Suppression of the eighth-order space-charge-induced resonance

The eighth-order space-charge-induced resonance $8{\nu }_y=171$ was identified as the source of beam losses and suppressed in the main ring synchrotron of the Japan Proton Accelerator Research Complex. Identification and interpretation were performed using investigations of the lost particles, Poincaré map simulations, and analytical calculations. In addition, the dependencies of the simulated beam loss on the vertical phase advance and resonance potential ${\widehat{U}}_{0,8,171}$ were very similar. Besides validating our assumption, this relationship indicates that the vertical phase advance can be exploited for controlling the beam loss. We, therefore, developed new beam optics by changing the vertical phase advance and successfully reduced the resonance width by a factor of 4. In an experimental study, the new optics reduced the beam loss during the beam injection period by 20% from that of the present optics.

Figure 1

Tune spread distribution obtained by the space charge tracker (sctr) [13, 14] simulation and major resonances (lines) in FX operation. The individual tunes were obtained via Fourier analysis using the data from the 5000th to the 5511th turn. The used parameters are listed in Table 2.

Figure 2

Simulated longitudinal distribution of the beam (upper) and beam loss (lower). The beam distribution in the upper panel was captured at 24,100 turns (130 ms) from the beam injection. The lower image shows the particles lost between the 500th and the 24,100th turns.

Figure 3

Simulated phase space distributions of the lost particles with the present FX optics in action-angle coordinates. The left and right images show the phase space positions of the lost particles with $J_x>J_y$ and $J_x<J_y$, respectively. The last 15 turn positions before the particles were lost are plotted.

Figure 4

Simulated tune distributions of the lost particles with $J_x>J_y$ (left) and $J_x<J_y$ (right) in the MR with the present FX optics. The tunes were calculated with one turn phase advance before the loss. The red solid lines represent the resonance $8{\nu }_y=171$, and the magenta dashed lines represent other space charge resonances up to the eighth order.

Figure 5

The regions of actions influenced by the resonances from analytical calculations. The solid colored lines and dashed lines show the actions of the on-resonance particles $(J_{xR},J_{yR})$ at $\lambda ={\lambda }_{\max }$, and $\lambda ={\lambda }_{\min }$, respectively. The red, blue, and green lines correspond to the resonances $8{\nu }_y=171,2{\nu }_x+6{\nu }_y=171$, and $4{\nu }_x+4{\nu }_y=171$, respectively. The two additional lines at $2J_x=60\pi \mathrm{mm}\mathrm{mrad}$ and $2J_y=60\pi \mathrm{mm}\mathrm{mrad}$ indicate the collimator aperture.

Figure 6

Simulated Poincaré map in the vertical plane using the present FX parameters. The horizontal action was set to $J_x=0$. The red line plots the analytically solved vertical action of the on-resonance particles $2J_{yR}$.

Figure 7

Calculated potential ${\widehat{U}}_{0,8,171}$ corresponding to the resonance $8{\nu }_y=171$. The green square and blue triangle represent the result of the present FX optics and new optics, respectively.

Figure 8

Simulated numbers of lost particles from turns 500 to 24,100. The green square and blue triangle represent the result of the present FX optics and new optics, respectively.

Figure 9

Simulated Poincaré map in the vertical plane with the new optics. The horizontal action was set to $J_x=0$. The red line represents the analytical solution of the vertical action of the on-resonance particles $2J_{yR}$.

Figure 10

Simulated phase space distributions of the lost particles with $J_x>J_y$ (left) and $J_x<J_y$ (right) with the new optics, displayed in action-angle coordinates. The last 15 turn positions before the particles were lost are plotted.

Figure 11

Simulated tune distributions of the lost particles with $J_x>J_y$ (left) and $J_x<J_y$ (right) with the new optics. The tunes were calculated with one turn phase advance before the loss. The red solid lines represent the resonance $8{\nu }_y=171$, and the magenta dashed lines represent other space charge resonances up to the eighth order.

Figure 12

Beam survival ratios in the MR with the present FX optics (red) and the new optics (blue), measured by the DCCT.

Figure 13

Simulated Poincaré map in the horizontal plane with the present FX parameters. The vertical action was set to $J_y=0$.

Figure 14

Simulated Poincaré map in the horizontal plane with the new optics. The vertical action was set to $J_y=0$.