A generalization of the Froissart-Stora formula to piecewise-linear spin-orbit resonance crossings

Abstract

Spin-polarized beams are important for some nuclear and high-energy physics experiments, such as those planned for the future Electron-Ion Collider (EIC). However, maintaining polarization during the acceleration of a charged-particle beam is difficult because the periodic nature of circular accelerators leads to spin-orbit resonances where the spin-precession frequency is a sum of integer multiples of the orbital frequencies. Usually, the dominant depolarization mechanisms are first-order spin-orbit resonances and the depolarization associated with crossing such a resonance can be computed using the Froissart-Stora formula. However, accelerating polarized hadron beams to high energy requires special magnet structures called Siberian snakes. When these are implemented to maintain a spin-precession frequency of one-half the revolution frequency, there will be no first-order spin-orbit resonance crossings. The dominant depolarization mechanisms are then higher-order spin-orbit resonances. The Froissart-Stora formula can be applied to higher-order resonances when the slope of the amplitude-dependent spin tune is constant. However, the slope of the amplitude-dependent spin tune often changes at the moment of resonance crossing. This work introduces a generalization of the Froissart-Stora formula which is applicable when the slope changes in this manner. The applicability of this formula is demonstrated through tracking simulations of a higher-order resonance crossing in both a toy model and the Relativistic Heavy Ion Collider (RHIC). It is additionally shown that the Froissart-Stora formula is mathematically equivalent to the Landau-Zener formula for the diabatic transition probability in two-level systems with a linearly increasing energy gap and constant coupling. This work therefore also extends the Landau-Zener formula to the case of changing slope.

Figures (7)

• Figure 1: A pair of third-order spin-orbit resonances across which the slope of the ADST $\nu$ changes drastically. The spin-orbit resonance crossings are characterized by discontinuities in $\nu$ and the dashed lines indicate the approximate slopes.
• Figure 2: Comparison of numerical integration of the SRM with piecewise-linear $\nu_0$ and the asymptotic value of $s_3$ predicted by Eq. \ref{['eq:Two_Slope_FS']}. The parameters are $\alpha_1=0.01$, $\alpha_2=1$, $\epsilon=0.1$, $Q=0.7$, and $\kappa=Q$.
• Figure 3: A pair of third-order resonances in the DRM with two Siberian snakes and parameters $Q=0.16$, $\kappa_1=382+Q$, $\kappa_2=482-Q$, $\epsilon_1=\epsilon_2=0.15$, and $\Delta\phi = \pi/4$. The resonances are characterized by a discontinuity in $\nu$ across the resonance line $-3Q$ and corresponding dips in $P_\mathrm{lim}$. Left: The maximum time-averaged polarization, computed by stroboscopic averaging. Right: The ADST, computed by averaging the rotation angle around an ISF from stroboscopic averaging Vogt.
• Figure 4: Comparison of the final spin action $J_S(\theta_f)$ from tracking through a higher-order resonance in the DRM with two Siberian snakes and the asymptotic spin action predicted by Eq. \ref{['eq:Two_Slope_FS']}. Top: Spin action after crossing the left resonance in Fig. \ref{['fig:DRM_FS_ADST']}. Bottom: Spin action after crossing the right resonance in Fig. \ref{['fig:DRM_FS_ADST']}.
• Figure 5: Properties of spin motion for particles on the ellipse with normalized action $J_y \approx 32 \, \text{mm mrad}$ in RHIC. A third-order resonance is characterized by a discontinuity in $\nu$ across the resonance line $88-3Q_y$ and a corresponding dip in $P_\mathrm{lim}$. Left: The maximum time-averaged polarization, computed by stroboscopic averaging. Right: The ADST, computed by Fourier analysis of the turn-by-turn spin vector Quasiperiodic.