Preamble

Effect of spin-orbit interaction on structure and two-proton radioactivity of Yanzhao Wang( Jie Li( , Wenhao Zhang( , Yueqing Li( , and Jianzhong Gu( 1 Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China Institute of Applied Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China Hebei Research Center of the Basic Discipline Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China Hebei Key Laboratory of Physics and Energy Technology, North China Electric Power University, Baoding 071000, China China Institute of Atomic Energy, P. O. Box 275 (10), Beijing 102413, China School of Physics, Xi’an jiaotong University, Xi’an 710049, China (Dated: November 10, 2025) The influence of spin orbit coupling strength ( ) on the structure and two-proton (2 ) radioac- tivity of Mg is investigated using the spherical Skyrme Hartree Bogoliubov (SHFB) approach with the SLy4 interaction and a mean-field cluster potential model.

Our calculations reveal that increasing enhances the splitting of the single-proton 1 orbitals.

Concurrently, the 2 proton state evolves from a weakly bound state into a resonance in the continuum. increases, the occupation probability of the 2 proton state decreases, and its density distribution near the nuclear surface becomes less diffuse.

Furthermore, both the spectroscopic factor and the decay energy for 2 radioactivity gradually decrease with increasing resulting in a longer half-life. When is held constant, the half-life is significantly enhanced by . At the mean time, it is found that the depth of the diproton cluster potential well increases , while the corresponding becomes smaller, indicating that the diproton cluster is considerably looser than the -cluster. Additionally, a clear linear correlation is observed between , as well as between log . The logarithmic half-lives, both with and without inclusion of , exhibit good linear relationships with , respectively. Finally, using the experimental value of Mg, the optimal is determined to be 1.15 MeV fm

Keywords

radioactivity; spin-orbit coupling strength; SHFB theory; mean-field cluster potential approach

INTRODUCTION

The 2 radioactivity is a rare decay mode observed in nuclei near the proton drip line, which was predicted by Zel’dovich and Goldansky in the 1960s [1–3]. Owing to the pairing correlation effect, the direct (true) 2 radioactivity occurs in even- nuclei. Because the one-proton emission is energetically forbidden while the 2 radioactivity is allowed [4], the true 2 radioactivity was first observed from the ground state of Fe at the beginning of the 21st century [5, 6].

Later, more events of the 2 radioactivity from the ground states were discovered from Ni [7], Zn [8], Kr [9] and Mg [10]. Moreover, the 2 emission may occur from the excited states populated in decay, which was usually named as the -delayed 2 radioactivity [11–20]. Besides this, some 2 emitters, such as O [21], Ne [22–26], Mg [27, 28] and S [29, 30], were discovered from the excited states fed by nuclear reactions. In addition, the 2 emission from the 21 isomer state of Ag was reported by the group of Mukha [31].

To understand the mechanism of the 2 radioactivity, quite a few microscopic and phenomenological models have been developed [32–63]. Generally, the 2 radioactivity mechanism includes the following three types: (i) Strongly correlated emission. This is due to the attraction of the two protons, also called the diproton emission. So, the emission undergoes through the penetration of a preformed He cluster.

The emitted two protons have a small angular distribution and share the same energy [32–48]. (ii) Three-body simultaneous emission. The simultaneously emitted two protons distribute isotropically but usually have similar energies [45–63]. (iii) Two-body sequential emission. A parent nucleus first emits a proton and decays to an intermediate state, which itself then emits another proton to the final state [45–63].

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A few years ago, the four proton radioactivity was discovered from the ground state of Mg [64].

In fact, it can be seen as two sequential 2 emissions through the intermediate nucleus To understand its 2 decay mechanism, the structure of Mg was investigated using a three-body Gamow coupled-channel method [65]. The results showed that the ground state of Mg is significantly influenced by the continuum, resulting in a significant -wave component. Recently, we explored the tensor force effect on the 2 radioactivity of Mg in the framework of the spherical SHFB theory and mean-field cluster potential approach [66, 67]. It showed that a 2 halo-like structure appears in Mg and its spectroscopic factor becomes larger with the increase of the 2 halo-like size caused by the tensor force. Furthermore, the decay mode prefers to the He emission [66, 67]. Although the 2 emission of has been investigated by a series of models, its mechanism has not yet been fully understood [48, 65–67]. Therefore, investigating the 2 emission mechanism of Mg remains a topic of significant interest.

