Multiple spin resonance crossing in accelerators.

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

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

4. M.Froissart, R.Stora, 1960 initial vertical spin component final vertical spin component normalized resonance strength Generalized F.S. formula SAI magnitude becomes equal to vertical spin component magnitude only in the limit | |: Generalized F.S. formula Dependence of spin tune n on Gg 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 case of SAI-flip The SAI keeps sign

8. Comparison with Results ofA.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. spin rotation angle in radial dipole length of central dipole spin rotation angle in solenoid length of one solenoid Scheme for shifting spin tune 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 forNuclotron, 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.