Hamiltonian approach to slip-stacking dynamics

Hamiltonian dynamics has been applied to study the slip-stacking dynamics. The canonical-perturbation method is employed to obtain the second-harmonic correction term in the slip-stacking Hamiltonian. The Hamiltonian approach provides a clear optimal method for choosing the slip-stacking parameter and improving stacking efficiency. The dynamics are applied specifically to the Fermilab Booster-Recycler complex. The dynamics can also be applied to other accelerator complexes.

Figure 1

Comparison of derived compensation second-harmonic ratio $\lambda$ with empirical values obtained by Eldred through simulations. Our determination is shown in red.

Figure 2

The slip-stacking parameter ${\alpha }_s$ versus the Recycler rf voltage $V_{\mathrm{rf}}$.

Figure 3

Top: phase-space structure with slip-stacking parameter ${\alpha }_s=4.1$ without second-harmonic rf compensation, i.e. $\lambda =0$. Bottom: phase-space structure when the second-harmonic rf compensation is turned on, or $\lambda =-4/{\alpha }_s^2=-0.238$. The interaction of two primary buckets is so strong that the phase space between them becomes chaotic. Only the cores of the fifth-order resonance islands are stable in the top plot. Once the compensation rf is included, the chaotic region becomes smaller. There are stable tori outside the fifth-order resonance islands and the stable bucket area increases, i.e., a larger injection bunch area can be accepted.

Figure 4

Phase-space structure with slip-stacking parameter ${\alpha }_s=6.0$ is shown in the top plot with no second-harmonic rf compensation or $\lambda =0$, and in the lower plot with second-harmonic rf compensation or $\lambda =-4/{\alpha }_s^2=-0.111$. It is clear that the compensation reduces the chaotic region between the upper and lower buckets with, however, only small increase in the phase-space areas of the slip-stacking buckets.

Figure 5

Phase-space structure with slip-stacking parameter ${\alpha }_s=3.5$ is shown in the top plot with no second-harmonic rf compensation or $\lambda =0$, and in the lower plot with second-harmonic rf compensation or $\lambda =-4/{\alpha }_s^2=-0.327$. The compensation restores some of the structure resonances, but they are still embedded in the chaotic sea. The compensation does not cancel resonances associated with each slip-stacking bucket.

Figure 6

The synchrotron tune and its harmonics of a stationary rf system versus ${\varphi }_{\max }$, the maximum phase amplitude of synchrotron oscillation. The horizontal dashed lines are modulation tunes at various slip-stacking parameters. When they cut through an $n$th harmonic of the synchrotron tune, the $n$th-order resonance will occur at that phase-space amplitude. These are the resonances visible in Figs. 3, 4, 5.

Figure 7

Left: bucket area (red line) and bucket height (black line) versus rf voltage $V_{\mathrm{rf}}$ for the Recycler. The typical operational Recycler rf voltage is 60 kV. Data points are obtained from numerical simulation without (black) and with (blue) second-harmonic rf compensation, where the dip at about 90 kV arises from the fifth-order resonance inside each slip-stacking bucket. The fourth-order resonance occurs around 150 kV Recycler rf voltage. Right: bunch height of the Booster beam at bunch areas 0.1, 0.15, and 0.2 eVs versus Booster rf voltage at extraction with the Booster rf sum of about 350 kV at extraction [10].