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Effective interactions for multistep processes
Nuclear Physics A 693 (2001) 616–629 www.elsevier.com/locate/npe
Effective interactions for multistep processes M. Avrigeanu a,∗ , A.N. Antonov b , H. Lenske c , I. Ste¸ ¸ tcu a,d a “Horia Hulubei” National Institute for Physics and Nuclear Engineering, PO Box MG-6, 76900 Bucharest,
Romania b Institute of Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria c Institute für Theoretische Physik, Universität Giessen, D-35392 Giessen, Germany d Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA
Received 24 January 2001; accepted 13 February 2001
Abstract The reliability of realistic M3Y effective NN interactions to describe multistep direct (MSD) processes is proved by analyzing the corresponding real optical potentials. This trial is done in order to overcome the uncertainties of the effective NN-interaction strength V0 obtained by direct fit to the experimental data. The microscopic potential for the nucleon–nucleus scattering at energies lower than 50 MeV has been calculated by using nucleonic and mesonic form factors. It has been analyzed through (i) a comparison with phenomenological optical potentials, and (ii) its use for description of nucleon elastic-scattering angular distributions. It results that the strongly simplified model interactions usually involved within MSD reaction theory, e.g. 1 fm range Yukawa (1Y) term, neglect important dynamical details of such processes. An 1Y-equivalent V0 strength of a realistic effective NN interaction is determined by corresponding optical-potential volume integrals, and involved within Feshbach–Kerman–Koonin theory calculations. 2001 Elsevier Science B.V. All rights reserved. PACS: 24.60.Gv; 25.40.-h; 24.10.Ht; 21.30.Fe
1. Introduction The quantum mechanical formalism developed by Feshbach, Kerman and Koonin (FKK) [1] for the multistep processes has been extensively used to describe a large amount of experimental data covering a broad energy range. The assumptions and simplifying approximations considered in the application of the FKK theory have been analyzed and important refinements of calculations have been made [2–7]. One of the important assumptions concerns the nucleon–nucleon (NN) effective interaction which is taken as * Corresponding author.
E-mail address: [email protected] (M. Avrigeanu). 0375-9474/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 1 ) 0 0 8 1 0 - 7
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a single Yukawa term with 1 fm range, its strength V0 being considered as the only free parameter of the FKK model. However, it should be noted that, even when a consistent standard parameter set has been used as well as several other effects have been taken into account, the systematics of the phenomenological V0 values still show discrepancies especially at low incident energies (Ei < 50 MeV). Such uncertainties of the phenomenological effective NN-interaction strengths may reflect the eventual scaling of V0 compensating for some effects which have been neglected and should be added to the theory. Thus, the necessity for using a more realistic effective NN interaction, which should be consistent with the corresponding real part of the optical-model potential (OMP), has been stated as one of the open problems in the theoretical description of the multistep direct nuclear reactions [8]. The adoption of the M3Y realistic effective NN interaction is among the main points of this new approach [8,9], its reliability being studied by means of the comparison between the corresponding microscopic potential and a phenomenological OMP. In this respect an analysis of the M3Y effective NN interactions based on the g-matrix elements of the Paris NN potentials [10] through the corresponding microscopic optical potentials [11] is given in this work. Actually, it is obvious that results derived for elastic nucleon–nucleus scattering cannot be taken over immediately within the multistep processes description. The inherent complications on how medium effects, induced interactions and the whole set of manybody effects contribute to statistical reactions leading to highly excited nuclei are open questions. This is the main reason why in statistical direct reaction theory still schematic interaction models are used. One of the aims of the present paper is to analyze the microscopic OMPs corresponding to the realistic M3Y effective NN interactions, in order to provide a reliable interaction strength to be used within the FKK theory instead of the free parameter V0 . Moreover, a first insight can be achieved through the comparison between the ‘equivalent’ strengths of the simple 1 fm range Yukawa interaction (1Y) corresponding to the volume integrals of the microscopic potentials, and the V0 values obtained as free parameter of FKK calculations. In this way we expect to overcome the uncertainties in fitting an effective interaction directly to multistep processes data. Furthermore, the trial to find a realistic effective NN interaction suitable for the description of multistep processes at low incident energies is completed in this work by using the corresponding microscopic OMP within the nucleon elastic-scattering cross-section analysis. The purpose has been neither the improved description of the data nor their fit by adjustment of parameters but the analysis of the realistic effective NN interactions used for derivation of the optical potential, and its further applications. One important aspect of using effective NN interactions is related to their regularization at short distances. It is known that the meson theory breaks down in the very short range due to the extended structure of the nucleons [12]. For that reason, in most meson models the one-boson-exchange potentials are usually regularized to remove the singularities at the origin by introducing a form factor in a phenomenological way. Usually monopole-, dipole-, exponential-, Woods–Saxon or other form factors (e.g., [13–15]) have been used accounting for the finite size of the nucleons and pions. It has been established (e.g., [16]) that the most important feature is the principal range of the form factors, while their
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detailed analytical structure does not have a large impact on NN processes and the overall results should be insensitive to the details of how the form factors have been chosen. The effective NN interactions in the medium, which are usually parameterized in terms of meson-exchange-type propagators of Yukawa shape (e.g., M3Y-type interactions), are clearly missing meson–nucleon vertex form factors. In folding calculations for opticalmodel and transition potentials an averaging over the intrinsic momentum distribution of the target nucleus is carried out. In these conditions the use of “on-shell” mesonexchange propagators might lead to spurious effects since contributions from large offshell momenta can be overestimated. Such uncertainties could be avoided by introducing nucleonic form factors depending on the off-shellness of the nucleon and suppressing large off-shell energies and momenta which probe subnucleonic structures (e.g. [17]). Thus one may exclude contributions from small distances where the composite structure of mesons and nucleons would become visible. As an appropriate approach we choose the folding model [18–20]. The effects of the off-shell momentum structure of a realistic interaction are investigated through a nucleonic cutoff form factor in Section 2. A comparison with other cutoff form factors is also given together with the folding formalism, the parameterization of the M3Y effective NN interactions, and the dispersion corrections taken into account for the elastic-scattering angular distribution calculations. The microscopic potential thus obtained is compared with the phenomenological one in Section 3, and the analysis of the angular distributions data is shown. Next, a V0 strength of the 1Y effective NN interaction which is equivalent to a realistic effective NN interaction [11] is involved within MSD calculations with the twofold aim of (i) performing multistep studies without any free parameter V0 , and (ii) having a first insight on the corresponding replacement of the 1Y effective NN interaction by a more realistic one. The conclusions of the work are given in Section 4.
