Configuration transition effect in heavy-ion fusion reactions with deformed nuclei

Nuclear structure is found to play an important role in heavy-ion fusion reactions [1–9], one such example is that the nuclear deformation effect leads to an enhancement of sub-barrier capture and fusion cross sections as compared to those for similar spherical nuclei [8–10]. Such fusion enhancement is extremely important in studying the syntheses of heavy and super heavy elements (SHE) [11] and can be explained with a distribution of fusion barriers. Extensive studies [12–16] indicate that such barrier distribution shows strong dependence on the nuclear deformation and the relative orientation of the deformed nuclei. Because most nuclei in the nuclear chart, especially in the actinide region and far from the β-stability line, have large deformations, detailed investigations of the deformation effect on fusion reactions thus are called for.

In most theoretical studies up to now, the barrier penetration concept and the approximation that the effective interaction potential for nonzero angular momentum orbits is simply treated as the sum of the interaction potential for head-on collisions and the corresponding centrifugal potential [17] are used. In head-on collisions, the barrier that the system "sees" in the tip(-tip) configuration is lower than that of the side(-side) configuration and the capture cross sections of the tip(-tip) configuration thus should be larger in these models. One successful such model in studying fusion reactions is the coupled-channels model as discussed in the review [18], which takes account of nuclear deformation, collective vibration and neutron transfer, and can well reproduce the fusion excitation functions of many reactions, especially with deformed nuclei. The shortcoming of this model, however, is that the final results strongly depend on the phenomenological interaction potential adopted and the channels included. Besides, Dasgupta et al. considered [19] the current scheme of coupled-channels calculations to be inadequate, because decoherence associated with dissipation should be considered to consistently describe nuclear fusion.

Recently, Umar et al. developed the density-constrained time-dependent Hartree-Fock (DC-TDHF) method [20] which is based on the quantum microscopic many-body TDHF theory and constrained HF techniques, and applied it together with the incoming-wave boundary condition (IWBC) method to study the deformation effect in fusion reactions [21]. Such studies and the related conclusions, however, are confined to head-on collisions. In our previous study [22], we found that the noncentral effect of the interaction potential and the mass parameter had an important role in fusion reactions at near-barrier energies with spherical nuclei, which results from the density rearrangement effect. In fusion reactions with deformed nuclei, the situations in noncentral collisions might be much different from those in head-on collisions. The aim of this letter is to investigate both the deformation and the noncentral effects on fusion reactions.

Here we concentrate on the fusion reactions of ¹⁶O targeting on a well prolate deformed $^24\text{Mg}\ (\beta_{2}\approx 0.4)$ at various incident energies above the barrier and at various impact parameters for four typical coplanar orientations of $^24\text{Mg}{:}\ \beta=0$ (tipward), $\pi/4$, $\pi/2$ (sideward) and $3\pi/4$, which are defined at the initial distance of TDHF calculations. The Coulomb reorientation effect [23,24] is neglected since it changes the orientation in the approaching stage and affects the total fusion cross sections only, which is beyond the scope of this paper. The TDHF method is used to study the collision dynamics and the classical sharp cutoff approximation is adopted to get the fusion cross sections. The DC-TDHF method is then adopted to get the interaction potential and the mass parameter and so on to analyse the fusion excitation functions.

The TDHF method is a pure mean-field theory based on the independent-particle picture [25,26] in which the N-particle wave functions are constrained to be a Slater determinant at any time. This theory has been successfully applied to study low-energy reactions [27–29]. Current three-dimensional TDHF calculations are performed with more accurate numerical methods, with no symmetry assumptions and with modern Skyrme energy density functionals (EDF). The only free parameters, those of the EDF, are adjusted to static properties only and are not fitted to the collision situations. Here, the Sky3D code [30] with the SLy6 parameterization [31] is used to get the ground states of the reacting nuclei and also to get the time evolution of the collision, which thus allows for a fully consistent treatment of nuclear structure and dynamics. The zero-point motion in the collective coordinates is ignored like in all TDHF calculations performed up to now. In principle the deformed nucleus should be represented by a coherent sum of states with different orientation of the axis. The nonlinearity of mean-field does not allow such a superposition, so that we rely on the assumption that when the interaction starts the different orientations become decoherent very rapidly, so that a spherical nucleus touching the tip of the deformed nucleus at a certain angle does not coherently add up with the situation say at $\pi/2$. There may be still some spread in the orientation angle present. The validity of this assumption has not really been proven, but we think that this will not affect the main points of this work.

