Numerical investigation of space charge effects on the positions of beamlets for transversely split beams

The use of transversely split beams was proposed some years ago as a means to perform extraction from a circular particle accelerator over multiple turns. In the course of studies carried out to increase understanding of the beam behavior, space charge effects have been probed. The experimental results showed a dependence of the beamlets’ positions on the total beam intensity. In this paper detailed numerical simulations studies are reported, which clearly indicate that the observed behavior is due to indirect space charge effects. The analysis includes configurations, which have not yet been experimentally probed, in order to better understand the complex interplay between nonlinear single-particle and intensity-dependent effects.

Figure 1

Summary of the experimental results of intensity-dependent effects with split beams from [13]. Upper: horizontal beam profiles measured with a wire scanner at the end of the splitting process with different total beam intensities. Lower: position of the peaks of the beamlets after splitting as a function of the total beam intensity.

Figure 2

Sketch of the PS main magnet (upper) and of the five circuits used to control tunes and chromaticities (lower), from [30].

Figure 3

Layout of the PS ring. The main dipoles are shown (blue and red shapes), including their focusing and defocusing half-units and the way they are installed, i.e., with the yoke inside or outside the circumference. The key elements to perform beam splitting are also shown, the number representing the name of the straight section in which they are installed, and the letter the magnetic type. The wire scanners are indicated together with the plane of measurement.

Figure 4

Iterative process to determine the self consistent position of the beamlets. The vertical axis describes the distance of a beamlet’s position from the closed orbit found after particle tracking including self-fields.

Figure 5

Upper left: Phase space portrait corresponding to the PS configuration used in the experimental studies without space charge effects, with $h_{2,0}<0$ and $Q_x=6.255$. Upper right: Phase space portrait corresponding to the configuration of upper left plot including five equally charged beamlets for a total intensity of $3.28\times {10}^{13}$ p. Lower left: Phase space portrait corresponding to a special PS configuration used only for the numerical simulations without space charge effects, with $h_{2,0}>0$ and $Q_x=6.245$. Lower right: Phase space portrait corresponding to the configuration of the lower left plot including five equally charged beamlets for a total intensity of $1.31\times {10}^{13}\mathrm{p}$. Parallel plates electric boundary conditions are included. The beam momentum is $14\mathrm{GeV}/c$.

Figure 6

Superposition of the transverse phase space corresponding to a horizontal tune of 6.255 and $h_{2,0}<0$ (red) and a tune of 6.245 and $h_{2,0}>0$ (blue). In both cases no intensity-dependent effects are taken into account.

Figure 7

Evolution of the position of one of the fixed points as a function of the horizontal tune for the two cases with negative (blue) and positive (red) value of $h_{2,0}$. In both cases no intensity-dependent effects are taken into account.

Figure 8

Beamlet position as a function of the total beam intensity. The cases corresponding to $h_{2,0}<0,Q_x=6.255$ and $h_{2,0}>0,Q_x=6.245$ are shown in the upper and lower plot, respectively. The contribution of the various sources of intensity-dependent position shifts is indicated by the various curves. Parallel plate electric boundary conditions are included. The beam momentum is $14\mathrm{GeV}/c$.

Figure 9

Dependence of a beamlet’s position on the total intensity. Several values of the horizontal tune, generating different beam positions for zero intensity, are probed. The cases corresponding to $h_{2,0}<0$ and $h_{2,0}>0$ are shown in the upper and lower plots, respectively. For $Q_x=6.247$ and 6.245, the fixed points disappear for the largest intensity reported in the plot. Parallel plates electric boundary conditions are included. The beam momentum is $14\mathrm{GeV}/c$.

Figure 10

Beamlet’s position as a function of the total beam intensity. The impact of the different electric boundary conditions is small, but visible. The cases corresponding to $h_{2,0}<0$, $Q_x=6.255$ and $h_{2,0}>0$, $Q_x=6.245$ are shown in the upper and lower plot, respectively. The contribution of the various sources of intensity-dependent position shifts is indicated by the various curves. The beam momentum is $14\mathrm{GeV}/c$.

Figure 11

Beamlet’s position as a function of the beam momentum (the total beam intensity is $3.28\times {10}^{13}$ p) for $h_{2,0}<0$ and $Q_x=6.255$ and parallel plates electric boundary conditions.

Figure 12

Beamlet’s position as a function of MTE efficiency with $h_{2,0}<0$ and $Q_x=6.255$ (the total beam intensity is $3.28\times {10}^{13}\mathrm{p}$) and parallel plates electric boundary conditions.

Figure 13

Beamlet’s position for a beam kicked into a stable island. Two different electric boundary conditions have been used. In this case $h_{2,0}<0$ and $Q_x=6.262$. The beam momentum is $14\mathrm{GeV}/c$.

Figure 14

Contour plot of the electric potential $\varphi$ for a unit charge density located in $(x_0,0)$. The boundary surfaces are shown in blue.

Figure 15

Contour plot of the electric potential $\varphi$ for a unit charge density located in $(x_0,0)$. The boundary surfaces are shown in blue.