High precision beam momentum determination in a synchrotron using a spin resonance method
P. Goslawski, A. Khoukaz, R. Gebel, M. Hartmann, A. Kacharava, A. Lehrach, B. Lorentz, R. Maier, M. Mielke, M. Papenbrock, D. Prasuhn, R. Stassen, H.J. Stein, H. Stockhorst, H. Ströher, C. Wilkin
aa r X i v : . [ phy s i c s . acc - ph ] F e b High precision beam momentum determination in a synchrotron using aspin-resonance method
P. Goslawski, ∗ A. Khoukaz, R. Gebel, M. Hartmann, A. Kacharava, A. Lehrach, B. Lorentz, R. Maier, M. Mielke, M. Papenbrock, D. Prasuhn, R. Stassen, H.J. Stein, H. Stockhorst, H. Str¨oher, and C. Wilkin Institut f¨ur Kernphysik, Universit¨at M¨unster, D-48149 M¨unster, Germany Institut f¨ur Kernphysik and J¨ulich Centre for Hadron Physics,Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany Physics and Astronomy Department, UCL, London WC1E 6BT, United Kingdom (Dated: September 16, 2018)In order to measure the mass of the η meson with high accuracy using the dp → He η reaction, themomentum of the circulating deuteron beam in the Cooler Synchrotron COSY of the Forschungszen-trum J¨ulich has to be determined with unprecedented precision. This has been achieved by studyingthe spin dynamics of the polarized deuteron beam. By depolarizing the beam through the use of anartificially induced spin resonance, it was possible to evaluate its momentum p with a precision of∆ p/p < − for a momentum of roughly 3 GeV/ c . Different possible sources of error in the appli-cation of the spin-resonance method are discussed in detail and its possible use during a standardexperiment is considered. PACS numbers: 29.27.Bd, 29.27.Hj
I. INTRODUCTION
For numerous high precision experiments, knowing thebeam momentum in an accelerator with the greatest ac-curacy is essential. Obvious examples of this are investi-gations of production reactions very close to the thresh-olds as well as particle mass determinations on the basisof reaction kinematics. Here we present a technique thatallows one to determine the momentum of a deuteronbeam which is suitable for use in a precise measurementof the mass of the η meson.Measurements of the mass of the η meson performedat different experimental facilities over the last decadehave resulted in very precise results which differ by upto 0.5 MeV/ c , i.e., by more than eight standard devia-tions. The experiments that are no longer considered inthe PDG tables [1] generally involve the identification ofthe η as a missing-mass peak produced in a hadronic reac-tion. In order to see whether this is an intrinsic problem,and to clarify the situation more generally, a refined mea-surement of the dp → He η reaction was proposed [2] atthe Cooler Synchrotron COSY of the ForschungszentrumJ¨ulich [3].After producing the η mesons through the dp → He η reaction using a hydrogen cluster-jet target [4], the Hewould be detected with the ANKE magnetic spectrom-eter [5] that is located at an internal-target position ofthe storage ring. Provided that the reaction is cleanly iso-lated, the η mass can be extracted from pure kinematicsthrough the determination of the production threshold.This requires one both to identify the threshold and tomeasure accurately the associated beam momentum. ∗ Electronic address: [email protected]
We have previously proved that ANKE has essentially100% acceptance for the dp → He η reaction for excessenergies Q below about 10 MeV [6], though in that exper-iment the deuteron beam was continuously ramped frombelow the threshold up to Q ≈
11 MeV. However, al-though the threshold was well identified, the correspond-ing value of the beam momentum was only known in theexperiment with a relative accuracy of about 10 − .For the new η mass proposal [2], the decision wastaken to measure at thirteen fixed energies in the range1 < Q <
10 MeV as well as Q = − m η <
50 keV/ c [1],the associated beam momenta have to be fixed with anaccuracy of ∆ p/p < − . This requires the thirteenbeam momenta in the range of 3100 − c to bemeasured to better than 300 keV/ c .Generally at synchrotron facilities like COSY, the ve-locity of the beam particles, and hence the beam momen-tum, is determined from the knowledge of the revolutionfrequency combined with the absolute orbit length. Theaccuracy that can be reached using this technique is lim-ited by the measurement of the orbit length by, e.g., beamposition monitors. This is in the region of ∆ p/p ≈ − and so an order of magnitude improvement is needed forthe η mass experiment. Because of the technical limi-tations of such a macroscopic device, it is not feasibleto obtain the necessary increase in accuracy by simplyscaling up the number of beam pick-up electrodes. Thebeam momentum must therefore be determined in someother way.The method proposed for electron colliders more thanthirty years ago to overcome this problem [7, 8] has beenvery successfully applied at the VEPP accelerator of theBINP at Novosibirsk to measure the masses of a widevariety of mesons from the φ to the Υ [9]. The