Microscopic analysis of {}^{10,11}Be elastic scattering on protons and nuclei and breakup processes of {}^{11}Be within the {}^{10}Be+n cluster model

Microscopic analysis of Be elastic scattering on protons and nuclei and breakup processes of Be within the Be+ cluster model

Abstract

The density distributions of Be and Be nuclei obtained within the quantum Monte Carlo (QMC) model and the generator coordinate method (GCM) are used to calculate the microscopic optical potentials (OPs) and cross sections of elastic scattering of these nuclei on protons and C at energies MeV/nucleon. The real part of the OP is calculated using the folding model with the exchange terms included, while the imaginary part of the OP that reproduces the phase of scattering is obtained in the high-energy approximation (HEA). In this hybrid model of OP the free parameters are the depths of the real and imaginary parts obtained by fitting the experimental data. The well known energy dependence of the volume integrals is used as a physical constraint to resolve the ambiguities of the parameter values. The role of the spin-orbit potential and the surface contribution to the OP is studied for an adequate description of available experimental elastic scattering cross section data. Also, the cluster model, in which Be consists of a -halo and the Be core, is adopted. Within the latter, the breakup cross sections of Be nucleus on Be, Nb, Ta, and U targets and momentum distributions of Be fragments are calculated and compared with the existing experimental data.

pacs:
25.40.Cm, 24.10.Ht, 25.60.Gc, 21.10.Gv

I Introduction

The discovery of halo nuclei (1) has been related to the measured interaction cross sections of nuclei like He, Li, Be isotopes with various target nuclei (2); (2); (3); (4); (5); (6). The evidences of the existence of an extended halo in neutron-rich nuclei are based on the observed unusually narrow momentum distribution of a core fragment and enhanced reaction cross section. The first example was the breakup of Li at high energies (7); (8); (9); (10) by observing the large interaction reaction cross section (2) and the narrow momentum distribution of Li in the breakup of Li, e.g., in the reaction Li+C at MeV/nucleon (7). Here we should mention also the results of the experiments at lower energies ( MeV/nucleon) of scattering of Li on Be, Nb and Ta (11) and of Li on a wide range of nuclei from Be to U (12). As shown in Ref. (13), not only scattering but also the breakup of Be in the collisions with the target nuclei Nb, Ta, and U play a decisive role when studying the internal cluster structure of Be. Indeed, the narrow peak of the momentum distributions of the breakup fragments of such a neutron-rich nucleus reflects the very large extension of its wave function, compared to that of the core nucleus Be, and thus evidences the existence of the nuclear halo (14); (15); (16); (17); (18); (19); (20). As was concluded in (18), namely the longitudinal component of the momentum (taken along the beam or the -direction) provides the most accurate information on the intrinsic properties of the halo, being insensitive to details of the collision and the size of the target. In addition, recent measurements of the charge radii of Be pointed out that the average distance between the halo neutrons and the Be dense core of the Be nucleus is around 7 fm (21). Thus, the halo neutron is about three times as far from the dense core as is the outermost proton because the core itself has a radius of only 2.5 fm.

An important finding when investigating reactions with Be and Be nuclei, in particular the Be+ breakup of Be, is the effect of the deformed Be core on the two-body cluster structure of Be. In fact, in the Be nucleus the inversion of the and orbitals predicted by Talmi and Unna (22) and confirmed by Alburger et al. (23) leads to a 1/2 ground state. Also, the probability of the E1 transition from this ground state to the 1/2 first excited state of Be located at 320 keV excitation energy is the largest ever measured in light nuclei (24); (25). The effects of the core deformation on the breakup of Be on protons have been studied in several works. For example, in addition to a two-body cluster structure with an inert Be(0) core and a valence neutron used in Ref. (26) in the continuum discretised coupled-channels (CDCC) calculations of elastic and inelastic proton scattering on Be, the authors have also discussed the necessity to account for contributions from configurations involving excited states of the Be core to the Be+ continuum of Be. Crespo et al. (27) have found that the core excitation +Be(0) +Be(2) provides a significant contribution to the breakup cross section of Be on the proton target at 63.7 MeV/nucleon incident energy.

