Nuclear stability and the ratio of kinetic to potential energy

Nuclear stability stands as a cornerstone research theme in nuclear physics, governed by nucleon-nucleon interactions and manifested through critical observables including binding energy systematics, magic number configurations, decay modes, and half-life patterns. This fundamental property not only dictates the synthesis pathways of new isotopes and superheavy elements (SHEs) but also serves as a stringent testbed for nuclear models. The nuclear shell model initially proposed in the 1960s predicted post-$ ^{208} $Pb magic numbers at $ Z=114 $ and $ N=184 $ [1−3], suggesting an "island of stability" centered at ($ Z=114 $, $ N=184 $) where superheavy nuclei might exhibit enhanced stability. However, modern self-consistent mean-field calculations reveal competing predictions, proton shell closures at $ Z=120 $ or 126 and neutron shell closures spanning $ N=172-184 $ [4, 5]. These theoretical divergences highlight the critical role of SHE stability studies in validating nuclear many-body approaches. Through fusion-evaporation reactions, superheavy nuclei with $ 104 \leq Z\leq 118 $ have already been successfully synthesized in experiments [6−12]. Up to now, the synthesized heaviest nucleus $ ^{294} $Og is neutron-deficient due to the limit of available projectile-target combinations and facilities in experiments and its half-live is at the order of millisecond. The determination of nuclear stability exhibits mass-dependent characteristics. For light and medium nuclei ($ A<150 $), β-decay predominance makes binding energy minimization along isobaric chains the key stability indicator. For heavy and super-heavy nuclei ($ A>200 $), multi-mode decay competition disrupts simple energy-based predictions. For example, although the ground state energy of $ ^{214} $Po is smaller than that of $ ^{214} $Pb by 2.7 MeV, the half-life of $ ^{214} $Pb is much larger than that of $ ^{214} $Po, due to different decay modes: $ ^{214} $Pb via β-decay whereas $ ^{214} $Po via α-decay. It is therefore necessary to develope new stability metrics beyond traditional binding energy considerations, particularly for SHEs. The enduring mystery surrounding the existence and location of the "island of stability" continues to drive interdisciplinary efforts combining nuclear experiments, astrophysical observations, and exascale computational modeling.

The binding energy of atomic nuclei predominantly arises from the dynamic equilibrium between the nuclear strong force and Coulomb repulsion [13], which governs their β-decay stability. Quantitatively, it manifests as the net energy difference between the absolute values of the attractive nuclear potential energy U and the repulsive kinetic energy T. Notably, the $ T/U $ ratio constitutes a fundamental parameter in nuclear astrophysics [14, 15], as it correlates with two critical phenomena: nuclear matter compressibility and stability mechanisms. The ratio $ T/U $ provides insights into the stiffness of nuclear matter, a key determinant of neutron star interior structure and equation of state. In neutron stars, gravitational binding stability requires the dominance of gravitational potential energy $ U_{\rm grav} $ over rotational kinetic energy $ T_{\rm rot} $. A significant imbalance ($ |U_{\rm grav}|\gg T_{\rm rot} $) may trigger catastrophic collapse into black holes. For atomic nuclei, the $ T/U $ ratio reflects the interplay between nucleon-nucleon interaction strength (governed by nuclear force saturation) and nucleon motion intensity (Fermi energy-driven). Higher $ T/U $ values typically indicate reduced binding coherence, thereby influencing fission barriers and decay modes. This intrinsic connection between the $ T/U $ ratio and multi-scale stability (from femtometer-scale nuclei to kilometer-scale neutron stars) underscores its significance in unified nuclear many-body theories. Systematic investigations of this ratio across nuclear chart regions could unveil universal constraints on hadronic matter under extreme conditions.

The Skyrme Hartree-Fock-Bogoliubov (HFB) model is a versatile microscopic theoretical framework widely used in nuclear physics research, with which nuclear structure at ground state such as binding energies, charge radii, deformation and shape coexistence can be investigated self-consistently [16−20]. In this work, we will perform systematic calculations for 2318 even-even nuclei with the Skyrme HFB model to analyze nuclear stability and the energy components of nuclei.

