Threshold anomalies in low-energy reactions with loosely-bound projectiles

Recent results concerning data for fusion and elastic scattering of low-energy neutron-halo 6He and weakly-bound 6Li projectiles, colliding with medium mass targets, are briefly reviewed. The extended optical model (EOM) was used, which includes two complex polarization potentials, one associated to fusion couplings and the other one associated to direct reaction couplings. Four systems having 6Li projectiles and one with 6He were analyzed. The extracted threshold anomalies are discussed.


Introduction
Much interest within the nuclear physics community has been generated by the study of reactions with loosely bound nuclei, stable or radioactive [1,2].Among the stable weakly bound nuclei, the most studied one is perhaps 6 Li, which in addition maintains a strong current interest [3][4][5][6][7][8].As for radioactive projectiles, reactions induced by the neutron-halo nucleus 6 He have proved to be a most fruitful area of research for over two decades [2,[9][10][11].
The aim of the present work is to describe some interesting threshold anomalies that have been shown recently in reactions induced by low-energy projectiles of weakly-bound 6 Li and neutronhalo 6 He.Energies near or below the Coulomb barrier have been analyzed.The term threshold anomaly (TA) [12,13] refers to a peculiar energy behavior of the corresponding optical model polarization potentials.This is usually associated to the normal closing down of direct reaction channels as the energy approaches the Coulomb barrier, which causes the potential strength to start decreasing at an energy around the barrier and eventually go to zero at even lower energies, when all reaction channels are finally closed.However, for the case of weakly-bound projectiles some direct-reaction channels may remain open even below the Coulomb barrier, which has led to the observation of a different kind of threshold anomaly, the so called breakup threshold anomaly (BTA) [14,15].The intention of the present work is to present a summary of some recent findings within this context.
Optical model potentials have been obtained [3,11] which can describe simultaneously both the elastic scattering data and the fusion data reported for a given system.All systems analyzed correspond to medium-mass targets.For  [3] while for the halo projectile only the 64 Zn target was analyzed [11].The formalism known as Extended Optical Model (EOM) [16,17] was used, where the potential is given by U T OT includes a real nuclear part V bare as well as the Coulomb potential V Coul .It considers also a purely imaginary potential interior to the barrier W int , as well as two complex polarization potentials, U F and U D , the first one associated to fusion couplings and the latest one associated to direct reaction couplings.One very interesting feature of this model is that, once the terms of the potential are determined, it is possible to calculate not only the total reaction cross section, in the usual manner, but also the fusion cross section can be calculated, through the respective absorption in the interior and the fusion polarization potentials.The corresponding expression is [16] where v is the relative velocity and χ (+) is the distorted wave function that satisfies the Schrödinger equation corresponding to the full optical potential of Eq. 1.
The bare potential, V bare , was taken as the Sao Paulo Potential (SPP) [18], which has no fitting parameters, and for the Coulomb potential V Coul , a radius of 1.2 fm was used; W int is a volume Woods-Saxon potential interior to the barrier, with depth, radius and diffuseness of 50 MeV, 1.0 fm and 0.2 fm, respectively.U F (V F + iW F ) and U D (V D + iW D ) are complex polarization potentials that can be associated to fusion and direct couplings, respectively.
For V F and W F a volume Woods-Saxon form was used with common radius and diffuseness (r F = 1.4 fm, a F = 0.43 fm).The respective strengths (V F 0 , W F 0 ) were taken as fitting parameters.The direct polarization potentials were surface potentials, taken as derivative Woods-Saxon form, whose strengths (V D0 , W D0 ) were fitting parameters.Both the real and the imaginary parts of U D were supposed to have common values of radius and diffuseness, independent of energy, whose actual values for each system were given in Refs.[3,11], respectively.For the case of the 6 Li + ( 58 Ni, 59 Co, 64 Ni, 64 Zn) systems, the values of (r D , a D ) were (1.61, 0.68), (1.57, 0.84), (1.57, 0.91), and (1.60, 0.85) fm, respectively, while for 6 He + 64 Zn the corresponding values were (1.51, 1.22) fm.
In summary, the real and imaginary strengths of both the Direct and the Fusion polarization potentials are the only free parameters, so there are 4 free parameters.These parameters were varied to obtain a simultaneous fit to the elastic scattering angular distribution and the respective fusion cross section for each measured energy, always making sure about the validity of the Dispersion Relation (DR) (see Refs. [3,11]) for the obtained potentials.

