References - Bogoliubov Laboratory of Theoretical Physics - JINR

2 The spin motion description under nonstationary
conditions
The special analysis is required to determine beam polarization behavior under nonstationary
conditions. Thus, in the traditional accelerator the spin field � W depends on particles
energy and is changed during acceleration. Orbital motion of particles is changed
during acceleration too. However, a wide class of problems can be solved in adiabatic
approximation when the “nonstationary” parameter is changed relatively little. For example,
the energy deviation of a particle per a turn in the accelerator is rather small.
In this approximation action variables of orbital motion remain constant during all cycle
of acceleration. It is necessary to pay attention only to small region of “critical” energy
during acceleration in which orbital action variables Ii can be changed and synchrotron
tune is tending to zero.
Dependence of a spin field � W on time under nonstationary conditions can be described
as a function from additional parameter λ(θ) which is slowly changing during acceleration:
�W (θ, Ψi,Ii,λ).
The beam polarization behavior is determined by means of solution for an precession
axis and the generalized tune which are found under stationary conditions. Procedure
is the following: if λ = const aspinfield� W , as well as under stationary conditions,
is a periodic function of all phases Ψi and azimuth θ. We find spin bases
{ �ℓ1(θ, Ψi,Ii,λ), �ℓ2(θ, Ψi,Ii,λ),�n(θ, Ψi,Ii,λ)} with periodic properties of each phase and
azimuth for each value of λ. Under this conditions the spin action variable J and the
generalized spin tune ν is a function of λ too: J = J(λ), ν = ν(Ii,λ).
From definition of the generalized spin tune the basic statement follows: tune ν, aswell
as natural system of orts { �ℓ1, �ℓ2,�n}, is a continuous function of λ parameter. That follows
from continuous dependence of a field � W on λ parameter. “Points” of spin resonances
mean only, that near these points spin motion is very sensitive to spin field value on
particles orbits.
ThespinactionvariableJremains constant during the changing of λ(θ) in adiabatic
approximation, if the speed of λ changing is small enough i.e. spin field changing is
relatively small during characteristic time of the precession axis changing. Characteristic
time of �n changing is determined by the nearest to ν combination from orbital motion
tunes νk, i.e. by the nearest spin resonance. Therefore, the condition of a spin action
variable J conservation while λ(θ) is changing, will be the following:
�
�
�
dλ ∂
� dθ ∂λ [(ν − νk)
�
�
�n] �
� ≪ (ν − νk) 2 , (3)
where νk = � k i νi is the nearest combination of orbital motion tunes. In this case spin,
rotating around �n axis, “has time” to follow up �n(λ) direction changing and the vector of
polarization is still determined by an average precession axis over distribution of particles
inabeamaccordingtotheformula(2).
The condition (3) indicates that there are only small regions of parameter in which
condition of a spin action variable conservation can be violated. These are the regions of
spin resonances of rather small strength.
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