Change in neutron skin thickness after cluster-decay

For more than a century, the disintegration of the heavy nuclei into more stable lighter ones has been experimentally and theoretically investigated. The study of nuclear decays is fundamental for many crucial problems in modern nuclear physics and astrophysics. Just as the discovery of α-decay [1, 2] was a real beginning of nuclear physics, the discovery of nuclear fission [3, 4] was a landmark. In addition to spontaneous binary fission [3–5], other rare fission modes were observed in the 1940s, such as ternary [6, 7] and quaternary fission [7]. Unlike other decay modes, cluster-decay was theoretically predicted [8] four years before it was confirmed empirically [9]. Practically detected cluster decays included emission of ¹⁴C, ^18,20O, ²³F, ^22,24–26Ne, ^28–30Mg and ^32,34Si clusters, leaving daughter nuclei with Z and N around the doubly magic nucleus ²⁰⁸Pb [10, 11].

The correlations between nuclear structure and the α and heavier cluster decay modes have been confirmed by dozens of studies [12–28]. Accurate knowledge of nuclear density distributions is essential to understand the fundamental properties of nuclear matter and the nature of nuclear force. Nuclear charge distributions of stable isotopes [29–32] and short-lived isotopes [33, 34] were accurately measured by electron scattering. Experimental efforts on the neutron distributions are mainly directed to measure the distribution root-mean-square (rms) radii [35–37] and the neutron-skin thickness [35, 38]. In general, the distributions of protons and neutrons are qualitatively and quantitatively different, and the concept of 'skin thickness' is especially important for this reason because it abstracts the difference between the two distributions as the difference between their rms radii.

In conjunction with the experimental studies, several theoretical approaches have been developed to study the nuclear structure and the density distributions. There are currently two common approaches for calculating the nuclear structure of heavy and superheavy nuclei, namely, the self-consistent Hartree–Fock–Bogolyubov (HFB) approach [39, 40] and the semi-microscopic approach [26, 41–44]. The semi-microscopic approaches show remarkable success in the calculation of nuclear masses, deformation [26, 41–44], and nuclear decays [45–48]. However, the self-consistent models outperform other models in calculating the distributions of protons and neutrons as the sum of single-particle densities, and this justifies density fluctuation in practical charge distributions that cannot be described by common analytical formulae [49, 50].

Numerous studies have indicated the importance of neutron- and proton-skin thickness in many fundamental problems in nuclear physics and astrophysics. Considering the neutron-skin thickness significantly improved the calculations [51–53]. It has also been reported that the change in the neutron- or proton-skin thickness from the parent to daughter nuclei is significantly related to the observed half-lives and determines the relative stability of parent nuclei [54]. To our knowledge, the influence of neutron-skin thickness on the cluster decay has not been investigated yet as with α-decay. In this study, we investigate the change in the thickness of neutron-skin from parent to daughter involved in the cluster decay process. The small number of confirmed cluster-emitters, which do not exceed a few tens of observed cluster decays, may be an advantage for the present study, as these few observations are hallmarks of the cluster-decays of the highest probabilities.

In the present work, we consider the self-consistent HFB calculations to extract the neutron skin thickness, and the preformed cluster approach to deduce the cluster-decay calculation. The studied decays cover the whole set of experimentally observed cluster-decay modes, in addition to investigating the decays of the U isotopic chain in order to understand the difference between the observed and the less likely decays. We organize the paper in four sections. The theoretical approaches used to obtain the neutron-skin thickness and to calculate the half-life against cluster decay are outlined in section 2. Both the structure and the necessary decay calculations are performed using mean-field methods based on the same Skyrme-SLy4 NN interaction. In section 3, we present and discuss the obtained results. Section 4 is devoted for the drawn conclusions.

The calculations in the current study are divided into two main parts. The first part includes the calculation of the nuclear structure to infer the neutron skin thickness, which is done in the framework of Skyrme HFB approach. The second part includes the cluster-decay calculations and it is performed within the cluster model.

