Unexpected enhancements and reductions of rf spin resonance strengths

We recently analyzed all available data on spin-ﬂipping stored beams of polarized protons, electrons, and deuterons. Fitting the modiﬁed Froissart-Stora equation to the measured polarization data after crossing an rf-induced spin resonance, we found 10–20-fold deviations from the depolarizing resonance strength equations used for many years. The polarization was typically manipulated by linearly sweeping the frequency of an rf dipole or rf solenoid through an rf-induced spin resonance; spin-ﬂip efﬁciencies of up to 99 : 9% were obtained. The Lorentz invariance of an rf dipole’s transverse R Bdl and the weak energy dependence of its spin resonance strength E together imply that even a small rf dipole should allow efﬁcient spin ﬂipping in 100 GeVor even TeV storage rings; thus, it is important to understand these large deviations. Therefore, we recently studied the resonance strength deviations experimentally by varying the size and vertical betatron tune of a 2 : 1 GeV =c polarized proton beam stored in COSY. We found no dependence of E on beam size, but we did ﬁnd almost 100-fold enhancements when the rf spin resonance was near an intrinsic spin resonance.


I. INTRODUCTION
Polarized beam experiments have become an important part of the programs in storage rings such as the IUCF Cooler Ring [1], AmPS at NIKHEF [2], the MIT-Bates Storage Ring [3], COSY [4], LEP at CERN [5], RHIC at BNL [6] and HERA at DESY [7,8]. To reduce their systematic errors, polarized scattering experiments require frequent spin-direction reversals (spin flips) while the polarized beam is stored. Spin resonances [9][10][11] induced by either an rf solenoid or rf dipole can produce these spin flips in a well-controlled way [12 -25].
In any flat storage ring or circular accelerator with no horizontal magnetic fields, each beam particle's spin precesses around the vertical fields of the ring's dipole magnets. The spin tune s , which is the number of spin precessions during one turn around the ring, is proportional to the particle's energy s G; (1) where G g ÿ 2=2 is the particle's gyromagnetic anomaly G p 1:792 847; G e 1:159 67 10 ÿ3 , and G d ÿ0:142 987 and is its Lorentz energy factor. The vertical polarization can be perturbed by an rf magnet's horizontal rf magnetic field. This perturbation can induce an rf depolarizing resonance, which can flip the spin direction of stored polarized particles [12 -25]; the reso-nance's frequency is where f c is the particle's circulation frequency and k is an integer.
Ramping an rf magnet's frequency through f r can flip each particle's spin. The modified [15,19] Froissart-Stora equation [9] relates the beam's initial polarization P i to its final polarization P f after crossing the resonance, the parameter is the limiting spin-flip efficiency and the ratio f=t is the resonance crossing rate, where f is the ramp's frequency range during the ramp time t. Equation (3) is valid if f is larger than the spin resonance's width. We will later discuss how one obtains the spin resonance strength E FS from Eq. (3). One can also obtain the resonance strength E Bdl due to an rf solenoid or rf dipole using where e is the particle's charge, p is its momentum, and R B rms dl is the rf magnet's rms magnetic field integral in its rest frame [26 -28]. These equations are for an ideal flat circular accelerator with a point rf magnet causing the only perturbation of the spin motion. Other authors [29,30] derived a similar equation that differed by a factor of 2 in the denominator

II. ANALYSIS OF EXISTING DATA
To understand this factor of 2 difference, we made a compilation of all available experimental data [31] that allowed a simultaneous evaluation of the spin resonance strength E Bdl , obtained from Eq. (4) or (5), and the spin resonance strength E FS obtained from Eq. (3), the modified Froissart-Stora formula. This compilation indicated that for many experiments E Bdl and E FS disagree with both [26,27] and [29,30] by factors of 0.1, 10, or more.
The resonance strength E FS can be obtained by first measuring the final beam polarization P f after ramping an rf magnet's frequency by a range f during a time t through a spin resonance; then P f is plotted vs the different t values and the data are fit to Eq. (3) with E FS as a fit parameter. Examples of these curves are shown in Figs. 1 and 2 for protons [23] and deuterons [25], respectively. For protons, using Eq. (5) gave E Bdl of 80 4 10 ÿ6 , while E FS was 580 10 10 ÿ6 , which is about 7.3 times larger. For deuterons, E Bdl was 16 1 10 ÿ6 , while E FS was 1:17 0:01 10 ÿ6 , which is about 14 times smaller. Note that these two experiments were both done with the same rf dipole at COSY; thus, these large strength deviations could not be due to incorrect calibrations of R Bdl.
We examined all available publications and logbooks on spin-flipping experiments [31] that had both: information on the rf magnet's R Bdl, and data for a polarization vs ramp time t curve. We also checked in each experiment for any other spin resonances near the rf-induced resonance. Fits to Eq. (3), the modified Froissart-Stora equation, gave the measured spin resonance strength for each experiment and its error. We also calculated the spin resonance strength from each rf magnet's R Bdl using Eq. (4) or (5). For the experiments at IUCF and MIT, there were no precise measurements of the rf magnet's R Bdl; thus, we assumed a 10% error. For the more recent experiments at COSY, the rf magnet's R Bdl was known with a 5% uncertainty.
Then we took the ratio of the two spin resonance strengths, E FS =E Bdl , for each experiment and obtained its error using simple error propagation formulas. Figure 3 shows these ratios plotted vs the ramp time t at which P passes through 0 in each t curve. The dashed [26 -28] and solid [29,30] lines show the predicted ratios.

