Resonant beam behavior studies in the Proton Storage Ring

We present studies of space-charge-induced beam profile broadening at high intensities in the Proton Storage Ring (PSR) at Los Alamos National Laboratory. We investigate the profile broadening through detailed particle-in-cell simulations of several experiments and obtain results in good agreement with the measurements. We interpret these results within the framework of coherent resonance theory. With increasing intensity, our simulations show strong evidence for the presence of a quadrupole-mode resonance of the beam envelope with the lattice in the vertical plane. Specifically, we observe incoherent tunes crossing integer values, and large amplitude, nearly periodic envelope oscillations. At the highest operating intensities, we observe a continuing relaxation of the beam through space charge forces leading to emittance growth. The increase of emittance commences when the beam parameters encounter an envelope stop band. Once the stop band is reached, the emittance growth balances the intensity increase to maintain the beam near the stop band edge. Additionally, we investigate the potential benefit of a stop band correction to the high intensity PSR beam.

Figure 1

Benchmark of vertical experimental profile measurements with PIC simulations. The solid (red) curve is the experimental profile. The dotted (blue) curve is the PIC result with space charge. The (pink) dashed line is the PIC result without space charge.

Figure 2

Benchmark of longitudinal profile measurements with PIC simulations. The solid (red) curve is the experimental profile. The dashed (blue) curve is the PIC result with longitudinal space charge and impedance. The (pink) dotted line is the PIC result with longitudinal space charge but without other impedance.

Figure 3

Experimental vertical beam profiles after the accumulation of the full intensity PSR beam (red solid curve), the half intensity beam (blue dotted curve), and the quarter intensity beam (pink dashed curve). The three profiles have been normalized to give identical areas under the curves.

Figure 4

Vertical rms emittance (in units of $\pi \mathrm{mm}\mathrm{mrad}$) evolution of the full intensity beam (solid red curve), the half intensity beam (dashed green line), and the quarter intensity beam (dotted blue line), obtained from PIC simulations.

Figure 5

Incoherent particle tune shifts versus longitudinal coordinate after the accumulation of the full intensity PSR beam (red points), the half intensity beam (blue points), and the quarter intensity beam (green points), obtained from PIC simulations.

Figure 6

Vertical second moment of the beam, normalized to $\sqrt{{\beta }_y{ϵ}_y}$, taken over one turn near the end of accumulation of $4.37\times {10}^{13}$ protons. The curves correspond to longitudinal slices of beam at approximately $0.01{\sigma }_l$ (red solid line), $0.25{\sigma }_l$ (blue dashed line), and ${\sigma }_l$ (pink dotted line).

Figure 7

Strength of the ${\nu }_e=4.0$ envelope harmonic versus longitudinal coordinate. The harmonic strengths are obtained via FFT of the second moments of the beam at specific longitudinal locations.

Figure 8

Envelope tune (as calculated by an envelope model) versus turn number for the full intensity (red +’s), half intensity (blue $*$’s), and quarter intensity (green ×’s) PSR beam.

Figure 9

PIC simulated maximum envelope oscillations (normalized to $\sqrt{{\beta }_y{ϵ}_y}$) during accumulation of $4.37\times {10}^{13}$ protons in the PSR ring (colored points).

Figure 10

Experimental longitudinal profile measurements. The blue curve is the profile of the beam without the longitudinal notch, and the red curve is the profile of the beam with the longitudinal notch.

Figure 11

Experimental transverse vertical profile measurements of the full intensity beam. The (blue) dashed curve is the profile of the beam without the longitudinal notch, and the (red) solid curve is the profile of the beam with the longitudinal notch.

Figure 12

Experimental transverse vertical profile measurements of the half intensity beam. The (blue) dashed curve is the profile of the beam without the longitudinal notch, and the (red) solid curve is the profile of the beam with the longitudinal notch.

Figure 13

Incoherent particle tunes taken from PIC simulation versus longitudinal coordinate for the high intensity beam. The blue points represent the beam without the longitudinal notch, and the red points represent the beam with the longitudinal notch.

