Low energy proton-deuteron versus neutron-deuteron breakup in four configurations: implications for Coulomb-force effects

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 603 (1996) 161-175

Low energy proton-deuteron versus neutron-deuteron breakup in four configurations: implications for Coulomb-force effects R. Grogmann a, G. Nitzsche a, H. Patberg a, L. Sydow a, S. Vohl a, H. Paetz gen. Schieck a, J. Golak h H. Witata b, W. G16ckle c, D. Hfiber c Institutfiir Kernphysik, Universitdltzu KOln, D-50937 Cologne, Germany b Institute of Physics, Jagellonian University, PL-30059 Cracow, Poland c lnstitutfiir Theoretische Physik 11, Ruhr-Universitiit Bochum, D-44780 Bochum, Germany a

Received 13 December 1995; revised 15 February 1996

Abstract We have measured cross sections of the pd breakup reaction at a laboratory proton energy of 10.5 MeV in four kinematically complete arrangements comprising space-star, collinearity, final-state interaction and quasi-free scattering conditions. We present our results and compare them to the predictions of rigorous three-nucleon Faddeev calculations using different realistic nucleon-nucleon potentials. We find cases of good agreement but also cases of clear discrepancies. The inclusion of the 2~'-exchange Tucson-Melbourne three-nucleon force does not remove these discrepancies. Considering also pd data at 13 and 19 MeV and nd data at 10.5 and 13 MeV, we draw tentative conclusions about possible Coulomb-force effects in these configurations. In the quasi-free scattering, space-star, and collinear configurations, the Coulomb-force effects might possibly be responsible for at least part of the observed discrepancies. For the np FSI configuration the Coulomb-force effects are likely to be small. PACS: 21.45.+v; 21.30.-x; 25.40.-h; 25.10.+s; 13.75.Cs Keywords: NUCLEAR REACTION 2H(p, pp)n, E = 10.5 MeV; measured tr(E3, E4, ~93,~94). Faddeev calculations, nucleon-nucleon potentials, three-nucleon force. Comparison with 2H(n, nn)p at 10.5 (10.3), 13.0 MeV. Deduced Coulomb effects.

I. Introduction For some time rigorous solutions o f three-nucleon ( 3 N ) Faddeev equations with modern nucleon-nucleon ( N N ) interactions have been available [ 1,2], even including 3N forces. Nowadays, they form a standard basis for the theoretical interpretation o f existing 0375-9474/96/$15.00 (~) 1996 Elsevier Science B.V. All fights reserved PII S0375-9474(96) 00108-X

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nd and pd breakup data [ 3]. A notorious drawback of present-day Faddeev breakup calculations, drastically coming to light in pd breakup data analysis, is their inability to include the Coulomb interaction between the two protons in the same rigorous manner as the short-range NN forces are included. Recently the first step to improve the situation in this respect has been done in momentum space, using a scheme with a screened proton-proton (pp) Coulomb force and taking the limit of vanishing screening [4]. Unfortunately, the NN force used is only a rank-one separable Yamaguchi force. However, in this case the calculations were very involved and the analytical nature of such a specific strong force was a much needed aid to the successful completion of the task. The resulting Coulomb-force effects for the breakup cross sections were in some cases quite large, but the question remains whether those effects will persist when realistic NN forces and higher partial waves are included. Until this problem is solved exactly and there are rigorous solutions of 3N Faddeev equations with the Coulomb force included exactly, all present day pd breakup data analysis will have to live with uncertainties about the possible Coulomb-force effects. Therefore, right now, the only way to learn about the magnitude of the Coulomb-force effects in the pd breakup process is a direct comparison between nd and pd data in the same kinematically complete configurations. In this work we present new cross-section data for the proton-induced deuteron breakup at 10.5 MeV proton laboratory energy. Cross sections, parametrized by the arc length S of the kinematical curve, were measured in four kinematically complete configurations which comprise the following conditions for the final-state momenta: the neutron-proton (np) final-state interaction (FSI) with equal momenta of the neutron and proton; the pp collinearity (COL) with the undetected neutron being at rest in the c.m. system; the space-star geometry (SST) where all the momenta of the outgoing nucleons lie in the same plane in the c.m. system, perpendicular to the beam, forming an equilateral triangle; finally the pp quasi-free scattering (QFS) with the undetected neutron being at rest in the lab system. This study together with results of our two previous pd measurements performed at 13 and 19 MeV [19,21] form a pd data basis for the four "classical" geometries. A comparison with the data of corresponding nd breakup measurements delivers information on the magnitude of the Coulomb-force effects in these configurations. In Section 2 we present briefly our theoretical scheme for solving the 3N Faddeev equations with 2N and 3N forces, but excluding the Coulomb force entirely. The experimental setup is described in Section 3 and details of the data analysis in Section 4. The comparison of the present data to the theory follows in Section 5. There the comparison is also made between of our pd data and nd data for the same configurations, and finally implications for the magnitude of Coulomb-force effects are discussed. A summary is given in Section 6.

