Particle-core model of spinning beam

The analytical simulations to observe particle trajectories with 4σ =360^◦ fourth-order particle reso- nance of non-spinning and spinning beams are done by using particle-core model. For the particle-core model, we assumed that every single particle has the same initial canonical angular momentum, such that⟨Pˆ_θ⟩=Pˆ_θ, and it remains constant along the linacs owing to the conservation theory of canonical angular momentum. Also, RMS emittance remains constant andσ⊥0 is kept constant along the lattice periods. Here, it should be noted that the RMS emittance and phase advances in longitudinal space are assumed to be extremely small, so that the coupling between transverse and longitudinal planes are not considered for 2D coasting beam simulation studies reported in this section.

Figure 6.4: Poincar´e-section plots in real frame whenσ⊥0=100^◦ andσ⊥=72^◦ of solenoid focusing channel. (a) ⟨Pˆ_θ⟩=0 is for non-spinning beam and (b) ⟨Pˆ_θ⟩=0.2 is for spinning beam. The four resonance structures become blurred for spinning beam.

Figure 6.4 shows Poincar´e-section plots in real frame when σ⊥0=100^◦andσ⊥ =72^◦of solenoid focusing channel, for (a)⟨Pˆ_θ⟩=0 non-spinning beam and (b)⟨Pˆ_θ⟩=0.2 spinning beam. Here,⟨Pˆ_θ⟩is in arbitrary units in the analytical calculation and x-axis of phase space plots is normalized by RMS beam sizeσr. The first plots in Fig. 6.4 are from many test particles with different initial conditions. For non- spinning beam, particles which have tune 90^◦/360^◦=0.25 show the fourth-order resonance structures that have a shape of cross on the phase space. Among them, we show a single particle trajectory which has tune 0.25 with the initial condition ofr(0)/σr=2.5 in the middle plot of Fig. 6.4 (a). Further, the last plots on x−yphase plane show the separate particle orbits away from the center (x=0,y=0), which eventually evolve to beam halos and unstable motions after passing through many lattice periods repeatably. On the other hand, for spinning beam as shown in Fig. 6.4 (b), the shape of cross onx−x^′ phase plane becomes blurred than that of Fig. 6.4 (a). Also, it is shown in a single particle trajectory that the resonance structure is distorted and the resonance particles are getting closer to the center on the x−yphase plane.

Figure 6.5: Poincar´e-section plots in Larmor frame when (a)σ⊥0=100^◦ andσ⊥=72^◦ and (b)σ⊥0= 100^◦ andσ⊥ =86^◦ of solenoid focusing channel for non-spinning and spinning beams with different values of⟨Pˆ_θ⟩. As⟨Pˆ_θ⟩increases, four separate islands are getting blurred and merge into the center of Larmor phase space.

Figure 6.5 shows Poincar´e-section plots in Larmor frame when (a)σ⊥0=100^◦ andσ⊥=72^◦and (b)σ⊥0=100^◦andσ⊥=86^◦ of solenoid focusing channel for non-spinning and spinning beams with different values of⟨Pˆ_θ⟩. For non-spinning beam having⟨Pˆ_θ⟩=0, the fourth-order particle resonance is observed by four separate islands of tune 0.25 particles. In this case, there is no coupling in X andY directions from Eq. (6.2). By contrast, for spinning beams in both Fig. 6.5 (a) and (b), the coupling between X and Y occurs in Eq. (6.2) because of the non-linear Gaussian space-charge fields.

As⟨Pˆ_θ⟩increases, four separate islands become blurred and the separatrix around the central elliptical orbits merge together vanishing the particle resonance trajectories. This phenomena indicates that the

resonance particles do not go farther away from the resonance structures but instead, get into the stable central orbits and avoid from halo formation along the linac.

Figure 6.6: Poincar´e-section plots in Larmor frame whenσ⊥0=100^◦andσ⊥=72^◦of solenoid focusing channel for non-spinning and spinning beams with different values of ⟨Pˆ_θ⟩. As⟨Pˆ_θ⟩ increases, four separate islands are getting blurred and merge into the center of Larmor phase space.

