Analyzing power in elastic scattering of 6He from polarized proton target at 71 MeV/nucleon
S. Sakaguchi, Y. Iseri, T. Uesaka, M. Tanifuji, K. Amos, N. Aoi, Y. Hashimoto, E. Hiyama, M. Ichikawa, Y. Ichikawa, S. Ishikawa, K. Itoh, M. Itoh, H. Iwasaki, S. Karataglidis, T. Kawabata, T. Kawahara, H. Kuboki, Y. Maeda, R. Matsuo, T. Nakao, H. Okamura, H. Sakai, Y. Sasamoto, M. Sasano, Y. Satou, K. Sekiguchi, M. Shinohara, K. Suda, D. Suzuki, Y. Takahashi, A. Tamii, T. Wakui, K. Yako, M. Yamaguchi, Y. Yamamoto
aa r X i v : . [ nu c l - e x ] J un APS/123-QED
Analyzing power in elastic scattering of He from polarized proton target at71 MeV/nucleon
S. Sakaguchi, ∗ Y. Iseri, T. Uesaka, M. Tanifuji, K. Amos, N. Aoi, Y. Hashimoto, E. Hiyama, M. Ichikawa, Y. Ichikawa, S. Ishikawa, K. Itoh, M. Itoh, H. Iwasaki, S. Karataglidis, T. Kawabata, T. Kawahara, H. Kuboki, Y. Maeda, R. Matsuo, T. Nakao, H. Okamura, † H. Sakai, Y. Sasamoto, M. Sasano, Y. Satou, K. Sekiguchi, M. Shinohara, K. Suda, D. Suzuki, Y. Takahashi, A. Tamii, T. Wakui, K. Yako, M. Yamaguchi, and Y. Yamamoto Center for Nuclear Study, University of Tokyo, Tokyo 113-0001, Japan Chiba-Keizai College, Chiba 263-0021, Japan Science Research Center, Hosei University, Tokyo 102-8160, Japan School of Physics, University of Melbourne, Melbourne, Australia RIKEN Nishina Center, Saitama 351-0198, Japan Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan Cyclotron & Radioisotope Center, Tohoku University, Miyagi 980-8578, Japan Department of Physics, University of Tokyo, Tokyo 113-0033, Japan Department of Physics, Saitama University, Saitama 338-8570, Japan Department of Physics, University of Johannesburg,P.O. box 924, Auckland Park, 2006 South Africa Department of Physics, Toho University, Chiba, Japan Research Center for Nuclear Physics, Osaka University, Osaka 567-0047, Japan Graduate School of Medicine, Gunma University, Gunma 229-8510, Japan Tsuru University, Yamanashi 402-8555, Japan (Dated: November 5, 2018)The vector analyzing power has been measured for the elastic scattering of neutron-rich He frompolarized protons at 71 MeV/nucleon making use of a newly constructed solid polarized protontarget operated in a low magnetic field and at high temperature. Two approaches based on localone-body potentials were applied to investigate the spin-orbit interaction between a proton and a He nucleus. An optical model analysis revealed that the spin-orbit potential for He is characterizedby a shallow and long-ranged shape compared with the global systematics of stable nuclei. A semi-microscopic analysis with a α + n + n cluster folding model suggests that the interaction betweena proton and the α core is essentially important in describing the p + He elastic scattering. Thedata are also compared with fully microscopic analyses using non-local optical potentials based onnucleon-nucleon g -matrices. PACS numbers: 24.10.Ht, 24.70.+s, 25.40.Cm, 25.60.Bx, 29.25.Pj
I. INTRODUCTION
Spin-orbit coupling in atomic nuclei is an essential fea-ture in understanding any reaction and nuclear struc-ture related to it. One of the direct manifestations ofthat spin-orbit coupling in nuclear reactions, is the po-larization phenomenon in nucleon elastic scattering [1–3]. Characteristics of the spin-orbit coupling betweena nucleon and stable nuclei have been well establishedby analyses of measured vector analyzing powers in theelastic scattering of polarized nucleons on various targetsover a wide range of incident energies [4–7].On the other hand, the spin-orbit coupling of a nu-cleon with unstable nuclei might be considerably differ-ent from that with the stable nuclei. Some neutron-rich ∗ Present address: Department of Physics, Kyushu University,Fukuoka 812-8581, Japan;Electronic address: [email protected] † Deceased. nuclei with small binding energies are known to have veryextended neutron distributions [8]. Since the spin-orbitcoupling is essentially a surface effect, it is natural to ex-pect that the diffused density distribution of a neutron-rich nucleus may significantly effect the radial shape anddepth of the spin-orbit potential. The purpose of thiswork is to investigate the characteristics of the spin-orbitpotential between a proton and He; a typical neutron-rich nucleus.Experimental determination of the spin-orbit poten-tial strongly owes to measurements and analyses of thevector analyzing powers. However, until recently, ana-lyzing power data were not obtained in the scatteringwhich involves unstable nuclei. This was mainly due tothe lack of a polarized proton target that is applicable toradioactive ion (RI) beam experiments. RI-beam exper-iments induced by light ions are usually carried out un-der inverse-kinematics conditions, where energies of recoilprotons can be as low as 10 MeV. Conventional polarizedproton targets [9, 10], based on the dynamic nuclear po-larization method, require a high magnetic field and lowtemperature such as 2.5 T and 0.5 K, respectively. It isimpossible to detect the low-energy recoil protons withsufficient angular resolution under these extreme condi-tions. For the application in RI-beam experiments, wehave constructed a solid polarized proton target whichcan be operated under low magnetic field of 0.1 T andat high temperature of 100 K [11–15]. The electron po-larization in photo-excited aromatic molecules is used topolarize the protons [16, 17]. A high proton polarizationof about 20% can be achieved in relatively “relaxed” op-erating conditions described above, since the magnitudeof the electron polarization is almost independent of themagnetic field strength and temperature.We have measured the vector analyzing power for the p + He elastic scattering at 71 MeV/nucleon [18] usingthe solid polarized proton target, newly constructed forRI-beam experiments. He is suitable for the presentstudy since it has a spatially extended distribution dueto a small binding energy. In addition, from an experi-mental viewpoint, the p + He elastic scattering measure-ment is relatively easy to perform since He does nothave a bound excited state. This allows us to identify theelastic-scattering event only by detecting He and a pro-ton in coincidence. The analyzing powers thus measuredare the first data set that can be used for quantitativeevaluation of the spin-orbit interaction between a protonand an unstable He nucleus. The essence of these mea-surements has been published in Ref. [18] together withtwo kinds of theoretical analyses by folding models; oneassumes a fully antisymmetrized large-basis shell modelfor He with the g -matrix interaction and the other an α + n + n cluster model for He with a p - n effective in-teraction and a realistic p - α static potential. The mainpurpose of the present paper is to give more details ofthe experiment and present an additional analysis of theexperimental data using a one-body p - He optical poten-tial. The analysis exhibits remarkable characteristics forthe spin-orbit part of that potential. Then it becomesimportant to investigate if such a potential can be de-rived theoretically from any model of He. As the firstapproach we examined the α + n + n folding potential inmore detail, since important contributions of the α clus-ter are suggested by the fact that the measured A y for He is similar to that for He [18], when plotted versus themomentum transfer of the scattering. To identify effectsof the clusterization, we also calculated the p - He foldingpotential for a 2 p +4 n non-cluster model of He and com-pared the results with those of the α + n + n cluster model.Hereafter, they are referred to as the αnn cluster folding(CF) model and nucleon folding (NF) one, respectively.In addition, the data are also compared with fully mi-croscopic calculations using non-local optical potentials.In this model, non-locality of the p - He interaction, aconsequence of the Pauli principle leading to nucleon ex-change scattering amplitudes, is taken into account ex-plicitly. Three sets of single-particle wave functions, aswell as the required one-body density matrix elementsdetermined from a large-basis shell model for He, have been used in these calculations.The present paper is subdivided as follows. In Section2, details of the experimental method are described. InSection 3, the method of the data reduction is presented.Section 4 deals with the phenomenological optical modelanalysis. Section 5 is devoted to the details of the αnn cluster folding calculation and the nucleon folding cal-culation. In Section 6, the data are compared with theanalysis by the non-local g -folding optical potentials. Fi-nally, a short summary of the obtained results is given inSection 7. II. EXPERIMENTA. Experimental setup
