Formation of transfermium elements in reactions with ²⁰⁸Pb

The primary method for producing superheavy atomic nuclei (SHN) has been heavy-ion fusion reactions dullman15:a ; rudolph16:a . When relatively light reaction partners reach their Coulomb barrier the configuration is more compact than that of the fission saddle shape. Therefore, a compound nucleus is automatically formed. However, for heavier reaction partners the Coulomb barrier is situated outside the fission saddle. To avoid a re-separation, referred to as quasi-fission (QF), the shape of the composite system must diffuse over the inner barrier swiatecki81:a ; swiatecki82:a ; bjornholm82:a .

Thus the production of a SHN in a fusion reaction requires a heavy reaction system and the process can be divided into three stages: (i) Contact: the two reaction partners overcome their mutual Coulomb repulsion and achieve contact; (ii) Formation: evolving in competition with QF, the combined system achieves a compact shape well inside the fission barrier; (iii) Survival: the compact system deexcites towards the ground state in competition with fission.

The heaviest known elements are produced in fusion with targets heavier than ²⁰⁸Pb at rather large excitation energies. However, fusion reactions with ²⁰⁸Pb are more accessible experimentally and thus serve as a good starting point for theoretical studies and we shall focus this first study on reactions with the projectiles ⁴⁸Ca, ⁵⁰Ti, ⁵⁴Cr, and ⁵⁸Fe on a ²⁰⁸Pb target. For three of these reactions, the formation probability was obtained experimentally by extracting the fusion-fission component from the total cross section banerjee19:a .

The dynamics in the formation stage is not yet well understood and several models have been developed to describe the process. They can broadly be classified into two groups: time-dependent Hartree-Fock methods godbey20:a and models based on the stochastic Langevin framework swiatecki03:a ; swiatecki05:a ; zagrebaev15:a .

In this study we investigate the formation process with a Langevin-type treatment. During the formation process, the nuclear shape is described by five parameters and the associated five-dimensional (5D) potential-energy surface includes energy-dependent shell and pairing effects. The shape evolution is simulated in a Monte-Carlo manner so an ensemble of events is generated, thus allowing the extraction of both fluctuations of the observables and correlations between them.

After the reacting nuclei have made contact, the early part of the formation stage is assumed to be dominated by the dissipation of the initial radial kinetic energy, leading the shape along an effectively one-dimensional path towards smaller overall elongations. With the same assumption of overdamped shape evolution as invoked in the fusion-by-diffusion (FBD) model swiatecki03:a ; swiatecki05:a , the subsequent shape evolution is then diffusion dominated. A preliminary version of this treatment was published in Ref. albertsson_thesis .

Section II describes the shape coordinates employed to represent the composite system and the relevant structures in the potential-energy landscape, such as the different valleys. Section III discusses the dynamics of the formation process and the calculated formation probabilities are presented in Sec. IV. Finally, Sec. V provides a summary and a discussion.

The shape of the evolving nuclear system is described within the three-quadratic-surface parametrization in which two spheroidal end parts are connected smoothly by a hyperbolic surface of revolution nix69:a . This shape family has five parameters: the overall elongation measured by the dimensionless quadrupole moment, $q_{2}$; the degree of reflection asymmetry, $\alpha$; the radius of the neck connecting the two parts, $c_{\rm neck}$; the deformations of the two spheroidal end parts, $\epsilon_{\rm P}$ and $\epsilon_{\rm T}$. Collectively, these five shape parameters are denoted by $\bm{\chi}$. The parameter $q_{2}$ is defined from the quadrupole moment of a uniformly charged sharp-surface volume, $Q_{2}$, as $q_{2}\equiv(4\pi Q_{2})/(3ZR_{A}^{2})$ with $R_{A}=1.2A^{1/3}$ fm.

For each shape, the potential energy $U(\bm{\chi})$ is calculated by the macroscopic-microscopic method using the finite-range liquid-drop model moller09:a , assuming that the system has a uniform charge-to-mass ratio. The associated 5D shape lattice is set up as described in Ref. albertsson20:b .

The resulting potential-energy profile $U_{\rm min}(q_{2})$, i.e. the minimum energy of $U$ for a specified $q_{2}$, is shown in Fig. LABEL:fig:epot1d for the reactions at angular momentum $I$=0 induced by the projectiles ⁴⁸Ca, ⁵⁰Ti, ⁵⁴Cr, and ⁵⁸Fe on a ²⁰⁸Pb target, leading to the compound nuclei ²⁵⁶No (a), ²⁵⁸Rf (b), ²⁶²Sg (c), and ²⁶⁶Hs (d), respectively.

