Evaluation of the CERN Super Proton Synchrotron longitudinal impedance from measurements of the quadrupole frequency shift

Longitudinal instabilities are one of the main limitations in the CERN Super Proton Synchrotron (SPS) to reach the beam parameters required for the High Luminosity LHC project. In preparation to the SPS upgrade, possible remedies are studied by performing macroparticle simulations using the machine impedance model obtained from electromagnetic simulations and measurements. To benchmark the impedance model, the results of simulations are compared with various beam measurements. In this study, the reactive part of the impedance was probed by measuring the quadrupole frequency shift with intensity, obtained from bunch length oscillations at mismatched injection into the SPS. This method was applied over many last years to follow up the evolution of the SPS impedance, injecting bunches with the same bunch length. A novel approach, giving significantly more information, consists in varying the injected bunch length. The comparison of these measurements with macroparticle simulations allowed us to test the existing model, identify some missing SPS impedance and to obtain its possible dependence on frequency.

Figure 1

The present SPS longitudinal impedance model (top: total absolute impedance, middle and bottom: resistive part and reactive part of the main contributions).

Figure 2

Examples of bunch length oscillations at SPS injection measured in the Q26 optics for different average bunch lengths ${\tau }_{\mathrm{av}}$ and intensities $N_b$.

Figure 3

Measurements of the quadrupole synchrotron frequency shift at $26\mathrm{GeV}/\mathrm{c}$ in the SPS since 1999 [10].

Figure 4

Schematic presentation of the quadrupole oscillations. The upper figure represents the distribution in the phase space ($\tau$, $\mathrm{\Delta }E$), the lower one the corresponding longitudinal profile $\lambda (\tau )$, which is the projection of the bunch distribution in the $\tau$ dimension. The distribution is shown at two different times, in blue when the bunch length is the biggest and in orange when it is the smallest. The matched area (grey) does not contribute to the quadrupole oscillations, the mismatched area is in blue/orange.

Figure 5

Line densities (left) and corresponding spectra (right) for different $\mu$ values and the same rms bunch length ${\sigma }_{\mathrm{rms}}$. The black line indicates the bunch length ${\tau }_{4\sigma }$.

Figure 6

The synchrotron frequency distribution $f_{s,\mathrm{inc}}^{(0)}(\widehat{\tau })$ for different values of $\mu$ with the same ${\tau }_{4\sigma }=0.7\mathrm{ns}$, using a pure reactive impedance $\mathrm{Im}\mathcal{Z}/n=3\mathrm{\Omega }$ and $N_b=5\times 10^{10}\mathrm{ppb}$ for the Q26 optics parameters in Table 1. The dots represents the values at $\widehat{\tau }=0$ and $\widehat{\tau }={\tau }_{4\sigma }/2$.

Figure 7

The relative incoherent synchrotron frequency shift as a function of bunch length for particles with $\widehat{\tau }\approx 0$ (left) and $\widehat{\tau }={\tau }_{4\sigma }/2$ (right), using a pure reactive impedance $\mathrm{Im}\mathcal{Z}/n=3\mathrm{\Omega }$ and $N_b=5\times 10^{10}\mathrm{ppb}$ for the Q26 optics parameters in Table 1.

Figure 8

The incoherent synchrotron frequency shift calculated for particle distributions with different $\mu$ as a function of the bunch length ${\tau }_{4\sigma }$, for the Q26 optics parameters and $N_b=5\times 10^{10}\mathrm{ppb}$ using the full SPS impedance model. The upper plots correspond to the incoherent frequency shift for small (left) and large (right) particle oscillation amplitudes $\widehat{\tau }$. The bottom plots (c) and (d) are the equivalent reactive impedances ${(\mathrm{Im}\mathcal{Z}/n)}_{\mathrm{eq}}$ corresponding to cases (a) and (b).

Figure 9

The equivalent impedance ${(\mathrm{Im}\mathcal{Z}/n)}_{\mathrm{eq}}$ of the main SPS impedance sources (shown in Fig. 1), taken separately for a bunch distribution with $\mu =1$, and for small (left) and large (right) particle oscillation amplitudes $\widehat{\tau }$.

Figure 10

Contribution of the incoherent shift $2\mathrm{\Delta }{f}_{\mathrm{inc}}$ and the coherent shift $\mathrm{\Delta }{f}_{\mathrm{coh},2}$ to the total synchrotron frequency shift $2\mathrm{\Delta }{f}_{\mathrm{inc}}+\mathrm{\Delta }{f}_{\mathrm{coh},2}$ as a function of bunch length, for a pure inductive impedance $\mathrm{Im}\mathcal{Z}/n=3\mathrm{\Omega }$ (left) and the full SPS impedance (right). The Q26 optics parameters in Table 1 were used with $N_b=5\times 10^{10}\mathrm{ppb}$.

Figure 11

The equivalent impedance ${(\mathrm{Im}\mathcal{Z}/n)}_{\mathrm{eq}}$ of the main SPS impedance sources in Fig. 1 taken separately. The solid lines correspond to the equivalent impedances defined only from the incoherent shift $\mathrm{\Delta }{f}_{\mathrm{inc}}$ as in Fig. 9, the dashed lines correspond to the equivalent impedances taking also into account the coherent shift $\mathrm{\Delta }{f}_{\mathrm{coh},2}$.

Figure 12

Measured quadrupole frequency $f_{s2}$ as a function of bunch length and intensity in both Q20 (left) and Q26 (right) optics. Each point corresponds to a single measurement, and the colored surface corresponds to the quadrupole frequency (gaps were filled using interpolation).

Figure 13

Examples of measured quadrupole frequency $f_{s2}$ as a function of intensity for selected average bunch lengths ${\tau }_{\mathrm{av}}$ (within $\pm 50\mathrm{ps}$) in the Q20 optics. The lines correspond to a linear fit.

Figure 14

The fitted origin $a$ (top), slope $b$ (middle) and their expected scaling (green), from Eqs. (4) and (19). The Eq. (26) was used to obtain the corresponding equivalent impedance ${(\mathrm{Im}\mathcal{Z}/n)}_{\mathrm{eq}}$ (bottom) of the quadrupole frequency shift with intensity, as a function of the average bunch length ${\tau }_{\mathrm{av}}$, in the Q20 (left) and Q26 (right) optics.

Figure 15

Equivalent impedance ${(\mathrm{Im}\mathcal{Z}/n)}_{\mathrm{eq}}$ as a function of bunch length obtained from measurements (blue) and simulations (red) using the full SPS impedance model in the Q20 (left) and Q26 (right) optics.

Figure 16

Measured equivalent impedance ${(\mathrm{Im}\mathcal{Z}/n)}_{\mathrm{eq}}$ (black) in the Q20 optics compared with simulations (colored lines) adding a variable amount of inductive impedance in the range $\mathrm{Im}\mathcal{Z}/n=(0-2)\mathrm{\Omega }$ to the full SPS impedance model.

Figure 17

Missing inductive impedance $\mathrm{Im}\mathcal{Z}/n$ as a function of bunch length needed to get a perfect agreement between measurements and simulations shown in Fig. 15 for both Q20 (left) and Q26 (right) optics.

Figure 18

Equivalent impedance ${(\mathrm{Im}\mathcal{Z}/n)}_{\mathrm{eq}}$ in measurements and simulations after adding an extra resonator with $f_r=350\mathrm{MHz}$, $R/Q=3\mathrm{k}\mathrm{\Omega }$ and $Q=1$ to the SPS impedance model to compensate for deviations from measurements in both Q20 (left) and Q26 (right) optics.