Measurement of differential cross sections for deuteron-proton breakup reaction at 160 MeV
W. Parol, A. Kozela, K. Bodek, A. Deltuva, M. Eslami-Kalantari, J. Golak, N. Kalantar-Nayestanaki, G. Khatri, St. Kistryn, B. Kłos, J. Kuboś, P. Kulessa, A. Łobejko, A. Magiera, H. Mardanpour, J.G. Messchendorp, I. Mazumdar, R. Skibiński, I. Skwira-Chalot, E. Stephan, A. Ramazani-Moghaddam-Arani, D. Rozpędzik, A. Wilczek, H. Witała, B. Włoch, A. Wrońska, J. Zejma
aa r X i v : . [ nu c l - e x ] A p r Measurement of differential cross sections for deuteron-proton breakup reaction at160 MeV
W. Parol, ∗ A. Kozela, K. Bodek, A. Deltuva, M. Eslami-Kalantari, J. Golak, N. Kalantar-Nayestanaki, G. Khatri, St. Kistryn, B. K los, J. Kubo´s, P. Kulessa, A. Lobejko, A. Magiera, H. Mardanpour, J.G. Messchendorp, I. Mazumdar, R. Skibi´nski, I. Skwira-Chalot, E. Stephan, A. Ramazani-Moghaddam-Arani, D. Rozp¸edzik, A. Wilczek, H. Wita la, B. W loch, A. Wro´nska, and J. Zejma Institute of Nuclear Physics Polish Academy of Sciences, PL-31342 Krak´ow M. Smoluchowski Institute of Physics, Jagiellonian University, PL-30348 Krak´ow, Poland Institute of Theoretical Physics and Astronomy, Vilnius University, Vilnius, Lithuania Department of Physics, School of Science, Yazd University, Yazd, Iran KVI-CART, University of Groningen, NL-9747 AA Groningen, The Netherlands CERN, CH-1211 Geneva 23, Switzerland Institute of Physics, University of Silesia, PL-41500 Chorz´ow, Poland Forschungszentrum Juelich IKP, DE-52425 Juelich, Germany Tata Institute of Fundamental Research, Mumbai 400 005, India Faculty of Physics, University of Warsaw, PL-02093 Warsaw, Poland Departments of Physics, Faculty of Science, University of Kashan, Kashan, Iran (Dated: April 7, 2020)Differential cross sections for deuteron breakup H ( d, pp ) n reaction were measured for a large setof 243 geometrical configurations at the beam energy of 80 MeV/nucleon. The cross section data arenormalized by the luminosity factor obtained on the basis of simultaneous measurement of elasticscattering channel and the existing cross section data for this process. The results are comparedto the theoretical calculations modeling nuclear interaction with and without taking into accountthe three–nucleon force (3NF) and Coulomb interaction. In the validated region of the phase spaceboth the Coulomb force and 3NF play an important role in a good description of the data. Thereare also regions, where the improvements of description due to including 3NF are not sufficient. I. INTRODUCTION
One of the most basic topics in modern nuclear physicsis the nature of the forces acting between nucleons. Exactknowledge of all features of the two–nucleon (NN) sys-tem dynamics should provide a basis for understandingof properties and interactions in heavier systems. Thispresumption has been verified by applying models of theNN interaction to describe systems composed of threenucleons (3N). Theoretical predictions of observables areobtained by means of the rigorous solution of Faddeevequations [1–4], including NN interaction as so-called re-alistic potential models, based on the meson exchangetheory, originally proposed by Yukawa [5] and confirmedby Occhialini and Powell [6]. Early stage of experimen-tal studies of the deuteron-proton elastic scattering inthe range of intermediate energies and theoretical efforts[7] have proven the dominant, but not sufficient, roleof the pairwise NN interaction. The missing piece ofthe dynamics, referred to as three–nucleon force (3NF),also contributes. The effects of this force, much smallerthan parwise NN contribution, arise in systems consist-ing of at least three nucleons. Modern NN potentialslike Argonne V18 (AV18) [8], CD Bonn (CDB) [9], andNijmegen I and II [10] have yielded a remarkably goodagreement (with a χ of around 1) between the predic- ∗ [email protected]; [email protected] tions of the calculations with the experimental data fortwo-nucleon systems. To describe three–nucleon systemsthese realistic NN potentials are used in Faddeev equa-tions together with present models of 3NF like UrbanaIX [11] or Tucson-Melbourne [12]. In another approach,three–nucleon interaction can be introduced within thecoupled-channel (CC) framework by an explicit treat-ment of the ∆-isobar excitation [13–15]. Alternatively,contributions of NN and 3NF to the potential energy ofa 3N system can be calculated within the Chiral Pertur-bation Theory [16, 17]. Here, the many-body interactionsappear naturally at higher orders (non-vanishing 3NFat next-to-next-to leading order). Modern calculationsinclude also other ingredients of few–nucleon dynamicssuch as Coulomb interactions [18, 19] or relativistic ef-fects [20, 21]. Predicted effects in differential cross sec-tions emerge in various parts of the phase space of thedeuteron-proton breakup reaction with different magni-tude. Existing experimental data [22–28] demonstratequite sizable 3NF and Coulomb effects, and confirm theirimportance for the correct description of differential crosssections for the deuteron breakup reaction at energiesabove 65 MeV/nucleon and below 400 MeV/nucleon.The present work is a continuation of experimentalcampaign focusing on the investigation of contributionsfrom various dynamical ingredients (3NF, Coulomb forceand relativistic component) of nuclear interaction viameasuring various observables in few-nucleon systems forlarge parts of the phase space. Measured differentialcross sections at 80 MeV/nucleon enlarges the system- FIG. 1. Schematic side view of BINA detection system atic database for the deuteron–proton breakup reactionat intermediate energies. Produced feedback allows forfurther validation of available and future theoretical mod-els of nuclear interaction.In Section II the experimental setup is described. Sec-tion III gives an overview of the data analysis, while insection IV the obtained results are presented. Section Vsummarizes the main outcome of the presented studies.
II. EXPERIMENT
The experiment was performed at Kernfysisch Ver-sneller Instituut (KVI) in Groningen, the Netherlands(currently KVI-CART). The deuteron breakup reac-tion, H ( d, pp ) n , was measured simultaneously with elas-tic scattering of deuterons on liquid hydrogen target.Deuteron beam of 80 MeV/nucleon energy was pro-vided by the cyclotron AGOR (Accelerateur Groningen-ORsay) [29], while charged reaction products weredetected by the BINA (Big Instrument for Nuclear-polarization Analysis, [30]) setup. The BINA detectionsystem is characterized by: high angular acceptance (al-most 4 π ), good (in forward region) and moderate (inbackward part) angular resolution, the ability to identifyand to provide complete kinematical information for twoor more charged particles in the final state. All thesefeatures make the BINA detector an excellent tool forstudying the systems of few nucleons in the intermediateenergy range.The BINA detection system, Fig. 1, consist of two mainparts, forward Wall and backward Ball. The liquid hy-drogen target cell is positioned in the center of the Ball,which served in this experiment as the scattering cham-ber only. The front part, Wall, consists of three detectorelements positioned in planes perpendicular to the beamline: a multi-wire proportional chamber (MWPC) and two scintillator hodoscopes, forming a set of 120 ∆E–Evirtual telescopes. The Wall is optimized for detectingprotons and deuterons in the energy ranges of 20–130MeV and 25–200 MeV, respectively.Precise measurement of scattering angles is accom-plished by MWPC [31] positioned directly behind thethin vacuum window as the first detector intersected bythe reaction products. It consists of three active planes: aplane measuring x-coordinate with vertical wires, a planemeasuring y-coordinate with horizontal wires, and a diag-onal plane U with wires inclined by 45 degrees. All wireswithin a plane are spaced by 2 mm and combined in pairsto form 118, 118 and 148 separate detector channels forX, Y and U planes, respectively. The active area of theMWPC is 38 ×
38 cm . It forms a pixel system allowingto precisely determine the crossing point of a chargedparticle, and thus to reconstruct the emission angles ofthe outgoing reaction products. The angular acceptanceof the detector in polar angles is ϑ ∈ (10 ◦ , ◦ ), with thefull azimuthal coverage up to 30 ◦ .The ∆E transmission detector consists of 24 verticalstrips of a 2 mm thick plastic scintillator (BICRON typeBC-408 [32]). The signals from each ∆E stripe are readby one photomultiplier tube (PMT) coupled through alight guide modeled for optimal light collection. As thesignals are proportional to the specific energy loss ofcharged particles they play a crucial role in particle iden-tification.The E detector is made of horizontally–arranged120 mm thick scintillator slabs (BICRON type BC-408[32]). In order to minimize particle cross-overs betweenneighboring scintillators central ten elements of E detec-tor follow a cylindrical symmetry, with the cylinder cen-ter at the target position (see Fig. 1). The additionalten elements attached from the top and bottom to thecylindrical part were not used in the present experiment.Energy deposited by particles in the E slab was convertedto scintillation light registered by two PMTs, attached toboth ends of each detector. This allows to compensatefor the light attenuation along the scintillator resultingin a position independent output.Other details concerning the setup as well as electronicand read-out systems used in the experiment can befound in Ref. [33]. Data acquisition system was basedon GSI Multi-Branch System (MBS) [34]. III. DATA ANALYSIS
The data analysis started with the selection of timeperiods characterized by stable operation of the cyclotronand all elements of the detector. The selection was basedon the scaler rates recorded for all individual channels ofthe detector. In order to minimize random coincidences,additional time gates rejecting particles not correlatedwith the trigger signal had been set.
