Skyrme Energy Density Formalism Research Articles

Exploring fission and quasi-fission in 31Al + 197Au: probing energies near and beyond Coulomb barrier

An idea to understand the fragmentation of a formed nuclear entity (i.e. compound nucleus), induced via heavy-ion reactions, is an interesting task especially at incident energies near, above and far away from the Coulomb barrier. In the present work, we are studying the decay dynamics of a formed nuclear entity i.e. 228U* compound nucleus (CN), via the radioactive beam of Aluminium-31 (31Al) striking on the stable Gold-197 (197Au) target nucleus, for incident energies E c.m. = 106.2–163.2 MeV. A comprehensive knowledge regarding the probable decaying fragments from 228U* has been accumulated using the collective clusterization approach of the Dynamical Cluster-decay Model (DCM) in terms of mass and charge dispersion, which built up on the Quantum Mechanical Fragmentation Theory (QMFT). Through both of these dispersion cases, we have identified the decaying fragments in the fission and heavy-mass regions, at different scale of incident energies. Further, the fission cross-sections ( σfisTheo ) have been obtained for 31Al + 197Au → 228U* reaction at the above said E c.m. (MeV) and compared with the experimental data. This study has been carried out using two different sets of Skyrme force parameters (SIII and SSk), which are employed within the semiclassical approach of Skyrme Energy Density Formalism. Unlike for energies below and near the barrier, a significant difference has been observed between calculated ( σfisTheo ) and experimental ( σfisExpt ) cross-sections for above barrier energies, with independent choice of Skyrme forces. Such discrepancy noticed has been addressed with the presence of quasi-fission (QF) event. Similar phenomena has also been seen for 19F-induced reaction forming the same CN (228U*).

Effect of double spin-orbit parameters of spin-orbit strength in reactions involving spherical-prolate, spherical-oblate, oblate-prolate and oblate-oblate configurations

The present work is carried out to see the role of double spin-orbit parameters (W1 and W2) on reactions having spherical-prolate, spherical-oblate, oblate-prolate and oblate-oblate configurations. For the study, semi-classical Skyrme energy density formalism is used by employing SkIx (x=2,3,4) and SAMi Skyrme forces. The calculations suggest that the presence of double spin-orbit strength imparts significant influence on spin-dependent potential (VJ) as maximum barrier height of VJ (VJB) is observed with SAMi force (W1≠W2) and minimum with SkI2 (W1=W2) i.e., VJB(SAMi)>VJB(SkI4)>VJB(SkI3)>VJB(SkI2). The maxima and minima in VJB are further related to the contribution from the isoscalar and isovector parts of the spin-orbit potential. As far as shape of projectile and(or) target is concerned, the higher VJB is obtained for oblate-prolate configuration and lower when spherical-prolate configuration is used. The chosen Skyrme forces also modify the fusion barrier height VB of different target-projectile combinations in the similar way as the spin-orbit barrier height (VJB) does. These Skyrme forces are further applied to address the experimental fusion excitation functions and fusion barrier distributions of the considered reacting partners. The contribution of VJ is also estimated in fusion cross-section of considered reactions.

The study of angular momentum on the fusion of [formula omitted]Si+[formula omitted]Si, using SEDF in semiclassical extended Thomas-Fermi approach

The total fusion cross-sections are calculated for energies both above and below the potential barrier by using the properties of the interaction potential in Hill-Wheeler approximation for a positive Q-value system 1428Si+1428Si. The Coulomb and centrifugal parts of the potentials are added directly to the nuclear proximity potential to calculate the total interaction potential, where nuclear proximity potential is obtained in semiclassical extended Thomas-Fermi approach of Skyrme energy density formalism for Skyrme SIV parametrization. The angular momentum spectrum of total cross sections is obtained and calculated by using angular momentum dependent properties of the total interaction potential. The properties of total interaction potential, which are usually treated as independent of angular momentum, i.e. barrier height, barrier position and its width are calculated and found to depend sharply on the angular momentum value at a fixed center of mass energy. It effect the fusion cross-section sufficiently above the barrier energy. However at energies below the barrier energy the addition of large angular momentum values have negligible effects on fusion cross-section. Further, the maximum in fusion cross-section is observed at a particular value of angular momentum, which is not equal to the maximum value of angular momentum.