The spin-orbit force is a key ingredient in the nucleon-nucleon interaction, which plays a crucial role in nuclear structure and radioactivity [68–71]. For the nuclei around the drip-line, the spin-orbit coupling strength is not only a core parameter in nuclear models but also a key to understanding exotic nuclear structures, decay mechanisms and shell evolution [68–71]. However, its magnitude has not yet been precisely determined. Therefore, it is essential to investigate the influence of on nuclear structure and decay processes. Very recently, we explored the effect of on the -decay of Po within the spherical SHFB theory. It was found that the localization of -cluster and -decay energy are enhanced obviously by the rise of [72]. This drives us to think whether the 2 radioactivity is also influenced substantially by . Moreover, the mechanism how the spin-orbit interaction affects the 2 radioactivity has not been understood clearly. And the observables of 2 emission can be used to constrain the magnitude of Driven by the above mentioned motivation, in this article we will explore how the structure and 2 emission of Mg are influenced by the spin-orbit interaction within the spherical SHFB theory and mean-field cluster potential approach. This article is organized as follows. In Sec. II, the theoretical framework is presented. Sec. III shows the calculated results and discussions. And some conclusions are drawn in the last section.

THEORETICAL FRAMEWORK SHFB theory It is well known that the SHFB theory is a powerful tool to describe the structure of nuclei [73–76]. It can provide a unified and self-consistent description of both the mean-field and the pairing field in terms of the Bogoliubov quasi-particles. A series of properties of nuclei are described successfully with the SHFB approach.

The Skyrme interaction Skyrme is written as [73–76]

V Skyrme = t 0 (1 + x 0 P σ ) δ ( r 1 − r 2 )

where =1,2,3) and are the parameters of the interaction, is the spin-exchange operator, and =1,2) are the Pauli matrices. The operator acts on the right and on the left.

In the particle-particle (pairing) channel, a density-dependent interaction is usually employed, whose form is [74] where corresponds to the mixed pairing force. is the pairing strength parameter. Usually, its value is determined by the empirical pairing energy gap [77–79]. is the isoscalar local density.

In the framework of the SHFB approach, the expressions of the effective mass, abnormal effective mass, particle-hole field, particle-particle field and Coulomb field can be found from Refs. [73–76]. As to the spin-orbit coupling fields, whose forms are expressed as

B q = − 1 8 ( t 1 x 1 + t 2 x 2 ) J + 1

� B q = [ t ′ 2 2 (1 + x ′ 2 ) + W ′ ] � J q . (4)

In Eqs. (3-4), are the particle and spin-current vector densities. is the corresponding abnormal spin- current vector density in the particle-particle channel. For the SLy4 interaction, equal to zero. The subscript stands for neutrons (protons).

The bulk and microscopic properties can be obtained by solving the following SHFB equation in the coordinate representation

) = E ( u 1 u 2

where is the first order derivative with respect to radial coordinate Furthermore, the 2 decay energy ( ) is calculated by where ) and ) are the binding energies of a parent nucleus and its daughter nucleus, respectively. is the 2 separation energy.

Mean-field cluster potential approach with Skyrme interaction For the 2 radioactivity, the angular momentum carried by the He-cluster is zero. So, the centrifugal potential vanishes. Based on the SHFB theory, the di-proton cluster potential can be decomposed as [80, 81]

V ( r ) = V N ( r ) + V C ( r ) , (7)

where ) and ) are the nuclear and Coulomb potentials, respectively, between the di-proton cluster and the core. ) and ) are expressed approximately by

V N ( r ) = λ [ U p ( r ) + B p ( r )] , (8)

V C ( r ) = 2 U c ( r ) , (9)

where is a folding factor; ) and ) refer to the single-proton potential without the spin-orbit potential, the proton spin-orbit potential and the single-proton Coulomb potential generated by the core, respectively.

These expressions can be found from Refs. [73–76].

For the folding factor , it can be determined by the following Bohr-Sommerfeld quantization condition

2 µ ℏ 2 | Q 2 p − V 2 p ( r ) | = (2 n + 1) π

2 = ( G + 1) π

where ( and later) is the classical inner (and outer) turning point obtained by is the reduced mass for the system of the cluster and daughter nucleus, and is the node number of the radial wave function of the cluster motion within the potential. A quasibound state of the cluster orbiting the core is characterized by a global quantum number , whose value is estimated by the Wildermuth rule

i =1 g i , (11)

G = 2 n =

where is the oscillator quantum number of a cluster nucleon orbiting the core.

The width of the 2 radioactivity can be calculated by

Γ 2 p = S ′ 2 p F ℏ 2

where is the spectroscopic factor.

0 W

Mg versus different values within the SLy4 interaction. The Fermi energies are depicted by the red dashed lines.