In HFB approach, both the mean-field and the pairing field are treated self-consistently in an iterative optimization procedure. With Skyrme forces, the HFB energy has the form of a local energy density functional,

$$\begin{equation}E\left[\rho ,\tilde {\rho }\right]=\int {\mathrm{d}}^{3}\mathbf{r}\mathcal{H}\left(\mathbf{r}\right),\end{equation} \tag{ 1 }$$

where the energy density $\mathcal{H}\left(\mathbf{r}\right)$ is the sum of the mean-field and pairing energy densities,

$$\begin{equation}\mathcal{H}\left(\mathbf{r}\right)=H\left(\mathbf{r}\right)+\tilde {H}\left(\mathbf{r}\right)\hspace{-1pt}.\end{equation} \tag{ 2 }$$

The mean-field energy density H(r) is a function of the nucleon densities ρ_q (r), the kinetic energy densities τ_q (r) and the spin densities J_q (r) [55], where q = n or p labels the neutron or proton densities, respectively. The total matter density is ρ = ρ_n + ρ_p and the total kinetic density is τ = τ_n + τ_p. H(r) includes the kinetic energy, nuclear interaction and Coulomb interaction energy parts,

$$\begin{equation}H\left(\mathbf{r},{\rho }_{q},{J}_{q}\right)=\frac{{\hslash }^{2}}{2m}\tau \left(\mathbf{r},{\rho }_{q}\right)+{H}_{\mathrm{S}\mathrm{k}}\left(\mathbf{r},{\rho }_{q},{J}_{q}\right)+{H}_{\mathrm{C}}\left(\mathbf{r}\right),\end{equation} \tag{ 3 }$$

where ${H}_{\mathrm{C}}\left(\mathbf{r}\right)$ is the Coulomb interaction, including direct and exchange parts [55]. The nuclear interaction is considered in the framework of Skyrme effective interaction ${H}_{\mathrm{S}\mathrm{k}}\left(\mathbf{r}\right)$ including several terms, namely, the zero-range, finite range, density dependent, effective mass, spin–orbit terms, in addition to tensor coupling with spin and gradient [56]. The pairing energy density $\tilde {H}\left(\mathbf{r}\right)$ is given by,

$$\begin{equation}\tilde {H}\left(\mathbf{r}\right)=\frac{1}{2}{V}_{0}\left[1-{V}_{1}{\left(\frac{\rho }{{\rho }_{0}}\right)}^{\gamma }\right]\sum _{q}{\tilde {\rho }}_{q}^{2}.\end{equation} \tag{ 4 }$$

The pairing strength (V₀) and saturation density (ρ₀) in addition to the parameters V₁ and γ depend on the Skyrme force [57, 58]. For the SLy4 force, considered in the present study, V₀ = −244.72 MeV fm³, ρ₀ = 0.16 fm⁻³, V₁ = 0.5, and γ = 1.0. Taking the derivatives of the energy $E\left[\rho ,\tilde {\rho }\right]$ with respect to the local particle density ρ(r) and the local pairing density $\tilde {\rho }\left(\mathbf{r}\right)$ gives the self-consistent particle field (h) and pairing field ($\tilde {h}$), respectively. In the framework of independent quasiparticle approach, the Skyrme HFB equation can be written as [59]:

$$\begin{equation}\left(\begin{matrix}\hfill h-\lambda \hfill & \hfill \tilde {h}\hfill \\ \hfill \tilde {h}\hfill & \hfill -h+\lambda \hfill \end{matrix}\right)\left(\begin{matrix}\hfill {\varphi }_{1,k}\hfill \\ \hfill {\varphi }_{2,k}\hfill \end{matrix}\right)={E}_{k}\left(\begin{matrix}\hfill {\varphi }_{1,k}\hfill \\ \hfill {\varphi }_{2,k}\hfill \end{matrix}\right),\end{equation} \tag{ 5 }$$

where E_k is the kth state quasiparticle energy, φ_1,k and φ_2,k are the upper and the lower components of the quasiparticle wavefunction. Since the Bogoliubov transformation does not conserve the number of particles, Lagrange multiplier λ is introduced to guarantee that the average particle number is equal to the actual number of particles in the system. The self-consistent solution of Skyrme HFB in transformed-harmonic-oscillator (THO) basis [40] generates an orthogonal set of quasiparticle wavefunctions, from which one obtains the particle and pairing density matrices as,

