1. Introduction
The concept of considering the nucleons in a nucleus as independent particles moving on relatively undisturbed single-particle orbits has proven to be a suitable approach for describing various nuclear properties. This understanding is mainly because of the impact of the Pauli exclusion principle and the uncertainty principle, which prohibit the nucleons from occupying the same states. Hence, the concentration of the nucleus is not particularly excessive. Extensive research has established that the nucleon–nucleon force possesses an extremely repulsive core. Embracing the notion of independent particles, it is logical to envision that the motion of these particles is governed by an average potential generated by all the nucleons within the nucleus. However, it should be observed that the behavior of the nucleons varies significantly in the interior area of the nucleus, where the force acting on them is minimal, compared to the surface area where the Pauli principle no longer applies, resulting in a confining force that maintains the particles within the nucleus [1–7].
The recognition of specific numbers, particularly 2, 8, 20, 28, 50, 82, and 126, that possess distinct characteristics in relation to protons and neutrons, has been a compelling motivating factor in the advancement of the nuclear shell model. These numbers, known as magic numbers, bear similarities to the concept of shell closure observed in electron shells of atoms [8–12].
The theoretical foundation of the shell model lies in the widely recognized harmonic oscillator. Although this model may not fully capture the complexity of a realistic potential, it has demonstrated its effectiveness in explaining numerous phenomena within the context of many-body nuclear problems. By employing the oscillator length as a parameter for each specific isobar, the model can be successfully adapted. Nevertheless, the harmonic oscillator model might not be able to accurately replicate certain experimental data, owing to its inherent simplicity [13–29]. Consequently, various endeavors have emerged to improve the predictive capabilities of the model. In this paper, we will explore an alternative approach to enhance the shell model calculations.
Acknowledging the strength and peculiarities of the harmonic oscillator, our aim is to introduce a slight modification by incorporating another term, leading us to the well-known Davidson potential. Specifically, we seek to include a centrifugal term that accounts for the non-penetrability property of the nucleons. The existence of this term in different studies [30–33] and its impact on the potential have been thoroughly examined, and now we aim to implement it within the shell model framework.
To give a transparent framework for our presentation, we have divided the paper into numerous sections. In section 2 , we will provide a concise review of the harmonic oscillator bases and subsequently propose the new bases for the shell model. Once the new bases have been introduced, we will delve into their properties and derive the corresponding eigenvalues and eigenfunctions in section 3 , which are crucial for further applications. Examination of the new basis is discussed in great detail in section 4 . It has four subsections as follows. As a preliminary application of the new bases, we will demonstrate their utility for one-particle nuclei in section 4.1 . Furthermore, we will extend these calculations to encompass one-hole nuclei in section 4.2 . Continuing the exploration initiated in the preceding sections, section 4.3 presents a captivating application of Davidson bases, specifically in unraveling the intricate dynamics of β-decay in one-hole nuclei. This section unveils the remarkable ability of Davidson bases to dissect the underlying mechanisms governing β-decay processes, providing profound insights into the nuclear structure and interactions that orchestrate these decay events. Finally, we will present our findings in section 5 , summarizing the main conclusions and emphasizing potential avenues for future research.