In the earlier works (e.g., Ref. (28)) the elastic scattering cross sections of Be on protons have been calculated using phenomenological OPs of given forms with numerous fitting parameters of their real (ReOP) and imaginary (ImOP) parts. However, in the further calculations the more physically motivated microscopic folding models were applied (see, e.g., (29); (30); (31); (32)). In many works (30); (31); (32) the folding procedure was explored for the real part of the OP. Within the latter procedure the direct and exchange parts of the ReOP with effective nucleon-nucleon forces are calculated. At the same time the P is usually taken in a phenomenological form. Many successful applications of this model have been made for the proton- and nucleus-nucleus collisions (see, e.g., cycles of works (31); (34); (33)). This model was also explored in Refs. (35); (36) for scattering of Be+, where the exchange part of the folded ReOP is taken in the form of the zero-range prescription and the density distribution of Be has the Gaussian-oscillator form. In the recent work (37) the authors account for the full exchange part of the ReOP, while the ImOP was calculated using the folding HEA formula from Refs. (38); (39). In Refs. (37); (40) a surface term to the ImOP was added to improve the agreement with the data at lower energies.

In our present work, as well as in our previous works considering processes with exotic He and Li isotopes (41); (42); (43); (44), we use microscopically calculated OPs within the hybrid model (38). In the latter the ReOP is calculated by a folding of a nuclear density and the effective NN potentials (32) (see also (45)) and includes both direct and exchange parts. The ImOP is obtained within the HEA model (46); (47). There are only two or three fitting parameters in the hybrid model that are related to the depths of the ReOP, ImOP and the spin-orbit part of the OP. Along with some phenomenological density distributions for He and Li isotopes we have used in our works realistic microscopic density obtained within the large-scale shell model (LSSM) (48); (49). In the present work devoted to processes with Be nuclei we use the density distribution for Be obtained within the QMC model (50); (51) and also the densities of Be and Be obtained within the GCM (52).

The main aim of our work is twofold. First, we study the elastic scattering of the neutron-rich exotic Be and Be nuclei on protons and nuclei at energies MeV/nucleon using microscopically calculated in our work real and imaginary parts of the optical potentials. Second, we estimate important characteristics of the reactions with Be, such as the breakup cross sections and momentum distributions of fragments in breakup processes. To this end we use the model in which Be consists of a core of Be and a halo formed by a motion of a neutron in its periphery (e.g., Refs. (53); (54); (55)). The latter model is justified by the small separation energy KeV of a neutron from the ground state of Be (56) and on the observed quite large total interaction cross sections of Be with target nuclei caused by the main contribution from the breakup of Be on Be and a neutron. The important role of the periphery is confirmed also by the experiments on scattering of Be on the heavy nucleus of Pb (57), where the prevailing mechanism is the direct breakup due to the long-range Coulomb force of the nucleus. Also we should mention the important observation of the narrow peak in the momentum distribution of the Be fragments at the breakup of Be scattering on the C nucleus (13), that is, as mentioned above, a consequence of the large extension of the wave function of the relative motion in the Be system related to the small neutron separation energy. By means of such a cluster model of Be one can calculate the OPs for scattering of Be on protons or nuclear targets. To this end one should use the known + potential and calculate using the microscopic model the optical potentials of Be+ (or Be+A and the +A potentials). Then the sum of these potentials are folded with a density probability of the relative motion of the core Be and the neutron. Also, in the framework of this cluster model one can calculate the momentum distribution of Be fragments from the breakup reactions Be+Be, Be+Nb, Be+Ta, and Be+U for which experimental data are available.

The structure of the paper is as follows. The theoretical scheme to calculate microscopically within the hybrid model the ReOP, ImOP, the spin-orbit part of the OP, the surface component of OP, as well as the results of the calculations of the elastic scattering cross sections of Be+ and Be+C are presented in Sect. II. The basic expressions to estimate the breakup of Be and to calculate the cross sections and the fragment momentum distributions of Be in the diffraction and stripping processes of Be on Be, Nb, Ta, and U are given in Sect. III. The summary and conclusions of the work are included in Sect. IV.

Ii Elastic scattering of Be on protons and C at MeV/nucleon

ii.1 Hybrid model of the microscopic optical potential

In the present work we calculate the microscopic OP that contains the volume real () and imaginary parts (W), and the spin-orbit interaction (). This OP is used for calculations of elastic scattering differential cross sections. We introduce a set of weighting coefficients , , and that are related to the depths of the corresponding parts of the OP and are obtained by a fitting procedure to the available experimental data. Details of the constructing of the OP are given in Refs.(30); (31); (32); (45). The OP has the form:

(1)

where fm with the squared pion Compton wave length fm. Let us denote the values of the ReOP and ImOP at by and . We note that the spin-orbit part of the OP contains real and imaginary terms with the parameters and related to and by the and , correspondingly. Here and (and and have to be negative. The ReOP is a sum of isoscalar () and isovector () components and each of them has its direct ( and ) and exchanged ( and ) parts.

