Figure 3.5: Envelope instability stop band in tune depression space (σ⊥/σ⊥0) under (a) solenoid and (b) FODO focusing channel. In the region of tune depression in which a growth factor is not 1, the envelope instabilities of B and Q modes are manifested. Here, zero-current phase advance isσ⊥0=115◦andηis the focusing factor as shown in Fig. 2.2.
parametric resonance of B mode.
When σ⊥0<90◦, the phase advance of stable B mode is σ+=2σ⊥02 +2σ⊥2, and that of stable Q mode isσ−=σ⊥02 +3σ⊥2, which correspond to the relations of Eqs. (3.9) and (3.13). Ifσ⊥0>90◦, the envelope instability can be manifested and the phase advances of B and Q modes are shifted because of the resonances. The plots of mode phase advanceσnand the envelope instability stop bands are shown in Fig. 3.6. In the region of parametric resonances, the mode phases have 180◦, which are shown in Fig. 3.6 (b) and (c). In the region of confluent resonances, the mode phases of B and Q modes are equal as shown in Fig. 3.6 (c).
The excitation of the envelope instability for the different zero-current phase advances under the solenoid focusing lattice are shown in Fig. 3.7. Forσ⊥0=89◦, there is no instability. Forσ⊥0>90◦in Fig. 3.7 (b), (c), and (d), the envelope instability is observed and the range of stop bands increase asσ⊥0
increases.
Figure 3.6: Plots of phase advances and growth factors of B and Q modes in (a) stable condition for σ⊥0=80◦, and instabilities under (b) solenoid and (c) FODO focusing channels forσ⊥0=115◦. Within the envelope instability stop bands, the phase advances of the envelope modes are shifted because of the resonances.
halo in the multi-particle simulations.
In this section, we will see the nth-order coherent resonances of envelope perturbation, which occurs forσ⊥0<90◦, and the condition of beam parameters in which the higher-order resonances are generated.
By considering the smooth focusing approximation of the solenoid focusing lattice, we can use the equilibrium of Eq. (3.4) and the breathing mode of Eq. (3.14). The phase advances of matched beam are calculated by
σ⊥0= Z S
0
kβ0ds=S·kβ0, σ⊥=ε⊥
Z S
0
ds r2b =ε⊥S
r2b
(3.22)
→ σ⊥
σ⊥0
= ε⊥
kβ0r2b. (3.23)
By combining with Eq. (3.4), it becomes (σ⊥
σ⊥0
)2+Kb 1 ε⊥kβ0
σ⊥
σ⊥0
−1=0
→ σ⊥
σ⊥0
=1
2[−Kb 1 ε⊥kβ0+
s Kb2
ε⊥2k2β0+4] = [( Kb
2ε⊥kβ0)2+1]1/2− Kb
2ε⊥kβ0.
(3.24)
For the normalization dimensionless envelope equations, we assume thatS=1 andε⊥=1. Then, from Eq. (3.23), Eqs. (3.14) and (3.24) are modified to [22]
k22=k2β0+Kb rb2 + 3
rb4 =σ⊥02 +Kbσ⊥+3σ⊥2, σ⊥
σ⊥0
= [( Kb
2σ⊥0
)2+1]1/2− Kb
2σ⊥0
(3.25)
Figure 3.7: Envelope instability stop band for different values ofσ⊥0. (a) For σ⊥0=89◦, there is no instability. Forσ⊥0>90◦in (b), (c), and (d), the envelope instability is observed and the range of stop bands increase asσ⊥0increases.
=⇒ k2(σ⊥0,K) = [4σ⊥02 +Kb2−Kb(4σ⊥02 +Kb2)1/2]1/2. (3.26) If beam current is negligibly small (large amplitude oscillation; large phase advance), the mode wave numberk2in Eq. (3.26) is approximately,k2=k2(σ⊥0,Kb→0) =2σ⊥0. The mode oscillation for σ⊥0<90◦under the smooth approximation (small amplitude oscillation; small phase advance) has the wave number,k2in Eq. (3.26.) Therefore, an arbitrary amplitude envelope oscillation has wave number k which satisfies the inequalityk2(σ⊥0,K)≤k≤2σ⊥0.
The nth-order resonance of perturbed envelope means that the wave number isk=2πln (l=1,2, ...), where n indicates the order of resonance. Figure 3.8 shows the Poincar´e-section plots of perturbed envelope oscillations for σ⊥0=45.5◦ in radial phase space (r−r′) under solenoid focusing channel.
