Systematic study on the static Robinson instability in an electron storage ring

The static Robinson instability and associated coherent synchrotron frequencies were systematically studied at the Photon Factory (PF) 2.5-GeV storage ring at KEK. Under machine parameters close to an anticipated Robinson limit, we observed beam dumps at which the beam and related signals were recorded. The evolution of the beam phase and the threshold condition at the beam dumps were consistent with theoretical values calculated based on the uniformly filled point-bunch model. However, we found remarkable discrepancies from theoretical predictions with additional coherent-frequency measurements. Under conditions close to the Robinson limit, we observed an unpredicted sharp peak at a frequency of approximately 5.2 kHz, together with discrepancies in both the frequency and damping rate of the coherent oscillation. These results indicate the importance of further experimental and theoretical studies on the static Robinson instability.

Figure 1

Solutions of the characteristic Eq. (1) for the parameters of the PF storage ring given in Table 1. Upper and lower graphs show the real and the imaginary parts of the solutions, respectively, as functions of the beam current when the total cavity voltage is fixed at $V_{\mathrm{c}}=1.70\mathrm{MV}$. The solid (blue and orange) lines show the solutions, and gray dashed lines indicate the detuning frequency ($\mathrm{\Delta }\omega$) of the cavity under optimum tuning or the damping rate ($\mathrm{\Gamma }$) of the cavity. The dashed-dotted lines show the incoherent synchrotron frequency (${\omega }_{\mathrm{s}0}$).

Figure 2

Solutions of Eq. (1) as functions of the cavity voltage with a fixed beam current of 350 mA. The solid lines show the solutions, the dashed lines show the cavity detuning frequency or the damping rate of the cavity, and the dashed-dotted lines show the incoherent synchrotron frequency.

Figure 3

The experimental setup used to observe the static Robinson instability.

Figure 4

The cavity voltage and average beam current during the experiment. In this example, the beam dumped at a cavity voltage between 1.23 and 1.15 MV.

Figure 5

Waveforms recorded with the oscilloscope 1 in Fig. 3 at the moment of beam dump. The yellow and green lines indicate the I and Q components of the beam signal, respectively. The blue line indicates the “rf trip trigger” signal, where high indicates the cutoff rf power, which triggered the recording. The scales are $500\mathrm{mV}/\mathrm{division}$ (div.) for the I and Q signals, and $2\mathrm{V}/\mathrm{div}$. for the rf trip trigger in the vertical axis. The horizontal scale is $50\mathrm{\mu }\mathrm{s}/\mathrm{div}$. The sampling rate was 1 GHz.

Figure 6

The I and Q components of the beam signal after offline low-pass-filter processing. In the lowest frame, the detected signals of the cavity pickup, klystron output, and reflected rf from the cavity are indicated, where all signals have negative polarity. At the time $t=0$, the rf power to the cavities was cut off due to the beam dump.

Figure 7

The amplitude (upper graph) and phase (lower graph) of the beam signal, respectively. These data were deduced from the measured IQ signals in Fig. 6.

Figure 8

The beam phase measured with the oscilloscope 2. The data were collected at the same time as those in Fig. 6, but with a longer time scale before the rf trip (at time $t=0$). The blue line indicates the measured phase, whereas the dashed line shows a fitted curve. The vertical dotted line shows an estimated onset time.

Figure 9

Experimental setup for measuring the beam responses to the rf phase modulation.

Figure 10

Beam responses measured under different beam currents. The total cavity voltage was fixed at 1.70 MV. To separate the traces, an offset of 30 dB was appended between neighboring traces. The center frequency corresponds to an rf frequency of 500.106 46 MHz at this measurement. Each trace was obtained by a single sweep without averaging. The resolution bandwidth (RBW) of the spectrum analyzer was 300 Hz. The phase-modulation amplitude was 0.23° p-p. Measurement date: May 23, 2019.

Figure 11

Beam responses measured under different cavity voltages. The beam current was fixed at 350 mA. An offset of 30 dB was appended between neighboring traces. The center frequency corresponds to an rf frequency of 500.113 80 MHz at this measurement. Each trace was obtained by averaging over ten sweeps. The RBW was 100 Hz and the phase-modulation amplitude was 0.20° p-p. Measurement date: March 11, 2021.

Figure 12

Coherent frequencies (upper graph) and damping rates (lower graph) as functions of the average beam current. The cavity voltage was 1.70 MV. Closed circles indicate the data obtained by fitting the measured traces. The error bars were determined by convoluting the fitting error and resolution bandwidth of the spectrum analyzer. The solid lines indicate the numerical solutions of Eq. (1) at the cavity voltage of 1.70 MV. Note that the angular frequency was divided by $2\pi$ in the upper graph to indicate the real frequency.

Figure 13

Measured coherent frequency and damping rate as the functions of the cavity voltage. The beam current was 350 mA. The closed circles and the triangles indicate the fitted results for the broad and narrow peaks, respectively. The solid lines show the numerical solutions of Eq. (1) for the beam current of 350 mA.

Figure 14

Variations of the amplitude and frequency of the unexpected peak as functions of the phase modulation amplitude. The cavity voltage was 1.26 MV and the beam current was 350 mA. The fill pattern was a single train of 250 bunches.

Figure 15

(a) Beam responses measured under different sets of beam current and cavity voltage and (b) frequency of the unexpected peak as a function of the beam currents. The fill pattern was four trains of 47 bunches. The error bars were determined by convoluting the fitting error and the resolution bandwidth of the spectrum analyzer.

Figure 16

Beam oscillation observed by a dual-sweep streak camera. Longitudinal profiles of every four bunches are shown along the vertical axis, whereas these profiles are separated along the horizontal axis. The beam oscillation was excited by the phase modulation at a frequency of 5.2 kHz with an amplitude of 0.8° p-p. The cavity voltage was 1.27 MV, and the beam current was 350 mA with a fill pattern of a single train of 250 bunches.

Figure 17

Measured amplitude of the center-of-bunch beam oscillation as a function of the phase-modulation amplitude. The error bars show fitting errors. The dotted line shows the linear fit between the oscillation amplitude and modulation amplitude.

Figure 18

Measured coherent synchrotron frequencies at a low beam current of 5 mA as a function of the cavity voltage. Solid line indicates the calculated incoherent frequency.

Figure 19

Comparison between the theoretical solutions and measurement of the complex frequency as functions of the cavity voltage. The beam current is fixed at 350 mA. The dashed lines show the numerical solutions of the sixth-order characteristic Eq. (b1), whereas the solid lines indicate the solutions of the fourth-order Eq. (1). The measured data are the same as those shown in Fig. 13.