techniquewas further developed at DORIS in Hamburg [10] andCESR in Cornell [11] as well as LEP at CERN [12].The spin of a polarized beam particle precesses aroundthe normal to the plane of the machine, which is generallyhorizontal. The spin can be perturbed by the applica-tion of a horizontal rf magnetic field from, for example,a solenoid. The beam depolarizes when the frequency ofthe externally applied field coincides with that of the spinprecession in the ring. The usefulness of the techniquerelies on the fact that a frequency f can be routinelymeasured with a relative precision of ∆ f /f = 10 − . Fur-thermore, the position of the depolarizing resonance de-pends purely upon the revolution frequency of the ma-chine and the kinematical factor γ = E/mc , where E and m are the particle total energy and mass, respec-tively. The measurements of the revolution and depolar-izing frequencies together allow the evaluation of γ andhence E and the beam momentum p .There is no in-principle reason why the induced-depolarization approach should not be equally applica-ble to other beam particles with an intrinsic spin, suchas protons or deuterons. In fact, the effects have re-cently been confirmed at COSY in studies of the spinmanipulation of both polarized proton [13] and deuteronbeams [14]. This is the methodology that we are pursu-ing at COSY for the measurement of the η mass. Forthe first time in 2007 it was possible in a test run toreach an accuracy in the beam momentum calibration of∆ p/p < − using the technique with a coasting beambut no internal target [15]. In the present paper we de-scribe how the method can be used in a standard beamtime under normal experimental conditions in the pres-ence of a thick internal target.In Sec. II we describe the physical principles under-lying the spin-resonance method. After discussing thebehavior of a vector polarized deuteron beam in COSY,we show how to induce an artificial spin resonance to depolarize the beam. The experimental conditions thatallow one to determine the two critical observables areexplained in Sec. III. The revolution frequency f ismeasured via the Schottky noise of the beam and thespin-resonance frequency f r using the rf solenoid andthe EDDA detector as a beam polarimeter [16]. Thedeuteron beam results are presented in Sec. IV, wherethe estimated uncertainties are discussed in some detail.Our conclusions are summarized in Sec. V. II. THEORETICAL BACKGROUND OF THESPIN-RESONANCE METHODA. Spin in synchrotrons
In contrast to the case of a spin-half fermion suchas an electron or proton, the deuteron is a spin-oneboson that can be placed in three magnetic sub-states m = − , , +1, and the resulting polarization phe-nomenology is more complex. Eight independent param-eters are necessary to characterize a spin-one beam, threefor the vector polarization and five for the tensor [17].However, only the vector polarization P V = ( N + − N − ) /N , (1)is used in the present experiment for the spin-resonancemethod since it can be measured with the beam polarime-ter to a higher precision than the tensor. Here N m is thenumber of particles in state- m and N = N + + N − + N is the total number of particles.The motion of the spin vector ~S , defined in the restframe of the particle, in a circular accelerator, syn-chrotron or storage ring, is given by the Thomas-BMTequation [18]: d~Sdt = eγm ~S × " (1 + γG ) ~B ⊥ + (1 + G ) ~B || + (cid:18) Gγ + γγ + 1 (cid:19) ~E × ~βc , (2)where ~B ⊥ and ~B || are the transverse and longitudinalcomponents of the magnetic fields of the accelerator inthe laboratory frame and ~E represents the electric field.The velocity of the particle is ~βc , in terms of which γ =1 / p − β .In a synchrotron without horizontal magnetic fieldsand where the electric field is always parallel to theparticle motion, the spin motion only depends on thefirst term, i.e., is a function of the transverse magneticfields ~B ⊥ of the accelerator. The deuteron spin precessesaround the stable spin direction, which is given by the vertical fields of the guiding dipole magnets of the syn-chrotron. The number of spin precessions during a singlecircuit of the machine, the spin tune ν s , is proportional tothe particle energy. In the coordinate basis of the movingparticle, the spin tune is given by ν s = G γ , (3)whereas, taking into account the extra rotation associ-ated with a single circuit of the machine, this becomes ν s = 1 + G γ in the laboratory frame. Here G = ( g − / g isthe gyromagnetic factor. For deuterons the gyromag-netic anomaly, G d = − . ± . B. Artificially induced depolarizing resonances