The isoscalar component has the form

(2)

where is the vector between two nucleons, one of which belongs to the projectile and another one to the target nucleus.

In the first term of the right-hand side of Eq. (2) the densities of the incident particle and the target nucleus are sums of the proton and neutron densities. In the second term and are the corresponding one-body density matrices. In our work we use for them the approximations for the knock-on exchange term of the folded potential from Refs. (58); (59) (see also (41); (43)). In Eq. (2) is the local momentum of the nucleus-nucleus relative motion and and are the direct and exchange effective NN potentials. They contain an energy dependence usually taken in the form and a density dependence with the form for the CDM3Y6 effective Paris potential (32)

(3)

with =0.2658, =3.8033, =1.4099 fm, and =4.0 fm. The effective NN interactions and have their isoscalar and isovector components in the form of M3Y interaction obtained within -matrix calculations using the Paris NN potential (31); (32). The isovector components of the ReOP can be obtained by exchanging in Eq. (2) the sum of the proton and neutron densities in by their difference and using the isovector parts of the effective NN interaction. In the case of the proton scattering on nuclei Eq. (2) contains only the density of the target nucleus.

The ImOP can be chosen either to be in the form of the microscopically calculated () or in the form obtained in Ref. (38); (39) within the HEA of the scattering theory (46); (47):

(4)

In Eq. (4) are the corresponding formfactors of the nuclear densities, is the amplitude of the NN scattering and is the averaged over the isospin of the nucleus total NN scattering cross section that depends on the energy. The parametrization of the latter dependence can be seen, e.g., in Refs. (60); (41). We note that to obtain the HEA OP (with its imaginary part in Eq.( 4)) one can use the definition of the eikonal phase as an integral of the nucleon-nucleus potential over the trajectory of the straight-line propagation and has to compare it with the corresponding Glauber expression for the phase in the optical limit approximation. In the suggested scheme we use the nuclear densities and NN cross sections known from other sources and also the already used NN potentials and amplitudes. In this way, the only free parameters in our approach are the parameters s that renormalize the depths of the OPs components. In the spin-orbit parts of the OP the functions (r) () correspond to WS forms of the potentials with parameters of the real and imaginary parts , , , [ and ], as they are used in the DWUCK4 code (61) and applied for numerical calculations. We determine the values of these parameters by fitting the WS potentials to the microscopically calculated potentials (r) and W(r).

ii.2 Results of calculations of elastic scattering cross sections

In the calculations of the microscopic OPs for the scattering of Be on protons and nuclei we used realistic density distributions of Be calculated within the quantum Monte Carlo model (50); (51) and of Be from the generator coordinate method (52). In general, the QMC methods include both variational and Green’s function Monte Carlo methods. In our case within the QMC method the proton and neutron densities of Be have been computed with the AV18+IL7 Hamiltonian (51). As far as the GCM densities are concerned, in Ref. (52) the Be wave functions are defined in the harmonic oscillator model with all -shell configurations. The Be wave functions are described in terms of cluster wave functions, relative to Be and to the external neutron. Thus, both microscopic densities effectively account for the non-ordinary nuclear structure peculiarities of Be (26); (27) and their use is physically justified. The QMC and GCM densities are given in Fig. 1. It can be seen that they have been calculated with enough accuracy up to distances much larger than the nuclear radius. In both methods the densities of Be occur quite similar up to fm and a difference between them is seen in their asymptotics. In the calculations of the OPs for Be+C the density of C was taken in symmetrized Fermi form with radius and diffuseness parameters fm and fm, respectively (62). The results of the calculations are compared with the available experimental data. All calculations of elastic scattering using the obtained OPs are performed by using the DWUCK4 code (61).

Figure 1: (Color online) Point-proton (normalized to ) and point-neutron (normalized to and , respectively) densities of Be and Be obtained in the GCM (52) and in the QMC method (50); (51).

Elastic scattering cross sections of Be+

On the basis on the scheme presented in subsection II.A. we calculated the elastic scattering cross sections of Be+ and compared them with the available experimental data.

It is accepted that the elastic scattering of light nuclei is rather sensitive to their periphery, where transfer and breakup processes also take place. Therefore, investigating the elastic scattering, one must bear in mind that virtual non-elastic contributions can also take part in the process. It has been pointed out in our previous papers (43); (42), as well as in Refs. (36); (40); (37), that the inclusion of a surface imaginary term to the OP [Eq. (1)] leads to a better agreement with the experimental data. As known, this contribution can be considered as the so-called dynamical polarization potential, which allows one to simulate the surface effects caused by the latter. In fact, the imaginary part of the term in our OP [see Eq. (1)] plays effectively this role. However, sometimes one needs to increase the absorption in the surface region and thus, one adds a derivative of the ImOP (surface term):

(5)

where is also a fitting parameter.

