The strength of a first-order spin-orbit resonance is defined as the amplitude of the corresponding Fourier component of the spin-precession vector. However, it is possible to obtain the resonance strength without computing the Fourier integral directly. If a resonance is sufficiently strong, then to a good approximation, one can neglect all other depolarizing effects when near the resonance. Such an approximation leads to the single resonance model (SRM), for which many aspects of spin motion are analytically solvable. In this paper, we calculate the strength of first-order resonances using various formulae derived from the SRM, utilizing spin tracking data, the direction of the invariant spin field, and jumps in the amplitude-dependent spin tune. Examples are drawn from the RHIC Blue ring.

The model of spin depolarization invoking isolated resonances hinges on a closed-form solution of energy ramping with the Single Resonance Model by Froissart and Stora. However, for non-resonant orbital tunes, resonant depolarization by single resonance crossing is impossible in the SRM while using a pair of Siberian Snakes since the amplitude-dependent spin tune is then fixed to one-half. Polarization loss in RHIC demonstrates that the isolated resonances model is not a good approximation of polarization dynamics with two Siberian Snakes. We therefore extend the model in which a pair of resonances in close proximity push the amplitude-dependent spin tune away from one-half in the presence of Siberian Snakes, allowing the crossing of higher-order spin resonances associated with depolarization. We present results from applying Machine Learning methods that establish spin transport models with two overlapping resonances from tracking data.

The strength of a first-order spin-orbit resonance is defined as the amplitude of the corresponding Fourier component of the spin-precession vector. However, it is possible to obtain the resonance strength without computing the Fourier integral directly. If a resonance is sufficiently strong, then to a good approximation, one can neglect all other depolarizing effects when near the resonance. Such an approximation leads to the single resonance model (SRM), for which many aspects of spin motion are analytically solvable. In this paper, we calculate the strength of first-order resonances using various formulae derived from the SRM, utilizing spin tracking data, the direction of the invariant spin field, and jumps in the amplitude-dependent spin tune. Examples are drawn from the RHIC Blue ring.