Ref. [53].

Figure 5.1 shows the relative emittance growth (i.e., final RMS emittance/initial RMS emittance over 200 periods) and maximum mismatch factors with cross sections of beam distribution on x−x^′ phase planes in tune depression spaceσ⊥/σ⊥0. The numerical simulation has done with initially well- matched Gaussian beams under the solenoid focusing fields over 200 lattice periods. Note thatσ⊥0 is kept constant along the periods.

The fourth-order particle resonance is observed by the four-fold structures when σ⊥0 >90^◦ and σ⊥<90^◦. As beam current increases (tune depression decreases), the resonance particles having phase

Figure 5.1: Relative emittance growth and maximum mismatch factors with cross sections of beam distribution onx−x^′phase planes in tune depression spaceσ⊥/σ⊥0. (a) Forσ⊥0=94^◦, there is no pink region and the maximum mismatch factors are lower than 0.1 for all tune depression values. (b) For σ⊥0=100^◦, a blue arrow corresponds to the case of Fig. 4.4 (b). A red arrow corresponds to the case of Fig. 4.4 (c).

tune 0.25 decreases, and indeed the four-fold structure gets weaker. When the emittance grows from the fourth-order particle resonance particles, it causes the beam mismatch and accordingly induce the enve- lope instability after long periods. The regions in which the envelope instability is finally induced fol- lowing the fourth-order particle resonance are represented by pink regions in the upper plots of Fig. 5.1.

Figure 5.1 (a) shows for σ⊥0 =94^◦ where there is no pink region. It means that although the fourth-order particle resonance is manifested, the envelope instability is not induced. It is because the mismatch factors generated from the fourth-order particle resonance is not large enough to excite the envelope instability. The lower plots of Fig. 5.1 (a) shows the mismatch factors lower than 0.1 for all tune depression values. Figure 5.1 (b) shows for σ⊥0=100^◦. The stop band in the tune depression space represented by pink region is 0.7<σ⊥/σ⊥0<0.82. Among them, a blue arrow corresponds to the case of Fig. 4.4 (b). It has the maximum relative emittance growth up to 2.27 and the mismatch factor is 0.43, which is within the pink region. The four-fold structures are dominantly observed in cell 20 and they evolve to envelope instability after long periods accompanied by large emittance growth and large mismatch factor. A red arrow corresponds to the case of Fig. 4.4 (c). The emittance growth and mismatch factor from the fourth-order particle resonance are too small to excite the envelope instability, which is not included in the pink region.

Figure 5.2: Relative emittance growth for various values of transverse zero-current phase advances in the tune depression space under (a) solenoid and (b) quadrupole focusing lattice. Emittance growth occurs only for σ⊥0 >90^◦. As σ⊥0 increases, the range of stop band and the maximum emittance growth increases.

Figure 5.2 (a) shows the relative emittance growth for various values of transverse zero-current phase advances in the tune depression space under solenoid focusing lattice. In accordance with the results of analytical studies, the emittance growth is observed only forσ⊥0>90^◦. Asσ⊥0increases, the range of stop band and the maximum emittance growth increases. Also, Fig. 5.2 (b) shows the relative emittance growth for various values of transverse zero-current phase advances in the tune depression space under quadrupole focusing lattice. Similar to the Fig. 5.2 (a), the emittance growth is observed only forσ⊥0>90^◦ and asσ⊥0increases, the range of stop band and the maximum emittance growth increases.

Interestingly,σ⊥0 dependency on the emittance growth in Fig. 5.2 is reminiscent of the stop bands of the envelope instability in K-V beam. In 2015, Chao Li published [24] about the emittance growth from PIC simulation with uniform K-V beams, compared to the envelope instability stop bands under solenoid and quadrupole focusing lattices. Compared to that, we deal with the Gaussian beams on the emittance growth originated from the fourth-order particle resonance. In addition, we examine the relationship between the emittance growth generated from the envelope instability and the stop bands of the envelope instability.

