Spin manipulating stored 1.85 GeV / c vector and tensor polarized spin-1 bosons

We recently studied the spin manipulation of 1 : 85 GeV =c vertically polarized deuterons stored in the COSY cooler synchrotron. We adiabatically swept an rf dipole’s frequency through an rf-induced spin resonance and observed its effect on the deuterons’ vector and tensor polarizations. After optimizing the resonance crossing rate and maximizing the rf dipole’s voltage, we measured spin-ﬂip efﬁciencies of 97 (cid:1) 1% and 98 : 5 (cid:1) 0 : 3% in two separate runs. We also conﬁrmed at higher energy the striking behavior of the spin-1 tensor polarization recently found at IUCF.


I. INTRODUCTION
Many polarized beam experiments are now operating in storage rings such as the MIT-Bates storage ring [1], COSY [2], RHIC at Brookhaven [3] and HERA at DESY [4,5]. Frequent spin direction reversals can significantly reduce the systematic errors in such spin asymmetry measurements. Beams of spin- 1 2 particles have been successfully spin flipped: horizontally polarized electrons at the MIT-Bates storage ring [6], and horizontally and vertically polarized protons at the IUCF cooler ring [7][8][9][10][11][12][13] and COSY [14]. Stored spin-1 deuterons were also spin flipped at the IUCF cooler ring [15]. We recently extended the studies of their interesting behavior by spin flipping 1:85 GeV=c vector and tensor polarized deuterons stored in the COSY cooler synchrotron.
A beam of spin-1 bosons has a more complex polarization behavior than spin-1 2 fermions; their spin component, m, along the vertical axis can have three values: m 1; 0; ÿ1. Describing the polarization of a beam of spin-1 particles requires three vector polarization components and five independent second-rank tensor components [16,17]. The magnitudes of a vertically polarized beam's vector and tensor polarizations are given by [16]: where N , N 0 , and N ÿ are the number of particles in m 1, 0, and ÿ1 states, respectively, and N N N 0 N ÿ is the total number of particles. In any flat circular accelerator or storage ring, with no horizontal magnetic fields, each deuteron's spin precesses around the vertical fields of the ring's bending magnets.
The spin tune s , which is the number of spin precessions during one turn around the ring, is proportional to the deuteron's energy s G; (2) where G g ÿ 2=2 ÿ0:142 987 is the deuteron's gyromagnetic anomaly and is its Lorentz energy factor. The deuteron's polarization can be perturbed by the horizontal rf magnetic field from either an rf solenoid or an rf dipole. At a resonant frequency the perturbations can add coherently to induce an rf spin resonance, which can flip the spin of stored polarized particles [7][8][9][10][11][12][13][14]18], such as deuterons [15]. The rf-induced spin resonance's frequency is where f c is the deuterons' circulation frequency and k is an integer. Adiabatically ramping the rf magnet's frequency through f r can rotate the deuterons' polarization direction by an angle around a horizontal axis. Under this rotation, the spin-1 vector and tensor polarizations, P V and P T (sometimes called P z and P zz [15]), respectively, transform as where P i V and P i T are the initial vector and tensor polarizations, respectively. When there is a vector spin flip: P V ÿP i V ; while the tensor polarization is unchanged P T P i T . At mid-spin-flip, when 2 , note that P V 2 0 while P T 2 ÿ 1 2 P i T .
PHYSICAL REVIEW SPECIAL TOPICS -ACCELERATORS AND BEAMS 8, 061001 (2005) 1098-4402=05=8(6)=061001(9) 061001-1  2005 The American Physical Society The Froissart-Stora equation [19] relates the beam's vector polarization P V , after ramping an rf dipole's frequency through a resonance, to the beam's initial vector polarization P i V and the ramp's frequency range f and ramp time t: where f=t is the resonance crossing speed. The resonance strength is given by [20] where e is the deuteron's charge, p is its momentum, and R B rms dl is the rf dipole's rms magnetic field integral.