In the Hartree-Fock-Bogoliubov (HFB) calculations, the Hamiltonian of a system with interacting fermions is written in terms of the annihilation and creation operators $ (c,c^{\dagger}) $ as [16, 17]

in which $ T_{\mu\mu'} = -\dfrac{\hbar^2}{2m} \langle \mu|\nabla^{2}|\mu' \rangle $ and $ \tilde{V}_{\mu\nu\mu'\nu'} = \langle \mu\nu|V(r_{12})|\mu'\nu'-\nu'\mu'\rangle $ are the single-particle kinetic energy matrix elements and the anti-symmetrized two-body interaction matrix elements respectively. In the HFB model, the particle operators are transformed into quasiparticle operators through the Bogoliubov transformation to deal with the pairing correlation

where U and V are the transformation matrices. The quasiparticle vacuum state is defined as the HFB ground state $ |\Phi \rangle $, satisfying the condition $ \alpha_{\mu} |\Phi \rangle = 0 $. By applying Wick's theorem, the expectation value of the Hamiltonian H can be expressed as a function of the hermitian density matrix ρ and the pairing tensor κ [16]

where $ \rho_{\mu\mu^{\prime}}=\langle\Phi|c^{\dagger}_{\mu^{\prime}}c_{\mu}|\Phi\rangle $ and $ \kappa_{\mu\mu^{\prime}}=\langle\Phi|c_{\mu^{\prime }}c_{\mu}|\Phi\rangle $.

In coordinate space, the operators $ c_{{\bf{r}}\sigma q} $ and $ c^{\dagger}_{{\bf{r}}\sigma q} $ refer to annihilate and create nucleons at the point $ {\bf{r}} $ with spin $ \sigma = \pm \dfrac{1}{2} $ and isospin $ q = \pm \dfrac{1}{2} $, which are rewritten with the quasiparticle operators as

The density matrix and the pairing tensor are expressed as

κ can be replaced by the pairing density matrix via $ \tilde{\rho}({\bf{r}} \sigma q,{\bf{r}}' \sigma' q') = -2 \kappa({\bf{r}} \sigma q,{\bf{r}}' -\sigma'q') $ for the convenient to describe time-even quasiparticle states when both ρ and $ \tilde{\rho} $ are hermitian and time-even [16].

With the Skyrme force which is a zero-range and non-local interaction, the HFB energy in Eq.(3) can be expressed as the volume integral of the energy density

$ {\cal{H}}({\bf{r}})={\cal{H}}_{S}({\bf{r}})+{\cal{\tilde{H}}}_{P}({\bf{r}}) + {\cal{H}}_{C}({\bf{r}}) $ includs the energy densities of Skyrme interaction $ {\cal{H}}_{S}({\bf{r}}) $, the pairing $ {\cal{\tilde{H}}}_{P}({\bf{r}}) $ and the Coulomb interaction $ {\cal{H}}_{C}({\bf{r}}) $ for protons. The mean field with Skyrme force $ {\cal{H}}_{S}({\bf{r}}) $ is a funcional of the particle density $ \rho = \sum_{q}\rho_{q} $, the kinetic-energy density $ \tau = \sum_{q}\tau_{q} $ and the spin-current tensor $ {\bf{J}}_{ij} = \sum_{q}{\bf{J}}_{q,ij} $, the index $ q=p,n $ stands for proton and neutron correspondingly:

in which $ \varepsilon_{ijk} $ is the Levi-Civita symbol with $ i,j,k=(1,2,3) $ and $ (x_{0},x_{1},x_{2},x_{3},t_{0},t_{1},t_{2},t_{3},W_{0},\alpha) $ are the parameters of Skyrme force. The pairing energy density is used as

with $ V_{0} $ the pairing strength and $ \rho_{0} $ the saturation density determined by the Skyrme parameters. The Coulomb interaction should be added into the energy densities for the case of proton

The Skyrme HFB equations can be obtained by the variation of the HFB energy in Eq.(6) with respect to ρ and $ \tilde{\rho} $. By referring to literatures [16−18], one can obtain the detailed expressions. Self-consistently solving the Skyrme HFB equations, the ground-state density matrix ρ and the pairing density matrix $ \tilde{\rho} $ are obtained, then the kinetic energy

and the total energy $ E_{\rm tot} $ at the ground state can be obtained. The contribution of the non-local term of the Skyrme interaction to the kinetic energy is incorporated via solving the density matrix under the single-particle potential. The potential energy U is calculated by subtracting the kinetic energy from the total energy

which includes contributions from the mean field, pairing energy, and Coulomb energy. With the predicted kinetic energy and potential energy for a nucleus, the ratio $ T/U $ for the nucleus at its ground state can be simultaneously obtained.