In the present work, the static HF calculations are performed on $28\times 28\times 28$ Cartesian grids with 0.8 fm grid spacing in all three directions. ²⁴Mg is prepared with the symmetry axis parallel to the z-axis, ground states of ²⁴Mg at different orientations are obtained through rotations in both coordinate and spin spaces. A similar method can be found in [32]. In dynamical calculations, the meshes in x- and z-directions are extended to 48 grid points with the same grid spacing as in the static HF. The initial separation distance between the mass centers of the two nuclei in the z-direction is set as 16 fm. The density constraint calculations are performed every 25 time steps.

The polarized fusion cross sections of the four typical initial orientations of ²⁴Mg are obtained by the sharp cutoff method and the maximum impact parameter $b_{\max}$ at which the system can fuse is determined within 0.1 fm for each $E_{\text{c.m.}}$. Note here that for such light systems, the compound nucleus is automatically formed once the two nuclei are captured. The results are presented and compared with the experimental data [33] in fig. 1. The lines serve to guide the eye. Remarkable features can be found: 1) There are clear crossings between the curves for different orientation, of which the most important three are around 17 MeV and 26.4 MeV. For convenience in the following discussions we denote the two specific energies as E_X1 and E_X2. 2) When $E_{\text{c.m.}} <E_{X1}$, ${\sigma_{\text{fus}}(\beta=0)>\sigma_{\text{fus}}(\beta=\pi/2)}$ and note that at $E_{\text{c.m.}}=15\ \text{MeV}$, head-on collision of $\beta=\pi/2$ cannot lead to fusion. 3) When $E_{\text{c.m.}}>E_{X1}$, ${\sigma_{\text{fus}}(\beta=0) <\sigma_{\text{fus}} (\beta=\pi/2)}$. 4) ${\sigma_{\text{fus}}(\beta=\pi/4)}>\sigma_{\text{fus}}(\beta=3\pi/4)$ is fulfilled in the whole energy range considered here. And when $E_{\text{c.m.}}<E_{X2} \ \text{MeV}$, the curves of $\beta=\pi/4$ and $3\pi/4$ are not located between those of $\beta=0$ and $\pi/2$.

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The above results are different from predictions of the theoretical models mentioned above, such as the code CCDEF [34] —based on the coupled-channels model and DC-TDHF together with the IWBC method [21]. In those models, the barrier height and width for $\beta=\pi/2$ are larger than those of $\beta=0$ in head-on collisions at all energies above the barrier if the nucleus is prolate deformed. Due to the fact that the potential at nonzero angular momentum is treated as the one in head-on collision plus a centrifugal potential in those models, the fusion probability of $\beta=\pi/2$ should be smaller than that of $\beta=0$ at all impact parameters. Then $\sigma_{\text{fus}}(\beta=\pi/2)<\sigma_{\text{fus}}(\beta=0)$ is consequently obtained. The results for $\beta=\pi/4$ should be equal to those of $\beta=3\pi/4$ as the two configurations are geometrically symmetric in head-on collisions. And the corresponding fusion cross sections should be located between $\sigma_{\text{fus}}(\beta=0)$ and $\sigma_{\text{fus}}(\beta=\pi/2)$ at all energies since the barrier height and width of them are smaller than those of $\beta=\pi/2$ and larger than $\beta=0$.