The experiment was carried out at the RIKEN Ac-celerator Research Facility (RARF). The He beam wasproduced through the projectile fragmentation of a Cbeam with an energy of 92 MeV/nucleon bombarding aprimary target. As that primary, we used a rotating Betarget [19] to avoid heat damage by the beam. A thick-ness of the target was 1480 mg/cm . The He particleswere separated by the RIKEN Projectile-fragment Sepa-rator (RIPS) [20] based on the magnetic rigidity and theenergy loss of fragments. The energy of the He beamwas 70.6 ± ◦ –71 ◦ (hor-izontal) and ± . ◦ (vertical) in the laboratory system.Their position resolution and detection efficiency werefound to be 2.6 mm (FWHM) and 99.3%. For the mea-surement of the total energy of protons, we used CsI(Tl)scintillation detectors. They were placed just behindthe SWDCs. Light output from the CsI(Tl) crystal wasdetected by photo-multiplier tubes. The front side ofthe CsI(Tl) scintillator was covered by the thin carbon-aramid film with a thickness of 12 µ m. Material thicknessof the film, the SWDC, and air between the detectors was24 mg/cm in total. Energy loss of 10 MeV protons in CsI (Tl)SWDCMWDC Plastic scintilatorsSolid polarized proton target He beam Recoil protonHe beam TargetcrystalBeam monitor & stopperAr-ion laser Cooling chamberCold N gasVacuum chamber He ScatteredHe ScatteredC-type magnet
FIG. 1: (Color online) Experimental setup of the secondary target and detectors is shown. these materials is 1.2 MeV, which does not prevent thedetection.A multi-wire drift chamber (MWDC) was used to re-construct the trajectories of scattered particles. Scatter-ing position on the secondary target was determined fromthe reconstructed trajectory. The MWDC was placed at880 mm downstream of the target. It has a sensitive areaof 640 mm (horizontal) ×
160 mm (vertical) and coveredan angular region of ± ◦ ×± ◦ in the laboratory system.The configuration of the planes of the MWDC is X-Y-X’-Y’-X’-Y’-X-Y, where “X(Y)-plane” has anode-wiresoriented along the vertical (horizontal) axis. The planeswith primes are displaced with respect to the “unprimed”planes by half the cell size. The cell size is 20 mm ×
20 mmfor the X-plane and 10 mm ×
10 mm for the Y-plane. Thematerial of the anode wire is gold-plated tungsten witha diameter of 30 µ m. Negative high voltages were ap-plied to the cathode and potential wires: − − H (50%) was used. Po-sition resolution and detection efficiency of the MWDCwere found to be 0.2 mm (FWHM) and 99.8%. For iden-tification of scattered particles, we used a plastic scin-tillation detector array placed just behind the MWDC.The first and second layers with thicknesses of 5 mm and100 mm provided information of the energy loss and thetotal energy of scattered particles. The total number ofthe beam particles was counted with a beam monitorplaced between the secondary target and the MWDC. A50 mm H ×
50 mm V ×
10 mm D plastic scintillator was usedfor the beam monitor. A beam stopper made of a copper block was placed just behind the beam monitor. B. Solid polarized proton target
The solid polarized proton target, used in the measure-ment, can be operated in a low magnetic field of 0.1 T andat a high temperature of 100 K. These relaxed operationconditions allow us to detect low-energy recoil protonswithout losing angular resolution. This capability is in-dispensable to apply the target to scattering experimentscarried out under the inverse kinematics condition. Theproton polarization of about 20% has been achieved [15]under such relaxed conditions by introducing a new po-larizing method using electron polarization in tripletstates of photo-excited aromatic molecules [16, 17]. Asingle crystal of naphthalene (C H ) doped with a smallamount of pentacene (C H ) is used as the target ma-terial. Protons in the crystal are polarized by repeatinga two-step process: production of electron polarizationand polarization transfer. In the first step, pentacenemolecules are optically excited to higher singlet states.A small fraction of them decays to the first triplet statevia the first excited singlet state by the so-called inter-system crossing. Here, electron population difference isspontaneously produced among Zeeman sublevels of thetriplet state [16]. In the second step, the electron popu-lation difference between two Zeeman sublevels, namelyelectron polarization, is transferred to the proton polar-ization by the cross-relaxation technique [17].As the target material, we used a single crystal of naph-thalene doped with 0.005 mol% pentacene molecules.The crystal was shaped into a thin disk whose diame-ter and thickness are 14 mm and 1 mm (116 mg/cm ),respectively. The number of hydrogens per unit area was4.29 ± × /cm . In order to reduce the relaxationrate, the target crystal was cooled down to 100 K ina cooling chamber with the flow of cold nitrogen gas.The cooling chamber was installed in another chamberas shown in Fig. 1. Heat influx to the cooling chamberwas reduced by the vacuum kept in the intervening spacebetween these two chambers. Each chamber has one win-dow (6 µ m-thick Havar foil) on the upstream side for theincoming RI-beam, two glass windows for the laser irra-diation, and three windows (20 µ m-thick Kapton foil) onthe left, right, and downstream sides for the detection ofrecoil and scattered particles.A static magnetic field was applied on the target crys-tal by a C-type electromagnet to define the polarizingaxis. The gap and the diameter of the poles were 100 mmand 220 mm, respectively. The strength of the magneticfield in the present experiment was 91 mT; a value muchhigher than that of the crystal field ( ≈ He particles and protons were sufficiently small (about0.07 ◦ and 0.2–0.8 ◦ , respectively), they were properly cor-rected in the data analysis.The target crystal was irradiated by the light of twoAr-ion lasers with a power of 25 W each in the multi-line mode. Wavelengths of main components of the lightwere 514.5 nm (10 W) and 488.5 nm (8 W). The laserlight was pulsed by a rotating optical chopper. Typi-cally the pulse width and repetition rate were 12–14 µ sand 1 kHz. Microwave (MW) irradiation and a magneticfield sweep are required in the cross-relaxation method.For the MW irradiation, the target crystal was installedin a resonator. In order to detect low-energy recoil pro-tons, we employed a thin cylindrical loop-gap resonator(LGR [21]) made of 25 µ m-thick Teflon film. Copperstripes with a thickness of 4.4 µ m were printed on bothsides of the film. The MW frequency was 3.40 GHz. TheLGR was surrounded by a cylindrical MW shield madeof 12 µ m-thick aluminum foil. For the cross-relaxation,the magnetic field was swept from 88 mT to 94 mT atthe rate of 0.36 mT/ µ s, simultaneously with the MW ir-radiation, by applying a current to a small coil placed inthe vicinity of the target material.Proton polarization was monitored during the experi-ment by the pulse NMR method. A radio-frequency (RF)pulse with a frequency and a duration of 3.99 MHz and2.2 µ s was applied to a 19 mm φ NMR coil covering thetarget crystal. The free induction decay (FID) signal wasdetected by the same coil. We carried out the absolutecalibration to relate the FID signal to the proton polar-ization by measuring the spin-asymmetry in the p + Heelastic scattering. Details of the calibration procedureare described in Appendix. A.Devices located near to the target, namely the LGR,the MW shield, the field sweeping coil, and the NMR coil, were fabricated with hydrogen-free materials to pre-vent production of background events. Table I showsthe material thicknesses of the devices that recoil pro-tons penetrate. Energy losses of the 20 MeV protons inthese materials are sufficiently small for the detection assummarized in Table I.The target polarization during the experiment is shownin Fig. 2 as a function of time. The polarization wasbuilt up for the first 40 hours and reached the maxi-mum value of 20.4 ± He beam for 55 hours, bya 80 MeV/nucleon He beam for the following 25 hours,and again by the He beam for 60 hours. The magni-tude of average polarization was found to be 13.8 ± − before the experiment to Γ = 0.295(4) h − after the beamirradiation. The direction of the target polarization wasreversed three times during the measurement to cancelspurious asymmetries. The 180 ◦ pulse NMR method wasused here. Reversal efficiency of 60–70% was achieved. FIG. 2: Target polarization is shown as a function of time.The open squares, closed circles, and open circles indi-cate the target polarization during the polarization buildup,the p + He elastic-scattering measurement, and the p + Heelastic-scattering measurement, respectively.