In Skyrme Hartree-Fock-Bogoliubov code HFBTHO (v2.00d) [17], the axially deformed solution of the Skyrme HFB equations can be obtained by using the transformed harmonic oscillator basis [16−18, 21−23]. A series of Universal Nuclear Energy Density Functional (UNEDF) [24] are proposed to provide a more accurate description of the properties of the ground state nuclei. The code HFBTHO (v2.00d) also implements HFB calculations adopting the UNEDF functional. In this work, the UNEDF0 functional, and the standard Skyrme functionals SLy4 [25], SkM* [26] and SIII [27] are adopted. The number of oscillator shells is set as $ N_{\text{max}}^{\text{shells}}=20 $ and the total number of states in the basis is $ N_{\text{states}}=1771 $. The default value $ b_{0}=-1.0 $ fm is taken for the oscillator length, which means that the code automatically sets $ b_{0}=\sqrt{\hbar /m\omega } $ with $ \hbar \omega =1.2\times 41/A^{1/3} $ [17]. The pairing force is assumed to be $ V_{\text{pair}}^{\text{n,p}}=V_{0}^{\text{n,p}}\left ( 1-\frac{1}{2} \frac{\rho \left ( \vec r \right ) }{\rho _{0}} \right ) \delta ( \vec r-\vec{r'} ) $ with $ \rho _{0}=0.16\; \rm{fm}^{-3} $, where a pre-defined pairing strength $ V_{0}^{\text{n,p}} $ is used for each Skyrme force for simple. The default value of the quasi-particle cutoff energy $ E_{\text{cut}}=60.0 $ MeV is adopted.

Based on the Skyrme HFB code HFBTHO (v2.00d) [17], we systematically investigate the ground state properties of even-even nuclei with $ 8\leq Z \leq 130 $. We firstly check the binding energies calculated by adopting different Skyrme parameters set. In Fig. 1, we show the calculated ground state energies $ E_{\rm{tot}} $ for the isobaric chains with $ A=40 $, 100,150,208,228 and 256 by using UNEDF0, SLy4, SkM* and SIII. The experimental data from AME2020 [28] are also presented for comparison. One can see that the data can be reasonably well reproduced for medium mass nuclei with all the four Skyrme forces. For heavy nuclei with $ A > 200 $, the uncertainties of the predictions from these Skyrme forces significantly increase, especially for neutron rich nuclei. We note that the calculated results with UNEDF0 are the best for nuclei shown in Fig. 1. UNEDF0 demonstrates better predictive capabilities in the region of heavy nuclei, possibly due to its refined treatment of isospin dependence. We therefore adopt UNEDF0 in the following analysis of the total energies and ratios of kinetic energy to potential energy $ T/U $.

Figure 1.  (color online) Nuclear ground state energies $ E_{\rm tot} $ for nuclei with $ A=40 $, 100,150,208,228,256. The curves denote the calculated results with UNEDF0, SLy4, SIII and SkM*, respectively. The red circles denote the experimental data [28].

The nucleus with the minimal energy for a certain light and medium isobars could be considered as the most stable nuclide on this isobaric chain as mentioned previously. According to the available data [28], one can see from Fig. 1 that the corresponding known stable nuclei (e.g. $ ^{40} $Ar, $ ^{40} $Ca, $ ^{100} $Ru, $ ^{100} $Mo, $ ^{150} $Nd, $ ^{150} $Sm and $ ^{208} $Pb) are located around the valleys of the ground state energies for the isobaric chains with $ Z\le 82 $. In addition to nuclear ground state energy $ E_{\rm tot} $, we simultaneously investigate the corresponding ratio $ T/U $. In Fig. 2, we show the ground state energy $ E_{\rm tot} $ and the ratio $ T/U $ for isobaric chains with $ A=40 $, 48,208,298, 90 and 120. One can see that the nuclei with minimal energies are generally those with maximal values of $ T/U $, which indicates that the ratio $ T/U $ also has a close relationship with nuclear stability in addition to nuclear binding energy. The values of $ T/U $ essentially reflects the relationship between the average kinetic and potential energies per nucleon, reflecting the relevance of nuclear force strength to the intensity of nucleonic motion. A larger average kinetic energy per nucleon $ T/A $ may indicate the more vigorous nucleon motion. Similarly, a larger absolute value of the average potential energy per nucleon $ |U|/A $ indicates a stronger attractive potential acting on the nucleons. Nuclear stability is therefore significantly influenced by the balance of T and U.