A. Tracking of individual particles
1. Track Reconstruction
Charged particles passing through the Wall detectordeposit their energy successively in MWPC, ∆E and E.In the simplest case, for a single particle only three wires(one wire per plane) in MWPC give a signal while inreality, clusters of two or more wires are observed. Asno signal amplitude from MWPC was collected, in suchcases the hit position is represented by the center of thecluster.In order to accept the track information from theMWPC several different strategies may be used. Inthe analysis published in several earlier papers e.g. inRef. [35], a coincidence of all three planes was required,with the X and Y planes defining ( x, y ) coordinates of theintersection of the track with the MWPC and the U-planewas used to validate this intersection. In the data analy-sis of deuteron-deuteron scattering presented in Ref. [36]we accepted also events with only two planes hit, with thecondition that no other hits are present in MWPC andthe resulting position information is correlated with hitsin ∆E and E detectors. This kind of tracks, which arefurther referred to as ”weak–tracks” (as opposed to ”full–tracks” indicating 3-plane coincidences), is important forconsideration of systematic effects such as e.g. energyand position dependent MWPC efficiency. The analysispresented in this paper is based on full–tracks and, inaddition, we take advantage from the position informa-tion supplied by the U-plane. In such a case, the finalposition is given by the center of a circle inscribed in thetriangle defined by corresponding cluster centroids pro-jected (from the target center direction) onto a commonplane (see Fig. 2). Assuming equal position resolution ofall planes, this algorithm improves the final angular reso-lution for the polar angle ϑ up to 0 . ◦ and the azimuthalangle ϕ to 0 . ◦ − . ◦ (depending on the polar angle).It is also clear that weak–tracks involving U-plane fea-ture lower position resolution in one direction than thosedefined by X and Y plane. It is important to mentionthat 3-dimensional track parameters, in this case polarand azimuthal angles, can be obtained by the followingformulas: ϑ = arctan p x + y Z Y ! , (1) ϕ = atan2( y, x ) , (2)under the assumption that the corresponding particle wasemitted from the target center ( Z Y is the distance of theprojection plane from this center). The atan2() functioncalculates the principal value of the arctan (cid:0) yx (cid:1) , using thesigns of the two arguments to determine the quadrant ofthe result [37]. Smearing of the reaction point due to tar-get thickness and beam size is included into systematicuncertainty of the final results. Having those parame-ters one may check if the track coincides, within given FIG. 2.
Panel a:
Geometrical reconstruction of the ( x, y )coordinates of the full–track intersection with the Y-plane.Lines X, Y and U represent centroids of clusters in the re-spective planes projected onto Y-plane. Reconstructed ( x, y )coordinates for all three types of weak–tracks are shown asgreen dots.
Panel b:
Distribution of d (defined in the panela) for the whole dataset. Limit of 7 mm for d corresponds tothe ∼ σ of the fitted gaussian distribution (red solid line). position resolution, with hits in the E and ∆E detec-tor elements – only such events are considered in furtheranalysis. In order to combine the hits in the individualplanes into the full–track event, a cut has been imposedon the distance between the centroid of the cluster re-constructed in the U-plane and the cross-point betweencentroids in the X and Y planes ( d -variable in Fig. 2).
2. Particle identification
Neglecting traces of heavier ions from beam interac-tions with the target frames, the particle identificationcan be reduced in this experiment to simple distinctionbetween protons and deuterons, while the later ones comeexclusively from the elastic scattering. The identificationis based on the linearization technique applied to the ∆E-E spectra [38]. It allows for identification of reactionproducts by analytically-determined conditions. Follow-ing simplified consideration based on Bethe-Bloch for-mula one introduces a new variable e E = ( E +∆ E ) κ − E κ ,the value of which is constant for each type of particlesin wide energy range [39]. The index κ characterizesthe detector material, its internal structure (variations oftransparency, quality of the surface) and geometry andis determined for each virtual telescope separately. As aresult, one-dimensional distribution of e E variable is ob-tained in which protons and deuterons are visible as dis-tinct peaks (Fig. 3). In order to improve the sensitivityof this method, the fine tuning of κ (as well as of the pa-rameters: µ p , σ p , µ d , σ d corresponding to the centroidsand widths of proton and deuteron peaks, respectively)was performed for each virtual ∆E-E telescope. For thispurpose a sample of the data with well balanced numberof protons and deuterons (from dd scattering experimentat the same beam energy) has been used. The κ indexhas been varied to get maximal separation between pro-ton and deuteron peaks. The obtained final values of κ FIG. 3. Example of the identification spectra for a chosenvirtual telescope (∆E=13, E=8).
Panel a: ∆E-E signal dis-tribution. The violet line separates the proton and deuteronbands and corresponds to the vertical line indicated in thedistribution on the right side.
Panel b:
Projection of thelinearized spectrum onto the e E variable. The two peaks cor-respond to the proton and deuteron bands. The 2 σ ranges ofthe fitted Gaussian functions are shown as red and blue linesfor protons and deuterons, respectively. range from 1.63 to 1.85, while according to Bethe-Blochrule κ = 1 .
73 is expected for an ideal scintillator. Thismethod allows for controllable selection of different eventsamples not biased by subjective cuts.
3. Energy Reconstruction
Energy calibration gives a relation between the reg-istered ADC channel and the deposited energy ( E D )in a given scintillator element. Since 2 mm thick ∆Estripes remove a relatively small fraction of particle en-ergy and, furthermore, this information is strongly bi-ased by light attenuation along the scintillator and thelight guide, only E detector was used for reconstruction ofparticles energies. The calibration was carried out usingprotons from elastic scattering and Monte-Carlo simu-lations including full detector geometry implemented inthe GEANT4 simulation package [40, 41]. The detectoris characterized by noticeable variation of the PMT sig-nal amplitude depending on the point of the interactionalong the scintillator. This dependence, caused by lightattenuation and losses, can be significantly suppressedby applying geometrical mean of responses of the leftand right PMTs ( C = √ c L · c R ). For the two middle Eslabs, partially cut in the center in order to accommodatean opening for the beam pipe, plain sum of the signals( C = c L + c R ) was applied. The remaining small depen-dence of the signal on position is taken into account inthe position dependent energy calibration.In order to extend the calibration over energies of pro-tons from the breakup reaction, a dedicated measurementhas been performed using energy degraders, placed be-tween the ∆E and E detectors. Degraders were made ofsteel plates of precisely-defined thicknesses, which weremounted in several configurations allowing for satisfac- FIG. 4. Energy calibration.
Panel a:
Example of the correla-tion between experimentally obtained centroids of the distri-bution of the variable C and of the corresponding distributionof the simulated energy deposited in E detector together withthe fitted function defined in Eq. (3). Panel b:
Set of poly-nomials transforming energy deposited by proton, E D ( ϑ ), toits initial kinetic energy at the reaction point. tory coverage of the energy range. The elastically scat-tered protons were selected according to kinematical con-ditions: co-planarity and ϑ p vs. ϑ d relations. The detec-tor plane was divided into 180 sectors, each of them la-beled by the side ( s = lef t, right ), E-scintillator elementnumber ( N = 0 , , . . . ,
9) and the polar angle bin number( ¯ ϑ = 0 , . . . , E Ds,N, ¯ ϑ ( C ) = a s,N, ¯ ϑ C + b s,N, ¯ ϑ √ C, (3)where take out: E Ds,N, ¯ ϑ ( C ) stands for the energy de-posited in this particular detector element as a functionof variable C , i.e. the combination of signals from leftand right PMTs defined above. An example of the fit isshown in Fig. 4a.Deuteron energy calibration was based on that for pro-tons and corrected for different light output correspond-ing to the same energy deposited by particles of differ-ent mass. Particle-dependent light output for the knownscintillator material has been taken from Bicron datasheets [32], and additionally validated in the dedicatedstudies (see Ref. [42] for the details).To reproduce the initial kinetic energy of a particleat the reaction point ( E i ), the conversion formula hasbeen found based on the energy loss of simulated mono-energetic protons and deuterons on their way to and in-side the E detector: E i ( ϑ ) = P i,ϑ ( E Di ( ϑ )) , (4)where subscript i stands for particle type (proton ordeuteron) and { P i,ϑ } is a set of 8 th –order polynomialswith factors calculated from deposited-to-initial energyrelations obtained from GEANT4 simulations of the ex-periment (see Fig. 4b).Final energy resolution reaches 2.1% for 123 MeV pro-tons, and deteriorates with energy in accordance withphoton statistics.
10 20 30 40 [deg] p ϑ [ deg ] d ϑ a)
10 20 30 40 [deg] p ϑ [ M e V ] p E b) FIG. 5.
Panel a:
Kinematic relation between polar angles ofcoincident coplanar particles with ± σ cut around the theo-retical kinematics of dp elastic scattering (black line). Panelb:
Energy distribution of particles identified on the basis ofcoplanarity and polar angle cut (shown in the left panel) aselastically scattered protons.
E [arb. units] E [ a r b . un i t s ] ∆ a) E [arb. units] E [ a r b . un i t s ] ∆ b) FIG. 6. Relation of particle energies deposited in chosen∆E-E telescope (E=1, ∆E=9).