The results for the elastic Be+ and Be+ scattering cross sections are given in Figs. 2 and 3, respectively, and compared with the data at energies 39.1 MeV/nucleon (63) and MeV/nucleon (28) for Be and 38.4 MeV/nucleon (63) and MeV/nucleon (28) (see also (64)) for Be. In general, our analysis points out that more successful results are obtained in the case when the ImOP is taken from HEA: [Eq. (4)]. We note that in the fitting procedure of the theoretical results to the data for elastic scattering cross sections for Be+ (and also for Be+C) there arises an ambiguity in the choice of the optimal curve among many of them that are close to the experimental data. Due to this we impose a physical constraint, namely choosing those ReOP and ImOP that give volume integrals which have a correct dependence on the energy. The volume integrals have the forms

(6)
(7)
(8)

where and are the mass numbers of the projectile and the target, respectively. In Eq. (8) we added also the integral over the surface term of the OP (5). It is known (65) that the volume integrals (their absolute values) for the ReOP decrease with the increase of the energy, while for the ImOP they increase up to a plateau and then decrease. The values of the s parameters from the fitting procedure and after imposing the mentioned constraint are given in Table 1. It can be seen that the tendency (the decrease of and the increase of ) is generally confirmed.

The calculated differential cross sections of Be+ elastic scattering at energies MeV/nucleon and MeV/nucleon are presented in Fig. 2. First, it is seen from the upper panel that the inclusion of only the volume OP is not enough to reproduce reasonably well the data in the small angles region. Then, after adding the spin-orbit component to the OP the agreement with the data becomes better, in particular for the angular distributions calculated using the GCM density at energies MeV/nucleon and MeV/nucleon for angles less than 20 and 30, correspondingly, as illustrated in the middle panel of Fig. 2. However, a discrepancy at larger angles remains. At the same time for the cross sections with the account for the interaction and using the QMC density we obtain fairly good agreement with the data at both energies and only a small discrepancy is seen at small angles at energy MeV/nucleon. Further improvement is achieved when both - and surface terms are included in the calculations. In this case, as it can be seen from the bottom panel of Fig. 2, the discrepancy between the differential cross sections for the GCM density and the experimental data at larger angles is strongly reduced.

Figure 2: Be+ elastic scattering cross sections. Upper panel: without term; middle panel: with term; bottom panel: with both and surface terms. Solid lines: calculations with GCM density of Be; dashed lines: calculations with QMC density of Be. Experimental data for 39.1 MeV/nucleon and 59.4 MeV/nucleon are taken from Refs. (63) and (28), respectively.
Nucleus Model
Be GCM 39.1 0.983 0.267 0.000 0.000 0.000 292.12 389.408 116.600 116.600
without and QMC 1.153 0.295 0.000 0.000 0.000 311.36 411.344 130.806 130.806
surface terms GCM 59.4 1.001 0.802 0.000 0.000 0.000 341.18 333.739 263.540 263.540
QMC 1.188 0.856 0.000 0.000 0.000 356.98 354.606 283.464 283.464
Be GCM 39.1 1.493 0.492 1.000 0.476 0.000 372.50 591.440 216.480 216.480
with and QMC 1.163 0.318 0.557 0.000 0.000 323.96 414.911 141.004 141.004
without surface GCM 59.4 1.294 0.804 0.190 0.000 0.000 355.29 431.427 264.197 264.197
terms QMC 1.014 0.527 0.940 0.000 0.000 287.68 302.669 174.516 174.516
Be GCM 39.1 0.995 0.266 0.095 0.082 0.004 298.65 394.161 117.040 122.321
with and QMC 1.194 0.260 0.075 0.025 0.018 333.71 425.971 115.286 139.235
surface terms GCM 59.4 0.970 0.000 0.365 1.000 0.373 400.26 323.404 0.000 367.802
QMC 1.043 0.281 0.000 1.000 0.270 389.27 311.325 93.053 361.343
Be GCM 38.4 0.824 0.659 0.000 0.000 0.000 459.05 339.388 293.493 293.493
without and 49.3 0.793 0.805 0.000 0.000 0.000 423.52 296.301 301.184 301.184
surface terms
Be GCM 38.4 0.787 0.799 0.000 0.507 0.000 458.63 324.148 355.844 355.844
with and 49.3 0.793 0.867 0.123 0.316 0.000 426.85 296.301 301.184 301.184
without surface
terms
Be with GCM 38.4 0.849 0.106 0.102 0.380 0.152 493.01 349.685 47.208 269.903
and surface 49.3 0.801 0.000 0.213 0.394 0.200 436.46 299.280 0.000 246.162
terms
Table 1: The renormalization parameters , , , , and , the total reaction cross sections (in mb), and the volume integrals , , and (in MeV.fm) as functions of the energy =39.1 and 59.4 MeV/nucleon for the Be+ and =38.4 and 49.3 MeV/nucleon for the Be+ elastic scattering

