Linear and nonlinear magnetic error measurements using action and phase jump analysis

“Action and phase jump” analysis is presented—a beam based method that uses amplitude and phase knowledge of a particle trajectory to locate and measure magnetic errors in an accelerator lattice. The expected performance of the method is first tested using single-particle simulations in the optical lattice of the Relativistic Heavy Ion Collider (RHIC). Such simulations predict that under ideal conditions typical quadrupole errors can be estimated within an uncertainty of 0.04%. Other simulations suggest that sextupole errors can be estimated within a 3% uncertainty. Then the action and phase jump analysis is applied to real RHIC orbits with known quadrupole errors, and to real Super Proton Synchrotron (SPS) orbits with known sextupole errors. It is possible to estimate the strength of a skew quadrupole error from measured RHIC orbits within a 1.2% uncertainty, and to estimate the strength of a strong sextupole component from the measured SPS orbits within a 7% uncertainty.

Figure 1

(a) Simulation of a RHIC first turn trajectory with every dot representing a BPM measurement, in the presence of a magnetic kick with strength ${\theta }_z$ at $s_{\theta }=1241\mathrm{m}$. (b) RHIC optics with short bars representing dipoles and long bars representing quadrupoles. (c) Action analysis of the simulated first turn trajectory. (d) Phase analysis of the same trajectory.

Figure 2

Horizontal magnetic kick error ${\theta }_x$ and vertical magnetic kick error ${\theta }_y$ as functions of the horizontal particle position $x(s_{\theta })$. The typical dependence between the horizontal particle position $x(s_{\theta })$ and the vertical particle position $y(s_{\theta })$ is also shown. Eight simulated first turn trajectories with different amplitudes have been used to obtain this plot.

Figure 3

The measured horizontal quadrupole kick error ${\theta }_x$ as a function of the horizontal position of the particle $x(s_{\theta })$, in response to a gradient error of $1\times {10}^{-3}{\mathrm{m}}^{-1}$ intentionally introduced in the lattice at $s_{\theta }=1241\mathrm{m}$. The slope of the curve, $-0.99969\times {10}^{-3}{\mathrm{m}}^{-1}$, is very close to the expected value.

Figure 4

The error of the method $ϵ$ as a function of the gradient error strength, predicting the method error to be less than 0.04% for RHIC gradient errors larger than ${10}^{-3}{\mathrm{m}}^{-1}$.

Figure 5

Simulations predict that the action and phase jump method determines skew quadrupole errors within an uncertainty as small as 0.01%.

Figure 6

The horizontal kick error ${\theta }_x$ is a nonlinear function of the trajectory excursion $x(s_{\theta })$ when a sextupole error is present. A quadratic fit of the plot gives the sextupole strength.

Figure 7

The action and phase jump method determines sextupole errors within a 3% uncertainty in the range of strengths of interest for RHIC sextupoles.

Figure 8

(a) One of the first difference orbits obtained from beam in RHIC in the year 2000. Each dot represents a BPM measurement. (b) Lattice representation of RHIC. (c) Action plot obtained by applying Eq. (13) to all pairs of adjacent BPM measurements in the ring. (d) Phase plot obtained by applying Eq. (14) to all pairs of adjacent BPM measurements in the ring.

Figure 9

Relation between the skew quadrupole errors intentionally introduced in the accelerator and the measured errors obtained from action and phase analysis of RHIC difference orbits.

Figure 10

Phase analysis of a SPS turn by turn difference orbit. The sextupoles that were intentionally introduced in the accelerator can be clearly identified by the jumps in phase (taken from Ref. 17).

Figure 11

Measured horizontal kick error as a function of the horizontal beam position for sextupole 63 in the SPS ring. Optimal turns of turn by turn difference orbits with seven different amplitudes were used to obtain this curve. A quadratic fit gave an integrated sextupole strength in good agreement with the actual set strength value (taken from Ref. 17).

Figure 12

Action and phase analysis of a RHIC difference orbit taken during the 2002 proton run. The difference is built out of two separated first turn orbits named bi8-qs3.-0.0010.8h.sdds and baseline.1.1.8h.sdds.

Figure 13

Action and phase jump analysis of a RHIC difference orbit. This time the difference is built by substrating orbit in turn 17 from orbit in turn 15 of the turn by turn trajectory named bi8-qs3.-0.0010.8h.sdds.

Figure 14

Skew quadrupole errors estimated from the conventional difference orbits (circles) and skew quadrupole errors estimated from the new difference orbits (squares). This figure was taken from Ref. 18.