III. DATA REDUCTIONA. Data analysis
In principle, elastic-scattering events of the He fromprotons can be identified by the coincidence detection ofscattered He particles and recoil protons, since the Hedoes not have a bound excited state. Note that the firstexcited state of He, which is the 2 + state at 1.87 MeV, is Material Thickness (mg/cm ) Energy loss (MeV)Target crystal (Naphthalene) 0 – 336 0 – 9.5LGR (Teflon, Cu foil) 9.3 0.2 – 0.4Microwave shield (Al foil) 3.2 0.05 – 0.1Cooling gas (N ) 13.5 0.3 – 0.6Window (Kapton film) 20 0.5 – 1.0Total 46 – 382 1.1 – 11.6TABLE I: Thicknesses of the materials of target devices and energy losses of 20 MeV recoil protons in them. above the two-neutron breakup threshold (0.975 MeV).Thus, any excited He particles decay into α + n + n sys-tems before reaching the detectors.Scattered particles were identified by the standard ∆ E - E method. Figure 3 shows a two-dimensional plot ofthe total energies of scattered particles E versus theirenergy losses ∆ E , where loci of tritons, He, He, and Liare found. Tritons and Li are the contamination in thesecondary beam. Most of He particles were produced by He dissociation in the secondary target. However, someoriginated from He reactions in the plastic scintillators.So to count all of the p + He elastic-scattering events, theparticle identification gate includes most of the He locusas shown by solid curves in Fig. 3. The contribution ofthe dissociation reaction, which is not excluded by thisgate, was subtracted using a kinematics relation. This isdescribed after the response of the recoil proton detectorsis considered.
X10 ∆ E (channel)
X10 E ( c h a nn e l ) Contour step =10000 t He He Li FIG. 3: Two-dimensional plot of the total energies of scat-tered particles versus their energy losses. Solid curves indicatethe particle-identification gate.
Figure 4 shows a two-dimensional scatter plot of theproton energies versus their scattering angles in thecenter-of-mass system, θ c.m. . The kinematic locus of theelastic scattering is clearly identified, while backgrounds from other reaction channels such as p ( He, p He) arealso evident. The kinematic locus of elastic-scatteringevents shows that the recoil protons were properly de-tected outside of the target. It should be noted that thiscorrelation was not used for the event selection, since itwould cause a loss of the events at forward angles.
30 40 50 60 70 80 90 100 θ c.m. (deg) P r o t o n e n e r g y ( a r b ) p+ He elasticscattering p( He, p He) FIG. 4: Two-dimensional plot of the proton energies versustheir scattering angles.
To discriminate elastic scattering from the background,we used the correlation of the azimuthal angles of pro-tons φ p with those of scattered particles φ scatt. . In thecase of the elastic scattering, a scattered He and a recoilproton stay within a well defined reaction plane since thefinal state is a binary system. Thus, the difference of az-imuthal angles ∆ φ = φ p − φ scatt. makes a narrow peak ataround 180 ◦ . This back-to-back correlation holds even ifthe scattered He is dissociated in the plastic scintillator.In the case of other reactions, however, the azimuthal an-gle difference is more spread since their final states consistof more than two particles.Figure 5 shows the distribution of the azimuthal angledifference ∆ φ fitted by a double-Gaussian function. Thenarrower component is reasonably identified as that ofthe elastic-scattering events. The peak width of 3.5 ◦ insigma is consistent with the detector resolution of 3.1 ◦ .We selected the events of | ∆ φ − ◦ | < . ◦ . Thebackground remaining in the gate was evaluated fromthe broader component and was subtracted. Contribu-tions of the inelastic scattering and other reactions suchas breakup were removed in this way without losing theelastic-scattering yields. Figure 6 shows a background-subtracted two-dimensional plot of scattering angles inthe center-of-mass system versus angles of scattered par-ticles. Center-of-mass scattering angles were deducedfrom recoil angles of the protons in the laboratory system,since the resolution of scattering angles of He particles isinsufficient due to the kinematic focusing. In Fig. 6, clearpeaks of elastic-scattering events lie along the solid curvesindicating the kinematics of the p + He elastic scattering.Small peaks at | θ He | ≈ ◦ originated from the ambigu-ity in the background subtraction. Yields of the p + Heelastic scattering were obtained by counting the eventsof the elastic-scattering peaks in the typical width of 4 ◦ in θ He . FIG. 5: Azimuthal angle difference between scattered parti-cles and recoil protons, fitted by a double-Gaussian function.FIG. 6: Scattering angle correlation between scattered par-ticles and recoil protons. The solid curves indicate the kine-matics of the p + He elastic scattering.
The present work demonstrates the applicability of thesolid polarized proton target in the RI-beam experiment.The relaxed operation condition of the target, i.e. a lowmagnetic field of 0.1 T and high magnetic field of 100 K,enables us to detect the low-energy recoil protons. Asdescribed in the data analysis above, information on thetrajectory of recoil proton is indispensable both in identi-fying the elastic-scattering events (Fig. 5) and in deduc-ing the scattering angle (Fig. 6).
B. Experimental data
The dσ/d
Ω of the p + He elastic scattering measured at71 MeV/nucleon are summarized in Table II. In the back-ward region, the uncertainty mainly results from statis-tics and from the ambiguity in the background subtrac-tion. In the forward angular region, θ c.m. < ◦ , the maincomponent of the uncertainty in dσ/d Ω is the systematicuncertainty in the number of the incident particles (10%).The target was hit by only a fraction of beam particlessince the size of the secondary beam was comparable tothat of the target. The percentage of the beam particlesincident on the target was determined from the beamprofile and was found to be 65 ±
7% of those counted bythe beam monitor. The beam profile was measured withthe MWDC by removing the beam stopper. Stability ofthe beam profile was confirmed by several measurementscarried out before, during, and after the elastic-scatteringmeasurement.The analyzing power A y is deduced with the standardprocedure as A y = 1 P L − RL + R , L = q N ↑ L · N ↓ R , R = q N ↓ L · N ↑ R , where P denotes the target polarization. The values N ’s represent the yield of the elastic-scattering eventswhere subscripts and superscripts denote the scatter-ing direction (left/right) and the polarization direction(up/down), respectively. The statistical uncertainty isexpressed by∆ A y A y = LRL − R s N ↑ R + 1 N ↓ R + 1 N ↑ L + 1 N ↓ L . This procedure allows us to minimize the systematic un-certainties originating from unbalanced detection efficien-cies and misalignment of detectors. The obtained A y aresummarized in Table III. It must be noted that there isan additional scale error of 19% resulting from the un-certainty in the target polarization P (see Appendix. A).Figure 7 shows dσ/d Ω and A y for the p + He elasticscattering at 71 MeV/nucleon (closed circles: presentwork, open circles: Ref. [22]), those for the p + He at72 MeV/nucleon (open squares: Ref. [23]), and those forthe p + Li at 72 MeV/nucleon (open triangles: Ref. [24]).The present data are consistent with the previous ones in θ c.m. (deg) ∆ θ c.m. (deg) dσd Ω (mb/sr) ∆ dσd Ω (mb/sr)42.1 2.5 5.02 0.5247.1 2.5 2.03 0.2252.1 2.5 0.796 0.09857.4 2.5 0.454 0.05962.3 2.5 0.360 0.04667.3 2.5 0.226 0.03172.3 2.5 0.172 0.02377.3 2.5 0.127 0.01882.2 2.5 0.064 0.01387.2 2.5 0.038 0.012TABLE II: Differential cross sections for the p + He elasticscattering at 71 MeV/nucleon. The ∆ θ c.m. denotes the binwidth. The ∆ dσd Ω denotes the quadratic sum of the statisticaland systematic uncertainties. θ c.m. (deg) ∆ θ c.m. (deg) A y ∆ A y − − p + He elastic scatteringat 71 MeV/nucleon. ∆ A y denotes the statistical uncertainty.Note that there is an additional scale error of 19% resultingfrom the uncertainty in the target polarization. The ∆ θ c.m. denotes the bin width. Ref. [22] in an overlapping angular region of θ c.m. = 40–50 ◦ . We extended the data to the backward angles of θ c.m. ≈ ◦ . It is found that the dσ/d Ω of p + He arealmost identical with those of p + Li at θ c.m. = 20–90 ◦ ,while they have a steeper angular dependence than thoseof p + He. In good contrast to the similarity found in dσ/d Ω, A y data are widely different between p + He and p + Li. The A y of p + Li increase as a function of thescattering angle in an angular region of θ c.m. = 40–70 ◦ and take large positive values. This behavior is com-monly seen in proton elastic scattering from stable nu-clei at the present energy region [25]. Unlike this globaltrend, A y of p + He decreases in θ c.m. = 37–55 ◦ , which israther similar to those of p + He. While the large errorbars prevent us from observing the difference between A y of p + He and of p + He, it is clearly seen that the angu-lar distribution of the A y in p + He deviates from that of p + Li.
