Dirac coupled-channels analysis of inelastic scattering of 800 MeV polarized protons from 16O, 24Mg and 26Mg

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DIRAC C O U P L E D - C H A N N E L S ANALYSIS OF INELASTIC SCATTERING OF 800 MeV POLARIZED PROTONS FROM 160,

27 October 1988

24MgAND 26Mg

R. DE SWINIARSKI, D.L. PHAM lnstitut des Sciences NuclOaires (1 N2 P3 and Universitk J. Fourier), 53 avenue des Martyrs, F-38026 Grenoble Cedex, France

and 3. RAYNAL Service de Physique Th~orique, CEN-Saclay, F-91191 G if-sur- Yvette Cedex, France

Received 2 June 1988;revised manuscript received 8 August 1988

Coupled-channels calculations using the Dirac phenomenologyhave been performed to analyse cross sections and analyzing power data for inelastic scattering of 800 MeV protons from low-lyingstates in ~60, 24Mgand 26Mg.Large negative real scalar potentials, large positive real vector potentials and large negative imaginary vector potentials were obtained. Considerable improvements were obtained using the Dirac equation comparedto the classical Schr6dingercalculations. These best fits have been found compatiblewith an imaginaryscalar potential equal to zero ( U~= 0) and therefore with less parameters (nine) than usually required in the classical calculations (twelve).

The nonrelativistic SchrSdinger equation has until recently been the basis for almost all calculations in traditional nuclear physics. At least, cross sections obtained from inelastic scattering of low energy 100200 MeV protons for light or heavy targets have been reasonably well reproduced by the distorted wave born approximation (DWBA) or coupled-channels (CC) calculations, using the Schr6dinger equation [1]. Good agreement has also been obtained between such calculations and 800 MeV (~, p' ) scattering data from LAMPF [ 2 ]. The only spin-observable data available for this energy range, namely the analyzing power A (0) obtained by inelastic scattering of polarized protons, have also been fairly well reproduced, mainly in the macroscopic collective model with the Schr6dinger equation, when the Full-Thomas form for the deformed spin-orbit potential has been used together with a general increase of the spin-orbit deformation parameter (ilLS) compared to the central potential deformation flc(flLs/flc=2>~ 1 ) which has been found to be often necessary mainly at energies below ~ 500 MeV [3]. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

Since about 1981, new capabilities at the Los Alamos Meson Physics Facility (LAMPF) have made it possible to measure the complete set of data obtained by elastic scattering of polarized protons off spin-zero nuclei at an energy up to 500 MeV, namely cross section a ( O ) , analyzing power A (0), and a new parameter, the spin-rotation function Q (0). At these energies one expected the elastic scattering, to be well described by the impulse approximation (IA) calculations from the free nucleon-nucleon scattering amplitude ( N - N ) (well known at these energies) in the conventional non-relativistic (NR) Schrrdinger equation formalism but including appropriate relativistic kinematics. Indeed once the N - N amplitudes are known and the proton or neutron densities are fixed, (the proton densities are obtained from empirical charge distributions, while neutron densities are assumed to be given by some appropriate model-independent form) there are no free parameters. Even though this NR approach has met with considerable success in reproducing proton-nucleus elastic data at low and high energies [4 ], it was found inadequate to reproduce 247

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the spin observables [A(0), and mainly Q ( 0 ) ] [5] for small momentum transfers. This apparent breakdown of the model was quickly supposed to lie not in the IA, but rather in the Schrt~dinger equation itself due to the importance of relativistic effects. These conclusions led Clark and Arnold and collaborators [6 ] to analyze proton-nucleus scattering in a purely relativistic way based on a phenomenological approach employing the Dirac equation. The relativistic optical model used consists of a Lorentz scalar potential, Us, and the time-like component, Uv, of a Lorentz vector potential. The real and imaginary parts of both Us and Uv have been chosen, by these authors, to be independent. Each employs two parameters (plus the strength parameter) and Fermi-function shapes. This twelve-parameter (the same as in the usual non-relativistic treatment) phenomenological Dirac approach was able to reproduce well the spin observables (Q, A ) for the 500 MeV ~ + 4°Ca data where the NR Schr~Sdinger treatment has failed [6]. Another great achievement of this Dirac phenomenology was that it gave a clear explanation of the unorthodox potential shape (wine bottle bottom) observed for + 4°Ca scattering at 180 MeV [ 6 ]. The Dirac phenomenology was thereafter quickly used in a relativistic impulse approximation (RIA) to describe proton-nucleus elastic scattering at intermediate energy [7 ]. The major ingredients were the free N - N amplitudes, determined by nucleon-nucleon scattering experiments and the nuclear density involving Hartree scalar Ps and vector pv terms. Very good descriptions of ~ + nucleus elastic scattering [ but mainly for the spin observables A (0), and Q(O) ] were also obtained for Eo~>400 MeV [7]. Recently the Dirac IA framework (DIA) has been extended to include also the inelastic excitation of collective degrees of freedom [8,9] assuming a vibrating (or "deformed") density model ( V D M ) for the transition densities. In this extension, both the elastic and inelastic transition potentials (including scalar parts Us and vector parts Uv) are obtained by folding ground state densities with the N - N invariant amplitudes obtained from the free N - N interaction as determined by phase shift analysis of such data. As in the IA for elastic scattering analysis, there are 248