In general, the account for the spin-orbit term in the volume OP gives a trend of an increase of the cross sections at larger angles, that seems to be related with the change of the form of the total OP at its periphery. If we evaluate the quantities of the two densities of Be on the basis of the values of the parameter (comparing which ones are closer to unity), our conclusion is that in the calculations without interaction the GCM density works better, while in the case with term in the OP the QMC density gives better results. A fair agreement between the calculated Be+ angular distributions and the experimental data is obtained only when both - and surface contributions to the OP are included.

In Fig. 3 are given and compared with the empirical data elastic cross sections for the scattering of Be on protons at energies 38.4 and 49.3 MeV/nucleon applying the fitting procedure for the parameters s. All of them are calculated using GCM density of Be. The different curves drawn in Fig. 3 correspond to those given in Fig. 2 with accounting for different contributions to the OP. One can see a discrepancy at small angles () that seems to be related to the contributions from the surface region of interactions, where breakup processes play an important role. Similarly to the results for the Be+ elastic scattering cross sections (see Fig. 2), the account for both spin-orbit and surface terms to the OP leads to a better agreement with the Be+ data in the region of small angles. In Table 1 are given the corresponding values of the parameters and whose values deviate from unity of about that points out that the hybrid model for the O*P can be used successfully in such calculations.

Figure 3: Be+ elastic scattering cross sections. Calculations are performed with GCM density of Be. Solid line: OP with both and surface terms [Eqs. (1) and (5)]; dashed line: OP with term [Eq. (1)]; dotted line: the volume part of OP from Eq. (1). Experimental data for 38.4 MeV/nucleon and 49.3 MeV/nucleon are taken from Refs. (63) and (28), respectively.

We would like to emphasize the fact that when considering the case of the total OP [Eqs. (1) and (5)], the values of the parameters drop down sufficiently in comparison with their values coming out from the two other cases. They are compensated in the most cases by the non-zero values of , , and parameters. Here we would like to note that the term used in our calculations (with both real and imaginary parts) plays the similar role as the surface term applied in Ref. (37), where, however, the imaginary term is disregarded. From our analysis made for the elastic scattering of Be and Be on protons we conclude also that the surface imaginary part of the OP is less necessary to fit the data of proton elastic scattering on the stable nucleus Be, but it is important to give an agreement with the proton elastic-scattering data of the halo nucleus Be. This is mainly due to the specific halo structure of the Be density distribution and its large rms radius.

For a more complete analysis of the elastic scattering cross sections, we extend the incident energy region to lower energies on the example of the scattering of Be on protons that has been recently studied by Schmitt et al. (66). Moreover, this could be a test of our hybrid model at low energies. In Ref. (66) proton energies of 6, 7.5, 9, and 10.7 MeV were selected to measure the elastic scattering cross sections for protons with Be beams in inverse kinematics in order to provide constraints on optical potentials for reaction studies with light neutron-rich nuclei. The calculated results for the differential cross sections, shown as a ratio to Rutherford scattering, are given and compared with the data (66) in Fig. 4 for energies of 7.5 and 10.7 MeV. The values of the s parameters from the fitting procedure and the corresponding total reaction cross sections and volume integrals are listed in Table 2. The results shown in Fig. 4 when including in the calculations only the term demonstrate a fairly good agreement with the data. The values of the parameters deduced from the fitting procedure for both energies in the case of GCM density of Be are quite large that indicates for the specific peculiarities of the elastic scattering at low energies with account for the spin-orbit term. We also calculated the Be+ elastic scattering cross sections at the same proton energies taking into account the surface term [Eq. (5)]. In this case, only the QMC density of Be was tested that has been also used in Ref. (37), where the two other energies of 6 and 9 MeV were considered. The results illustrate that the inclusion of the surface contribution does not affect the good agreement obtained without it. Here we note that in Ref. (66) no single optical potential had been found to reproduce well the proton elastic scattering data over this range of energies. At the same time, it is seen from the left panel of Fig. 4 a deviation of the results of our model with both densities beyond 55. Therefore, it would be desirable to measure the elastic channel in this angular range to constrain the Be optical potential.