2. A brief review of the proper basis
The harmonic oscillator basis stands out as a particularly suitable choice for representing quantum systems due to its inherent simplicity and remarkable accuracy. Its derivation is grounded in the assumption that the potential energy governing the system's behavior adheres to the specific form of the Schrödinger equation's potential. This choice of basis proves to be highly effective in capturing the essential features of a wide range of quantum phenomena. The harmonic oscillator basis is derived assuming the potential of Schrödinger equation as [1–7]
$\begin{eqnarray}V(r)=\displaystyle \frac{1}{2}{m}_{N}^{* }{\omega }^{2}{r}^{2},\end{eqnarray}$
with the nucleon mass denoted by ${m}_{N}^{* }$ and the angular frequency represented by ω, the wave function takes the form $\begin{eqnarray}{\rm{\Psi }}({\boldsymbol{r}})={{\rm{R}}}_{{nl}}(r){{\rm{Y}}}_{{lm}}(\theta ,\phi ),\end{eqnarray}$
$\begin{eqnarray}{{\rm{R}}}_{{nl}}(r)={N}_{{nl}}{r}^{l}\exp \left(\displaystyle \frac{-{r}^{2}}{2{b}^{{\prime} 2}}\right){L}_{n}^{l+1/2}\left(\displaystyle \frac{{r}^{2}}{{b}^{{\prime} 2}}\right),\end{eqnarray}$
with the wave function for a particle in a three-dimensional harmonic oscillator potential can be expressed in terms of spherical harmonics, Ylm, the number of radial nodes, n, the angular momentum quantum number, l, the z-component of the angular momentum quantum number, m, Laguerre polynomials, L, the length of the oscillator $\begin{eqnarray}b^{\prime} =\sqrt{\displaystyle \frac{{\hslash }}{{m}_{N}^{* }\omega }},\end{eqnarray}$
and the normalization constant $\begin{eqnarray}{N}_{{nl}}={\left({\int }_{0}^{\infty }{{\rm{R}}}_{{nl}}^{2}(r){r}^{2}\,{\rm{d}}r\right)}^{-1/2}.\end{eqnarray}$
In our pursuit of enhanced accuracy and predictive capabilities, we propose to augment the harmonic oscillator potential with an additional physical term. While the harmonic potential has proven to be a valuable tool in various scientific disciplines, we recognize the potential for further refinement. To address this, we introduce a centrifugal term into the harmonic oscillator potential. This modification aims to capture the nuances of physical systems more effectively, leading to a more robust and reliable representation of their behavior. By incorporating this additional term, we anticipate achieving a more comprehensive and accurate description of various nuclear shell model problems. This enhanced potential will not only improve the precision of our calculations but also open new avenues for exploring the intricate dynamics of physical systems. Thus we have
$\begin{eqnarray}V(r)={a}_{0}{r}^{2}+\displaystyle \frac{{b}_{0}}{{r}^{2}},\end{eqnarray}$
where a0, b0 are real constants determined for each nucleus separately. This form of the potential is known as Davidson potential. Existence of this term brings some important advantages such as:| • | The introduction of this term reveals the existence of a repulsive core within the potential energy landscape. This core, often referred to as a ‘hardcore', acts as a physical barrier that prevents nucleons, the fundamental building blocks of atomic nuclei, from approaching each other beyond a certain distance. The presence of this hardcore is attributed to the inherent repulsion between nucleons at close range, arising from the strong nuclear force. This repulsive force effectively limits the degree of proximity between nucleons, maintaining the stability and structure of the nucleus. |
| • | The strengths of this interaction, represented by the parameter a0 and b0, can be precisely tuned by selecting an appropriate value for a0 and b0. This flexibility allows us to tailor the potential energy landscape to match the observed behavior of physical systems with remarkable accuracy. In essence, the a0 and b0 parameter act as a versatile tool for fine-tuning the theoretical predictions to align seamlessly with experimental results. This ability to precisely adjust the potential energy landscape enables us to gain a deeper understanding of the underlying physical mechanisms governing the behavior of various systems. |
| • | The introduction of this new term, while altering the mean-field potential significantly, does not fundamentally disrupt the inherent harmonic oscillator character of the potential. The inclusion of this term preserves the essential features of the harmonic oscillator potential, maintaining its underlying structure and functionality. Despite the substantial modification, the modified potential retains the distinctive characteristics that define the harmonic oscillator, allowing for continued application and analysis within the framework of harmonic oscillator theory. |
In pursuit of a deeper understanding of Davidson's potential, we seek to determine its eigenfunctions and eigenvalues. By solving the Schrödinger equation for the Davidson potential, we seek to uncover the unique wave functions corresponding to the potential's discrete energy levels. Simultaneously, the eigenvalues, representing the corresponding energy levels, paint a picture of the potential's energetic landscape, revealing the energy requirements for transitions between different states.