Similar to the Fig. 6.5, Fig. 6.6 shows more detail phase space plots between non-spinning and spinning beams with many test particles of different initial conditions. As shown in Figs. 6.4 and 6.5, the fourth-order resonance structures get blurred and particles are getting into the central region for the spinning beams. The mitigation effects on the 4σ =360^◦ fourth-order particle resonance of spinning beams are generated from the coupling in X andY directions for non-zero average canonical angular momentum. We verified that the spinning beam has an intrinsic characteristic to mitigate the particle resonances by using particle-core model and it evidently proves the theoretical results which we derived from Eq. (6.8).

6.3.3 2D Multiparticle simulations of spinning beam

Motivated by the analytical interpretation of spinning effects on the fourth-order particle resonance, we perform numerical simulations by using TraceWin PIC code in this section. We investigate emit- tance growth and halo parameters for non-spinning and spinning beams generated from the fourth-order particle resonance and the following envelope instability along the solenoid focusing fields of initially well-matched Gaussian 2D coasting beam. The numerical simulation studies give more clearer evi- dence for the spinning effects on the mitigation of the 4σ =360^◦ fourth-order particle resonance and the associated envelope instability.

The average initial canonical angular momentum of spinning beam is calculated over 100,000 macro-

particles in TraceWin simulation. Also, for our numerical simulations, σ⊥0 is kept constant along the lattice periods. As shown in Fig. 6.3, a spinning beam is generated as it gets out of a solenoid without any initial rotating term. The external magnetic field of a solenoid makes charged particles to be azimuthal rotating around the propagation axis and by means of conservation of the canonical angular momentum, the outcome beam becomes ‘spinning beam’ having non-zero average canonical angular momentum. For a non-spinning beam used in our simulation isεrms,⊥=εth,⊥=6.85 mm-mrad. If we simply assume that θ^′ are the same for all particles, the rotating angle per lattice periodthetawould be approximately 30^◦, 45^◦, 60^◦, and 90^◦for⟨Pˆ_θ⟩=3.75, 7.5, 11, and 18.7 mm-mrad, respectively. The initial RMS emittances for the spinning beams areεrms,⊥=7.28, 7.97, 9, and 11.7 mm-mrad, respectively. Here, longitudinal RMS emittance and phase advance are assumed to be very small, so that the coupling between transverse and longitudinal spaces are not considered.

Figure 6.7: (a) Relative emittance growth during 50 lattice periods forσ⊥0=100^◦, in which the 4σ= 360^◦ fourth-order particle resonance is dominantly manifested. As ⟨Pˆ_θ⟩ increases, emittance growth rates become smaller and constant over tune depression space. Particle phase space plots forσ⊥=72^◦ of (b) non-spinning and (b) spinning beam at cell 20. The four-fold structure is suppressed for the spinning beam.

Figure 6.7 shows the relative emittance growth during 50 lattice periods forσ⊥0=100^◦ with dif- ferent values of average canonical angular momentum. Here, the relative emittance growth is defined by the ratio of the final RMS emittance to the initial RMS emittance (εf/εi), and the units of ⟨Pˆ_θ⟩ is mm-mrad. Along the 50 lattice periods (which is short propagation), only the 4σ =360^◦ fourth-order particle resonance is dominantly manifested within the fourth-order resonance stop band, σ⊥0 >90^◦ andσ⊥<90^◦. Therefore, the relative emittance growth shown in Fig. 6.7 (a) is affected only by the fourth-order particle resonance. As⟨Pˆ_θ⟩increases, the relative emittance growth becomes smaller and constant over tune depression space (see the cyan arrow). This results imply that the emittance growth induced from the fourth-order particle resonance reaches a certain limit for spinning beams because the fourth-order resonance is mitigated and the corresponding emittance growth is shortened for the spin-

ning beams. It is also proven by the phase space plots in Fig. 6.7 (b) and (c). For⟨Pˆ_θ⟩=0 beam, we can see the four-fold structure at cell 20 and the maximum emittance growth along 50 periods is up to 4%.

For⟨Pˆ_θ⟩=7.5 mm-mrad beam, however, the four-fold structure is evidently shortened at cell 20 and the maximum emittance growth along 50 periods is shortened to 2%. We found that the fourth-order particle resonance in Fig. 6.7 (c) is mitigated with fewer particles populating the fourfold structure compared to those in Fig. 6.7 (b).

