The tensor force is a crucial component of the nuclear interaction that plays a significant role in determining the energy levels of nuclei, especially for nucleons in high-angular-momentum states and in nuclei far from the stability (23). In the nuclei far from stability or with high isospin asymmetry, the neutrons and protons can occupy different orbitals. Since the tensor component of the nuclear force arises primarily from the exchange of pions ( π -mesons) between nucleons, the exchange process contributes dominantly to the monopole part of the tensor force, which is much stronger for the proton–neutron ( T = 0 ) interaction, and is approximately twice as strong as the ( T = 1 ) interaction. The tensor force causes the effective interactions between the proton orbital with j > = ℓ + 1 / 2 (or j < = ℓ − 1 / 2 ) and neutron orbitals j < ′ (or j > ′ ) to be more attractive, whereas j > and j > ′ (or j < ′ and j < ′ ) repel each other. This effect accumulates as the proton–neutron asymmetry increases, and the shell evolution occurs consequently.

It is, therefore, natural to expect that the neutron SO splittings evolve with the change in the proton number. As the proton fills the j > orbitals, the SO-splitting decreases, and vice versa, which is supported by experimental data. For example, in the Ca isotopes, it was shown that the proton 0 d 3 / 2 is attracted (lowered in energy), while 0 d 5 / 2 is repelled (raised in energy) due to the neutron filling of the 0 f 7 / 2 orbit [13]. Similarly, in the Sb isotopes, as more neutrons occupy 0 h 11 / 2 , the protons 0 h 11 / 2 and 0 g 7 / 2 move apart [14]. This trend is also consistent with a decrease in the nuclear SO interaction.

Since the SO interaction is majorly a surface term, it could be modified in neutron-rich nuclei away from stability, where neutrons may have a diffuse surface density distribution due to weak binding. Hamamoto et al. [15] predicted the SO splittings of weakly bound orbits in light, neutron-rich nuclei to decrease due to the extended radial wavefunctions of neutron orbits, with no reduction in the SO potential strength.

By approximating SO potential to a δ function at the nuclear surface, a simple evaluation of the SO-splitting was established in Reference [16], Δ S O ∝ V s o ℓ ⋅ s r 0 2 R Ψ 2 R , (2) where V s o is the SO potential strength, Ψ ( R ) is the radial wavefunction, r 0 is the scaling parameter for the radius of nuclei (usually taken as 1.2 fm), and R is the radial distance from the center of nuclei. Figure 1A plots the radial 1 p 3 / 2 wavefunctions multiplied by the radius under different binding energies, showing that the SO-splitting decreases as the corresponding orbitals become less bound.

Due to the saturation and short-range nature of the nuclear force, it is natural to expect that the density in the center of nuclei is constant. However, there have been many theoretical studies supporting the existence of central depletion in ³⁴Si [17, 18]. ³⁴Si is a candidate for a so-called “bubble” nuclei, providing a valuable test case for the SO potential in the center of nuclei. The prediction of central depletion in ³⁴Si arises from its doubly magic characteristic ( N = 20 and Z = 14 ), which results in an extremely low proton occupancy number in the 1 s 1 / 2 orbital. This occupancy was determined to be between 0.17 and 0.24 in the proton knockout reaction [19]. As a large fraction of the radial part of the 1 s 1 / 2 orbital peaks in the center of the nucleus, the lack of 1 s 1 / 2 naturally induces a central density depletion. Despite no direct proof of such central depletion, experimental developments in electron scattering measurements, ideally suited for such studies, of radioactive isotopes are being made [20].

Since the SO-splitting is proportional to the derivative of the density distribution (see Equation 1), it is expected to change due to the presence of density depletion. The one-neutron adding reaction is useful for determining the angular momentum transfer ℓ and spectroscopic factors through comparison to the reaction models, and the population strength indicates the single-particle strength in each state. Therefore, the SO splittings can be mapped out with the addition and removal of single-particle strengths and the corresponding binding energies [21], E j = ∑ G j + E j + + ∑ G j − E j − , (3)

with G j + + G j − = 1.0 . For the case in which the single-particle removal strengths were not measured, the energy centroid can be used to determine the single-particle energies E j = ∑ G j + E j + , (4) with G j + = 1 .

