Preamble

Spectroscopic factors single proton capture silicon phosphorus isotopes Xuxia Hantao Cheng wei Dong

1 Department

Physics, North University China,030051, Taiyuan, People’s Republic of China Proton capture reactions silicon phosphorus crucial hydrogen burning during explosions.

However, experimental measurements reactions involving radioactive isotopes extremely short-lived nuclei exceptionally challenging.

Calculating spectroscopic factors these reactions provide guidance cross-section estimates serve essential reference experimental studies. work, employ nuclear shell model compute spectroscopic factors associated silicon isotopic chain, specific roton apture processes phosphorus nuclei.

results

reactions agreement existing experimental previous theoretical calculations.

Additionally, predict proton spectroscopic factor ground state approximately words: spectroscopic factors; direct proton capture; silicon isotopes

INTRODUCTION

Capture reactions are key determinants of reac- tion rates in explosive nucleosynthesis within high- temperature, high-density stellar environments [ They not only influence stellar energy output but also directly govern the evolution of elemental abundances, serving as a fundamental driving force behind com- plex nuclear reaction networks [ In the rapid proton capture process(rp-process), the reac- tion is particularly critical [ ]. For proton-unstable nuclei, their proton capture rates may approach or even exceed the -decay rates, thereby dominating explosive hydrogen combustion. Moreover, determin- ing the cross sections of these astrophysically impor- tant reactions remains a challenging issue in nuclear astrophysics, and direct measurements are the most reliable method [ ]. However, the reaction cross sections in the astrophysically relevant energy range are on the order of µb to pb. For most isotopes of interest, radioactive ion beams available from cur- rent facilities have low beam intensities, making di- -reaction measurements extremely difficult.

Therefore, it is important to employ indirect methods ], such as using proton-transfer reactions like , to identify which states play significant roles in direct proton capture, or assess them theo- retically by determining proton spectroscopic factors Nuclear spectroscopic factors, which link nuclear structure and nuclear reactions, are regarded as one of the most important inputs in nuclear astrophysics models [ ]. Quantities such as the proton width and the capture cross section are proportional to the spectroscopic factors, after accounting for the single- particle width and the relevant Coulomb/centrifugal penetration factors.

Spectroscopic factors derived from direct proton-capture studies are generally con- sidered more reliable than those inferred from proton- transfer reactions [ In nuclear astrophysics, proton capture reactions are particularly crucial, especially during stellar ex- plosive hydrogen burning, where proton capture reac- tions on silicon isotopes play an important role [ ]. For example, the P reaction is crucial at this stage and represents one of the key pathways ], while the P rate modu- lates the P branching ratio and thus indirectly affects the final Si abundance observed in classical novae [ In this work, we calculate the spectroscopic factors for direct proton capture reactions along the isotopic chain of silicon using the nuclear shell model. The out- comes for the proton spectroscopic factors of show favorable consistency with both available exper- imental data and prior theoretical computations. We then examine the spectroscopic factors in relation to the excitation energies of the residual nuclei. Addi- tionally, we find that certain excited states make sig- nificant contributions to the spectroscopic factors and, under specific conditions, are more likely to undergo proton capture than the ground state.

This find- ing provides critical parameters for refining reaction networks involving silicon isotopes in nuclear astro- physics and facilitates a more accurate understanding of elemental abundance evolution in stellar interiors.

THEORETICAL BASIS Spectroscopic factors were first introduced into nu- clear reaction theory approximately sixty years ago, marking a significant milestone in the advancement of nuclear physics [ ]. This concept has since be- come a cornerstone in the study of nuclear structure and reactions, as it provides a quantitative measure of the overlap between nuclear states before and after a reaction [ ]. By comparing spectroscopic factors derived from experimental data with those calculated using various nuclear models, researchers are able to probe the internal structure of atomic nuclei and as- sess the validity of theoretical frameworks. This com- parative approach has proven instrumental in refin- ing our understanding of nuclear forces and configu- rations. Consequently, the study of spectroscopic fac- tors has remained a central theme in both theoretical