Poincar´e-section plotis to plot all points in every S lattice period with different initial conditions for over 300 lattice periods. On the phase space of envelope oscillations, the fixed point corresponds to the matched beam of which the radius becomes the equal at every lattice period S. Figure 3.8 (a) shows the envelope oscillations of different initial conditions for Kb=0 andη =1/6. The envelope modes are having the same phase advance because of the zero space-charge effect. They have stable elliptical trajectories around the matched beam. Figure 3.8 (b) shows the envelope oscillations of different initial conditions for Kb=3 and η=1/6. The 4 separate elliptical islands denote the 4th-order resonance of which the wave number isk4= 2πl4 . Furthermore, 5 separate elliptical islands in the vicinity of the fixed point denote the 5th-order resonance of which the wave number isk5=2πl5 . The beam radius in one of the five islands will move from one to another island until it comes back to the starting point after 5 lattice periods. For the conditions of σ⊥0 =45.5◦ and Kb=3, both 4th-order and 5th-order resonances can be generated as following the resonance condition in Ref. [22]. If with sufficiently large
Figure 3.8: Poincar´e-section plots of perturbed envelope oscillations for σ⊥0=45.5◦ in radial phase space (r−r′) under solenoid focusing channel. The fixed point corresponds to the matched beam with initial conditionrb(0) =1.16 andr′b(0) =0. (a) ForK=0, mismatched beams oscillate with elliptical trajectory around the matched beam. (b) For K=3, both 4th-order and 5th-order resonances are ob- served.
space-charge perveance, the envelope instabilities can occur and envelope modes become chaotic under the mismatch or nth-order resonance conditions.
3.2.1 Particle-core model
The analytical model for describing the particle trajectories which oscillate with beam envelope is a particle-core modelthat we plot test particles by solving both equations of motion (test particle) and envelope equations (beam core). In real multiparticle system, many single particles compose a beam envelope and their deviations change the envelope radius or the beam emittance which is the averaged value of particle distribution. Instead, in particle-core model, we do not consider the self-consistent effects of multiparticle system but we deal with the single particle motions and the envelope oscillations separately. The single particle motions are affected by envelope mode frequencies, which affect the linear space-charge term for uniform density beam, and accordingly affect the wave number of a single particle.
Figure 3.9 shows plots of particle trajectories on the normalized phase space (x/xrb−x′) by using particle-core model. All particles in Figs. 3.9 (a) and (b) are plotted in every S lattice period by solving the equations of motion and the envelope equations under the different envelope mode oscillations.
Figure 3.9 (a) is for the matched envelope oscillation in which single particles have stable elliptical orbits on the phase space. Figure 3.9 (b) is for mismatched envelope perturbation as shown by the
Figure 3.9: Particle-core model on the normalized phase space (x/xrb−x′) of (a) matched beam, (b) mismatched beam, and (c) 5th-order envelope resonance. All particles in (a) and (b) are plotted in every S lattice period. For the 5th-order envelope resonance, particles are plotted in every 5 lattice periods.
Figure 3.10: Particle-core model on the normalized phase space (x/xrb−x′) for 1:2 resonance. The single particle wave number is a half the wave number of envelope mode oscillation,kparticle=kmode/2.
elliptical orbits in Fig. 3.8. Under the mismatched condition, when the particle wave number is a half the wave number of envelope mode oscillation (kparticle=kmode/2), there is a1:2 resonancebetween the single particle and the perturbed envelope motion [16–19]. The 1:2 resonance appears as the separatrix in the vicinity of the fixed points on the (x/xrb−x′) phase plane [see Fig. 3.10].
Figure 3.9 (c) shows particle trajectories for the 5th-order envelope resonance of which the wave number of the envelope isk5=2πl5 . The test particles are plotted in every 5 lattice periods to observe the trajectories of particles with the resonant envelope mode perturbation. When the test particles are within the beam envelope boundary, they experience the linear space-charge force which is dependent on the beam size. As particle approaches the beam core, it is de-focused by the space-charge force, and on the other way, as particle leaves the beam core, it is focused by the space-charge force; Frsc∝r/r2b forr(s)<rb(s). However, if the test particles are outside the beam envelope boundary, they experience
the non-linear space-charge force which does not depend of the beam size;Frsc∝1/rforr(s)>rb(s).
Therefore, the unstable envelope perturbations of mismatched beam and nth-order resonances induce the non-linear effects on a single particle and generate chaotic motions and halo formations.