Mg, we assume that the two emitted protons originate from the 2 orbitals. The value is calculated by a BCS model [82]

] = 1 , (14)

) cos with the wave number ) calculated by

k ( r ) =

Finally, the 2 radioactivity half-life is calculated by

T 1 / 2 = ℏ ln 2 Γ 2 p . (16)

RESULTS AND DISCUSSIONS In the SHFB calculations, the HFBRAD code with the SLy4 interaction is employed [74]. Specifically, the spherical box and mesh sizes are selected as 20 fm and 0.1 fm, respectively. The quasiparticle energies are cut off at 60 MeV.

The maximum angular momenta of the quasi-neutron and quasi-proton are set to be , respectively. All the calculations of this article are converged with these conditions.

Within the mixed pairing force, the single-proton energy spectra near the Fermi energies of Mg are calculated is taken in the range of 0-1.4 MeV fm with an interval of 0.2 MeV fm , which are plotted in Fig. 1 [FIGURE:1].

1 E

, 1.0 and 1.4 MeV fm within the SLy4 interaction. TABLE I: The calculated values of Mg with different values. is the unoccupation probability of the 2 proton state for is the spin-orbit coupling strength of the SLy4 interaction, whose value is

123 MeV fm

. The experimental value of Mg is 3.440 MeV [64]. (MeV fm (MeV fm (MeV)

1.15 W

Note that is the spin-orbit coupling strength of the SLy4 interaction and equals to 123 MeV fm . In calculations, the corresponding each value is determined by the empirical proton pairing gap of Mg. This gap is obtained by the following way: Firstly, the value is determined by fitting the experimental neutron pairing gap (1.31 MeV) of Sn. Next, within the SHFB theory by inputting the determined value we can extract the proton pairing gap of Mg, whose value is 1.921 MeV. The values for different values determined by the proton pairing gap of (1.921 MeV) are shown in column 2 of Table I.

In Fig. 1, the single-proton energies refer to those in the canonical basis. From Fig. 1, one can see that the splitting of the 1 orbital becomes increasingly pronounced with the increase of . Moreover, the 2 energy levels are below the Fermi energies for , 0.2 , 0.4 and 0.6 , which indicates that the 2 states are weakly bound. However, with the enhancement of the 1 orbital splitting the 2 energy level exceeds the 1 orbital and is located in the resonant states of the continuum energy region when . No matter whether the 2 state is weakly bound or not, a certain number of protons are occupied on it. For the 2 state, its centrifugal barrier vanishes. So, the valence protons can extend to a large region and then a 2 halo or 2 halo-like structure appear. This situation is similar to the formation of the neutron halo [83, 84]. Therefore, the contribution of the state to the full density in the large region is dominant. This can be seen clearly from Fig. 2 [FIGURE:2], which shows

= - 0 . 0 0 4 4 W - 0 . 1 0 1 7

versus the corresponding values. (b) Same as 3(a), but for the values. the density distribution of each orbit near the Fermi energy as a function of , 1.0 and 1.4 . Our recent studies with the T31 and SLy5 interactions suggested that the deformation of Mg is very weak [66, 67].

Furthermore, relevant studies indicate that the 2 halo-like structure rather than a deformation is responsible for the correlation and the mechanism of 2 emission [85–89]. So, the microscopic single-proton energy spectra are enough to examine the 2 halo or 2 halo-like structure of With the enhancement of the 1 orbital splitting, less protons occupy the 2 orbital. As shown in column 3 of state ( ) decreases with increasing . As a result, the tail of the density distribution of the 2 orbit is less extended, which can be seen clearly from Fig. 2. Hence, the 2 halo-like sizes of Mg are depressed by the increase of . Similarly, the occupation probability of the 2 state of its daughter nucleus Ne falls with the rise of . It means that the unoccupation probability increases with the rise of , which can be seen from column 4 of Table I. Using Eq. (13), it is easy to obtain the values, which are listed in column 5 of Table I. As can be seen from column 5 of Table I, on the whole, the value is getting smaller and smaller with the increase of . Nevertheless, when is weak the value changes very little with . Note that the diproton cluster preforms more easily under the condition that the density on the nuclear surface is low. The weakening sizes of the 2 halo or 2 halo-like structure in Fig. 2 suggest that the density distribution on the nuclear surface is less extended with the enhancement of . As a result, the value becomes smaller and smaller. Therefore, the value may be correlated strongly to the 2 halo or 2 halo-like size. To study this kind of correlation, in our recent work, we had to use a virtual charge radius to represent the 2 halo-like size of Mg because it is difficult to distinguish a halo from a gaslike structure for an unbound nucleus [66, 67]. We found that there exists a pronounced linear correlation between the and the 2 halo-like size [66, 67]. So, we will not discuss the correlation between the and the 2 halo or 2 halo-like size in this article.

