References - Bogoliubov Laboratory of Theoretical Physics - JINR

3 The conditions for transparent spin resonance
crossing
Taking into account the above conception one can get a complete compensation of depolarization
degree at resonance crossing. The conditions for transparent crossing are
given by
J(λ−) =±J(λ+) . (4)
where λ− and λ+ are λ values before (θ = θ−) andafter(θ = θ+) resonance crossing.
Let us consider the simple example of circular accelerator with only one Fourier harmonic
of the field � W due to vertical betatron oscillations for the purposes of illustration:
where �e1(Ψz) =�ex cos Ψz+�ey sin Ψz, Ψz is the phase, νz =Ψ ′ z
�W = ν0 �ez + w�e1(Ψz) (5)
is the vertical betatron tune,
�ex, �ey, �ez denote the corresponding radial, longitudinal and vertical orts of “accelerator”
basis, ν0 = γG = γ(g − 2)/2 is spin tune in ideal accelerator at w =0.
Thus, the “natural” spin basis is:
�ℓ1 = − w ε
�ez+
h h �e1(Ψz)
�
, �ℓ2 = �n × � �
ℓ1 = �ey cos Ψz−�ex sin Ψz , �n(Ψz) = ε w
�ez+
h h �e1(Ψz) ,
here are h = √ ε 2 + w 2 , ε = ν0 − νz — spin detuning.
The general spin tune ν is equal to h in this natural basis { � ℓ1, � ℓ2,�n}, near the isolated
spin resonance.
Ν
Ν0�Νk
Ν0�ΓG
Figure 1: The general spin
tune dependence vs tune ν0 (vs
the energy)
1
�1
� �
n�e
z
Ν0�Νk
Ν0
Figure 2: The vertical component
of precession axis plotted
vs the tune ν0
1
�1
� �
n�e
1
Ν0�Νk
Figure 3: The transverse component
of precession axis plotted
vs the tune ν0
The dependencies of general spin tune as well as vertical and transverse �n axis projections
on “energy” are shown in Fig. 1-3. The mirror-reflected decision {�n, ν} →{−�n, −ν}
is shown by a dashed line.
The variable λ coincide with the ν0 = γG value at the particles accelerations. The condition
(3) can be fulfilled at any moment of adiabatic crossing if (ν − νk = h, w = const)
|dν0/dθ| ≪w 2 . (6)
The condition (6) can be violated for very small resonance strength (at w → 0 ). Spin
action variable J(λ) can be changed only in the resonance region. The beam’s polarization
is decrease in this case. The condition for transparent crossing is the restoration of the
spin action variable value after spin resonance crossing.
408
Ν0