$$\begin{equation}\rho \left(\mathbf{r}\sigma ,{\mathbf{r}}^{\prime }{\sigma }^{\prime }\right)=\sum {\varphi }_{2,k}\left(\mathbf{r}\sigma \right){\varphi }_{2,k}^{{\ast}}\left({\mathbf{r}}^{\prime }{\sigma }^{\prime }\right),\end{equation} \tag{ 6 }$$

$$\begin{equation}\tilde {\rho }\left(\mathbf{r}\sigma ,{\mathbf{r}}^{\prime }{\sigma }^{\prime }\right)=-\sum {\varphi }_{2,k}\left(\mathbf{r}\sigma \right){\varphi }_{1,k}^{{\ast}}\left({\mathbf{r}}^{\prime }{\sigma }^{\prime }\right).\end{equation} \tag{ 7 }$$

The local particle density can be determined from the scalar part of the density matrix at which r = r' [60] as,

$$\begin{equation}\rho \left(\mathbf{r}\right)=\rho \left(\mathbf{r},\mathbf{r}\right)=\sum _{\sigma }\rho \left(\mathbf{r}\sigma ,\mathbf{r}\sigma \right)\hspace{-1pt}.\end{equation} \tag{ 8 }$$

The proton and neutron blocks are separately considered in the calculation. The resulting rms radii of neutron (R_n) and proton (R_p) distributions are then used to obtain the neutron-skin thickness (Δ_n),

$$\begin{equation}{{\Delta}}_{\mathrm{n}}={R}_{\mathrm{n}}-{R}_{\mathrm{p}},\end{equation} \tag{ 9 }$$

where

$$\begin{equation}{R}_{q}={\left[\frac{\int {r}^{2}{\rho }_{q}\left(\mathbf{r}\right)\mathrm{d}\mathbf{r}}{\int {\rho }_{q}\left(\mathbf{r}\right)\mathrm{d}\mathbf{r}}\right]}^{1/2},\enspace q=\mathrm{n},\mathrm{p}.\end{equation} \tag{ 10 }$$

The change of the neutron-skin thickness (δ_n) after the cluster decays is determined as,

$$\begin{equation}{\delta }_{\mathrm{n}}={\left({{\Delta}}_{\mathrm{n}}\right)}_{\text{daughter}}-{\left({{\Delta}}_{\mathrm{n}}\right)}_{\text{parent}}.\end{equation} \tag{ 11 }$$

The Skyrme HFB calculations can be performed using THO basis with zero-range pairing interaction [40]. The approximate particle number projection can be also considered by the Lipkin–Nogami (LN) method in the pairing interaction approximation [40], where the particle-number symmetry is approximately restored then it followed by an accurate particle number projection after the variation (THO + LN). The Skyrme–Hartree–Fock calculations can be also performed with schematic pairing force within the Bardeen–Cooper–Schrieffer (BCS) approximation (HF + BCS) [61]. It is worth noting that the LN method performs better than the BCS method when the neutron–proton pairing interaction is considered [62], and below the critical value of the pairing force at which BCS approach fails [63].

In the cluster model [64, 65] based on the Gamow two-step mechanism of α-decay, the light nucleus emitted in a cluster decay is assumed to be formed at first as a distinguished entity, within the surface region of the parent nucleus [66], with a certain preformation probability (S_c). Then the formed cluster periodically knocks against the produced cluster-core Coulomb barrier, with a particular frequency, in a trial to tunnel through it. Eventually, it penetrates the barrier with a specific penetrability. The knocking frequency (ν) and the penetration probability (P) can be calculated using the semi-classical Wentzel–Kramers–Brillouin (WKB) method as,

$$\begin{equation}\nu ={\left[{\int }_{{R}_{1}}^{{R}_{2}}\frac{2\mu }{\hslash k\left(r\right)}\mathrm{d}r\right]}^{-1},\end{equation} \tag{ 12 }$$

and

$$\begin{equation}P=\mathrm{exp}\left(-2{\int }_{{R}_{2}}^{{R}_{3}}k\left(r\right)\mathrm{d}r\right),\end{equation} \tag{ 13 }$$

with the wave number,

$$\begin{equation}k\left(r\right)=\sqrt{\frac{2\mu }{{\hslash }^{2}}\left\vert {V}_{\mathrm{T}}\left(r\right)-{Q}_{\mathrm{c}}\right\vert }.\end{equation} \tag{ 14 }$$