Using Eq. (6), we can obtain different values when different values are selected, whose values are shown in column 6 of Table I, where it is shown that the values are gradually decreasing with the growth of . Previous studies have suggested a dependence of [4, 48, 50, 51, 90]. Therefore, we plot the logarithm versus as well as that versus in Figs. 3(a) and 3(b), respectively. From Fig. 3 FIGURE:3, a good linear correlation between with the Pearson coefficient = 0.9975 is found. Similarly, a less pronounced linear correlation between log = -0.8899 is seen from Fig. 3(b).

To see the effect on the diproton cluster potentials , we calculate the with different values using the mean-field cluster potential approach. The values with , 1.0 and 1.4 are shown in Fig. 4 FIGURE:4.

= 0 . 0 3 7 8 5 W - 2 5 . 5 6 3 8

= 0 . 0 2 8 5 3 W - 2 4 . 9 9 2 2

, 1.0 and 1.4 . (b) Same as 4(a), but for the potentials in the nuclear surface region. (c) The logarithm half-lives without and with versus the corresponding values.

The red dashed line represents the logarithm half-lives without The red solid line means that with ). (d) Same as 4(c), but for the corresponding values.

To examine the potential in the nuclear surface region influenced by , the curves in the surface region with , 1.0 and 1.4 are shown in Fig. 4(b). From Figs. 4(a) and 4(b), one can see that the potential well becomes deeper and deeper with the increase of . However, the potential barriers in the surface region grow gradually with the rising . Studies on -decay showed that the preformation probability of -cluster is enhanced with the increasing potential well depth [72, 91]. Nevertheless, for the of the diproton cluster of Mg, it gets smaller and smaller when the potential well becomes deeper and deeper. It implies that the diproton cluster is different largely from the -cluster and should be much looser than the -cluster.

Next, we calculate the 2 radioactivity half-lives without and with ) using the mean-field cluster potential approach by inputting the values with different values. The values with different values are listed in column 7 and the last column of Table I. For the cases of , the value exceeds the height of the diproton potential barrier, thus the half-life cannot be calculated within the tunneling model and is not listed. Besides, the logarithm half-lives without and with and log ) versus and those versus are shown in Figs. 4(c) and 4(d), respectively. From Figs. 4(c) and 4(d), we can see that log show good linear relationships with , respectively. As the picture of the tunneling effect, the decreasing values may result in longer half-lives, which can be seen from Table I, Fig. 4(c) and Fig. 4(d).

Moreover, from Table I, Fig. 4(c) and Fig. 4(d) we can see that the logarithm half-lives are enhanced by . For the case of 0.8 , the half-life has increased by a factor of 2.375. When , the half-life has been extended by 11.496 times. value of Mg was measured as 3.440 MeV [64].

So, using the experimental value the value can be determined. We found that when =1.15 , the calculated value exactly equals to the corresponding experimental one. Besides, with =1.15 the corresponding values are calculated. These values are listed in the last line of Table I, from which it is easy to obtain that is 5.181 times as large as

CONCLUSIONS

In this article, the impact of the spin-orbit coupling strength on the structure and 2 radioactivity of has been investigated within the spherical SHFB theory and mean-field cluster potential approach using the SLy4 interaction. Specifically, the single-proton spectra, density distributions, spectroscopic factors, diproton cluster po- tentials and half-lives of 2 radioactivity have been calculated when the value is taken in the range of 0-1.4 The following conclusions could be drawn: (i) The splitting of the 1 proton orbital becomes more and more evident with the increase of . Moreover, the proton state is weakly bound for the cases of , 0.2 , 0.4 and 0.6 . However, the 2 state becomes a resonant state in the continuum energy region when (ii) With the enhancement of the 1 orbital splitting, the occupation probabilities of the 2 state decrease gradually. As a result, the tail of the density distribution of the 2 orbital is less extended. Correspondingly, the spectroscopic factor is getting smaller and smaller. (iii) The diproton cluster potential well becomes deeper and deeper with the increase of . Moreover, the deeper the diproton potential well, the smaller the . It implies that the diproton cluster is much looser than the -cluster. (iv) The declining values caused by the increasing lead to the longer half-lives of 2 radioactivity.

Moreover, the half-life is enhanced by keeps as a constant. (v) A pronounced linear correlation between log and that between log are found. Besides, and log show good linear relationships with , respectively. (vi) Using the experimental value of Mg, the optimal value is constrained to be 1.15 ACKNOWLEDGEMENTS We thank professor Ligang Cao, professor Jianmin Dong and professor Yibin Qian for helpful suggestions and comments. This work was supported by the S&T Program of Hebei (Grant No. 236Z4601G); Scientific Research Foundation for the Introducing Returned Overseas Chinese Scholars of Hebei Province (Grant No. C20230360) and Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy of Sciences (Grant No. IMPKFKT2021002).

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