To understand the results in fig. 1, we calculate the nucleus-nucleus interaction potential and the dynamical mass parameter for various impact parameters and four orientations with the DC-TDHF method. The mass parameter is obtained in the same way as mentioned in ref. [22] where the excitation and rotational energies are subtracted. We choose $E_{\text{c.m.}}=19.5\ \text{MeV}$ as an example to demonstrate the results since the trends are similar for $E_{\text{c.m.}}>E_{X1}$. At this energy, $b_{\max}$ for $\beta=0, \pi/4, \pi/2$ and $3\pi/4$ are 4.4, 5.1, 4.8 and 3.8 fm, respectively. In fig. 2 we show the nucleus-nucleus interaction potential for the four orientations at b = 0 and 3.6 fm. For head-on collisions, one can clearly see from fig. 2(a) that both the barrier height and width for $\beta=0$ are smaller than those for ${\beta=\pi/2}$. The potential pocket for $\beta=\pi/2$ is much deeper. The potentials for $\beta=\pi/4$ and $\beta=3\pi/4$ are identical and located between $V(R,\beta=0)$ and $V(R,\beta=\pi/2)$. This result is in accordance with other studies [1,13]. Comparing fig. 2(a) and (b) one can find that: 1) Both the barrier height and width of $\beta=0$ and $\beta=3\pi/4$ increase whereas those of $\beta=\pi/4$ and $\beta=\pi/2$ decrease with increasing impact parameter. 2) The difference between the potentials in central and noncentral collisions are large. The difference of the barrier height can be up to several MeV. We also check that the above two trends persist at very high energies. These phenomena are very different from the results obtained for spherical systems [22], for which: 1) Both the barrier height and width decrease monotonically with the increase of the impact parameter. 2) The difference between the potentials in central and noncentral collisions are small. 3) The noncentral effect just plays a key role at energies around the barrier. It has been concluded [22] that the above results in spherical systems are caused by the density rearrangement effect which takes place at low energies. Although density rearrangement is also observed at low energies for this reaction, from above comparisons we can suspect that the trends in fig. 2 may predict a novel feature that is different from the noncentral effect observed in spherical systems. We will show in the following that the complicated phenomena presented in figs. 1 and 2 are caused by the geometrical configuration transition effect in the dynamical process.

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To this end and to simplify the problem without losing generality, here we choose $\beta=0$ and $\pi/2$ to do qualitative analysis. On the basis of the obtained effective mass parameter and by adopting the coordinate scale transformation [35], we get the scaled potential $U(\bar{R})$ at typical impact parameters for $E_{\text{c.m.}}=16$ ($b_{\max}$ are 3.1 and 2.4 fm for $\beta=0$ and $\pi/2$) and 19.5 MeV (the two specific energies are denoted by arrows in fig. 1). Note that only the width of the barrier but not the height will change with this transformation. The results are presented in fig. 3. At $E_{\text{c.m.}}= 16\ \text{MeV} <E_{X1}$, one can easily find that in both central and noncentral collisions, the height and width of the barrier for $\beta=\pi/2$ are larger than those of $\beta=0$ which makes the fusion probability smaller than for $\beta=0$ at all impact parameters. Here we should claim that although the trend show that the barrier height and width of $\beta=0$ will increase and that of $\beta=\pi/2$ decrease with the impact parameter, because we cannot deal with the case of $b>b_{\max}$ in the framework of TDHF approximation and also the fusion probability will decrease rapidly when b approaches $b_{\max}$, the conclusion that $\sigma_{\text{fus}}(\beta=\pi/2)<\sigma_{\text{fus}} (\beta=0)$ at this energy should be justified. While at $E_{\text{c.m.}}=19.5\ \text{MeV}> E_{X1}$, the scaled barrier height is the lowest for $\beta=0$ in head-on collisions, which is similar with that at $E_{\text{c.m.}}=16\ \text{MeV}$. But due to the fact that this energy is about 4.5 MeV above the barrier, the fusion probabilities for both $\beta=0$ and $\pi/2$ are unity in this case. At $b=4.4\ \text{fm}$, it is interesting to find that the barrier heights of $\beta=0$ and $\pi/2$ are almost equal. The barrier width of $\beta=0$, however, is much larger than that of $\beta=\pi/2$. And it is clear that the disparity between them will increase with the impact parameter (see the trends in fig. 2). The rotational energies of these two cases with respect to the impact parameter are quite similar. So the effective interaction potentials have the same properties with $U(\bar{R})$. The penetration probability and thus the fusion cross sections of $\beta=\pi/2$ are larger than that of $\beta=0$ at this energy.