Panel a: spectrum for parti-cles registered in coincidence with any other particle in Wall;
Panel b: the same with an additional condition that coinci-dent particles meet angular kinematical relations of the elasticscattering.
B. Process identification
Neglecting small admixtures of electromagnetic pro-cesses, deuteron–proton interaction in the studied en-ergy range may result in elastic scattering or breakup.A deuteron in the exit channel uniquely identifies the re-action as elastic scattering, while two protons have tobe identified in order to identify the breakup reaction.This goal can be accomplished for most of the recordedevents using the PID procedure described in Sec. III A 2.On the other hand, precise measurement of angles andstrict kinematic relations of the scattering angles and en-ergies in the elastic deuteron-proton scattering (Fig. 5)allow for correct identification of the process even whenthe regular PID method fails. As a consequence two–track events can be identified as deuteron-proton elasticscattering even when one, or both particles underwenthadronic interaction in the thick scintillator, which in-fluences the energy measurement and prevents successfulapplication of ∆E-E–based the PID method (Fig. 6).
C. Detector efficiency
Determination of a true number of events of a giventype from the number of events registered by the detec-tor requires knowledge of the detector efficiency. Sincesuccessful registration of a single particle is conditionedwith complete information from three detectors (MWPC,∆E and E), the total detection efficiency, ε ( x, y ), can beconsidered as a product of individual efficiencies of thosedetectors. ε ( x, y ) = ε MWPC ( x, y ) · ε ∆ E ( x, y ) · ε E (5)The efficiency of the E detector ( ε E ) has been assumedto be 100%. This is justified by a very tight fitting of thedetector slabs. The gap between adjacent elements is ofthe order of 50 microns as compared to 10 cm width ofthe front face of each element. Therefore, the problemof efficiency reduces to particle-type dependent energythreshold.Since the main sources of inefficiency for both ∆E andMWPC detectors are well localized, proper accountingfor them required construction of position-dependent ef-ficiency maps. This was possible using the position in-formation from MWPC.The efficiency of the ∆E detector, ε ∆ E ( x, y ) (Fig. 7d),is calculated directly on the basis of a single particleevents according to the following formula: ε ∆ E ( x, y ) = N ref+∆ E ( x, y ) N ref ( x, y ) , (6)where the reference number of events, N ref , correspondsto the number of all particles registered by MWPC witha correlated hit in the E scintillator, regardless of the∆E information, while for N ref+∆ E , additional matchingwith information from the ∆E detector was required.Position dependent efficiency maps of the multi-wireproportional chamber have been obtained from singleplane efficiencies ( ε X , ε Y , ε U ) according to the followingformulas: ε MWPC ( x, y ) = ε X ( x, y ) · ε Y ( x, y ) · ε U ( x, y ) , (7) ε weakMWPC ( x, y ) = ε MWPC ( x, y ) (8)+ ε X ( x, y ) · ε Y ( x, y ) · (1 − ε U ( x, y ))+ ε X ( x, y ) · (1 − ε Y ( x, y )) · ε U ( x, y )+ (1 − ε X ( x, y )) · ε Y ( x, y ) · ε U ( x, y ) . The first one (Eq. (7)) corresponds to full–tracks, whenthe coincidence of all 3 planes was required, and the sec-ond (Eq. (8)) to the analysis in which also weak–trackswere accepted (see section III A 1). Efficiencies of individ-ual planes have been calculated using position informa-tion from the remaining two planes, and requiring infor-mation from both scintillator hodoscopes, in a way anal-ogous to the one already introduced for ∆E. It is clearthat the efficiency in the approach allowing for weak–tracks is much less sensitive to local defects (dead chan-nels) present in each plane. Average efficiency in thiscase was as high as 98% (see Fig. 7c), as compared to85% for full–tracks (see Fig. 7b). In order to better un-derstand systematics associated with MWPC efficiency,full analysis of cross section has been performed with andwithout accepting weak–tracks. The good agreement ob-tained strengthens our confidence in the final results ofefficiency calculations [43]. The data analysis is based onthe full–tracks due to better angular resolution achievedin this way.The MWPC efficiency depends also on the particletype and energy and this effect is visible in efficiencymaps for full–tracks. This in fact can be traced back tothe dependence on the relative energy loss of a given par-ticle within the detector gas-mixture. In order to accountfor this effect, a new variable was introduced: E loss ∼ q mE k , (9)where q and m are the particles’ charge and mass, while E k denotes their kinetic energy. This variable was nor-malized in a way that for the most energetic among allthe registered particles, elastically-scattered protons, itequals to one. Fig. 7a presents distribution of the E loss (regardless the particle type) and the average efficiencyobtained for various bins in E loss . In the final analy-sis, in order to retain acceptable statistics in each binof the efficiency map, only three satisfactory populatedranges of E loss have been defined, as shown in Fig. 7a.The final efficiency map for minimum ionizing particles(Fig. 7b) registered in this experiment (region markedas (1) in E loss distribution) is compared to the efficiencymaps constructed for ε weakMWPC (Fig. 7c). Because of con-tact problems at hardly accessible places, certain elec-tronics channels of MWPC did not work. There arequite numerous inefficient regions in MWPC, especiallyfor full–tracks corresponding to region (1) of E loss . Thecorrection defined in Eq. (7) is not effective for crossinginefficient wires in two or more planes. In any such casesor in more general cases of low final detector efficiency ε ( x, y ) < .
1. Configurational Efficiency
When two particles enter the same detector elementthe reconstructed information is distorted. This leadsto false energy reconstruction and may corrupt particleidentification. Such effect, further referred to as configu-rational efficiency, depends strongly on geometry of thefinal state and of the detector, and can be accounted forby Monte–Carlo simulations.Due to coplanarity condition, the loss of events cor-responding to elastic scattering due to configurationalefficiency is practically negligible. For the breakup reac-tion the efficiency strongly depends on breakup kinemat-ics, defined by polar angles of emission of both protons, − − − X [mm] − − − Y [ mm ] b) Arbitrary Energy Loss E ff i c i en cy (1) (2) (3) a) − − − X [mm] − − − Y [ mm ] c) − − − X [mm] − − − Y [ mm ] d) FIG. 7.
Panel a:
The average MWPC efficiency (blue dotsconnected by line) as a function of the energy loss in thegas mixture, Eq. 9. The distribution of events is shown ingray.
Panels b and c:
Position dependence of the MWPCefficiency shown only for the range (1), separately for full-(b) and weak–tracks (c).
Panel d:
Analogous map of the ∆Eefficiency. ϑ and ϑ , and relative azimuthal angle, ϕ , betweenthem. Due to axial symmetry of the cross section, theso-defined configuration is rotated around the beam axis.The configurational efficiency is determined by the anal-ysis of a set of breakup events simulated with the use ofGeant4 framework with the Wall detector geometry in-cluded. Since a good statistical accuracy of such correc-tion has been ensured, the only significant uncertaintymay originate from the inaccuracies of the experimen-tal setup (detector or beam geometry) and the appliedmodel of an event generator. In the following, the uni-form 3-body breakup phase space distribution has beenused, which is well justified in the case of narrow angularranges applied in defining the configuration. The configu-rational efficiency for a given geometry ( ϑ , ϑ , ϕ ) is de-fined as the ratio of the number of events for which bothparticles were registered by separate detector elements,to the number of all simulated events. As expected, theconfiguration efficiency rises with increase of ϕ , withpronounced local minima reflecting the structure of theE detector (Fig. 8). Due to much finer granularity of theMWPC as compared to the hodoscopes, the contributionof this detector to the configurational efficiency is verysmall. In the simulation, the distribution of cluster sizesobserved in the experiment was used to account for hitlosses due to coalescence of clusters produced by differ-ent particles. The final correction for acceptance lossestakes into account losses due to two particles registeredin the same element and, earlier-discussed regions of low
20 40 60 80 100 120 140 160 180 [deg] ϕ ) ξ ( c ε FIG. 8. Configurational efficiency for a set of breakup proton-proton configurations characterized with ϑ = 27 ◦ , ϑ = 21 ◦ and ∆ ϕ = 10 ◦ . General trend, observed also for other com-binations of polar angles, shows the decrease of the efficiencywith decreasing relative azimuthal angle between protons, ϕ , with local minima determined by the geometry of theE detector. efficiency. It is calculated as follows: ε c ( ξ ) = N rec ( ξ ) N tot ( ξ ) , (10)where ξ defines the geometry of the reaction products: ξ = { ϑ , ϑ , ϕ } for the breakup reaction and ξ = { ϑ p } for the elastic-scattering channel, N tot is the total num-ber of coincidences generated for this configuration and N rec counts only those events which are successfully reg-istered by the virtual BINA detector.The total correction factor related to the efficienciesfor registering of a number ( N ) of coincident events inthe chosen configuration ξ can be written as: ε ξN = N ε c ( ξ ) X ε ( x i , y i ) · ε ( x j , y j ) − , (11)where ǫ ( x i , y i ) is the single particle efficiency defined inEq. 11 and < i, j > symbolizes the set of N coincidentpairs.
2. Hadronic reactions
The calibration and particle identification proceduresfail when the particle undergoes a hadronic reaction withlarge momentum transfer on its way to- or inside the Edetector. In such cases a part of particle energy is lost,less light is produced in the scintillator, and as a con-sequence, the reconstructed kinetic energy is underesti-mated, leading to event rejection due to the PID cut.The amount of affected events has been estimated basedon experimental ∆E–E spectra gated by kinematical con-ditions defining dp elastic scattering (Fig. 9a) in order to E [MeV] E [ a r b . un i t s ] ∆ a) [MeV] p E R b) FIG. 9.