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2.

163

Theory

The theoretical results presented in this work are based on rigorous solutions of the 3N Faddeev equations using different realistic NN interactions. In the following we give a short presentation of our formalism and the numerical realization. When only NN interactions are active and neglecting the long-range Coulomb force, we solve the Faddeev equation for the T-operator, T = tP + tPGoT,

(2.1)

where Go is the free three-body propagator and t is the two-body off-shell t-matrix. P denotes a sum of cyclic and anticyclic permutations of three nucleons. After solving Eq. (2.1) the breakup-transition operator U0 follows by quadrature: U0 = ( i + P)T.

(2.2)

In the case when the potential energy of the 3N system contains, in addition to the pure NN interaction, a three-nucleon force (3NF), we introduce the operator t4 = I/4 + V4Got4, driven by the three-nucleon interaction I,~. Now in the transition operator U0 a new term T4 on top of T appears. Both T and T4 fulfill the following set of coupled equations: T = t P + tGoT4 + tPGoT,

(2.3)

T4 = (1 -t- P)t4 + (1 + P)t4GoT.

(2.4)

These equations are solved in a perturbative approach in powers of I/4 and the different orders are then summed up by the Pad6 method. The breakup amplitude is then given by U0=(1 + P ) T + T 4 .

(2.5)

The physical content of Eqs. (2.1) and (2.3), (2.4) is revealed after iterating them. The resulting multiple-scattering series describes contributions from scattering processes where three nucleons interact with two- or three-body forces with free propagation in between. For more details of the theoretical formulation and numerical performance we refer to Refs. [ 1,2,5] and references therein. We solved Eq. (2.1) using the following realistic NN potentials: AV18 [6], CD Bonn [7], and Nijmegen 93, I and II [8]. All these interactions are charge dependent in isospin t = 1 states having thus inherently built in the difference in the ZS0 force component for the np, pp and nn systems. Also they are practically equivalent on-shell and describe the 2N data with impressively good quality, characterized by X 2 ~ 1 per degree of freedom. In all calculations the charge-independence breaking (CIB) of the NN interaction in the state Is0 was treated exactly by including an admixture of total isospin T = 3/2 [9]. Such CIB requires different IS0 NN interactions in the two-body subsystems (pp and np) of the pd system, which are provided in the AV18, CD Bonn, Nijmegen 93, I and II interactions. For the remaining isospin t = 1 states the 2/3 - 1/3 rule was applied [9].

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In order to gain some insight into possible 3NF effects we took the Bonn B potential [10] in conjunction with the Tucson-Melbourne (TM) three-nucleon force [11,12]. From our previous study [2] we know that the effects of this 3NF depend strongly on the value of the cut-off parameter A,~ appearing in the ~rNN form factor. In the present calculations we used the "recommended" value for this parameter A,r = 5.8m,~ (m~ = 139.6 MeV). To get the proper binding energy of the triton with the Bonn B potential and the TM-3NF, a value of A,~ = 4.55m~ should be used. Such a decrease of A~ reduces the effects of the TM 3NF significantly. Therefore the 3NF effects for the breakup configurations presented in this work are an overestimation of the effects caused by this 3NF model. In all calculations performed with 2N interactions only we included all 3N states with total angular momenta of the two-nucleon subsystem j ~< 3. In the case of the calculation with the TM 3NF included, we restricted them to j ~< 2 because of computer limitations.