Systematic study of probable target-projectile combinations for the synthesis of Z = 120 isotopes using the Skyrme energy density formalism

The synthesis of superheavy elements is an interesting field of research for both theoretical as well as experimental physicists. Till now superheavy element from Z=104 to 118 has been synthesized in various laboratories using cold and hot fusion reactions. Recent predictions of various theoretical models state that Z=120 and N=184 seems most probable doubly magic pair in the island of stability. Hence, we aim to the predict most probable target-projectile combinations of 300,302,304120 superheavy nuclei, so that relevant inputs may be imparted for future synthesis of superheavy nuclei. The target-projectile combinations are taken in reference to the minima's of the fragmentation potential calculated using the Quantum Mechanical Fragmentation Theory (QMFT). In addition, the target-projectile combinations from literature are also considered for which attempts were made for the synthesis of Z=120. The barrier height (VB) of the considered target-projectile combinations are calculated using the spherical and deformed choices of decay fragments. The percentage change in the barrier height ΔVB is explored for the hot and cold fusion approaches. The nuclear interaction potential is obtained within the Skyrme energy density formalism (SEDF) using the GSkI force parameters. For the feasible target-projectile (t-p) combinations, the capture cross-sections of these three isotopes 300,302,304120 are calculated using ℓ-summed Wong model and their comparative analysis is worked out. The probability of formation of compound nucleus (PCN) is determined using the energy dependent function. Among the thirty-six considered target-projectile combinations, the Ti-based reactions with Cf targets and also Cr-based reactions with Cm targets are identified as most suitable combinations for the synthesis of the superheavy element with Z=120. For 300120 isotope are 48Ti+252Cf, 50Ti+250Cf, 50Cr+250Cm, 52Cr+248Cm; whereas 50Ti+252Cf, 54Cr+248Cm, and 54Cr+250Cm form most probable target-projectile pair respectively, for 302120 and 304120 superheavy nuclei.

Investigating the effect of entrance channel mass asymmetry on fusion reactions using the Skyrme energy density formalism

In the present investigations, the fusion cross-sections for the formation of 200Pb compound nucleus (CN) using 16O + 184 W, 30Si + 170Er, and 40Ar + 160Gd nuclear reactions at energies above the Coulomb barrier were calculated to understand the effect of entrance channel mass asymmetry (α) on the fusion reactions; the Skyrme energy density formalism (SEDF) was used for this calculation. The SEDF uses the Hartree–Fock–Bogolyubov (HFB) computational program with Skyrme forces such as SkM*, SLy4, and SLy5 to obtain the nucleus-nucleus potential parameters for the above reactions. Using the SEDF model with SkM*, SLy4, and SLy5 interaction forces, the theoretical fusion cross-sections were determined above the barrier energy and compared with the available experimental fusion cross-sections. The results show a close agreement between the theoretical and experimental values for all selected systems at energies well above the barrier. However, near the barrier energies, the theoretical values were observed to be higher than the experimental values.

Analysis of the barrier characteristics and related fusion properties using 2pF, 3pF and 3pG density functions

The role of three-parameter Fermi (3pF) and three-parameter Gaussian (3pG) density functions on the fusion dynamics has been explored in comparison to the two-parameter Fermi (2pF) within the semiclassical Skyrme Energy Density Formalism. For the case of the three-parameterized density functions (3pF and 3pG), an additional parameter ‘w’ influences the tail and nearby region, and hence the barrier characteristics get modified accordingly. In view of this, the role of 2pF, 3pF and 3pG density functions has been analyzed by calculating the fusion barrier height VB (and barrier position RB) for a variety of nuclear reactions with the mass-asymmetry spread from symmetric to extreme asymmetric reaction partners, and the results are compared with the available data. In addition to this, the fusion cross-sections for 58Ni-based reactions are also calculated using extended ℓ-summed Wong model and the collective effect of VB, RB and ħωB (barrier height, position and curvature respectively) for the use of above mentioned density distributions has been examined. It is found that 2pF density functional is more suitable for symmetric reactions, whereas 3pF and 3pG seems to be appropriate for asymmetric reactions.