and experimental nuclear physics, inspiring extensive research over the decades and resulting in a substan- tial body of research on the topic [ The spectroscopic factor is closely related to the occupancy probability of a single-particle orbital. It reflects both the probability that a specific orbital is occupied in the target nuclear state and the overlap integral between this orbital and the single-particle state involved in the transfer reaction. Therefore, the higher the occupancy probability of an orbital, the larger its corresponding spectroscopic factor tends to Cole et al. calculated spectroscopic factors for single-nucleon transfer reactions in nuclei with mass numbers based on the shell model [ Specifically, the nuclear wave functions are calcu- lated within the full or truncated -shell configura- tion space, and an appropriate creation or annihilation operator is applied to the target wave function (an an- nihilation operator in the case of pick-up reactions).

The spectroscopic factor is then obtained by squaring the overlap integral between the resulting wave func- tion and that of the final state.

According to the nuclear shell model, a spectro- scopic factor depends on the overlap integral between the final state and the state formed by coupling the target state with the transferred particle to a coupled- channel state, which is defined as [ ������ �����

S A if =

where is the wave function of the final-state nu- clei, is the wave function of the initial-state nuclei, A + 1 is the mass number of the nucleus, and the generating operator.

By the precise definition, usually includes the Clebsch-Gordan coefficient and thus is therefore written as [ ������ ����� ����� ������

C 2 S =

where denotes the isospin quantum number of the initial nucleus Si, and is its corresponding isospin -component. The quantities represent the isospin and isospin -component of the incoming pro- ton, respectively. denotes the isospin quantum number of the final nucleus, while corresponds to the isospin -component of the final nucleus P after the proton is captured.

In the process of proton emission [ ], the decay can be expressed as where denotes the parent nucleus, repre- sents the daughter nucleus, and p is the emitted pro- The Coulomb barrier involved in this process can typically reach values as high as 15 MeV. The to- tal half-life of proton emission is determined by the decay width (or resonance width), which is related through [

T 1 / 2 = ℏ ln 2 C 2 S Γ

= ℏ ln 2 Γ 0

where is the decay width, and denotes the effec- tive decay width corrected by the spectroscopic factor.

Both quantities have the dimension of energy and are commonly expressed in MeV. The decay width is con- nected to the transition amplitude via

Γ = 2 π | T A+1 , Z+1;A , Z | 2 (5)

Within distorted-wave approximation (DWBA), the transition amplitude takes the form Z+1;A

where ψ Ap denotes the incoming spherical wave that describes the relative motion between the proton and the daughter nucleus. The initial wave function of the proton–daughter system is given by

Ψ Ap = Φ A Φ p (7)

being the intrinsic functions of the daughter nucleus and the proton, respectively. tailed formulas can be found in the references [ DETAILS OF THE CALCULATION In this paper, calculations are performed in the full (sd) shell model space using the W, CW, and CWH effective interactions, with the shell- model code NuShellX, which performs diagonaliza- tion [ ]. It is a core-shell model code, an important part of which is its ability to predict transition rates, spectroscopic factors, cluster spectroscopic factors, etc W interaction refers to the two-body matrix ele- ments derived from a linear fit [ Under the as- sumption of mass-independent two-body matrix ele- ments, Chung and Wildenthal introduced two differ- ent sets of two-body matrix elements, CW interaction for the A = 17-28 region and CWH interaction for the A = 28-39 region [ ]. The spin and parity data required in the calculation process come from Ref [ Since the mass numbers of the silicon and phospho- rus isotopes fall near the applicable ranges of these effective interactions, their use is reasonable. thermore, as these interactions have been validated in previous studies and show good agreement with ex- perimental data [ ], their adoption in this work is reliable.

In the theoretical description of proton emission from a nucleus, the total spectroscopic factor is gen- erally the sum of contributions from all accessible single-particle orbitals, such as . Each or- bital contributes a partial spectroscopic factor, which can be interpreted as the probability of emitting a proton from that specific orbital.