Figure 4: Be+ elastic scattering cross sections as a ratio to Rutherford scattering at proton energies of 7.5 MeV (left panel) and 10.7 MeV (right panel). The solid and dashed lines show the results with QMC and GCM density of Be, respectively, and with term in OP. The dotted lines show the QMC results obtained by accounting for both the - and surface terms in OP. Experimental data are taken from Ref. (66).
Nucleus Model
Be GCM 7.5 2.287 0.473 0.000 0.425 0.000 906.19 1215.283 527.749 527.749
with and QMC 1.244 0.056 0.065 0.103 0.000 330.03 603.634 62.966 62.966
without surface GCM 10.7 2.232 1.129 0.000 0.759 0.000 804.23 1144.742 1151.009 1151.009
terms QMC 1.915 0.247 0.963 0.307 0.000 722.54 895.179 253.766 253.766
Be QMC 7.5 1.483 0.000 0.442 0.208 0.044 306.28 719.605 0.000 148.453
with and QMC 10.7 1.354 0.098 0.178 1.000 0.193 636.50 632.936 100.685 695.676
surface terms
Table 2: The renormalization parameters , , , , and , the total reaction cross sections (in mb), and the volume integrals , , and (in MeV.fm) as functions of the proton energy =7.5 and 10.7 MeV for the Be+ elastic scattering

Elastic scattering cross sections of Be+C

The calculated within the hybrid model elastic scattering cross sections of Be+C (their ratios to the Rutherford one) at the same energies, as for Be+ scattering, are given in Figs. 5 and 6 and compared with the experimental data (see also (64)). In comparison with the case of Be+, the experimental data (28); (63) for the scattering on C demonstrate more developed diffractional picture on the basis of the stronger influence of the Coulomb field. It can be seen in Fig. 5 that in both cases of calculations of OPs with QMC or GCM densities the results are in a good agreement with the available data. It is seen also from the figures that it is difficult to determine the advantage of the use for the ImOP or , because the differences between the theoretical results start at angles for which the experimental data are not available. The values of the parameters and (the depths of ReOP and ImOP) are given in Table 3. From the comparison of these values, when GCM or QMC densities are used, one can see that in the case of GCM densities the values of the parameters are closer to unity. In this way, we may conclude that as in the Be+ case without term of OP, the GCM density can be considered as a more realistic one.

Figure 5: Be+C elastic scattering cross sections. Solid lines: ; dashed lines: . Left panel: calculations with GCM density of Be; right panel: calculations with QMC density of Be. Experimental data for 39.1 MeV/nucleon and 59.4 MeV/nucleon are taken from Refs. (63) and (28), respectively.
Figure 6: Be+C elastic scattering cross sections. Solid lines: ; dashed lines: . For Be GCM density was used. Experimental data for 38.4 MeV/nucleon and 49.3 MeV/nucleon are taken from Refs. (63) and (28), respectively.
Nucleus Model
Be GCM 39.1 0.939 0.708 104.539 255.156 283.037
0.816 0.465 105.958 221.733 126.355
59.4 1.013 1.010 101.052 238.122 302.581
0.884 0.577 102.635 207.798 135.633
Be QMC 39.1 0.888 0.620 105.332 245.613 249.769
0.782 0.434 106.878 216.294 120.041
59.4 0.970 0.887 101.616 231.953 267.782
0.849 0.534 103.035 203.019 127.694
Be GCM 38.4 0.769 0.711 127.123 216.879 287.235
0.708 0.521 126.825 199.676 146.937
49.3 0.820 0.883 124.406 213.754 300.193
0.743 0.574 123.302 193.682 149.628
Table 3: The renormalization parameters and , the total reaction cross sections (in mb), and the volume integrals and (in MeV.fm) as functions of the energy =39.1 and 59.4 MeV/nucleon for the Be+C and =38.4 and 49.3 MeV/nucleon for the Be+C elastic scattering

