Transverse Schottky and beam transfer function measurements in space charge affected coasting ion beams

Transverse Schottky spectra and beam transfer functions (BTFs) of coasting ion beams were measured in the heavy ion synchrotron SIS-18 in order to study the impact of space charge on the transverse beam dynamics. The particle number in the beam was varied to investigate the intensity dependence of the space-charge effect. No cooling was applied to the beams throughout the experiment. The expected deformation of the Schottky spectra and BTFs is observed. An analytic model with linear space charge is employed to describe the deformed Schottky and BTF signals. In this model, the incoherent space-charge force and the coherent forces due to impedances are treated separately. Using the model, the space-charge induced tune shift is evaluated both from the position and the form of the signals. The data are well described by the model, only in the high-intensity BTFs deviations are observed. The stability diagrams are shifted according to the space-charge parameter obtained from the BTFs. In addition, the tune shift is estimated by virtue of measured beam profiles and particle numbers. The estimated tune shift is of the same order of magnitude but smaller than the measured one. Possible explanations for deviations between the measurements, the model, and the estimation are discussed.

Figure 1

The hardware for Schottky and BTF diagnostics. The dashed arrow refers to BTF measurements with a network analyzer (NA) only. In this case only the difference ($\Delta$) signal from the detector is useful. For Schottky measurements, a spectrum analyzer (SA) is connected to the difference or sum ($\Sigma$) signal of the pickup.

Figure 2

Lower Schottky sideband at $m=50$ measured with different ion numbers and the corresponding fits. The corresponding $\Delta U_{\mathrm{sc}}$ are 0., 0.9, and 1.8, respectively. The curves are rescaled for a better prospect.

Figure 3

Amplitude of the BTF for different currents. The settings are the same as in Fig. 2. $\Delta U_{\mathrm{sc}}$ found for these data are 0., 0.9, and 2.7. Also the fits are shown.

Figure 4

BTF phases corresponding to the amplitudes shown in Fig. 3 with the fits. A vertical shift was added to separate the lines.

Figure 5

Stability diagrams obtained from the BTF shown in Figs. 3, 4.

Figure 6

Comparison of the space-charge parameters $\Delta U_{\mathrm{sc}}$ obtained from Schottky (blue diamonds) and BTF (green triangles) measurements, and estimated from the beam parameters (red crosses) versus ion number. Least mean square fits to the estimated and measured points are shown, too.

Figure 7

Fractional part of the betatron tune $Q_f$ from the Schottky measurement (green triangles) and estimation (red crosses) for varying ion number. In addition, the reconstructed machine tune $Q_f+\Delta Q_{\mathrm{sc}}$ is plotted (blue diamonds). As it is always $\approx Q_{f,0}$ (horizontal line), only space charge is responsible for the deformation. The nonhorizontal lines are least mean square fits to the effective tunes.