1. Multiple spin resonance crossing in accelerators. A.M. Kondratenko (1), M.A. Kondratenko (1) and Yu. N. Filatov (2) (1) GOO “Zaryad”, Novosibirsk (2) JINR, Dubna In this paper the method of beam depolarization compensation during spin resonance crossing is developed. This method is based on the restoration of the spin adiabatic invariant after the spin resonance crossing. The case when the restoration of spin adiabatic invariant occurs due to multiple crossing of the resonance effective area is considered. At the same time it is necessary to provide the certain difference of spin phases between the moments of crossing of the resonance effective area. The conditions of beam polarization preservation when the resonance effective area is repeatedly crossed are obtained. Numerical examples are presented.

2. Spin Motion at Circular Accelerator In ideal accelerator Thomas-BMT equation [Ya.S. Derbenev, A.M. Kondratenko, A.N. Skrinsky (1970)] : spin precession tune Spin vector rotate around n-axis: periodical axis of precession If The spin equilibrium closed orbit Not equilibrium orbit spin deviation gyromagnetic anomaly Spin Adiabatic Invariant spread of n-axises spread of spin tune vector of polarization, degree of depolarization particle azimuth act-phase variables of betatron motion

3. Spin precession near the spin resonance when Spin adiabatic invariant (SAI) is blue arrows red arrows effective resonance region Adiabatic crossing Intermediate crossing Fast crossing

4. Generalized F.S. formula M.Froissart, R.Stora, 1960 normalized resonance strength initial vertical spin component final vertical spin component Generalized F.S. formula SAI magnitude becomes equal to vertical spin component magnitude only in the limit  |  |   : Dependence of spin tune on G  in accelerator with and without resonance. is spin tune In ideal accelerator

5. Single crossing of the spin resonance

6. Two times crossing of the spin resonance The spinor transformation matrix is

7. The condition of depolarization compensation (2 times crossing) The SAI keeps signThe case of SAI-flip

8. Comparison with Results of A.W. Chao “Spin Echo in Synchrotrons”, 2005 Condition when spin echo is not present after two spin detuning jumps: This result of A..Chao is similar to our result for the case

9. Three times crossing of the spin resonance The spinor transformation matrix is

10. Depolarization compensation condition stable in resonance strength and spin detuning deviation SAI-flip

11. Conclusions We found simple depolarization compensation conditions for cases of multiple resonance crossing. This method allows one to significantly reduce necessary resonance crossing rate in comparison with known methods. This method can also be applied to intrinsic resonances.

13. Scheme for shifting spin tune spin rotation angle in radial dipole spin rotation angle in solenoid length of central dipole length of one solenoid Spin tune shift: Total length of insertion: Orbit’s maximum vertical deviation: Betatron tune shift:

14. Example of crossing spin resonance using depolarization compensation technique for Nuclotron, JINR (Dubna) Spin detuning change pattern used in this example

15. Conclusions Technique for shifting spin tune would allow one to preserve polarization at resonance crossing without substantially perturbing beam’s betatron motion. Depolarization compensation technique allows one to substantially reduce required magnetic field integrals in insertion, which controls spin tune shift.