IV. PHENOMENOLOGICAL OPTICAL MODELANALYSISA. Optical potential fitting
The aim of this section is to extract the gross character-istics of the spin-orbit interaction between a proton and
FIG. 7: (Color online) Differential cross sections and analyz-ing powers of the p + He at 72 MeV (open squares: Ref. [23]),of the p + Li at 72 MeV (open triangles: Ref. [24]), and ofthe p + He at 71 MeV (open circles: Ref. [22], closed circles:present work). He. For this purpose, we determined the optical modelpotential that reproduces the experimental data of bothdifferential cross sections and analyzing powers. The op-tical model potential obtained in this phenomenologicalapproach will be compared with the semi-microscopic cal-culations in Section V.We adopted a standard Woods-Saxon optical potentialwith a spin-orbit term of the Thomas form: U OM ( R ) = − V f r ( R ) − i W f i ( R )+ 4 i a id W d ddR f id ( R )+ V s R ddR f s ( R ) L · σ p + V C ( R ) (1)with f x ( R ) = (cid:20) (cid:18) R − r x A / a x (cid:19)(cid:21) − (2)( x = r, i, id, or s ) . Here, R is the relative coordinate between a proton anda He particle (see Fig. 12 (b)), L = R × ( − i ~ ∇ R ) isthe associated angular momentum, and σ p is the Paulispin operator of the proton. The subscripts r , i , id , and s denote real, volume imaginary, surface imaginary, andspin-orbit, respectively. V C is the Coulomb potential ofuniformly charged sphere with a radius of r A / fm( r = 1 . χ values of dσ/d Ω, and second the parameters of the spin-orbit termby fitting A y . These two steps were iterated alternatelyuntil convergence was achieved. Such a procedure is fea-sible since the contribution of the spin-orbit potential to dσ/d Ω is much smaller than those of the central terms. Inthe fitting, we used the data in Ref. [22] and the presentones. Uncertainties of dσ/d
Ω smaller than 10% were ar-tificially set to 10% in order to avoid trapping in an un-physical local χ minimum. The fitting was carried outusing the ECIS79 code [26]. A set of parameters for the p + Li elastic scattering at 72 MeV/nucleon [24], labeledas Set-A in Table IV, was used as the initial values in thesearch of the p - He potential parameters.The parameters obtained for the p + He elastic scat-tering are labeled as Set-B in Table IV. The reduced χ values for dσ/d Ω and A y were 0.95 and 0.96, respectively.Uncertainties of the parameters of the spin-orbit poten-tial, r s , a s , and V s , are evaluated in the following man-ner. Figure 8 shows the contour map of the deviation of χ value for A y from that calculated by the Set-B (as in-dicated by the point-P), ∆ χ A y , on the two-dimensionalplane of r s and a s after projecting with optimized V s at each point of the plane. In the figure, a simultane-ous confidence region for r s and a s is presented by thesolid contour indicating ∆ χ A y = 1. In this region, theoptimum V s ranges between 1.15 MeV (at the point-Q)and 2.82 MeV (at the point-R). In the r s - a s - V s space,a surface that ∆ χ A y = 1 touches planes that are ex-pressed by r s = 1 . ± .
13 fm, a s = 0 . ± .
17 fm, and V s = 2 . ± .
87 MeV, which gives a rough estimationfor uncertainties of the parameters.
B. Characteristics of spin-orbit potential
In Fig. 9, the results of calculations of the observablesmade with the optical potentials of Set-A, -B, and -C inTable IV are shown together with the experimental data.Set-C was taken from Ref. [27], where a phenomenologi-cal optical model potential that reproduced only the pre-vious dσ/d
Ω data of the p + He at 71 MeV/nucleon [22]was reported. The radial dependences of the p - He opti-cal potentials (Set-B and Set-C) are shown in Fig. 10 bysolid and dashed lines, respectively.The calculation with the potential Set-C reasonablyreproduces the present dσ/d
Ω data, whereas it largelydeviates from the A y data at θ c.m. & ◦ . It should benoted that the A y data were unavailable when the poten-tial Set-C was sought. The calculation with the potentialSet-B reproduces both dσ/d Ω and A y over whole angularregion except for the most backward data point of A y .Similarity of the dσ/d Ω calculated with Set-B and Set-
FIG. 8: Contour map of the ∆ χ A y values (see the text for thedefinition) on the two-dimensional plane of r s and a s . Thesolid, dashed, and dotted curves indicate ∆ χ A y =1, 3, and5, respectively. The point that gives the best-fit parameters,Set-B in Table IV, is indicated by the point-P. See the textfor the points-Q and -R.FIG. 9: (Color online) Differential cross sections and analyz-ing powers calculated by the phenomenological optical poten-tials are shown together with the experimental data. The dot-dashed curves denote calculations of the Set-A in Table IV,the solid curves those of set-B, and the dashed curves thoseof Set-C. Solid circles are present data and open circles arefrom Ref. [22]. TABLE IV: Parameters of the optical potentials for p + Li at 72 MeV/nucleon [24] and p + He at 71 MeV/nucleon ([27] andpresent work). V r r a r W r i a i W d r id a id V s r s a s (MeV) (fm) (fm) (MeV) (fm) (fm) (MeV) (fm) (fm) (MeV) (fm) (fm)Set-A p + Li [24] 31.67 1.10 0.75 14.14 1.15 0.56 — — — 3.36 0.90 0.94Set-B p + He (Present) 27.86 1.074 0.681 16.58 0.86 0.735 — — — 2.02 1.29 0.76Set-C p + He [27] 30.00 0.990 0.612 14.0 1.10 0.690 1.00 1.76 0.772 5.90 0.677 0.630FIG. 10: (Color online) Radial dependences of the phe-nomenological optical potential (Set-B and Set-C in Ta-ble IV). Solid and dashed curves in the upper panel representthe real and imaginary parts of the central term. The lowerpanel shows the spin-orbit potential.