3. Experiment The measurements were carried out at the Cologne FN tandem Van-de-Graaff accelerator facility. The proton beam was produced by a sputter source and accelerated to a laboratory energy of 10.5 MeV. The beam was then focused into an ORTEC 2800 scattering chamber, which was modified by a hemispherical 2~r extension in order to measure the non-coplanar space-star configuration. We obtained a typical beam current on target of 300 nA within a beam spot diameter of 1-2 mm. The focusing of the beam was controlled using a quartz plate with a center hole. The target foils used in our measurements consisted of solid polyethylene (CD2)n with thicknesses of about 1503 0 0 / z g / c m 2. Thermal stability of the targets was achieved by additional carbon layers ( 3 0 / z g / c m 2) evaporated on both sides of the foils. Furthermore the lifetime of the foils was extended using rotating target holders. This technique reduced the deuterium loss to approximately 15% in 24 h. For the detection of the breakup protons we used silicon surface-barrier detectors with thicknesses of 1.5 to 2 mm, cooled to ~ 3°C. They were positioned at the laboratory scattering angles ( O3, O4, At~34) = (54.9 °, 54.9 °, 180 °), ( 37 °, 63 °, 180 °), (37 °, 37 °, 180°), (49.4°,49.4 °, 120 °) for the configurations COL, FSI, QFS and SST, respectively. The solid angles were determined by diaphragms with angular apertures of AO = 0.5°-1 ° and A O = 1.5°-2.5 °. The configurations FSI and SST were studied twice. In the FSI case the second detector arrangement was obtained by a 180 ° rotation around the beam axis and for the SST case by a corresponding 60 ° rotation. These symmetric arrangements increased the breakup counting rates by a factor of two. One additional detector at a lab angle of O = 30 ° served as a monitor for the absolute normalization of the breakup cross sections. A typical monitor spectrum is shown in Fig. 1. The signals from all detectors were processed simultaneously by standard fast-slow coincidence electronics and then recorded in list mode on magnetic tape using the

R. Groflmann et al./Nuclear Physics A 603 (1996) 161-175

x

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Fig. I. Typical monitor spectrum at Ep = 10.5 MeV and olah = 30 o. We used the reaction D ( p , p ) D for the absolute normalization of our coincidence spectra.

Cologne FERA analyzer system [ 13]. For each coincidence event the stored information consisted of a logical status word indicating the kinematical configuration, the energies and the time-of-flight differences of the detected particles. In order to control the experiment, complete online analysis was provided, including two-dimensional energy spectra, and background-corrected S-curve projections, but without dead-time corrections.

4. Data analysis To compare the experimental results of our kinematically complete three-body breakup measurements (p] + d2 ~ P3 + p4 "+"n5 ) with the corresponding theoretical predictions, we transformed the measured breakup events into one-dimensional S-curve spectra. The S curve represents the kinematically allowed positions of the breakup events in the (E3, E4) plane by the arc length S of the corresponding kinematical curve [ 15]. In this work we take S -- 0 at that point of the kinematical curve which corresponds to the