In all four cases the compound nucleus has a prolate ground-state shape with $q_{2}\approx 0.7$, corresponding to a quadrupole deformation of the overall shape of $\varepsilon_{2}\approx 0.21$. The inner saddle point protecting the compound nucleus from fissioning is located at $q_{2}\approx 1.5$, as determined by the immersion method moller09:a . The saddle point is almost symmetric in all cases: $\alpha$=0.02 for projectiles ⁴⁸Ca, ⁵⁰Ti and ⁵⁴Cr, and $\alpha$=0.1 for ⁵⁸Fe. Different valleys are found by identifying local minima for each $q_{2}$ value in the energy surface and smoothly following these minima as $q_{2}$ increases. The ground states are connected to valleys that are mainly symmetric and these are denoted as the respective fission valleys (blue circles). Being initially reflection symmetric, the fission valleys bifurcate into asymmetric shapes at $q_{2}\approx 3.5$ (see Fig. 2).

For each case, in addition to the fission valley, there is a valley (denoted as the fusion valley) having a mass asymmetry similar to that of the projectile-target system (red circles). The ridge separating the two valleys is found using the immersion method and is shown as a black curve. The shapes along the fusion valley are highly asymmetric with $\alpha\approx$ 0.5–0.6 for $q_{2}\approx 2.5$ but become less asymmetric for smaller $q_{2}$ (see Fig. 2).

During the approach stage, while still well separated, the nuclei interact only via the Coulomb potential, which is shown for two spheres in Fig. LABEL:fig:epot1d as dashed green curves. However, when the nuclei come close, the nuclear interaction will start to have effect and it is energetically more favourable to develop a neck. This occurs at the contact elongation $q_{2}^{\text{cont}}$ (shown as black arrows in Fig. LABEL:fig:epot1d).

The neck radius along the fusion path is shown in Fig. 3, where the neck is seen to develop at $q_{2}$=4.6 and 5.1 for the reactions ⁵⁰Ti+²⁰⁸Pb and ⁵⁸Fe+²⁰⁸Pb, respectively. This rapid increase in neck radius is also in agreement with the “neck zip” concept discussed in the FBD model swiatecki03:a ; swiatecki05:a . The nuclear interaction leads to a substantial lowering of the potential energy. The shapes at contact and shortly after are also displayed in Fig. 3 and it can be seen how the neck formation causes a dramatic shape change of the projectile-like part, from being nearly spherical to having a large prolate deformation. Similar changes of the elongation of the projectile was found in the investigation of fusion barriers in Ref. ichikawa05:a .

The development of the optimal shapes along the fusion path is illustrated in Fig. 4. While the target-like part retains a constant and slightly oblate shape, the projectile-like part acquires a substantial and variable prolate shape, with that of the titanium-like part building up quickly and that of the iron-like part deforming more gradually.

The numerical calculations have been performed for different values of excitation energy $E_{\rm CN}^{*}$ and angular momentum $I$. The excitation energy of the compound nucleus is given by

where $E_{\rm cm}$ is the relative kinetic energy and

with $M_{\rm P}$, $M_{\rm T}$, and $M_{\rm CN}$ being the masses of projectile nucleus, target nucleus, and corresponding compound nucleus, respectively. The masses are calculated within the same macroscopic-microscopic model that is used to obtain the potential-energy surfaces moller04:a .

Generally, the resulting compound nucleus inherits the angular momentum $I$ brought in by the colliding projectile-target system. To a good approximation, this can be taken into account by adding a centrifugal potential, leading to an effective potential

where $\mathcal{J}_{\perp}(\bm{\chi})$ is the rigid-body moment of inertia perpendicular to the symmetry axis. The rotational term corresponds to an increase of the potential energy at smaller $q_{2}$ relative to larger $q_{2}$. For example, for $I$=60$\,\hbar$ in ⁵⁰Ti+²⁰⁸Pb, the centrifugal potential is 5.8 MeV at contact, while it is 9.9 MeV at the fission saddle.

After establishing contact, the two colliding nuclei continue their inward relative motion while experiencing a frictional force that gradually converts the relative kinetic energy into intrinsic excitation. After this early drift-dominated stage has brought the relative motion to a halt, the further shape evolution takes on a diffusive character in the full 5D shape space. The system may then eventually become as compact as the ground-state shape inside the inner barrier (which we shall denote as fusion) or it may re-divide into two fragments (which is denoted as QF).

Section III.1 describes the contact configurations. Then the drift stage is discussed in Sec. III.2 and the diffusive stage is presented in Sec. III.3.

