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Proton elastic scattering from light nuclei at intermediate energies Van Oers, W. T. H. 1977

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T R I U M FPROTON ELASTIC SCATTERING FROM LIGHT NUCLEI AT INTERMEDIATE ENERGIESW.T.H. van OersDepartment of Physics University of ManitobaMESON FACILITY OF:UNIVERSITY OF ALBERTA SIMON FRASER UNIVERSITY UNIVERSITY OF VICTORIA UNIVERSITY OF BRITISH COLUMBIATRI-77-3TRIUMFPROTON ELASTIC SCATTERING FROM LIGHT NUCLEI AT INTERMEDIATE ENERGIESW.T.H. van OersDepartment of Physics University of ManitobaBased on a talk presented at the 1976 TRIUMF Users Group Annual Meeting, December 1976Postal address:TRIUMFUniversity of British Columbia Vancouver, B.C.Canada V6T 1W5 December 1977Recent years have seen a wealth of information presented on intermediate-energy proton elastic scattering from very light nuclei (2H, 3H, 3He, and 4He) . It must be said that the energy range studied has a lower limit of approximately 400 MeV. The dif­ferential cross-section angular distributions can be thought of as divided into threeregions: a forward angular region where one expects the multiple scattering model of Glauber1 to be valid, this region being characterized by a minimum and secondary maximum caused by the interference of single-scattering and double-scattering amplitudes; aregion of intermediate angles where one expects a minimum caused by the interference ofdouble- and triple-scattering amplitudes; and finally the region of backward angles which quite generally exhibits peaking at 180° caused by nucleon or mu 11i-nucleon ex­change processes, which may include N*'s or A's, and pion emission and reabsorption pro­cesses in the 3,3 resonance region.For orientation Fig. 1 shows do/dt as a function of the four-momentum transfer variable t [t = -2k2 (1-cos9)] in the energy range 150 to 1150 MeV. The solid dotsrepresent data obtained at Saclay with the proton synchrotron SATURNE I and the SPES Ispectrometer.2 These data are now considered to be final. The open circles are data obtained with the Berkeley 184-inch synchrocyclotron by a UCLA-LBL-University of Texas collaboration.3 The filled— in square are the results obtained at 600 MeV with the CERN synchrocylotron by a group from Clermont and Lyon.4 The open squares, which represent the results obtained at Saclay using an incident a-particle beam and a 1arge-momentum spectrometer,5 span the entire allowed range of t-values. Finally, the filled tri­angles give the data obtained with the Orsay synchrocyclotron.6 One should first of all note that the ratio (R) of the values of the cross-sections at the first minimum [at t ~  -0.25 (GeV/c)2] and the second maximum is most pronounced at about 700 MeV (if one excludes for the moment the 600 MeV CERN results). This ratio increases again at higher energies as will be shown below. There is perhaps some indication of a change of slope at around t = -1.0 (GeV/c)2 where one expects double- and triple-scattering amplitudes to interfere destructively. At the higher energies the figure does not give the complete range of allowed four-momentum transfer.At extreme forward angles there is a Coulomb-nuclear interference region which changes from destructive interference to constructive interference because of a change in sign of the real part of the nuclear forward scattering amplitude. This effect is more clearly seen in Fig. 2, which shows the p-4He elastic scattering differential cross-section angular distributions measured prior to 1975 (with the exception of the 156 MeV Orsay data).7-9 The change in sign of a, the ratio of the real part to the imaginary part of the nuclear scattering amplitude at 0°, can only be determined prop­erly from extreme forward-angle scattering data which contain the interference region. Accurate measurements of the total cross-sections provide a check since the imaginary part of the 0° scattering amplitude is directly related to the total cross-section via the optical theorem [im f(0°) = (k/AnOay]. One expects a to change sign around 400 MeV. Note that the figure shows a difference between the 1 GeV Brookhaven data