2. The effective NN interactions and corresponding optical potentials The real optical potential for the nucleon–nucleus scattering at lower energies (Ei < 50 MeV) which corresponds to a M3Y effective NN interaction has been calculated including nucleonic and mesonic form factors by a folding approach [18,19]. The effective NN interaction involved within multistep reaction calculations should account for nonelastic processes, e.g. inelastic scattering and nucleon knock-out reactions, so that the off-shell momentum structure of a realistic interaction has to be taken into account. Equally important is a reliable treatment of the nonlocal exchange contributions from the antisymmetrization. Therefore, in order to avoid the complications of a nonlocal optical potential, we have used the local momentum approximation while the effects of the nuclear and nucleon density distribution and the average relative momentum on the folded potential were also considered. Since the details of the folded optical potential are given elsewhere [11,21,22] only the basic points are resumed in the following. In order to regularize the Yukawa-type function we followed the formalism given in Ref. [13]. Thus the regularized Yukawa function φC (r) is obtained from the Fourier
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transform to configuration space by introducing a momentum cutoff form factor F (p): eipr d3 p 4π F (p) = φC (r). (1) m (2π)3 (p2 + m2 ) The right-hand side of Eq. (1) returns to the familiar Yukawa function φC0 (r) = e−mr /mr for F (p) = 1. Frequently used cutoff form factors are either the monopole-, dipole-, and exponential form factors [13] or that of Pandharipande [15]. The experimental information on the electric nucleon form factor GE (p), available at momentum transfer for about p2 20 GeV2 , could be also used. We have used in this work as a cutoff form factor the nucleon form factor [22] F (p) = dr 1 e−ipr 1 ρN (r 1 ), (2) derived from the experimental proton charge density distribution ρN (r) given by Chandra and Sauer [23]. After some transformations we obtain for the regularized Yukawa function φC (r) the expression e−m|r−r 1 | . (3) φC (r) = dr 1 ρN (r 1 ) m|r − r 1 | Thus, an interaction v of Yukawa form can be written in the regularized form as: V (R + r 2 ) = dr 1 ρN (r 1 )v(s = R + r 2 − r 1 ).
(4)
The folding accounting for the target density distribution ρ(r) makes it possible to obtain the real part of the OMP on the basis of a renormalized effective NN interaction in the form [18] (5) U (R) = dr 2 ρ(r 2 )V (R + r 2 ). The effects of different cutoff form factors applied to a simple 1Y function are shown in Fig. 1. The monopole- , dipole-, and exponential form factors are taken from Ref. [13]. The regularized interactions obtained by using either the present nucleonic cutoff form factor or the previous approaches evidence that they have been obtained by similar vertex renormalizations. Therefore we expect to take into account typical features of realistic NN potentials, e.g. [12,13,16,24], of which the regularized Reid93 potential [13] includes explicitly a dipole form factor, as can be seen in Fig. 1. We have calculated (5) in the momentum space [18–20] by means of the detailed procedure developed by Khoa et al. [19] as well as the following approximations. The effective NN-interaction has included the isoscalar and isovector components of the direct D , v D and exchange parts v EX , v EX of the M3Y interaction based on the results of the v00 01 00 01 g-matrix calculations using Paris NN potential [19]. Moreover, the energy dependence of the M3Y-Paris effective interaction has been taken to be a linear function of the incident energy per nucleon of the form g(E) = 1 − 0.003E, so that D(EX) D(EX) (E, r) = g(E)v00(01) (r). v00(01)
(6)
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Fig. 1. The comparative analysis of the regularized Yukawa-type function by the monopole-, dipoleand exponential cutoff form factors [13] and by the present nucleon cutoff form factor (see text for details).
Finally, since the M3Y effective interaction is characteristic for the nuclear density around 1/3 of the normal matter density [19], its choice could be considered appropriate for the nucleon–nucleus scattering at lower incident energies which are localized in the nuclear surface [25]. The approximation used for the calculation of the knock-on exchange term of the folded potential proposed by Campi and Bouyssy [26], preserves the first term of the expansion given by Negele–Vautherin [27] for the realistic density-matrix expression s s (7) ρ(R, R + s) = ρ R + jˆ1 kav R + s , 2 2 where jˆ1 (x) = 3(sin x − x cos x)/x 3 , and kav defines the average relative momentum [26] 1/2 5 1 2 kav (r) = . (8) τ (r) − ∇ ρ(r) 3ρ(r) 4 The latter quantity is a function of the density distribution ρ(r) and the approximation used for the kinetic-energy density τ (r). We have used the Fermi-type density for the target nucleus and the Negele parameterization [28], as well as the modified Thomas–Fermi (MTF) approximation [29] of the kinetic-energy density τ (r) [11,22]: 5|∇ρ(r)|2 5∇ 2 ρ(r) 1/2 MTF − (r) = kF (r)2 + , (9) kav 12ρ 2(r) 12ρ(r) where kF (r) = [3π 2 ρ(r)/2]1/3 is the local Fermi momentum. Furthermore, the spin–orbit potential has been calculated as the derivative of the microscopic real potential [30] 2 1 dU (r) h¯ σ L. (10) USO = 2mc r dr
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The angular distributions thus obtained were found very close to the results provided by the spin–orbit term of Vautherin and Brink [31] with the W0 parameter value of 120 MeV fm5 [32]. We adopted the imaginary component of a phenomenological OMP in order to carry on the elastic-scattering data analysis. The global parameter set established by Walter and Guss [33] for the nucleon scattering on target nuclei with A > 53 at incident energies up to 80 MeV has been considered in this respect. Moreover, the dispersion relation which connects the real and imaginary parts of the average nucleon–nucleus potential [34,35] have been taken into account. Finally, the real potential UR (r; E) has been obtained by adding the dispersive correction U(r; E) determined from the imaginary optical potential to the folded potential U (r; E) [35]: UR (r; E) = U (r; E) +
U (r; E).