The beam polarization can be perturbed by a horizon-tal magnetic field in the synchrotron and, if the frequencyof the perturbation coincides with the spin-precession fre-quency, the beam depolarizes. One kind of first-orderresonance is the imperfection resonance. If the spin tuneis an integer, then the horizontal imperfection fields ofthe synchrotron can interact resonantly with the particlespin, building up effects coherently turn by turn. Thepositions in momentum of the depolarizing resonancesdepend on the gyromagnetic anomaly of the particle. Incontrast to the case of protons, where the first imper-fection resonance occurs at a momentum of 464 MeV/ c ,the first for deuterons is at 13 GeV/ c , which is well out-side the COSY momentum range. Furthermore, in thepresent experiment the spin tune remains in the regionof ν s = 0 . − . rf field from a solenoid can lead to rf -induced depolarizing resonances. Depending on the formof the field, these can be used to depolarize the beam, tomeasure the spin tune, or even to flip the spin directionof the beam particles. The spin-resonance frequency fora planar accelerator where there are no horizontal fieldsis given by [8] f r = ( k + γ G ) f , (4)where f is the revolution frequency of the beam, γ G isthe spin tune, and k is an integer. If the rf frequencyof the perturbation is close to f r then the polarizationof the beam is maximally influenced. Horizontal mag-netic fields in the accelerator lead to modifications ofEqs. (3) and (4) [12, 20]. To avoid this complication,all solenoidal and toroidal magnets in the COSY ring,those of the experiment as well as those of the electroncooler, were switched off. Residual shifts in the resonancefrequency arising from field errors and vertical orbit dis-tortions were estimated and found to be negligibly small.These effects are discussed in more detail in Sec. IV C.It is important to note that Eq. (4) is only valid if, as isthe case for the present experiment, there is no full or par-tial Siberian snake. The resonance with k = 1 was usedas this matches the frequency range of the rf solenoidinstalled at COSY. The kinematic γ -factor, and thus the beam momentum, can be determined purely by measur-ing both the revolution and spin-resonance frequencies. III. EXPERIMENTAL CONDITIONS
The COSY accelerator facility is presented in Fig. 1.After pre-acceleration in the Cyclotron JULIC, COSYcan provide unpolarized and polarized proton anddeuteron beams in the momentum range of 300 − c . For the present experiment, two of thefour internal facilities were used, viz. ANKE with a thickhydrogen cluster-jet target and EDDA [16] as beam po-larimeter [21]. The beam was accelerated with the rf cavity and the barrier bucket ( bb ) cavity was used tocompensate for the energy losses incurred through thebeam-target interactions (see Sec. III A). The positionof the rf solenoid is also shown. The integrated value ofthe solenoid’s maximum longitudinal rf magnetic field is R B rms dl = 0 .
67 T mm at a rf voltage of 5.7 kV rms.Its frequency range is 0.5–1.5 MHz. A. The rf cavity system For a high precision experiment it is crucial that thebeam momentum remains stable throughout the wholeaccelerator cycle. In a typical cycle of a standard scat-tering experiment at ANKE, the beam is first injectedinto COSY and accelerated to the nominal momentum.The rf cavity is then switched off to provide a coastingbeam that fills the ring uniformly. This then gives con-stant count rates, which reduces the dead time of the dataacquisition system (DAQ). But, because of the energylosses of the charged beam particles through electromag-netic processes as the beam passes repeatedly throughthe target, the momentum changes and this leads to ashift in the revolution frequency [22]. For a deuteronbeam and a hydrogen cluster-jet target with a density of ρ = 1 × cm − , the revolution frequency would changeby up to 103 Hz over a 180 s long cycle, correspondingto a shift in beam momentum of 2.2 MeV/ c .To compensate for this effect and to guarantee a con-stant beam momentum over the whole data-taking cycle,a second cavity, the barrier bucket ( bb ) cavity [23], wasswitched on after the rf cavity was switched off. In thisway a beam with a constant momentum over the wholecycle could be produced that filled roughly 80 – 90% ofthe ring homogenously and thus achieved the necessaryreduction in the dead time of the DAQ. B. The cycle timing and the supercycle
The thirteen closely spaced energies studied near the η threshold were divided into two so-called supercyclesthat involved up to eight different COSY machine set-tings. The first and the second supercycle each consisted S t o c h a s t i c C oo l i n g FIG. 1: (Color online) The COSY accelerator facility. The cy-clotron JULIC provides both unpolarized and polarized pro-ton and deuteron beams for injection into the COSY ring,where they are accelerated and stored. COSY operates in themomentum range of 300 − c . The position of theANKE spectrometer with the thick internal hydrogen cluster-jet target is shown, as are those of the rf solenoid to depolarizethe deuteron beam, the barrier bucket cavity to compensatebeam-target energy losses, and the EDDA detector that wasused as a beam polarimeter. of seven different energies where, to allow comparisonbetween the two sets, the first energies of the two su-percycles were chosen to be identical. Data at thirteendifferent