and the 1.05 GeV Saclay data. After renormalization and a small angular shift (-0.5°) both angular distributions can be made to agree except for the three data points at the position2of the minimum, which I gathered were the last three points measured in the Brookhaven experiment. It was stated above that the ratio of the cross-sections at the first mini­mum and at the second maximum increases again for incident proton energies well above 1 GeV. This is demonstrated in Fig. 3, which shows preliminary results of an experiment at 2k GeV using the CERN proton synchrotron.10 The ratio R has become approximately equal to A.It is interesting to note that the first minimum becomes more pronounced at ener­gies where spin and isospin effects are probably small. At 2k GeV there are only small differences between the p-p and n-p scattering parameters. If spin effects contribute significantly then one should have some indication from a p-4He polarization contour diagram. Such a contour diagram based on the existing p-4He polarization data11 is shown in Fig. k. One observes that the contour lines tend to close towards 700 MeV (where R as discussed above exhibits a relative maximum). Very large polarizations exist in p-4He elastic scattering, in particular in the energy region of 200 to 300 MeV where polarization values of near +1 and -1 are attained in the forward hemisphere. The ongo­ing TRIUMF experiment will determine if this pattern persists in the backward hemisphere. (Preliminary results indicate that this is indeed the case.) The current experiment using the ZGS at Argonne National Laboratory will determine if the p-4He polarizations become once more pronounced at higher incident proton energies (i.e., at energies around 3 GeV).The experimental situation regarding p-3He and p-3H elastic scattering is much more sketchy. The existing differential cross-sections, again plotted versus the four- momentum transfer variable t, are shown in Fig. 5. The open circles present 1.0A GeV data obtained at Saclay with the SPES I spectrometer,12 the solid dots present 600 MeV data obtained at CERN,4 the open triangles present 582 MeV data obtained at SREL9 while the filled triangles present 156 MeV data obtained at Orsay.13 There exists also p-3H data at 156 MeV13 and at 600 MeV4 obtained using gaseous tritium targets containing up to 1000 Ci of tritium. One observes again the characteristic minimum and second maximum due to the interference of single-scattering and double-scattering amplitudes. Note also that there is a break in the slope of the distribution near t = -1.0 (GeV/c)2 .In the multiple-scattering eikonal model of Glauber1 the elastic nucleon-nucleus amplitude takes the form:F(q) - £  f  d3r1...«3rA p<A>(ri,..TA) s ( Z  ?,). / f t ,  . # [ t - n ( ,  - ^  f  ,J(,)) (1)In this expression fj represents the two-particle amplitude for the scattering of the projectile from the jth target nucleon; kQ and k are the incident momentum in the nucleon-nucleon and nucleon-nucleus centre-of-mass system, respectively; and q is the momentum transfer. The nuclear structure information is contained in the many-body3ground state density p    .... r^) of the target nucleus; the vector Sj representsthe projection of the position rj of the nucleon in the plane normal to the incident d i rect i on.One usually assumes that the target nucleon overlap is negligible and consequent­ly the phases are additive, which in turn allows one to introduce the experimentally determined nucleon-nucleon amplitudes fj(q). But this is an assumption which really is not very good, since the inter-nucleon spacing is of the order of 1 fm (well within the range of the strong interaction, of course), and thus one requires a knowledge of the nucleon-nucleon off-energy-shel1 interaction. One also assumes that the average Coulomb phase, calculated from the total charge distribution, can simply be added to the strong interaction phases. The Coulomb interaction has a noticeable influence not only at extreme forward angles but also at the diffraction minima where it interferes according to the sign of the real part of fj.The most general representation of