C potentials originates from that of the central terms asshown in the upper panel of Fig. 10. The reliability ofthe potential obtained in the present work is supportedby the fact that two independent analyses yielded thesimilar results for the central terms. In contrast to thecentral terms, the spin-orbit terms of these two potentialsare quite different, resulting in a large difference in A y asshown in Fig. 9. Note that the present data are sensitiveto the optical potential in a region of R & R . V s and large values of r s and a s of Set-B compared with those of Set-C. Thephenomenological optical model analysis suggests thatthe A y data can be reproduced only with a shallow andlong-ranged spin-orbit potential. The parameters of our spin-orbit potential are com-pared with those of neighboring even-even stable nucleiand with global potentials in Table V. Phenomenologicaloptical potentials for the p + O at 65 MeV and p + Cat 16–40 MeV are taken from Ref. [25] and Refs. [28, 29],respectively. In addition to these local potentials, wealso examined the parameters of global optical poten-tials: CH89 [4] and Koning-Delaroche (KD) [5], of whichapplicable ranges are E = 10–65 MeV, A = 40–209 and E = 0.001–200 MeV, A = 24–209, respectively. Whilethey are constructed for nuclei heavier than He, it isworthwhile comparing them, since the mass-number de-pendence of the parameters is relatively small. For ex-ample, the mass-number dependence appears only in r s in the case of CH89 [4] as: V s = 5 . ,r s = 1 . − . A − / fm ,a s = 0 . . Table V includes the parameters of these potentials forthe nuclei within the applicable range. Incident energiesof E = 65 MeV and E = 71 MeV were assumed here forCH89 and KD, respectively.Firstly, we focus on r s and a s to discuss the radialshape of the spin-orbit potential. Combination of differ-ent values of r s and a s can provide similar results of A y since the observable is sensitive to the surface regionof the spin-orbit potential. We thus compare these pa-rameters on the two-dimensional plane of r s and a s asshown in Fig. 11. Parameters for the stable nuclei aremostly distributed in a region of r s = 0 . . a s = 0 . . He is located in theupper right side of the figure. These large r s and/or a s values indicate that the spin-orbit potential betweena proton and a He has a long-ranged nature comparedwith those for stable nuclei. The depth parameter V s was also compared with the global systematics. The V s value of p - He potential was found to be 2.02 MeV forthe best-fit potential (Set-B) and ranges between 1.15and 2.82 MeV in the simultaneous confidence region for r s and a s . On the other hand, those of stable nucleiare mostly distributed around 5 MeV as shown in Ta-ble V. Comparing these values, the depth parameter ofthe spin-orbit potential between a proton and a He isfound to be much smaller than those of stable nuclei.The phenomenological analysis indicates that the spin-orbit potential between a proton and He is characterized0 V s (MeV) r s (fm) a s (fm) p + He, E = 71 MeV (Set-B) 2.02 1.29 0.76 p + C, E = 40 MeV [28] 6.18 1.109 0.517 p + C, E = 16–40 MeV [29] 6.4 1.00 0.575 p + O, E = 65 MeV [25] 5.793 1.057 0.5807CH89, E = 65 MeV, A =40–209 [4] 5.9 ± ± E = 71 MeV, A =24–209 [5] 4.369–4.822 0.961–1.076 0.59TABLE V: Parameters of the spin-orbit term of phenomenological and global optical potentials.FIG. 11: Two-dimensional distribution of r s and a s ofphenomenological OM potentials for p + He (closed circle), p + C (closed triangle: Ref. [29], closed diamond: [28]), and p + O (closed square: [25]). The solid contour indicates thesimultaneous confidence region for the r s and a s values forthe p + He as displayed in Fig. 8. Parameters of global OMpotentials [4, 5] are also shown by solid lines which representthe A -dependence of r s . by large r s /a s and small V s values yielding shallow andlong-ranged radial dependence. Intuitively, these char-acteristics can be understood from the diffused densitydistribution of He. However, its microscopic origin cannot be clarified by the phenomenological approach. Toexamine the microscopic origin of the characteristics ofthe p - He interaction, microscopic and semi-microscopicanalyses are required. Section V describes one of suchanalyses based on a cluster folding model for He.
V. SEMI-MICROSCOPIC ANALYSES
In this section, we examine two kinds of the folding po-tential, the cluster folding (CF) and the nucleon folding (NF) ones. They are compared with the phenomeno-logical optical model (OM) potential determined in thepreceding section. The results of calculations of observ-ables made by these potentials are compared with theexperimental data.In the CF potential, we adopt the αnn cluster modelfor He and fold interactions between the proton and thevalence neutrons, V pn , with the neutron density in Heand those between the proton and the α core, V pα , withthe α density in He. In the NF potential, we decomposethe α core into two neutrons and two protons and foldthe interactions between the incident proton and the fourneutrons, V pn , with the neutron density in He and thosebetween the incident proton and two target protons, V pp ,with the proton density in He.The detailed expressions of such folding potentials aregiven in the following subsection, where the Coulomb in-teraction is considered in the p - p and p - α interactionsrespectively when compared with the corresponding scat-tering data but finally it is assumed to act between theproton and the He target with r = 1 .
400 fm [27].
A. Folding potentials
Denoting two valence neutrons by n and n , the CFpotential U CF is given as U CF = Z V pn ρ CF n ( r ) d r + Z V pn ρ CF n ( r ) d r + Z V pα ρ CF α ( r α ) d r α , (3)where r , r , and r α are the position vectors of n , n ,and the α core from the center of mass of He, respec-tively. The neutron and α densities, ρ CF n and ρ CF α , arecalculated by the αnn cluster model for He [30, 31],where the condition r + r + 4 r α = 0 is considered asusual.In the present work, we specify the potentials in theright hand side of Eq. (3) by the central plus spin-orbit(LS) type: V pn i = V pn ( | r pn i | ) + V LS pn ( | r pn i | ) ℓ pn i · ( σ p + σ n i ) , where i = 1 , V pα = V pα ( | r pα | ) + V LS pα ( | r pα | ) ℓ pα · σ p . (4)1 FIG. 12: (Color online) Coordinate systems for the clusterfolding model.
Here, r pn , r pn , and r pα are defined in Fig. 12 (a), and ℓ pn = r pn × ( − i ~ ∇ pn ), etc.In the following, we transform the set of coordinates( r pn , r pn , r pα ) to that of ( ξ , ζ , R ), which are defined inFig. 12 (b), to describe the angular momenta ℓ pn i and ℓ pα in terms of L . The transformation is r pn = − R − ζ − ξ , r pn = − R − ζ + 12 ξ , r pα = − R + 13 ζ , (5)and consequently ∇ pn = − ∇ R − ∇ ζ − ∇ ξ , ∇ pn = − ∇ R − ∇ ζ + ∇ ξ , ∇ pα = − ∇ R + ∇ ζ . (6)These relations lead to, for example, ℓ pn = ( − R − ζ − ξ ) × [ − i ~ ( − ∇ R − ∇ ζ − ∇ ξ )] . (7)Here, ∇ ξ and ∇ ζ can be neglected, because these arethe momenta for the internal degrees of freedom of Heand their expectation values are zero for a sphericallysymmetric nucleus [32]. Using ξ + ζ = − r , we get ℓ pn = 16 [ L − r × ( − i ~ ∇ R )] , (8)which is independent of the special choice of the Heinternal coordinates, ξ and ζ . To L , r can contributeby its component along the R direction [32], then ℓ pn = 16 L (1 − r · R R ) . (9)Similar expressions are obtained for ℓ pn and ℓ pα . Setting ( σ n + σ n ) = 0 and considering other quantities toappear in symmetric manners on 1 and 2, we obtain the p - He potential as U CF = U CF0 ( R ) + U CFLS ( R ) L · σ p , (10) with U CF0 ( R ) = 2 Z V pn ( | r − R | ) ρ CF n ( r ) d r + Z V pα ( | r α − R | ) ρ CF α ( r α ) d r α (11)and U CFLS ( R ) = 13 Z V LS pn ( | r − R | ) (cid:26) − r · R R (cid:27) ρ CF n ( r ) d r + 23 Z V LS pα ( | r α − R | ) (cid:26) − r α · R R (cid:27) ρ CF α ( r α ) d r α . (12)In a way similar to the above development, we get theNF model potential U NF . In this case, the relative coor-dinates between the incident proton and six nucleons inthe He nucleus are transformed to the proton- He rela-tive coordinate R and a set of five independent internalcoordinates of He. The obtained U NF , which is indepen-dent on the choice of the set of the internal coordinates,is written as U NF = U NF0 ( R ) + U NFLS ( R ) L · σ p , (13)with U NF0 ( R ) = 2 Z V pp ( | r − R | ) ρ NF p ( r ) d r + 4 Z V pn ( | r − R | ) ρ NF n ( r ) d r (14)and U NFLS ( R ) = 13 Z V LS pp ( | r − R | ) (cid:26) − r · R R (cid:27) ρ NF p ( r ) d r + 23 Z V LS pn ( | r − R | ) (cid:26) − r · R R (cid:27) ρ NF n ( r ) d r , (15)where ρ NF n and ρ NF p denote point neutron and proton den-sities, respectively. B. Numerical evaluation of p - He potentials
To evaluate the p - He folding potentials as specifiedin the preceding section, we have to fix the followingelements; the p - α interaction V pα , the p - p and p - n in-teractions V pp and V pn , and the densities in He, ρ α , ρ p and ρ n . These are discussed in the following subsections,respectively. p - α interactions For V pα used in the CF potential, we assume the stan-dard WS potential such as given in Eq. (1). The pa-rameters involved are searched so as to fit the data of dσ/d Ω and A y in the p + α scattering at 72 MeV/nucleon2[23]. Particular attention was given to reproducing theobservables in the forward angular region, since overallagreements with the data are not found in spite of thecareful search of the parameters. Two typical parametersets, with and without the volume absorption term, arelabeled as Set-1 and Set-2 in Table VI. The results of cal-culations made with these potentials are compared withthe data in Fig. 13, where the solid and dashed lines showthose by Set-1 and Set-2 potentials, respectively. Bothcalculations describe the data up to θ ≈ ◦ but donot reproduce those at backward angles, θ & ◦ . Suchdiscrepancies between the calculated results and the mea-sured data at the backward angles suggest participationof contributions of other reaction mechanisms, such asknock-on type exchange scattering of the proton withtarget nucleons. Such possible extra mechanisms willbe disregarded at present since we are concerned withthe p - α one-body potential. In our CF calculations, weadopt the potentials with the above parameter sets as V pα . However, the validity of the CF potential thus ob-tained is limited to forward scattering angles, a low mo-mentum transfer region, of p + He scattering. The realand imaginary parts of V pα and the real part of V LS pα forthe above parameter sets are displayed in the upper andlower panels of Fig. 14. Although Set-1 (dashed) andSet-2 (solid) potentials have rather different r pα depen-dence, as shown later, this difference is moderated in thefolding procedure so yielding similar CF potentials. p - p and p - n interactions For V pn and V pp used in the CF and NF potentials,we adopt the complex effective interaction, CEG [33–35],where the nuclear force [36] is modified by the mediumeffect which takes account of the virtual excitation of nu-cleons of the nuclear matter up to 10 k F by the g -matrixtheory. The nuclear force is composed of Gaussian formfactors and the parameters contained are adjusted to sim-ulate the matrix elements of the Hamada-Johnston po-tential [37]. The CEG interaction has been successful inreproducing dσ/d Ω and A y measured for the proton elas-tic scattering by many nuclei in a wide incident energyrange, E p = 20–200 MeV, in the framework of the fold-ing model [33–35]. It has been shown that the imaginarypart of the folding potential given by the CEG interac-tion is slightly too large to reproduce experimental N-Ascattering [33, 35]. In the present calculation, therefore,we adopt the normalizing factor N I = 0 . N I = 1 .