Accordingly, the

TABLE I: The spectroscopic factors ( ) for the possible proton orbitals of the P isotopic chain calculated by program NuShellX with different interactions (W, CW, and CWH) at different energies ( ) are listed in comparisons with the experimental data and the theoretical data from references. (CW/CWH)

Experiment

Theory

25 Si

0.3600 [ 0.6257 [

26 Si

0.5271 [

27 Si

0.5345 [

28 Si

0.3600 [ 0.5000 [ 0.6400 [ 0.5900 [ 0.0600 [

30 Si

0.1100 [ partial half-life for each orbital can be calculated to re- flect the tunneling probability and barrier penetrabil- ity associated with that single-particle configuration.

The overall proton-emission half-life of the nucleus is then determined by the combination of these partial half-lives, allowing the decay branching structure to be accurately taken into account and enabling direct comparison between theoretical predictions and ex- perimental measurements. Although the calculation procedure is relatively intricate, it can be systemat- ically implemented in existing computational frame- works to handle contributions from different orbitals.

In this work, we calculate the total proton-emission half-life and the corresponding proton width Consequently, the contributions from individual or- bitals are directly connected to the overall decay prop- erties, allowing for a consistent comparison between theoretical calculations and experimental results.

RESULTS AND DISCUSSION We calculated the spectroscopic factors for the sil- icon isotope chain, selected nuclear states with spec- troscopic factors greater than 0.1 for analysis, and systematically compared the results with theoretically predicted values and experimentally measured data reported in existing literature, as detailed in Table . The analysis revealed a high degree of consistency among the results obtained through different interac- tions within the same shell model. And the calculated spectroscopic factors, which range from 0 to 1, closely matched the literature values, indicating the accuracy of the shell model calculations. And for the ground state ( = 0 MeV), the proton spectroscopic factors P are between 0.44 and 0.63. Furthermore, we found that the proton spectroscopic factor of the P ground state is relatively large, approximately 0.6, with a value of 0.63 under the W interaction and 0.60 under the CW interactions.

W and CW/CWH. These results are compared with all available theoretical values and experimental mea- surements reported in the literature listed in Table , and the relative deviations between the calculated values and the corresponding averages are analyzed.

Through the plotting of error bars, it can be clearly observed that the overall error range is small, and the shell model calculation results maintain good consis- tency with both theoretical data points and experi- mental data points. Specifically, for the nuclear states , and , the calcula- tion results of the shell model in this study exhibit a higher degree of agreement with the experimental measurement values. However, for the

Comparison between theoretical values and experimental uncertainties of spectroscopic factors ( ) for the phosphorus isotope chain, calculated using different interactions.

TABLE II: Proton-radioactive properties of the selected nuclides P and P: energies in MeV, half-lives in seconds, and proton widths in MeV.

Nuclide Proton emission energy/MeV Half-life/s Proton width/MeV

29 P

clear state, there is a relatively large deviation be- tween the calculated values and the experimental val- ues. Overall, our calculation results demonstrate high reliability, which validates the effectiveness of the shell model method.

We selectively extracted the data for the isotopes P, which have attracted considerable interest, and plotted the spectroscopic factors as a function of excitation energy, as shown in Figure In most of the analyzed reactions, the pro- ton spectroscopic factor of the ground state of P is typically more than twice that of the excited states.

However, several notable exceptions emerge, in which the proton spectroscopic factors of certain excited states are comparable to, or even exceed, those of the corresponding ground states.

For instance, in P reaction (Figure (b)), the proton spectroscopic factor of the excited state MeV is 0.41, which is very close to the ground state value of 0.46.

Similarly, in the P reaction (Figure (d)), the proton spec- troscopic factor of the excited state at MeV is 0.66, exceeding that of the ground state (0.45). Furthermore, as shown in Table , the proton spectroscopic factor of the excited state at MeV in the P reaction is 0.35, which is comparable to that of the ground state (0.44).

A similar situation is observed in the P re- action, where the proton spectroscopic factor of the excited state at MeV is 0.54, close to the ground state (0.51).

These results further highlight the importance of consider-

The proton spectroscopic factors of P at different energy levels ( /MeV), calculated using the W effective interaction. ing not only ground states but also excited states in nuclear reaction models. Taking into account the tab- ulated data as a whole, we also observe that nuclei with higher binding energies generally exhibit larger spectroscopic factors in their excited states, as exem- plified by P isotopes. Overall, the present results show good agreement with existing experimental or theoretical data, demonstrating the reliability of the employed computational approach.