Figure 14

Simulated vertical rms beam emittances for the notched and non-notched beams during accumulation of full and half intensity beams. The red solid and orange dashed lines are the notched beams at full and half intensity, respectively, and the blue dotted and turquoise dashed lines are the non-notched beams at full and half intensity, respectively.

Figure 15

Experimentally measured vertical beam profiles for tunes in the range of $({\nu }_x=3.19,{\nu }_y=2.09,\dots ,2.19)$, where the profile widths increase as ${\nu }_y$ decreases from 2.19 to 2.09 (2.19, 2.15, 2.13, 2.11, 2.09).

Figure 16

Experimentally measured beam loss during accumulation versus vertical bare tune.

Figure 17

Envelope tune (as calculated by a PCM model) versus turn number for ${\nu }_y=2.19$ (red +’s), ${\nu }_y=2.13$ (green ×’s), and ${\nu }_y=2.09$ (blue $*$’s).

Figure 18

PIC simulated maximum envelope oscillations (normalized to $\sqrt{{\beta }_y{ϵ}_y}$) versus turn number for ${\nu }_y=2.19$ (red +’s), ${\nu }_y=2.13$ (green ×’s), and ${\nu }_y=2.09$ (blue $*$’s).

Figure 19

Vertical emittance evolution for ${\nu }_y=2.19$ (red solid line), ${\nu }_y=2.13$ (green dashed line), and ${\nu }_y=2.09$ (blue dotted line).

Figure 20

Envelope tune (as calculated by a PCM model) versus turn for the lattice setting $({\nu }_x,{\nu }_y)=(3.19,2.13)$ for the regular PSR beam injection rate (red +’s), and for twice the regular injection rate (blue $*$’s).

Figure 21

Simulated vertical rms emittances versus turn for the full accumulation of the beam, followed by 1000 turns of beam storage. The solid (red) line shows the case for ${\nu }_y=2.19$, the dashed (green line shows the case for ${\nu }_y=2.13$, and the dotted (blue) line shows the case for ${\nu }_y=2.09$. The dashed vertical line marks the end of accumulation at 3423 turns.

Figure 22

Experimentally measured vertical beam profiles for two different painting schemes of a low intensity beam. The red curve is the beam with a 17.51 mm injection offset, and the blue curve is the beam with a 12.61 mm injection offset.

Figure 23

Experimentally measured vertical beam profiles for two different painting schemes of a high intensity beam at the end of accumulation. The (red) solid curve is the beam with a 17.51 mm injection offset, and the (blue) dashed curve is the beam with a 12.61 mm injection offset.

Figure 24

Top panel: Simulated vertical rms emittances during the accumulation of a high intensity beam with two different injection painting schemes. The solid (red) curve represents the beam with a 17.51 mm injection offset, and the dashed (blue) curve represents the beam with a 12.61 mm injection offset. Bottom panel: Calculated envelope tune versus turn for the 17.51 mm injection offset (red +’s), and for the 12.61 mm injection offset (blue $*$’s).

Figure 25

Simulated vertical rms emittances during the accumulation of a high intensity beam in a PSR phase-advance-equivalent, uniform focusing lattice.

Figure 26

Top panel: Harmonic spectrum of the vertical beta function for the real PSR lattice. Bottom panel: Harmonic spectrum of the vertical beta function for a perfectly symmetric, PSR-like lattice. Both lattices have $({\nu }_x,{\nu }_y)=(3.19,2.19)$.

Figure 27

Top panel: Vertical rms emittance versus turn number during the accumulation of a high intensity beam in the real PSR lattice. Bottom panel: Vertical rms emittance versus turn number during the accumulation of a high intensity beam in the symmetric PSR-like lattice. In both plots, the solid (red) curves represent the beam with space charge, and the dashed (blue) curves represent the beam without space charge.

Figure 28

Tune footprints of the beam in the real PSR lattice (blue points), and the symmetric, PSR-like lattice (red points). The bare tune for both cases is $({\nu }_x,{\nu }_y)=(3.19,2.19)$.