3. Densities of α , p and n in He The densities ρ CF n and ρ CF α for the CF calcula-tion are obtained by applying the Gaussian expansion −1 He elastic at 72 MeV/u d σ / d Ω A y ( m b / s r) θ c.m. (deg) Exp. dataSet−1Set−2
FIG. 13: (Color online) Angular distribution of thecross section and A y for the p + He elastic scattering at72 MeV/nucleon. The solid and dashed lines are the opticalmodel calculations with Set-1 and Set-2 parameters, respec-tively. The experimental data are taken from Ref. [23]. method [30, 31] to the αnn cluster model of He. Thismethod has been successful in describing structures ofvarious few-body systems as well as He [30, 31]. Asfor the n - n interaction, we choose AV8’ interaction [38].It is reasonable to use a bare (free space) n - n interac-tion between the two valence neutrons in He as theyare dominantly in a region of low density. As for the α - n interaction, we employ the effective α - n potential inRef. [39], which was designed to reproduce well the low-lying states and low-energy-scattering phase shifts of the α - n system. The depth of the α - n potential is modi-fied slightly to adjust the ground-state binding energy of He to the empirical value. In Fig. 15(a), the densitiesobtained are shown as functions of r , the distance fromthe center of mass of He, where ρ CF α is localized in arelatively narrow region around the center, while ρ CF n isspread widely.The NF calculation depends on the assumptions madefor the densities of the two protons and four neutronsin He as well as those made for the p - p and p - n in-teractions [18]. At present, to see the essential role ofclustering the four nucleons into the α -particle core, weuse the densities of the proton and the neutron in the α obtained by decomposing the density of the point α , ρ CF α , to the densities of the constituent nucleons with aone-range Gaussian form factor with range 1.40 fm. The3 TABLE VI: Parameters for the optical potentials for p + He at 72 MeV/nucleon. V r r a r W r i a i W d r id a id r V s r s a s (MeV) (fm) (fm) (MeV) (fm) (fm) (MeV) (fm) (fm) (fm) (MeV) (fm) (fm)Set-1 64.13 0.7440 0.2562 6.338 1.450 0.2089 46.23 1.320 0.1100 1.400 2.752 1.100 0.2252Set-2 54.87 0.8566 0.09600 — — — 31.97 1.125 0.2811 1.400 3.925 0.8563 0.4914 −60−40−2000 1 2 3 4−10−50 Set−1Set−2 p − H e P o t e n ti a l ( M e V ) CentralSpin−Orbit
Real Imag. r p α (fm) Real
FIG. 14: (Color online) Optical potentials for the p + He elas-tic scattering at 72 MeV/nucleon. The thick dashed (solid)lines are for the real part of Set-1 (Set-2) potential and thethin dashed (solid) lines are for the imaginary part. total nucleon densities of He, ρ NF p , and ρ NF n , where thelatter includes the contribution of the valence neutrons,are displayed in Fig. 15(b). The neutron density ρ NF n hasa longer tail than the proton one ρ NF p due to the presenceof the valence neutrons. In Refs. [40, 41] the nucleon den-sities of He were calculated in a more sophisticated way.They produced densities similar to the present ones forthe protons and neutrons. These two kinds of nucleondensities provide similar results in the NF calculation of dσ/d
Ω and A y of the p + He scattering. Thus, in the fol-lowing, we will discuss U NF as formed using the densitiesshown in Fig. 15(b). p - He folding potentials
In Fig. 16, the resultant p - He potentials, U CF and U NF , are compared with each other as well as with theoptical model potential U OM . The CF potentials cal-culated by the two sets of V pα in Table VI, say U CF-1 and U CF-2 , are shown by long-dashed and solid lines inFig. 16(a), respectively. The folding procedure gives sim-ilar results in both cases. The contribution of V pn is dis-played by short-dashed lines in the figure, which is foundto be mostly small. Especially, in the spin-orbit poten- −5 −4 −3 −2 −1 −5 −4 −3 −2 −1 H e d e n s i t y ρ α CF ρ nCF ( f m − ) CF model H e d e n s i t y ρ pNF ρ nNF R (fm)
NF model (a)(b) ( f m − ) FIG. 15: (Color online) The densities of α , p , and n in Heused in folding models. The ρ CF α and ρ CF n in panel (a) areused in the CF calculation. The ρ NF p and ρ NF n in panel (b) are used in the NF calculation. All densities are normalizedas 4 π R ρ x r dr = 1, where x = α , n , and p . tial, the contribution from V pn is one order of magnitudesmaller than that from V pα . The main contribution to U CF arises from the interaction V pα except for the cen-tral real potential at R & V pn contribution. This is supposed to be the reflec-tion of the extended neutron density shown in Fig. 15and produce significant contributions to the observablesas discussed later.In Fig. 16(b), U CF due to Set-2 of V pα , U NF , and U OM are shown by solid, dot-dashed, and short-dashed lines,respectively. First we consider the central part of thepotentials. For small R , the real part of U CF0 is deeperthan those of U NF0 and U OM0 , while for large R , U CF0 isshallower than other two. In the imaginary part, themagnitude of U CF0 is much bigger than those of the othertwo potentials. This will compensate the deficiency of thereal part of U CF0 at large R , for example in the calculationof the cross section. On the other hand, the spin-orbitpart of U OM has larger magnitude for R & −50−40−30−20−100 −30−20−1000 1 2 3 4 5−2−10 −50−40−30−20−100 −30−20−1000 1 2 3 4 5−2−10 CF−1CF−2 2n part
Central
Real Imag.
Spin−Orbit
Real p − H e P o t e n ti a l ( M e V ) R (fm)
CF−2NFOMCF−2 (add ∆ U LS ) Central
RealImag.