The dynamical calculations described above are carried out for a range of angular momenta $I$. For a given energy and a particular $I$ the formation probability is calculated as the number of formation events divided by the total number of events,

For each energy and angular momentum we simulate $10^{5}$ events. For a given energy, reactions having a higher angular momentum $I$ will come to a stop further out in $q_{2}$, because the effective potential energy $U_{I}(R)$ from the contact elongation and inwards grows ever steeper at larger $I$. As a consequence, the subsequent diffusion process will be less likely to lead to compact shapes and the formation probability $P_{\rm form}(E^{\ast}_{\rm CN},I)$ will thus decrease as $I$ is increased.

An effective formation probability can be obtained by averaging over all angular momenta up to the maximum value $I_{\rm max}$ (see Fig. 5), weighting each $I$ by the geometric factor $2I+1$,

Figure 8 shows $(2I\!+\!1)P_{\rm form}(E^{\ast}_{\rm CN},I)$ for the reaction ⁵⁰Ti+²⁰⁸Pb at various energies. For each energy, $E^{\ast}_{\rm CN}=15$, 20 and 40 MeV, the contribution to $\langle P_{\rm form}(E_{\rm CN}^{*})\rangle$ is shown by the shaded region. This should be compared to the corresponding area under the $2I+1$ line which represents a 100% formation probability. The ratio between these two areas, which thus constitutes the effective formation probability in Eq. (14), decreases with increasing energy, with the values being 0.285, 0.270, 0.128, respectively. These values may be compared to the formation probabilites for $I=0$, which are found to be 0.286, 0.276, 0.162, respectively. Thus, the inclusion of angular momentum implies a decrease in the formation probability of only 0.3%, 2.2%, and 21.0%, respectively. This shows that in our calculations, there is a much higher sensitivity to the excitation energy, and a rather weak dependence on the angular momentum.

Figure LABEL:fig:form_tot shows the calculated effective formation probability $\langle P_{\rm form}(E^{\ast}_{\rm CN})\rangle$ as a function of energy for the four considered reactions. We note that the data of Naik et al. naik07:a for ⁵⁰Ti+²⁰⁸Pb exhibit a maximum in the formation probability at $E^{\ast}_{\rm CN}\approx 25$ MeV. The data of Banerjee et al. banerjee19:a , which represent the upper limit of the formation probabilities, instead show a steady decrease with energy.

Because the model does not include tunneling, formation is only possible if the total energy is sufficient to allow the system to move up the fusion valley and over the inner saddle. The corresponding thresholds in $E_{\rm CN}^{*}$ are 12.6, 10.6, 9.2, and 6.7 MeV for the four reactions, respectively. For all four cases, the formation probability initially increases steeply above the threshold energy and then reaches a maximum at $E^{*}_{\rm CN}\approx 15$ MeV, followed by a moderate decrease. This behaviour arises from the competition between two effects. As the energy is increased, the early inward motion stops at ever more compact shapes, thereby increasing the probability that the subsequent diffusive evolution succeeds in forming a compound nucleus. However, it also becomes ever easier for the shape to find more favorable paths on the potential-energy landscape and cross the ridge from the fusion valley to the fission valley, leading to QF and thus decreasing the formation probability. As already noted, the angular momentum does not have a significant effect on these results.

In Fig. LABEL:fig:form_tot it is seen that the formation probability decreases as the projectile mass is increased. The calculated maximal formation probability is shown in Fig. 10 versus the projectile mass, together with the available data. Ranging from about 60 percent for ⁴⁸Ca to just a few percent for ⁵⁸Fe, the calculated values agree very well with the measured results. The large formation probabilty for ⁴⁸Ca is partly due to the contact configuration being relatively compact and partly due to the slightly downward sloping potential energy in the fusion valley as one moves towards smaller values of $q_{2}$ (see Fig. LABEL:fig:epot1da). Both of these effects cause the early inward motion to proceed further, so the diffusion process starts at more compact shapes and therefore needs fewer steps to form a compound nucleus. For heavier projectiles, the contact configuration becomes more elongated and a gradual steeper upward slope in the fusion valley (see Fig. LABEL:fig:epot1d) leads to a steady reduction in the maximal formation probability.

At higher energies, from about 40 MeV and up, the calculated formation probabilities increase steadily. This is due to the increasing temperature and the associated disappearance of the shell and pairing effects which allows the diffusion process to explore larger regions of the deformation parameter space, thereby also making it easier to attain more compact shapes, counted as fusion events. However, the walks that do venture inside the inner saddle would relatively quickly cross the saddle again and proceed towards QF, so an equilibrated compound nucleus is less likely to form. Still, the comparison to the data of Ref. banerjee19:a is meaningful, because the shapes in the registered formation events have temporarily been very compact. During such a compact stage, the memory of the original direction of the symmetry axis will be lost and, upon separation, the angular distribution will then be that of a fusion-fission event. However, in experimental attempts to produce SHN, the increased formation probability at high energies will be of little importance because the very excited compound system will promptly fission and the event will be registered as fusion-fission.