A significant reduction in SO-splitting is predicted for ³⁴Si compared to other N = 20 isotones due to central density depletion. This prediction seems to be supported by the nearly 50 % reduction in the SO-splitting in ³⁴Si compared to ³⁶S, as determined using the dominant single-particle component [19, 22] (see Figure 2A). However, this assertion was questioned because only dominant single-particle strength was considered, instead of including the fragmented components of the ℓ = 1 single-particle strength as in Equation 3, which may result in overestimation of SO splittings. After taking them into account, a smooth reduction from ⁴¹Ca via ³⁹Ar and ³⁷S to ³⁵Si was shown (see Figure 2A), which was explained by the finite binding energies of the neutron states [23]. So far, the interpretation remains highly debated. There is an ongoing investigation into whether the observed changes in the 1 p SO-splitting are driven by the weak-binding effect or by the weakening of the two-body SO potential in this region [6, 24]. This motivated the recent measurement of the N = 19 isotones.

In order to enhance our understanding of the microscopic origins of the SO interaction, studying the SO interaction near the S and Si isotopes is crucial. The evolution from Si to S is particularly important since only the 1 s 1 / 2 proton orbital is filled between these two nuclei. Consequently, the resulting proton–neutron interaction involves no tensor component because it vanishes for ℓ = 0 ; only the SO part of the nuclear force plays a role.

For ³²S to ³⁰Si ( N = 16 ) , the proton 1 s 1 / 2 occupancy changes from 1.35 to 0.65 (not 2.0 to 0.0) based on the proton knockout reaction data [25], making ³⁰Si not an ideal candidate to study the proton central depletion. However, for ³²Si, the neighboring even–even isotope of ³⁴Si, both density functional theory and shell model calculation predict a very small proton 1 s 1 / 2 occupancy ( ∼ 0.3) compared to ³⁴S, where 1 s 1 / 2 is almost fully occupied. Furthermore, density functional theory calculations of ³²Si predict a depletion similar to that of ³⁴Si in the proton density distribution, as well as a sudden reduction in SO-splitting in ³²Si compared to ³⁴S (see Figures 2C, D). It provides another testing ground for investigating if there is a sudden reduction in SO-splitting due to proton depletion. It should also be noted that one major difference in ³²Si is that its neutrons are more deeply bound than ³⁴Si, so it should be less influenced by the weak binding effect.

The single-particle energies of shell-model orbitals in N = 19 isotones (³³Si, ³⁵S, and ³⁷Ar) can be mapped out with the addition and removal of single-particle strengths using Equations 3, 4. The neutron addition data of the N = 19 isotone ³⁷Ar and ³⁵S can be found in Refs. [26–29]. With these data, the weighted average values of the 0 f 7 / 2 and 1 p 1 / 2,3 / 2 orbitals were obtained and are plotted in Figure 1C. It was found that the location of the weighted average is clearly different from the dominant strength, showing that considering the fragmented strength is important. The single-particle removal strength of these orbitals was also considered where one-neutron removal data exist for ³⁷Ar and ³⁵S. Only the 1 p 3 / 2 and 0 f 7 / 2 single-particle energies of ³⁷Ar have been shifted downward by approximately 100 and 250 keV, respectively. The p f -shell orbitals of ³⁵S have been shifted less than 50 keV. However, no such previous addition or removal data exist for ³³Si.

In order to quantitatively determine the SO-splitting, a measurement of ³²Si ( d , p ) 33 Si cross-sections was carried out at the ReA6 beamline in FRIB using the newly constructed solenoid spectrometer SOLARIS in the silicon array mode [30]. The solenoid spectrometer is capable of measuring the transfer reactions, in particular the one-neutron adding ( d , p ) reactions with high resolution. The experimental spectroscopic factors and the single-particle energies of the 1 p 3 / 2,1 / 2 and 0 f 7 / 2 orbitals are plotted in Figure 1C and compared with its S and Ar N = 19 isotones.

