Periodic transient beam loading effect with passive harmonic cavities in electron storage rings

In our previous study of bunch lengthening for the planned Hefei Advanced Light Facility (HALF) storage ring, a periodic transient beam loading effect was found in the presence of passive harmonic cavities for some cases, which can dramatically affect the bunch lengthening as well as the beam stability [T. L. He et al., Bunch lengthening of the HALF storage ring in the presence of passive harmonic cavities, in Proceedings of the IPAC2021, Campinas, SP (JACoW, Campinas, SP, Brazil, 2021), pp. 2082–2085, TUPAB265]. In this paper, we continue to use HALF as an example, via both tracking simulation and semianalytical calculation, to comprehensively study the characteristics of this periodic transient effect, analyze its dependence on some relevant factors and investigate the impact of fill pattern on its periodicity. It is found that there exits a threshold current for this periodic transient effect, which can be increased by the reduction of $R/Q$ of harmonic cavity (HC) and the increase of main cavity (MC) voltage, HC detuning, momentum compaction, and energy spread, as well as the inclusion of short-range wake. In addition, the periodicity of bunch centroid oscillation can be affected by the damping time and MC beam loading, and completely interrupted due to the nonuniform fill pattern, but this is not helpful for improvement of bunch lengthening.

Figure 1

Centroid positions (top) and rms bunch lengths (bottom) vs tracking turns, obtained with the stable code. Eight equally spaced bunches out of a total of 800 are displayed and distinguished by colors.

Figure 2

HC voltage amplitude (top) and phase (bottom) vs bunch number at 260 mA, given by the semianalytical calculation. The left column is the stage where the HC voltage difference continues to grow with the number of iterations, and the right column is the stage where saturation is reached. The legend gives the iteration number.

Figure 3

Comparison between the tracking and semianalytical method. Centroid positions (top) and rms bunch lengths (bottom). The legend gives the tracking turn number (blue and black dots) and numerical iteration number (green and red dots).

Figure 4

Simplified diagram of the iterative scheme for bunch profiles by semianalytical method.

Figure 5

Tracking results given by the stable code. (a) Centroid positions, (b) rms lengths, (c) HC voltage amplitude ratio, and (d) HC voltage phase. The legend gives the bunch number. The tracking simulation is successive for the three cases of 265, 300 and 350 mA. Ten equally spaced bunches out of a total of 800 are displayed and distinguished by colors.

Figure 6

Charge density profiles at different iteration steps for the case of 350 mA, obtained by the semianalytical calculation. The legend gives the centroid positions. “N-max” and “P-max” represent the negative and positive maximum of centroid position, respectively. “Zero” denotes the zero position, and “From P-max to N-max” means the centroid position moves from the positive maximum to the negative maximum.

Figure 7

The product of threshold current and $R/Q$-value as a function of the main cavity voltage.

Figure 8

RMS bunch lengths (top) and centroid positions (bottom) vs bunch number, obtained with the semianalytical method for the two cases of 350 mA (left column) and 500 mA (right column). The legend gives the detuning frequency, and $\mathrm{\Delta }f=17.2\mathrm{kHz}$ for the case of 350 mA is given by the near-optimum lengthening condition, as well as $\mathrm{\Delta }f=24.57\mathrm{kHz}$ for the case of 500 mA.

Figure 9

Effect of the radiation damping time on PTBL. The oscillation frequency (left) is computed by taking the reciprocal of the oscillation period. The oscillation amplitude (right) is taken by half of the oscillation range.

Figure 10

Effect of the short-range wake on PTBL, obtained with the stable code. The legend gives the shunt impedance of the broad-band resonator in $k\mathrm{\Omega }$ and the radiation damping time in ms.

Figure 11

Centroid position vs bunch number. The results of lines are obtained with the semianalytical method, while the dots with the STABLE code. The yellow line and black dots are corresponding to the case of with MC beam loading, and the red line and blue dots are for the case of without MC beam loading. Note that the resulting distributions have been shifted to coincide for easy comparison.

Figure 12

RMS bunch length (top) and centroid position (bottom) vs bunch number, obtained by the semianalytical method. The legend gives the beam current in mA. Note that for the cases of 265 and 270 mA, the resulting distributions are periodically varied with the number of iterations, while only the results of the final iteration are plotted.

Figure 13

RMS bunch lengths (top) and centroid positions (bottom) vs bunch number, obtained by the semianalytical method. The legend gives the beam current in mA. Note that for the cases of 270, 300, and 350 mA with 1% charge error, the resulting distribution changes periodically with the number of iterations, while only the results of the final iteration are illustrated.

Figure 14

Centroid position vs tracking turns for the case of 270 mA with 1% Gaussian charge error. The legend gives the bunch number.