Figure 6.8: (a) Relative emittance growth during 200 lattice periods forσ⊥0=100^◦, in which envelope instability is generated following the fourth-order resonance. (b) Envelope instability stop band of B and Q modes under solenoid focusing channel. Particle phase space plots forσ⊥=72^◦of (c) non-spinning and (d) spinning beam at cell 150. The envelope instability is also suppressed for the spinning beam.

Figure 6.8 shows the relative emittance growth during 200 lattice periods for initially well-matched Gaussian beams with different values of average canonical angular momentum, and the envelope insta- bility stop band of the equivalent K-V beam forσ⊥0=100^◦under solenoid focusing channel. Along the 200 periods, the relative emittance growth is affected by the fourth-order particle resonance as shown in Fig. 6.7 (a) and the envelope instability which is induced following the fourth-order particle resonance after long periods.

Whenσ⊥<90^◦, the fourth-order particle resonance is predominantly excited as observed in Fig. 6.7.

The resonant particles excite beam mismatch which accordingly leads to the envelope instability accom- panied by emittance growth [see the black line in Fig. 6.8 (a)], within the envelope instability stop band as indicated by|growth factor| ̸=1 in Fig. 6.8 (b). The envelope instability stop band is defined for a fixedσ⊥0as a function ofσ⊥/σ⊥0, which is independent of values of⟨Pˆ_θ⟩. For non-spinning beam, the emittance growth from the envelope instability is evidently observed within the B-mode instability stop

band, with the maximum value near the lower boundary of the B-mode stop band and sharply drops to 1 outside the stop band [53, 68].

Compared to that, beam mismatch generated by the fourth-order particle resonance is mitigated along the short periods as discussed in Fig. 6.7 for spinning beams. Then the following envelope insta- bility is also mitigated along the long periods with lower emittance growth as shown in Figs. 6.8 (a) and (d). In the region ofσ⊥/σ⊥0≤0.82 (left side of the cyan arrow), the fourth-order particle resonance is mitigated and the beam mismatch and corresponding emittance growth are not sufficiently strong to develop the envelope instability, even though the tune depression lies within the envelope instability stop band. As a result, the stop bands of the relative emittance growth in Fig. 6.8 (a) shift to right and become narrower as⟨Pˆ_θ⟩increases (see the cyan arrow). The maximum relative emittance growth decreases and the tune depression value in which the maximum emittance growth occurs shifts to right.

The detailed beam distributions inx−x^′phase plane are plotted in Figs. 6.8 (C) and (d) forσ⊥=72^◦. For spinning beam in Fig. 6.8 (d), particles gather close to the stable central region with smaller emit- tance growth which is affected by the mitigation effect of spinning beam on the fourth-order resonance, and indeed mitigation on the envelope instability after all.

Figure 6.9: Particle phase space plots for σ⊥0=100^◦ andσ⊥ =86^◦. The four-fold structure is sup- pressed, but Q-mode instability occurs because of the tune shift for spinning beam.

Whenσ⊥/σ⊥0=0.84∼0.86, the relative emittance growth of spinning beam is still considerable despite low values of emittance growth for non-spinning beam. The corresponding phase space plots are shown in Fig. 6.9. This is because beams are affected by Q-mode instability owing to the shifted stop bands for high values of tune depressions as discussed in Fig. 6.8 (a). Becauseσ⊥=0.84 is outside the envelope instability stop band, the fourth-order particle resonance persists over 200 periods and the relative emittance growth remains 1.09 for non-spinning beam. However, for⟨Pˆ_θ⟩=18.7 mm-mrad in Fig. 6.9 (b), the relative emittance growth is 1.02 at cell 20 and 1.45 at cell 150. It is affected by Q-mode instability and the emittance growth at cell 150 is larger than that in the case of ⟨Pˆ_θ⟩=0. As beam

currents decreases further such thatσ⊥/σ⊥0>0.9, both fourth-order particle resonance and envelope instability are not observed.