Results for 6 Li projectiles
Fig. 1 shows the elastic scattering angular distributions for the four systems having 6 Li projectiles, along with the respective predictions from EOM.The data are well reproduced for all systems.
The reported fusion cross sections, represented by the black circles in Fig. 2, are mostly well reproduced by the calculations indicated by the red diamonds; but there are some exceptions, for the two lowest energy points of 6 Li + 58 Ni and 6 Li + 64 Zn.These points are underpredicted by the calculations.This underprediction is a consequence of forcing the fusion polarization potentials to satisfy the DR.However, if this condition is relaxed, one gets highly unphysical fusion polarization potentials [3].Our hypothesis was that the data may have been originally Elastic scattering angular distributions for 6 Li + ( 58 Ni, 59 Co, 64 Ni, 64 Zn).The data are from Refs.[19][20][21][22], respectively, and the curves represent the predictions of the EOM.overpredicted, which is quite feasible because they were measured as inclusive data.Exclusive measurements are needed to corroborate this quite plausible hypothesis.

Figure 2.
Measured and predicted fusion cross sections for 6 Li + ( 58 Ni, 59 Co, 64 Ni, 64 Zn).The data are from Refs.[23][24][25][26], respectively, and the red diamonds represent the predictions of the EOM.Fig. 3 shows the strengths obtained for the real (red circles) and imaginary (black squares) polarization potentials in the case of 6 Li projectiles.The validity of the Dispersion Relation was always checked by approximating the imaginary parts by straight line segments which were used in the integral of the DR to obtain the predicted real parts, indicated by the dashed curves.The good consistency of these curves with the points extracted from the optimization corroborates the validity of the DR.
In the case of the points enclosed by the blue rectangles, for 6 Li + 58 Ni and 6 Li + 64 Zn, these points were forced to satisfy the Dispersion Relation, as mentioned above, causing an underprediction of the respective fusion cross sections.However, there are good reasons to believe that this procedure is correct.
It can be pointed out that the direct polarization potentials (left panel) are all consistent with the Breakup Threshold Anomaly.One can see a tendency for the W D to increase with decreasing energy for energies around or below the barrier.Also, the real parts, V D , are mostly repulsive (negative) for all systems.These features characterize the BTA.
On the other hand, the real parts of the fusion polarization potentials, V F , are attractive and the imaginary parts, W F , start decreasing around the barrier for all systems.This means that the fusion polarization potentials are all consistent with the TA rather than the BTA.

Results for 6 He projectiles
Switching now to the neutron-halo system, 6 He + 64 Zn, Fig. 4 presents the respective elastic scattering angular distributions.The curves describe the results of the EOM fits, which reproduce the data quite well.
In figure 5, the black filled circles represent the experimental fusion cross sections and the red empty circles refer to the respective calculated results.Considering the reported error bars, one can say that the calculations reproduce the data reasonably well.The blue empty squares represent the total reaction cross sections, calculated from the fitted potentials.
An interesting feature is that such empty squares show a local maximum, at 12.3 MeV, Elastic scattering angular distributions for 6 He + 64 Zn.Data are from Refs.[10,27] and the curves represent EOM calculations.
which has been remarked in Fig. 5 by enclosing the involved points within an ellipse.This is unexpected, because total reaction cross sections normally grow larger with increasing energy.One can notice that the fusion cross sections don't show such a local maximum, indicating that it must originate from the respective direct reaction mechanisms.This will be corroborated below by looking at the behavior of the respective threshold anomalies. 6He + 64 Zn.Data (filled symbols) are from Refs.[27,28] and the empty symbols represent EOM calculations.