Panel a: sample of ∆E-E spectrum for chosen polarangle of elastically-scattered protons ϑ p = 31 ◦ ± ◦ . Hor-izontal band extending on the left from elastic spot corre-sponds to hadronic interactions lowering the registered en-ergy. Events below dashed line are used in the calculationof tail-to-peak ratio R . Panel b:
Obtained tail-to-peak ratio(points) compared to theoretical calculations at lower energies[44] (lines) and to simulations [35] (triangles). Solid line rep-resents predictions of a simple model based on the materialcomposition of the scintillator [44] extrapolated to energiesabove 100 MeV. reject the breakup band. In these spectra hadronic inter-actions are visible as horizontal band protruding on thelow-energy side from the elastic-scattering spot. Numberof events integrated within this band has been normal-ized to the number of events inside the elastic peak (tail-to-peak ratio R ). Results obtained for several energiesare in satisfactory agreement with theoretical predictionsbased on the effective inelastic cross section model forprotons in a combination of materials building the plasticscintillator [44] extrapolated to energies above 100 MeV(Fig. 9b). Therefore, the R values corresponding to thesolid line in Fig. 9, considered as validated, were used infurther analysis with up to 3.8% systematic uncertainty.In order to account for the loss of breakup–originatedprotons due to hadronic interactions, a dedicated correc-tion factor ( η ( E p )) has been introduced on the basis of R : η ( E p ) = 1 + R. (12)In the case of elastic scattering protons and deuterons areidentified on the basis of kinematical relations of angles.As a consequence, events were lost only if hadronic in-teractions had occurred before the particle reached scin-tillators and the corresponding correction was negligible. D. Luminosity
Direct measurement of absolute differential cross sec-tions requires precise knowledge of the beam current, tar-get thickness and scattering angles of the reaction prod-ucts. In the present experiment, neither target cell sur-face density nor very low beam current (in the range
10 20 30 40 [deg] p ϑ ] - [ m b s r l ab σ Interpolated 80 MeVCDB + TM99 65 MeV/nucleon 77 MeV/nucleon108 MeV/nucleon120 MeV/nucleon135 MeV/nucleon150 MeV/nucleon170 MeV/nucleon190 MeV/nucleon
FIG. 10. Interpolated cross section for deuteron-proton elasticscattering at the energy of 80 MeV/nucleon (violet) in thelaboratory frame in comparison to the theoretical calculationsbased on CDB+TM99 potential [49]. The experimental dataused for this interpolation are shown in black. of few pA) were known sufficiently precisely. The nor-malization factor had to be obtained on the basis ofsimultaneously-measured elastic dp scattering and thecorresponding cross section derived from previous exper-iments. Since no published cross section data for the dp elastic scattering process at 80 MeV/nucleon exist,model independent interpolation has been done basedon all existing experimental data in the range of 65–190 MeV/nucleon [45–48]. The obtained absolute val-ues of the differential cross section, σ lab , agree very wellwith theoretical calculations based on the Charge De-pendent Bonn potential supplemented with the TM99three–nucleon force [49] (Fig. 10). The reference datawere subsequently used to calculate experimental inte-grated luminosity according to: L ( ϑ p ) = N pd ( ϑ p ) σ lab ( ϑ p ) · ∆Ω · ε ( ϑ p ) , (13)where ∆Ω is the solid angle and N pd ( ϑ p ) is the num-ber of elastically-scattered protons registered in certainbin in polar angle of proton emission. To reduce theinfluence of acceptance losses resulting from reachingedges of a square-like detector, the elastically-scatteredparticles were collected from limited space close to di-agonals of MWPC defined by azimuthal angle ϕ = { ◦ , ◦ , − ◦ , − ◦ } with tolerance ± ◦ .Obtained values are, as expected, consistent withone another within experimental uncertainties (Fig. 11).As the final integrated luminosity value, the average¯ L = (19 . ± . stat ± . syst ) · [mb − ] was taken.
20 25 30 [deg] p ϑ × ] - L [ m b -1 mb ⋅ ± ± = (19.68 L FIG. 11. Luminosity integrated over time determined on thebasis of elastic scattering, independently for each proton polarangle. Statistical errors are negligible, while gray shade cor-responds to the range of systematic errors of individual datapoints. The average value of the luminosity (violet dashedline) is known with accuracy dominated by systematic uncer-tainty (dashed black lines).
E. Differential Cross Sections for Breakup Process
The breakup cross section has been determined for 243angular configurations. The angular range for integrationof events has been chosen to 2 ◦ in polar and 20 ◦ in az-imuthal angles. For each configuration, breakup eventsare placed on the ( E , E ) plane, in which they groupalong the corresponding kinematical curve (Fig. 12a).The width of the distribution in the direction perpendic-ular to the curve, D –coordinate, depends on the energyresolution and spread of kinematics corresponding to thesize of the angular bin. Arc-length measured along thekinematic curve is used to define S -coordinate [50].The five-fold differential cross section for the deuteronbreakup reaction has been calculated according to thefollowing formula: d σ ( ξ, S ) d Ω d Ω dS = N BR ( ξ, S ) · η ( E ) · η ( E ) ε ξN BR · L · ∆Ω · ∆Ω · ∆ S , (14)where L is the luminosity integrated over time and N BR corresponds to the number of events falling into thechosen geometry ξ within the ranges of integration de-fined below. ε ξ is the total detector efficiency as definedin Eq. (11), and η ( E ) , η ( E ) account for the energy-dependent hadronic interaction corrections. Ordering ofprotons in the case of analysis of symmetric configura-tions ( ϑ = ϑ ) is random, while in the case of asymmet-ric configurations ( ϑ = ϑ ), the proton scattered at a E1 [MeV] E [ M e V ] S ∆ DS ° =23 ϑ ° =17 ϑ ° =40 ϕ a)
60 80 100 120 140
S [MeV] ] - s r - [ m b M e V d S Ω d Ω d σ d exp. points2N2N+TM99AV18+CAV18+UIX+C+C ∆ CDB+ ° =23 ϑ ° =17 ϑ ° =40 ϕ b) FIG. 12.
Panel a:
Distribution of events for a selectedbreakup configuration together with the corresponding kine-matic curve; definitions of variables S and D are presentedin a graphic way. Panel b:
Measured cross section distribu-tion as a function of S -variable compared to the theoreticalpredictions for this configuration. See Sec. IV for details oftheoretical models specified in the legend. larger polar angle is marked as the first one ( ϑ > ϑ ).Determining the cross section starts from the mea-sured number of events ( N BR ) from the deuteron breakupchannel. All the accepted events are classified intokinematic configurations defined by scattering angles( ϑ , ϑ , ϕ ). The adopted grid assumes 9 intervals inthe relative azimuthal angle (with width of ∆ ϕ = 20 ◦ )and 27 combinations of intervals in ϑ and ϑ angles, 2 ◦ wide. Centers of these intervals are given by the formula: ϑ , = 17 ◦ + 2 ◦ k, k = 0 , , , . . . , ϕ , = 20 ◦ j, j = 1 , , . . . , E vs. E dis-tribution is constructed and events falling within a singlebin in S with ∆ S = 8 MeV (see e.g. a hatched rectanglein Fig. 12) are projected onto the axis locally perpendic-ular to the S -curve ( D -coordinate). For each S bin, agaussian function is fitted to the distribution of eventsalong D and integrated over ± σ . The resulting N BR value is normalized according to Eq. (14) and the crosssection distribution as a function of S is obtained (seeFig. 12). F. Experimental Uncertainties
The elastic scattering and breakup reactions were mea-sured simultaneously, by the same detector and under thesame experimental conditions, like beam current, trig-gers, dead–time etc.. Though some of the systematic ef-fects are the same, the clear differences between both pro-cesses (coplanarity of elastic scattering kinematics anddifferent particle types in the exit channels) lead to dif-ferent balance of systematic uncertainties which will bediscussed separately for each reaction channel.For the elastic scattering, systematic errors were calcu-lated in bins of ϑ p . These systematic factors bias the lu-minosity and, as a consequence, the global normalization Discrepancy [%] N u m be r o f D a t a P o i n t s FIG. 13. Distribution of relative difference between breakupcross section values obtained with 2 σ and 3 σ cut on protonPID peak. of the breakup cross section. Estimation of systematicerrors of differential cross sections for deuteron breakupwas performed separately for each configuration definedby ξ or for an individual data point. Global results arepresented in Table I, while the individual uncertaintiesof data points are shown as bands in the Figs. 18–26.The methods adopted to reconstruct physical parame-ters of the registered particles were studied as one of thepotential sources of systematic errors. Among them, un-certainties associated with the reconstruction of anglesbased on the assumption of a point-like target were de-termined. For that purpose, reaction point was variedwithin the volume of beam–target intersection and a cor-responding range of variation of angles was determined.The analysis was repeated with all angles shifted withintheir uncertainty and the resulting change of cross sectionwas included into the systematic uncertainty.The effect of the particle-identification method basedon linearization of the E–∆E spectra has been estimatedby data analysis for different ranges of accepted pro-tons around the corresponding peaks in the e E variable(See Sec. III B). The shape of PID peaks is not exactlygaussian, and corresponding factors have been calculatedfrom the real distribution. In an ideal case applying thesecorrection factors should result in the same cross sectionvalues for any range of PID peak used in analysis, withonly statistical uncertainties affected. As the PID-relateduncertainty, the maximal deviation from this behaviorwas taken, while the proton acceptance range was variedbetween 2 σ and 3 σ of the corresponding peaks. Observedpercentage discrepancy for most measured cross sectionpoints lie below 2% (see Fig. 13) which is adopted asPID-related systematic uncertainty. In the case of elasticscattering the effect of PID was limited to only one parti-cle, since the coincident deuteron was identified based onthe strict kinematical relation of elastic scattering. Here,PID affects additionally the value of luminosity by about0.7%.0 TABLE I. Evaluation of total systematic errors.
