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

Joseph P. Devlin1,*, Georg H. Hoffstaetter1,2, and Desmond P. Barber3,4

  • *Contact author: jpd285@cornell.edu
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Phys. Rev. Accel. Beams 29, 074702 – Published 27 July, 2026

DOI: https://doi.org/10.1103/sxkc-zrh5

Abstract

Spin-polarized beams are important for some nuclear and high-energy physics experiments, such as those planned for the future Electron-Ion Collider. 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 (ADST) is constant. However, the slope of the ADST 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. 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.

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Phys. Rev. Accel. Beams 29, 074702– Published 27 July, 2026
  • Received 25 March 2026
  • Accepted 10 June 2026
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