Space charge effects on the third order coupled resonance

The effect of space charge on bunched beams has been the subject of numerous numerical and experimental studies in the first decade of 2000. Experimental campaigns performed at the CERN Proton Synchrotron in 2002 and at the GSI SIS18 in 2008 confirmed the existence of an underlying mechanism in the beam dynamics of periodic resonance crossing induced by the synchrotron motion and space charge. In this article we present an extension of the previous studies to describe the effect of space charge on a controlled coupled (2D) third order resonance. The experimental and simulation results of this latest campaign shed a new light on the difficulties of the 2D particle dynamics. We find striking experimental evidence that space charge and the coupled resonance create an unusual coupling in the phase space, leading to the formation of an asymmetric halo. Moreover, this study demonstrates a clear link between halo formation and fixed-lines.

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Figure 1

Top: Experimental tune scan at 2 GeV kinetic energy. The color scale is proportional to the measured normalized beam loss. The red color indicates maximum beam loss. Solid lines indicate 2nd order, dashed lines 3rd order, and dash-dotted lines 4th order resonances. The resonance $2Q_x+Q_y=19$ was found to be strongly excited by natural nonlinear errors of the machine. Grey areas indicate the absence of measurement data [23]. Bottom: Detail of the tune diagram shown in Fig. 1, with identical colour scale. The sextupolar resonance $Q_x+2Q_y=19$ is shown with a white solid line and the working points used for the systematic study are indicated with the white crosses.

Figure 2

Magnetic cycle of the PS as used for the measurements. The first plateau starting at 170 ms corresponds to injection kinetic energy at 1.4 GeV and the long plateau starting at 250 ms to 2 GeV. For each working point the evolution of the transverse profiles was recorded between 300 ms and 1400 ms as indicated by the dashed lines. The final acceleration is required to extract the beam towards the beam dump.

Figure 3

Measured emittance growth and beam intensity as a function of the horizontal tune. Error bars are obtained from the statistical fluctuation resulting from three consecutive measurements.

Figure 4

Comparison between final and initial profiles at $Q_{x0}=6.104$ for the horizontal (a) and the vertical plane (b). Figures (c) and (d) show the corresponding pictures for $Q_{x0}=6.129$.

Figure 5

(a) Emittance growth and beam survival as computed with MAD-X (adaptive mode). (b) Simulation results with MICROMAP (frozen mode). The quantities retrieved from the experimental data are shown with dashed lines in both cases.

Figure 6

Comparison between experimental and simulated final profiles for $Q_{x0}=6.104$ in the (a) horizontal and (b) vertical plane.

Figure 7

Resonance detuning $\mathrm{\Delta }{Q}_{\mathrm{sc},\mathrm{x}}+2\mathrm{\Delta }{Q}_{\mathrm{sc},\mathrm{y}}$ and ${\mathrm{\Delta }}_r$ as a function of amplitudes $(0,Y)$ (black curve), and $(X,0)$ (red curve). The amplitudes are computed at the location of the vertical wire scanner (SS64).

Figure 8

Pictures (a) and (b) show the horizontal and vertical beam profiles in simulations with the chromaticity set to zero. The working point is $Q_{x0}=6.104,Q_{y0}=6.476$. The blue curves are the initial profiles while the black curves are the final profiles. Pictures (c) and (d) represent the same settings, but including the chromatic effect. These beam distributions are computed at the location of the vertical wire scanner. In this case the final profiles are shown in red.

Figure 9

Simulated growth of horizontal and vertical emittance with the chromaticity set to zero (solid lines), and including natural chromaticity (dashed lines). The simulations are done with MICROMAP.

Figure 10

Resonance detuning $\mathrm{\Delta }{Q}_{\mathrm{sc},\mathrm{x}}+2\mathrm{\Delta }{Q}_{\mathrm{sc},\mathrm{y}}$, and ${\mathrm{\Delta }}_r$ as a function of amplitudes $(0,Y)$ (black curve), and $(X,0)$ (red curve) including chromaticity. The working point is $Q_{x0}=6.104$; the vertical line corresponds to the halo edge of $Y\sim 9{\sigma }_y$. The amplitudes are computed at the location of the vertical wire scanner (SS64).

Figure 11

Resonance stop-band.

Figure 12

Space charge tune-spread. The solid line is the third order coupled resonance $Q_x+2Q_y=19$. The circles enclose the tunes of the two resonant orbits shown in Fig. 13.

Figure 13

(a) Resonant orbits of two particles (black & red) at the longitudinal position $z=z^{\prime }=0$. (b) Resonant orbits for two particles (black & red) at longitudinal position $z=1.5{\sigma }_z$ and $z^{\prime }=0$. The synchrotron motion is kept frozen.

Figure 14

Resonant orbits for particles having longitudinal momentum $\delta p/p=2{\sigma }_{\delta p/p}$, and $z=0$. In the simulation the longitudinal motion is kept frozen.

Figure 15

(a) Evolution of the horizontal Courant-Snyder invariant of a test particle over 430 synchrotron oscillations. (b) Evolution of the horizontal invariant over a single synchrotron period, experiencing four resonance-induced kicks.

Figure 16

(a) Evolution of the vertical Courant-Snyder invariant of a test particle over 430 synchrotron oscillations. (b) Evolution of the vertical invariant over a single synchrotron period, experiencing four resonance-induced kicks.

Figure 17

(a) Sequence of fixed-lines constructed by a particle moving from $z=-2{\sigma }_z$ to $z=0$. (b) Same as in (a) but with the natural chromaticity included in the simulation. In both cases the red curves are obtained from Eq. (4) with one value of $\alpha$.

Figure 18

Cross section of one MU of the PS. The reference point between the two poles corresponds to the location of the closed orbit. The circuits of the PFW, which are encapsulated by an epoxy resin, are situated directly on top of the poles. Furthermore, the main coils and the F8L are visible [33].

Figure 19

Modeling of the PS MU. Even though the junction between the two half-units is included as SBEND, it is actually just a drift space.

Figure 20

Typical set of initial transverse profiles measured with the wire scanners. The offset of the baseline is clearly visible in both planes. The horizontal profile also exhibits a position offset due to the closed orbit distortion at the location of the wire scanner. Both offsets were removed prior to further treatment of the signals.

Figure 21

This profile corresponds to the horizontal one of Fig. 20; however, only data points within the restricted interval $[-6{\sigma }_1+{\mu }_1,6{\sigma }_1+{\mu }_1]$ are shown. The shaded areas indicate data points, which are considered for the evaluation of the baseline using ${\mathcal{F}}_2(z)$.

Figure 22

After removing the contribution of the baseline and the horizontal offset, the profile depicted in blue is obtained. For comparison the horizontal raw data is shown as well.

Figure 23

In order to compute the statistical moments only data points above the defined threshold were considered.