While $\mu =\frac{{m}_{\mathrm{c}}{m}_{\mathrm{D}}}{{m}_{\mathrm{c}}+{m}_{\mathrm{D}}}$ defines the reduced mass of the cluster (m_c)-daughter (m_D) system, the Q_c-value represents the energy released in the spontaneous cluster decay, in MeV. The three classical turning points R_i=1,2,3 (fm) are defined along the radial path of the cluster-core system at which ${V}_{\mathrm{T}}\left(r={R}_{i=1,2,3}\right)={Q}_{\mathrm{c}}$. The total interaction potential ${V}_{\mathrm{T}}\left(r\right)$ between the emitted cluster and the core daughter nucleus, at a radial separation distance r, includes nuclear (V_N), Coulomb (V_C), and centrifugal (V_ℓ ) contributions,

$$\begin{equation}{V}_{\mathrm{T}}\left(r\right)={\lambda V}_{\mathrm{N}}\left(r\right)+{V}_{\mathrm{C}}\left(r\right)+{V}_{\ell }\left(r\right).\end{equation} \tag{ 15 }$$

The nuclear part of the potential is usually normalized by the factor λ to guarantee a quasi-stationary state for the binary cluster-cores system [67, 68]. One can determine a suitable value of λ using the Bohr–Sommerfeld quantization conditions [69, 70],

$$\begin{equation}{\int }_{{R}_{1}}^{{R}_{2}}k\left(r\right)\mathrm{d}r=\left(2n+1\right)\frac{\pi }{2}.\end{equation} \tag{ 16 }$$

The quantum number n represents the number of nodes of the internal part of the quasibound radial wave function for the cluster-core system [21, 71]. In the present work, the nuclear part of the interaction potential is evaluated using the Hamiltonian energy density approach [28, 72–74] based on the Skyrme energy functional with the Skyrme-SLy4 parameterization of the nucleon–nucleon force [56].

$$\begin{align}\hfill {V}_{\mathrm{N}}\left(r\right)& =\int \left\{H\left[{\rho }_{\mathrm{p}\mathrm{c}}\left( \overrightarrow {x}\right)+{\rho }_{\mathrm{p}\mathrm{D}}\left(r, \overrightarrow {x}\right),{\rho }_{\mathrm{n}\mathrm{c}}\left( \overrightarrow {x}\right)+{\rho }_{\mathrm{n}\mathrm{D}}\left(r, \overrightarrow {x}\right)\right]\right.\hfill \\ \hfill & \quad \left.-{H}_{\mathrm{c}}\left[{\rho }_{\mathrm{p}\mathrm{c}}\left( \overrightarrow {x}\right),{\rho }_{\mathrm{n}\mathrm{c}}\left( \overrightarrow {x}\right)\right]-{H}_{\mathrm{D}}\left[{\rho }_{\mathrm{p}\mathrm{D}}\left( \overrightarrow {x}\right),{\rho }_{\mathrm{n}\mathrm{D}}\left( \overrightarrow {x}\right)\right]\right\}\mathrm{d} \overrightarrow {x}.\hfill \end{align} \tag{ 17 }$$

Here, H, H_c and H_D define the energy density functionals, given by equation (3) without the Coulomb part, for the compound system, emitted cluster (c), and daughter nucleus (D), respectively. The same Skyrme-SLy4 NN interaction has been used to compute the neutron-skin thickness of the involved parent and daughter nuclei. The Coulomb potential (${V}_{\mathrm{C}}\left(r\right)$) in equation (15) is calculated by the double-folding model using the multipole expansion method [75, 76], in terms of the charge densities of the involved nuclei and using. The centrifugal part of the potential is calculated in terms of the minimum orbital angular momentum ℓℏ carried by the emitted cluster to conserve the angular momentum and parity in the decay process, ${V}_{\ell }\left(r\right)=\ell \left(\ell +1\right){\hslash }^{2}/2\mu {r}^{2}$. More details regarding the method of calculating the different parts of the total potential are outlined in references [28, 73, 75].

