Excitation and suppression of synchrobetatron resonances in high-intensity hadron linacs

The space-charge potential can be a source of various resonances in high-intensity hadron bunches traveling in linear accelerators at relatively low energy. Those natural resonances, which may occur even without external driving fields, have to be avoided carefully to minimize undesired beam loss. We perform self-consistent multiparticle simulations systematically to locate major resonance stop bands in the tune diagram, starting from equipartitioned and nonequipartitioned beams. Serious excitation of synchrobetatron resonances is confirmed, depending on the emittance condition at injection. It is demonstrated that a synchrobetatron difference resonance of any order can be suppressed strongly by controlling the ratio of the transverse and longitudinal projected emittances. The equipartitioned linac design is shown to broaden the usable operating area in the tune space, though that has almost nothing to do with thermodynamic effects.

Figure 1

Sinusoidal focusing model.

Figure 2

Transverse and longitudinal rms envelopes matched to (a) the standard FODO lattice, and (b) the sinusoidal lattice in Fig. 1. The operating phase advances are set at $({\mu }_{0\perp },{\sigma }_{0\parallel })=(55°,55°)$ in both cases. As an example, we have assumed an equipartitioned proton beam propagating at the kinetic energy of 10 MeV through a unit cell of 1 m long. The transverse rms emittance and tune depression are adjusted to ${ϵ}_{\perp }=1.0\pi \text{mm}·\mathrm{mrad}$ and ${\eta }_{\perp }=0.7$.

Figure 3

Linear-mode stability predicted by the 3D rms envelope equations. Red lines indicate the positions of the second-order ($m=2$) resonances expected from Eq. (6) with $C_2=0.7$. In the left panel (a), the equipartitioning condition is applied over the whole tune space with the transverse tune depression ${\eta }_{\perp }$ fixed at 0.8. In the right panel (b), the beam intensity (perveance) is kept unchanged. ${\eta }_{\perp }$ thus depends on the operating point. The intensity is determined so as to achieve ${\eta }_{\perp }=0.8$ at $({\mu }_{0\perp },{\sigma }_{0\parallel })=(55°,55°)$.

Figure 4

Warp simulation result obtained with the equipartitioned waterbag distribution. PIC simulations were done at $3721(=61\times 61)$ different operating points uniformly distributed in the range $36°<{\mu }_{0\perp },{\sigma }_{0\parallel }<144°$, which enables us to identify instability bands wider than about 2°. The emittance-growth rates evaluated at the exit of the 100th AG cell are color-coded in the tune diagram. ${\eta }_{\perp }$ is adjusted to 0.8 over the whole tune space. The transverse emittances are also fixed at injection, while we choose the initial longitudinal emittance so as to meet the equipartitioning condition. Black lines indicate the positions of low-order coherent resonances predicted by Eq. (6) with $C_2=0.7$, $C_3=0.8$, and $C_4=0.9$. Three numbers beside each line represent $(n_{\perp },n_{\parallel },n^{\prime })$.

Figure 5

Warp simulation results obtained with (a) the Gaussian beam and (b) the waterbag beam. The emittance growth is evaluated at the exit of the 100th AG cell. The initial value of ${\eta }_{\perp }$ is fixed at 0.7 everywhere. The transverse rms emittance ${ϵ}_{\perp }$ is also maintained in both cases, but the longitudinal parameters ${\eta }_{\parallel }$ and ${ϵ}_{\parallel }$ are determined at each operating point so as to meet the equipartitioning condition. The locations of the noncoupling sextupole ($m=3$) and octupole ($m=4$) resonances expected from Eq. (6) with $C_3=0.8$ and $C_4=0.9$ are indicated by dashed lines. Three numbers in the picture represent $(n_{\perp },n_{\parallel },n^{\prime })$. The phase-space distributions of the waterbag beam at the operating points (A) and (B) are exhibited in Fig. 6.

Figure 6

Horizontal phase-space distributions after the waterbag beam goes through 60 AG cells at the operating points (A) and (B) in Fig. 5. As an example, we have assumed a 10 MeV proton bunch whose horizontal emittance is $0.1355\pi \text{mm}·\mathrm{mrad}$ at injection.

Figure 7

Warp simulation result obtained with a nonequipartitioned Gaussian distribution. The transverse and longitudinal emittance growth rates at the exit of the 100th AG cell are plotted separately in the left panel (a) and the right panel (b). The beam intensity is fixed at the value that gives ${\eta }_{\perp }=0.7$ at the operating point $({\mu }_{0\perp },{\sigma }_{0\parallel })=(57.6°,57.6°)$.