Fusion Dynamics of Compound Nuclei with ACN = 166–202 Investigated Using Different Projectiles on 154Sm Target

The effect of mass, energy, and orientation of the nuclei of projectile beams is analyzed in reference to the dynamics of heavy-ion induced reactions spread across the Coulomb barrier. A set of reactions comprising of different projectiles such as 12C, 16O, 28Si, and 48Ca interacting with 154Sm target nucleus are considered in the present analysis. The calculations are performed using the extended Wong model by taking quadrupole deformations of the interacting nuclei into account. The nuclear interaction part is determined using the Skyrme energy density formalism. The fusion cross section of the considered reactions is estimated using the conventional SIII Skyrme force, which depicts that fusion hindrance persists in the sub-barrier region and it increases with an increase in the mass of projectile. The deviation of theoretical cross sections with respect to experimental data is addressed via (i) the cold orientation approach and (ii) different forms of nuclear potentials, such as GSkI Skyrme force and the proximity potential mod Prox 88. Further, the impact of different projectile beams has also been analyzed on the corresponding barrier profile of interaction potentials, which suggests that GSkI force is suitable for the heavier symmetric reactions. Also, the relative effect of spin-orbit independent (VP) and spin-orbit dependent (VJ) on total nuclear potential (VN) is examined for SIII and GSkI Skyrme forces.

Effect of different nuclear density approximations on fusion dynamics within Skyrme Energy Density Formalism

The role of different density approximations such as frozen, sudden and modified sudden, investigated within the Skyrme Energy Density Formalism (SEDF), is compared with the phenomenological relaxed-density approach for Ni-based reactions. These approximations give different ways in which individual densities of projectile and target can be added for obtaining compound nucleus density. For the 58Ni + 58Ni reaction, unlike sudden, the frozen density approximation is known to give the realistic barrier parameters (Nucl. Phys. A 870-871, 42 (2011)) and addresses the experimental data with the use of the extended $ \ell$ -summed Wong model. In view of this, the barrier parameters within the sudden approximation are obtained, by modifying the central density term as an independent function of the surface thickness factor, which in turn, fits the data nicely at sub barrier energies. In comparison to this, the largest barrier height of the phenomenological potential of relaxed-density advocates the possibility of fusion hindrance. Additionally, for the 18O and 40Ca + 58Ni reactions, the frozen and modified sudden approximations address the data at below barrier-energies. For the case of a heavier-mass system, the barrier parameters cannot be extracted due to the disappeared pocket for the case of sudden and modified sudden density approximations. In such situation, the frozen and relaxed-density approaches behave in a similar manner and show a better fit to the data using the extended $ \ell$ -summed Wong formula. The $ \ell$-values are extracted using the sharp cut-off model for above-barrier energies and extrapolated at lower energies. Among the chosen approximations, the frozen approximation seems to provide a better option for the reactions considered from different mass regions. Further, on the basis of frozen and phenomenological approaches, the reduced cross-sections are calculated and compared with the Universal Fusion Function (UFF) for symmetric and asymmetric choices of nuclear partners.

Impact of spin-orbit density dependent potential in heavy ion reactions forming Se nuclei

The Skyrme energy density formalism is employed to explore the effect of spin-orbit interaction potential by considering a two nucleon transfer process via various entrance channels such as 23Na + 49V , 25Mg + 47Ti , 27Al + 45Sc , 29Si + 43Ca and 31P + 41K , all forming the same compound system 72Se $ ^{\ast}$ , using both spherical as well as quadrupole deformed ( $ \beta_{2}$ nuclei. For spherical nuclei, the spin-orbit density part $ V_{J}$ of nuclear potential remains unaffected with the transfer of two nucleons from the target to the projectile, however, show notable variation in magnitude after inclusion of deformation effects. Likewise, deformations play an important role in the spin-orbit density independent part $ V_{P}$ , as the fusion pocket start appears, which otherwise diminish for the spherical nuclei. Further, the effect of an increase in the N/Z ratio of Se is explored on $ V_{J}$ as well as $ V_{P}$ and results are compared with transfer channels. In addition to this, the role of double spin-orbit parameters ( $ W_{0}$ and $ W'_{0}$ with relative contribution of the isoscalar and isovector parts of spin-orbit strength is explored in view of SkI2, SkI3 and SkI4 Skyrme forces. Beside this, the decay path of 72Se $ ^{\ast}$ nucleus formed in 27Al + 45Sc reaction is investigated within the framework of dynamical cluster decay model (DCM), where the nuclear proximity potential is obtained by both Skyrme energy density formalism (SEDF) and proximity pocket formula. The fusion hindrance in the 27Al + 45Sc reaction is also addressed via the barrier lowering parameter $ \Delta V_{B}$ . Finally, the contribution of spin-orbit density dependent interaction potential is estimated for the 27Al + 45Sc reaction using single ( $ W_{0}$ or $ W_{0}'$ and double spin-orbit parameters ( $ W_{0}$ and $ W_{0}'$ .