The partial half-life of the parent nucleus against the cluster decay can be then determined in terms of the decay width (νP) of the decay process and the cluster preformation factor S_c as,

$$\begin{equation}{T}_{\mathrm{c}}=\frac{\mathrm{ln}\enspace 2}{{S}_{\mathrm{c}}\nu P}.\end{equation} \tag{ 18 }$$

One can estimate the preformation factor S_c(A_c) of a specific cluster of mass number A_c inside the parent nucleus in terms of the α-preformation factor S_α inside it [78, 79] by,

$$\begin{equation}{S}_{\mathrm{c}}={\left({S}_{\alpha }\right)}^{\frac{{A}_{\mathrm{c}}-1}{3}}.\end{equation} \tag{ 19 }$$

The α-preformation factor can be extracted from the observed half-life against α-decay (${T}_{\alpha }^{\mathrm{exp}}$) and the calculated α-decay width (ν_α P_α ), ${S}_{\alpha }=\mathrm{ln}\enspace 2/\left({T}_{\alpha }^{\mathrm{exp}}{\nu }_{\alpha }{P}_{\alpha }\right)$ [17, 80]. For the parent nuclei which have no observed α-decay, accurate empirical formulas can be used to find the α-preformation factor [77].

Most of the known U isotopes have shown observed α-decay modes. Only ^{237,239,241–243}U did not show α-decays yet. Figure 1(a) shows the Q-values of the probable ²⁸Mg and ^24,26Ne decay modes of the different ^215–243U isotopes. These nuclei are the most particles heavier than α-particle that spontaneously observed from the U isotopes. Five cluster decay modes of ²⁸Mg and four modes of ²⁴Ne were observed from ^232–236U and ^{232–234,236}U isotopes, respectively [10]. Also, two ²⁶Ne-decay modes were experimentally indicated for ^234,236U [10, 81]. Other singular decays of ^20,22,25Ne and ³⁰Mg were observed from ^230,235,236U [10].The Q-values presented in figure 1(a) are determined from the masses of the nuclei [82] involved in each decay. The solid symbols in figure 1(a) represent the isotopes of observed ²⁸Mg and ^24,26Ne cluster decays. The minimum Q-values for observed ²⁸Mg and ^24,26Ne of U isotopes are those of ²³⁶U(²⁸Mg, Q = 70.733 MeV), ²³⁶U(²⁴Ne, 55.945 MeV) and ²³⁶U(²⁶Ne, 56.692 MeV). However, as seen in figure 1(a), the ^222–231U(²⁸Mg, ²⁴Ne), ^231–233U(²⁶Ne) and ²³⁵U(^24,26Ne) decays show larger Q-values than the mentioned minimum Q-values. Then, those ²⁸Mg and ^24,26Ne decays are expected to be energetically favorable than the ²³⁶U(²⁸Mg, ^24,26Ne) decays. Figure 1(b) confirms the preference of these decay modes over some of the observed half-lives against cluster decays. In figure 1(b) we display the calculated partial half-lives of the ^{220–238,240}U isotopes against their probable ²⁸Mg and ^24,26Ne cluster decays. The corresponding calculated half-lives of the other ^{215–219,239,241–243}U isotopes presented in figure 1(a) are extremely large (>10³³ years). The calculations presented in figure 1(b) are carried out in the framework of the preformed cluster model based on the Skyrme-SLy4 NN interaction, equations (12)–(19). The cross symbols in figure 1(b) represent the experimentally observed partial half-lives. Figure 1(b) shows that the ²⁸Mg decay modes of the three ^226,228,230U isotopes exhibit estimated partial half-lives shorter than the observed half-life of the ²⁸Mg decay of ²³³U(²⁸Mg, 3.9 × 10²⁷ s), which is the largest half-life of an observed ²⁸Mg decay of U isotopes. Also, the ²⁴Ne and ²⁶Ne decays of the ^{223,226,228–231}U and ^228–233U isotopes, respectively show estimated half-lives shorter than the observed half-life of the ²⁴Ne decay of ²³⁶U(²⁴Ne, >1.9 × 10²⁶ s) and the ²⁶Ne decay of ²³⁴U(²⁶Ne, 8.6 × 10²⁵ s), respectively, which are the largest confirmed half-lives of ^24,26Ne decays of U isotopes.

