Precision measurements of differential cross sections and analyzing powers in elastic deuteron-deuteron scattering at 65 MeV/nucleon
R. Ramazani-Sharifabadi, A. Ramazani-Moghaddam-Arani, H.R. Amir-Ahmadi, C. D. Bailey, A. Deltuva, M. Eslami-Kalantari, N. Kalantar-Nayestanaki, St. Kistryn, A. Kozela, M. Mahjour-Shafiei, H. Mardanpour, J.G. Messchendorp, M. Mohammadi-Dadkan, E. Stephan, E. J. Stephenson, H. Tavakoli-Zaniani
EEPJ manuscript No. (will be inserted by the editor)
Precision measurements of differential cross sections andanalyzing powers in elastic deuteron-deuteron scattering at 65MeV/nucleon
R. Ramazani-Sharifabadi , , A. Ramazani-Moghaddam-Arani , H.R. Amir-Ahmadi , C. D. Bailey , A. Deltuva ,M. Eslami-Kalantari , N. Kalantar-Nayestanaki , St. Kistryn , A. Kozela , M. Mahjour-Shafiei , H. Mardanpour ,J.G. Messchendorp , M. Mohammadi-Dadkan , , E. Stephan , E. J. Stephenson , and H. Tavakoli-Zaniani , Department of Physics, University of Tehran, Tehran, Iran KVI-CART, University of Groningen, Groningen, The Netherlands Department of Nuclear Physics, Faculty of Physics, University of Kashan, Kashan, Iran Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47408 USA Institute of Theoretical Physics and Astronomy, Vilnius University, Vilnius, Lithuania Department of Physics, School of Science, Yazd University, Yazd, Iran Institute of Physics, Jagiellonian University, Krak´ow, Poland Institute of Nuclear Physics, PAS, Krak´ow, Poland Department of Physics, University of Sistan and Baluchestan, Zahedan, Iran Institute of Physics, University of Silesia, Chorz´ow, PolandReceived: date / Revised version: date
Abstract.
We present measurements of differential cross sections and analyzing powers for the elastic H( (cid:126)d, d ) d scattering process. The data were obtained using a 130 MeV polarized deuteron beam. Crosssections and spin observables of the elastic scattering process were measured at the AGOR facility atKVI using two independent setups, namely BINA and BBS. The data harvest at setups are in excellentagreement with each other and allowed us to carry out a thorough systematic analysis to provide the mostaccurate data in elastic deuteron-deuteron scattering at intermediate energies. The results can be used toconfront upcoming state-of-the-art calculations in the four-nucleon scattering domain, and will, thereby,provide further insights in the dynamics of three- and four-nucleon forces in few-nucleon systems. Key words. deuteron-deuteron scattering – elastic channel – vector and tensor analyzing powers – nuclearforces
PACS.
Understanding the degrees of freedom that describe nu-clear forces is of great importance to make progress in nu-clear physics. The first major breakthrough came in 1935when Yukawa presented the description of the nucleon-nucleon force by the exchange of massive mesons [1] inanalogy to the exchange of massless photons describ-ing successfully the electromagnetic interaction. More re-cently, various phenomenological nucleon-nucleon (NN)potentials have been derived based on Yukawa’s idea.Some of these potentials were successfully linked to the un-derlying fundamental theory of quantum chromodynam- a reza [email protected] b [email protected] c Present address: American Physical Society, 1 Physics El-lipse, College Park, MD 20240 USA ics [2,3]. Precision measurements obtained from nucleon-nucleon scattering data are strikingly well described bythese modern NN potentials [4].It is compelling to apply the high-precision NN po-tentials to systems composed of at least three nucleons.Rigorous Faddeev calculations of the binding energy ofthe simplest three-nucleon system, triton, underestimatethe experimental data [5]. This observation shows thatcalculations based solely on NN potentials are not suffi-cient to describe systems that involve more than two nu-cleons. This has led to the notion of the three-nucleonforce (3NF), a concept that was introduced already inthe early days of nuclear physics by Primakoff and Hol-stein [6]. Green’s function Monte Carlo calculations basedon the AV18 NN potential complemented with the IL7three-nucleon potential demonstrated the necessity of the3NF to describe the experimental data for the binding a r X i v : . [ nu c l - e x ] J a n R. Ramazani-Sharifabadi et al.: Title Suppressed Due to Excessive Length energies of light nuclei [7]. Moreover, rigorous Faddeevcalculations based on modern NN potentials show largediscrepancies with cross section data in elastic nucleon-deuteron scattering. The inclusion of 3NF effects partlyresolves these deficiencies [8]. There are, by now, a largenumber of evidences revealing the importance of 3NF ef-fects.In the last decades, many nucleon-deuteron elas-tic [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27] and breakup [28,29,30,31,32,33,34,35,36,37] scat-tering experiments at various energies below the pion-production threshold have provided an extensive databasefor the study 3NF effects. The addition of 3NF effects,in particular the role of the ∆ resonance, reduces signifi-cantly the discrepancies between differential cross-sectiondata and corresponding calculations excluding 3NF ef-fects. The situation for spin observables is vastly different.For instance, the inclusion of 3NF effects for the vectoranalyzing power of the elastic channel at the intermedi-ate energies gives a better agreement between data andtheory, while for the tensor analyzing power, Re ( T ), thediscrepancies are not removed by adding 3NF effects inthe model [39]. The inclusion of 3NF effects even deteri-orates the agreement between model predictions and thedata for the vector analyzing power of the proton in theproton-deuteron breakup reaction at configurations thatcorrespond to small relative energies between the two out-going protons [39]. These observations imply that spin-dependent parts of 3NF effects are not yet well under-stood [38,39].Although the three-nucleon (3N) system is the clean-est system to study 3NF effects since only NN and 3Nforces can contribute and observables can be calculated inan ab-initio manner, the influence of 3NF effects are ingeneral small in a 3N system. Only at specific parts of thephase space in three-nucleon scattering processes, 3NF ef-fects become significant. A well-known example of sucha phase space appears at scattering angles correspondingto the minimum of the differential cross section in elas-tic N d scattering [8,40]. In spin observables, a significant3NF effect can also be seen for pd break-up configura-tions corresponding to small relative energies between thetwo outgoing protons [39]. Alternatively, and this is thefocus of this paper, one may investigate the four-nucleon(4N) system in which 3NF effects could be significantlyenhanced [39]. Deuteron-deuteron scattering, as a 4N sys-tem, is a rich laboratory to study 3NF effects because of itsvariety of final states, observables, and kinematical con-figurations. Compared to the amount of available data inthe 3N scattering domain, the database in the 4N systemis very limited. Most of the 4N data cover the very low-energy regime, below the three- and four-body breakupthreshold [41,42,43]. Although, calculations at these lowenergies are very reliable, the effect of the 3NF is verysmall. Therefore, the low-energy realm is not the most at-tractive regime to study rigorously the dynamics of 3NFs. Ab-initio theoretical calculations for four nucleonsystems are still limited to beam energies below 40MeV [44,45,46,47,48,49,50,51]. At intermediate energies, below the pion-production threshold, the 4N experimentaldatabase is very scarce [52,53,54]. Despite the fact that ab-initio calculations are still in development in this en-ergy regime, the prospects of studying the structure of 3Nforces, and possibly higher-order four-nucleon force effects,look promising [55,56]. Recent theoretical approximationsfor deuteron-deuteron scattering are able to reasonablypredict the experimental results in the quasi-free (QF)regime [57,58]. However, one should consider the final-state interactions of spectator neutrons to identify the QFlimit correctly [34]. Besides, charge symmetry breakingstudies (CSB) in d + d → He + π reaction reveal thenecessity of theoretical calculations in dd elastic scatter-ing process to provide an unambiguous formulation of theinitial-state interaction. In this energy regime, a single-scattering approximation is used in which one nucleonscatters from the opposite deuteron before it recombinesto reform the original deuteron [59,60].This paper presents measurements of the differen-tial cross section and spin observables in the H( (cid:126)d, d ) d elastic scattering process for a deuteron-beam energy of65 MeV/nucleon. The data were obtained by making useof a vector- and tensor-polarized deuteron beam that wasprovided by the AGOR facility at KVI in Groningen,the Netherlands. Two experimental equipments, locatedat two different beam lines, were used to measure inde-pendently the various observables in H( (cid:126)d, d ) d scatter-ing, namely the Big-Bite Spectrometer (BBS) and theBig Instrument for Nuclear-Polarization Analysis (BINA).These setups bring complementary features: one (BINA)covering large phase space, particularly in φ , using a liquiddeuterium target leading to less background. The other(BBS) possesses an excellent momentum resolution, butwith moderate coverage, using a solid CD target withmore precise knowledge on the target thickness at the costof a larger background. These two sets of measurementscombined have provided a good experimental databasethat can be used as a benchmark for future ab-initio cal-culations. This paper addresses the analysis of these twoindependent datasets. The results presented here are themost precise and accurate data of the H( (cid:126)d, d ) d process atintermediate energies. This experiment was performed with two different setups,BINA and BBS. In the following, details of both setupsrelevant for the present paper will be presented. Detaileddiscriptions are presented in [61,62], respectively.