Figure 6.10: Relative emittance growth after (a) 30 lattice periods and (b) 200 lattice periods for non- spinning beams with increased initial thermal emittances. There is no mitigation effect for non-spinning beams when the RMS (thermal) emittance is increased only without canonical angular momentum. Here, σ⊥0=100^◦is fixed.

We want to emphasize that the mitigation effect coming from a coupling between X and Y associated with non-zero canonical angular momentum is evidently different from the case with an increase of initial thermal emittance only. In fact, the initial RMS emittances of spinning beams are becoming larger than that of non-spinning beam because a non-zero canonical angular momentum is added from the initial thermal emittance without spinning ;εrms,⊥=q

ε_th,⊥² +⟨Pˆ_θ⟩²/4.

However, the mitigation effect on the 4σ =360^◦fourth-order particle resonance is associated with the non-zero canonical angular momentum. If the initial RMS emittances of non-spinning beams are equal to those of spinning beams by increasing the initial thermal emittance only, there is no mitigation impact on the fourth-order particle resonance in the absence of spinning.

Figure 6.10 shows the relative emittance growth for non-spinning beams of which the initial thermal

emittances increase as the same RMS emittance values of spinning beams. The upper plot shows the relative emittance growth after 30 lattice periods, in which only the fourth-order particle resonance is revealed. The lower plot shows the relative emittance growth after 200 lattice periods, in which the envelope instability is induced after the fourth-order particle resonance. Because the envelope instability stop bands are independent of the initial RMS emittances, emittance growth phenomena within the envelope instability stop band for a fixedσ⊥0is the same, and there is no mitigation effect in the absence of spinning.

Therefore, it is verified that the intrinsic characteristic of the spinning beams on mitigating the 4σ =360^◦fourth-order particle resonance is from non-zero average canonical angular momentum, not affected by the absolute values of thermal emittances without spinning.

Chapter 7

3D beam dynamics

So far, we deal with the 2D coasting beams of uniform density K-V beam and Gaussian density profile in a cross section of transverse plane. The longitudinal beam parameters were assumed to be very small, so that we can ignore the coupling effect such as emittance exchange between the longitudinal and transverse spaces. 2D beam dynamics of space-charge dominant high-intensity beams have been studied and considered as the simplest way to implement the particle motions on phase space in which we can easily observe the particle trajectories under periodic focusing lattice with high space-charge effects. It has been done by using both analytical and numerical simulations. Especially, particle-core model method for plotting particle trajectories by solving equations of motion and envelope equations in 2D space makes possible to investigate the halo mechanisms of parametric instabilities and particle resonances at an intuitive point of view. Furthermore, multiparticle simulation studies by using PIC simulation tools show emittance growth and halo formations along the lattice periods from the self- consistent space-charge mechanisms accompanied by parametric instabilities and non-linear particle resonances.

From now, we will expand our study for the space-charge dominant beam dynamics in 3D bunched beams, which is more realistic. As shown in Fig. 2.4, the realistic beam is a single bunch which has the elliptical shape in transverse and longitudinal directions similar to the shape of rugby ball. The space-charge effects in longitudinal space are excited as the same way as the transverse case.

In Sec. 7.1, we investigate the envelope instability stop bands indicated by growth factor in the tune depression space of 3D bunched beams under a periodic lattice with solenoid and RF cavity for transverse and longitudinal focusing.

In Sec. 7.2, we introduce a fast acceleration of 3D bunched beam along the linacs and the mitigation of the envelope instability by using fast acceleration.

In Sec. 7.3, we verify the 4σ=360^◦fourth-order particle resonance in longitudinal space generated by the longitudinal Gaussian space-charge fields, which is dominant over the non-linear fields of RF cavities. The observation of the space-charge driven fourth-order particle resonance in longitudinal space has not been studied so far, and we first report the research in Ref. [69].

Furthermore, in Sec. 7.4, we apply a novel approach of using spinning beams to mitigate the 4σ=

360^◦ fourth-order particle resonance of Gaussian 3D bunched beams. We found that the fourth-order particle resonance in transverse space can be mitigated by spinning beams even with acceleration along the short periods in linacs under both solenoid and quadrupole focusing channels. On the contrary to the case of transverse plane, the spinning effects on the mitigation of the fourth-order resonance is not obtained in longitudinal plane because of the azimuthal rotating property of the spinning beams without coupling between transverse and longitudinal directions.

