CERN Proton Synchrotron booster space charge simulations with a realistic model for alignment and field errorsn1^*

The CERN Proton Synchrotron booster (PSB) is one of the machines of the LHC injector chain which will be upgraded within the LHC Injectors Upgrade (LIU) project. The injection energy of the PSB will be increased to 160 MeV in order to mitigate direct space charge effects, considered to be the main performance limitation, aiming to double the brightness for the LHC beams. In order to better predict the gain to be expected, space charge simulations are being carried out. As a first step, benchmarking between simulations and measurements is needed. Efforts to establish a realistic modeling of field and alignment errors aim at extending the basic model of the machine toward a more realistic one. Simulations of beam dynamics with strong space charge and realistic errors are presented and analyzed in this paper.

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Figure 1

Transverse and longitudinal alignment errors and tilts of triplet quadrupoles and bending magnets in Ring 2, from May 2013 tunnel survey.

Figure 2

Relative errors (percent) of triplet quadrupole magnet strengths in Ring 2, determined by LOCO analysis.

Figure 3

The COD without (a) and with (b) measured COD matching for the lowest ($Q_y=4.09$) vertical working point.

Figure 4

Half-integer (long bunch simulations): PTC-Orbit simulations vs measurements. Legend: (A) Simulation without errors but with SC; (B) Simulation with errors but no SC; (C) Simulation with only quadrupolar field errors (matching to $Q_y=4.53$) and SC; (D) Simulation with SC and quadrupolar field and misalignment errors; (E) Simulation with only quadrupolar field errors (matching to $Q_y=4.525$) and SC.

Figure 5

Half-integer (long bunch): measured (left) and simulated (right) waterfall color plot in the time window [450–620] ms. The color code is the particle density (blue, low density—red, high density).

Figure 6

Measured (top) and simulated (bottom) horizontal (a) and vertical (b) profiles in [450–620] ms for the long bunch case, normalized with respect to the max value.

Figure 7

Half-integer (short bunch). PTC-Orbit simulations vs measurements. Grey error bands are measurements, the red solid line are simulations with misalignment and quadrupolar field errors.

Figure 8

Half-integer (short bunch): measured (left) and simulated (right) waterfall color plot in the time window [450–605] ms. The color code is the particle density (blue, low density—red, high density). The comparison of the behavior with the vertical dashed black lines puts in evidence the bunch shortening.

Figure 9

Losses for different random errors. Legend: (A) Measurements; (B) Simulation with quadrupolar field and misalignment errors; (C) Simulation with only quadrupolar field errors (matching to $Q_y=4.525$); (D) Simulation with only quadrupolar field errors (matching to $Q_y=4.535$); (E) Simulation with random quadrupolar field errors and relative $\mathrm{beta}\text{−}\mathrm{beating}=3.46\%$ rms; (F) Simulation with random quadrupolar field errors and relative $\mathrm{beta}\text{−}\mathrm{beating}=19.51\%$ rms. All the simulations include SC.

Figure 10

Loss rate vs vertical beta beating at $130\times 10^{10}\mathrm{p}$ for the different loss profiles. The curves are normalized with respect to the slowest loss rate. Colored points are the values of the loss rate for two different beta-beating seeds as shown in Fig. 9. The gray points are all the other simulated seeds.

Figure 11

Simulated position of the lost particles in the longitudinal phase space. The color code is the tracking time.

Figure 12

The synchrotron period variation in a double harmonics rf bucket.

Figure 13

Half-integer (long bunch simulations). The loss rate (red) vs the synchrotron period of the lost particles (blue). One can observe the different regimes before and after 558 ms, where the bunch shortening starts to interest prevalently the particles inside the inner separatrix (with smaller synchrotron period).

Figure 14

The PTC-Orbit simulated tune footprint (averaged over 1500 turns) at 485 ms (a) and 565 ms (b), with particle density color code (blue, low density—red, high density). In (b), note the narrow peak on the $2Q_y=9$ line.

Figure 15

A particle tracked at 485 ms [on the solid magenta line in Fig. 4] before getting vertically lost by scattering. The turn-by-turn tracking is represented with blue dots the vertical phase space (a) and in the tunes plane (b).

Figure 16

A particle trapped into quadrupolar islands at 565 ms [on the solid magenta line in Fig. 4]. The turn-by-turn tracking is represented with blue dots in the vertical phase space (a) and in the tunes plane (b).

Figure 17

The dynamic approach to $Q_y=4$: horizontal (a) and vertical (b) tunes vs time.

Figure 18

The dynamic approach to $Q_y=4$: intensities vs time without COD correction. Legend: (A) Measurements; (B) Simulation without space charge; (C) Simulation with direct space charge; (D) Simulations error bar with direct space charge and random quadrupolar field errors distributions; (E) Simulation with measured quadrupolar fields errors and direct space charge.

Figure 19

The simulated vertical profiles with uncorrected COD, without (a) and with space charge (b). The dashed lines are the Gaussian fits.

Figure 20

The dynamic approach to $Q_y=4$ in case of corrected COD: intensities vs time with vertical boundaries at $\pm 29.5\mathrm{mm}$. Legend: (A) Measurement; (B) Simulation without space charge; (C) Simulation with direct space charge; (D) Simulation with space charge (with boundaries) without field errors; (E) Simulation with space charge (with boundaries) and measured quadrupolar field errors; (F) Simulation with space charge (with boundaries) and random quadrupolar field errors distributions.

Figure 21

The vertical profiles in case of COD correction: (a) Measurements, (b) Simulations including the direct space charge and the boundaries. Legend—left: (A) Measurements at 500 ms; (B) Measurements at 560 ms. Legend—right: (C) Simulation: initial vertical profile at 500 ms; (D) Vertical profile at 560 ms (with direct space charge); (E) Vertical profile at 560 ms (SC with vertical boundaries at $\pm 40\mathrm{mm}$); (F) Vertical profile at 560 ms (SC with vertical boundaries at $\pm 29.5\mathrm{mm}$).

Figure 22

The Gaussian shape is lost already after 1000 turns (1 ms).

Figure 23

The dynamic approach to $Q_y=4$: the normalized horizontal (a) and vertical (b) rms emittances behavior for the space charge simulations including and excluding the boundaries ($\pm 61\mathrm{mm}\times \pm 29.5\mathrm{mm}$). The simulations represent the case with the vertical orbit correctors ON. Legend—top: (A) Measurements (Gaussian fit of wire-scanner profiles); (B) Simulated statistical emittance (with SC with boundaries); (C) Simulated statistical emittance (direct space charge). Legend—bottom: (A) Measurements (Gaussian fit of wire-scanner profiles); (B) Simulated statistical emittance (with SC with boundaries); (C) Simulated emittance from Gaussian fit (with SC with boundaries); (D) Simulated emittance from Gaussian fit (with direct SC); (E) Simulated statistical emittance (with direct SC).

Figure 24

Dynamic integer with COD correction: simulations with different chamber heights.

Figure 25

Vertical coherent tunes simulations with COD correction. From bottom to top the following space charge modules have been used: vertical boundaries $\pm 40\mathrm{mm}$; vertical boundaries $\pm 29.5\mathrm{mm}$; direct space charge. The magenta dots represent the peaks in the FFT of the first order momenta.

Figure 26

Vertical coherent tune peaks. The dots are the maxima of the first order vertical momenta of the beam (magenta points of Fig. 25).

Figure 27

Dynamic integer without COD correction: simulations with different chamber heights of intensity (top), normalized statistical emittance (center) and vertical centroid (bottom).