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Figures 1(a) and (b) highlight the predominant role of the shell effect in the cluster decay phenomena, which would explain the non-detection of ²⁸Mg and/or ²⁴Ne cluster decay modes of ^226,228,230U and ^{223,226,228–231}U isotopes, respectively. As shown in figures 1(a) and (b), the observed ²⁸Mg and ²⁴Ne cluster decay modes of the U isotopes start with ²³²U, leading to the neutron magic ²⁰⁴Hg (N = 124) and the doubly magic ²⁰⁸Pb (Z = 82, N = 126) daughter nuclei. Also, the observed ²⁶Ne decays start with that of ²³⁴U, leading to the doubly magic nucleus ²⁰⁸Pb. Generally, among the observed 34 cluster decays [10], there are 17, 8 and 4 decays modes produce ^208–212Pb (N = 126–130), ^206–208Hg (N = 126–128) and ²⁰⁷Tl (N = 126) daughter nuclei, respectively. Five single decays were observed with ²¹¹Bi (N = 128), ^207,215Pb (N = 125, 133) and ^204,205Hg (N = 124, 125) daughter nuclei. Except three, these daughter nuclei have neutron numbers at or beyond the closed shell N = 126. The appearance of the three daughter nuclei with N = 124 and 125 can be understood in view of arrangement of the neutron levels just below N = 126 with the ν3p_1/2 orbital at the top, followed by the ν2f_5/2, ν3p_3/2 and ν1i_13/2 orbitals [83]. The last one or two neutrons forming the emitted cluster in the ^232,233U(²⁸Mg)^204,205Hg and ²²¹Ra(¹⁴C)²⁰⁷Pb decays are then coming from the ν3p_1/2 state.

Now, we turn to the change (δ_n) of the neutron-skin thickness (Δ_n) after the cluster decays. The neutron-skin thicknesses of the involved nuclei are obtained using HFB calculations based on the Skyrme-SLy4 NN interaction and the transformed harmonic oscillator states along with the LN scheme (THO + LN) to consider the pairing treatment [40], as discussed in section 2. In principle, it is useful to employ the same Skyrme-SLy4 NN interaction to perform both the structure and half-life calculations for the investigated nuclei on an equal footing. The Skyrme-SLy4 NN interaction effectively estimate the nuclear matter and nuclear structure properties, including the mass, binding energy, the proton- and neutron skin thicknesses, and the other isospin properties from the β-stability line to the drip lines [56, 84–90]. It is also successfully used to study the α and cluster decays [15, 26, 77, 91–94] of heavy nuclei, as well as their reactions [74, 95–99]. Figure 2 shows the change in the neutron skin thickness (equation (11)) and in the isospin asymmetry (δI = I_D − I_P), from the parent to daughter nuclei, for the investigated cluster decay modes presented in figures 1(a) and (b). The isospin asymmetry coefficient is defined as I = (N − Z)/A. For the examined cluster decays of ^215,231U, the δ_n change is very small (δ_n ⩽ 0.021 fm) and the corresponding change in the isospin asymmetry is less than it for the observed cluster decays. As we mentioned above, these decays are of less probability upon the dominant influences of the shell effect and/or the Q-values. For the ^232–243U isotopes, δ_n starts with small values, δ_n(²³²U(²⁸Mg)) ≈ 0.020 fm, δ_n(²³²U(²⁴Ne)) ≈ 0.017 fm, and δ_n(²³³U(²⁶Ne)) ≈ 0.005 fm, and it increases with increasing the mass number, up to δ_n ≈ 0.047 fm for the decay modes of ²⁴³U. For the experimentally observed decays, the maximum estimated change in Δ_n is δ_n(²³⁶U(²⁸Mg)) = 0.028 ± 0.002 fm. The associated change in the isospin asymmetry (δI) steadily increases with increasing the mass number of the parent nucleus. The observed cluster decays often yield daughter nuclei of larger isospin asymmetry than their parent nuclei (0.001 ⩽ δI ⩽ 0.012). Only the three observed ²⁶Ne decays of ²³²Th (δI = −0.0008) and ^234,236U (−0.0013, −0.0021) and the ²³F decay of ²³¹Pa (−0.0006) yield daughter nuclei of slightly less isospin asymmetry.