(11)
The dispersive correction which takes into account the fact that the target does not remain in its ground state during the elastic-scattering process, is the sum of the corresponding volume and surface components (q = v, s) Uq (r; E) =
2 (EF − E) π
∞ 0
dE
Wq (r; E + EF ) , (E − EF )2 − E 2
(12)
where EF is the Fermi energy defined as lying half-way between the first particle state E+ > EF and the first hole state E− < EF of the n + A system EF = 12 (E+ + E− ).
(13)
3. Results and discussion 3.1. OMP real component The real OMPs for incident protons on 93 Nb and 90 Zr, at energies around 22 MeV, calculated in this work by means of Eq. (5) are shown in Fig. 2. First, the microscopic real potential U (r; E) is compared with the real part of the phenomenological OMP of Walter and Guss [33] in Figs. 2(a) and 2(c). One may consider that the real potentials calculated in this work are in rather good agreement with the phenomenological predictions for the both targets. The M3Y microscopic potentials are deeper at the center of the nucleus and less diffuse in the nuclear surface region with respect to the phenomenological real OMPs, while their volume integrals are rather similar. Second, it is illustrated the completion of the real parts of the microscopic OMPs by the dispersive corrections. The volume Uv (r; E) and surface Us (r; E) components as well as the total dispersive corrections U (r; E) obtained by using the imaginary part of the Walter–Guss potential are shown in Figs. 2(b) and 2(d). The volume dispersive correction leads to the total real potentials which are deeper at r = 0, and increasing for higher incident energies [35]. At the same time, the corresponding surface component which
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Fig. 2. (a,c) The comparison of the radial dependence of the optical potential obtained by using the M3Y effective NN interactions in the MTF approximation, for protons incident on 93 Nb and 90 Zr respectively, without (dotted curves) and including the dispersion component (solid curves), and the real component of the phenomenological OMP [33] (double-dot-dashed curves); (b,d) the radial dependence of the volume- (dashed curves) and surface components (dotted curves) of the dispersive correction (solid curves) obtained by using the Walter–Guss imaginary potential [33] for 22.2 MeV and 22.5 MeV protons incident on the same nuclei.
is strongly damped for nucleon energies such that |E − EF | > 20 MeV [34] lowers the dispersive correction towards the nuclear surface. The corresponding real potential, UR (r) Eq. (11), is presented in Figs. 2(a) and 2 (b) by solid curve. 3.2. The angular distributions analysis The comparison of the elastic-scattering angular distributions calculated by using the microscopic OMPs given in the previous section and the experimental data is shown in Fig. 3. The former have been obtained by using the real microscopic optical potential UR (r, E) with the dispersive correction (11), and the imaginary component of the Walter– Guss phenomenological potential [33]. It can be concluded that the experimental elasticscattering angular distributions are satisfactory well described by the microscopic potential over the whole angular range. We underline that no adjustable parameter or normalization constant has been involved within this analysis in order to couple the real and imaginary parts of the OMP (e.g., [38] and references therein). Our aim has been neither the elastic-scattering data analysis nor the
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Fig. 3. The comparison of the elastic-scattering angular distributions of protons on (a) 93 Nb and on (b) 90 Zr calculated (full curves) by using the microscopic potentials and phenomenological OMP parameter set [33], and the experimental data of (a) Fulmer [36], and (b) Ball et al. [37].
improvement of the microscopic real potentials but validating a suitable realistic effective NN interaction to be used in the multistep nucleon emission. From this point of view one may find a good agreement between measured and calculated differential cross sections which generally proves that microscopic potentials built on the M3Y effective NN interactions are able to describe rather well the proton interaction with the given target nuclei. 3.3. Equivalent 1Y NN interaction strength eq
The 1Y-equivalent effective NN-interaction strength V0 of the M3Y effective NN interaction is determined hereafter by corresponding volume integrals of the related optical potentials J1Y = JU ,
(14)
where J1Y corresponds to the use within Eq. (5) of the 1Y effective NN interaction eq eq with the strength V0 . Thus, V0 is energy- and mass-dependent, in agreement with the phenomenological predictions [4–6]. This quantity should not be considered as a trial for continuing to use the crude 1Y model in FKK calculations but as a check point for the phenomenological V0 values. Its usefulness could be shown by a comparison between the phenomenological real potentials [33] and the calculated results obtained by using either the realistic effective NN interaction or the usual 1Y interaction with phenomenological V0 strength values (Fig. 4). The potentials which correspond to the V0 values obtained by Watanabe et al. [6] are deeper by up to 50% with respect to the phenomenological ones. These large differences should emphasize the possibility of having some effects neglected in FKK calculations and corresponding scaling of the free parameter V0 . eq The energy dependence of V0 is compared in Fig. 5(a) with the phenomenological V0 values for (p, p ) processes. Our calculated values for the 98 Mo target nucleus are compared with the phenomenological V0 values obtained within MSD studies of protons incident on target nuclei in the mass range A ∼ 90, i.e. 90 Zr [4,5,7], 93 Nb and 98 Mo [6], as well as
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Fig. 4. The radial dependence of the microscopic real potential calculated by using the M3Y effective NN interactions (solid curves), the 1Y interaction and the phenomenological V0 values [6] (dashed curves), and of the phenomenological OMP [33] (double dotted-dashed curves), for protons incident on (a) 93 Nb, and (b) 90 Zr.
eq
Fig. 5. (a) The energy dependence of V0 of the M3Y Paris effective NN interaction for protons on 98 Mo, compared with the phenomenological V0 values for protons on 90 Zr (open circle [4], square [5], diamonds [7]), 93 Nb and 98 Mo (triangles) [6], and the phenomenological global value eq (solid circle) [39]. (b) The target mass-dependence of V0 of the M3Y Paris effective NN interactions 95,97,100 Mo. for 25.6 MeV protons on
the earlier phenomenological result of Austin [39]. The large differences between the phenomenological values found at low incident energies by Watanabe [6] and by Koning and Chadwick [7] have been discussed in terms of the remaining uncertainties of the optical potentials and the single-particle level scheme involved in the FKK calculations. At the same time, the nonlocality correction missing in the DWBA calculations may be responsible especially at low energies for the low V0 values found by Koning and Chadwick [6,40,41]. In eq these conditions, the agreement of the V0 values with the phenomenological systematics could be considered satisfactory. The increased slope of the phenomenological V0 values at low energies can be explained by the surface effects present at these energies [25].