Figure 8

Warp simulation results obtained with nonequipartitioned beams. The initial distribution is the Gaussian type that has the rms emittance ratio of (a) ${ϵ}_{\parallel }/{ϵ}_{\perp }=3$ and (b) ${ϵ}_{\parallel }/{ϵ}_{\perp }=10$. The bunch intensity and all projected emittances at injection are fixed over the whole tune space. The color is chosen based on largest emittance growth of the three directions at the exit of the 100th AG cell. The solid and dashed lines are obtained from Eq. (6) for $m=2$ and 3. Three numbers in the left picture represent $(n_{\perp },n_{\parallel },n^{\prime })$.

Figure 9

Rms tune depressions in (a) the transverse direction and (b) the longitudinal direction, under the condition assumed in Fig. 8.

Figure 10

Warp simulation results obtained under the initial conditions (a) ${ϵ}_{\parallel }/{ϵ}_{\perp }=1$ and (b) ${ϵ}_{\parallel }/{ϵ}_{\perp }=0.5$. The largest emittance growth of the three directions is taken to choose the color at each operating point. The result in Fig. 7 is replotted in the left panel (a) for comparison. We have changed the initial emittance ratio ${ϵ}_{\parallel }/{ϵ}_{\perp }$ from 1.0 to 0.5 in the right panel (b), maintaining the beam intensity.

Figure 11

Warp simulation results obtained with the initial emittance ratios (a) ${ϵ}_{\parallel }/{ϵ}_{\perp }=1$ and (b) ${ϵ}_{\parallel }/{ϵ}_{\perp }=2$. The simulation conditions are the same as in Fig. 10 except for the value of ${ϵ}_{\parallel }/{ϵ}_{\perp }$ in the right panel.

Figure 12

Contour plots of $\mathrm{\Lambda }(n_{\perp },n_{\parallel })$ in an equipartitioned case. The same initial beam conditions as considered in Fig. 4 are taken to calculate the $\mathrm{\Lambda }$-value at each operating point with (a) $(n_{\perp },n_{\parallel },n^{\prime })=(1,-1,0)$, (b) $(n_{\perp },n_{\parallel },n^{\prime })=(2,-1,0)$, and (c) $(n_{\perp },n_{\parallel },n^{\prime })=(1,-2,0)$. A number on each contour line is the value of $\mathrm{\Lambda }(n_{\perp },n_{\parallel })$ divided by a normalization constant, i.e., the value of $\mathrm{\Lambda }(1,1)$ at the operating point $({\mu }_{0\perp },{\sigma }_{0\parallel })=(60°,60°)$. Black solid lines are obtained from Eq. (6) with $C_2=0.7$ or $C_3=0.8$.

Figure 13

Time evolution of the projected rms emittances of a 10 MeV proton beam at the operating point $({\mu }_{0\perp },{\sigma }_{0\parallel })=(60°,60°)$. The beam initially has the equipartitioned Gaussian distribution whose transverse tune depression is adjusted to ${\eta }_{\perp }=0.6$. Black curves correspond to the case where the phase-space configuration is in the pseudoequilibrium, while red curves to the case of rms matching.

Figure 14

Initial and final particle distributions in the horizontal phase space, corresponding to the Warp simulations in Fig. 13. The transverse rms emittance has been assumed initially to be about $0.05\pi \text{mm}·\mathrm{mrad}$. In the upper panel (a), the beam is initially in a pseudoequilibrium state while in the lower panel (b), rms matched to the lattice at injection.

Figure 15

Warp simulation results with the bare synchrotron tune ${\sigma }_{0\parallel }$ fixed at 55°. The initial phase-space distribution of particles in the bunch is the Gaussian type. The projected rms emittances in the three spatial directions are equalized everywhere in the diagram. The beam intensity is maintained at the value that gives ${\eta }_{\perp }=0.8$ at the operating point $({\mu }_{0\perp },{\sigma }_{0\parallel })=(60°,60°)$. Solid and dashed lines in the picture indicate the locations of the possible second-order and third-order stop bands obtained from Eq. (b1) with $C_2=0.7$ and $C_3=0.8$. Four integers written beside each line represent $(n_x,n_y,n_{\parallel },n^{\prime })$.