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Now it is clear that the anomalous results in fig. 1 are caused by the complicated dependence of the interaction potential on the impact parameter (see fig. 2) due to the effects of deformation and orientation of the deformed nucleus and noncentrality. And we reach the place to understand such complicated dependence. Here as an example we choose three specific cases where the interaction potentials are close to each other but the initial configurations are totally different. The three cases are $\beta=0, b=0\ \text{fm}$, $\beta=\pi/4, b=3.8\ \text{fm}$ and $\beta=\pi/2, b=4.6\ \text{fm}$. The interaction potentials for these are presented in the left panel of fig. 4. We labeled seven typical reaction stages by arrows and give the corresponding density contour plots of the system in the right panels of fig. 4. One can find that when the two nuclei are apart, the configuration of the system has little effect on the interaction potential since the Coulomb interaction plays the dominant role in the approaching stage. Around the barrier centroid, the configurations evolve close to each other and the potentials are not much different. When the two nuclei are largely overlapped, the configurations are similar to each other and one cannot even distinguish the initial configurations from the final ones which make the potentials similar too. Thus one can find that the interaction potential obtained with DC-TDHF is directly related to the geometrical configuration of the system since the local densities at low energies have nonviolent changes during the collision and the lowest energy state of the system is obtained in DC-TDHFs. So it is clear that such configuration transition takes place in the whole energy range in this work.

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At energies much above the barrier, such as ${E_{\text{c.m.}}>E_{X2}}$, the result that $\sigma_{\text{fus}}(\beta=\pi/2)$ is the largest can also be understood from a classical point of view. In sideward collisions, the contact zone of the two nuclei achieved in the x-direction is the largest since ²⁴Mg elongates in this direction. While in tipward collisions, the radius of ²⁴Mg in the x-direction is the smallest.

It is surprising that the complicated results in fig. 1 can be explained by the configuration transition effect based on a simple geometric concept. We hope that such results could provide new insights into the investigations of the dynamics and mechanisms of fusion reactions with deformed nuclei, especially into the study of syntheses of SHE, where the colliding nuclei are deformed in general. It should be noted here that the calculated fusion cross sections in fig. 1 are larger than the experimental data. This might be caused by the shortcomings of the TDHF theory which does not consider the nucleon-nucleon collisions and pairing effect. To include pairing correlations, a full Skyrme EDF within the time-dependent Hartree-Fock-Bogolyubov theory has been done in [36] and a simplified version of it named TDHF+BCS approximation has been applied to heavy-ion collisions alyeady [37].

In conclusion, we use the TDHF theory in this work and investigate nuclear deformation, initial orientation and noncentral effects in fusion reactions of ¹⁶O + ²⁴Mg. The fusion excitation functions for different orientations of ²⁴Mg show very interesting properties which are different from other theoretical predictions. Furthermore, we use a density-constrained method to get the interaction potentials at typical impact parameters for different orientations and find that the interaction potentials show complicated dependence on the impact parameter. The characteristic results of the interaction potentials and the fusion excitation functions are caused by the geometrical configuration transition effect which is essentially different from a density rearrangement effect. To our knowledge such an effect has never been investigated in other TDHF studies.

This work is partly supported by the National Nature Science Foundation of China under Grant No. 10975019, the Fundamental Research Funds for the Central Universities and by the German Ministry BMBF under Grant No. 05P12RFFTG.