In the relativistic mean field (RMF) calculation with the DD-ME2 interaction [31], ³²Si was predicted to exhibit a depletion in central density, similar to ³⁴Si, due to low 1 s 1 / 2 proton occupancy. This calculation predicts a sudden reduction of the neutron 1 p -shell SO-splitting in 33 Si compared to ³⁵S, similar to the N = 21 isotones. However, as observed from the present measurement, the SO-splitting in ³³Si is similar to that of ³⁵S, in contradiction to the RMF calculation (see Figure 2B). The mismatch of this calculation might be attributed to the fact that the proton–neutron quadrupole correlations are not taken into account in the RMF calculation. Therefore, this study does not support the existence of a sudden reduction in SO-splitting associated with a proton bubble.

To explore this weak binding effect on SO splittings, the calculation was carried out with a Woods–Saxon (WS) potential. Figure 4 of Reference [30] shows the binding energy of 1 p 1 / 2 and 1 p 3 / 2 orbitals from existing experimental data, together with the WS calculation, using the radius and diffuseness parameters r 0 = 1.2 fm, a 0 = 0.7 fm, r s o = 1.3 fm, a s o = 0.65 fm, and SO strength V s o = 6 MeV. The depth of the potential was chosen to reproduce the binding energies of these two orbitals with a χ 2 minimization method. The SO strength is not varied in the calculation.

It can be seen immediately that the SO-splitting and single-particle energies of the 1 p orbitals have been reproduced by the calculation without changing the SO potential strength. The good agreement with the calculation with WS formalism indicates that the evolution of the p -shell single-particle energies was described by the behavior of the wavefunctions resulted from the geometric effect (a large radius or diffuseness) of the low- ℓ orbitals as they become less bound. This was achieved without inducing a weakening of the SO potential strength or other additional effects.

From Equation 2, it is seen that the SO-splitting depends on the term R Ψ ( R ) if the strength of the SO potential V s o remains unchanged. In Figure 1A, this term is plotted as a function of R. The radius of the nucleus R 0 was taken as 1.25 fm × A 1 / 3 = 4.05 fm. It is clearly seen that the term R Ψ ( R ) reduces as the binding energies approach to 0, diminishing to more than 60 % of its original value. This indicates that the reduction observed in the 1 p -orbital SO-splitting can be fully accounted for by the evolution of the wavefunctions toward weak binding.

³²Si should have a similar 1 s 1 / 2 occupancy as ³⁴Si, according to the latest safe Coulomb excitation measurement [32], as also supported by the theories. It is noted that there is yet no experimental measurement informing on the proton occupancy. Related measurements to determine its proton occupancy in the 1 s 1 / 2 orbital are being planned with the Active-Target Time Projection Chamber (AT-TPC) [33] coupled with the HELIOS solenoid. Using the proton addition or removal reaction, the proton occupancy of ³²Si in the s 1 / 2 orbital will be determined.

The discussion above mostly focuses on the SO-splitting of the 1 p -shell orbitals. One may wonder if the weak binding or central depletion effect may be revealed in the SO-splitting of the 0 f orbitals. The radial wavefunction of the 0 f orbital is compared with that of the 1 p orbital in Figure 1B. In addition, Equation 1 shows that the changes in the wavefunction at the smaller radius would have a larger impact on the SO potential. Therefore, some may expect that there would be a sudden reduction in the SO-splitting in case of a central depletion. However, the 0 f orbital wavefunction seems to have very little sensitivity to the change in the potentials in the very center of nuclei ( R < 2 fm), where the depletion was presented. Consequently, the central depletion should have very little impact on the SO-splitting of the 0 f orbitals.

On the other hand, the weak binding effect may still impact the SO-splitting of the 0 f orbitals, although much less than the 1 p orbital. According to a calculation with the WS potential, the change in the SO-splitting from binding energy is approximately 50 % less compared to that of 1 p orbitals. However, this effect will still be clearly seen based on the usual uncertainties of approximately 100–200 keV for determining the single-particle energies from the transfer reactions. Future experiments to measure the 0 f orbital SO splittings in Si and S under weak binding would be important to further study whether the weak binding effect or the central density depletion plays a major role.