27 October 1988

here no free parameters except the deformation length ( f i r ) acting as a "normalization factor". Analysis of 800 MeV 54Fe (~, p' ) scattering data for some low-lying states (2 +, 3 r ) using this DIA led to excellent agreement with the cross section but however only fair agreement with the analyzing power [ 8 ] data. The quality of the fits obtained using the DIA were moreover of the same order as the standard collective model DWBA [ 10 ] obtained by deforming the phenomenological optical model potential using the Schr6dinger equation. The DIA has also been applied recently [ 9 ] to analyze 58Ni (~, p' ) scattering data at 500 MeV using the VDM or the standard non-relativistic VPM (NR VPM) model with deformed collective form factors (NRIA). It was observed that cross sections for most low-lying states in 58Ni obtained by 500 MeV(~, p' ) scattering were better reproduced by NRIA calculations than by those of the DIA while Dirac IA predictions for&. are somewhat better than those of the nonrelativistic VPM. Therefore, even though the Dirac phenomenology has been shown to give improved fits compared to standard SchriSdinger equation treatments in analyzing elastic scattering data for polarized protons between 400 and 800 MeV, the need for a Dirac treatment in inelastic proton scattering analyses is less obvious. Moreover, no such Dirac calculations have yet been reported for strongly deformed nuclei where multistep processes are known to be important and therefore require proper coupled-channels (CC) calculations. We present in this letter CC calculations using a new program with the Dirac phenomenology [ 11 ] ~ to analyse cross sections and analyzing powers for inelastic scattering of 800 MeV polarized protons from both spherical (~60) and for the first time also from strongly deformed nuclei (24Mgand 26Mg). These data have already been analysed through DWBA or CC Schr6dinger deformed optical potential and have been already published [ 12,13 ]. This new CC [ 11 ] program uses the generalized Dirac equation written as follows:

"~ The code used for the present calculations (ECIS 87 ) is a preliminary version of the program of ECIS 88.

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27 October 1988

{o~-p+ ~[m + e~(r) ] - [ E - Uv(r) ] + (ih/2m)flot. [VUx(r)l}~u(r) = O,

( 1)

where Us(r), Uv(r), Ux(r) are three complex potentials of which the real and the imaginary parts are approximated by a Woods-Saxon potential but can be replaced by any other form factor given by a theory. For each potential we have

~5~5 I

I

U= U~f(r, r~, a~) +iU)r(r, ri, ai) , where 1 f(r, rs, as) = 1 + e x p [ ( r - r s A I/3)/as] '

(2)

and U ~and U ~are the real and imaginary strengths of the potentials. Us, Uv, are the Lorentz scalar, Lorentz vector potentials and UT(r) is a small "tensor" potential. The Coulomb potential is included in Uv (r) and in Uv (r). The tensor potential UT(r), although always present due to the interaction of the anomalous magnetic moment of the projectile with the charge distribution of the target, has been found to be always very small

~77

[14]. For a strongly deformed target or a target with strong collective states these potentials are treated in the same way as the central potential in the non-relavistic Schr6dinger equation; the potentials Us(r), Uv(r), UT(r) are deformed or include creation and annihilation operators of bosons for a spherical nucleus. In the new CC program (ECIS 87) used in this study, the coupled Dirac equations are solved by iteration using the integral version of the Ecis method described elsewhere [ 11 ] ~1 Since tensor potentials have been found to be small [ 14 ] compared to scalar or vector potentials, this potential has been neglected in eq. ( 1 ) at least in the initial search procedure. There are therefore twelve arbitrary parameters to fit the scattering data as in the conventional Schr6dinger equations for intermediate-energy proton scattering analysis. The data investigated included a(0) +A (0) for the 0 + (ground state), the 2 + (6.92 MeV) and the 3i(6.13 MeV) in ~60 as well as for the low-lying 0 +, 2 +, 4 + states in 24Mg and 26Mg belonging to the ground state rotational band (GSRB) in these nu-