Figure 2.  (color online) Comparison of the ground state energy $ E_{\rm tot} $ and the ratio $ T/U $ for isobaric chains with $ A=40 $, 48,208,298, 90 and 120. The dashed lines are to guide to the eyes.

From Fig. 2, we also note that for the isobars with $ A=40 $ and 48, the maximal values of $ T/U $ are located at the double magic nuclei $ ^{40} $Ca and $ ^{48} $Ca, respectively. Whereas the corresponding values of the minimal $ E_{\rm tot} $ are found at the nuclides $ ^{40} $Ar and $ ^{48} $Ti. To investigate the influence of shell closure on $ E_{\rm tot} $ and $ T/U $, we further analyze the $ T/U $ and $ E_{\rm tot} $ for nuclei around $ ^{208} $Pb. The predicted ground state energy $ E_{\rm tot} $ and the corresponding ratio $ T/U $ for isobaric chains with isobaric chains with $ A=202 $, 204,206 and 210 are shown in Fig. 3. One can see that the peaks of $ T/U $ for these isobars are all located at proton magic number $ Z=82 $. For $ ^{210} $Pb with proton shell closure and $ ^{210} $Po with neutron shell closure, we note that the predicted values of $ T/U $ are very close to each other. It seems that the known magic numbers can be more evidently observed from the ratio $ T/U $, which could be helpful for exploring the magic numbers in super-heavy region.

Figure 3.  (color online) The same as Fig. 2, but for isobaric chains with $ A=202 $, 204,206 and 210.

To further analyze the difference between $ E_{\rm tot} $ and $ T/U $, we systematically investigate the ratios $ T/U $ for even-even nuclei with $ 8\leq Z \leq 130 $. In Fig. 4, we show the predicted nuclei with maximal $ T/U $ on a given isobaric chain from the HFBTHO calculations with UNEDF0. The solid curve denotes the predicted β-stability line according to Green's formula [29], $ Z=A/2[1-0.4A/ (A+200)] $. One can see that in the regions with known stable nuclei, the predicted nuclei with maximal $ T/U $ are generally located around the β-stability line. In addition, the calculated nuclei with minimal total energy $ E_{\rm tot} $ are also presented for comparison. One sees that in super-heavy mass region, the numbers of nuclei obtained from $ T/U $ are much more than those from $ E_{\rm tot} $ at $ N=184 $. Simultaneously, we also note that the numbers of nuclei with $ Z=108 $ and $ Z=118 $ obtained from $ T/U $ are much more than that of the neighboring nuclei.

Figure 4.  (color online) Nuclear β-stability line. The gray squares denote the known even-even stable nuclei. The circles and the red squares denote the calculated even-even nuclei with maximal $ T/U $ and those with minimal $ E_{\rm tot} $, respectively. The solid curve denotes the results of Green's formula.

It is known that the central position of the "island of stability" is closely related to the microscopic shell correction energies of nuclei which are usually estimated by subtracting the smooth macroscopic part from $ E_{\rm tot} $. In this work, we simultaneously study the corresponding microscopic energies of nuclei by combining the modified Bethe-Weizsäcker mass formula [30, 31]

Here, the parameters in Eq.(12) are determined by fitting the calculated ground-state energies of 2138 even-even nuclei with UNEDF0. In Fig. 5, we show the contour plot of the calculated microscopic energies $ E_{\rm tot}-E_{\rm LD} $ for nuclei with $ Z\ge 90 $. In the region $ N>126 $, one can see two islands: one is located around $ N=152-162 $, $ Z=102-108 $, the other is located around $ N=184 $, which is consistent with the predicted ratios $ T/U $. The predicted microscopic energy for the super-heavy nucleus ($ Z=120, N=184 $) is about 7 MeV, which is comparable with the results of macroscopic-microscopic mass models [33, 34].

