References - Bogoliubov Laboratory of Theoretical Physics - JINR

and depends on a choice of � ℓ1 and � ℓ2 orts in a perpendicular to the �n axis plane. It is
shown, that � ℓ1(θ, Ψi,Ii) and � ℓ2(θ, Ψi,Ii) orts can be chosen with similar to field properties
of periodic dependence phases Ψi and azimuth θ. Moreover, there is a “natural” system of
orts { � ℓ1, � ℓ2,�n} when the generalized spin tune doesn’t depend on phases Ψi and azimuth
θ and remains constant at each particles trajectory: ν = ν(Ii) =const.
The spin, directed along �n axis, is stable to small field deviations. On the contrary,
motion of perpendicular to an �n axis spin components is unstable to small field deviations
which cause a small deviation of spin tune ν.
In natural system of orts { � ℓ1, � ℓ2,�n}, spin rotates in a constant field:
� h(Ii) =ν(Ii) �n.
The generalized spin tune which depends on periodic orts �ℓ1 and �ℓ2 can be changed
for an integer combination from orbital motion tunes νk =Ψ ′ k = � ki νi. The generalized
spin tune is changed for value νk (ν → ν − νk) if periodic orts �ℓ1 + i �ℓ2 are replaced by
periodic orts ( �ℓ1 + i �ℓ2) exp(iΨk). Under stationary conditions the investigation of beam polarization behavior in the
accelerator is reduced to search of “natural” system orts which can be found, for example,
by numerical methods.
The vector of polarization � P is determined by average meaning of spin � S over particles
distribution and is a function of azimuth θ:
� � �
�P = �S = J�n + √ 1 − J 2 �
Re ( �ℓ1 + i � ��
ℓ2) exp(−i (νθ + α)) ,
where J = � S·�n = const is a spin variable of action or spin adiabatic invariant which is kept
under stationary conditions. In this formula the spin is normalized to unit. Averaging
over the distribution of particles under stationary conditions is reduced to independent
averaging over all orbital phases Ψi and orbital actions Ii. Because of spin tune ν(Ii) and
orbital tunes Ψ ′ i spreading the beam polarization under stationary conditions relaxes to
following value: 1
�P = 〈J〉 〈�n〉 . (2)
The greatest possible degree of polarization is achieved at the maximum value of spin
adiabatic invariant, i.e. when J = ±1 (forexample,foraparticlewith1/2 spin the
adiabatic invariant in dimensional units is equal ±�/2). In this case � Pmax = ±〈�n〉. The
deviation of | � Pmax| value from unit is connected with spread of precession axes directions
at particles trajectories. This “dynamical” beam depolarization is reversible. The beam
polarization can be restored by means of the organization of an appropriate spin field
in the accelerator. The � Pmax value depends on azimuth θ and in a place of carrying
out of experiment can be equaled to unit. For example, in an opposite section of the
accelerator with one Siberian Snake precession axes of all particles have a longitudinal
direction (| � Pmax(0)| = 1). In other places of an orbit there is a spread of precession
axes and consequently in these places there is the reduction of a degree of polarization
(| � Pmax| < 1).
1 Here the effects connected with loss of beam polarization at injection in accelerator are not considered.
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