Figures 5.3, 5.4, 5.5, and 5.6 show the comparison between the relative emittance growth from PIC simulation and the envelope instability stop bands of B and Q modes from analytical simulations under the solenoid focusing and quadrupole focusing channels, respectively. For σ⊥0 <90^◦, there are no envelope instabilities nor fourth-order particle resonance.

The lower plots of Figs. 5.3 and 5.4 show the envelope instability stop bands of B and Q modes under the solenoid focusing channel. Beam is axisymmetric under the solenoid focusing fields and only lattice resonance can occur in which one of the B and Q modes are resonant with the structural oscillations.

In contrast, confluent resonance between B and Q modes are observed under the quadrupole focusing channel because both B and Q modes can be contributed simultaneously as shown in the lower plots of

Figure 5.3: Relative emittance growth and the envelope instability stop bands for (a) σ⊥0=89^◦, (b) σ⊥0=92^◦, and (C)σ⊥0=94^◦under solenoid focusing channel.

Figs. 5.5 and 5.6.

In the upper plots of relative emittance growth in the tune depression space, both 4σ=360^◦fourth- order particle resonance and envelope instability are included from PIC multiparticle simulations. After the fourth-order particle resonance is predominantly manifested for the initially well-matched Gaussian beams, the resonance particles incur the RMS emittance growth. As beam propagates through the lat- tice periods, depressed phase advanceσ⊥ increases because of the increased RMS emittance (i.e., tune depression increases for constantσ⊥0). If the initial value of tune depression is designed to be close to the lower boundary of the envelope instability stop bands, the resonance particles make envelope mis- matches and emittance growth that enter into the region of envelope instability stop band. In that sense, forσ⊥0>90^◦, emittance growth within and near the envelope instability stop band is observed from the numerical simulations.

Especially for largeσ⊥0as in Figs. 5.4 and 5.6, pink regions exist in which emittance growth is larger than 20 % from the envelope instability. Within the pink regions, the fourth-order particle resonance is manifested along short periods and the envelope instability following the fourth-order particle resonance is induced after long periods with large emittance growth. Here, pink regions exist within the B mode instability stop bands under solenoid focusing [see Fig. 5.4], while pink regions exist within the B+Q mode instability stop bands under quadrupole focusing [see Fig. 5.6].

A blue arrow and a red arrow in Fig. 5.4 (c) correspond to the cases of Figs. 4.4 (b) and (c), respec- tively. Because the tune depression indicated by the red arrow is far from the boundary of the envelope instability stop band, beam cannot evolve to the instability. On the other way, the tune depression indi- cated by the blue arrow is near the boundary of the B mode instability stop band. The tune depression of the beam increases and finally evolve to the envelope instability as it enters the stop band region.

Even for σ⊥0>90^◦, there are no pink regions in Figs. 5.3 (b), (c) and 5.5 (C). In these cases, the

Figure 5.4: Relative emittance growth and the envelope instability stop bands for (a) σ⊥0=96^◦, (b) σ⊥0=98^◦, and (C)σ⊥0=100^◦under solenoid focusing channel. Within the pink regions, the fourth- order particle resonance is manifested along short periods and the envelope instability is induced after long periods only within the envelope instability stop bands of B mode.

fourth-order particle resonance is still manifested but its effects on the beam mismatch are weaker and the regions of the envelope instability stop bands are too small to stay within the stop bands as beam propagates through a linac. Also, the fourth-order particle resonance occurs at shorter periods under the quadrupole focusing lattice than that of the solenoid focusing lattice. Phase space plots in Fig.5.6 (c) show that the four-fold structures are clearly observed at cell 16 and it evolves to the envelope instability after cell 50.

In summary, we investigated the general stop band of the 4σ=360^◦fourth-order particle resonance, which isσ⊥0>90^◦andσ⊥<90^◦. The fourth-order particle resonance is originated from the non-linear space-charge fields in realistic beam distribution along the linac. In general, parametric instabilities have been well-known for the emittance growth and beam halo mechanisms in high-intensity accelerators.