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eq
In addition, the target-mass dependence of V0 of the M3Y Paris effective NN interaction for protons incident on Mo isotopes is shown in Fig. 5(b). Obviously, neither eq the mass- nor the energy-dependence of V0 could explain the broad range of the phenomenological V0 values. The uncertainties in the model calculations may explain this behavior. Therefore, an independent estimation of this parameter value prior to any comparison between the MSD calculations and experimental data may enlighten the correctness of various assumptions undertaken in practical FKK calculations. The eq comparison shown in Fig. 5(a) between V0 of the M3Y effective NN interaction and the phenomenological V0 values could be considered encouraging in this respect. 3.4. The MSD calculations The FKK calculations of multistep direct-reaction cross sections have been carried out by means of the recent computer codes MSD22NEWDW-NON, MSD22NEWDW [42], and MULTI21NON [43], in agreement with the latest achievements in the field [8]. Since the main goal of this work concerning the MSD calculations was the attempt for eq replacement of the V0 free parameter at least by V0 of a realistic effective NN interaction, the default options of the codes have been used. They are mainly the Dilg et al. [44] parameterization for the pairing corrections included in the particle–hole state densities, the nonlocality correction factor of 0.85 [45], and the use of Walter and Guss [33] distorted optical potential for the calculations of the DWBA matrix elements. In order to focus our study on the MSD contribution to the double-differential cross sections, we have analyzed the (p, n) reactions. Thus, we have avoided the strong collective contribution present in (n, n ) or (p, p ) processes. We have also selected for this analysis only high outgoingenergy data which does not include a significant contribution of the multistep compound (MSC) process, so that the MSD contribution remains the only important [7]. The MSD double-differential cross sections of the (p, xn) reaction on 90 Zr and 94−98,100 Mo isotopes at 25.6 MeV incident energy, calculated by using V eq of the 0 M3Y Paris effective NN interactions, are shown in Fig. 6. One may note a general good agreement except the MSD calculations underestimate the experimental angular distributions at backward angles for 90 Zr and 94,96 Mo. Data corresponding to lower (14 MeV) neutron emission from the 90 Zr(p, n) reaction show that there are still present contributions from the MSC emission and pure neutron evaporation. Following the calculation of the corresponding cross sections by using the computer code EMPIRE [46], the total angular distribution is in a good agreement with the experimental data. The available data for the Mo isotopes are rather well reproduced especially for higher target–nucleus asymmetry. The high value of the outgoing energy of 17 MeV prevents in these cases the MSC channel contribution. Finally, the general good agreement between eq the experimental data and the MSD results obtained by using V0 may be considered a support for the implementation of the realistic effective NN interaction in the frame of the FKK theory. This could make possible also further improvements concerning the surface effects present at low incident energies [25] and taken into account by appropriate particle– hole state density formalisms (e.g., [50]).
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Fig. 6. Comparison of experimental double-differential cross sections for the (p, xn) reactions on 90 Zr [47,48] (squares and circles, respectively) and 94−98,100 Mo [49] at 25.6 MeV incident energy, and MSD cross sections (solid curves). MSC and evaporation contributions (dotted curves) and the total cross section (dotted-dashed curve) are also shown for 90 Zr.
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4. Conclusions The analysis of the M3Y effective NN interaction has been carried on by means of the corresponding microscopic potential, in order to suggest a way for improving the FKK description of multistep processes. The present procedure involving the nucleon cutoff form factor for the regularization of the M3Y Paris effective NN interactions leads to real optical potential at lower energies which is in a good agreement with the phenomenological OMP. In addition, the microscopically obtained real optical potential has been applied also to the analysis of proton elastic-scattering data. In order to complete this analysis, the imaginary component of the phenomenological OMP parameter set of Walter and Guss [33] has been adopted, and the effect of the dispersion relations on the real folded optical potential has been taken into account. The satisfactory agreement between measured and calculated differential cross sections proves the applicability of the microscopic potential. No parameter fit or adjustment has been undertaken in order to improve the agreement quality since our aim has been to check the realistic effective NN interactions for further use in practical MSD calculations. The comparison of the microscopic real potential which is obtained by using a realistic force with the ones corresponding to the strongly simplified single Yukawa description shows the restrictions of the simple parameterization. In these circumstances we have eq found the 1Y-equivalent strengths V0 of the M3Y Paris effective NN interactions reliable to provide an alternative procedure for the MSD calculations without free parameters. Moreover, it is shown that the realistic M3Y Paris effective NN interactions could not explain the broad range of the V0 phenomenological values, so that an independent estimation of this quantity prior to comparison between the MSD calculations and experimental data may enlighten the correctness of various model assumptions. The MSD double-differential cross sections of the (p, n) reaction on 90 Zr and 94−98,100 Mo at 25.6 MeV incident energy have been calculated by using the V eq values of 0 the M3Y Paris effective NN interaction. Thus it has been proved as suitable the replacement eq of the V0 free parameter by V0 of a realistic effective NN interaction. The practical FKK calculations could become independent of any free parameter, whose phenomenological values may compensate for the questionable handling of other quantities involved in the FKK formalism such as the particle–hole state density, the optical potential for distorted waves, the DWBA matrix elements, the coupling between MSD and MSC emission chains [7,8]. Therefore, the replacement of the 1Y effective NN interaction by the M3Y Paris is foreseen at this stage. The further consistent use of the corresponding microscopic real potential for the distorted waves in the FKK framework is also intended. Next, consideration of, e.g., DDM3Y density dependent effective NN interactions [21] should improve the realistic description of the multistep nucleon emission at higher incident energies. Indeed, the density independent M3Y interaction has been used successfully in the folding model calculations involving the nucleon–nucleus scattering or light heavy-ion scattering at relatively low energies, where the data are sensitive only to the potential at the nuclear surface [20]. Increasing the incident energy over 50A MeV, the interaction moves towards higher nuclear density region and the density dependence of the
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effective NN interaction has to account for the reduction of its strength that occurs as the density of the surrounding medium increases [19,20].