Finally, in Sec. 7.5, we will see the emittance exchange between transverse and longitudinal spaces under the coupling regions in x-z motions. It has been known as the Hofmann chart that indicates the coupling stop bands in various range of beam parameters. We will compare the effects of emittance exchange between x and z planes and the fourth-order particle resonance along linacs.

7.1 Envelope instability of 3D bunched beam

Figure 7.1: Periodic lattice period with solenoid magnet for transverse focusing and RF cavity for lon- gitudinal focusing of 3D bunched beam. The length of a focusing period isS.

In our analytical simulation studies of 3D bunched beams, we use solenoid magnets for the transverse focusing and RF cavities for the longitudinal focusing [35] along the linacs as shown in Fig. 7.1. The length of a focusing period isSand the periods of transverse focusing and longitudinal focusing fields are the same.

In this section, we investigate the envelope instability stop bands indicated by growth factor in the tune depression space as discussed in Sec. 3.1 by using 3D bunched beams under the periodic focusing lattice. The analytical calculation of envelope instability stop bands can be obtained by solving perturbed equations from the envelope equations of well-matched 3D bunched beam [see Eq. (2.78)]. In this case, the transfer matrix of 3D bunched beams are expressed by 6×6 phase space of (x,x’,y,y’,z,z’) [25].

Figure 7.2 shows the envelope instability stop bands of 3D bunched beams with various range of zero-current phase advances. σ⊥0 andσ⊥ represent the zero-current phase advance and the depressed phase advance in transverse plane, respectively.σ_z0represents the zero-current phase advance in longi- tudinal plane. The envelope instability stop bands are depicted in the transverse tune depression space (σ⊥/σ⊥0) with fixedσ⊥0andσz0along 200 lattice periods. As discussed in Sec. 3.1, the tune depression

Figure 7.2: Envelope instability stop bands of 3D bunched beams with various range of zero-current phase advances. No envelope instability for (a)σ⊥0<90^◦ andσ_z0<90^◦. Only lattice resonance is excited for (b) and (c). Both lattice and confluent resonances are excited for (d)σ⊥0>90^◦andσ_z0>90^◦. range in which the absolute value of growth factor is not 1 indicates the envelope instability stop band.

Figure 7.2 (a) is forσ⊥0=80^◦with σ_z0=30^◦, 60^◦, and 80^◦. Because both transverse and longi- tudinal zero-current phase advances are less than 90^◦, there is no envelope instability over the whole tune depression values. However, in Figs. 7.2 (b), (c), and (d), we can see the excitation of the envelope instability related to the lattice resonance and the confluent resonance. For the case ofσ⊥0<90^◦and σz0>90^◦ in Fig. 7.2 (b), the lattice resonance is excited from the longitudinal perturbed modes which resonate with a linac structure. By contrast, for the case ofσ⊥0>90^◦andσ_z0<90^◦in Fig. 7.2 (c), the lattice resonance is excited from the transverse perturbed modes as similar to the case of 2D coasting beam shown in Fig. 3.7. Under the solenoid focusing fields, only the lattice resonance occurs for the envelope instability.

Interestingly, for the case of σ⊥0 >90^◦ and σ_z0>90^◦ in Fig. 7.2 (d), both lattice resonance and confluent resonance are excited even under the solenoid focusing channel. This is because the perturbed modes in transverse and longitudinal directions can resonate each other along the linac.

The more detailed plots of envelope instability stop bands are shown in Fig. 7.3. According to the results of Fig. 3.7, the envelope instability is not observed when the zero-current phase advance is less than 90^◦. However, from the Fig. 7.3 (a), the lattice resonance is observed even for σ⊥0 =80^◦ because the instability is generated in the longitudinal space, instead. Further, if both zero-current phase advances in transverse and longitudinal planes are larger than 90^◦, the confluent resonance betweenxand zdirectional envelope perturbations can be induced. Especially in Fig. 7.3 (c), we can see the envelope instability of confluent resonance [between x (or y) and z] along with the lattice resonance [of y (or x)]

at the same time.

In the tune depression space, the envelop instability stop bands do not depend on the initial val-