The two experiments were conducted using AGOR facilityat KVI. The measurement on BINA took place the weekafter the BBS data taking. BINA has the ability to iden-tify and measure all reaction channels of the deuteron-deuteron scattering process simultaneously, while BBS . Ramazani-Sharifabadi et al.: Title Suppressed Due to Excessive Length 3 measures the hadronic channels with particles emerg-ing from the two-body final states. Vector- and tensor-polarized (unpolarized) deuteron beams were producedby the atomic Polarized Ion Source (POLIS) [63,64] withnominal polarization values between 60-80% of the the-oretical values and accelerated by the AGOR cyclotronto energies of 65 MeV/nucleon. The polarization of thedeuteron beam was monitored for different periods of theexperiment and found to be stable within statistical un-certainties [65].
Figure 1 shows a sketch of BINA. The setup consistsof two parts, a forward wall and a backward ball.The forward wall consists of a multi-wire proportionalchamber (MWPC) to determine the scattering anglesof the particles, twelve-vertically mounted plastic ∆ E-scintillators with a thickness of 2 mm, and ten horizontallymounted E-scintillators with a thickness of 12 cm. TheE-scintillators are placed in a cylindrical shape where thetarget is positioned on the axial symmetry of the cylinder.Although, the ∆ E-E hodoscope provides the possibility toperform particle identification, the information from the ∆ E detector was not used in this experiment. In a visualinspection after the experiment, these scintillators wereobserved to be damaged. Therefore, the ∆ E-E detectorscould not provide the PID information for all scatteringangles. Photomultiplier tubes (PMTs) were mountedon both sides of each E-scintillator. Signals from thesePMTs are used to extract the energy and time-of-flight(TOF) of each scattered particle. The TOF informationis used to perform PID. The MWPC covers scatteringangles between 10 ◦ and 32 ◦ with a full azimuthal anglecoverage and up to 37 ◦ with a limited azimuthal anglecoverage. The MWPC has a resolution of 0 . ◦ for thepolar angle and between 0 . ◦ and 2 . ◦ for the azimuthalangle depending on the scattering angle. The detectionefficiency of the MWPC for deuteron with energiescorresponding to the reaction of interest is typically98 ±
1% [69]. The backward ball of BINA is made of 149phoswich scintillators that were simultaneously used asdetector and scattering chamber with a scattering-anglecoverage between 40 ◦ and 165 ◦ and nearly full azimuthalcoverage. For more details on BINA, we refer to [61,66].The deuteron beam, with a typical current of 4 pA,bombarded a liquid-deuterium target that was mountedinside the scattering chamber of BINA [67]. The thicknessof the target cell was 3.85 mm with an uncertainty of5%. The scattering angles, energies, and (partly) timeof flights of the final-state deuterons were measuredby the multi-wire proportional chamber (MWPC) andscintillators of BINA. A Faraday cup was mounted atthe end of the beam line to monitor the beam currentthroughout the experiment. The current meter of theFaraday cup was calibrated using a current source withan uncertainty of 2% [65]. A small offset in the readoutof the current was observed with a value around 0.28 ± Forward WallTargetBeamBackward Ball EMWPC ΔΕ Fig. 1.
A sketch of the various components of the BINA setup.The elements on the right side show a side view of the forwardpart of BINA, including the multi-wire proportional cham-ber (MWPC), an array of twelve thin plastic ∆ E-scintillatorsfollowed by ten thick segmented E-scintillators mounted in acylindrical shape. On the left side, the backward part of BINAis depicted composed of 149 phoswich scintillators glued to-gether to form the scattering chamber.
The Big-Bite Spectrometer (BBS) is a QQD-type mag-netic spectrometer with a K -value of 430 MeV and a solidangle of up to 13 msr. By changing the position of thequadrupole doublet with respect to the dipole magnet,while the distance between the object (target) and thedipole remains the same, the momentum-bite acceptancecan be changed from 13 to 25%, the solid angle changesfrom 13 to 7 msr, simultaneously. The BBS consists of ascattering chamber containing a target ladder, a large slitwheel containing several entrance apertures (including asieve slit for angle reconstruction), two sets of quadrupolemagnets for beam focusing, a large dipole magnet for mo-mentum selection, two sets of x-u plane wire-chamber de-tectors, and a scintillator plane which is used to generatethe event trigger. A diagram of the BBS is shown in Fig. 2.In the BBS setup, different thick or thin sets of CD and carbon targets were used for different ranges of lab an-gles. The carbon targets were used in the forward range ofspectrometer angles to be able to subtract the backgroundgenerated by deuterons elastically scattered from carbonin the CD target. For large angles ( ≥ ◦ ), several layersof solid CD were combined, resulting in a total thicknessof 45.15 ± . For small angles (4 ◦ and 6 ◦ ) R. Ramazani-Sharifabadi et al.: Title Suppressed Due to Excessive Length (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
MWPCs D1−D4Carbon analyzer CScintillators S1, S2VDCs 1 and 2 2107 621 270324 270
FPPFPDSQ2 D c m target Q1 Fig. 2.