Spin−Orbit
Real ( M e V ) R (fm) (a) (b) ×5×5×10
FIG. 16: (Color online) Potentials between proton and He. (a) : The long-dashed (solid) lines are the CF calculation with Set-1(Set-2) parameters for V pα . The short-dashed lines show the contribution of the valence two neutrons to the CF potential. (b) :The solid lines are the CF calculation with Set-2 parameters, the dot-dashed lines are the NF one and the short-dashed linesare the phenomenological optical potential. The CF spin-orbit potential corrected by ∆ U LS term is shown by the long-dashedline (see text for detail). two potentials. Such long-range nature of the spin-orbitinteraction is a characteristic feature of the spin-orbitpart of U OM as described in Section IV. This is discussedlater in more detail with relation to A y . C. Comparison between experiments andcalculations in p + He scattering
In the following, the dσ/d
Ω and A y for p - He elas-tic scattering calculated using U CF , U NF , and U OM arecompared with the data taken at 71 MeV/nucleon. InFig. 17(a) the results obtained using the two CF poten-tials, U CF-1 and U CF-2 , are shown by long-dashed andsolid lines, respectively. Both results are very similarto each other and well describe the data of dσ/d
Ω, ex-cept for large angles where the calculations overestimatesthe data by small amounts. The calculations also de-scribe the angular dependence of the measured A y upto θ ≃ ◦ . These successes basically support the CFpotential as a reasonable description of the scattering.The discrepancies at large angles, i.e. a large momentumtransfer region, may be related to the limitation of thevalidity of V pα used in the folding, as discussed in thesubsection V B.In Fig. 17(a), the results of the calculation made usingthe NF potential are shown by dot-dashed lines. Theydo not reproduce the data well. The calculation gives andeep valley around θ ≃ ◦ in the angular distributionof dσ/d Ω and a large positive peak at the correspond-ing angle of the A y angular distribution. These features do not exist in the data. Since the present nucleon den-sities originated from the CF model ones, the essentialdifference between the CF and NF potentials will be pro-duced by the use of the different interactions. Thus, theCF calculation will owe its successes to the inclusion ofthe characteristics of the realistic p - α interaction into the p - He potential.It is interesting to examine if the α core in He is some-what diffused compared with a free α -particle, due to theinteractions from the valence neutrons. For that purpose,we increased the radius and diffuseness parameters, r and a , in V pα potential as r to 1 . r and a to a + 0 . dσ/d Ω at large angles.In Fig. 17(b), the contributions of the valence neu-trons are demonstrated for the CF-2 calculation. As isspeculated from the analyses of the form factors of thepotential in Fig. 16(a), the dominant contribution to theobservables in the CF calculation arises from V pα dis-played by the dot-dashed lines in Fig. 17(b). However,the valence neutrons produce indispensable correctionsto the observables. That is, the pn central interactiondecreases dσ/d Ω at large angles, giving remarkable im-provements of the agreement with the data as shown bythe dashed lines. The pn interaction also contributes to A y by a considerable amount through the central part.A detailed examination of the calculation revealed thatsuch corrections were due to the V pn part of the folding5 −2 −1 CF−1CF−2NFCF−2 (increase r , a) p + He elastic at 71 MeV/u d σ / d Ω A y ( m b / s r) θ c.m. (deg) Ref. [22]Present (a) −2 −1 cent+so −cent+so centcent+so cent+so p + He elastic at 71 MeV/u d σ / d Ω A y ( m b / s r) θ c.m. (deg) Ref. [22]Present (b) Vp α Vpn −2 −1 CF−2CF−2 (add ∆ U LS )OM p + He elastic at 71 MeV/u d σ / d Ω A y ( m b / s r) θ c.m. (deg) Ref. [22]Present (c)
FIG. 17: (Color online) Angular distribution of the cross section and A y for the p + He elastic scattering at 71 MeV/nucleon.The experimental data are denoted by circles (present) and squares (Ref. [22]). (a) : The long-dashed (solid) lines are the CFcalculation in which Set-1 (Set-2) parameters are used for V pα . The dot-dashed lines are the NF calculation. (b) : The dashedlines and the solid ones include the V pn interaction, where the formers neglect the spin-orbit part of V pn . The dash-dotted linesinclude only V pα interaction. (c) : The solid lines are the CF calculation with Set-2 parameters for V pα and the short-dashedlines are the OM calculation. The long-dashed lines are the CF calculation with Set-2 parameters for V pα where ∆ U LS is added(see text for detail). R region between R =2 fm and 4 fm(see Fig. 16(a)). The spin-orbit part of V pn gives almostno effect to the observables as shown by the solid lines inFig. 17(b). This is consistent with the result shown byCrespo et al. [42] in a study at higher incident energy.In Fig. 17(c), we compared the results of the CF-2 cal-culation (solid lines) with those of the OM calculationin the preceding section (long-dashed lines) as well aswith the data. In the optical model analysis, the experi-mental data can be reproduced only with a shallow andlong-ranged spin-orbit potential. Compared with this po-tential, the spin-orbit part of CF-2 potential has a shorterrange as displayed by a solid line in the lower panel ofFig. 16(b). To investigate the role of the long tail inthe spin-orbit potential, we calculated the observablesby adding a weak but long-range spin-orbit interaction∆ U LS to the CF interaction. This correction is assumedto be the Thomas type as∆ U LS ( R ) = v add R ddR h n ( R − / r add ) /a add oi − . (16)For simplicity, we adopt v add = 1 MeV, r add = 1 . a add = 0 . r add and a add are consistent with the characteristics of themagnitudes of r s and a s of the OM potential discussedin Sec. IV. The calculated observables are displayed inFig. 17(c) by short-dashed lines, where dσ/d Ω is littleaffected but A y receives a drastic change, i.e. the angu-lar distribution of A y is now similar to that by the OMcalculation in a global sense showing qualitative improve-ments in comparison with the data. To see the contri-bution of ∆ U LS to the potential, we plot U CFLS + ∆ U LS in Fig. 16(b) by long-dashed lines, where the new spin-orbit potential becomes very close to that of the OMpotential at R & . A y , while itsmicroscopic origin is still to be investigated. When somecorrections which increase the range of the spin-orbit in-teraction are found, they will be effective for improvingthe CF calculation. VI. MICROSCOPIC MODEL ANALYSES
In this section, we will describe the theoretical analysisof the present data by a microscopic model developed inRef. [7]. In this model, one can predict the scatteringobservables such as cross sections and analyzing pow-ers with one run of the relevant code (DWBA98) withno adjustable parameter. Complete details as well asmany examples of use of this coordinate space micro-scopic model approach are to be found in the review [7].Use of the complex, non-local, nucleon-nucleus opticalpotentials defined in that way, without localization ofthe exchange amplitudes, has given predictions of differ-ential cross sections and spin observables that are in goodagreement with data from many nuclei ( He to
U) and for a wide range of energies (40 to 300 MeV). Cru-cial to that success is the use of effective nucleon-nucleon(
N N ) interactions built upon
N N g -matrices. The ef-fective
N N interactions are complex, energy and densitydependent, admixtures of Yukawa functions. They havecentral, two-nucleon tensor and two-nucleon spin-orbitcharacter. The
N A optical potentials result from fold-ing those effective interactions with the one-body densitymatrix elements (OBDME) of the ground state in the tar-get nucleus. Antisymmetrization of the projectile withall target nucleons leads to exchange amplitudes, makingthe microscopic optical potential non-local. For brevity,the optical potentials that result are called g -folding po-tentials. Another application has been in the predictionof integral observables of elastic scattering of both pro-tons and neutrons, with equal success [43]. Thus, themethod is known now to give good predictions of bothangular-dependent and integral observables.It is important to note that the level of agreement withdata in the g -folding approach depends on the quality ofthe structure model that is used. Due to the characterof the hadron force, proton scattering is preferentiallysensitive to the neutron matter distributions of nuclei;a sensitivity seen in a recent assessment, using protonelastic scattering, of diverse Skyrme-Hartree-Fock modelstructures for Pb [44].