3. Eigenfunctions and values of the Davidson potential
In this section, we delve into the process of determining the eigenfunctions and eigenvalues of the Davidson potential, as expressed in equation (2.6 ). By employing the wave function ${\rm{\Psi }}({\boldsymbol{r}})={\rm{R}}(r){{\rm{Y}}}_{{lm}}(\theta ,\phi )=\tfrac{u(r)}{r}{{\rm{Y}}}_{{lm}}(\theta ,\phi )$, we derive the corresponding Schrödinger equation, which serves as the foundation for our investigation. This equation encapsulates the interplay between the particle's kinetic energy and the Davidson potential, providing a mathematical framework for exploring the potential's energy landscape, as
$\begin{eqnarray}\displaystyle \frac{{{\rm{d}}}^{2}u(r)}{{\rm{d}}{r}^{2}}+\left[\varepsilon -{{ar}}^{2}-\displaystyle \frac{b+l(l+1)}{{r}^{2}}\right]u(r)=0,\end{eqnarray}$
where to facilitate the analysis of the Schrödinger equation and simplify the mathematical treatment, we introduce the dimensionless energy variable $\varepsilon =\tfrac{2{m}_{N}^{* }}{{{\hslash }}^{2}}E$ and the dimensionless parameter $a=\tfrac{2{m}_{N}^{* }}{{{\hslash }}^{2}}{a}_{0},b=\tfrac{2{m}_{N}^{* }}{{{\hslash }}^{2}}{b}_{0}$, where ${m}_{N}^{* }$ denotes the effective nucleon mass, ℏ is the reduced Planck constant, E is the total single-particle energy, and l is the angular momentum quantum number. These redefinitions simplify the form of the differential equation and enable a more tractable analysis.Furthermore, we introduce a new variable r2 = x to transform the differential equation into a more manageable form. This substitution allows us to express the differential equation in terms of the new variable x, leading to a simplified expression that facilitates further investigation.
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{{\rm{d}}}^{2}u(x)}{{\rm{d}}{x}^{2}}+\displaystyle \frac{1}{2}\displaystyle \frac{{\rm{d}}u(x)}{{\rm{d}}x}\\ \,+\displaystyle \frac{1}{4{x}^{2}}\left[-{{ax}}^{2}+\varepsilon x-b-l(l+1)\right]u(x)=0.\end{array}\end{eqnarray}$
This differential equation can be solved using the well-known Nikiforov–Uvarov (NU) method [34, 35]. By applying the NU method, we can derive the eigenfunctions and eigenvalues as $\begin{eqnarray}\begin{array}{l}{{\rm{R}}}_{{nl}}(r)={N}_{{nl}}\,\displaystyle \frac{{r}^{\tfrac{1}{2}\left(1+\sqrt{4b+{\left(1+2l\right)}^{2}}\right)}}{r}\\ \,\times \,\exp \left(\displaystyle \frac{-\sqrt{a}}{2}{r}^{2}\right){L}_{n}^{\tfrac{1}{2}\sqrt{4b+{\left(1+2l\right)}^{2}}}\left(\sqrt{a}{r}^{2}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\varepsilon =\sqrt{a}\left(4n+2+\sqrt{4b+{\left(1+2l\right)}^{2}}\right).\end{eqnarray}$
In the context of the harmonic oscillator, it is evident that as the parameter b approaches zero, the conventional harmonic oscillator results are regained. The influence of this parameter on the outcomes will now be examined. Without this parameter, the top argument of Laguerre polynomials is solely determined by the angular momentum value, but this is no longer the case. The overlap between two given states can be manipulated using this parameter. However, orthogonality is not a concern because it will persist for any given value of the parameter.The introduction of a new basis with similar characteristics to the harmonic oscillator, but incorporating an additional adjustable parameter, presents a promising avenue for further exploration. This new basis will be subjected to a thorough examination in the subsequent section, where its predictions will be compared against both experimental data and those generated by the harmonic oscillator model.
4. Numerical results
This section consists of four subsections, each of which aims to showcase the applications of the Davidson basis and compare the results with its counterpart in the harmonic oscillator basis. Before presenting the results, the relevant theoretical information is presented, both to lay the foundation for the calculations and to facilitate the comparison of the two bases.
4.1. Examination of new bases for one-particle nuclei
To initiate an assessment of the newly proposed basis sets, we will focus on a selection of one-particle nuclei. The primary objective is to replicate the electromagnetic transitions observed in these nuclei, as documented in experimental reports. This replication will be carried out utilizing both the harmonic oscillator basis sets and the Davidson bases set. Subsequently, a comparative analysis of the predictions generated by these different basis sets will be conducted, followed by a comprehensive discussion of the results obtained. The explanations provided in the subsequent sections will hold true for both one-particle and one-hole nuclei. Therefore, we will limit our detailed explanations to one-particle nuclei and merely present the final outcomes for one-hole nuclei.