A sketch of the main features of the BBS setup. the CD target thickness was 10.49 ± . Thethickness of the carbon target for large angles ( ≥ ◦ )was 46.80 ± ; for small angles it was 14.2 ± . The scattering chamber of the BBS con-sisted of a large cylindrical chamber containing the targetsand essentially forming the pivot point around which thedevice covers the scattering angles between 4 ◦ and 48 ◦ during the data taking. For the beam integration, a largecopper Faraday cup was used for the angles larger than15 ◦ , where the unscattered beam could hit the wall of thescattering chamber. For small scattering angles (less than15 ◦ ), the unscattered beam was within the acceptance ofBBS and entered the region of the quadrupole magnets.Therefore, a separate Faraday cup is mounted between thequadrupole magnets Q1 and Q2 for the beam integrationin this region. A detailed description of the BBS setup ispresented in [68]. In this section, the analysis procedures of both experi-ments related to the BINA and BBS setups are describedseparately. Detailed discriptions are presented in [61,62],respectively.
During data taking with BINA, various hardware trig-gers with different down-scale factors were implementedthat were dedicated to a specific hadronic final state in deuteron-deuteron scattering. To select events originat-ing from elastically scattered deuterons, two triggers wereof importance. The first one, referred to as the coinci-dence trigger, registered events for which there was atleast one signal from the forward wall scintillators in coin-cidence with at least one signal originating from the back-ward wall. This trigger was down-scaled by a factor two.The hardware thresholds for detection of a particle weretypically set around 1 MeV. Although, with the coinci-dence trigger we were able to cover a large part of theangular distribution of the H( (cid:126)d, d ) d reaction, wherebyboth deuterons in the final state were detected, we ob-served a significant drop in the detection efficiency forlow-energetic deuterons that scatter towards the backwardball due to energy losses of those particles in the liquid-deuterium target. The data selected with the coincidencetrigger were used to extract the spin observables, sincedetection inefficiencies cancel out in the analysis. To ex-tract the differential cross section, we exploited a secondtrigger the so-called, “single trigger”. This trigger, down-scaled by a factor 256, was built from a logical OR of allthe discriminated signals of the scintillators of BINA, and,thereby, not biased on the response of the backward ball.The data from the coincidence trigger were calibratedand further preprocessed by requiring that the relative an-gles of the reconstructed particles hitting the forward walland backward ball match the correlation that is expectedfrom kinematical considerations for the elastic deuteron-deuteron scattering process. Cuts were applied to meet arelative opening angle of 83 ◦ and a coplanar configura-tion with respect to the azimuthal angles. After applyingthese angular cuts with a window of ± ◦ , a major reduc-tion (around 75%) of backgrounds from other hadronic fi-nal states, such as breakup and nucleon-transfer reactions,was obtained. Figure 3 shows the correlation between en-ergy and scattering angle of deuterons detected in the for-ward wall of BINA after the aforementioned event selec-tion. The solid line represents the expected kinematical lo-cus for the elastic deuteron-deuteron scattering. As seen,elastically scattered deuterons can easily be observed anddistinguished from background channels. The data belowthe elastic events reveal another clear correlation whichhas been identified as events belonging to the neutron-transfer channel, H( (cid:126)d, H) p .To count the number of events that originate fromthe elastic process, the center-of-mass energy for each re-constructed particle is calculated from its energy depositand scattering angle, and a corresponding histogram isgenerated in intervals of 2 ◦ of the scattering angle andseparated for the various polarization states of the beam.Figure 4 depicts the center-of-mass energy distributionthat has been obtained using the single trigger. The upperspectrum shows the raw response after calibration and forparticles that scatter to 26 ± ◦ but without any furtherconditions. For the lower spectrum, a coincidence with thebackward ball was required in addition using the kine-matical cuts discussed earlier but from data taken withthe single trigger. The solid lines are the result of a fitthrough the data based on a Gaussian-distributed signal . Ramazani-Sharifabadi et al.: Title Suppressed Due to Excessive Length 5 [deg] θ
15 20 25 30 35 [ M e V ] d E Counts
Fig. 3.
The correlation between the reconstructed energyand scattering angle of the particles that were detected in theforward part of BINA with a coincidence requirement with thebackward ball. The solid line represents the kinematical locusfor the elastic deuteron-deuteron scattering process. [MeV] c.m.
E24 26 28 30 32 34 36 C oun t s [ ~ M e v / b i n ] Fig. 4.