Spectroscopic factors play a crucial role in both nu- clear structure and nuclear reactions. Accurate spec- troscopic factors facilitate more precise calculations of capture cross sections, which are essential for under- standing the probability of nuclear reactions and for improving both theoretical predictions and the inter- pretation of experimental results. Similarly, reliable spectroscopic factors allow for more accurate evalua- tions of proton widths, which are critical for under- standing the stability of excited nuclear states and their decay processes[ ]. Given that nuclear reaction data often require comparison with theoret- ical predictions, the spectroscopic factors obtained in this study can provide a valuable reference for the fur- ther refinement of relevant theoretical models, thereby providing some potentially valuable guidance and sup- port for future experimental investigations.

In addition, the half-life and proton width of the proton emission from Si were calculated us- ing the DWBA method [ ]. As shown in Table the proton-decay properties of P and P exhibit dis- tinct behaviors. For P, the proton-emission energy is 0.941 MeV, corresponding to a half-life of s and a relatively small proton width of only 0.009 MeV. In contrast, P displays a pronounced multi- channel decay pattern with three accessible final states , 2.323, and 4.249 MeV. A clear correlation be- tween decay energy and decay probability is observed: as the proton-emission energy increases, the half-life decreases drastically from s, and further to s. This trend is consis- tent with the corresponding increase in proton width from 0.006 MeV to 0.114 MeV and 0.135 MeV, indi- cating enhanced barrier penetrability in the higher- energy channels. Consequently, the effective lifetime P is mainly governed by the highest-energy de- cay mode and is significantly shorter than that of P. These results reveal a general rule that, in nuclei with multiple emission pathways, higher decay ener- gies typically correspond to larger proton widths and faster decay rates. With increasing proton-emission energy, the rapid reduction of the half-life and the significant enhancement of the proton width manifest

the characteristic quantum-tunneling behavior, fur- ther confirming the shell model’s capability in accu- rately describing the proton-emission process in light nuclei such as phosphorus.

SUMMARY

In this study, we systematically calculated the pro- ton spectroscopic factors for the phosphorus isotopic chain and compared them with existing experimental and theoretical values.

Our analysis revealed a high degree of consistency across results obtained using various nuclear interac- tions within the same shell model framework. The cal- culated spectroscopic factors, which are constrained to the range 0–1, exhibit strong agreement with experi- mental data and published theoretical values, thereby validating the reliability and accuracy of our shell model approach. A detailed examination of the data showed that, for the ground state (with excitation en- = 0 MeV), the proton spectroscopic factors of isotopes P through P generally lie in the range of 0.44 to 0.63. Furthermore, we observed that ex- cited states make non-negligible contributions to the total spectroscopic strength, and the tabulated data reveal that the spectroscopic factors of stable nuclei are larger than those of unstable nuclei. This indi- cates that including these excited states is essential for a comprehensive understanding of nuclear struc- ture.

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These calculated spectro- scopic factors not only enhance our understanding of the nuclear structure of silicon isotopes but also serve as a valuable reference for modeling astrophysical phe- nomena and nuclear synthesis processes, particularly in proton-rich environments.

Finally, we calculated the proton emission half-lives and proton widths of P and P using the calculated spectroscopic factors in combination with the DWBA method. The results show that P has a relatively low proton emission energy, a longer half-life, and a smaller proton width, whereas for P, among multi- ple decay channels, the higher-energy channels corre- spond to shorter half-lives and larger proton widths.

As the decay energy increases, the proton width be- comes larger and the decay rate faster, reflecting the quantum tunneling effect. Overall, these results illus- trate the applicability of the shell model in describing proton emission processes in light nuclei.

ACKNOWLEDGMENTS

The authors are very grateful to Prof. B. A. Brown, Michigan State University, for providing us with the computer program NuShellX. This work is supported by National Natural Science Founda- tion of China (Grant No.12205257, 11647085, and 11647086), the Shanxi Province Science Foundation for Youths (Grant No.201901D211252), Fundamen- tal Research Program of Shanxi Province (Grant No. 202203021221095, 20210302124025).

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