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To provide an estimate of the uncertainty in the calculated neutron-skin thickness, either related to the used structure model and/or the used NN interaction, we compare in the first panel of figure 2 the calculated δ_n due to the ²⁸Mg decays of the U isotopes based on the mean field calculations in terms of the SLy4 [56], SLy230a [84], and SLy5 [56] parameterizations of the Skyrme NN interaction. Moreover, different schemes of the mean field method have been used to calculate δ_n(²⁸Mg), namely the THO, THO + LN, and HF + BCS schemes, mentioned in section 2. As shown in figure 2, the different methods and interactions considered in calculating the change in the neutron-skin thickness give the same characteristic behavior of δ_n as a function of A_P, with the same minima and maxima. The different considered schemes yield a systematic small uncertainty (<0.006 fm) in the calculated δ_n. This uncertainty reduces around the obtained minimal value of δ_n(²⁸Mg) to be less than 0.002. Figure 2 shows that the observed ^24,26Ne and ²⁸Mg decays of U isotopes correspond to the minimum visual spacing between the increase of neutron skin thickness and the change in the isospin asymmetry. This indicates that the cluster decay most likely proceeds to minimize the increase in the neutron skin thickness relative to the corresponding increase (slight decreases) of the isospin asymmetry. Although the estimated decays of some of the U isotopes lighter than ²³²U exhibit change in the neutron skin thickness and Q-values that are comparable to those obtained for experimentally observed decays, they are not preferred due to the controlling role of the shell effect.Examples are the ^222–231U (²⁸Mg), ^222–231U (²⁴Ne) and ^231–233U (²⁶Ne) decays displayed in figure 1, which produce the ^194–203Hg (N = 114–123), ^198–207Pb (N = 116–125) and ^205–207Pb (N = 123–125) daughter nuclei, respectively. The supposed daughter nuclei in these decays will have one or a few neutron holes at the orbits below N = 126, instead of a few valence neutrons outside the closed shell of N = 126 in their parent nuclei, which suppress the likelihood of such decays.

The change in the neutron-skin thickness after the more probable cluster decay modes of the ²³⁶U nucleus is displayed in figure 3(a) as a function of the atomic number of the emitted cluster (Z_c), and as a function of the Q_c-value in figure 3(b). The decay modes shown in figure 3(a) are picked out upon determining the maximum Q_c-value for each Z_c. Also, we added in figure 3 the other probable cluster decays (Q_c > Q_α ) of Z_c = 8, 10, 12, 16 and 20, for which the emitted cluster is lighter than the indicated most probable A_c(Z_c).The probability of emitting clusters heavier than the most probable A_c(Z_c), with a less Q_c-value, strongly decreases with increasing A_c. The experimentally observed decays among the more probable decay modes presented in figure 3 are marked by black solid symbols. δ_n(Q_Max(Z_c = 2, 10, 12)) appear in figure 3 as black-red solid symbols because they correspond to Q_Max(Z_c) and they are also experimentally observed. Figures 3(a) and (b) show that the more probable cluster decay modes tend to exhibit less changes in the neutron-skin thickness compared with the less probable cluster decay modes,which have the same Z_c but smaller A_c. All the more probable cluster decay modes indicated in figure 3 show δ_n ⩽ 0.035 fm, even with the wide range of the corresponding Q_c-value, which extends from 9 to 117 MeV.

