In order to overcome the operational limit in high-intensity accelerators which isσ⊥0should be less than 90◦, we must avoid the stop bands of space-charge driven parametric instability and particle resonance or mitigate the excitation of them. In Ref. [70], Qiang reported that the envelope instability can be mitigated with a high accelerating gradient under periodic solenoid or quadrupole and RF focusing channels in high-intensity linacs.
The external focusing force of solenoid and quadrupole magnetκqis inversely proportional toγ2β2 andγ β, respectively. Also, the external focusing force of RF cavityκr f =λ2π
r f
qE0T sinφs
γ3β3mc2 scales as 1/γ3β3. This means that as beam accelerates through a linac, the focusing forces decreases and it makes zero- current phase advance decreases with constant magnetic or electric fields strength. If we increase the magnetic or electric fields strength so that the zero-current phase advance stays constant, depressed phase advance increases as beam accelerates, because 3D space-charge perveance cs= 32γ2βNq2mc2 2
1 5√ 5
scales as 1/γ2β2. In that sense, beam parameters change rapidly under a high accelerating gradient and the beam passes through envelope instability stop band faster than the beam without acceleration [70].
Figure 7.7 shows the transverse phase advances, relative emittance growth, and halo parameter with
Figure 7.7: (a) Transverse phase advances, (b) relative emittance growth, and (c) halo parameter with and without acceleration under the solenoid focusing lattice. σ⊥0=100◦ is constant over 100 lattice periods and beam current is 10 mA. Envelope instability is mitigated with a high accelerating gradient (purple lines) because beam passes through envelope instability stop band faster than the beam without acceleration (green lines).
and without acceleration under the solenoid focusing lattice. The transverse zero-current phase advance σ⊥0is kept constant for 100◦and the initial transverse depressed phase advance isσ⊥=70◦, in which the envelope instability and the 4σ =360◦ fourth-order particle resonance can be manifested in the transverse plane. The initial longitudinal zero-current phase advance isσz0=30◦. Here, the halo pa- rameterHi(iforx,y, andz) is defined as the ratio between the fourth-order and second-order moments of the beam in 2D phase space(qi,pi)[71],
Hi≡ q
3⟨q4i⟩⟨p4i⟩+9⟨q2ip2i⟩2−12⟨qip3i⟩⟨q3ipi⟩
2⟨q2i⟩⟨p2i⟩ −2⟨qipi⟩2 −2. (7.1) For the uniform KV distribution,Hi=0, and for the Gaussian distribution,Hi=1. A significant halo population in the multiparticle simulations corresponds toHi>1.
Without acceleration as denoted by green lines in Fig. 7.7, beam remains within the fourth-order particle resonance stop band over 40 periods and generates an envelope mismatch, which accordingly induces a large emittance growth over 50∼80 periods up to 4.5 growth rate and more than 5 times the initial halo parameter.
If there is an accelerating RF field as denoted by purple lines in Fig. 7.7, σ⊥ increases as beam accelerates through the linac faster than the case of without acceleration. This causes the beam pass through the envelope instability stop band faster than the beam without acceleration and the accelerating beam does not have sufficient time to evolve to the envelope instability. The relative emittance growth and halo parameter with acceleration increases very little compared to those without acceleration. The accelerating RF gradient is 5 MV/m and the beam current is 10 mA of initially well-matched Gaussian bunched beam in our numerical simulation.
It is confirmed that we can suppress the envelope instability observed after long lattice periods through the linac. By applying a fast acceleration of space-charge dominant beams, beam parame- ters change fast enough to escape from the excitation of the envelope instability as it passes through the instability stop bands.
Even in this situation, however, the 4σ =360◦ fourth-order particle resonance is manifested along 5∼40 lattice periods with the noticeable halo formations as shown in Fig. 7.7 (c) by a purple line. The fourth-order particle resonance is still the important source for emittance growth and halo evolution even with acceleration whenσ⊥0>90◦andσ⊥<90◦.
7.3 4σ = 360
◦Fourth-order particle resonance in longitudinal space
In recent studies in Refs. [72, 73] for lattice structures with more than one RF gap per external focusing period, it was reported that the 90◦restriction needs not to be applied to longitudinal space as the same way as to transverse one. However, for the periodic lattices in which the transverse focusing magnets and RF gap have the same lattice period, we expect the space-charge driven fourth-order particle resonance to occur in longitudinal space under the conditions ofσz0>90◦andσz<90◦, similar to the transverse case. Here,σz0(σz)is the longitudinal zero-current (depressed) phase advance per lattice period.
Multiparticle simulations are performed within the fourth-order particle resonance stop band in lon- gitudinal space without acceleration of initially well-matched Gaussian bunched beams. Figure 7.8 shows the depressed phase advances, relative emittance growth, and phase space plots on phase (deg)· energy (MeV) planes under the solenoid focusing lattice without acceleration. The initial longitudinal phase advances areσz0=100◦andσz=77◦, which is within the fourth-order particle resonance stop band. Here, σz0=100◦ sets to be constant along 100 periods. The initial transverse phase advances areσ⊥0=85◦andσ⊥=50◦, where there is no coupling effect between the transverse and longitudinal spaces. Beam energy is 5 MeV and the beam current is 14.5 mA.
Similar to the phenomena occurred in transverse plane, the fourth-order particle resonance in longi- tudinal space is predominantly manifested along 40 periods. The longitudinal depressed phase advance remains almost constant over 40 periods [see green line in Fig.7.8 (a)] and the four-fold structures clearly appear at cell 20 and 30 as shown in Fig.7.8 (c). After the fourth-order particle resonance is excited, the beam becomes mismatched and grows to the envelope instability asσzincreases over 90◦accompanied by large perturbed motion in phase advance and RMS emittance (green lines). At cell 80 and cell 100, four-fold structures vanish and the particles have unstable trajectories on phase (deg)·energy (MeV)
Figure 7.8: (a) Depressed phase advances, (b) relative emittance growth, and (c) plots on phase·energy planes under the solenoid focusing lattice without acceleration. The initial longitudinal phase advances areσz0=100◦andσz=77◦. The fourth-order particle resonance is manifested over 40 periods and the envelope instability occurs over 100 periods.
planes.
Actually, when σz0>90◦, non-linear fields of RF cavities can excite the non-linear resonances as well as non-linear space-charge driven particle resonances [35, 74]. Nevertheless, the RF non-linear field resonance is neglected in our multiparticle simulations because it develops much slower than the space-charge effects and it occurs only for large beam size inzdirection which is not in the range of consideration of our simulation study.
Figure 7.9 shows the comparison between the non-linear space-charge driven fourth-order particle resonance and RF non-linear field resonance in longitudinal space on phase (deg)·energy (MeV) planes.
While the space-charge driven fourth-order particle resonance is clearly observed at cell 10 in Fig. 7.9 (a), the resonance structures generated from RF non-linear fields are observed to be much weaker and appear after cell 100 as shown in Fig. 7.9 (b). In that reason, we focus on the space-charge driven par- ticle resonance and envelope instability for the beam dynamics study in longitudinal space of Gaussian density beams.
Figure 7.9: Plots of (a) Space-charge driven fourth-order particle resonance and (b) RF non-linear field resonance in longitudinal space on phase·energy planes. RF non-linear field resonance is much weaker than space-charge driven particle resonance and begins to appear after cell 100.