Effect of deformation and orientation on spin orbit density dependent nuclear potential

Role of deformation and orientation is investigated on spin-orbit density dependent part VJ of nuclear potential (VN =VP +VJ ) obtained within semi-classical Thomas Fermi approach of Skyrme energy density formalism. Calculations are performed for 24-54 Si+30 Si reactions, with spherical target 30 Si and projectiles 24-54 Si having prolate and oblate shapes. The quadrupole deformation β 2 is varying within range of 0.023 ≤ β 2 ≤0.531 for prolate and -0.242 ≤ β 2 ≤ -0.592 for oblate projectiles. The spin-orbit dependent potential gets influenced significantly with inclusion of deformation and orientation effect. The spin-orbit barrier and position gets significantly influenced by both the sign and magnitude of β 2 -deformation. Si-nuclei with β2 2 2 >0. The possible role of spin-orbit potential on barrier characteristics such as barrier height, barrier curvature and on the fusion pocket is also probed. In reference to prolate and oblate systems, the angular dependence of spin-orbit potential is further studied on fusion cross-sections.

Comparative study of fusion barriers using Skyrme interactions and the energy density functional

Using different Skyrme interactions, we have carried out a comparative analysis of fusion barriers for a wide range of interacting nuclei in the framework of semiclassical Skyrme energy density formalism. The results of our calculations reveal that SVI, SII, and SIII Skyrme forces are able to reproduce the empirical values of barrier heights with higher accuracy than the other considered forces in this formalism. It is also shown that the calculated nucleus-nucleus potentials derived from such Skyrme interactions are able to explain the fusion cross sections at energies near and above the barrier.

The Analytical Parametrization of Fusion Barrier by Using the Skyrme Energy-Density Function Model

Using the skyrme energy density formalism, a pocket formula is introduced for barrier heights and positions of 95 fusion reactions (48 ≤ ZP ZT ≤ 1520) with respect to the charge and mass numbers of the interacting nuclei. It is shown that the parameterized values of RB and VB are able to reproduce the corresponding experimental data with good accuracy. Moreover, the absolute errors of our formulas are less than those obtained using the analytical parametrization forms of the fusion barrier based on the proximity versions. The ability of the parameterized forms of the barrier heights and its positions to reproduce the experimental data of the fusion cross section have been analyzed using the Wong model.

On the role of deformed Coulomb potential in fusion using energy density formalism

Using the Skyrme energy density formalism, the effect of deformed Coulomb potential on fusion barriers and fusion cross-sections is studied. Our detailed study reveals that the fusion barriers as well as fusion probabilities depend on the shape deformation (due to deformed Coulomb potential) of the colliding nuclei. However, this dependence due to deformed Coulomb potential is found to be very weak.