Spectrum of the center-of-mass energy of particles hit-ting the E detectors of the forward wall. Data are obtainedusing the single trigger. The scattered particles are confinedto polar angles of 26 ± ◦ . For the lower spectrum, a coinci-dence condition is imposed in the event selection. The solidlines show the results of a least- χ fit based on a Gaussian(signal) and a 5 th -order polynomial (background) distribution.The background contribution is indicated by the dashed lines.The χ /64 of the fit is 1.4 for the upper spectrum and 1.2 forthe lower one. combined with a 5 th -order polynomial representing thebackgrounds. The background component of the fit is indi-cated by the dashed lines. A clear peak can be observed inboth cases, corresponding unambiguously to the channelof interest. The difference between the integrals of the sig-nal distributions before and after applying the coincidencecondition excluding inefficiencies of the ball is less than2%. The coincidence requirement reduces significantly thebackground contribution. Monte Carlo simulations showedthat the remaining background is mostly due to hadronicinteractions of elastically scattered deuterons in the scin-tillator.To extract cross sections, the number of counts pass-ing the kinematical criteria has been corrected for effi-ciencies of the system such as live-time, MWPC efficien-cies, hadronic interactions, and the down-scale factor thatcomes from triggers. The average live-time of the data ac- quisition of BINA is around 40%.Events from the elastic reaction that suffered fromhadronic interactions do not give a clear peaking struc-ture in the energy spectrum, and are, therefore, not eas-ily separated from other background channels, we did notcount these events and corrected for their loss. Using aGEANT3-based Monte Carlo simulation, we estimated aloss of 16 ±
2% for the energy range of interest. The crosssections are corrected for this effect accordingly.Vector and tensor polarized beams make it possible tomeasure spin observables. Using parity conservation, thecross section for H( (cid:126)d, d ) d reaction is given by the follow-ing equation [72]: σ ( θ, φ ) σ ( θ ) = k (cid:34) p Z A y ( θ ) cos( φ ) − p ZZ A zz ( θ )+ 14 p ZZ (cid:16) A zz ( θ ) + 2 A yy ( θ ) (cid:17) cos(2 φ ) (cid:35) , (1)where θ and φ are polar and azimuthal angles of the scat-tered deuteron, respectively. A y is the vector analyzingpower, while A zz and A yy are the tensor analyzing powers. p Z ( p ZZ ) represents the vector (tensor) polarization ofthe beam. σ ( σ ) is the effective cross section obtained fordata taken with (un)polarized beam. These effective crosssections correspond to the number of counts normalizedby the accumulated and dead-time corrected charge.Please note that in first order, the efficiencies cancelby taking the ratio between σ ( θ, φ ) and σ ( θ ). Finally, k is a normalization factor and should be equal to onein the ideal case. Considering k as a free parameter, itfluctuates around one with a value of k = 1 . ± .
03 thatis considered as a systematic uncertainty for the normal-ization procedure. Experimentally, however, we evaluatedpossible systematical differences in the extraction of theeffective cross sections σ ( θ, φ ) and σ ( θ ) accommodatedin k . These may be due to small differences in detectionefficiencies or beam-current measurements between datataken with unpolarized and polarized beams. For theextraction of the analyzing powers, we analyzed datataken with the coincidence trigger and enforcing theselection criteria as described above. We note that thebackground using the coincidence conditions is verysmall.We extracted the analyzing powers with two differentmethods which both lead to compatible results withinthe uncertainties. In the first method; we assume that thebeam polarization is purely vector ( p Z (cid:54) = 0 and p ZZ = 0)or purely tensor ( p Z = 0 and p ZZ (cid:54) = 0). Therefore,in Eq. 1, the corresponding terms are kept and theother terms are set to zero. As can be seen in Eq. 1, theasymmetry ratio of polarized to un-polarized cross sectionis a function of cos φ (cos 2 φ ) for the case of pure vector(tensor) polarized beam, see Fig. 5. Therefore, vectoranalyzing power, A y , is extracted from the amplitude ofcos φ . In the same way, tensor analyzing powers of A zz and A yy are extracted from the off-set of cos 2 φ from one andits amplitude, respectively. To estimate the systematicuncertainty due to the possible impurity in the vector- R. Ramazani-Sharifabadi et al.: Title Suppressed Due to Excessive Length σ / σ [deg] φ σ / σ Fig. 5.
