Closed orbit change induced by nonzero dispersion rf cavities

The particle motion in storage rings is coupled between the longitudinal and the transverse planes in the presence of nonzero dispersion rf cavities. We found that the particle motion can be modeled separately with a redefined closed orbit. The closed orbit can be described by a Green’s function, which was confirmed in the simulation and in the experiment. The pathlength is calculated from the redefined closed orbit, and we found that the longitudinal phase slip is related not only to the momentum, but also to the rf phase of the particle. The effect on the longitudinal motion becomes significant if the phase slip caused by the rf cavities is large or if the momentum compaction factor is small, such as in the lower alpha-$c$ lattice which is intended to produce shorter bunches.

Figure 1

The calculated closed-orbit change at the APS storage ring due to rf cavities. The parameters are as follows: total rf voltage $U_{rf}=9.2\mathrm{MV}$; synchronous phase ${\varphi }_{rf}=144.{36}^{°}$.

Figure 2

(Color online) The $C_{rf}\left(s\right)$ function of the APS storage ring. It changes value only in the straight sections of sectors 36, 37, 38, and 40, where rf cavities are placed.

Figure 3

(Color online) The comparison of the orbit change between the simulation (the diamonds) and the prediction (the solid line) in part of the ring outside the rf zone. The agreement is similar in the part outside the rf zone.

Figure 4

(Color online) The measured closed-orbit change at BPM S2B:P1 while $\Delta V$ is varied. The squares indicate the experimental data and the line is the linear fit. The total gap voltage was kept constant and the voltage step size was fixed for each cavity. The slope obtained by fitting is discussed in Fig. 5.

Figure 5

(Color online) The comparison of the slopes of $d{x}_{c.o.}∕d\Delta V$ outside the rf zone. The squares were obtained from the experiment and the line is from the calculation.

Figure 6

(Color) The closed-orbit change inside the rf zone. The red diamonds are from the simulation, the blue line is the calculated $\zeta \left(s\right)\sin {\varphi }_{rf}$ difference only, and the black line is calculated from Eq. (36), including the dispersion effect.

Figure 7

(Color online) The slope of $d{x}_{c.o.}∕d\Delta V$ inside the rf zone. The squares with an error bar are obtained from the experiment. The dispersion term was subtracted from the original orbit data change, and the slope is obtained by fitting the difference with $\Delta V$. The line is the slope from the calculation.

Figure 8

$\zeta \left(s\right)∕\rho \left(s\right)$ in part of the ring.

Figure 9

(Color) The coordinate system in the rf cavity. The tilt angle between $s$ and $z$ is ${\rm Y}=x_{c.o.}^{\prime }=(D\delta +\zeta \sin \varphi {)}^{\prime }$.