the nucleon-nucleon scattering amplitude con­sists of five amplitudes:M(q) = A(q) + C(q)(a0+aj) • n + M ' (q)(a0*n)(aj *n) + G(q)[(a0 *P)(aj*P)+ (a0-k)(a0*k)] + H(q)[(a0*P)(aj • P) - (a0 •k)(aj• k) ] (2)using standard notation, see for instance Ref. \k, and n, P, and k are unit vectors in the direction of £0 x £p, i<0 + £p, and £0 - £p. Of these one usually retains only two ampli tudesM(q) —  A(q) + C(q)(o0+aj) • n. (3)The amplitudes A and C are parametrized in a form suggested by the optical theorem for diffractive scattering and frequently used at high energiesA(q) = °T k° (i + a) exp (~B q2/2) (A)Airfor the spin-independent part andC (q) = — -— — i l ~ T  (i + as)Ds exp d2/2) (5)Air \ km*for the spin-dependent part. In these expressions aj is the total cross-section, a(as) the ratio of real to imaginary parts of the amplitude, B(BS) the slope parameter, Ds the relative strength of the spin-dependent amplitude, and m is the nucleon mass. Note that in these expressions one assumes that the real part and the imaginary part of the ampli­tude have the same q dependence.It is obvious that one would prefer the use of a more complete parameterization of the nucleon-nucleon scattering amplitudes. However, the lack of precise nucleon- nucleon scattering data, especially at forward angles (which is the region of importance for nucleon-nucleus scattering) and in particular at energies above 500 MeV, at presentprevents the deduction of the corresponding scattering amplitudes in a reliable manner, would like to draw attention to the current uncertainties with regard to a as extracted from forward-angle p-p scattering data at energies up to roughly 1 GeV,15 not to mention the situation with regard to a for the n-p system. The analysis of the nucleon-nucleon scattering data is furthermore increasingly complicated as a result of the increasing inelasticity. Figure 6 shows a, the ratio of the real part to the imaginary part of the forward p-p non-spin-flip scattering amplitude as deduced in the phase-shift analyses of Bystricky and Lehar (shaded bands).16 Curve a) represents a as deduced from the energy- dependent phase-shift analysis of MacGregor et al. , and curves b) and c) are dispersion relation calculations of Soding and Dumbrais, respectively (see Ref. 15). Figure 7 shows the behaviour of app and anp as a function of energy.17 The values of app were obtained by fitting the forward-angle p-p elastic scattering data with an expression of the formda (zZa)2^ 2 zZaP  R OTN -------- exp(bt) -   EN (sin26 + aDD cos26)exp(bt)dt 62 |t|2 3 |t|+ Ti7 aK' + apP) ^ exp(bt)> (6)where a is the fine structure constant, Be is the velocity of the incident particle, z = Z = 1, b is the form factor slope parameter, 6 is the Bethe phase, 6 = (-^ in |t| -Jin b + in 2-y) Zfa , and y is Euler's constant.pThe nuclear structure enters the problem either through the N-body density p(N) ("?!,..."r^ ) or the one-body density p(r). One usually proceeds through a one-body dens i ty,P(r) = N (e~Kl r2 - C e~K2 r*) , (7)and determines the parameters by a fit to the charge form factor.The results obtained by Auger, Gillespie and Lombard,18 which include correctionsdue to target-nucleon overlap and charge exchange, give rather good agreement with the p-^He differential cross-sections but rather poor agreement with the p-^He polarizations Figures 8 and 9 show the results of their calculations at 1.15 GeV and 2h GeV. It should be remarked that before one can get to the details of the nuclear target wave function many of the approximations made need a great deal more investigation.Rule and Hahn19 made an effective channel analysis of p-^He elastic scattering.In their approach the intermediate-energy proton-nucleus scattering is formulated in terms of coupled equations in which the effects of inelastic processes, which are speci­fically associated with the excitations of the target system during the collision, are represented by an average inelastic channel. The theory incorporates approximately the effects of nonlocality, energy dependence, rescattering and absorption of all the in­elastic channels. Fair agreement with the p-^He differential cross-section angular distributions has been obtained. Spin-dependent effects have been ignored.5Phenomenological optical-model analyses of p-4He elastic scattering have most recently been made by Arnold et at.20 The optical potential used has the formU = (V + iW) f(r), (8)with the shape function f(r) given by a three-parameter Fermi distribution1 + wr2/c2 . .f(r) =  p; — r .  (9)1 + expL(r-c)/z]The potential is inserted in the Dirac equation as the fourth component of a four-vector, with the vector part of the potential taken to be zero. Note that spin-dependent effects are not considered. Very good fits to the forward differential cross-section data are obtained, as shown in Fig. 10. The volume integral per nucleon of the central potential, as defined byf U(r)d3rJ AJ/A = /  : , (10)has the characteristic energy dependenceJr/A = J0/A + B£nTp , (11)also found from optical-model analyses of proton elastic scattering at intermediate energies from various nuclei throughout the periodic table.21 The energy dependence of J/A for p-4He obtained by Arnold et al. is shown in Fig. 11. The real part of the potential changes sign at around AOO MeV. This compares very favourably with what ob­tained from the analyses of proton elastic scattering from heavier nuclei (see Fig. 12). Note the difference in the sign convention. For a nucleus as light as 4He one has to restrict the data to be fitted to the forward angular region where contributions from the exchange amplitude are small.Considerable attention has been given to the anomalies in the backward hemisphere of the angular distributions of protons elastically scattered from 2H, 3He and 4He.These anomalies are easily recognizable in the 180° excitation functions where they appear as a secondary maximum at ^500 MeV in the case of 2H(p,p)2H and as pronounced changes in the slopes in the case of 3He(p,p)3He and 4He(p,p)4He. Figure 13 shows the p-d 180° excitation function.11 The open circles present neutron data while the solid dots represent proton data. A simple neutron exchange mechanism fails to explain the p-d backward angle distributions around the energy of the second maximum even after both 3Sj and 3LA components of the deuteron wave function have been taken into account. It should be noted that, to first order, the backward angle differential cross-sections are proportional to | <j> (| q| ) [ A where |q| is the magnitude of the momentum transfer. Thus the backward angle differential cross-section is extremely sensitive to the details of the wave function at large momenta. The p-d backward-angle distributions have been explained using various hypotheses, i.e., the exchange of nucleon resonances,22 and high-order multiple scattering.23 Gurvitz, Alexander and Rinat24 have formulated a noneikonal6approach to hadron-nucleus scattering valid for all angles but for incident momenta greater than 1 GeV/c or in the case of proton scattering for incident energies greater than approximately 500 MeV. They obtain agreement with the experimental p-d data over a large part of the differential cross-section angular distributions and also backward peaking, but there is no quantitative agreement. Unfortunately, their theory is not valid in the energy region where the 180° excitation function shows a relative maximum,i.e., around 500 MeV. Thus, no detailed quantitative agreement with the experimental data has been presented as yet, and the possible existence of nucleon resonances in nuclei is still an intriquing problem indeed.A polarization contour diagram (Fig. 14), based on the existing data of the polar­ization of protons elastically scattered from deuterium,25’9 reveals rapid changes towards the energy where the relative maximum in the 180° excitation function occurs. At the resonant energy, due to the large value of the cross-section for the reaction p + N -*■ d + TN the two-step process, shown in the insert in Fig. 15, may become important. In this two-step process the incident proton interacts with one of the nucleons of the deuteron initiating the reaction p + d -> d + TNS The pion is successively reabsorbed by the other nucleon. Using this model predictions of