Iii Breakup reactions of Be

iii.1 The Be+ model of Be

In this section we consider the characteristics of breakup processes of the Be nucleus, namely diffraction and stripping reaction cross sections and the momentum distributions of the fragments. We use a simple model in which Be consists of a core of Be and a halo of a single neutron (see, e.g., (54)). In this model the density of Be has to be given. As in Sect. II we use the QMC (50) and GCM (52) density distributions of Be. The hybrid model is applied to calculate the OP of the interaction of Be with the target, as well as OP for the +target interaction. In the final step of the procedure the sum of these potentials is folded with the respective density distribution corresponding to the relative motion wave function of the clusters in Be. The latter is obtained by solving the Schrödinger equation with the Woods-Saxon potential for a particle with a reduced mass of two clusters. The parameters of the WS potentials are obtained by a fitting procedure, namely, to reach the neutron separation energy KeV. They have the following values for state in which the valence neutron in Be is mainly bound (see Refs. (16); (67)): fm, fm and MeV. The rms radius of the cluster formation is obtained to be 6.87 fm.

The -state () of the relative motion of two clusters has the form:

(9)

The corresponding density distribution is the probability of both clusters to be at a mutual distance :

(10)

Within the Be+ cluster model, in order to calculate the Be breakup in its collision with the protons and nuclear targets, one should calculate two OPs of Be+(or ) and +(or ) scattering:

(11)

In Eq. (11) and give the distances between the centers of each of the clusters and the target, and determines the relative distance between the centers of the two clusters. and are the distances between the centers of Be and each of the clusters, correspondingly. The respective OPs for the Be+ and + scattering are calculated within the microscopic model of OP from Sect. II.A.

In the case of the Be breakup on the proton target the + potential is taken in the form (68) (in MeV):

(12)

with

(13)

where .

For calculations of breakup cross sections and momentum distributions of fragments in the Be+ breakup model we give here briefly the eikonal formalism, namely the expressions of the -matrix (as a function of the impact parameter ):

(14)

where

(15)

is the OP. For negative and one can write

(16)

and, correspondingly,

(17)

In our case is the imaginary part of the microscopic OP [Eq. (11)]. gives the probability that after the collision with a proton () (in the Be+ scattering), the cluster or the neutron with impact parameter remains in the elastic channel ():

(18)

The probability a cluster to be removed from the elastic channel is . The probability of the case when both clusters ( and ) leave the elastic channel is . As shown in the next subsection, Eqs. (14)-(18) take part in the calculations of the diffraction breakup and stripping reaction cross sections.

iii.2 Momentum distributions of fragments

The necessary quantity to calculate the diffraction breakup and absorption scattering cross sections (differential and total) and momentum distributions is the probability function of the -momentum distribution of a cluster in the system of two clusters as a function of the impact parameter (16):

(19)

In Eq. (19) is given by the products of two -functions and [Eqs. (14)-(18)] of the core Be and the neutron, is the continuum wave function, is the relative momentum of both clusters in their center-of-mass frame, and the vector is the projection of the relative coordinate between the centers of the two clusters on the plane normal to the -axis. The bound-state wave function of the relative motion of two clusters is given for the -state by Eq. (9). As to the wave function in the final state , we will neglect its distortion and, thus, replace it by in the case of the -state. Then, following Ref. (16), the probability function has the form

(20)

where

(21)

and , being given by Eq. (9).

Hence, the diffraction breakup cross section has the form

(22)

In Eq. (22) is given by Eq. (20). The integrals over and mean integration over the impact parameter of the neutron with respect to the target.

The cross sections of the stripping reaction when the neutron leaves the elastic channel is (16):

(23)

Equation (23) is obtained when the incident nucleus has spin equal to zero and for the -state of the relative motion of two clusters in the nucleus with , , and

(24)

coming from , where is the projection of on the plane normal to the -axis along the straight-line trajectory of the incident nucleus.

In the end of this subsection we note that the real and imaginary parts of the OPs taking part in Eq. (11) and in the -matrices [Eqs. (14)-(18)] are used for calculations of the cross sections [Eqs. (19)-(24)] in the cases of scattering and breakup of Be on protons and nuclei that will be considered in the following part of our work. They are calculated microscopically within the hybrid model given in subsection II.A.

iii.3 Results of calculations of breakup reactions

In this subsection we perform calculations of the breakup cross sections of Be on the target nucleus Be and heavy nuclei, such as Nb, Ta, and U, and compare our results with the available experimental data (13). The densities of these heavy nuclei needed to compute the OPs are taken from Ref. (69). The diffraction and stripping cross sections (when a neutron leaves the elastic channel) for reactions Be+Be, Be+Nb, Be+Ta, and Be+U are calculated from Eqs. (22) and (23). The obtained results are illustrated in Figs. 7, 8, 9, and 10, respectively. We note the good agreement with the experimental data from light and heavy breakup targets. The obtained cross sections for the diffraction and stripping have a similar shape. The values of the widths are around MeV in agreement with the experimental ones. Our results confirm the observations (e.g., in Refs. (11); (12)) that the width almost does not depend on the mass of the target and as a result, it gives information basically about the momentum distributions of two clusters. Here we note that due to the arbitrary units of the measured cross sections of the considered processes it was not necessary to renormalize the depths of our OPs of the fragments-target nuclei interactions.

