New Method for a Continuous Determination of the Spin Tune in Storage Rings and Implications for Precision Experiments

A new method to determine the spin tune is described and tested. In an ideal planar magnetic ring, the spin tune—defined as the number of spin precessions per turn—is given by ${\nu }_{\mathrm{s}}=\gamma G$ ($\gamma$ is the Lorentz factor, $G$ the gyromagnetic anomaly). At $970\mathrm{MeV}/c$, the deuteron spins coherently precess at a frequency of $\approx 120\mathrm{kHz}$ in the Cooler Synchrotron COSY. The spin tune is deduced from the up-down asymmetry of deuteron-carbon scattering. In a time interval of 2.6 s, the spin tune was determined with a precision of the order ${10}^{-8}$, and to $1\times {10}^{-10}$ for a continuous 100 s accelerator cycle. This renders the presented method a new precision tool for accelerator physics; controlling the spin motion of particles to high precision is mandatory, in particular, for the measurement of electric dipole moments of charged particles in a storage ring.

Figure 1

(a) Counts $N_U$ and $N_D$ after mapping the events recorded during a turn interval of $\mathrm{\Delta }n={10}^6$ turns into a spin phase advance interval of $4\pi$. (b) Count sums $N_{U,D}^{+}({\phi }_{\mathrm{s}})$ and differences $N_{U,D}^{-}({\phi }_{\mathrm{s}})$ of Eq. (6) with ${\phi }_{\mathrm{s}}\in [0,2\pi )$ using the counts $N_U({\phi }_{\mathrm{s}})$ and $N_D({\phi }_{\mathrm{s}})$, shown in panel (a). The vertical error bars show the statistical uncertainties, the horizontal bars indicate the bin width.

Figure 2

Measured asymmetry $\epsilon ({\phi }_{\mathrm{s}})$ of Eq. (7) fitted with $\epsilon ({\phi }_{\mathrm{s}})$ of Eq. (8) to extract amplitude $\tilde{\epsilon }$ and phase $\tilde{\phi }$, using the yields $N_{U,D}^{+,-}({\phi }_{\mathrm{s}})$ of Fig. 1 for a single turn interval of $\mathrm{\Delta }n={10}^6$ turns at a measurement time of $2.6\mathrm{s}<t<3.9\mathrm{s}$.

Figure 3

(a) Phase $\tilde{\phi }$ as a function of turn number $n$ for all 72 turn intervals of a single measurement cycle for $|{\nu }_{\mathrm{s}}^{\mathrm{fix}}|=0.160\hspace{0.17em}975\hspace{0.17em}407$, together with a parabolic fit. (b) Deviation $\mathrm{\Delta }{\nu }_{\mathrm{s}}$ of the spin tune from ${\nu }_{\mathrm{s}}^{\mathrm{fix}}$ as a function of turn number in the cycle. At $t\approx 38\mathrm{s}$, the interpolated spin tune amounts to $|{\nu }_{\mathrm{s}}|=(16\hspace{0.17em}097\hspace{0.17em}540\hspace{0.17em}628.3\pm 9.7)\times {10}^{-11}$. The error band shows the statistical error obtained from the parabolic fit, shown in panel (a).

Figure 4

Walk of the spin tune during eight consecutive cycles with alternating initial vector polarization ${p_{\xi }}^{+}$ (black) and ${p_{\xi }}^{-}$ (gray). The third cycle is depicted in Fig. 3 as well. Cycles with unpolarized beam that followed the ${p_{\xi }}^{-}$ state are not shown.