II. APPARATUS
The apparatus used for this experiment, including the COSY storage ring [21][22][23][24], the EDDA detector [25], the low energy polarimeter, the injector cyclotron, and the polarized ion source [26 -28] are indicated in Fig. 1, along with the rf dipole. The dipole consisted of an 8-turn ferritecore copper coil with the spacing between its turns optimized to produce a uniform radial magnetic field; it was part of an LC resonant circuit, which normally ran at about 2.8 kV rms producing an R B rms dl of 0:54 0:03 T mm at f 916 kHz. Then using Eq. (6) for 1:85 GeV=c deuterons this gives a resonance strength 16 1 10 ÿ6 : An air-core rf dipole at COSY was used earlier to spin-flip polarized protons [14] and polarized deuterons [29] with lower spin-flip efficiency. The beam emerging from the polarized D ÿ ion source was accelerated by the cyclotron to COSY's 75.6 MeV injection energy. Then the low energy (LE) polarimeter monitored the beam's polarization before injection into COSY to monitor the stable operation of the ion source and cyclotron. We measured the polarization in COSY using the EDDA detector [2,25] as a polarimeter. The beam parameters are given in the Table I.
The deuteron vector and tensor polarizations were obtained using the EDDA detector. No published data were available at 1850 MeV=c for the vector and tensor analyzing powers of p-d elastic scattering; thus, they were determined in an earlier calibration run [33], as described below. First one measured the deuteron vector and tensor polarizations both below and above the experimental energy, at 1042 and 2430 MeV=c, where the analyzing powers were published [34,35]. These measured polarizations agreed within errors. Thus, there was no depolarization during acceleration between 1042 and 2430 MeV=c, which includes 1850 MeV=c. These two polarization data sets were used to obtain the vector and tensor analyzing powers for p-d elastic scattering at 1850 MeV=c [33].
We later used these elastic analyzing powers to measure the polarizations for each spin state as shown in Fig. 2. Note that the two polarizations obtained from the analyzing powers obtained from the 1042 MeV=c and the 2430 MeV=c calibrations agree within errors. We simultaneously calibrated the effective analyzing powers and detector efficiencies of the EDDA detector, which was at the same time operating as a simple 4-quadrant polarimeter using scalers to record all scattering events in each quadrant [33]. We also confirmed these calibrations to a precision of better than 1% during the main experiment [36]. We used these detector efficiencies and effective analyzing powers to obtain the vector and tensor polarizations from the measured scaler asymmetries during the main experiment.

IV. SPIN MANIPULATION
We first determined the approximate position of this spin depolarizing resonance by sweeping the rf dipole's frequency initially by 10 kHz around this f r ; then we continued by narrowing the frequency sweeping range into those half-ranges that caused spin flip. This technique was described earlier [6,12,13]. The resulting data indicated that the resonance was located near 916.9 kHz.
We then more precisely determined f r by measuring the polarization, after running the rf dipole at different fixed frequencies near 916.9 kHz. The measured vector polarization ratios for the four nonzero polarization states are plotted against the rf dipole's frequency in Fig. 3  We next maximized the deuteron polarization's spin-flip efficiency. We flipped them by linearly ramping the rf dipole's frequency from f r ÿ 0:1 to f r 0:1 kHz, with various ramp times t; we measured the polarizations after each frequency ramp. These measured vector and tensor polarizations are plotted against the ramp time in Fig. 4. The vector polarization data are fit to the empirically modified [6,11] Froissart-Stora formula [19] where V is the vector resonance strength. The parameter l V is defined as the upper limit of the achievable spin-flip efficiency V when the exponential approaches zero. This limit could be due to many depolarizing mechanisms such as: f being too small to completely cover the resonance width, or any weak nearby resonance. This fit gave l V 100:4 2:0%; V 1:167 0:011 10 ÿ6 : (10) To fit the tensor data in Fig. 4, we use Eqs. (4) and (9) as in [15] to obtain where we introduced l T as the tensor spin-flip efficiency and T as the tensor resonance strength. Fitting the tensor polarization data in Fig. 4 and averaging for the three states then gives l T 100:2 2:1%; T 1:144 0:022 10 ÿ6 : (12) Note that the V and T obtained from the extended Froissart-Stora formula, Eqs. (9) and (11), agree quite well with each other. However, both are more than 10 times smaller than the obtained from the measured R B dl in Eq. (7). We recently started an experiment to try to understand the discrepancy [20].
For each of the three nonzero tensor polarization states, the two data points nearest the t value in Fig. 4 where P V 0 agree fairly well with the fit curves for the tensor ratio P T =P i T . This agrees very well with the prediction of Eqs. (4) and (11) that P T =P i T is ÿ 1 2 when P V 0. Moreover, the and values shown in Eqs. (10) (vector) and (12) (tensor) are all consistent with each other. This confirms [15] that one l and one describe both the vector and tensor polarizations during a resonance crossing.
For the data in Fig. 4, we also calculated the ratios of the vector and tensor polarizations to their initial values obtained from the fits; these ratios are plotted against the frequency ramp time in Fig. 5 for the nonzero states. This allowed us to fit all nonzero vector and tensor data to Eqs. (9) and (11), respectively. Notice the good agreement of the data points for different states with each other and with the fit curves. Fitting the vector and tensor data sets, each with a single curve, gave a better precision results for l and than Eqs. (10) and (12) l V 100:6 1:5%; V 1:166 0:009 10 ÿ6 ; l T 100:2 1:8%; T 1:152 0:022 10 ÿ6 : (13) We next spin flipped the deuterons while varying the rf dipole's frequency range f, with its t at 50 s and its R B rms dl at 0:54 T mm. The data in Fig. 6 shows a broad maximum in the spin-flip efficiency near f=2 75 Hz. The Froissart-Stora equation is only valid for sufficiently wide frequency ranges covering the spin resonance completely; thus, in Fig. 6, we fit the vector and tensor data to Eqs.  Table II. The rf dipole's frequency halfrange f=2 was 100 Hz, and its R B rms dl was 0:54 T mm. The top and bottom curves are fits to all states' data using Eqs. (9) and (11), respectively. The arrows mark a 90 rotation where P V 0 and P T ÿ 1 2 P i T . Below f=2 75 Hz in Fig. 6, we used hand-drawn curves to guide the eye. For the data in Fig. 6, we also calculated the ratios of the vector and tensor polarizations to their initial values obtained from the fits; these ratios are plotted against the frequency range in Fig. 7 for the nonzero states. Notice that the data points for different states agree with each other within errors. We again fit all nonzero vector and tensor data, at and above f=2 75 Hz, to Eqs. (9) and (11), respectively. These fits gave We showed the fits as solid lines to compare the results of Eqs. (9) and (11) with the experimental data both above and below f=2 75 Hz. Notice that above f=2 75 Hz the vector data is described very well by Eq. (9) while below f=2 75 Hz the vector data's behavior (shown by the dashed curve) is very different from Eq. (9). However, the tensor data seem consistent with Eq. (11) for all frequency ranges. A possible explanation for this behavior involves the beam's spin-resonance frequency spread due to its momentum spread. When the frequency range is smaller than the resonance frequency spread, only those particles, with resonance frequencies within the frequency range, are spin flipped according to the Froissart-Stora formula, while the rest of the beam's polarization remains unchanged. The beam's total polarization is the average polarization of the flipped and nonflipped parts. Below f=2 75 Hz, the vector and tensor polarizations of the flipped part are about ÿP i V and P i T , respectively. Combining these with the P i V and P i T of the nonflipped part implies that the vector polarization should return from ÿP i V to its initial value as f becomes smaller, while the tensor polarization should stay near P i T . This is observed in Figs. 6 and 7. Thus, the polarized deuteron behaves like a mixture of two polarized spin-1 fluids each obeying Eq. (4), one spin flipped and one unflipped, with the spin-flipped fraction decreasing as f decreases.