However, recent studies have been found that the fourth-order particle resonance is the important issue to be focused for operating high-intensity accelerators. It is because the fourth-order particle resonance is manifested at short lattice periods even for initially well-matched beam conditions before the envelope instability occurs. It has been verified by both analytical calculations and numerical simulations that the fourth-order particle resonance is predominantly manifested with four-fold structures in four separate islands on the phase space. It can evolve to emittance growth and beam mismatches that accordingly induce the envelope instability. The stop band of the 4σ=360^◦fourth-order particle resonance is wider than the envelope instability stop bands of B and Q modes, and the envelope instability accompanied by large emittance growth is induced only within the envelope instability stop band following the fourth- order particle resonance.

In addition to the Fig. 3.11, a brief comparison between coherent resonance and incoherent par-

Figure 5.5: Relative emittance growth and the envelope instability stop bands for (a) σ⊥0=89^◦, (b) σ⊥0=90^◦, and (C)σ⊥0=92^◦under quadrupole focusing channel.

ticle resonance is summarized in Fig. 5.7. The condition of space-charge driven single particle reso- nances [26] is

lσ_x+mσ_y+∆σ_coh=n360^◦, (5.1)

wheren=1,2,3, ..andl+mindicates the order of resonance. ∆σ_cohis added from the zero-current con- dition,lσ_x0+mσ_y0=n360^◦, because of a phase shift generated from space-charge effects. The differ- ence between the parametric instability and the single particle resonance is shown in RHS of Eqs. (3.28) and (5.1). In Eq. (3.28), there is a ¹₂ factor from 360^◦phase, which is coming from the characteristic of the envelope oscillation that is a half of the single particle phase advance.

Figure 5.6: Relative emittance growth and the envelope instability stop bands for (a) σ⊥0=96^◦, (b) σ⊥0=98^◦, and (C)σ⊥0=100^◦under quadrupole focusing channel. Within the pink regions, the enve- lope instability is induced only within the envelope instability stop bands of B+Q mode.

Figure 5.7: A brief comparison between coherent resonance and incoherent particle resonance.

Chapter 6

Mitigation of the fourth-order particle resonance

6.1 σ

< 90

limit in high-intensity accelerators

As discussed in the previous section, the two halo mechanisms (coherent and incoherent resonances as summarized in Fig. 5.7) set the operational limit for high-intensity accelerators that zero-current phase advance σ₀ should be less than 90^◦. Over the past 50 years, charged particle beam dynamics to un- derstand and describe about halo formation mechanisms in high-intensity linear accelerators have been actively studied. Actually, only parametric instabilities were known to the beam dynamics community as a key concept that causes halo formations and beam losses along the linacs. The first experimental report for emittance growth due toσ₀>90^◦were published in 1985 [52]. In 2009, the 4σ=360^◦fourth-order particle resonance in linac was first discovered by D. Jeon [27] and was experimentally proved from the experiments in GSI UNILAC [28], Germany and SNS linac in USA [30]. Here,σ is depressed phase advance per cell. Since then it has been considered that the 4σ =360^◦fourth-order particle resonance together with the envelope instability set the operational limits ofσ0<90^◦.

Theσ₀<90^◦limit for operating high-intensity accelerators means that the external focusing strength should be less than a certain limit. Under the limited focusing strength, neither the beam current cannot be increased above a certain limit because the beam size can become too large to design the optimized beam pipe condition. In that reason, it is highly desirable to overcome these operational limitations associated with space-charge-driven resonances in order to expand the range of choices for focusing pa- rameters and beam currents in accelerator design. The efforts to surpass this operational limit have been actively in progress to achieve much higher beam current for the future applications such as intensity- frontier particle physics, generating high energy neutron sources, accelerator driven systems for nuclear waste transmutation, etc.

When σ₀>90^◦, envelope instability is triggered by perturbed envelope modes [i.e., breathing (B) and quadrupole (Q) modes]. Within the envelope instability stop band, emittance grows significantly over long lattice periods that leads to beam halos along the linacs. It has already been discussed mainly