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Figures 4(a) and (b) show the change in the neutron-skin thickness after emission for the whole set of cluster decay modes observed to date, and after their corresponding α decays. δ_n is displayed versus the mass number of the parent nucleus in figure 4(a) and versus the change in the isospin asymmetry from parent to daughter nuclei in figure 3(b). In addition to their dedicated symbols, the decays are further classified in figures 4(a) and (b) in a color code according to the neutron number N of the emitted cluster. In addition to the set of α decays (N = 2), the five cluster decay sets distinguished in figure 4(a) include the emitted cluster isotones of N = 8 and 10(¹⁴C, ¹⁸O, ²⁰Ne) in blue, N = 12(²⁰O, ²²Ne) in brown, N = 14(²³F, ²⁴Ne) in red, N = 15 and 16 (^25,26Ne, ²⁸Mg) in black, as well as N = 18 and 20(³⁰Mg, ³⁴Si) in green. Figure 4(a) shows that δ_n tends to increase with decreasing the neutron number of the emitted heavy cluster and with increasing the mass number of the parent nucleus. On contrary, α decays yield the minimum change in Δ_n, and remains almost constant with increasing A_P. This is because the change in the isospin asymmetry after α decays is almost constant (δI_α = 0.003–0.004) for the all observed cluster emitters, as shown in figure 4(b). Except the slightly larger value of δ_n[²³⁵U(²⁰Ne)²¹⁵Pb, δI = 0.020] = 0.055 fm, the change in Δ_n after the all observed cluster decays lies in the narrow range of the small values δ_n(δI ⩽ 0.012) = 0.007–0.033 fm. The change in Δ_n after the corresponding α decays lies within the range of δ_nα (δI_α = 0.003–0.004) = 0.009 ± 0.002 fm. Figure 4(b) clearly shows the increasing behavior of δ_n with increasing the isospin asymmetry of daughter nucleus relative to the parent nucleus. The decays of the clusters which have large isospin-asymmetry such as ²⁰O, ²³F and ²⁶Ne trivially increase the neutron-skin thickness of the daughter nucleus relative to that of the parent nucleus. Heavier clusters of a larger isospin-asymmetry would decrease the neutron-skin thickness.

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Figure 4(c) displays the observed partial half-lives (T_c) of the ¹⁴C, ²⁴Ne, ²⁸Mg and ³⁴Si cluster decays, as function of the corresponding change in Δ_n due to the decay. One can clearly see in figure 4(c) that the half-lives relatively increase for the decays yielding larger change in the neutron-skin thickness. This indicates that the cluster decay becomes more favorable to the daughter nuclei of less increase in Δ_n, relative to their parent nuclei. The nucleus then becomes more stable against the cluster decays that produce a relatively large increase in the neutron-skin thickness.

We studied the change in the neutron-skin thickness from parent to daughter nuclei, after cluster decays. To investigate the key factors influencing the cluster decays, we compared the more-probable decays which manifest themselves energetically with those already observed. We started by investigating the probable ²⁸Mg and ^24,26Ne decay modes of the all known ^215–243U isotopes. These light nuclei are the most particles heavier than α-particle that spontaneously observed from the U isotopes. We then investigated the more probable cluster decays that theoretically estimated for the ²³⁶U isotope, which shows 4 observed cluster decay modes in addition to its α-decay. All the cluster decay modes that have been observed to date are investigated. We found that this change in Δ_n is one of the key factors affecting the cluster decay process, after the shell effect and the released energy. The cluster decay most likely takes place along with the minimum possible increase of the neutron-skin thickness (δ_n) from parent to daughter nuclei. Generally, the more probable cluster decays of a certain cluster emitter, based on the released energy and the mass number of the emitted cluster, tend to show smaller change in the neutron-skin thickness compared with the less probable cluster decays of the same Z_c. This correlation between increasing the decay rate and the small change in the neutron-skin thickness is due to that they are both arising from the same nuclear-structure influences, which affect the probability of forming the emitted cluster within the surface region of the parent nucleus and its penetrability. For the emitted light clusters heavier than α-particle, Δ_n increases upon decreasing the neutron number of the emitted cluster, and consequently it decreases with increasing its isospin-asymmetry. Also, δ_n shows an increasing behavior with increasing the isospin asymmetry of daughter nucleus relative to the parent. The relative stability of a given radioactive nucleus and its partial half-life against a certain cluster decay increase if this decay produces a relatively large increase in the neutron-skin thickness.