166

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smallest value of E4 and move in a counter-clockwise direction. For this purpose we first performed a linear energy calibration of our coincidence spectra using the protons elastically and inelastically scattered from the target nuclei 2H and ~2C together with the recoil deuterons. For the assignment (projection) of the (E 3, E4) breakup events to the most probable point on the kinematical curve we assumed a two-dimensional Gaussian distribution of the data around this curve. This point was defined to be given by the shortest distance to the point-geometry kinematical curve. The background in our (E3, E4) matrices consisted mainly of random coincidences due to elastic scattering and reactions with the target nuclei 2H and inC. The backgroundcorrection procedure is based on methods similar to that described in [16]. We first calculated, for each list-mode event, a theoretical time-of-flight difference using the known distances between the detectors and the target, together with the measured energies, and assuming that the particle masses were those of the detected nucleons of the pd breakup. Then we constructed two-dimensional time spectra by sorting the events according to the calculated theoretical time-of-flight difference and the measured timeof-flight difference. The true breakup events, for which a characteristic relation between these two time-of-flight differences exists, produce a peak above the uniformly distributed random-coincidence background (Figs. 2a,b). Such a representation allows the determination of the random background originating from reactions on the target nuclei 2H and ~2C by interpolation. This interpolation uses regions in two-dimensional time spectra which contain purely random coincidences. However, in the FSI configuration the presence of true two-body coincidences prevented us from analyzing the complete kinematical curve by applying this method. The two-body coincidences produced intensities in the two-dimensional time spectra which partially overlapped the peak of the true breakup events. In practice, the background-correction procedure was carried out using analyzing software allowing for a graphical representation of the time spectra and the application of appropriate graphical software cuts [ 14]. The yields of our S-curve spectra were normalized using the relation d3o r dO3 d o n dS

(4.1)

_ (dtr/dg2)Md£2MRrNtr(dS~,)

N~ Ag23 Af~4 ASI~

'

where (do-/dO)M denotes the differential cross section of the monitor reaction, Ag/M is the solid angle of the monitor detector and N~ is the background and dead-time corrected monitor-peak intensity./~r denotes an average dead-time correction factor to get the true breakup intensity Ntr over an interval AS~, of the arc length S. zl/23 and A/24 are the solid angles of the coincidence detectors. We used the monitor reaction 2H(p,p)2H for which elastic scattering cross-section data have been published by Griiebler et ah [ 17]. The explicit value that we took for the elastic scattering cross section at our laboratory projectile energy of 10.5 MeV and our laboratory monitor angle of 30 ° is (d°')

7-hM= ( 2 3 7 + 7 ) m b / s r .

(4.2)

R. Groflmann et al./Nuclear Physics A 603 (1996) 161-175

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Fig. 2. (a,b) Two plots of a time matrix used for backgroundsubtraction.In (a) the time-of-flightdifferences are plotted versus the measured time-of-flightdifferencesas a contourplot in arbitrary units. The rectangular region is enlarged in a perspective view in (b) with the viewingdirectionindicatedby the arrow. In (c) and (d) plots of a calibrated (E3, E4) energy matrix in the COL configurationare shown. The background in these matrices has been subtracted using the conditionthat all displayedevents belong to the "time" peak of (a). The errors of our data have several origins. First there are systematic errors due to the normalization factors. They are composed of the error of the normalization cross section (4%) and the solid angle errors of our monitor detector (3%) and the coincidence detectors (1.5%). Another systematic error was induced by the projection procedure. Because of the finite angular resolution the breakup events are distributed around the kinematical loci for point geometry. Therefore we had to choose a maximum distance from the S curve ("S-curve cuts") within which true breakup events can be expected. By comparing cuts using different values for this maximum distance from the S curve we estimated the possible loss of true breakup events to be on the order of 1%. Finally a statistical error in the range of 2 - 4 % originated from the background-corrected absolute

168

R. Groflmann et al./Nuclear PhysicsA 603 (1996) 161-175

breakup yield.