Figure 5. Fusion and total reaction cross sections for
The ellipse emphasizes the presence of a local maximum in the total reaction cross sections.
The right hand side panel of Fig. 6 shows the real and imaginary strengths of the direct reaction polarization potentials obtained for 6 He + 64 Zn.The imaginary strengths are approximated by the straight line segments, which are used in the Dispersion Relation to calculate the corresponding real part, given by the dashed red curves.One can see that the DR is well satisfied.
A strikingly irregular BTA is observed in this case.It is striking because of two reasons.On one hand, at the peak the strength rises quite high above the value corresponding to the next Potential strengths obtained for 6 He + 64 Zn.Continuous curves approximate the imaginary parts while dashed curves represent corresponding calculations for the respective real parts, using the Dispersion Relation.
higher energy.On the other hand, the energy at which this peak occurs is quite high above the barrier, in contrast to the situation for the 6 Li systems, where the peak appeared at energies close or even below the barrier.
The peak in this case is actually correlated with the local maximum in the total reaction cross sections pointed out previously, which also occurred at about 12 MeV.The conclusion is that there is some kind of transition which occurs in the involved direct reaction mechanisms, at an energy close to 12 MeV.
The left panel of Fig. 6 shows the strengths obtained for the fusion polarization potential, the imaginary part being on top and the real part on the bottom.Once again, one can see that the results satisfy the Dispersion Relation and, in this case, the typical behavior of the normal threshold anomaly is observed.
A possible mechanism explaining the results can be sketched.The 6 He can be thought of as formed by an alpha core and a 2-neutron halo.The Coulomb force due to the charge of the target tends to keep the alpha particle away from it but it doesn't act on the neutrons of the halo, so these neutrons could get close to the target and thus enhance the mutual nuclear attraction, which could lead to an increase in the total reaction cross section.
The probability that this situation occurs depends on the ratio between the relative speed of colliding nuclei (v r ) and the average speed of outer neutrons in the projectile (v n ).If this ratio is low, the neutrons can make several turns around the core during the interaction time, in which case the neutrons may have enough time to interact with the target and actually affect the reaction cross section.In the opposite situation, the effect of neutron rearrangement on the reaction cross section would be weaker or null.
The hypothesis is then that the observed local maximum in the reaction cross sections is the result of a transition between an adiabatic and a nonadiabatic rearrangement of the neutrons.It would be interesting to have real theoretical calculations which describe the time of the loosely bound neutrons of 6 He collisions with Zn.
Additional observations can made, from semiclassical point of view.By doing a transformation from angle to distance of closest approach on a Rutherford trajectory, a different representation of the angular distributions of Fig. 1  for all energies in a same system are plotted together, it can be shown that they follow a single trajectory [3].This allows one to define a single interaction distance for each system, as the distance where the ratio to Rutherford starts decreasing from one [29][30][31].
For the case of the halo system, however, a large spread of points corresponding to the different energies appears in such a plot.Each energy must be analyzed individually in this case, which leads to different values of the interaction distance for each energy.Such values are plotted with red squares in Fig. 7, as a function of energy.
The transition mentioned above in direct reaction mechanisms is consistent with a transition from low to high values of the interaction distance when going from the assumedly non-adiabatic to the adiabatic region of energies.It is quite intuitive that in the adiabatic region, where the neutrons in the halo have more time to interact with the target, the interaction may occur at a larger distance.Measurements in an extended energy range would be desirable in order to further corroborate these results.

Conclusions
The results of EOM analyses for 6 Li + ( 58 Ni, 59 Co, 64 Ni, 64 Zn) and 6 He + 64 Zn are described.In all cases, the fusion part of the potential presented a normal Threshold Anomaly (TA), while the behavior of the direct reaction part was consistent with the Breakup Threshold Anomaly (BTA).
The Dispersion Relation was verified for all systems and it was pointed out that the constriction of having dispersive potentials is useful to identify seemingly unphysical data.For the particular case of the neutron-halo projectile, a strikingly irregular BTA was observed for the direct reaction part of the potential, which lead to the conclusion that an unexpected transition in the corresponding direct reaction mechanisms is present.

Figure 3 .
Figure 3. Strengths obtained for the direct (left hand side) and the fusion (right hand side) polarization potentials.The imaginary parts are approximated by the continuous lines and the dashed lines correspond to respective Dispersion Relation calculations.The blue rectangles are explained in the text.

Figure 6 .
Figure 6.Potential strengths obtained for6 He + 64 Zn.Continuous curves approximate the imaginary parts while dashed curves represent corresponding calculations for the respective real parts, using the Dispersion Relation.