Characterization of overlapping betatron resonances above the phase advance of 90° per cell

The space-charge-induced resonance band just above the betatron phase advance of 90° per cell is of great practical importance. The instability is closely related to the linear collective mode and can thus give rise to severe emittance growth at high beam density. In a circular machine, this type of second-order resonance occurs not only at half-integer tunes but also near quarter-integer tunes, depending on the lattice superperiodicity. Self-consistent numerical simulations are carried out to elucidate the resonance feature above the 90° cell tune. The present results suggest the existence of three different resonance mechanisms working there; namely, the fourth-order incoherent resonance in the beam tail, the second-order and fourth-order coherent resonances in the beam core. It is reconfirmed that no serious emittance growth occurs even if particles deep inside the core satisfy the incoherent resonance condition. The recently proposed stop-band diagram, free from the concept of incoherent tune spread, appears to be consistent with the numerical observations.

Figure 1

Stop-band diagrams corresponding to the current operating condition of the J-PARC MR. The rms tune depression is estimated to be (a) $\eta \approx 0.994$ at 3 GeV and (b) $\eta \approx 0.999$ at 10 GeV. The coherent resonance bands of up to the third-order ($m\le 3$) are plotted on the basis of Eq. (3) with $C_2=0.7$ and $C_3=0.8$. The third-order resonance driven by the normal sextupole magnets for orbit correction overlaps with the second-order resonance slightly above ${\nu }_{0x}=21$. The width of each stop band is defined by the approximate formula in Eq. (5). The safety factor ${\zeta }_{k\ell }$ is set equal to unity for all stop bands. We have ignored the effect of error fields that produce additional stop bands.

Figure 2

Stop-band diagram near the cell tune of 0.25 (the betatron phase advance of 90°) under the condition ${ϵ}_x={ϵ}_y$. The rms tune depression is assumed to be 0.9 everywhere in the diagram.

Figure 3

Emittance growth of a Gaussian beam propagating through a long AG transport channel. The transport distances are (a) 100 FD cells, (b) 200 FD cells, and (c) 1000 FD cells. The rms tune depression is fixed at 0.9 regardless of the operating cell tune ${\nu }_0$. Vertical lines show the boundaries of the second-order ($m=2$) and fourth-order ($m=4$) coherent resonance bands estimated from Eqs. (3) and (5) with $C_2=0.7$ and $C_4=0.9$. The phase-space configurations at the three operating tunes indicated by arrows in the middle picture are exhibited in Figs. 4, 5, 6.

Figure 4

Phase-space configurations at ${\nu }_0=0.257$ with three different tail-truncation factors ($\kappa =0$, 10, 20), observed at (a) the transport entrance, (b) 100th FD cell, and (c) 500th FD cell.

Figure 5

Phase-space configurations at ${\nu }_0=0.263$ with three different tail-truncation factors ($\kappa =0$, 10, 20), observed at (a) the transport entrance, (b) 100th FD cell, and (c) 500th FD cell.

Figure 6

Phase-space configurations at ${\nu }_0=0.275$ with three different tail-truncation factors ($\kappa =0$, 10, 20), observed at (a) the transport entrance, (b) 100th FD cell, and (c) 500th FD cell.

Figure 7

Time evolution of the rms emittance-growth rates in the three cases of Figs. 4, 5, 6. The operating bare tune per cell is fixed at (a) ${\nu }_0=0.257$, (b) ${\nu }_0=0.263$, and (c) ${\nu }_0=0.275$.

Figure 8

Incoherent tune spread of a Gaussian beam at ${\nu }_{0x}={\nu }_{0y}=0.290$ (red dot) where the emittance growth is negligible. The rms tune depression is adjusted initially to $\eta =0.9$, the same as in the case of Fig. 3.

Figure 9

Incoherent tune spread of a Gaussian beam after the rapid initial emittance growth due to the fourth-order resonance was nearly settled. The operating point is set at ${\nu }_{0x}={\nu }_{0y}=0.275$ (red dot) where non-negligible emittance growth occurs. The rms tune depression is adjusted initially to $\eta =0.9$.

Figure 10

Incoherent tune spread of a Gaussian beam at ${\nu }_{0x}={\nu }_{0y}=0.263$ (red dot) where the severe coherent envelope instability is expected. The conditions of the PIC simulation and Fourier analysis are the same as adopted in Fig. 9 (except for the operating bare tunes).

Figure 11

Horizontal phase-space configurations of a waterbag beam observed at (a) the transport entrance, (b) 400th FD cell, and (c) 1500th FD cell. The initial rms tune depression is chosen to be $\eta =0.8$. The operating point is set at ${\nu }_{0x}={\nu }_{0y}=0.192$ where the coherent sextupole ($m=3$) parametric resonance is expected from Eq. (3).

Figure 12

Incoherent tune spread of the waterbag beam exhibited in Fig. 11, after the initial particle distribution collapsed due to the coherent core resonance. The vertical and horizontal solid lines in the lower panel represent ${\nu }_{0x(0y)}=1/6$.