the p-d backward angle polarizations have been made.26 These predictions are depicted as solid dots in the figure. Prelim­inary values for the p-d polarization at 630 MeV, obtained at Dubna,26 are shown as opencircles. The curve is drawn only to guide the eye. The asymmetry angular distribution at 630 MeV would indicate even more pronounced changes than shown in the polarization contour diagram (Fig. 15). It should be noted that the same model fails to give any resemblance with the experimental polarization angular distribution at 425 MeV.The p-^He backward angular distributions plotted as function of cos0 are shown inFig. 16. Note that the angular distributions for increasing energy first show an increasetowards 180°, then a slowly decreasing or flat behaviour followed by another increase towards 180° at the two highest energies. The data at 2 9 8 , 438, 648, and 840 MeV wereobtained at Saclay using an a-particle beam incident on a hydrogen target and the large-momentum spectrometer.5 The horizontal error bars present the angular acceptance in the centre-of-mass system. In the laboratory the a-particles are restricted to a forward cone with half-angle of ^14.6°. The resulting 180° excitation function is shown in Fig. 17-27 Here the anomalous behaviour of the angular distributions corresponds to a change in slope at around 200 MeV. The shoulder at about 40 MeV is caused by a particular interplay of the p-wave and d-wave p-^He phase shifts. Lesniak, Lesniak and Tekou28 have calculated backward p-^He angular distributions in the framework of a triton exchange model including absorption in the initial and final states. They found that absorption can diminish the cross-sections obtained in the Born approximation by one or two orders of magnitude. The calculated angular distributions give fair agreement with the experi­mental data at 2 9 8 , 438, and 648 MeV, as shown in Fig. 18. These authors predict addi­tional structure in the angular distributions around 240 MeV, with possibly a minimum in the excitation function.The p-^He backward angular distributions plotted as function of cosQ are shown in Fig. ig.29,7,13 Note that all measured angular distributions show peaking towards 180°7but the decrease in the value of the differential cross-sections is non-monotonic. The data at *tl5> 600, and 800 MeV were obtained at Saclay with the SPES I spectrometer. The 180° excitation function shows a shoulder around b00 MeV (Fig. 20). The curve is drawn to guide the eye and has no further significance. In summary, it is apparent that the backward angle anomalies in p-d, p-3He and p-^He elastic scattering require a great deal further experimental and theoretical attention, including measurements of the asymmetries using an incident polarized proton beam, before a quantitative description of the under­lying physical process can be given.References1. R.J. Glauber, in Lectures in Theoretical Physics, ed. by W.E. Brittin and L.G. Dunham (interscience, New York, 1959), vol 1, p.315-2. E. Aslanides, T. Bauer, R. Bertini, R. Beurtey, A. Boudard, F. Brochard, G. Bruge,A. Chaumeaux, H. Catz, J.H. Fontaine, R. Frascaria, D. Garreta, P. Gorodetzky,J. Guyot, F. Hibou, M. Matoba, Y. Terrien, and J. Thirion, contribution IV.A.lAtoVI Int. Conf. on High-Energy Physics and Nuclear Structure, Santa Fe and Los Alamos, 1975; D. Legrand, private communication.3. S.L. Verbeck, J.C. Fong, G.J. Igo, C.A. Whitten, Jr., D.L. Hendrie, Y. Terrien,V. Perez-Mendez, and G.W. Hoffmann, Phys. 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Lett. 66B, 329 (1977).1975 ((spun AjDJjiqjD) ip/-°p(js/qiu) UP/-op6cnv (degrees) -t (GeV/c)Fig. 2 Fig. 3Fig. 5Momentum (MeV/c)OIdKinetic energy (MeV)Fig. 7a Fig. 7bdcr/dXl (cm2)Fig. 8 Fig. 9§CDO“O\b“O-t (GeV/c)Fig. 10>a>u w j .  A 9 ^i)v/r(degrees) dcr/dH in mb/srFig. 13Tp (M e V ))28 6b- 6bt 630GeVt /cHp, 1975 E 66-JSb,616 GeVt /cHp, 1958A i i i i i i i i I i i i60  80  100 120 140 160^ c m  ( d e g r e e s )Fig. 15He(p,p) HeEO  OOI C GCO pO  IIC JCUD( js /qu j) (U P / -O P )(JS/quj)up/-op-  c\j ro ^ ^q  uq H  uq q0(js/qui)up/-°p(A9|AI)U1 jjjuo 1 oo♦

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