III. COSY EXPERIMENT
To better understand these deviations, we recently studied experimentally the dependence of an rf resonance's strength on the beam size and on its distance from an  intrinsic spin resonance. We did this by varying the size and the vertical betatron tune of a 2:1 GeV=c polarized proton beam stored in COSY.
The apparatus used for this experiment, including the COSY storage ring [32 -35], the EDDA detector [36], the electron Cooler [37], the low energy polarimeter, the injector cyclotron, and the polarized ion source [38][39][40], are indicated in Fig. 4, along with the rf dipole. The beam emerging from the polarized H ÿ ion source was accelerated by the cyclotron to COSY's 45 MeV injection energy. Then the low energy polarimeter measured the beam's polarization before injection into COSY to monitor the stable operation and polarization of the ion source. The electron Cooler reduced the beam's size and momentum spread at injection energy. A 24.5 keV electron beam of about 170 mA, with a diameter of about 25 mm, cooled the protons in a 2-m-long cooling section; it took 10 s to cool the proton beam to its equilibrium emittances in both the longitudinal and transverse dimensions. This decreased its initial momentum spread by about a factor of 10 to p=p 9 10 ÿ5 FWHM.
We manipulated the beam's polarization using a ferritecore rf dipole, which contained a copper coil of 8 turns; their spacing was optimized to produce a uniform radial magnetic field. The rf dipole was part of an LC resonant circuit, which operated near f r 902:6 kHz, at an rf voltage of 1 kV rms giving an rf R B rms dl of 0:19 0:01 T mm.
We measured the polarization in COSY using the EDDA detector [4,36]; we reduced its systematic errors by cycling the polarized source between the up and down vertical    (4) or (5). The dashed and solid lines show the ratios predicted by [26 -28] and [29,30], respectively. Note that points (a) and (b) overlap, as do points (e) -(h). The data for all points are listed in Table I. polarization states. The rf acceleration cavity was turned off and shorted during COSY's flattop; thus, there were no synchrotron sideband effects [14,41,42]. The measured flattop polarization, before spin manipulation, was typically 50% to 60%.
We determined the resonance's position and the upper limit on its width using the procedure described in [23]. Then, we obtained the resonance strength E FS , as in Figs. 1 and 2, for different fast quadrupole strengths. The fast quadrupole increased the beam's emittance at the start of COSY's flattop, before we manipulated the beam's polarization with the rf dipole. We measured beam's vertical profile for each fast quadrupole strength by moving a thin fiber target vertically through the beam, while measuring the current due to its scattering events as the fiber moved. The E FS =E Bdl ratios are plotted against the beam's vertical size (FWHM) in Fig. 5. We also plot deuteron data obtained in a similar way during a December 2004 run [25]. The proton and deuteron data both show no dependence on the beam's vertical size. The fits to horizontal straight lines give resonance strength ratios of 6:07 0:04 for protons and 0:0618 0:0002 for deuterons.
We also obtained the resonance strength E FS , as in Figs. 1 and 2, for different values of the vertical betatron tune y . We plotted the resulting E FS =E Bdl ratios against y in Fig. 6; we also plotted our 2004 deuteron data [25]. Notice the s 8 ÿ y first-order intrinsic spin resonance for protons, which let us study the rf resonance strength in its proximity; there were also four third-order resonances. Using references [43,44], one can obtain the dependence of E FS =E Bdl on the distance between the vertical betatron tune y and the rf spin resonance's tune r k f r =f c , where k is an integer; this dependence is Fitting the proton data in Fig. 6 to Eq. (7) gave A of 0:44 0:46, B of 0:50 0:03, and r of 3:6060 0:0005, which was very near the calculated value of 3:605 for the first-order intrinsic spin resonance. The parameter B depends on many details of the ring. The parameter A should give the predicted resonance strength ratio E FS =E Bdl of either 1 [26 -28] or 0.5 [29,30] when one is infinitely far from any intrinsic spin resonances. Clearly our data could not determine A with good precision. The deuteron data in Fig. 6 show no dependence on y ; its fit to a horizontal straight line gave a resonance strength ratio of 0:063 0:002, which agrees with the result from  7. (Color) Ratio of E FS to E Bdl is plotted vs the ramp time t giving P 0 in each t curve [31]. E FS is the resonance strength obtained by fitting the data in each t curve to Eq. (3); E Bdl is obtained using each data point's R Bdl in Eq. (4) or (5). The dashed and solid lines show the ratios predicted by [26 -28] and [29,30], respectively. Note that points (a) and (b) overlap, as do points (e) -(h). The data for all points are listed in Table I