While the formation probability generally decreases when the friction is stronger, the overall behavior of the curves in Fig. LABEL:fig:form_tot is rather insensitive to the friction strength (governed by the parameter $k$), The underestimate of the measured formation probability with the projectile ⁴⁸Ca (a) could be remedied by reducing the friction strength. Note, however, that the data points of Ref. banerjee19:a are upper limits. In turn, the overestimate obtained with ⁵⁴Cr (c) could be remedied by using a slightly stronger friction strength. Such adjustments of the friction strength are well within the current uncertainty on the nuclear dissipation.

We have calculated the probabilities for compound-nucleus formation in several reactions with ²⁰⁸Pb within the Langevin framework for the nuclear shape evolution. After the colliding nuclei have come into contact, the early evolution is dominated by an inward motion during which initial radial kinetic energy is being dissipated. The system then continues its shape evolution in a diffusive manner, until it either reaches the region of compact shapes near the ground state or redivides in a QF process.

The shape of the evolving system is described in the three-quadratic-surfaces parametrization. The associated 5D table of the effective potential-energy surface includes shell and pairing effects that subside as the energy is raised. During the early stage, the relative motion of the two parts of the system is subject to the one-body window friction while the subsequent diffusive shape evolution is simulated as a random walk in the shape parameter space. While no parameters were adjusted, it was studied how the results depend on the dissipation strength. Comparison with the experimental data suggests that it might increase with increasing system size.

Relative to previous treatments of SHN formation (see Refs. zagrebaev08:a ; cap22:a ) several refinements have been made. The full dynamical process from contact to formation or separation is followed, with the starting shape for the diffusive evolution being affected by energy-dependent nuclear structure effects obtained with well-established models moller09:a . The evolution of the system is obtained by Monte Carlo event-by-event simulation, which allows the extraction of a variety of correlation observables.

The calculated formation probabilities compare very well with available data, as illustrated in Figs.  LABEL:fig:form_tot and 10. As the energy is increased, two opposite effects give rise to a maximum in the formation probability: 1) at higher energy the early inward evolution leads to more compact shapes which increases the chance for the subsequent shape diffusion to reach the compact region inside the fission barrier and thus form a compound nucleus; 2) the higher energy also makes a larger shape domain accessible to the diffusive evolution, thereby facilitating the crossing of the ridge towards the fission valley, hence decreasing the formation probability. At even higher energies shell and pairing effects gradually subside and it becomes ever easier to enter the region of very compact shapes. However, as the ground-state minimum then becomes less pronounced, any compact shapes will quickly grow more elongated and proceed to fission, thus being ineffective for SHN formation.

The gradual decrease in formation probability with increasing projectile mass (see Fig. 10) was understood as being partly due to the fact that heavier projectiles lead to more elongated contact configurations and partly due to the changes in the slope of the potential energy along the fusion path (see Fig. LABEL:fig:epot1d).

Earlier studies with a diffusion-based model zagrebaev08:a show a steady increase of formation probability with energy. It was therefore suggested in Ref. banerjee19:a that diffusion is not the main mechanism that drives SHN formation in fusion reactions with ²⁰⁸Pb nuclei. This was addressed in the extended FBD model cap22:a where the decrease in formation probability with increasing energy was attributed to suppression of contributions of higher partial waves in the reactions. In contrast, the present results display only a marginal effect of the angular momentum. The decrease in formation probability with excitation energy seems instead to occur because more favorable QF paths become accessible in the potential-energy landscape.

There have been indications that multi-nucleon transfer and energy dissipation play a significant role already before contact is reached cook23:a . This would result in a lower remaining kinetic energy at contact and therefore more elongated shapes at the start of the diffusive stage. The effect might be effectively emulated by an increase in the friction strength. Multi-nucleon transfer would also result in a distribution of $N/Z$ ratios in the two reaction partners at contact; this degree of freedom is not yet taken into account.

Generally, the formation of the compound system requires the diffusive shape evolution to pass over the inner barrier. However, at energies only slightly above the threshold energy, shell and pairing effects in the saddle region may modulate the transmission probability because of the structure in the local density of states. In particular, the absence of excited states between 0 and 2 quasiparticle states is enhanced by the relatively large value of the pairing correlation in the barrier region, resulting in a non-monotonic energy dependence of the formation probability. This is an effect similar to the undulating behavior of the symmetric fragment yield in actinide fission, as discussed theoretically in Ref. Ward17 and also observed experimentally Chadwick11 ; Glendening81 . One might thus expect such structure effects to appear in the energy dependence of the formation probability just above the threshold energy.