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PHYSICS bo3

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27 October 1988

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has a large positive real part (200 to 400 MeV). On the other hand, the imaginary part o f the scalar potential turns out to be positive ( ~ 100 MeV), corresponding to particle production, while the imaginary part of the vector potential which accounts for absorption was found to be negative ( ~ - 100 MeV). Dirac optical model parameters obtained for 800 MeV (~, p) on 4°Ca [15] were used in the present study as starting parameters in the search procedure. They include a large negative real scalar potential ( ~ - 300 MeV), a large positive real vector potential ( ~ 200 MeV), a positive imaginary scalar potential

Volume 213, number 3

PHYSICS LETTERS B

27 October 1988

24 Mg (~, p)24Mg

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ecm(deg) Fig. 2. Coupled-channels calculations for the cross sections and analyzing powers for the 0 ÷, 2 + and 4 + states in 24Mg. Solid line: Dirac CC calculations; dashed line: Schr6dinger CC calculations. ( ~ 100 M e V ) , a n d a negative i m a g i n a r y vector potential ( ~ - 100 M e V ) with all radii close to 1.0 fm and diffuseness a r o u n d 0.7 fro. F o r the C o u l o m b potential, the radius (re) o f the charge distribution (with the diffuseness, ac equal to z e r o ) is set equal to that o f the real vector potential. P r e l i m i n a r y D i r a c CC calculations have been performed for 160 first, by coupling together the 0 ÷ (g.s.) a n d the 3 - (6.13 M e V ) states with b o t h dcr/d£2 a n d

A v d a t a and a d e f o r m a t i o n p a r a m e t e r o f 0.50 for the o n e - p h o n o n 3 - state in the f r a m e w o r k o f the vibrational model. In this initial step, a search was p e r f o r m e d allowing all 12 parameters to vary. About 100 partial waves were used and integration o f the differential equations was carried out between 0.1 fm and the matching radius o f 15 fm. Since little change was observed for U~ (the i m a g i n a r y scalar p o t e n t i a l ) in this pre251

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27 October 1988

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liminary search it was decided to try a search on all parameters but fixing the potential U~ to various constant values from 100 MeV to zero. Best fit X2 were obtained with decreasing values for U~-, the lowest X 2 being observed for U~ =0. Therefore a new search procedure was performed on all nine remaining parameters with the imaginary scalar potential set to zero. Very good fit with low X2 was quickly obtained. A search was then made by allowing the strength parameters U~, U~¢ and U~¢ to vary 252

together with either all radii or all diffusenesses. Final Dirac CC parameters were obtained by fixing the best radii and diffusenesses obtained and searching on the strengths and deformation parameters together. The same procedure was adopted for the 0 ÷ (g.s.) state coupled to the 2 + (6.92 MeV) state with an initial deformation parameter of 0.2 for this assumed vibrational state. Slightly different parameters were obtained for

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these calculations obviously due to the well known different structure of the 2 + state in J60 since this state belongs to a K = 0 + rotational band based on the (unresolved) 0 ÷ state at 6.05 MeV. These best parameters are presented in table I while fig. 1 presents the best Dirac coupled channels calculations for the 0 +, 2~- and 3? states in ~60 compared to the traditional CC Schr6dinger equation calculations. As can be seen from this figure, improved fits were obtained for both cross sections and analyzing powers with Z: generally decreased by more than a factor of two. The best optical parameters obtained for the 0 ÷2 ÷ coupling calculations for J60 were thereafter used as starting parameters to analyse the 24'Z6Mg(p, p' ) data. Calculations were performed by coupling the 0+(g.s.) to the 2~- and the 4+ states of the ground state rotational band (GSRB) in these nuclei [a(0) and A (0) simultaneously ] using the rotational model with starting deformation parameters//2 and//4 obtained from previous Schr6dinger CC analysis [ 13 ] and allowing all nine parameters to vary. Final fits were here also obtained by searching on the strengths, U~, U~/, U~, and the deformation parameters fl: and//4 and fixing the best radii and diffusenesses obtained previously. An additional check was made for the imaginary scalar potential Uk, by fixing this potential to various values between 0 and 50 MeV in 10 MeV steps, leaving all other parameters fixed to the best values obtained, and performing a CC Z z calculation without search. The best Z: was obtained for U~ =0. Z z increases dramatically as this potential increases, for example, by more than 50% for U~ = 10 MeV. The best Dirac CC and deformation parameters are presented in table 1 while figs. 2 and 3 present the corresponding Dirac CC calculations (solid lines) compared to the Schr6dinger CC calculations (dashed lines). As in the ~60 data analysis, considerable improvements were obtained for the fits for these deformed nuclei using the Dirac phenomenology with Z -~ decreased by as much as a factor of five compared to classical calculations. Interesting also is the observation that the best Dirac parameters obtained in the present analysis for the 24Mg and ~ lvlgt p, p' ) data are very close to each