Asymmetry ratio of cross section for polarized over un-polarized beam as a function of φ for a pure-vector polarizedbeam (top panel) and pure tensor polarized beam (bottompanel. Scattering angle of elastically scattered deuteron is 26 ± ◦ . The reduced χ for the top (bottom) panel is 1.04 (0.97). (tensor-)polarized beam, the second method is applied.In the second method, we suppose that the pure-vector(tensor) polarized beam is not actually a pure-vector(tensor), ( p Z (cid:54) = 0 and p ZZ (cid:54) = 0). In other words, thepure-vector (tensor) polarized beam is contaminated withanother polarization, say the tensor (vector) polarization.Therefore, Eq. 1 including all the terms is used to fit tothe asymmetry ratio of polarized to un-polarized crosssection beam for vector and tensor analyzing powers.As described before, the analyzing powers can again beextracted from the amplitudes of the cos φ and cos 2 φ functions as well as the offset of cos 2 φ function from one.The difference between the two results is considered asthe systematic uncertainty due to the possible impurityin the beam polarization. The results of the first methodis considered as the final results, see Sec. 5.To verify the procedure of extracting the differentialcross sections and analyzing powers of the H( (cid:126)d, d ) d reaction, we measured and analyzed the H( (cid:126)d, dp ) reactionas well. The same procedure was used to analyze the dataof the well-studied H( (cid:126)d, dp ) reaction which were obtainedusing a CH target and with the same setup and beamconditions as was applied in the study of the H( (cid:126)d, d ) d reaction.Differential cross sections and analyzing powers for thereaction H( (cid:126)d, dp ) are presented in Fig. 6. In each panel,the results of this analysis are represented by filled circles.The error bars indicate the statistical uncertainties andthe gray bands represent the systematical errors. A de-tailed description of the related systematic uncertaintiesis presented in [26]. The open triangles show the resultsof cross sections measured at RCNP [23]. The open cir- cles and filled triangles show the analyzing powers datataken at KVI, [26,22], and open rectangles are those takenat RIKEN [21]. The solid curves show the results of acoupled-channel calculation by the Hannover-Lisbon the-ory group based on the CD-Bonn potential including theCoulomb interaction and an intermediate ∆ -isobar [73].The dotted lines represent results of a similar calcula-tions by excluding the ∆ -isobar. We note that the 3NFeffects are predicted to be small and, therefore, the re-sults of the presented Faddeev calculations based on thehigh-precision NN potential are expected to accuratelydescribe the experimental data. In addition, the resultsare compared with the results of a rough approximationbased on the lowest-order terms in the Born series expan-sion of the Alt-Grassberger-Sandhas (AGS) equation fora three-nucleons interaction using CD-Bonn+ ∆ potential(the dashed lines). The comparison shows that the Bornapproximation is not very good in three-body systems atthis energy, and therefore, we do not expect that such anapproach will provide a good description in the four-bodyscattering process; see Sec. 5. It is worth noting that thequality of the Born approximation improves with increas-ing the energy and/or at small scattering angles as thelowest-order terms become dominant in all observables.Our measurements for the H( (cid:126)d, dp ) reaction are in excel-lent agreement with previously published data and withstate-of-the-art calculations, lending, thereby, confidencein the analysis procedure and our estimates of systematicuncertainties. In the following, the analysis procedure of the BBS datais described. Details of the analysis methods related theBBS data are presented in [62].The differential cross section and spin observables wereextracted at various scattering angles by counting elasti-cally scattered deuterons for various polarization states ofthe beam. To access different scattering angles, the spec-trometer was moved around the target. The quadrupoleand dipole fields were changed according to the kinemat-ics of the related reaction to focus and bend the particlesof interest and bring them to the detector plane. In thiscase, one focal point was produced via a combination ofquadrupole and dipole fields for a scattered particle witha given momentum. Therefore, the solid angle spanned byparticles, as they scatter from the target inside the scat-tering chamber, were determined by a defining aperture infront of the spectrometer. For this purpose, a “seive slit”,an aperture fitted into the slit wheel of the BBS contain-ing several pre-drilled holes, was used during several runsof the experiment. With this slit system, the optical coef-ficients of BBS were fitted and the system was, therefore,calibrated for various settings.The main background sources are the events includingdeuterons elastically or inelastically scattered from Car-bon. These events are appeared in the detector plane alongwith the events of interest. To subtract the background, weapplied two techniques. For the runs with no discernible . Ramazani-Sharifabadi et al.: Title Suppressed Due to Excessive Length 7 -0.4-0.20.00.2
CDBonn+C+CDBonn+CBorn Approx.BINARCNPRIKENKVI 2008KVI 2007 -0.2-0.10.00.10.2
50 60 70 80
50 60 70 80 c.m. [deg] d / d [ m b / s r ] A y A zz A yy Fig. 6.