Systematic analysis of hot Yb* isotopes using the energy density formalism

A systematic study of the spin-orbit density interaction potential is carried out, with spherical as well as deformed choices of nuclei, for a variety of near-symmetric and asymmetric colliding nuclei leading to various isotopes of the compound nucleus Yb*, using the semiclassical extended Thomas-Fermi formulation (ETF) of the Skyrme energy density formalism (SEDF). We observe that the spin-orbit density interaction barrier height ( $ V_{JB}$ and barrier position ( $ R_{JB}$ increase systematically with the increase in number of neutrons in both the projectile and target, for spherical systems. On allowing deformation effects with optimum orientations, the barrier-height increases by a large order of magnitude, as compared to the spherical case, in going from 156Yb* to 172Yb* nuclear systems formed via near-symmetric Ni+Mo or asymmetric O+Sm colliding nuclei, except that for the oblate-shaped nuclei, the $ V_{JB}$ is the highest and $ R_{JB}$ shifts towards a smaller (compact) interaction radius. The temperature does not change the behavior of spin-orbit density dependent ( $ V_{J}$ and independent ( $ V_{P}$ interaction potentials, except for some minor changes in the magnitude. The orientation degree of freedom also plays an important role in modifying the barrier characteristics and hence produces a large effect on the fusion cross section. The fusion excitation function of the compound nuclei 160, 164Yb* formed in different incoming channels, show clearly that the new forces GSkI and KDE0v1 respond better than the old SIII force. Among the first two, KDE0v1 seems to perform better. The fusion cross-sections are also predicted for a few other isotopes of Yb*.

Skyrme forces and the fusion-fission dynamics of the132Sn+64Ni→196Pt*reaction

The dependence of the fusion-fission process on Skyrme forces is studied by using the dynamical cluster-decay model (DCM) and the $\ensuremath{\ell}$-summed extended-Wong model in the ${}^{132}\mathrm{Sn}+{}^{64}\mathrm{Ni}\ensuremath{\rightarrow}{{}^{196}\mathrm{Pt}}^{*}$ reaction, where the nuclear proximity potential is obtained by using the semiclassical extended Thomas-Fermi (ETF) approach in the Skyrme energy density formalism (SEDF) under the frozen density approximation. The DCM gives an excellent fit to the measured fusion evaporation residue (ER) and the fission cross sections below and above barrier energies, with ER data needing ``barrier lowering'' at below-barrier energies for each Skyrme force (an in-built property of the DCM). The fission cross sections show a contribution of quasifission (qf) at the above-barrier two or three highest energies, depending on the Skyrme force. Calculations are illustrated for three Skyrme forces, GSkI, SSk, and SIII. Another interesting result is that there is a change of fission mass distribution from a predominantly asymmetric one to a symmetric one with a decrease in the $N/Z$ ratio of the compound nucleus, independent of the choice of nuclear interaction potential, which gives an opportunity to address the isospin effects in the Pt${}^{*}$ nucleus. Within the $\ensuremath{\ell}$-summed extended-Wong model we find that the GSkI and SSk forces fit the total fusion cross-section data exactly, whereas the SIII force needs ``barrier modification'' in order to fit the data at below-barrier energies. This happens because the isospin and neutron-proton asymmetry nature of GSkI and SSk forces is different from that of the SIII force, and because the center-of-mass energy ${E}_{\mathrm{c}.\mathrm{m}.}$ dependence of the barrier height for the SIII force and that of Blocki et al. [Ann. Phys. (NY) 105, 427 (1977)] differs strongly (by a constant amount of $\ensuremath{\sim}$7 MeV) from those for GSKI and SSk forces. Note that, because of the associated preformation factor with each fragment, the DCM has the advantage of treating various decay processes separately, whereas the Wong model describes only the total fusion cross section, a sum of cross sections due to all contributing processes.

Barrier modification in sub-barrier fusion reaction 64Ni+100Mo using Wong formula with Skyrme forces in semiclassical formalism

We obtain the nuclear proximity potential by using semiclassical extended Thomas Fermi (ETF) approach in Skyrme energy density formalism (SEDF), and use it in the extended ℓ-summed Wong formula under frozen density approximation. This method has the advantage of allowing the use of different Skyrme forces, giving different barriers. Thus, for a given reaction, we could choose a Skyrme force with proper barrier characteristics, not-requiring extra "barrier lowering" or "barrier narrowing" for a best fit to data. For the 64Ni+100Mo reaction, the ℓ-summed Wong formula, with effects of deformations and orientations of nuclei included, fits the fusion-evaporation cross section data exactly for the force GSkI, requiring additional barrier modifications for forces SIII and SV. However, the same for other similar reactions, like 58,64Ni+58,64Ni, fit the data best for SIII force. Hence, the barrier modification effects in ℓ-summed Wong expression depend on the choice of Skyrme force in semiclassical ETF method.