Normalization factor (luminosity):
Reconstruction of angles 0.24%Particle identification 0.7%Cross-section interpolation 4.1%
Breakup cross section:
Reconstruction of angles 0.15–0.24%Particle identification 0.12–7.6%Hadronic reactions 3.8%Configurational efficiency 0.1–16.3%
The systematic uncertainty related to the configura-tional efficiency has been studied and presented in de-tails in a former publication [36]. Discrepancies betweendifferent methods of calculations of configurational effi-ciency presented in that work allow to state that thisuncertainty is small for most of the configurations andit rises with decreasing relative azimuthal angles, ϕ .In particular, the largest contribution corresponds to afew selected configurations characterized by the small-est ϕ , while elastic scattering channel ( ϕ = 180 ◦ ) ispractically immune to this component. IV. RESULTS AND DISCUSSION
The measured points of differential cross sec-tions for 243 geometrical configurations of the deuteronbreakup at 160 MeV were used to validate modern theo-retical calculations.In order to account for possibly large variations ofthe theoretical calculations within the finite bin size,the comparison must include values of the cross sec-tion at the same angular range as in the experiment:( ϑ ± ∆ ϑ, ϑ ± ∆ ϑ, ϕ ± ∆ ϕ ), where ∆ ϑ = 1 ◦ and∆ ϕ = 10 ◦ . The final value representing the theoreti-cal cross section for a given configuration includes, besidethe value at the central point, also 26 points enclosing thecorresponding bin, all of them projected onto a commonrelativistic kinematics calculated for the central geome-try. The S –coordinate is defined individually for eachconfiguration, but the same step width, ∆ S of 8 MeV,has been set for all. As an example, the data compared tothe sample of raw and averaged predictions are presentedin the Fig. 14.The predicted values of cross sections were calculatedfor (by H. Wita la et al.) the set of nucleon–nucleon (2N)phenomenological potentials (CD Bonn [9], Argonne V18[8], Nijmegen I [10], Nijmegen II [10]) and for these poten-tials supplemented with the Tucson-Melbourne (TM99)[12] three–nucleon force (2N+TM99). The next groupof calculations (by A. Deltuva) is based on the ArgonneV18 potential in the variants with the added 3N forcemodel of Urbana IX (AV18 + UIX) [11] and taking intoaccount the Coulomb force (AV18+C, AV18+UIX+C).
50 100 150 200
S [MeV] ] - s r - [ m b M e V d S Ω d Ω d σ d > ° , 40 ° , 17 ° <17 ) ° , 30 ° , 16 ° (16 ) ° , 40 ° , 16 ° (16 ) ° , 40 ° , 17 ° (17 ) ° , 40 ° , 18 ° (18 ) ° , 50 ° , 18 ° (18
50 100 150 200
S [MeV] ] - s r - [ m b M e V d S Ω d Ω d σ d > ° , 160 ° , 17 ° <19 ) ° , 150 ° , 16 ° (18 ) ° , 160 ° , 16 ° (18 ) ° , 160 ° , 17 ° (19 ) ° , 160 ° , 18 ° (20 ) ° , 170 ° , 18 ° (20 FIG. 14. Effect of averaging (CDB+∆+C) calculations for aconfiguration given in region of large variations of the crosssections. Black solid line presents the average, dashed blueline is the prediction for central geometry while other dashedlines represent predictions for the limits of accepted angularrange, each as a function of S along the central kinematic.For comparison, the experimental data are also shown (redpoints). Another set of calculations (by A.Deltuva) is based onthe coupled channel formalism with Charge DependentBonn potential with intermediate ∆ creation (CDB+∆)[15], also taking into account the electromagnetic interac-tion (CDB+∆+C). Calculation for 2N phenomenologicalpotentials are presented in the form of bands, the widthof which reflects the range of predictions obtained withindividual potentials. The calculations for 2N+TM99 arepresented in a similar way, while all other calculations arepresented as individual lines. The complete set of exper-imental results is shown in Figs. 18–26. Each figure con-sists of three parts corresponding to three combinationsof polar angles, as specified in the legends. The data areshown as red dots (full circles) surrounded by gray bandsof systematic errors. Statistical errors are usually smallerthan data points. The most striking observation is thatthe theories overestimate the cross section values for theconfigurations with small relative angles ϕ , and under-estimate them for ϕ lager than 120 ◦ . This effect is vis-ible for all investigated polar angle combinations and isconsistent with the observations of dp breakup measure-ment at 130 MeV (65 MeV/nucleon) [23]. The studiesat 65 MeV/nucleon showed that including Coulomb in-teraction practically solved the problem of discrepancy,provided 3NF effects were also taken into account. Inthe present data, the effect is significantly reduced, butnot removed for models including Coulomb force. In thecertain areas of phase space the disagreement betweenthe experiment and theory is in general large and cannotbe accounted for by estimated systematical uncertainties.These uncertainties are in the most cases comparable oreven larger than the differences between theoretical pre-dictions given by the different calculations with or with-out three–nucleon force.In order to make a quantitative comparison of the dataand the theory and to conclude on the compatibility ofthe theoretical models with the obtained results, the χ analysis was carried out. The variable χ was calculated1 − − ° ° ° ° ° ° ° ° ° ° , 17 ° (17 ) ° , 17 ° (19 ) ° , 17 ° (21 ) ° , 17 ° (23 ) ° , 17 ° (25 ) ° , 17 ° (27 ) ° , 17 ° (29 ) ° , 19 ° (19 ) ° , 19 ° (21 ) ° , 19 ° (23 ) ° , 19 ° (25 ) ° , 19 ° (27 ) ° , 19 ° (29 ) ° , 21 ° (21 ) ° , 21 ° (23 ) ° , 21 ° (25 ) ° , 21 ° (27 ) ° , 21 ° (29 ) ° , 23 ° (23 ) ° , 23 ° (25 ) ° , 23 ° (27 ) ° , 23 ° (29 ) ° , 25 ° (25 ) ° , 25 ° (27 ) ° , 25 ° (29 ) ° , 27 ° (27 ) ° , 27 ° (29 a) − − ° ° ° ° ° ° ° ° ° ° , 17 ° (17 ) ° , 17 ° (19 ) ° , 17 ° (21 ) ° , 17 ° (23 ) ° , 17 ° (25 ) ° , 17 ° (27 ) ° , 17 ° (29 ) ° , 19 ° (19 ) ° , 19 ° (21 ) ° , 19 ° (23 ) ° , 19 ° (25 ) ° , 19 ° (27 ) ° , 19 ° (29 ) ° , 21 ° (21 ) ° , 21 ° (23 ) ° , 21 ° (25 ) ° , 21 ° (27 ) ° , 21 ° (29 ) ° , 23 ° (23 ) ° , 23 ° (25 ) ° , 23 ° (27 ) ° , 23 ° (29 ) ° , 25 ° (25 ) ° , 25 ° (27 ) ° , 25 ° (29 ) ° , 27 ° (27 ) ° , 27 ° (29 b) FIG. 15. Maps of χ for individual geometries of breakupreaction defined by polar ( ϑ , ϑ on vertical axis) and az-imuthal ( ϕ on horizontal axis) angles. Data are comparedto the center of the bands corresponding to 2N (panel a) and2N+TM99 (panel b) calculations. For details see the text. for each theoretical model and each geometrical configu-ration as follows: χ / d.o.f = 1 N − N X i =1 (cid:18) σ expi − σ thi ∆ σ expi (cid:19) , (16)where σ exp corresponds to the measured experimentalvalue of the cross section, ∆ σ expi is the total experimentalerror including both systematic and statistical uncertain-ties added in squares. σ th is the prediction of the theorybeing validated. In case of 2N and 2N+TM99 calcula-tions, σ th corresponds to the center of the band. Tab. IIpresents