Differential cross section and analyzing powers of the elastic channel of the reaction H( (cid:126)d, dp ) that were taken witha deuteron beam of 65 MeV/nucleon. In each panel, the data taken with BINA are indicated with filled circles whereby theerror bars are statistical. The open triangles show the cross section results obtained at RCNP [23]. The open circles and filledtriangles show the analyzing powers data taken at KVI [26,22], and open rectangles are those obtained at RIKEN [21]. The solidcurves show the results of a coupled-channel calculation by the Hannover-Lisbon theory group based on the CD-Bonn potentialincluding the Coulomb interaction and an intermediate ∆ -isobar [73]. The dotted lines represent results of a similar calculationby excluding the ∆ -isobar. The dashed lines represent the results of a rough approximation based on the lowest-order terms inthe Born series expansion of the Alt-Grassberger-Sandhas (AGS) equation using CD-Bonn+ ∆ potential. The gray band showsthe systematic error (2 σ ) in each panel, background structure due to the Carbon in the CH tar-get, the procedure of background subtraction is similar tothat described for the event selection in BINA. For theruns in which a clear background structure due to theCarbon in the CH target was present, the separate Car-bon data from the Carbon target which were taken duringthe experiment were used. For each of these runs the cor-responding Carbon data (i.e. data which were taken withexactly the same spectrometer settings and beam energy,but with a solid carbon target) were analyzed using thesame parameters as the reaction data of interest. Finally,to obtain the differential cross section for each of the fivebeam polarization states, the extracted number of countsafter background subtraction is corrected for the efficien-cies of the system such as live-time, and wire chamberefficiencies.By knowing the polarized and unpolarized cross sec-tions for each of the five beam polarization values, wecould then calculate the unpolarized cross section and an-alyzing powers A y and A yy using Eq. 1 through a simplematrix inversion. We have five equations and only threeunknowns: the unpolarized cross section σ , A y , and A yy .Therefore, the analyzing powers are obtained from the po-larized cross sections using a matrix inversion, and theirstatistical errors determined using standard error prop- agation techniques. Generally, there were almost alwaysfive good polarized cross sections available, and therefore,this was an over-determined system; however for a fewruns only three or four polarization states were available,in which case the matrix inversion was reduced to onlyinclude the existing polarized cross sections [62]. The common systematic uncertainties between the two ex-periments as well as those specifically for BINA and BBSsetups are separately presented in the following subsec-tions.
The main common source of systematic error comes fromthe uncertainty in the A y measurements in the H( (cid:126)d, dp )reaction to extract the polarization that is around 4.5%.The results of A yy measurements obtained from BBSare also used to estimate an offset in the readout of thecurrent. The offset was determined by minimizing thereduced χ whereby an offset in the current is introduced R. Ramazani-Sharifabadi et al.: Title Suppressed Due to Excessive Length as a free parameter using the comparison between theresults of the A yy from the elastic channel of dd scat-tering coming from the BINA and those coming fromthe BBS [62]. The error is obtained by evaluating the χ distribution as a function of the offset. The intersectionpoint of this distribution with a χ value that is one unitlarger than its minimum has been used to determine theuncertainty in the offset. The systematic error arisingfrom the measurement of the beam current using aFaraday cup leads to a small offset of 0.28 ± One of the systematic uncertainty in the cross sectionmeasurement is attributed to the thickness of the liquiddeuterium target. We estimated a corresponding errorof 5% due to the bulging of the cell. The size of bulgingwas first estimated via a measurement of the targetthickness as a function of pressure at room temperature.The actual target thickness was obtained by comparingthe cross section measurements at KVI between solidand liquid targets and the difference is considered as theuncertainty due to the thickness measurement. Othersystematic uncertainties come from the beam luminosityusing a precision current source (2%), the MWPCefficiency for deuterons which was obtained using anunbiased and nearly background-free data sample of the pd elastic scattering process (1%), and the errors in thecorrection factor for losses due to hadronic interactionsin the detector. For deuterons, this error is extractedfrom the difference between the measured and simulateddeposition of deuteron energy in the forward wall ofBINA (2%) [65]. The uncertainty of the extraction ofthe differential cross sections due to the offset currentis around 5%. To estimate the systematic uncertaintydue to the background model, we used the 3 th and 7 th orders of polynomial fit-functions instead of the 5 th orderpolynomial representing the backgrounds. The maximumdifference between the results are considered as thesystematic uncertainty due to the background modelwhich is around 4.5%.The polarization of deuteron beam was monitoredwith a Lamb-Shift Polarimeter (LSP) [71] at the low-energy beam line and measured with BINA after beamacceleration at the high-energy beam line by mea-suring the asymmetry in the elastic deuteron-protonscattering process [70]. The vector and tensor polariza-tions of the deuteron beam of BINA were found to be p Z = − . ± .
029 and p ZZ = − . ± . k factor in Eq. 1 as a free parameter.This error turned out to be around 3%. Moreover, themaximum shift in the results of A y , A yy , and A zz dueto the offset current is around 0.01, 0.035, and 0.08, re-spectively, while the measured values of these observablesvary between − .
07 to +0.35, − .
