References - Bogoliubov Laboratory of Theoretical Physics - JINR

RF dipole (transverse B): εBdl = 1
π2 √ 2
e(1 + Gγ) �
Brmsdl, (3)
p
where e is the particle’s charge, p is its momentum, and � Brmsdl is the rf magnet’s rms
magnetic field integral in its rest frame.
We analyzed all available data on spin-flipping stored beams of protons, deuterons and
electrons [3]. We calculated the rf-induced spin resonance strength ratios εFS/εBdl, where
εFS was obtained by fitting the measured polarization to Eq. (1); εBdl was calculated
using Eqs. (2) and (3). We found that εFS/εBdl was 7 times lower than predicted for
deuterons, and 12 to 170 times higher for protons. We systematically studied εFS/εBdl
ratios with vertically polarized protons and deuterons in COSY.
Identical apparatus was
used for both protons and
deuterons, including the
COSY ring, the EDDA
detector, the low energy
polarimeter, the electron
cooler, the injector cyclotron,
the polarized ion
source, and the rf dipole
or solenoid [3–5, 8, 10–12].
All available εFS/εBdl
data are shown in Fig. 1.
With an rf dipole, we
studied the dependence of
εFS/εBdl on the beam’s
ε FS / ε Bdl
100
10
1
0.1
Dec.04 (d, dipole, COSY)
Nov.05 (p, dipole, COSY)
May 06 (d, dipole, COSY)
May 07 (d, solenoid, COSY)
0.1 1
Δf (kHz)
10
a (p, dipole, COSY)
b (p, dipole, COSY)
c (p, dipole, COSY)
d (p, dipole, COSY)
e (p, dipole, IUCF)
f (p, dipole, IUCF)
g (p, dipole, IUCF)
h (p, dipole, IUCF)
i (p, dipole, IUCF)
j (p, dipole, IUCF)
k (p, solenoid, IUCF)
l (p, solenoid, IUCF)
m (d, dipole, COSY)
n (d, dipole, COSY)
o (d, solenoid, IUCF)
p (e, dipole, MIT)
Figure 1: Ratio of εFS to εBdl is plotted vs. rf magnet’s frequency
sweep range Δf.
size, momentum spread Δp/p, and distance from the nearest 1 st -order intrinsic spin resonance
for both protons [3] and deuterons [4], and on Δf for deuterons. We observed no
dependence of εFS/εBdl on the beam’s size or Δp/p for either protons or deuterons, and
no dependence of εFS/εBdl on Δf for deuterons. We varied the vertical betatron tune
νy near a 1st-order intrinsic spin resonance and observed an enhancement of εFS/εBdl
with a hyperbolic dependence on distance from the 1 st -order intrinsic spin resonance for
both protons and deuterons. This can explain the enhancement for protons; however it
can not explain the deuteron’s very small εFS/εBdl.
With an a rf-solenoid and stored polarized deuterons, we measured an εFS/εBdl of
1.02 ± 0.05 over a wide range of parameters [5]. These new data agree precisely with
Eq. (2). These rf-solenoid data together with earlier rf-dipole results [3, 4] indicate that
Eq. (2) is correct for longitudinal rf fields in an ideal accelerator, while Eq. (3) for radial
rf fields is incorrect. Moreover, Eq. (3) must be replaced by more complex calculations,
which depend on each ring’s properties. One must properly include the modification of
εBdl due to coherent beam oscillations caused by the rf dipole, which then interact with
all the ring’s radial and longitudinal magnetic fields [2, 6].
Equation (1) is only valid if Δf is significantly larger than the spin resonance’s width.
Chao’s new matrix formalism [7] deals with conditions where the FS formula is not valid.
The Chao formalism can be used to calculate the spin dynamics anywhere inside a piecewise
linear resonance crossing. Our measurements [8] of the deuteron’s polarization near
and inside the resonance agree with the Chao formalism’s predicted oscillations.
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