5. Results and discussion

In Figs. 3a-6a we present our new pd data and compare them to the point-geometry predictions using different NN potentials. A very good agreement between theory and FSI cross-section data is seen (Fig. 3a) with the exception of three experimental points in the immediate vicinity of the FSI condition. This discrepancy can be eliminated by taking into account the finite resolution of the experimental setup and by performing a proper averaging of the point-geometry predictions over the finite angular apertures of the two proton detectors. In the case of the collinear configuration a clear discrepancy between experiment and theory exists in the region of collinearity around S ,~ 7.5 MeV, where theory underestimates the data by about 20% (Fig. 4a). In the two maxima the theoretical cross sections overshoot the data by about 12%. These discrepancies cannot be accounted for by averaging the theory over finite angular resolutions. It would introduce only a minor modification to the point-geometry cross sections. The theoretical cross sections also clearly overestimate the pd data for the space-star configuration (Fig. 5a) and in the region of the QFS peak (Fig. 6a) by about 20% and 10%, respectively. This cannot be explained by finite angular resolution of the experimental setup which introduces only a slight change of the QFS peak cross section and does not significantly affect the SST cross section. It is important to note that the theoretical results are very stable against exchanges of one NN potential by another. Only in the case of the QFS configuration is some dependence on the choice of the NN force visible. This stability reflects the very good fine tuning of these potentials to 2N data and their nearly exact on-shell equivalence. The small differences for the QFS cross sections result from the small differences of the 3S1-3DI NN force components of the corresponding potentials. One possible source for the differences between the experimental and theoretical cross sections is the Coulomb force acting between the two protons. This force is neglected in our theory. As already mentioned in the introduction, up to the present time there does not exist an exact solution of the 3N Faddeev equations above the deuteron breakup threshold which includes this long-range interaction in an exact manner. Therefore the only way right now to learn about the magnitude of possible Coulomb-force effects in the pd breakup is the direct comparison of corresponding pd and nd breakup data. Such an approach assumes of course the reliability of both pd and nd data in the configurations under study. For the four configurations investigated here, several corresponding nd data are available (open circles and open squares in Figs. 3a-6a). Note that for the SST configuration the nd data were obtained at 10.3 MeV. In the case of the FSI configuration the nd data of Ref. [ 18] are clearly below our pd data in the region of the FSI peak, but in agreement in the wings. This would

R. Groflmann et al./Nuclear Physics A 603 (1996) 161-175

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Therefore we are tempted to assume that the nd measurements of Ref. [ 18] lead to a lower peak height because of the much larger angular averaging than in the pd case. Unfortunately we are not able to perform the required averaging calculations ourselves, since the necessary detailed knowledge of the experimental conditions is no longer available. Further we have estimated possible 3NF effects on the FSI cross sections. We used the TM 3NF choosing the cut-off parameter A~ = 5.8m~. This is also shown in Figs. 3-6. We see that the FSI peak heights are increased at 13.0 and 19.0 MeV by about 3% and 10%, respectively but at 10.5 MeV there is practically no effect. Using A~ = 4.55m~, which together with the Bonn B potential leads to the proper binding energy of the triton, reduces the 3NF effects dramatically, making them practically negligible. From all this we conjecture that Coulomb-force effects for the np FSI configurations displayed in Fig. 3 have to be quite small. In the collinear configuration one finds a drastic disagreement of the 10.5 MeV nd data of Ref. [ 23 ] in comparison to our theoretical cross sections over the entire S curve. (Fig. 4a). Our pd data differ significantly from nd ones too, lying about 30% above the nd data in the collinearity region and about 17% above the nd data in the two maxima. At 13 MeV (Fig. 4b) the pd data are about 10% above the purely nuclear theory in the region of collinearity and fall below theory in the two maxima (which are tails of FSI peaks located at nearby angles). The nd data [20] scatter appreciably, which makes it difficult to judge their relation to theory and to the pd data. Recent nd measurements at TUNL [25] taken under exactly the same central detector angles are much closer to the pd data than the ones of Ref. [20] but the data have not yet been finally analyzed. At 19 MeV (Fig. 4c) the discrepancy in the minimum between the pd data [21] and theory is reduced to about 5%. Because the TM 3NF effects are negligible in the region of collinearity, one is tempted to conclude that at the collinearity condition one can hypothesize Coulomb-force effects of 5-20% in magnitude (in the energy range discussed here), which raise the pd cross section in relation to purely nuclear theory but with diminishing importance as the incident beam energy increases. Also Coulomb-force effects of about 10% appear to exist in the two maxima. This is in qualitative agreement with the theoretical results of Ref. [4]. For the space-star configuration the pd data are always below our purely nuclear theory. That discrepancy decreases with energy (20%, 12% and 7% at 10.5, 13 and 19 MeV, respectively). At 65 MeV recent pd cross-section data [26] for the space star agree very well with purely nuclear theory. There are two sets of nd data at 10.3 MeV. One set [24] is in good agreement with our theoretical prediction, the other one [ 22] consists of strongly scattered data points and lie above theory on the average by about 20%, see Fig. 5a. These two sets of nd data are inconsistent. At 13 MeV (Fig. 5b) the nd data of Ref. [20] are also clearly above the theory by about 20%. Recent independent nd measurements performed at TUNL [25] support those nd data. If correct, this points to rather strong Coulomb-force effects, specifically at 13 MeV. The approximate theoretical treatment of Ref. [4], which uses a rank-one separable Yamaguchi interaction in the S waves only, provides, for that configuration, a decrease