04 to +0.22, and − . Systematic errors in the measurement of the differentialcross sections originate mainly from the errors in theknowledge of the target thickness and the calibration ofthe Faraday cup. As was already stated, the error in thetarget thickness for the elastic dd reaction was around 5%.The errors in the areal density measurements, which is themass of the material divided by its area with the unit ofmg / cm , come from both mass measurement errors andthose from the measurements of the size of the target. Theerror in the calibration of the Faraday cup was estimatedto be 0 . dd reaction).The polarization states of the deuteron beam of theBBS were measured with the Ion-Beam Polarimeter (IBP)and found to be as follows: vector plus (0 . ± . − . ± . . ± . − . ± . d + p reaction whileusing IBP. The polarization values for each state weremeasured at different beam energy ranges and found tobe consistent within the statistical uncertainties [62]. Themain sources of systematic error for the analyzing powersinclude the uncertainty due to beam polarization measure-ments ( p y and p yy ), and the total calibration error. Thecalibration errors for ( A y , A yy ) are found to be around(1 . , . Figure 7 shows the measured differential cross sectionsand analyzing powers for the elastic deuteron-deuteronscattering, H( (cid:126)d, d ) d . The results of BINA data are pre-sented as filled circles and the results of data taken by BBSsetup are shown as open circles [62]. The light (dark) grayband in each panel shows the systematic uncertainty of the . Ramazani-Sharifabadi et al.: Title Suppressed Due to Excessive Length 9 BINABBSCDBonn+BINA Sys Err (2 )BBS Sys Err (2 ) -0.10.00.10.20.335 40 45 50 55 60 65 70 75 c.m. [deg] d / d [ m b / s r ] A y A zz A yy Fig. 7.
Differential cross section and analyzing powers of the elastic channel of the reaction H( (cid:126)d, d ) d are shown with statisticalerrors for each point. The total systematic uncertainty related to BINA (BBS) results is shown with a light (dark) grayband for each panel. The results of BINA data are shown as filled circles and those for the BBS data are presented by opencircles [62]. The solid lines are the result of a calculation based on the lowest-order terms in the Born series expansion of theAlt-Grassberger-Sandhas equation for a four-nucleons interaction using CD-Bonn+ ∆ potential [60,57,58]. BINA (BBS) data, and the error bars represent the sta-tistical errors which are smaller than the symbol size formost of the data points. As discussed before, the resultsof the A yy measurement obtained from BBS were usedto normalize the offset of the current readout and hence,the corresponding systematic error is the same for both se-tups. Therefore, just one gray band is shown in Fig. 7. Thesolid lines are the results of a rough approximation basedon the lowest-order terms in the Born series expansion ofthe Alt-Grassberger-Sandhas equation for a four-nucleonsinteraction using CD-Bonn+ ∆ potential [60,57,58].The comparison between the results of the two exper-iments, namely data taken from BINA and BBS setups,indicates that both data sets are in very good agreementwithin the uncertainties. But, comparing the experimen-tal data with the theoretical approximation shows contra-dictions specially in the results of the analyzing powers.Aside from the normalization in the results of the differen-tial cross section, the theoretical prediction follows at leastthe shape of the experimental data. In the case of analyz-ing powers, the comparison shows contradictory results indicating defects in the spin parts of theoretical calcula-tions of the scattering amplitude. As already mentioned,the comparison between the results of exact calculationsand those coming from Born approximation in Fig. 6, in-dicates that this approximation is not very suitable for theH( (cid:126)d, dp ) reaction in this energy range, and therefore, weexpect to observe discrepancies between Born approxima-tion and the experimental data in the H( (cid:126)d, d ) d reactionin Fig. 7. In fact, Born approximation may provide a rea-sonable estimation for observables at higher energies andsmall angles, but, it is not reliable in the considered en-ergy and angle regime in this paper. It indicates that exacttheoretical calculations of four-body systems are a neces-sity to do a reasonable comparison with the experimentaldata. In summary, we have analyzed the elastic chan-nel of deuteron-deuteron scattering, H( (cid:126)d, d ) d , at
65 MeV/nucleon. Two experiments were performed withtwo independent setups, namely BINA and BBS, whichwere located at KVI in Groningen, the Netherlands. Crosssections and analyzing powers were obtained for a largeangular range of the phase space. An excellent agreementis found between the measured differential cross sectionsand analyzing powers of both experiments for the angularrange at which they overlap. The experimental resultsare also compared with a theoretical approximationbased on lowest-order terms in the Born series expansionusing CD-Bonn+ ∆ potential. The very poor agreementbetween the experimental data and theoretical approx-imations shows the necessity of ab-initio calculations inthe four-body systems at intermediate energies. The authors acknowledge the work by the cyclotron andion-source groups at KVI for delivering a high-qualitybeam used in these measurements. The present work hasbeen performed with financial support from the “Ned-erlandse Organisatie voor Wetenschappelijk Onderzoek”(NWO). This work was partly supported by Iran NationalScience Foundation (INSF) as a research project under No98028747.
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