global values of χ obtained for all presented − − AV18+UIX ° ° ° ° ° ° ° ° ° ° , 17 ° (17 ) ° , 17 ° (19 ) ° , 17 ° (21 ) ° , 17 ° (23 ) ° , 17 ° (25 ) ° , 17 ° (27 ) ° , 17 ° (29 ) ° , 19 ° (19 ) ° , 19 ° (21 ) ° , 19 ° (23 ) ° , 19 ° (25 ) ° , 19 ° (27 ) ° , 19 ° (29 ) ° , 21 ° (21 ) ° , 21 ° (23 ) ° , 21 ° (25 ) ° , 21 ° (27 ) ° , 21 ° (29 ) ° , 23 ° (23 ) ° , 23 ° (25 ) ° , 23 ° (27 ) ° , 23 ° (29 ) ° , 25 ° (25 ) ° , 25 ° (27 ) ° , 25 ° (29 ) ° , 27 ° (27 ) ° , 27 ° (29 a) AV18+UIX − − AV18+C ° ° ° ° ° ° ° ° ° ° , 17 ° (17 ) ° , 17 ° (19 ) ° , 17 ° (21 ) ° , 17 ° (23 ) ° , 17 ° (25 ) ° , 17 ° (27 ) ° , 17 ° (29 ) ° , 19 ° (19 ) ° , 19 ° (21 ) ° , 19 ° (23 ) ° , 19 ° (25 ) ° , 19 ° (27 ) ° , 19 ° (29 ) ° , 21 ° (21 ) ° , 21 ° (23 ) ° , 21 ° (25 ) ° , 21 ° (27 ) ° , 21 ° (29 ) ° , 23 ° (23 ) ° , 23 ° (25 ) ° , 23 ° (27 ) ° , 23 ° (29 ) ° , 25 ° (25 ) ° , 25 ° (27 ) ° , 25 ° (29 ) ° , 27 ° (27 ) ° , 27 ° (29 b) AV18+C − − AV18+UIX+C ° ° ° ° ° ° ° ° ° ° , 17 ° (17 ) ° , 17 ° (19 ) ° , 17 ° (21 ) ° , 17 ° (23 ) ° , 17 ° (25 ) ° , 17 ° (27 ) ° , 17 ° (29 ) ° , 19 ° (19 ) ° , 19 ° (21 ) ° , 19 ° (23 ) ° , 19 ° (25 ) ° , 19 ° (27 ) ° , 19 ° (29 ) ° , 21 ° (21 ) ° , 21 ° (23 ) ° , 21 ° (25 ) ° , 21 ° (27 ) ° , 21 ° (29 ) ° , 23 ° (23 ) ° , 23 ° (25 ) ° , 23 ° (27 ) ° , 23 ° (29 ) ° , 25 ° (25 ) ° , 25 ° (27 ) ° , 25 ° (29 ) ° , 27 ° (27 ) ° , 27 ° (29 c) AV18+UIX+C
FIG. 16. Same as in Fig. 15 but for calculations with AV18potential in combination with Coulomb interaction and/orwith Urbana IX force, as specified at the top of each panel. − − ∆ CDB+ ° ° ° ° ° ° ° ° ° ° , 17 ° (17 ) ° , 17 ° (19 ) ° , 17 ° (21 ) ° , 17 ° (23 ) ° , 17 ° (25 ) ° , 17 ° (27 ) ° , 17 ° (29 ) ° , 19 ° (19 ) ° , 19 ° (21 ) ° , 19 ° (23 ) ° , 19 ° (25 ) ° , 19 ° (27 ) ° , 19 ° (29 ) ° , 21 ° (21 ) ° , 21 ° (23 ) ° , 21 ° (25 ) ° , 21 ° (27 ) ° , 21 ° (29 ) ° , 23 ° (23 ) ° , 23 ° (25 ) ° , 23 ° (27 ) ° , 23 ° (29 ) ° , 25 ° (25 ) ° , 25 ° (27 ) ° , 25 ° (29 ) ° , 27 ° (27 ) ° , 27 ° (29 a) ∆ CDB+ − − +C ∆ CDB+ ° ° ° ° ° ° ° ° ° ° , 17 ° (17 ) ° , 17 ° (19 ) ° , 17 ° (21 ) ° , 17 ° (23 ) ° , 17 ° (25 ) ° , 17 ° (27 ) ° , 17 ° (29 ) ° , 19 ° (19 ) ° , 19 ° (21 ) ° , 19 ° (23 ) ° , 19 ° (25 ) ° , 19 ° (27 ) ° , 19 ° (29 ) ° , 21 ° (21 ) ° , 21 ° (23 ) ° , 21 ° (25 ) ° , 21 ° (27 ) ° , 21 ° (29 ) ° , 23 ° (23 ) ° , 23 ° (25 ) ° , 23 ° (27 ) ° , 23 ° (29 ) ° , 25 ° (25 ) ° , 25 ° (27 ) ° , 25 ° (29 ) ° , 27 ° (27 ) ° , 27 ° (29 b) +C ∆ CDB+
FIG. 17. Same as in Fig. 15 but for calculations with CDB+∆potential, optionally in combination with Coulomb interac-tion, as specified at the top of each panel. geometrical configurations.Data presented for a wide spectrum of geometries al-low to select the most reliable subset characterized byhigh experimental statistics, small value of estimatedsystematic error and flawless agreement between all thepresented theoretical models predictions. For the dataset selected in this way, the agreement between exper-iment and theory is significantly better (see Figs. 15–17). Calculations based on the 2N interactions alone (seeFig. 15a) provide very good description of the cross sec-tion data in the central part of the studied angular range.The quality of the description deteriorates significantly atlow ϕ values, where final state interactions (FSI) of theproton pair play a more important role. In the FSI re- TABLE II. Calculated global χ including all presentedbreakup geometries.d.o.f. 29442N 4.55 H. Wita la2N+TM99 3.92 GroupAV18+UIX 3.42AV18+C 3.89 A. DeltuvaAV18+UIX+C 2.60CDB+∆ 3.51CDB+∆+C 2.74 A. Deltuva gion Coulomb interaction between protons should not beneglected and, indeed, calculations including this ingre-dient (AV18+C in Fig. 16b) provide results, which aremuch closer to the data in this region. The dominance ofCoulomb interaction in the proton-proton FSI region wasalso observed in the breakup cross section at other ener-gies [28, 51, 52]. In extreme cases, like the configuration ϑ = 21 ◦ , ϑ = 21 ◦ , ϕ = 20 ◦ (see Fig. 22), Coulombrepulsion produces a dip in the middle of S distribution,in the point corresponding to equal proton energies. Al-though adding of the Coulomb force improves descriptionat low ϕ , it has no positive impact in other regionsof discrepancies and even deteriorates the agreement at ϕ ≥ ◦ (see Fig. 18). The remaining discrepanciescan be attributed either to 3NF or relativistic effects.Calculations including 3NF, like AV18+UIX (Fig. 16a),CDB+∆ (Fig. 17a) or 2N+TM99 (Fig. 15b) show sig-nificant improvement in the whole region of large ϕ ,but only calculations including both Coulomb and 3NF,AV18+UIX+C (Fig. 16c) and CDB+∆+C (Fig. 17b)provide fairly good description for majority of the stud-ied configurations. This success is also reflected in globalvalues of χ shown in Table II. We can conclude aboutthe importance of 3NF but, on the other hand, the im-provement is not always sufficient. Generally, there aretwo regions of remaining high χ values. In the firstone, at ϕ ≤ ◦ and largest studied polar angles, ϑ , ϑ ≥ ◦ , all the calculations are above the data andadding 3NF even increases χ . In the second one, at ϕ ≥ ◦ , improvements due to introducing the 3NFare significant, but not sufficient. V. SUMMARY
The measurement of H ( d, pp ) n at 80 MeV/nucleonenlarged existing dataset of differential cross sections by data points for 243 geometrical configurations cre-ating dense grid in solid angle limited by ϑ ∈ (17 ◦ , ◦ ).A set of models including contributions from two–nucleoninteraction combined or not with 3NF or Coulomb forcedynamics was validated via the χ –test method refer-3ring each model predictions to the cross section distribu-tions. In conclusion, evidence has been found that takingCoulomb and three-nucleon forces into account for mod-eling effective nuclear interaction globally improves thequality of predictions. Sensitivity of the differential crosssection to Coulomb and 3NF effects varies significantlyacross the studied phase space. There are also config-urations where none of the models provides satisfactorydescription of the data. Underestimation of the cross sec- tion data by theoretical calculations was also observed inthe corresponding phase space regions in measurementsof H ( p, pp ) n reaction at 135 and 190 MeV [51, 52] andrecently for H ( d, pp ) n reaction at 170 MeV/nucleon [53].This observation may suggest either important role ofrelativistic effects or problem with 3NF at higher en-ergies. Further experimental studies, as well as devel-opment of fully relativistic calculations with 3NF andCoulomb force included, are important for ultimate un-derstanding of the nature of observed discrepancies. ACKNOWLEDGMENTS