A. Structure of He used He is a two-neutron halo nucleus and has been de-scribed well by shell model calculations. In calculation ofthe g -folding potential for protons interacting with He,a complete (0+2+4) ~ ω shell model calculation has beenmade to specify the ground state OBDME. Essentiallythey are the occupation numbers which define the mat-ter densities of the nucleus.In the present study, we assume three sets of the single-nucleon (SN) wave functions for He. One is the oscil-lator wave functions with an oscillator length of 2.0 fm(HO set). However, a neutron-halo character of He cannot be given by the oscillator wave function whatever os-cillator length is used as shown by the dashed curve inthe lower panel of Fig. 18. Thus, we assume two sets ofSN wave functions defined in Woods-Saxon (WS) poten-tials. One of them is obtained by taking the geometryof the potential from that found appropriate in Ref. [7],where electron form factors and proton scattering from , Li are studied. That study provided a set of SN wavefunctions that we specify as WS nonhalo set since the p -shell nucleons were all reasonably bound. The extendedneutron matter character of He is found by choosingthe binding energy of the halo-neutron orbits to give thesingle-neutron separation energy (1.8 MeV) to the lowestenergy resonance in He. The set of SN wave functionsthat result are specified as WS halo set. The associateddensity profile has the extensive neutron density comingfrom the halo. Density profiles given by the various sets7of SN wave functions are shown in Fig. 18. The dashed,solid, and dot-dashed curves show the density distribu-tions of HO, WS halo, and WS nonhalo sets, respectively.The difference between proton distributions of WS haloand WS nonhalo sets can not be seen.
FIG. 18: The (model) proton and neutron densities are shownin the upper and lower panels, respectively. The solid, dot-dashed, and dashed curves represent those obtained with WShalo, WS nonhalo, and HO sets, repectively.
Use of WS halo set in analyses of 40.9 MeV/nucleondata [45] gave a value of 406 mb for the reaction crosssection, which is in good agreement with the measuredvalue. Additional evidence for WS halo set is given bythe root mean square (r.m.s.) radius of the matter dis-tribution, which is most sensitive to characteristics ofthe outer surface of a nucleus. Using WS nonhalo set ofthe SN wave functions gave an r.m.s. radius for He of2.30 fm, which is much smaller than the expected valueof 2.54 fm. On the other hand, using WS halo set gavean r.m.s. radius for He of 2.59 fm in good agreementwith that expectation.
B. Differential cross sections and analyzing powers
The cross sections and analyzing powers for the p + Heelastic scattering at 71 MeV/nucleon are shown in thetop and bottom panels of Fig. 19, respectively. The cal-culated results shown therein by the dashed lines werefound using the g -folding potential obtained with HO set of SN wave functions. This calculation does not give asatisfactory result; especially in the case of the analyzingpower. The solid curves show results found using WShalo set while those depicted by the dot-dashed curvesare those found with WS nonhalo set. Of these the halodescription gives the better match to data especially atthe larger scattering angles. This result is consistent withthe findings from analyses of lower energy scattering dataat 40.9 MeV/nucleon [45] and at 24.5 MeV/nucleon [46]. FIG. 19: Differential cross sections and analyzing powers ofthe p + He elastic scattering at 71 MeV/nucleon (open cir-cles: Ref. [22], closed circles: present work). Three curvesare results of g -matrix folding calculation with He densitiespresented in Fig. 18.
In fact, the differential cross sections calculated withWS halo set match the data so well that one does notneed to contemplate any adjustment. However, the storyis not so simple when one also considers the analyzingpower data. At forward scattering angles, both WS setsreasonably match the data. But neither WS result pro-duces the distinctive trend of small values found at largerscattering angles. Nonetheless the best result is thatfound on using WS halo set of SN functions. Given thatthe cross section values in the region of 60 ◦ to 90 ◦ is of anorder of 0.1 mb/sr, the limitations in the present micro-scopic model formulation of the reaction dynamics maybe the problem.8 VII. SUMMARY
The vector analyzing power has been measured forthe elastic scattering of He from polarized protons at71 MeV/nucleon to investigate the characteristics of thespin-orbit potential between the proton and the He nu-cleus. Measurement of the polarization observable wasrealized in the RI-beam experiment by using the newlyconstructed solid polarized proton target, which can beoperated in a low magnetic field of 0.1 T and at high tem-perature of 100 K. The measured dσ/d
Ω of the p + Heelastic scattering were almost identical to those of the p + Li. On the other hand, the A y were found to belargely different from those of the p + Li and rather sim-ilar to those of the p + He elastic scattering.To extract the gross feature of the spin-orbit interac-tion between a proton and He, an optical model po-tential was determined phenomenologically by fitting theexperimental data of dσ/d
Ω and A y . Compared with theglobal systematics of the potentials for stable nuclei, it isindicated that the spin-orbit potential for He is charac-terized by a small value of V s and large values of r s and a s , namely by a shallow and long-ranged radial shape.Such characteristics might be the reflection of the dif-fused density of the neutron-rich He nucleus.The cluster folding calculation was carried out to geta deeper insight into the optical potential, assuming the α + n + n cluster structure for He. In addition, nucleonfolding calculations were also performed by decomposingthe α core into four nucleons. The experimental datacould not be reproduced by the nucleon folding calcula-tion, whereas the αnn cluster folding calculation givesthe reasonable agreements with the data. Thus, this in-dicates that it is important to take into account of the α -clusterization in the description of p + He elastic scat-tering. The cluster folding calculation shows that thedominant contribution to the p - He potential arises fromthe interaction between the proton and the α core. Es-pecially, in the spin-orbit potential, the contribution ofthe interaction between the proton and valence neutronswas found to be much smaller than the α core contri-bution. However, the measured cross section at largeangles can not be understood without the contributionfrom the scattering by the valence neutrons. Comparisonof the phenomenological optical potential and the clusterfolding one indicates that the long-range nature of thespin-orbit potential is important in reproducing the A y data at large angles. The microscopic origin of such along tail is still to be investigated.The data were also compared with the predictions ob-tained from a fully microscopic g -folding model. Threesets of single nucleon wave functions were tried sinceother details of the calculation were predetermined. Themodel, which has been successful in analyzing p + Hescattering cross sections in the past [44], again gives goodreproduction of the data in the present case when thebound state wave functions specify that He has a neu-tron halo. However, the match to the data, in particular the analyzing power, is not perfect. This may indicatelimitation of the structure model used and/or of unac-counted reaction mechanisms that influence the largermomentum transfer results.This work has demonstrated the capability of the solidpolarized proton target in low magnetic field and hightemperature to probe the new aspects of the reactioninvolving unstable nuclei. Future polarization studies ofsuch kinds will provide us with valuable information onthe reaction and structure of unstable nuclei.
Acknowledgments
We thank the staffs of RIKEN Nishina Center and CNSfor the operation of the accelerators and ion source dur-ing the measurement. S. S. acknowledges financial sup-port by a Grant-in-Aid for JSPS Fellows (No. 18-11398).This work was supported by the Grant-in-Aid for Scien-tific Research No. 17684005 of the Ministry of Education,Culture, Sports, Science, and Technology of Japan.
Appendix A: Absolute measurement of targetpolarization
In the case of conventional solid polarized targets, theNMR signal usually is related to the absolute magni-tude of the polarization by measuring the target po-larization under the state of thermal equilibrium (TE).However, measurement of the TE polarization is quitedifficult in our target. The first reason for this is thatthe TE polarization is very small in a low magneticfield and at high temperature, since it is represented by P TE = tanh (cid:16) µB kT (cid:17) , where µ, B , and T are the magneticmoment of proton, the field strength, and the tempera-ture, respectively. The second reason is that the sensi-tivity of the present NMR system is not sufficiently highsince the target design is optimized for scattering exper-iments.One of the simple methods to measure the absolute tar-get polarization would be the measurement of the spin-dependent asymmetry ǫ = P y A y for the proton elasticscattering whose analyzing power A y are known. In thepresent study, we measured the spin-asymmetry for the p + He elastic scattering at 80 MeV/nucleon. The A y have already been measured by Togawa et al [47]. Theuse of the p + He scattering is profitable since we canmeasure the ǫ with the same experimental setup as thatfor the p + He measurement only by changing settingsof the fragment separator RIPS to produce a secondary He beam. The profile of the He beam on the targetwas tuned to be almost same as that of the He beam.Figure 20 shows A y of the p + He elastic scatteringat 80 MeV/nucleon. The open circles represent the pre-vious data [47], while the closed ones show the presentdata whose magnitudes are scaled to the previous ones.9From the scaling factor, the average polarization dur-ing the p + He measurement was determined to be P y =12 . ± . P y /P y , which was 19% in the present work, resultedfrom the statistics of the p + He scattering events. Fu-ture development of the NMR system would be requiredfor determining the absolute polarization more preciselywithout losing beam time.
FIG. 20: Analyzing powers of the p + He elastic scatteringat 80 MeV/nucleon. Closed circles indicate the present datawhere P y = 12 .
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