energies were therefore recorded. The differentmachine settings in the supercycles were imposed sequen-tially, after which the supercycle was repeated. Each su-percycle was used for five days of continuous Schottkydata taking to study the long term stability of COSYand to take data in parallel for the η meson mass deter-mination. The reason for choosing supercycles instead ofindependent measurements at fixed energies was to guar-antee the same experimental conditions for each of thebeam energies in one supercycle. In this way the system-atic uncertainties could be investigated in more detail, aswill be discussed in Sec. IV.Before starting each of the five day blocks, the individ-ual beam energies were measured using 36 s acceleratorcycle lengths. The timing structure of the acceleratorcycles is described in Table I. After the injection of the TABLE I: Cycle timings used to determine the spin-resonancefrequency spectrum with the polarized beam.Time (s) Process0 Start of cycle: injection0 – 3.7 Acceleration of the beam with rf cavity3.7 Switch off rf cavity4 Switch on bb cavity20 – 25 rf solenoid on25 – 30 Polarization measurement with EDDA36 End of cycle beam into COSY, the stored deuterons were acceleratedto the first nominal beam energy of the supercycle usingthe regular COSY rf cavity. At t = 3 . t = 4 s the bb cavity was broughtinto operation to compensate for the beam energy losses.At t = 20 s the amplitude of the depolarizing rf solenoidwas linearly ramped from 0 to 2.4 kV rms to produce a R B rms dl = 0 .
29 T mm in 200 ms, remained constantfor 5 s, and was then ramped down in 200 ms. Thiswas followed by a beam polarization measurement forfive seconds using the EDDA detector [16]. At t = 36 sthe cycle was terminated. This procedure was repeatedat the same beam energy but with different rf solenoidfrequencies in order to obtain the spin-resonance spec-trum. After completion of this first sub-measurement,the next beam energy of the supercycle was used and thecorresponding spin-resonance spectrum measured untilcomplete data was obtained at all the energies of the su-percycle.After measuring the spin-resonance spectrum, the su-percycle was switched on for five days of continuous datataking to investigate the long term stability of the COSYaccelerator. For this study the polarization measure-ments were omitted and total cycle lengths of 206 s wereused. After injection, acceleration and starting the bb cavity, Schottky measurements were performed over thetime interval of t = 14 −
196 s. The eight beam energies inone supercycle (the first one is installed twice) involveda total time of 1648 s, after which the supercycle wasrepeated. After the five days of data taking, the systemwas returned to the conditions of Table I to repeat themeasurement of the spin-resonance spectrum in order tocontrol systematic effects.The polarized ion source at the injector cyclotron ofCOSY currently gives a beam intensity that is about anorder of magnitude too low compared to that which is re-quired for the η mass proposal. It was therefore decidedto use this ion source only for the beam energy measure-ment before and after the supercycles. As a consequence,for the long term stability studies COSY was switched tothe unpolarized ion source, which allowed beam intensi-ties up to n d ≈ × . However, it had to be carefullychecked that the same COSY beam energies were ob-tained when using the polarized and the unpolarized ionsources. To ensure this, the complete settings of the cy-clotron, the beam injection, as well as COSY itself, werefixed when switching from one ion source to the other.The revolution frequencies f of the stored beam in thetwo cases matched to within ∆ f ≤ f waslimited in the present case by the experimental resolutionof the Schottky spectrum analyzer, though this could beimproved by better calibration.Both in the beam energy determination, as well as laterin the Schottky data-taking time, one had to be assuredthat the measurements within the cycles were startedsufficiently long after the ramping of the COSY dipolemagnets for the acceleration of the beam. Otherwise,the not-yet-stable magnetic fields would lead to devia-tions in the values determined for the beam momentum.Detailed measurements of the beam energy by the spin-resonance method as a function of the time in the cycleshowed that the experimental situation is already stableten seconds after the start of the cycle [24]. Therefore,the rf solenoid field and the Schottky data-taking werestarted 20 and 14 s after injection, respectively. As afurther check, measurements showed that the same beamenergy was observed close to the end of the cycle as atthe beginning [24] (see Sec. IV B). Thus it is valid toinvestigate the beam energy at one fixed time during thecycle and to take the resulting value as representative forthe whole cycle. C. Determination of the revolution frequency f via the Schottky noise measurements The revolution frequency f was measured by using theSchottky noise of the deuteron