27 October 1988

other. Indeed small changes in the Lorentz scalar U~- and the real and imaginary vector potentials U~, U~¢ were found to be sufficient to reproduce well the cross sections and analyzing power data, even though these data are very different due to the different nuclear structure of these nuclei. Conclusion. The Dirac phenomenology used in coupled-channels calculations for spherical nuclei employing a vibrational model and, for the first time, for strongly deformed nuclei, the rotational model, has been shown to give spectacularly improved fits to data obtained by inelastic scattering of 800 MeV polarized protons. The Z 2 is decreased by factors of two to five compared to classical Schr6dinger CC calculations. Contrary to previous DIA calculations, the improvement is obtained this time both for cross sections as well as the analysing power data. The best fits are consistent with an imaginary scalar potential set to zero ( U~ = 0) leaving therefore only nine arbitrary parameters (besides the deformation parameters) instead of the twelve for the usual Schr6dinger CC calculations at intermediate energy. The best Dirac "optical model" parameters obtained, show large negative real scalar potentials ( U ~ ~ - 2 0 0 to ~ - 3 0 0 MeV), large positive real vector potentials (U~¢~ 124 to ~ 164 MeV), and large negative imaginary vector potentials (U~¢ ~ - 7 4 to - 8 8 MeV), all increasing in absolute value from ~60 to 26Mg. Further, introduction of a tensor term Uv has been found to give negligible effects. Finally the deformation parameters obtained by these calculations are in general agreement with previous classical results although slightly higher ( ~ 5%). It would now be interesting to see ifa microscopic formulation of inelastic scattering using the Dirac equation would confirm the observations presented in this paper.

References

[ 1] D.K. Hasell et al., Phys. Rev. C 27 (1983) 482; G.S. Adamset al.. Phys. Rev. C 21 (1980) 2485. [2] M.L. Barlett et al., Phys. Rev. C 22 (1980) 1168; M.M. Gazzalyet al., Phys. Rev. C 25 (1982) 408; C. Glashausser et al., Phys. Lett. B 116 (1982) 215. [ 3] K. Amosel al., Nucl. Phys. A 413 (1984) 255; 253

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R. de Swiniarski and D.L. Pham, Z. Phys. A 296 (1980) 367; B.J.V. Verhaar, Lectures Notes in Polarization in nuclear physics, ed. J.Ehlers (Springer, Berlin, 1974). [4] P. Schwandt et al., Phys. Rev. C 26 (1982) 55, and references therein. [5] G.W. Hoffmann et al., Phys. Rev. Lett. 47 ( 1981 ) 1436. [6] L.G. Arnold and B.C. Clark, Phys. Lett. B 84 (1979) 46; L.G. Arnold, B.C. Clark and R.L. Mercer, Phys. Rev. C 19 (1979) 917; L.G. Arnold et al., Phys. Rev. C 23 ( 1981 ) 1949; B.C. Clark, S. Hama and R.L. Mercer, in: The interactions between medium energy nucleons and nuclei, ed. H.O. Meyer (American Institute of Physics, New York, 1983 ) p. 260; B.C. Clark, R.L. Mercer and P. Schwandt, Phys. Lett. B 122 (1983) 211. [ 7 ] B.C. Clark et al., Phys. Rev. Lett. 50 ( 1983 ) 1644; J.A. Mc Neil, J.R. Shepard and S.J. Wallace, Phys. Rev. Lett. 50 (1983) 1439;

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J.R. Shepard, J.A. Mc Neil and S.J. Wallace, Phys. Rev. Lett. 50 (1983) 1443. [8] E. Rost et al., Phys. Rev. C 29 (1984) 209. [ 9 ] N.M. Hintz et al., University of Minnesota Progress Report (1987) p. 78. [ 10] G.S. Adams el al., Phys. Rev. C 21 (1980) 2485. [ 11 ] J. Raynal, Workshop on Applied theory and nuclear model calculations for nuclear technology applications (ICTP Trieste, February-March 1988 ); Proc. Sixth Intern. Symp. on Polarisation phenomena in nuclear physics (Osaka, 1985 ) p.922. [ 12] G.S. Adams et al., Phys. Rev. Lett. 43 (1979) 421; R. de Swiniarski and D.L. Pham, Nuovo Cimento 99A (1988) 117. [ 13 ] G.S. Blanpied et al., Phys. Rev. C 37 ( 1988 ) 1987. [ 14 ] S.J. Wallace, LAMPF Users Group Proc. (Los Alamos, NM, 1983). [ 15] J. Raynal, Phys. Lett. B 196 (1987) 7.