Figure 7: Cross sections of diffraction breakup and stripping reaction in Be+Be scattering at MeV/nucleon. Experimental data are taken from Ref. (13).
Figure 8: The same as Fig. 7, but for Be+Nb scattering.
Figure 9: The same as Fig. 7, but for Be+Ta scattering.
Figure 10: The same as Fig. 7, but for Be+U scattering.

Iv Conclusions

In the present work the hybrid model is applied to study characteristics of the processes of scattering and reactions of Be and Be on protons and nuclei. In the model, the ReOP is calculated microscopically in a folding procedure of the densities of the projectile and the target with effective NN interactions related to the -matrix obtained on the basis of the Paris NN potential. The ReOP includes both the direct and exchange terms. The ImOP is calculated microscopically as the folding OP that reproduces the phase of scattering obtained in the high-energy approximation. The only free parameters in the hybrid model (s) are the coefficients that correct the depths of the ReOP, ImOP, and the spin-orbit parts of OP. Their values are obtained by a fitting procedure to the experimental data whenever they exist. Additionally, in some cases the surface absorption is accounted for by including another term to the OP that requires one more fitting parameter. The density distributions of Be obtained within GCM and QMC microscopic methods and of Be from GCM are used. The resulting within the hybrid model OPs are applied to calculate characteristics of various processes.

The results of the present work can be summarized as follows.

(i) Elastic scattering cross sections of Be and Be on protons and C are calculated using the microscopic OPs for energies MeV/nucleon and compared with the existing experimental data. In order to resolve the ambiguities of the magnitudes of the depths of the OPs, the well established energy dependence of the respective volume integrals of the OPs is taken into account. The theoretical approach gives a good explanation of a wide range of empirical data on the Be+ and Be+C elastic scattering. It was established that the obtained by fitting procedure values of the coefficients (depths of ReOP) are close to unity. The correction of the ImOP by factor is in some cases larger, e.g., for Be+ at energy MeV/nucleon in the case when the spin-orbit () component is not accounted for. The inclusion of a surface term to the OP leads to a better agreement with the experimental elastic scattering cross section data. We conclude that, in general, the hybrid model for microscopic calculations of the OPs gives the basic important features of the scattering cross sections and can be recommended and applied to calculate more complex processes such as breakup reactions, momentum distributions of fragments and others.

(ii) Apart from the usual folding model, we use another folding approach to consider the Be breakup by means of the simple Be+ cluster model for the structure of Be. Within this folding model we construct the OP of the interaction of Be with the target, as well as the +target interaction. Using the cluster OPs Be+(or ) and +(or ) the corresponding functions and (-matrices) for the core and neutron within the eikonal formalism are obtained.

(iii) The calculated and functions are used to get results for the longitudinal momentum distributions of Be fragments produced in the breakup of Be on different targets. This includes the breakup reactions of Be on Be, Nb, Ta and U at MeV/nucleon for which a good agreement of our calculations for the diffraction and stripping reaction cross sections with the available experimental data exist. The obtained widths of about 0 MeV/c are close to the empirical ones. Future measurements of such reactions are highly desirable for the studies of the exotic Be structure. The accurate interpretation of the expected data requires more refined theoretical approaches, for instance that of Ref. (70) within the CDCC method and its extensions to study the effects of the dynamic core excitation, especially its large contribution to nuclear breakup in the scattering of halo nuclei.

Acknowledgements.
The authors are grateful to S. C. Pieper for providing with the density distributions of Be nuclei calculated within the QMC method. The work is partly supported by the Project from the Agreement for co-operation between the INRNE-BAS (Sofia) and JINR (Dubna). Four of the authors (D.N.K., A.N.A., M.K.G. and K.S.) are grateful for the support of the Bulgarian Science Fund under Contract No. DFNI–T02/19 and one of them (D.N.K.) under Contract No. DFNI–E02/6. The authors V.K.L., E.V.Z., and K.V.L. thank the Russian Foundation for Basic Research (Grant No. 13-01-00060) for partial support. K.S. acknowledges the support of the Project BG-051P0001-3306-003.

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