This work was partially supported by the Polish National Science Center (NCN) from grants DEC-2012/05/B/ST2/02556, 2016/22/M/ST2/00173 and 2016/23/D/ST2/01703 and by the European Commission withinthe Seventh Framework Programme through IA-ENSAR (contract No. RII3-CT-2010-262010). Thanks also to theguys from AGOR and the source group who delivered a nice beam for the experiment. [1] F. D. Fadeev, Sov. Phys. JETP , 1014 (1961).[2] H. Wita la, T. Cornelius, and W. Gl¨ockle, Few Body-Syst. , 123 (1988).[3] D. H¨uber, H. Wita la, and W. Gl¨ockle, Few Body-Syst. , 171 (1993).[4] W. Gl¨ockle, H. Wita la, D. H¨uber, H. Kamada, andJ. Golak, Phys. Rep. , 107 (1996).[5] H. Yukawa, Proc. Phys.-Math Soc.Japan , 48 (1935).[6] G. P. S. Occhialini and C. F. Powell, Nature , 186(1947).[7] H. Wita la, W. Gl¨ockle, J. Golak, A. Nogga, H. Kamada,R. Skibi´nski, and J. Kuro´s- ˙Zo lnierczuk, Phys. Rev. C , 024007 (2001).[8] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys.Rev. C , 38 (1995).[9] R. Machleidt, Phys. Rev. C , 024001 (2001).[10] V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen, andJ. J. de Swart, Phys. Rev. C , 2950 (1994).[11] B. S. Pudliner, V. R. Pandharipande, J. Carlson, S. C.Pieper, and R. Wiringa, Phys. Rev. C , 1720 (1997).[12] S. A. Coon and H. K. Han, Few-Body Syst. , 131(2001).[13] A. Deltuva, K. Chmielewski, and P. U. Sauer, Phys. Rev.C , 34001 (2003).[14] A. Deltuva, R. Machleidt, and P. U. Sauer, Phys. Rev.C , 024005 (2003).[15] A. Deltuva, A. C. Fonseca, and P. U. Sauer, Phys. Lett.B , 471 (2008).[16] R. Machleidt and D. Entem, Phys. Rept. , 31 (20011).[17] E. Epelbaum and U.-G. Meißner, Annu. Rev. Nucl. Part.Sci. , 159 (2012).[18] S. Ishikawa, Few-Body Syst. , 229 (2003).[19] A. Deltuva, R. Fonseca, and P. U. Sauer, Phys. Rev. C , 057001 (2006).[20] H. Wita la, J. Golak, W. Gl¨ockle, and H. Kamada, Phys.Rev. Lett. , 054001 (2005).[21] R. Skibi´nski, H. Wita la, and J. Golak, Eur. Phys. J. A , 369 (2006).[22] S. Kistryn and E. Stephan, J. Phys. G Nucl. Part. Phys. , 014006 (2003).[23] S. Kistryn, E. Stephan, et al. , Phys. Rev. C , 044006(2005).[24] K. Sekiguchi, H. Sakai, H. Wita la, W. Gl¨ockle, J. Golak,K. Hatanaka, et al. , Phys.Rev. Lett. , 162301 (2005).[25] M. Eslami-Kalantari et al. , Mod. Phys. Lett. A , 839(2009).[26] H. Mardanpour et al. , Phys. Lett. B , 149 (2010).[27] S. Kistryn et al. , Few-Body Syst. , 235 (2011).[28] I. Ciepa l, B. K los, et al. , Few-Body Syst. , 665 (2015).[29] B. Sytze, The Superconducting Cyclotron AGOR: Accel-erator for Light and Heavy Ions (1987).[30] H. Mardanpour and A. Ramazani-Moghadam-Arani,
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100 150012 ° = 20 ϕ
100 150 ° = 40 ϕ
100 150 ° = 60 ϕ
100 150 ° = 80 ϕ
100 150 200
S [MeV] ° = 100 ϕ
100 150 200
S [MeV] ° = 120 ϕ
100 150 200
S [MeV] ° = 140 ϕ
100 150 200
S [MeV] ° = 160 ϕ
100 150 200
S [MeV] ° = 180 ϕ ° =17 ϑ ° =17 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d a) ] - s r - [ m b M e V d S Ω d Ω d σ d
100 15000.511.5 ° = 20 ϕ
100 150 ° = 40 ϕ
100 150 ° = 60 ϕ
100 150 ° = 80 ϕ
100 150
S [MeV] ° = 100 ϕ
100 150 200
S [MeV] ° = 120 ϕ
100 150 200
S [MeV] ° = 140 ϕ
100 150 200
S [MeV] ° = 160 ϕ
100 150 200
S [MeV] ° = 180 ϕ ° =17 ϑ ° =19 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d b) ] - s r - [ m b M e V d S Ω d Ω d σ d
60 80 100 120 14000.51 ° = 20 ϕ
100 150 ° = 40 ϕ
100 150 ° = 60 ϕ
100 150 ° = 80 ϕ
100 150
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150 200
S [MeV] ° = 140 ϕ
100 150 200
S [MeV] ° = 160 ϕ
100 150 200
S [MeV] ° = 180 ϕ ° =17 ϑ ° =21 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d c) ] - s r - [ m b M e V d S Ω d Ω d σ d FIG. 18. Differential cross section for polar angles ϑ , ϑ : 17 ◦ ,17 ◦ (a); 19 ◦ ,17 ◦ (b); 21 ◦ ,17 ◦ (c). Details in the text.
60 80 100 120 14000.20.40.6 ° = 20 ϕ
50 100 150 ° = 40 ϕ
100 150 ° = 60 ϕ
100 150 ° = 80 ϕ
100 150
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150 200
S [MeV] ° = 140 ϕ
100 150 200
S [MeV] ° = 160 ϕ
100 150 200
S [MeV] ° = 180 ϕ ° =17 ϑ ° =23 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d a) ] - s r - [ m b M e V d S Ω d Ω d σ d
60 80 100 12000.10.20.3 ° = 20 ϕ
60 80 100 120 140 ° = 40 ϕ
50 100 150 ° = 60 ϕ
100 150 ° = 80 ϕ
100 150
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150 200
S [MeV] ° = 160 ϕ
100 150 200
S [MeV] ° = 180 ϕ ° =17 ϑ ° =25 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d b) ] - s r - [ m b M e V d S Ω d Ω d σ d
60 80 100 12000.10.2 ° = 20 ϕ
60 80 100 120 ° = 40 ϕ
60 80 100 120 ° = 60 ϕ
50 100 150 ° = 80 ϕ
100 150
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150
S [MeV] ° = 160 ϕ
100 150
S [MeV] ° = 180 ϕ ° =17 ϑ ° =27 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d c) ] - s r - [ m b M e V d S Ω d Ω d σ d FIG. 19. Differential cross section for polar angles ϑ , ϑ : 23 ◦ ,17 ◦ (a); 25 ◦ ,17 ◦ (b); 27 ◦ ,17 ◦ (c). Details in the text.
60 80 10000.050.10.15 ° = 20 ϕ
60 80 100 ° = 40 ϕ
40 60 80 100 120 ° = 60 ϕ
60 80 100 120 140 ° = 80 ϕ
50 100 150
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150
S [MeV] ° = 160 ϕ
100 150
S [MeV] ° = 180 ϕ ° =17 ϑ ° =29 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d a) ] - s r - [ m b M e V d S Ω d Ω d σ d
100 15000.51 ° = 20 ϕ
100 150 ° = 40 ϕ
100 150 ° = 60 ϕ
100 150 ° = 80 ϕ
100 150
S [MeV] ° = 100 ϕ
100 150 200
S [MeV] ° = 120 ϕ
100 150 200
S [MeV] ° = 140 ϕ
100 150 200
S [MeV] ° = 160 ϕ
100 150 200
S [MeV] ° = 180 ϕ ° =19 ϑ ° =19 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d b) ] - s r - [ m b M e V d S Ω d Ω d σ d
60 80 100 120 14000.20.40.60.8 ° = 20 ϕ
60 80 100 120 140 ° = 40 ϕ
100 150 ° = 60 ϕ
100 150 ° = 80 ϕ
100 150
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150 200
S [MeV] ° = 140 ϕ
100 150 200
S [MeV] ° = 160 ϕ
100 150 200
S [MeV] ° = 180 ϕ ° =19 ϑ ° =21 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d c) ] - s r - [ m b M e V d S Ω d Ω d σ d FIG. 20. Differential cross section for polar angles ϑ , ϑ : 29 ◦ ,17 ◦ (a); 19 ◦ ,19 ◦ (b); 21 ◦ ,19 ◦ (c). Details in the text.
60 80 100 12000.20.40.6 ° = 20 ϕ
60 80 100 120 140 ° = 40 ϕ
60 80 100 120 140 ° = 60 ϕ
100 150 ° = 80 ϕ
100 150
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150 200
S [MeV] ° = 160 ϕ
100 150 200
S [MeV] ° = 180 ϕ ° =19 ϑ ° =23 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d a) ] - s r - [ m b M e V d S Ω d Ω d σ d
60 80 100 12000.10.20.3 ° = 20 ϕ
60 80 100 120 ° = 40 ϕ
60 80 100 120 140 ° = 60 ϕ
60 80 100 120 140 ° = 80 ϕ
100 150
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150 200
S [MeV] ° = 160 ϕ
100 150 200
S [MeV] ° = 180 ϕ ° =19 ϑ ° =25 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d b) ] - s r - [ m b M e V d S Ω d Ω d σ d
60 80 10000.10.2 ° = 20 ϕ
60 80 100 ° = 40 ϕ
60 80 100 120 ° = 60 ϕ
60 80 100 120 140 ° = 80 ϕ
100 150
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150
S [MeV] ° = 160 ϕ
100 150
S [MeV] ° = 180 ϕ ° =19 ϑ ° =27 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d c) ] - s r - [ m b M e V d S Ω d Ω d σ d FIG. 21. Differential cross section for polar angles ϑ , ϑ : 23 ◦ ,19 ◦ (a); 25 ◦ ,19 ◦ (b); 27 ◦ ,19 ◦ (c). Details in the text.
