Space charge induced resonance excitation in high intensity rings

We present a particle core model study of the space charge effect on high intensity synchrotron beams, with specific emphasis on the Proton Storage Ring (PSR) at Los Alamos National Laboratory. Our particle core model formulation includes realistic lattice focusing and dispersion. We transport both matched and mismatched beams through real lattice structure and compare the results with those of an equivalent uniform-focusing approximation. The effects of lattice structure and finite momentum spread on the resonance behavior are specifically targeted. Stroboscopic maps of the mismatched envelope are constructed and show high-order resonances and stochastic effects that dominate at high mismatch or high intensity. We observe the evolution of the envelope phase-space structure during a high intensity PSR beam accumulation. Finally, we examine the envelope-particle parametric resonance condition and discuss the possibility for halo growth in synchrotron beams due to this mechanism.

Figure 1

Vertical beam envelope versus distance over one turn of a PSR phase-advance-equivalent uniform-focusing lattice. The envelope is normalized to the zero-space charge envelope, $\sqrt{{\beta }_y{ϵ}_y}$.

Figure 2

The maximum normalized envelope amplitude versus the vertical space charge tune shift normalized to the fractional vertical bare tune (0.19). The solid (red) and dashed (blue) lines represent the predictions of Sacherer’s model for the horizontal and vertical planes, respectively, and the red $+$’s and blue $*$’s represent the horizontal (with $\Delta p/p=0$) and vertical predictions, respectively, of the particle core model implemented in the ACCSIM tracking code.

Figure 3

Vertical coherent resonance response curves for the PSR lattice (red $+$’s), and for an equivalent uniform-focusing approximation (blue $*$’s).

Figure 4

The harmonic spectrum of the PSR vertical beta function. Plot (a) shows a large portion of the spectrum, while plot (b) zooms in on the low-order harmonics.

Figure 5

Coherent resonance response curves in the horizontal direction (a) and the vertical direction (b) with and without energy spread. The (red) $+$’s represent the response without dispersion, and the (blue) $*$’s represent the response with dispersion.

Figure 6

Beam envelope versus distance along the PSR lattice, plotted with and without the dispersion term. The (red) solid line is the envelope plotted with the dispersion, and the (blue) dashed line is the envelope plotted without the dispersion.

Figure 7

(a) Stroboscopic map of the vertical envelope in the PSR lattice for a beam with intensity equal to one-quarter of the full PSR beam intensity listed in Table . The colors represent varying levels of mismatch. (b) Envelope tune versus the mismatch values used in (a).

Figure 8

(a) Stroboscopic map of the envelope in the PSR lattice for a beam with intensity equal to one-half of the full PSR beam intensity listed in Table . The colors represent varying levels of mismatch. (b) Envelope tune versus the mismatch.

Figure 9

(a) Stroboscopic map of the envelope in the PSR lattice for the full PSR beam intensity listed in Table  and emittances representative of the PSR beam at this intensity. The colors represent varying levels of mismatch. (b) Envelope tune versus mismatch.

Figure 10

(a) Stroboscopic map of the envelope in the PSR phase-advance-equivalent uniform-focusing lattice for a beam with intensity equal to one-quarter of the full PSR beam intensity listed in Table . The colors represent varying levels of mismatch. (b) Envelope tune versus the mismatch values used in (a).

Figure 11

(a) Horizontal stroboscopic map of a beam with $1.4\times {10}^{13}$ protons and $dp/p=0$. (b) Stroboscopic map of a beam with $1.4\times {10}^{13}$ protons and $dp/p=0.001$. (c) Stroboscopic map of a beam with $1.5\times {10}^{13}$ protons and $dp/p=0.001$. (d) Envelope tune versus mismatch parameter for the three cases above; the red $+$’s represent the beam with $1.4\times {10}^{13}$ protons and $dp/p=0$, the blue $*$’s represent the beam with $1.4\times {10}^{13}$ protons and $dp/p=0.001$, and the green $\times$’s represent the beam with $1.5\times {10}^{13}$ protons and $dp/p=0.001$.

Figure 12

(a) Stroboscopic map of the particle vertical phase space in the PSR lattice with zero envelope mismatch, $M=0$, and for a beam intensity equal to one-quarter of the full beam intensity listed in Table . (b) Map for $M=0.02$. (c) Map for $M=0.30$. The particle coordinates are normalized to the statistical Twiss parameters, such that the core boundary is represented by $Y^2+P_y^2=1$.

Figure 13

(a) Stroboscopic map of the particle vertical phase space in the PSR lattice with envelope mismatch $M=0.02$ and for a beam intensity equal to one-half of the full beam intensity listed in Table . (b) Map for $M=0.30$. The particle coordinates are normalized to the statistical Twiss parameters, such that the core boundary is represented by $Y^2+P_y^2=1$.

Figure 14

Stroboscopic map of the particle vertical phase space in the uniform-focusing lattice with envelope mismatch $M=0.300$ and for a beam intensity equal to one-half of the full beam intensity listed in Table . The coordinates are normalized to the mismatched envelope.

Figure 15

Separation of two points located on the separatrix of Fig. 13 versus longitudinal distance traveled by the beam.

Figure 16

The square root of the perturbed vertical betatron amplitude function (blue dashed line), for a beam with high intensity ($4.37\times {10}^{13}$ particles) in the PSR at LANL, is compared with the square root of the intrinsic betatron amplitude function (solid red line). The ratio of these two betatron amplitude functions, shown as the dashed (green) line, is the reduced envelope radius $R$ defined in Eq. (A3). Note that the average of $R$ is slightly larger than 1.