Figure 5.  (color online) Microscopic energy (in MeV) $ E_{\rm tot}-E_{\rm LD} $ based on the ground state energy from UNEDF0 and the smoothed liquid-drop energy $ E_{\rm LD} $ from the modified Bethe-Weizsäcker mass formula. The dashed lines are to guide to the eyes.

In Fig. 6(a), we show the peak values of $ T/U $ for each isobaric chains as a function of mass number A. One sees that for heavy nuclei, the maximum of $ T/U $ decreases linearly with A in general. The oscillations in the results from HFBTHO+UNEDF0 are mainly due to the microscopic shell effects. For light stable nuclei with $ A<50 $, the contribution of nuclear surface energy is evident from Eq.(12) and the average kinetic energy per nucleon is relatively low, which results in a larger $ T/U $ ratio comparing with the heavy nuclei. From Fig. 6(b), one can see that the average kinetic energy per nucleon significantly increases with the mass number A in the region $ A<150 $. In the heavy and superheavy mass region ($ A>150 $), the average kinetic energy per nucleon generally approaches to a constant due to the saturation of nuclear density. It is known that the Coulomb energy significantly increases with the charge number Z in heavy mass region, which results in the reduction of the total potential energy in absolute value. For stable nuclei in the medium-heavy mass region, the nearly linear dependence of $ T/U $ on the mass number A can be clearly observed in Fig. 6(a), which is mainly due to the competition between the significant increase of the Coulomb energy and the saturation behavior of the nuclear force.

Figure 6.  (color online) (a) Maximum of the ratio $ T/U $ for each isobaric chain as a function of mass number A. (b) The corresponding average kinetic energy per nucleon $ T/A $ (dashed curve) and the average potential energy per nucleon $ U/A $ (solid curve).

The virial theorem states that for an inverse square force field (such as Coulomb potential), the ratio of kinetic to potential energy is $ T/|U| = 0.5 $. According to the HFBTHO calculations, the absolute values of $ T/U $ for bound nuclei are about $ 0.65 \sim 0.75 $, which implies that the short-range character and complexity of nuclear force may lead to a different virial theorem. The linear dependence of $ T/U $ on mass number A for heavy nuclei could be helpful for further exploring the virial theorem of nuclear forces and the equation of state for neutron stars.

In this work, we have investigated nuclear ground state energies and the ratios of the kinetic energy T to the potential energy U by using the Skyrme Hartree-Fock-Bogoliubov (HFB) code HFBTHO. With the Skyrme energy density functional of UNEDF0, the ratios $ T/U $ for 2318 even-even nuclei are systematically calculated. We note that the nuclei with maximal value of $ T/U $ for a certain isobaric chain are generally stable nuclei or long-lived nuclei. We also note that the known magic numbers can be more evidently observed from the ratio $ T/U $ comparing with nuclear binding energy, particularly for the isobaric chains with semi-magic nuclei (with either proton or neutron magic numbers). Combining nuclear binding energies and the values of $ T/U $ from the Skyrme HFB calculations, the magic numbers in super-heavy mass region are simultaneously studied. In the super-heavy mass region, the neutron magic number $ N=184 $ can be clearly observed from the obtained ratios of $ T/U $ and the extracted microscopic energy of nuclei by using UNEDF0. We also find that the numbers of nuclei with $ Z=108 $ and $ Z=118 $ obtained from $ T/U $ are much more than that of the neighboring nuclei, which is helpful for exploring the shell structure of superheavy nuclei. For super-heavy nuclei with $ Z=118 $ and $ N=178-196 $, we note that the calculated deformations of nuclei are spherical or oblate in shapes, whereas the predicted $ ^{292-300} $Fl are all spherical in shape. It is therefore necessary to further investigate the influence of nuclear oblate deformations on the stability of super-heavy nuclei in the following work. The $ T/U $ peak value could be used as a supplementary indicator in selecting potential long-lived superheavy nuclei. In addition, we note that the obtained ratios $ T/U $ almost linearly decrease with the mass number A for heavy nuclei around the β-stability line, due to the competition between the strong Coulomb replusion and the saturation property of nuclear force, which implies that the short-range character and complexity of nuclear force may lead to a different virial theorem.