Acknowledgement The authors are grateful to H. Wolter for useful discussions and a critical reading of the manuscript.
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Effective interactions for multistep processes M. Avrigeanu a,∗ , A.N. Antonov b , H. Lenske c , I. Ste¸ ¸ tcu a,d a “Horia Hulubei” National Institute for Physics and Nuclear Engineering, PO Box MG-6, 76900 Bucharest,
Romania b Institute of Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria c Institute für Theoretische Physik, Universität Giessen, D-35392 Giessen, Germany d Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA
Received 24 January 2001; accepted 13 February 2001
Abstract The reliability of realistic M3Y effective NN interactions to describe multistep direct (MSD) processes is proved by analyzing the corresponding real optical potentials. This trial is done in order to overcome the uncertainties of the effective NN-interaction strength V0 obtained by direct fit to the experimental data. The microscopic potential for the nucleon–nucleus scattering at energies lower than 50 MeV has been calculated by using nucleonic and mesonic form factors. It has been analyzed through (i) a comparison with phenomenological optical potentials, and (ii) its use for description of nucleon elastic-scattering angular distributions. It results that the strongly simplified model interactions usually involved within MSD reaction theory, e.g. 1 fm range Yukawa (1Y) term, neglect important dynamical details of such processes. An 1Y-equivalent V0 strength of a realistic effective NN interaction is determined by corresponding optical-potential volume integrals, and involved within Feshbach–Kerman–Koonin theory calculations. 2001 Elsevier Science B.V. All rights reserved. PACS: 24.60.Gv; 25.40.-h; 24.10.Ht; 21.30.Fe
1. Introduction The quantum mechanical formalism developed by Feshbach, Kerman and Koonin (FKK) [1] for the multistep processes has been extensively used to describe a large amount of experimental data covering a broad energy range. The assumptions and simplifying approximations considered in the application of the FKK theory have been analyzed and important refinements of calculations have been made [2–7]. One of the important assumptions concerns the nucleon–nucleon (NN) effective interaction which is taken as * Corresponding author.
E-mail address: [email protected] (M. Avrigeanu). 0375-9474/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 1 ) 0 0 8 1 0 - 7
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a single Yukawa term with 1 fm range, its strength V0 being considered as the only free parameter of the FKK model. However, it should be noted that, even when a consistent standard parameter set has been used as well as several other effects have been taken into account, the systematics of the phenomenological V0 values still show discrepancies especially at low incident energies (Ei < 50 MeV). Such uncertainties of the phenomenological effective NN-interaction strengths may reflect the eventual scaling of V0 compensating for some effects which have been neglected and should be added to the theory. Thus, the necessity for using a more realistic effective NN interaction, which should be consistent with the corresponding real part of the optical-model potential (OMP), has been stated as one of the open problems in the theoretical description of the multistep direct nuclear reactions [8]. The adoption of the M3Y realistic effective NN interaction is among the main points of this new approach [8,9], its reliability being studied by means of the comparison between the corresponding microscopic potential and a phenomenological OMP. In this respect an analysis of the M3Y effective NN interactions based on the g-matrix elements of the Paris NN potentials [10] through the corresponding microscopic optical potentials [11] is given in this work. Actually, it is obvious that results derived for elastic nucleon–nucleus scattering cannot be taken over immediately within the multistep processes description. The inherent complications on how medium effects, induced interactions and the whole set of manybody effects contribute to statistical reactions leading to highly excited nuclei are open questions. This is the main reason why in statistical direct reaction theory still schematic interaction models are used. One of the aims of the present paper is to analyze the microscopic OMPs corresponding to the realistic M3Y effective NN interactions, in order to provide a reliable interaction strength to be used within the FKK theory instead of the free parameter V0 . Moreover, a first insight can be achieved through the comparison between the ‘equivalent’ strengths of the simple 1 fm range Yukawa interaction (1Y) corresponding to the volume integrals of the microscopic potentials, and the V0 values obtained as free parameter of FKK calculations. In this way we expect to overcome the uncertainties in fitting an effective interaction directly to multistep processes data. Furthermore, the trial to find a realistic effective NN interaction suitable for the description of multistep processes at low incident energies is completed in this work by using the corresponding microscopic OMP within the nucleon elastic-scattering cross-section analysis. The purpose has been neither the improved description of the data nor their fit by adjustment of parameters but the analysis of the realistic effective NN interactions used for derivation of the optical potential, and its further applications. One important aspect of using effective NN interactions is related to their regularization at short distances. It is known that the meson theory breaks down in the very short range due to the extended structure of the nucleons [12]. For that reason, in most meson models the one-boson-exchange potentials are usually regularized to remove the singularities at the origin by introducing a form factor in a phenomenological way. Usually monopole-, dipole-, exponential-, Woods–Saxon or other form factors (e.g., [13–15]) have been used accounting for the finite size of the nucleons and pions. It has been established (e.g., [16]) that the most important feature is the principal range of the form factors, while their
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detailed analytical structure does not have a large impact on NN processes and the overall results should be insensitive to the details of how the form factors have been chosen. The effective NN interactions in the medium, which are usually parameterized in terms of meson-exchange-type propagators of Yukawa shape (e.g., M3Y-type interactions), are clearly missing meson–nucleon vertex form factors. In folding calculations for opticalmodel and transition potentials an averaging over the intrinsic momentum distribution of the target nucleus is carried out. In these conditions the use of “on-shell” mesonexchange propagators might lead to spurious effects since contributions from large offshell momenta can be overestimated. Such uncertainties could be avoided by introducing nucleonic form factors depending on the off-shellness of the nucleon and suppressing large off-shell energies and momenta which probe subnucleonic structures (e.g. [17]). Thus one may exclude contributions from small distances where the composite structure of mesons and nucleons would become visible. As an appropriate approach we choose the folding model [18–20]. The effects of the off-shell momentum structure of a realistic interaction are investigated through a nucleonic cutoff form factor in Section 2. A comparison with other cutoff form factors is also given together with the folding formalism, the parameterization of the M3Y effective NN interactions, and the dispersion corrections taken into account for the elastic-scattering angular distribution calculations. The microscopic potential thus obtained is compared with the phenomenological one in Section 3, and the analysis of the angular distributions data is shown. Next, a V0 strength of the 1Y effective NN interaction which is equivalent to a realistic effective NN interaction [11] is involved within MSD calculations with the twofold aim of (i) performing multistep studies without any free parameter V0 , and (ii) having a first insight on the corresponding replacement of the 1Y effective NN interaction by a more realistic one. The conclusions of the work are given in Section 4.