V. MULTIPLE SPIN FLIPPING
Using Figs. 5 and 7, we optimized the spin-flip efficiency by setting t at 60 s and f at 150 Hz at our maximum R B rms dl. Then we more precisely determined the spin-flip efficiencies by simultaneously measuring, after n frequency sweeps, the vector and tensor polarizations P n V and P n T . To plot and fit these vector and tensor polarization data, we defined the measured spin-flip efficiencies V and T in terms of the measured P n V and P n T by taking the nth power of Eqs. (9) and (11): We then calculated the ratios of the vector and tensor polarizations to their initial values; these ratios are plotted against n in Fig. 8 for the nonzero states. As indicated in Table IV, we fit these vector and tensor data using Eq. (16) to obtain V 96:5 0:6%; T 98:3 1:0%; for the data in Fig. 8. While these two values are close to each other, they are slightly outside of their statistical errors. Thus, in averaging them, we increased the error on their average by 2 p to obtain a spin-flip efficiency 97 1%: We also had a more recent run where we were able to flip the spin 5 times with rather good precision. There were some problems in calibrating the effective tensor analyzing powers, which may be resolved in a later paper. Thus, we only present the vector results at this time. These more recent vector data are plotted in Fig. 9. As indicated in FIG. 7. (Color) The measured vector and tensor deuteron polarization ratios at 1850 MeV=c are plotted against the rf dipole's frequency half-range f=2. The P i V and P i T were obtained from fits to the data as shown in Table III. The rf dipole's ramp time t was 50 s, and its R B rms dl was 0:54 T mm. The top and bottom solid curves are fits to the data for all states at and above f=2 75 Hz using Eqs. (9) and (11), , respectively. The dashed lines are to guide the eye. The arrows mark a 90 rotation where P V 0 and P T ÿ 1 2 P i T .
Since the tensor calibration problem might cause some small second-order changes in V , we increased the error from its statistical value of 0.2% to 0.3%.

VI. SUMMARY
By adiabatically sweeping an rf dipole's frequency through an rf-induced spin resonance, we spinmanipulated the vector and tensor polarizations of a vertically polarized 1:85 GeV=c spin-1 deuteron beam. We experimentally demonstrated that a single spin-flip efficiency and a single resonance strength can describe both the vector and tensor polarizations. Their interesting behavior, first seen at IUCF [15] and now confirmed at COSY, can be understood in terms of the quantum mechanics of spin-1 rotations, Eq. (4), and the Froissart-Stora equation, Eq. (5).
By decreasing the frequency sweep range f, we found that the Froissart-Stora equation became invalid, as expected, when f was not significantly larger than the resonance width w. However, the behavior of the vector and tensor polarizations in this small f region can be described in terms of a simple model of a mixture of two polarized spin-1 fluids each obeying Eq. (4); one is spin flipped and one is unflipped, with the spin-flipped fraction decreasing as f decreases.
Multiple spin flips were used to determine more precisely the spin-flip efficiency of both the vector and tensor polarizations. By placing the spin-resonance crossing parameters, f and t, at their optimal values, we reached a measured spin-flip efficiency of 97 1% and 98:5 0:3% for two separate runs with 1:85 GeV=c vertically polarized spin-1 deuterons stored in the COSY ring. These were significantly higher than the 94:2 0:3% attained at IUCF [15].   All states in Fig. 8 (N ÿ 1 7) 0:965 0:006 9.1 All states in Fig. 9 (N