Spin ﬂipping with an rf dipole and a full Siberian snake

We recently used an rf dipole magnet to study the spin ﬂipping of a 120 MeV horizontally polarized proton beam stored in the presence of a nearly full Siberian snake in the Indiana University Cyclotron Facility Cooler Ring. We ﬂipped the spin by ramping the rf dipole’s frequency through an rf-induced depolarizing resonance. After optimizing the frequency ramp parameters, we used multiple spin ﬂips to measure a spin-ﬂip efﬁciency of 86.5 6 0.5% . The spin-ﬂip efﬁciency was apparently limited by the ﬁeld strength in the rf dipole. This result indicates that spin ﬂipping a stored polarized proton beam should be possible in high energy rings such as the Brookhaven Relativistic Heavy Ion Collider and HERA where Siberian snakes are certainly needed and only dipole rf-ﬂipper magnets are practical.

Polarized beam experiments are now a major component of the programs in storage rings such as the Indiana University Cyclotron Facility (IUCF) Cooler Ring [1], the MIT-Bates Storage Ring [2], the Brookhaven Relativistic Heavy Ion Collider [3], and HERA at DESY [4]. Frequent reversals of the beam polarization direction can significantly reduce the systematic errors in an experiment's spin asymmetry measurements. An rf solenoid was used earlier to spin flip a horizontally polarized proton beam stored in the Cooler Ring containing a Siberian snake [5] with 97 6 1% spin-flip efficiency [6]. However, a solenoid's spin rotation decreases linearly with energy because of the Lorentz contraction of its R B dl; thus, a solenoid is impractical for spin flipping in high energy rings. Fortunately, a dipole's spin rotation is energy independent. Therefore, we recently used an rf dipole to spin flip a 120 MeV horizontally polarized proton beam stored in the IUCF Cooler Ring with a nearly full Siberian snake.
In any flat circular accelerator or storage ring with no horizontal magnetic fields, each proton's spin precesses around the vertical fields of the ring's dipole magnets. The spin tune n s , which is the number of spin precessions during one turn around the ring, is proportional to the proton's energy where G ͑g 2 2͒͞2 1.792 847 is the proton's gyromagnetic anomaly and g is its Lorentz energy factor. This vertical spin precession can be perturbed by the horizontal rf magnetic field from either an rf solenoid or an rf dipole. This perturbation can induce an rf depolarizing *Also at Moscow State University, Moscow, Russia. † Also at Portland Physics Institute, Portland, OR 97201. resonance, which can be used to flip the spin direction of the ring's stored polarized protons [6,7]. The frequency f r , at which an rf-induced depolarizing resonance occurs, is given by where f c is the proton's circulation frequency and k is an integer. Sweeping the rf magnet's frequency through f r can flip the spin. The Froissart-Stora equation [8] relates the beam's polarization after crossing the resonance P f to its initial polarization P i , where e is the resonance strength and Df͞Dt is the resonance crossing rate, while Df is the frequency range during the ramp time Dt. The apparatus used for this experiment, including the rf dipole, the IUCF Cooler Ring, and the polarimeter, were discussed earlier [6,[9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and are shown in Fig. 1. The 120 MeV horizontally polarized proton beam in the Cooler Ring was obtained using the new cooler injector polarized ion source (CIPIOS) [24] and the new cooler injection synchrotron (CIS) [25]. The beam polarization after the 7 MeV linac was 59 6 2%. At 120 MeV, the circulation frequency in the Cooler Ring was f c 1.597 84 MHz.
With a nearly full Siberian snake in the ring, the spin tune n s is very near but not exactly equal to 1͞2. Therefore, at 120 MeV, Eq. (2) implies that two closely spaced rf depolarizing resonances should be centered around with their frequencies at Since our snake strength s was about 1.01, n s was about 0.505; thus, the f 2 r resonance should have a frequency slightly below 1.5f c .
We first determined the f 2 r resonance's frequency to be near 2.384 MHz by using the much stronger rf solenoid. Then we used a simple LC resonant circuit to increase the rf dipole's input voltage to 71 V rms corresponding to an R B dl amplitude of 0.06 T mm rms. We then measured, at different dipole frequencies, the radial polarization, which is about 89% of the total horizontal polarization at the position of our polarimeter at 120 MeV. This measured radial polarization is plotted against the rf dipole's frequency in Fig. 2. The curve is a second-order Lorentzian fit to the data with a resonance frequency of f 2 r 2 384 040 6 90 Hz and a width w of 970 6 20 Hz. The separation d of this f 2 r resonance from the central frequency of 1.5f c allows one to determine the spin tune n s , and thus the Cooler Ring's total snake strength 1 1 Ds, using the approximation [13,26] Ds 2͑n s 2 0.5͒ 2d͞f c 2͑2.396 76 2 2.384 04͒ 1.597 84 0.8% .
To study spin flipping, we crossed this rf-induced resonance by linearly ramping the rf-dipole's frequency, through the measured f 2 r , from f 2 r 2 5 to f 2 r 1 5 kHz, with various ramp times Dt, while measuring the beam polarization after each frequency ramp. The measured radial polarization is plotted against the ramp time in Fig. 3. Note that, after a very rapid change, the polarization's magnitude seems constant for ramp times above 200 msec. We fit this measured polarization to a modified [6] Froissart-Stora formula where h is the spin-flip efficiency; from this fit, we obtained an 87 6 2% spin-flip efficiency. We tried to further increase the spin-flip efficiency by varying the rf dipole's frequency range Df with its ramp time Dt set at 500 msec, its amplitude set at 0.06 T mm, After setting Dt and Df to maximize the spin-flip efficiency, we more precisely determined this efficiency by measuring the radial polarization after many spin flips. We varied the number of spin flips while keeping, for each spin flip, the ramp time, the frequency range, and the rf voltage all fixed. This radial polarization is plotted against the number of spin flips in Fig. 4. We fit this data using FIG. 4. The measured radial proton polarization at 120 MeV is plotted against the number of spin flips. The rf dipole's frequency ramp time Dt was 400 msec, its frequency range Df was 10 kHz, and its R B dl was 0.06 T mm. The curve is a fit to the data using Eq. (8).
where P n is the measured radial beam polarization after n spin flips, P i is the initial polarization, h is the spin-flip efficiency, and n is the number of spin flips. The best fit gave a spin-flip efficiency of 86.4 6 0.5%. The spin-flip efficiency was apparently limited by the strength of the rf dipole's field; we hope to further increase the spin-flip efficiency by further increasing the rf-dipole's R B dl. In summary, we used an rf dipole to spin flip a stored 120 MeV horizontally polarized proton beam with a nearly full Siberian snake in the IUCF Cooler Ring. Combining the data from Figs