beam. The origin of thiseffect is the statistical distribution of the charged parti-cles in the beam. This leads to random current fluctua-tions that induce a voltage signal at a beam pick-up in thering. The Fourier transform of this voltage-to-time signalby a spectrum analyzer delivers the frequency distribu-tion around the harmonics of the revolution frequency ofthe beam.For the measurement of the Schottky noise, the beampick up and the spectrum analyzer of the stochastic cool-ing system of COSY were used. The spectrum analyzer(standard swept-type model HP 8595E) is sensitive tothe Schottky noise current, which is proportional to thesquare root of the number N of the particles in the beam.To get the Schottky power spectra, which represent themomentum distribution [25], the amplitudes of the mea-sured distribution were squared. The spectrum analyzerwas operated in the range of the thousandth harmonicbut, because of the different flat top revolution frequen-cies, harmonics from 997 to 1004 were also measured.The Schottky spectra were recorded every 30 sthroughout the whole beam time so that altogethernearly 15000 distributions were collected and sorted byenergy, i.e., by flat top. This large number of Schottky measurements allows the study of the long term stabilityof the revolution frequency, which will be discussed inSec IV. From all the spectra taken over five days thatwere measured under the same conditions at a particularenergy, one mean spectrum was calculated, an exampleof which is presented in Fig. 2. [MHz] revolution frequency f i n t e n s i t y [ a r b . un i t s ] FIG. 2: Mean Schottky power spectrum extracted from mea-surements over five days at one energy. The statistical er-ror bars lie within the data points. By calculating theweighted arithmetic mean, an average revolution frequencyof f = 1403831 . ± .
12 Hz was deduced.
The full width at half maximum is in the region of40 −
50 Hz for all energies. The position of the meandistribution of the circulation frequency is stable for thewhole cycle time but, within the cycle, a small tail is seenat lower frequencies. This corresponds to beam particlesthat escaped the influence of the bb cavity but still circu-lated in COSY. By calculating the weighted arithmeticmean of the revolution frequency distribution, an aver-age revolution frequency was estimated. The statisticaluncertainty of the mean revolution frequency, which isbelow 0.2 Hz for all energies, depends on both the num-ber of measured Schottky spectra and on the distributionvariations. D. Determination of the spin-resonance frequency f r via an induced spin resonance For all thirteen energies the spin-resonance spectrumwas measured twice, once before and once after the fivedays of Schottky data taking, as described in Sec. III B.The polarization of the beam leads to an asymmetry inscattering from a carbon target, which was measuredwith the EDDA detector [26]. For our purposes abso-lute calibrations of this device at the different energieswere not required; a quantity merely proportional to thepolarization such as the left-right asymmetry is sufficient.An example of a spin-resonance spectrum at one en-ergy is shown in Fig. 3. This displays the non-normalized solenoid frequency [MHz] r e l a t i ve po l a r i z a t i on FIG. 3: Spin-resonance measurements at one energy (closedcircles). The cycle timings are described in Table I. The opensymbols represent results obtained for an extended cycle time,where the perturbing solenoid was switched on after 178 s. polarization (“relative polarization”) as a function of thesolenoid frequency. Far away from the spin resonanceat 1.0116 MHz and 1.0120 MHz, a high beam polariza-tion was measured. In contrast, when the frequency ofthe solenoid coincided with the spin-precession frequency,the beam was maximally depolarized. The full width athalf maximum was in the region of 80-100 Hz for all ener-gies. Unlike the earlier spin-resonance test measurementwith a coasting beam, i.e., no cavities and no internaltarget [15], the spin-resonance spectra are not smooth.The structures, especially the double peak in the center,are caused by the interaction of the deuteron beam withthe bb cavity. However, by comparing the spin-resonancespectra measured for an unbunched and bunched beamwith accelerating cavity with h = 1 or the barrier bucketcavity, it was found that the centers of gravity of thespectra were the same.To study the shapes of the spin-resonance spectra inmore detail, all 26 distributions were fitted with gaus-sians and then shifted along the abscissa so that themean value of each individual spectrum was zero. Inaddition, each spectrum was shifted along the ordinateso that the off-resonance polarization vanished. Finally,the data were scaled to a uniform height and displayedtogether in a single plot to allow a comparison of all thespectra. The resulting global spin-resonance spectrumshown in Fig. 4(a) is symmetric around zero and smooth,except for the structure at the center. This