40 60 80 10000.050.10.15 ° = 20 ϕ
40 60 80 100 ° = 40 ϕ
60 80 100 ° = 60 ϕ
60 80 100 120 ° = 80 ϕ
50 100 150
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150
S [MeV] ° = 160 ϕ
100 150
S [MeV] ° = 180 ϕ ° =19 ϑ ° =29 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d a) ] - s r - [ m b M e V d S Ω d Ω d σ d
60 80 100 120 14000.20.40.6 ° = 20 ϕ
60 80 100 120 140 ° = 40 ϕ
60 80 100 120 140 ° = 60 ϕ
100 150 ° = 80 ϕ
100 150
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150 200
S [MeV] ° = 140 ϕ
100 150 200
S [MeV] ° = 160 ϕ
100 150 200
S [MeV] ° = 180 ϕ ° =21 ϑ ° =21 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d b) ] - s r - [ m b M e V d S Ω d Ω d σ d
60 80 100 12000.20.4 ° = 20 ϕ
60 80 100 120 ° = 40 ϕ
60 80 100 120 140 ° = 60 ϕ
60 80 100 120 140 ° = 80 ϕ
100 150
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150 200
S [MeV] ° = 160 ϕ
100 150 200
S [MeV] ° = 180 ϕ ° =21 ϑ ° =23 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d c) ] - s r - [ m b M e V d S Ω d Ω d σ d FIG. 22. Differential cross section for polar angles ϑ , ϑ : 29 ◦ ,19 ◦ (a); 21 ◦ ,21 ◦ (b); 23 ◦ ,21 ◦ (c). Details in the text.
60 80 10000.10.20.3 ° = 20 ϕ
60 80 100 120 ° = 40 ϕ
60 80 100 120 ° = 60 ϕ
60 80 100 120 140 ° = 80 ϕ
100 150
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150
S [MeV] ° = 160 ϕ
100 150 200
S [MeV] ° = 180 ϕ ° =21 ϑ ° =25 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d a) ] - s r - [ m b M e V d S Ω d Ω d σ d
60 80 10000.10.2 ° = 20 ϕ
60 80 100 ° = 40 ϕ
60 80 100 ° = 60 ϕ
60 80 100 120 ° = 80 ϕ
60 80 100 120 140
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150
S [MeV] ° = 160 ϕ
100 150
S [MeV] ° = 180 ϕ ° =21 ϑ ° =27 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d b) ] - s r - [ m b M e V d S Ω d Ω d σ d
40 60 8000.050.10.150.2 ° = 20 ϕ
40 60 80 100 ° = 40 ϕ
60 80 100 ° = 60 ϕ
60 80 100 120 ° = 80 ϕ
60 80 100 120 140
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150
S [MeV] ° = 160 ϕ
100 150
S [MeV] ° = 180 ϕ ° =21 ϑ ° =29 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d c) ] - s r - [ m b M e V d S Ω d Ω d σ d FIG. 23. Differential cross section for polar angles ϑ , ϑ : 25 ◦ ,21 ◦ (a); 27 ◦ ,21 ◦ (b); 29 ◦ ,21 ◦ (c). Details in the text.
60 80 10000.20.4 ° = 20 ϕ
60 80 100 120 ° = 40 ϕ
60 80 100 120 ° = 60 ϕ
60 80 100 120 140 ° = 80 ϕ
100 150
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150 200
S [MeV] ° = 160 ϕ
100 150 200
S [MeV] ° = 180 ϕ ° =23 ϑ ° =23 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d a) ] - s r - [ m b M e V d S Ω d Ω d σ d
60 80 10000.10.20.30.4 ° = 20 ϕ
60 80 100 ° = 40 ϕ
60 80 100 120 ° = 60 ϕ
60 80 100 120 140 ° = 80 ϕ
100 150
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150
S [MeV] ° = 160 ϕ
100 150 200
S [MeV] ° = 180 ϕ ° =23 ϑ ° =25 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d b) ] - s r - [ m b M e V d S Ω d Ω d σ d
40 60 8000.10.20.3 ° = 20 ϕ
40 60 80 100 ° = 40 ϕ
60 80 100 ° = 60 ϕ
60 80 100 120 ° = 80 ϕ
60 80 100 120 140
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150
S [MeV] ° = 160 ϕ
100 150
S [MeV] ° = 180 ϕ ° =23 ϑ ° =27 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d c) ] - s r - [ m b M e V d S Ω d Ω d σ d FIG. 24. Differential cross section for polar angles ϑ , ϑ : 23 ◦ ,23 ◦ (a); 25 ◦ ,23 ◦ (b); 27 ◦ ,23 ◦ (c). Details in the text.
30 40 50 60 7000.10.20.3 ° = 20 ϕ
40 50 60 70 80 ° = 40 ϕ
40 60 80 100 ° = 60 ϕ
60 80 100 ° = 80 ϕ
60 80 100 120
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150
S [MeV] ° = 160 ϕ
100 150
S [MeV] ° = 180 ϕ ° =23 ϑ ° =29 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d a) ] - s r - [ m b M e V d S Ω d Ω d σ d
40 60 8000.10.20.30.4 ° = 20 ϕ
60 80 100 ° = 40 ϕ
60 80 100 ° = 60 ϕ
60 80 100 120 ° = 80 ϕ
60 80 100 120 140
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150
S [MeV] ° = 160 ϕ
100 150
S [MeV] ° = 180 ϕ ° =25 ϑ ° =25 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d b) ] - s r - [ m b M e V d S Ω d Ω d σ d
40 50 60 7000.10.20.30.4 ° = 20 ϕ
40 50 60 70 80 ° = 40 ϕ
60 80 100 ° = 60 ϕ
60 80 100 ° = 80 ϕ
60 80 100 120 140
S [MeV] ° = 100 ϕ
100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150
S [MeV] ° = 160 ϕ
100 150
S [MeV] ° = 180 ϕ ° =25 ϑ ° =27 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d c) ] - s r - [ m b M e V d S Ω d Ω d σ d FIG. 25. Differential cross section for polar angles ϑ , ϑ : 29 ◦ ,23 ◦ (a); 25 ◦ ,25 ◦ (b); 27 ◦ ,25 ◦ (c). Details in the text.
30 40 5000.10.20.30.4 ° = 20 ϕ
30 40 50 60 70 ° = 40 ϕ
40 60 80 ° = 60 ϕ
60 80 100 ° = 80 ϕ
60 80 100 120
S [MeV] ° = 100 ϕ
50 100 150
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150
S [MeV] ° = 160 ϕ
100 150
S [MeV] ° = 180 ϕ ° =25 ϑ ° =29 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d a) ] - s r - [ m b M e V d S Ω d Ω d σ d
30 40 50 6000.10.20.30.4 ° = 20 ϕ
40 50 60 70 ° = 40 ϕ
40 60 80 ° = 60 ϕ
60 80 100 ° = 80 ϕ
60 80 100 120
S [MeV] ° = 100 ϕ
60 80 100 120 140
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150
S [MeV] ° = 160 ϕ
100 150
S [MeV] ° = 180 ϕ ° =27 ϑ ° =27 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d b) ] - s r - [ m b M e V d S Ω d Ω d σ d
30 40 5000.20.4 ° = 20 ϕ
30 40 50 ° = 40 ϕ
40 50 60 70 ° = 60 ϕ
60 80 100 ° = 80 ϕ
60 80 100 120
S [MeV] ° = 100 ϕ
60 80 100 120 140
S [MeV] ° = 120 ϕ
100 150
S [MeV] ° = 140 ϕ
100 150
S [MeV] ° = 160 ϕ
100 150
S [MeV] ° = 180 ϕ ° =27 ϑ ° =29 ϑ ∆ CDB+ +C ∆ CDB+ ] - s r - [ m b M e V d S Ω d Ω d σ d c) ] - s r - [ m b M e V d S Ω d Ω d σ d FIG. 26. Differential cross section for polar angles ϑ , ϑ : 29 ◦ ,25 ◦ (a); 27 ◦ ,27 ◦ (b); 29 ◦ ,27 ◦ (c). Details in the text. (2019).[37] M. Kerrisk et al. , The Linux man-pages project (2004),http://man7.org/linux/man-pages/man3/atan2.3.html.[38] J. P loskonka et al. , Nucl. Instr. Meth. , 57 (1975).[39] W. Parol et al. , EPJ Web. Confs. , 7019 (2014).[40] A. Wilczek, PhD dissertation, University of Silesia, Ka-towice (2010).[41] W. Parol et al. , Acta. Phys. Pol. B , 527 (2014).[42] G. Khatri, PhD dissertation, Jagiellonian University,Krakw (2015).[43] I. Ciepa l, G. Khatri, K. Bodek, A. Deltuva, N. Kalantar-Nayestanaki, S. Kistryn, et al. , Phys. Rev. C , 024003(2019).[44] D. Measday and C. Richard-Serre, Nucl. Instr. Meth. ,45 (1969).[45] H. Shimizu et al. , Nucl. Phys. A , 242 (1982). [46] M. Davidson, H. Hopkins, L. Lyons, and D. Shaw, Nucl.Phys. , 423 (1963).[47] K. Sekiguchi, H. Sakai, H. Wita la, W. Gl¨ockle, J. Golak,M. Hatano, et al. , Phys. Rev. C , 34003 (2002).[48] K. Ermisch, H. R. Amir-Ahmadi, A. M. van den Berg,R. Castelijns, B. Davids, E. Epelbaum, et al. , Phys. Rev.C , 051001 (2003).[49] H. Wita la, W. Gl¨ockle, D. Huber, J. Golak, and H. Ka-mada, Phys. Rev. Lett. , 1183 (1998).[50] W. Parol, PhD dissertation, Jagiellonian University,Krak´ow (2016).[51] M. Eslami-Kalantari, PhD dissertation, University ofGroningen, Groningen (2009).[52] H. Mardanpuor-Mollalar, PhD dissertation, University ofGroningen, Groningen (2008).[53] B. K los et al.et al.