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of about 10% due to the Coulomb force. The TM 3NF effects for A~ = 4.55m~ are practically negligible. The effects evaluated for A~ = 5.8m~ and shown in Fig. 5 are energy dependent, lowering the cross section at 10.3 MeV and increasing it at 19 MeV. These effects are too small to lead to significant changes. Altogether it appears that the Coulomb-force effects lower the cross section. However, the situation at 13 MeV poses a puzzle, since the experimentally confirmed nd data are far above the purely nuclear theory. This may call for 3N forces whose nature is different from the one of the TM 3NF. For the pp QFS configuration the only existing nd data are at 10.5 MeV of Ref. [ 18]. They lie about 20% above our pd data and about 10% above theory, see Fig. 6a. Our pd data at 13 MeV (Fig. 6b) and 19 MeV (Fig. 6c) are overestimated by purely nuclear theory by about 13% and 25%, respectively. The theoretical result obtained in Ref. [4] at 14.1 MeV is, that, for QFS, the Coulomb force indeed lowers the cross section in the QFS peak area appreciably. Thus the systematic discrepancy shown in Fig. 6 could be caused by the Coulomb force. This is also supported by a comparison of purely nuclear theory predictions to existing pp QFS pd data over a wider range of energy [21,27]. We also note that at all three energies the TM 3NF shifts theory away from the pd data. Even for A~ = 4.55m,r some non-negligible effects are left.

6. Summary We have described new cross-section data in four kinematically complete configurations of the pd breakup reaction at 10.5 MeV. The data have been compared to predictions from different realistic NN interactions. These predictions were obtained by solutions of the 3N Faddeev equations without taking into account the Coulomb force between the two protons. We found some discrepancies between theory and data for collinear, space-star, and pp QFS configurations. In the case of np FSI geometry, a good description of our data was obtained. Comparing our new pd data, together with our older pd data taken at 13 and 19 MeV, with the corresponding nd breakup data and regarding the energy dependence of the discrepancies, we conjecture that the existing discrepancies are, at least to a very large extent, a result of Coulomb-force effects. This is also supported by the theoretical estimates of Ref. [4]. While the np FSI cross sections seem not to be affected by Coulomb-force effects, they clearly are present in the other three configurations. Especially large Coulomb-force effects are found for the space-star and the pp QFS cross sections. However, in order to justify our conclusions, rigorous solutions of 3N Faddeev equations are required in the breakup domain with the long-range Coulomb force included exactly. This should finally result in a reliable analysis of the pd breakup.

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Acknowledgements This w o r k was supported financially by the D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t ( P r o j e c t Nos. Pa 4 8 8 / 1 - 1 and Pa 4 8 8 / 1 - 2 ) , by the B u n d e s m i n i s t e r i u m fiir F o r s c h u n g und Techn o l o g i e ( P r o j e c t No. 06 B O 7 3 8 ( 7 ) )

and by the Polish C o m m i t t e e for Scientific Re-

search under Grant No. 2 P302 104 06. The numerical calculations have been p e r f o r m e d on the C R A Y Y - M P o f the H 6 c h s t l e i s t u n g s r e c h e n z e n t r u m in Jtilich, Germany. The authors thank Dr. J. K r u g ( B o c h u m ) and the Erlangen group for m a k i n g their n d data available in numerical form for comparison.

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