FIG. 8. (Color) Ratio of E FS to E
Bdl is plotted vs the dimensionless crossing rate ÿ giving P 0 in each t curve [31]. E FS is the resonance strength obtained by fitting the data in each t curve to Eq. (3); E Bdl is obtained using each data point's R Bdl in Eq. (4) or (5). The dashed and solid lines show the ratios predicted by [26 -28] and [29,30], respectively. Note that points (a) and (b) overlap, as do points (e) -(h). The data for all points are listed in Table I 9. (Color) Ratio of E FS to E Bdl is plotted vs the frequency ramp range f used for each t curve [31]. E FS is the resonance strength obtained by fitting the data in each t curve to Eq. (3); E Bdl is obtained using each data point's R Bdl in Eq. (4) or (5). The dashed and solid lines show the ratios predicted by [26 -28] and [29,30], respectively. Note that points (a) and (b) overlap, as do points (e) -(h). The data for all points are listed in Table I.   TABLE I. Data for Figs. 3 and 7-9 along with relevant experimental parameters for this experiment and earlier experiments [31]. The letters (a) -(p) in column 1 stand for the references listed in [31], while the letters p, d, and e in column 3 denote protons, deuterons, and electrons, respectively. We assumed a 10% error in the rf magnets' R Bdl for the experiments at IUCF and MIT, and a 5% error in the rf dipole's R Bdl for our experiments at COSY. Note that, for the bottom 12 COSY proton points, y was varied as explained in Figs. 5 and 6.
Ring Magnet type  Fig. 7: ramp time t where P 0 in each t curve, Fig. 8: crossing rate ÿ where P 0 in each t curve, Fig. 9: frequency ramp range f in each t curve. The data seem scattered randomly in both Figs. 7 and 8. There is much less scatter in Fig. 9: all data with small f have E FS =E Bdl values far below 1, while all data with large f have E FS =E Bdl values near or above 1.
Clearly there still is no complete understanding of what is causing these very large resonance strength deviations. Figure 6 indicates that the large deviations above the predicted values of 1 [26 -28] and 0.5 [29,30] are due to a nearby intrinsic spin resonance. Figure 9 suggests that the deviations below these lines could be due to the small f frequency ramp range. Figure 9 also shows that, for deuterons, all E FS =E Bdl ratios are far below 1 for rf dipoles, but closer to 1 for rf solenoids. This may suggest anomalous behavior of spin-1 deuterons when spin-manipulated by dipoles.
Studying these deviations helped inspire a theoretical paper [45], which may be the first generalization of the Froissart-Stora equation [9] since its publication in 1960.

IV. SUMMARY AND CONCLUSION
In summary, by compiling existing data and fitting it to the Froissart-Stora equation, we first found 10 -20-fold deviations from the spin resonance strength equation that has been used for many years. We recently studied these deviations experimentally at COSY. We found no dependence on the beam's vertical size; however, we did find even larger deviations, of almost 100, when the rf spin resonance was near an intrinsic spin resonance. This proximity to an intrinsic resonance probably explains the 100times-larger-than-predicted deviations. The 10 -20-timessmaller-than-predicted deviations for deuterons could be due to the following: the small f sweep used to flip the deuteron spin; the resulting very slow crossing rate (Fig. 8); or some unexpected behavior of spin-1 deuterons in an rf dipole. We plan to test these and other possibilities at COSY, to learn if any are valid.
By adiabatically ramping the frequency of an rf dipole or rf solenoid through an rf-induced spin resonance, one can flip the polarization of a stored beam of protons, electrons or deuterons with an efficiency of up to 99:9%. An rf dipole only slightly stronger than our 0:5 T mm dipole should allow efficient spin flipping of polarized protons in the 200 GeV RHIC and perhaps someday in the 7 TeV LHC.
Thus, these sometimes 100-fold deviations must be better understood both theoretically and experimentally.