2. The effective NN interactions and corresponding optical potentials The real optical potential for the nucleon–nucleus scattering at lower energies (Ei < 50 MeV) which corresponds to a M3Y effective NN interaction has been calculated including nucleonic and mesonic form factors by a folding approach [18,19]. The effective NN interaction involved within multistep reaction calculations should account for nonelastic processes, e.g. inelastic scattering and nucleon knock-out reactions, so that the off-shell momentum structure of a realistic interaction has to be taken into account. Equally important is a reliable treatment of the nonlocal exchange contributions from the antisymmetrization. Therefore, in order to avoid the complications of a nonlocal optical potential, we have used the local momentum approximation while the effects of the nuclear and nucleon density distribution and the average relative momentum on the folded potential were also considered. Since the details of the folded optical potential are given elsewhere [11,21,22] only the basic points are resumed in the following. In order to regularize the Yukawa-type function we followed the formalism given in Ref. [13]. Thus the regularized Yukawa function φC (r) is obtained from the Fourier
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transform to configuration space by introducing a momentum cutoff form factor F (p): eipr d3 p 4π F (p) = φC (r). (1) m (2π)3 (p2 + m2 ) The right-hand side of Eq. (1) returns to the familiar Yukawa function φC0 (r) = e−mr /mr for F (p) = 1. Frequently used cutoff form factors are either the monopole-, dipole-, and exponential form factors [13] or that of Pandharipande [15]. The experimental information on the electric nucleon form factor GE (p), available at momentum transfer for about p2 20 GeV2 , could be also used. We have used in this work as a cutoff form factor the nucleon form factor [22] F (p) = dr 1 e−ipr 1 ρN (r 1 ), (2) derived from the experimental proton charge density distribution ρN (r) given by Chandra and Sauer [23]. After some transformations we obtain for the regularized Yukawa function φC (r) the expression e−m|r−r 1 | . (3) φC (r) = dr 1 ρN (r 1 ) m|r − r 1 | Thus, an interaction v of Yukawa form can be written in the regularized form as: V (R + r 2 ) = dr 1 ρN (r 1 )v(s = R + r 2 − r 1 ).
(4)
The folding accounting for the target density distribution ρ(r) makes it possible to obtain the real part of the OMP on the basis of a renormalized effective NN interaction in the form [18] (5) U (R) = dr 2 ρ(r 2 )V (R + r 2 ). The effects of different cutoff form factors applied to a simple 1Y function are shown in Fig. 1. The monopole- , dipole-, and exponential form factors are taken from Ref. [13]. The regularized interactions obtained by using either the present nucleonic cutoff form factor or the previous approaches evidence that they have been obtained by similar vertex renormalizations. Therefore we expect to take into account typical features of realistic NN potentials, e.g. [12,13,16,24], of which the regularized Reid93 potential [13] includes explicitly a dipole form factor, as can be seen in Fig. 1. We have calculated (5) in the momentum space [18–20] by means of the detailed procedure developed by Khoa et al. [19] as well as the following approximations. The effective NN-interaction has included the isoscalar and isovector components of the direct D , v D and exchange parts v EX , v EX of the M3Y interaction based on the results of the v00 01 00 01 g-matrix calculations using Paris NN potential [19]. Moreover, the energy dependence of the M3Y-Paris effective interaction has been taken to be a linear function of the incident energy per nucleon of the form g(E) = 1 − 0.003E, so that D(EX) D(EX) (E, r) = g(E)v00(01) (r). v00(01)
(6)
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Fig. 1. The comparative analysis of the regularized Yukawa-type function by the monopole-, dipoleand exponential cutoff form factors [13] and by the present nucleon cutoff form factor (see text for details).
Finally, since the M3Y effective interaction is characteristic for the nuclear density around 1/3 of the normal matter density [19], its choice could be considered appropriate for the nucleon–nucleus scattering at lower incident energies which are localized in the nuclear surface [25]. The approximation used for the calculation of the knock-on exchange term of the folded potential proposed by Campi and Bouyssy [26], preserves the first term of the expansion given by Negele–Vautherin [27] for the realistic density-matrix expression s s (7) ρ(R, R + s) = ρ R + jˆ1 kav R + s , 2 2 where jˆ1 (x) = 3(sin x − x cos x)/x 3 , and kav defines the average relative momentum [26] 1/2 5 1 2 kav (r) = . (8) τ (r) − ∇ ρ(r) 3ρ(r) 4 The latter quantity is a function of the density distribution ρ(r) and the approximation used for the kinetic-energy density τ (r). We have used the Fermi-type density for the target nucleus and the Negele parameterization [28], as well as the modified Thomas–Fermi (MTF) approximation [29] of the kinetic-energy density τ (r) [11,22]: 5|∇ρ(r)|2 5∇ 2 ρ(r) 1/2 MTF − (r) = kF (r)2 + , (9) kav 12ρ 2(r) 12ρ(r) where kF (r) = [3π 2 ρ(r)/2]1/3 is the local Fermi momentum. Furthermore, the spin–orbit potential has been calculated as the derivative of the microscopic real potential [30] 2 1 dU (r) h¯ σ L. (10) USO = 2mc r dr
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The angular distributions thus obtained were found very close to the results provided by the spin–orbit term of Vautherin and Brink [31] with the W0 parameter value of 120 MeV fm5 [32]. We adopted the imaginary component of a phenomenological OMP in order to carry on the elastic-scattering data analysis. The global parameter set established by Walter and Guss [33] for the nucleon scattering on target nuclei with A > 53 at incident energies up to 80 MeV has been considered in this respect. Moreover, the dispersion relation which connects the real and imaginary parts of the average nucleon–nucleus potential [34,35] have been taken into account. Finally, the real potential UR (r; E) has been obtained by adding the dispersive correction U(r; E) determined from the imaginary optical potential to the folded potential U (r; E) [35]: UR (r; E) = U (r; E) +
U (r; E).
