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.

Spin Motion at Circular Accelerator In ideal accelerator Thomas-BMT equation[Ya.S. Derbenev, A.M. Kondratenko, A.N. Skrinsky (1970)] :The spin equilibrium closed orbitNot equilibrium orbit spin deviation gyromagnetic anomaly Spin Adiabatic Invariant spread of n-axises spread of spin tune vector of polarization, degree of depolarization particle azimuthact-phase variables of betatron motion

EMBED Equation.DSMT4

EMBED Equation.3

EMBED Equation.3

EMBED Equation.DSMT4

EMBED Equation.3

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Spin precession near the spin resonance when blue arrowsred arrowseffective resonance regionAdiabatic crossingIntermediate crossingFast crossing

Generalized F.S. formula Generalized F.S. formulaSAI magnitude becomes equal to vertical spin component magnitude only in the limit | | :Dependence of spin tune n on Gg in accelerator with and without resonance.is spin tuneIn ideal accelerator

Single crossing of the spin resonance

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

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

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

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

Depolarization compensation condition stable in resonance strength and spin detuning deviationSAI-flip

ConclusionsWe 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.

Scheme for shifting spin tuneSpin tune shift:Total length of insertion:Orbits maximum vertical deviation:Betatron tune shift:

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

1

14

0.2

0.3

0.52

4.7

0.6

7

0.1

0.6

0.26

4.7

0.6

2

0.06

1.0

0.13

1.9

1.2

3

14

0.07

0.3

0.17

4.7

0.2

7

0.035

0.6

0.086

4.7

0.2

2

0.02

1.0

0.045

1.9

0.4

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ConclusionsTechnique for shifting spin tune would allow one to preserve polarization at resonance crossing without substantially perturbing beams betatron motion.

Depolarization compensation technique allows one to substantially reduce required magnetic field integrals in insertion, which controls spin tune shift.