Explicit symplectic orbit and spin tracking method for electric storage ring

We develop a symplectic charged particle tracking method for phase space coordinates and polarization in all electric storage rings. Near the magic energy, the spin precession tune is proportional to the fractional momentum deviation ${\delta }_m$ from the magic energy, and the amplitude of the radial and longitudinal spin precession is proportional to $\eta /{\delta }_m$, where $\eta$ is the electric dipole moment for an initially vertically polarized beam. The method can be used to extract the electron electric dipole moment of a charged particle by employing narrow band frequency analysis of polarization around the magic energy.

Figure 1

Numerical closed orbit variation from turn to turn. mad represents $s$-domain tracking [11], and tim represents the $\tau$-domain tracking. The test lattice is TTX [20]. The number of integration step for all the magnetic elements is chosen to be 10 and 20, and represented by $10_{\text{Kicks}}$ and $20_{\text{Kicks}}$ respectively.

Figure 2

Poincaré surface plot of longitudinal motion on 12 superperiod focusing defocusing quadrupoles and other elements in between (FODO) ring. timtrack represent the $\tau$-domain tracking and elegant represent the $s$-domain tracking code elegant [12]. 6 particles of different initial ${\delta }_E$ are tracked. The subplot shows close look at the numerical error on closed orbit.

Figure 3

Poincaré surface plot of horizontal phase space. timtrack represents the 4th order symplectic $\tau$-domain tracking on electric FODO ring and elegant [12] represent the 4th order symplectic $s$-domain tracking on magnetic FODO ring.

Figure 4

The longitudinal spin precession for 10 million turns. Each particles are given nonzero initial off energy values of ${\delta }_E$ as listed. The computation time of tracking single particle for ten millions turns in a 12 superperiod FODO ring (21.6 m) was about 18 min in a normal PC.

Figure 5

The radial (top) and longitudinal (bottom) spin precession for one billion turns. 11 particles are prepared with different initial off energies and tracked in a 12 superperiod FODO ring.

Figure 6

The integration range for $n=10$. The shaded area is the integration range, and the horizontal and vertical axes are $n_1$ and $n_2$, respectively. From left to right, they correspond to $\mathcal{I}$, ${\mathcal{I}}_u$, and ${\mathcal{I}}_d$, respectively.

Figure 7

The analytic prediction of the radial (top) and longitudinal (bottom) spin precession for one billion turns.

Figure 8

Fast Fourier transformation (FFT) on one billion turns of longitudinal spin tracking data taken every 10 thousand turns. The peak at ${\delta }_m=3\times {10}^{-4}$ appears to be smaller due to insufficient FFT resolution. NAFF algorithm is used for better accuracy of tune and amplitude measurement in Table 2. While the tune shown in this plot legend is based on FFT.

Figure 9

FFT on 1 billion turns of longitudinal spin tracking data taken every 10 thousand turns. The rf cavity frequency is shifted so that the closed orbit energy is off the magic condition by ${\delta }_m=2.0\times {10}^{-4}$. Particles with different energies ${\delta }_E$ around the closed orbit are tracked. The particle on closed orbit is denoted by ${\delta }_E=0$. The tunes shown in the plot legend correspond to the maximum peaks found by FFT.

Figure 10

FFT on 1 billion turns of longitudinal spin tracking data taken every 10 thousand turns. The rf cavity frequency is shifted so that the closed orbit energy is off the magic condition by ${\delta }_m=1.0\times {10}^{-5}$. Particles with different energies ${\delta }_E$ around the closed orbit are tracked. The particle on closed orbit is denoted by ${\delta }_E=0$. The amplitudes of ${\delta }_E=3.0$, $4.0\times {10}^{-4}$ appears to be smaller because of insufficient FFT resolution.

Figure 11

FFT on 1 billion turns of longitudinal spin tracking data taken every 10 thousand turns. The design energy is off the magic energy by ${\delta }_m=2.3\times {10}^{-6}$. Particles with different energies ${\delta }_E$ around the design orbit are tracked. The particle on design orbit is denoted by ${\delta }_E=0$. The amplitudes of ${\delta }_E=0.0$, 3.0, $5.0\times {10}^{-4}$ appears to be smaller because of insufficient FFT resolution.