(11)
The dispersive correction which takes into account the fact that the target does not remain in its ground state during the elastic-scattering process, is the sum of the corresponding volume and surface components (q = v, s) Uq (r; E) =
2 (EF − E) π
∞ 0
dE
Wq (r; E + EF ) , (E − EF )2 − E 2
(12)
where EF is the Fermi energy defined as lying half-way between the first particle state E+ > EF and the first hole state E− < EF of the n + A system EF = 12 (E+ + E− ).
(13)
3. Results and discussion 3.1. OMP real component The real OMPs for incident protons on 93 Nb and 90 Zr, at energies around 22 MeV, calculated in this work by means of Eq. (5) are shown in Fig. 2. First, the microscopic real potential U (r; E) is compared with the real part of the phenomenological OMP of Walter and Guss [33] in Figs. 2(a) and 2(c). One may consider that the real potentials calculated in this work are in rather good agreement with the phenomenological predictions for the both targets. The M3Y microscopic potentials are deeper at the center of the nucleus and less diffuse in the nuclear surface region with respect to the phenomenological real OMPs, while their volume integrals are rather similar. Second, it is illustrated the completion of the real parts of the microscopic OMPs by the dispersive corrections. The volume Uv (r; E) and surface Us (r; E) components as well as the total dispersive corrections U (r; E) obtained by using the imaginary part of the Walter–Guss potential are shown in Figs. 2(b) and 2(d). The volume dispersive correction leads to the total real potentials which are deeper at r = 0, and increasing for higher incident energies [35]. At the same time, the corresponding surface component which
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Fig. 2. (a,c) The comparison of the radial dependence of the optical potential obtained by using the M3Y effective NN interactions in the MTF approximation, for protons incident on 93 Nb and 90 Zr respectively, without (dotted curves) and including the dispersion component (solid curves), and the real component of the phenomenological OMP [33] (double-dot-dashed curves); (b,d) the radial dependence of the volume- (dashed curves) and surface components (dotted curves) of the dispersive correction (solid curves) obtained by using the Walter–Guss imaginary potential [33] for 22.2 MeV and 22.5 MeV protons incident on the same nuclei.
is strongly damped for nucleon energies such that |E − EF | > 20 MeV [34] lowers the dispersive correction towards the nuclear surface. The corresponding real potential, UR (r) Eq. (11), is presented in Figs. 2(a) and 2 (b) by solid curve. 3.2. The angular distributions analysis The comparison of the elastic-scattering angular distributions calculated by using the microscopic OMPs given in the previous section and the experimental data is shown in Fig. 3. The former have been obtained by using the real microscopic optical potential UR (r, E) with the dispersive correction (11), and the imaginary component of the Walter– Guss phenomenological potential [33]. It can be concluded that the experimental elasticscattering angular distributions are satisfactory well described by the microscopic potential over the whole angular range. We underline that no adjustable parameter or normalization constant has been involved within this analysis in order to couple the real and imaginary parts of the OMP (e.g., [38] and references therein). Our aim has been neither the elastic-scattering data analysis nor the
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Fig. 3. The comparison of the elastic-scattering angular distributions of protons on (a) 93 Nb and on (b) 90 Zr calculated (full curves) by using the microscopic potentials and phenomenological OMP parameter set [33], and the experimental data of (a) Fulmer [36], and (b) Ball et al. [37].
improvement of the microscopic real potentials but validating a suitable realistic effective NN interaction to be used in the multistep nucleon emission. From this point of view one may find a good agreement between measured and calculated differential cross sections which generally proves that microscopic potentials built on the M3Y effective NN interactions are able to describe rather well the proton interaction with the given target nuclei. 3.3. Equivalent 1Y NN interaction strength eq
The 1Y-equivalent effective NN-interaction strength V0 of the M3Y effective NN interaction is determined hereafter by corresponding volume integrals of the related optical potentials J1Y = JU ,
(14)
where J1Y corresponds to the use within Eq. (5) of the 1Y effective NN interaction eq eq with the strength V0 . Thus, V0 is energy- and mass-dependent, in agreement with the phenomenological predictions [4–6]. This quantity should not be considered as a trial for continuing to use the crude 1Y model in FKK calculations but as a check point for the phenomenological V0 values. Its usefulness could be shown by a comparison between the phenomenological real potentials [33] and the calculated results obtained by using either the realistic effective NN interaction or the usual 1Y interaction with phenomenological V0 strength values (Fig. 4). The potentials which correspond to the V0 values obtained by Watanabe et al. [6] are deeper by up to 50% with respect to the phenomenological ones. These large differences should emphasize the possibility of having some effects neglected in FKK calculations and corresponding scaling of the free parameter V0 . eq The energy dependence of V0 is compared in Fig. 5(a) with the phenomenological V0 values for (p, p ) processes. Our calculated values for the 98 Mo target nucleus are compared with the phenomenological V0 values obtained within MSD studies of protons incident on target nuclei in the mass range A ∼ 90, i.e. 90 Zr [4,5,7], 93 Nb and 98 Mo [6], as well as
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Fig. 4. The radial dependence of the microscopic real potential calculated by using the M3Y effective NN interactions (solid curves), the 1Y interaction and the phenomenological V0 values [6] (dashed curves), and of the phenomenological OMP [33] (double dotted-dashed curves), for protons incident on (a) 93 Nb, and (b) 90 Zr.
eq
Fig. 5. (a) The energy dependence of V0 of the M3Y Paris effective NN interaction for protons on 98 Mo, compared with the phenomenological V0 values for protons on 90 Zr (open circle [4], square [5], diamonds [7]), 93 Nb and 98 Mo (triangles) [6], and the phenomenological global value eq (solid circle) [39]. (b) The target mass-dependence of V0 of the M3Y Paris effective NN interactions 95,97,100 Mo. for 25.6 MeV protons on
the earlier phenomenological result of Austin [39]. The large differences between the phenomenological values found at low incident energies by Watanabe [6] and by Koning and Chadwick [7] have been discussed in terms of the remaining uncertainties of the optical potentials and the single-particle level scheme involved in the FKK calculations. At the same time, the nonlocality correction missing in the DWBA calculations may be responsible especially at low energies for the low V0 values found by Koning and Chadwick [6,40,41]. In eq these conditions, the agreement of the V0 values with the phenomenological systematics could be considered satisfactory. The increased slope of the phenomenological V0 values at low energies can be explained by the surface effects present at these energies [25].