Fusion reaction cross-sections using the Wong model within Skyrme energy density based semiclassical extended Thomas Fermi approach

First, the nuclear proximity potential, obtained by using the semiclassical extended Thomas Fermi (ETF) approach in Skyrme energy density formalism (SEDF), is shown to give more realistic barriers in frozen density approximation, as compared to the sudden approximation. Then, taking advantage of the fact that, in ETF method, different Skyrme forces give different barriers (height, position and curvature), we use the l-summed extended-Wong model of Gupta and collaborators (2009) [1] under frozen densities approximation for calculating the cross-sections, where the Skyrme force is chosen with proper barrier characteristics, not-requiring additional “barrier modification” effects (lowering or narrowing, etc.), for a best fit to data at sub-barrier energies. The method is applied to capture cross-section data from 48Ca + 238U, 244Pu, and 248Cm reactions and to fusion–evaporation cross-sections from 58Ni + 58Ni, 64Ni + 64Ni, and 64Ni + 100Mo reactions, with effects of deformations and orientations of nuclei included, wherever required. Interestingly, whereas the capture cross-sections in Ca-induced reactions could be fitted to any force, such as SIII, SV and GSkI, by allowing a small change of couple of units in deduced lmax-values at below-barrier energies, the near-barrier data point of 48Ca + 248Cm reaction could not be fitted to lmax-values deduced for below-barrier energies, calling for a check of data. On the other hand, the fusion–evaporation cross-sections in Ni-induced reactions at sub-barrier energies required different Skyrme forces, representing “modifications of the barrier”, for the best fit to data at all incident center-of-mass energies Ec.m.ʼs, displaying a kind of fusion hindrance at sub-barrier energies. This barrier modification effect is taken into care here by using different Skyrme forces for reactions belonging to different regions of the periodic table. Note that more than one Skyrme force (with identical barrier characteristics) could equally well fit the same data.

Study of Fusion Dynamics Using Skyrme Energy Density Formalism with Different Surface Corrections

Within the framework of Skyrme energy density formalism, we investigate the role of surface corrections on the fusion of colliding nuclei. The coefficient of surface correction is varied between 1/36 and 4/36, and its impact is studied on about 180 reactions. The detailed investigations indicate a linear relationship between the fusion barrier heights and strength of the surface corrections. Our analysis of the fusion barriers advocate the strength of surface correction of 1/36.

Universal functions of nuclear proximity potential for Skyrme nucleus–nucleus interaction in a semiclassical approach

The universal function of the nuclear proximity potential is obtained for the Skyrme nucleus–nucleus interaction in the semiclassical extended Thomas–Fermi (ETF) approach. This is obtained as a sum of the spin–orbit-density-independent and spin–orbit-density-dependent parts of the Hamiltonian density, since the two terms behave differently, the spin–orbit-density-independent part mainly attractive and the spin-orbit-density-dependent part mainly repulsive. The semiclassical expansions of kinetic energy density and spin–orbit density are allowed up to second order, and the two-parameter Fermi density, with its parameters fitted to experiments, is used for the nuclear density. The universal functions or the resulting nuclear proximity potential reproduce the ‘exact’ Skyrme nucleus–nucleus interaction potential in the semiclassical approach, within less than ∼1 MeV of difference, both at the maximum attraction and in the surface region. An application of the resulting interaction potential to fusion excitation functions shows clearly that the parameterized universal functions of nuclear proximity potential substitute completely the ‘exact’ potential in the Skyrme energy density formalism based on the semiclassical ETF method, including also the modifications of interaction barriers at sub-barrier energies in terms of modifying the constants of the universal functions.

Isotopic dependence of fusion probabilities for neutron-deficient and -rich colliding nuclei

Isotopic dependence of the fusion dynamics is studied by analyzing the collision of a large number of isotopes of Ca and Ni with 0.6 \( \leqslant\)N/Z\( \leqslant\) 2. This study, which results from the Skyrme energy density formalism, reveals that the addition of neutrons favors fusion of reacting partners, whereas the reverse happens with the removal of neutrons. The fusion barrier heights and positions follow a non-linear second-order dependence on (\( {\frac{{{N}}}{{{Z}}}}\)-1 ), whereas fusion cross-sections can be parameterized by a straight line.-1