region isshown in greater detail in the insert. In order to improvethe visibility of the structures close to the minimum, thesize of the frequency bins was increased and the resultsdisplayed in Fig. 4(b).A structure with a symmetric double peak and an os-cillation is observed in the center of the spin resonance.However, it is important to note that the gaussian meanvalue, i.e., the spin-resonance frequency, is not influencedby this structure. This was checked by making a fit where r e l a t i ve po l a r i z a t i on -1.6-1.4-1.2-1-0.8-0.6-0.4-0.200.2 -0.04-0.02 0 0.02 0.04-1.6-1.4-1.2-1-0.8-0.6-0.4 solenoid frequency (normalized) [kHz] -0.3 -0.2 -0.1 0 0.1 0.2 0.3-1.4-1.2-1-0.8-0.6-0.4-0.200.2 -0.06 -0.02 0.02 0.06-1.2-1-0.8-0.6 FIG. 4: Panel (a): The spin-resonance spectra normalized bya gaussian. Panel (b): The same but with larger bins. Thespin-resonance shape is symmetric about zero and smoothexcept in the center, where a double peak structure is seen.The structures, especially the double peak in the center, arecaused by the interaction of the deuteron beam with the bb cavity. The inserts show the resonance valley in greater detail. the data points at the center were excluded. The spin-resonance frequencies f r were extracted from the spin-resonance spectra for all energies by making gaussian fits.These gave χ /ndf in the region of 2–3. The statisticaluncertainties of the spin-resonance frequencies are on theorder of 1–2 Hz at f r ≈ .
01 MHz.
IV. RESULTSA. Stability of the revolution frequency f The bb cavity compensates the effects of beam-targetenergy losses and should ensure that the revolution fre-quency remains constant. The large number of Schottkymeasurements allowed us to study the long term stabilityand to identify the magnitude of the variations of the rev-olution frequency at COSY. Therefore all the Schottkyspectra at one energy from one day were analyzed andthe mean revolution frequency of that day calculated, asdescribed in Sec III C.In addition, the revolution frequencies for these datawere calculated for every four hours to study the dailyvariation of the circulation frequency. The differences be-tween the revolution frequencies of every four hours andthe mean frequency of the day are presented in the upperpart of Fig. 5. To study the variation of the revolutionfrequency over the five days of data taking, the same pro-cedure was carried out for the Schottky data measuredover this period. The differences between the mean rev-olution frequencies of every day and the mean frequencyof the whole five days of data taking are presented inthe bottom part of Fig. 5. The horizontal bars representthe time intervals for which the revolution frequency wasevaluated. time [hour] [ H z ] - f hou r f -1-0.500.51 time [day] [ H z ] - f d ay f -1-0.500.51 FIG. 5: Stability of the revolution frequency f . In panel (a)the differences between the revolution frequencies for everyfour hours and the mean revolution frequency of the day areshown. In panel (b) the differences between the revolutionfrequencies for each day and the mean revolution frequencyof the five days of Schottky data taking are shown. Fromthese figures it is clear that the revolution frequency at COSYis very stable, with variations below 1 Hz at a circulationfrequency of f ≈ . The analysis shows that the revolution frequency atCOSY over one day and also over five days is very sta- ble. The variations of the revolution frequency are verysmall, being on the order of 1 Hz at a circulation fre-quency of f ≈ . f = 6 Hz dom-inated the precision, and this arose from the preparationof the Schottky spectrum analyzer used. A more refinedcalibration of this device could improve the systematicprecision of the circulation frequency measurement downto 1 Hz. B. Spin-resonance frequency f r It is important for the interpretation of the spin-resonance measurements to know to what extent the po-sitions of the observed spin-resonance frequencies are sta-ble over the finite accelerator cycle in the presence of athick internal target. Therefore, in a special measure-ment, the switch-on of the rf solenoid was delayed from20 s to 178 s in order to investigate the position of thespin-resonance frequency close to the end of a long cy-cle. The observed data (open symbols of Fig. 3) showeda resonance position which agreed with the data taken atthe beginning of the cycle to within 2 Hz. energy settings s p i n r es on a n ce f r e q . s h i ft [ H z ] -30-25-20-15-10-505101520 SuperCycle 1 SuperCycle 2 s h i ft o f o r b i t l e ng t h [ mm ] -6-5-4-3-2-101234 FIG. 6: (Color online) The spin-resonance frequencies weremeasured twice, once before and once after the five days ofdata taking. The red triangles present the shift of the spin-resonance frequency f r from the first to the second measure-ment. These shifts correspond to changes in the orbit length,which are shown as blue circles. For the first supercycle, thespin-resonance frequencies decrease between the two measure-ments by 4 −