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eq
In addition, the target-mass dependence of V0 of the M3Y Paris effective NN interaction for protons incident on Mo isotopes is shown in Fig. 5(b). Obviously, neither eq the mass- nor the energy-dependence of V0 could explain the broad range of the phenomenological V0 values. The uncertainties in the model calculations may explain this behavior. Therefore, an independent estimation of this parameter value prior to any comparison between the MSD calculations and experimental data may enlighten the correctness of various assumptions undertaken in practical FKK calculations. The eq comparison shown in Fig. 5(a) between V0 of the M3Y effective NN interaction and the phenomenological V0 values could be considered encouraging in this respect. 3.4. The MSD calculations The FKK calculations of multistep direct-reaction cross sections have been carried out by means of the recent computer codes MSD22NEWDW-NON, MSD22NEWDW [42], and MULTI21NON [43], in agreement with the latest achievements in the field [8]. Since the main goal of this work concerning the MSD calculations was the attempt for eq replacement of the V0 free parameter at least by V0 of a realistic effective NN interaction, the default options of the codes have been used. They are mainly the Dilg et al. [44] parameterization for the pairing corrections included in the particle–hole state densities, the nonlocality correction factor of 0.85 [45], and the use of Walter and Guss [33] distorted optical potential for the calculations of the DWBA matrix elements. In order to focus our study on the MSD contribution to the double-differential cross sections, we have analyzed the (p, n) reactions. Thus, we have avoided the strong collective contribution present in (n, n ) or (p, p ) processes. We have also selected for this analysis only high outgoingenergy data which does not include a significant contribution of the multistep compound (MSC) process, so that the MSD contribution remains the only important [7]. The MSD double-differential cross sections of the (p, xn) reaction on 90 Zr and 94−98,100 Mo isotopes at 25.6 MeV incident energy, calculated by using V eq of the 0 M3Y Paris effective NN interactions, are shown in Fig. 6. One may note a general good agreement except the MSD calculations underestimate the experimental angular distributions at backward angles for 90 Zr and 94,96 Mo. Data corresponding to lower (14 MeV) neutron emission from the 90 Zr(p, n) reaction show that there are still present contributions from the MSC emission and pure neutron evaporation. Following the calculation of the corresponding cross sections by using the computer code EMPIRE [46], the total angular distribution is in a good agreement with the experimental data. The available data for the Mo isotopes are rather well reproduced especially for higher target–nucleus asymmetry. The high value of the outgoing energy of 17 MeV prevents in these cases the MSC channel contribution. Finally, the general good agreement between eq the experimental data and the MSD results obtained by using V0 may be considered a support for the implementation of the realistic effective NN interaction in the frame of the FKK theory. This could make possible also further improvements concerning the surface effects present at low incident energies [25] and taken into account by appropriate particle– hole state density formalisms (e.g., [50]).
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Fig. 6. Comparison of experimental double-differential cross sections for the (p, xn) reactions on 90 Zr [47,48] (squares and circles, respectively) and 94−98,100 Mo [49] at 25.6 MeV incident energy, and MSD cross sections (solid curves). MSC and evaporation contributions (dotted curves) and the total cross section (dotted-dashed curve) are also shown for 90 Zr.
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4. Conclusions The analysis of the M3Y effective NN interaction has been carried on by means of the corresponding microscopic potential, in order to suggest a way for improving the FKK description of multistep processes. The present procedure involving the nucleon cutoff form factor for the regularization of the M3Y Paris effective NN interactions leads to real optical potential at lower energies which is in a good agreement with the phenomenological OMP. In addition, the microscopically obtained real optical potential has been applied also to the analysis of proton elastic-scattering data. In order to complete this analysis, the imaginary component of the phenomenological OMP parameter set of Walter and Guss [33] has been adopted, and the effect of the dispersion relations on the real folded optical potential has been taken into account. The satisfactory agreement between measured and calculated differential cross sections proves the applicability of the microscopic potential. No parameter fit or adjustment has been undertaken in order to improve the agreement quality since our aim has been to check the realistic effective NN interactions for further use in practical MSD calculations. The comparison of the microscopic real potential which is obtained by using a realistic force with the ones corresponding to the strongly simplified single Yukawa description shows the restrictions of the simple parameterization. In these circumstances we have eq found the 1Y-equivalent strengths V0 of the M3Y Paris effective NN interactions reliable to provide an alternative procedure for the MSD calculations without free parameters. Moreover, it is shown that the realistic M3Y Paris effective NN interactions could not explain the broad range of the V0 phenomenological values, so that an independent estimation of this quantity prior to comparison between the MSD calculations and experimental data may enlighten the correctness of various model assumptions. The MSD double-differential cross sections of the (p, n) reaction on 90 Zr and 94−98,100 Mo at 25.6 MeV incident energy have been calculated by using the V eq values of 0 the M3Y Paris effective NN interaction. Thus it has been proved as suitable the replacement eq of the V0 free parameter by V0 of a realistic effective NN interaction. The practical FKK calculations could become independent of any free parameter, whose phenomenological values may compensate for the questionable handling of other quantities involved in the FKK formalism such as the particle–hole state density, the optical potential for distorted waves, the DWBA matrix elements, the coupling between MSD and MSC emission chains [7,8]. Therefore, the replacement of the 1Y effective NN interaction by the M3Y Paris is foreseen at this stage. The further consistent use of the corresponding microscopic real potential for the distorted waves in the FKK framework is also intended. Next, consideration of, e.g., DDM3Y density dependent effective NN interactions [21] should improve the realistic description of the multistep nucleon emission at higher incident energies. Indeed, the density independent M3Y interaction has been used successfully in the folding model calculations involving the nucleon–nucleus scattering or light heavy-ion scattering at relatively low energies, where the data are sensitive only to the potential at the nuclear surface [20]. Increasing the incident energy over 50A MeV, the interaction moves towards higher nuclear density region and the density dependence of the
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effective NN interaction has to account for the reduction of its strength that occurs as the density of the surrounding medium increases [19,20].
Acknowledgement The authors are grateful to H. Wolter for useful discussions and a critical reading of the manuscript.
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