Enhanced dynamical stability with harmonic slip stacking

We develop a configuration of radio-frequency (rf) cavities to dramatically improve the performance of slip stacking. Slip stacking is an accumulation technique used at Fermilab to nearly double proton intensity by maintaining two beams of different momenta in the same storage ring. The two particle beams are longitudinally focused in the Recycler by two 53 MHz 100 kV rf cavities with a small frequency difference between them. We propose an additional 106 MHz 20 kV rf cavity with a frequency at the double the average of the upper and lower main rf frequencies. We show the harmonic rf cavity cancels out the resonances generated between the two main rf cavities and we derive the relationship between the harmonic rf voltage and the main rf voltage. We find the area factors that can be used to calculate the available phase space area for any set of beam parameters without individual simulation. We establish Booster beam quality requirements to achieve 99% slip stacking efficiency. We measure the longitudinal distribution of the Booster beam and use it to generate a realistic beam model for slip stacking simulation. We demonstrate that the harmonic rf cavity can not only reduce particle loss during slip stacking, but also reduce the final longitudinal emittance.

Figure 1

The Booster batch is represented by the circles and the Recycler (or Main Injector) is represented by the seven-sector wheel. (a–d) Six Booster batches injected each Booster cycle (boxcar stacking). (e) The rf frequency is gradually lowered in between the sixth and seventh batch injection. (f–i) Subsequent batches are injected into the gap left by the first six and gradually overlap. (j) When the first six and last six batches are aligned, the batches are extracted to the Main Injector (if needed) and both beams are accelerated as one.

Figure 2

Stability of initial coordinates for ${\alpha }_s=4.18$. The color shows the number of synchrotron periods a test particle survives before it is lost. The top plot shows conventional slip stacking and the bottom plot shows harmonic slip stacking.

Figure 3

(top) Slip stacking area factor $F$ as a function of ${\alpha }_s$ and $\lambda$. (bottom) Slip stacking area factor $F$ for case without a harmonic cavity ($\lambda =0$) shown as a single line and case with a harmonic cavity (optimal $\lambda$) shown as a double line.

Figure 4

(top) Modified slip stacking area factor $Z$ as a function of ${\alpha }_s$ and $\lambda$. (bottom) Modified slip stacking area factor $Z$ for case without a harmonic cavity ($\lambda =0$) shown as a single line and case with a harmonic cavity (optimal $\lambda$) shown as a double line.

Figure 5

Value of $\lambda$ which maximizes phase-space area, as a function of ${\alpha }_s$.

Figure 6

Balanced value of harmonic rf voltage has a quadratic dependence on main rf voltage. Bottom line shows case for a 15-Hz Booster (black) and top line for 20-Hz Booster (red).

Figure 7

Poincaré map for conventional slip stacking with a value of ${\alpha }_s$ corresponding to 100 kV main rf voltage and 1260 Hz rf frequency separation.

Figure 8

Poincaré map for conventional slip stacking with a value of ${\alpha }_s$ corresponding to 65 kV main rf voltage and 1680 Hz rf frequency separation.

Figure 9

Poincaré map for harmonic slip stacking with a value of ${\alpha }_s$ corresponding to 100 kV main rf voltage and a balanced value of $\lambda$ corresponding to 21 kV harmonic rf voltage.

Figure 10

Poincaré map for harmonic slip stacking with a value of ${\alpha }_s$ corresponding to 100 kV main rf voltage and an unbalanced value of $\lambda$ corresponding to 35 kV harmonic rf voltage.

Figure 11

The 99% admittance at 99% efficiency (at an optimal ${\alpha }_s$ and $\lambda$) as a function of aspect ratio. The solid double lines indicate the admittance for harmonic slip stacking constrained to 100 kV main rf voltage and 20 kV harmonic rf voltage. The dashed double lines indicate admittance for unconstrained rf voltage.

Figure 12

Optimal slip stacking parameter ${\alpha }_s$ for 99% admittance (at 99% efficiency) as a function of aspect ratio.

Figure 13

Optimal harmonic-main voltage ratio $\lambda$ for 99% admittance (at 99% efficiency) as a function of aspect ratio.

Figure 14

Tomography measurements of longitudinal distribution of beam injected into the Recycler. Distribution derived from an average across Booster batch. Measurement taken from a typical $2+6$ slip stacking cycle on May 27th 2015 at a bunch intensity of $5.1\times 10^{10}$ protons.

Figure 15

Particle loss as a function of Recycler main rf voltage and $\lambda$.

Figure 16

Longitudinal emittance after capture in Main Injector, as a function of Recycler main rf voltage and $\lambda$.

Figure 17

Optimal Main Injector capture voltage as a function of Recycler main rf voltage and $\lambda$.

Figure 18

Pareto fronts for particle loss rate and emittance after capture.