A Peculiarity Detected in a Tracy Routine

One of the first perturbations I wanted to apply was a kick on a horizontal or vertical corrector magnet. These correctors are built onto the ends of sextupole magnets and are used to give the passing particles a kick, i.e. a deviation of their angle. This leads to an oscillation of the beam around the unperturbed orbit and therefore to a new closed orbit. Due to coupling of the horizontal and vertical planes of motion (in a non-linear machine, i.e. when sextupole magnets are switched on), an oscillation following a kick in one plane will also be visible in the perpendicular plane. Figure 1 illustrates the oscillation of the beam in the same plane the kick (1mrad horizontal) was applied. Figure 2 shows the coupling effect in the other plane - when looking at the scale it has to be considered that the amplitude of the oscillation due to coupling is much smaller. It is also very important to keep in mind that measurement of the beam position can only be done at the 72 beam position monitors (BPM) along the storage ring. In the experiment it is not possible to observe the beam position in every element as here seen in the theoretical plots in figures 1-4.

Figure 1: Beam oscillation in horizontal plane after kick on a horizontal corrector magnet.

While simulating this effect in Tracy, I found an effect in beam position predictions which can only be understood when actually looking at the special approach of the Tracy code. Applying a vertical kick results in normal oscillation in the vertical plane. The oscillation in the horizontal plane however shows six spikes which are of much higher magnitude than the amplitude of the oscillation (figure 3). These six spikes were found to originate in the skew quadrupoles. Skew quadrupoles are basically quadrupoles which are tilted (rotation axis is the beam-line) at a certain angle. Calculation of the beam properties in these skew quadrupoles is done by transforming beam coordinates $(s, x, y)$ into the rotated system $(s, x', y')$, applying quadrupole fields and transforming back to the unrotated system. A kick given along one axis has two perpendicular non-zero components in this rotated system. If one looks at the beam position before transforming back to the unrotated system (as if one were sitting in the skew magnet system), it seems as if the original kick had a perpendicular component too, thus giving a much greater amplitude to the oscillation in the plane perpendicular to the plane of the original kick. If one takes into account that the beam position data calculated by the Tracy code is always indicated in the system of the measurement point, this leads to the spikes seen. Hence a rotated skew quadrupole magnet will make Tracy rotate the output at the position of the skew quadrupoles so that it is correct if seen in the skew quadrupoles system however not correct if compared to the other positions of unrotated elements. If the predicted position values of the skew quadrupoles are removed, the normal oscillation due to coupling is observed (figure 4) and shows that no change in the oscillation is generated by this ''position displacement'' due to the fact that it is only displaced when looked at from another system. Tracy calculations therefore remain correct.

Figure 2: Beam oscillation in vertical plane after kick on a horizontal corrector magnet.

Figure 3: Beam oscillation in horizontal plane after kick on a vertical corrector magnet.

Figure 4: Beam oscillation in horizontal plane after kick on a vertical corrector magnet (suppressing skew quadrupoles).