10 Hz, which corresponds to a increase in theorbit length in the range of 0 . − . −
17 Hz was observed, i.e., a decrease in the orbit lengthin the range of 2 . − . In Fig. 6 the shifts between the first and second spin-resonance measurements are shown as red triangles for allthirteen energies. The frequencies in the first supercycledecrease by between 4 and 10 Hz for all energies, whereasfor the second supercycle they increase in the range of12 −
17 Hz. These systematic shifts of the frequencies inthe same direction indicate slight changes in the COSYsettings. Because the revolution frequency is stable, asdescribed in Sec. IV A, the change is attributed to a shiftin the orbit length s .The velocity v of the particle is the product of therevolution frequency and the orbit length v = s f . Us-ing Eq. (4), the orbit length can be calculated from therevolution and the spin-resonance frequencies: s = c " f − (cid:18) G d f r − f (cid:19) , (5)which allows the orbit lengths to be extracted with aprecision better than 0 . . s/s × − . The uncertaintyis dominated by that of the spin-resonance frequency.The shift in the spin-resonance frequency corresponds toa change in the orbit length of up to 3 mm, which ispresented for all energies in Fig. 6 as blue circles. Theshifts of the spin-resonance frequencies of the first su-percycle suggest an increase in the orbit length in therange of 0 . − . . − . f r = 15 Hz was assumed. C. Accuracy and systematic shifts of the resonancefrequency
One obvious limitation on the spin-resonance methodis given by the uncertainty in the deuteron gyromagneticanomaly G d . However, this leads to a relative precisionin the beam momentum of ∆ p/p = 5 × − , which canbe safely neglected.The first order uncertainties in the momentum mea-surement depend on the accuracies to which the spin-resonance and revolution frequencies are determined.As described in Sec. IV B and Sec. IV A, these are15 Hz / .
01 MHz = 1 . × − and 6 . / .
40 MHz =4 . × − , respectively. The error therefore arises pri-marily from the measurement of the spin-resonance fre-quency.The intrinsic width of the spin-resonance may also im-pose a limit on the accuracy achievable. In this exper-iment, the integrated value of the solenoid’s maximumlongitudinal rf magnetic field gives a resonance strengthof about ǫ = 3 × − , which leads to a spin resonancewith a FWHM width ≈ TABLE II: Accuracy and possible systematic shifts of theresonance frequency f r .Source ∆ f r /f r Resonance frequency accuracy fromdepolarization spectra 1 . × − Spin tune shifts from longitudinal fields(field errors) 1 . × − Spin tune shifts from radial fields(field errors, vertical correctors) 6 . × − Spin tune shifts from radial fields(vertical orbit in quadrupoles) 4 . × − Radial and longitudinal fields in the accelerator maylead to a modification of Eq. (4) [20], i.e., to a sys-tematic shift of the resonance frequency. Even thoughall solenoidal and toroidal fields, which may act as par-tial Siberian snakes, were turned off for this experiment,field errors and vertical orbit distortions could generatesome net radial or longitudinal fields [9, 12]. These ef-fects were estimated for the current experimental con-ditions and found to be negligibly small. The typicalfield errors of the main magnets, ∆
B/B ≈ × − ,would lead to a shift in the spin-resonance frequency of∆ f r /f r < . × − . Similarly, the observed verticalorbit displacement of ∆ y rms < . f r /f r < . × − .The largest contribution to a systematic shift of theresonance frequency could come from the vertical closedorbit deviations in the quadrupole magnets of the ring.However, this contribution of ∆ f r /f r = 4 × − is com-parable to the in-principle limitation of the method aris-ing the knowledge of the deuteron G -factor. It is overtwo orders of magnitude below the accuracy achieved inthe experiment. D. Determination of the deuteron beam momenta p and the momentum smearing ∆ p/p The deuteron kinematic γ -factor and the beam mo-menta were calculated according to Eq. (6) γ = 1 G d (cid:18) f r f − (cid:19) p = m d β γ = m d p γ − f = 6 Hz, corresponding to one in the beam momen-tum of 50 keV/ c . The error in the determination of thespin resonance frequency ∆ f r = 15 Hz arises from thesmall variations of the orbit length and corresponds toan uncertainty in the beam momentum of 164 keV/ c .Because these systematic uncertainties are independent,they are added quadratically to give a total uncertainty∆ p/p × − , i.e., a precision of 170 keV/ c for beammomenta in the range of 3100 − c . This is overan order of magnitude better than ever reached before fora standard experiment in the COSY ring. An exampleof the reconstructed beam properties is presented in Ta-ble III for one typical energy setting. The measured beammomentum differed by ≈ c from the nominal re-quested momentum. TABLE III: Typical results for one beam setting.Nominal beam momentum 3150.5 [MeV/ c ]Revolution frequency 1403832 ± ±
15 [Hz]Orbit length 183 . ± . γ factor 1 . ± . . ± .
17 [MeV/ c ] Two further quantities, the beam momentum smear-ing δp/p and the smearing of the orbit length δs/s , canbe extracted from the spin-resonance spectra. As dis-cussed in Sec. IV C, the measured spin-resonance widthsof 80 to 100 Hz are dominated by the momentum spread.Assuming a gaussian distribution in the revolution fre-quency with a FWHM = 40 −
50 Hz, and neglecting othereffects, the width of the spin-resonance distribution re-quires a momentum spread of ( δp/p ) rms ≈ × − . Thisupper limit on the beam momentum width correspondsto a smearing of the orbit length of ( δs/s ) rms ≈ × − .The momentum spread could be checked from the fre-quency slip factor η , which was measured at each energy.Using δp/p = 1 /η × ( δf /f ), this leads for example at p nominal = 3 . c to ( δp/p ) rms = 1 . × − , which is consistent with the limit obtained from the resonancedistribution. V. CONCLUSIONS AND OUTLOOK
In this paper we have shown how to determine the mo-mentum of a deuteron beam in a circular accelerator withhigh precision using the spin-resonance technique devel-oped at the VEPP accelerator for electron beams. Wehave studied the depolarization of a polarized deuteronbeam at COSY through an induced spin resonance forthirteen different beam energies. This was done understandard experimental conditions, i.e., with cavities, inparticular the bb cavity, and a thick internal cluster-jettarget. The momenta and other beam properties werefound by measuring the position of the spin-resonanceand revolution frequencies.It was possible to determine the beam momenta withan accuracy of ∆ p/p × − , i.e., the thirteen mo-menta in the range 3100 − c were measuredwith precisions of ≈
170 keV/ c , a feat never beforeachieved at COSY. The actual precision was limited bythe systematic variations of the orbit length and the char-acteristics of the Schottky spectrum analyzer. The lattercould be improved significantly through the comparisonwith a calibrated frequency standard.The orbit length could be extracted from the revolu-tion and spin-resonance frequencies with an accuracy of∆ s/s × − . Thus for COSY, with a circumferenceof 183.4 m, the orbit length could be measured with aprecision below 0.3 mm. This may allow one to gain abetter knowledge of the orbit behavior in COSY.These results were achieved using a deuteron beam,but there are no in-principle reasons why the depolariza-tion technique should not be applicable to proton beamsat COSY with same success.In summary, the spin-resonance method is a power-ful beam diagnostic tool for circular accelerators, syn-chrotrons or storage rings without Siberian snakes to in-vestigate and determine beam properties. In our particu-lar case it should eventually allow the mass of the η mesonto be measured with a precision of ∆ m η
50 keV/ c . Acknowledgments
The authors wish to express their thanks to the othermembers of the COSY machine crew for producing suchgood experimental conditions and also to the other mem-bers of the ANKE collaboration for diverse help in theexperiment. The spin-depolarizing studies for deuteronsand protons were initiated at COSY by Alan Krisch andother members of the SPIN@COSY collaboration and wehave benefited much from their experience. This workwas supported in part by the HGF-VIQCD, and JCHPFEE.0 [1] C. Amsler et al. [Particle Data Group], Phys. Lett. B , 1 (2008).[2] A. Khoukaz, COSY proposal
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