A one-particle nucleus is characterized by the presence of a single particle outside of a closed nuclear shell [1–7]. This concept is exemplified by nuclei such as 17O, 17F, 41Ca, and 41Sc, which each possess either a proton or a neutron residing beyond the confines of their respective closed shells. This unique configuration leads to distinct properties for the states of these nuclei, which will be subsequently explored. So for these ones we have
$\begin{eqnarray}| \pi \rangle ={c}_{\pi }^{\dagger }| \mathrm{CORE}\rangle ,\quad | \nu \rangle ={{\rm{c}}}_{\nu }^{\dagger }| \mathrm{CORE}\rangle ,\end{eqnarray}$
where the particle creation operator is denoted by ${c}_{}^{\dagger }$, and the symbols π and ν represent protons and neutrons, respectively. Upon examining the experimental data for electromagnetic transitions in these nuclei, it is evident that 17O and 17F exhibit a 1/2+ → 5/2+ transition within the E2 mode, while 41Ca and 41Sc display a 3/2− → 7/2− transition within the same mode. The values for these transitions are provided in [36] with the exception of 41Sc. Our discussion will focus on the remaining nuclei. The objective is to replicate the experimental data values for the relevant electromagnetic transitions using the effective charges defined as [1–7]: $\begin{eqnarray}{e}_{\mathrm{eff}}^{{\rm{p}}}=(1+\chi )e,\end{eqnarray}$
$\begin{eqnarray}{e}_{\mathrm{eff}}^{{\rm{n}}}=\chi e,\end{eqnarray}$
where e represents the electric charge and χ denotes the polarization constant. While χ can be considered as χ = −Z/A, other values may also be applicable. For these nuclei, a preferred value of χ would lie between 0.1 and 0.3, as larger values indicate deformation properties of the nuclei, which are not expected given their proximity to closed shells.The reduced transition probability is [1–7]
$\begin{eqnarray}B(\sigma \lambda ;{\xi }_{i}{J}_{i}\to {\xi }_{f}{J}_{f})=\displaystyle \frac{1}{2{J}_{i}+1}{\left|({\xi }_{f}{J}_{f}\parallel {{ \mathcal M }}_{\sigma \lambda }\parallel {\xi }_{i}{J}_{i})\right|}^{2},\end{eqnarray}$
with the usual nations for the electric (Q) and the magnetic (M) operators $\begin{eqnarray}{{ \mathcal M }}_{{\rm{E}}\lambda }={{\boldsymbol{Q}}}_{\lambda },\qquad {{ \mathcal M }}_{{\rm{M}}\lambda }={{\boldsymbol{M}}}_{\lambda },\end{eqnarray}$
and the order of the transition λ. The single-particle matrix elements of the multipole operators are written as $\begin{eqnarray}({\xi }_{f}{J}_{f}\parallel {{ \mathcal M }}_{\sigma \lambda }\parallel {\xi }_{i}{J}_{i})={\hat{\lambda }}^{-1}\displaystyle \sum _{{ab}}(a\parallel {{ \mathcal M }}_{\sigma \lambda }\parallel b)({\xi }_{f}{J}_{f}\parallel {[{c}_{a}^{\dagger }{\tilde{c}}_{b}]}_{\lambda }\parallel {\xi }_{i}{J}_{i}),\end{eqnarray}$
where using the Baranger notation we have $\begin{eqnarray}{\tilde{c}}_{\alpha }={\left(-1\right)}^{{j}_{a}+{m}_{\alpha }}{c}_{-\alpha },\quad {c}_{-\alpha }={c}_{a,-{m}_{\alpha }},\end{eqnarray}$
and the single-particle matrix element is shown by $(a\parallel {{ \mathcal M }}_{\sigma \lambda }\parallel b)$. Using Wigner–Eckart theorem and some straightforward calculation, it would be easily shown that $\begin{eqnarray}({\xi }_{f}{J}_{f}\parallel {{ \mathcal M }}_{\sigma \lambda }\parallel {\xi }_{i}{J}_{i})=(a\parallel {{ \mathcal M }}_{\sigma \lambda }\parallel b).\end{eqnarray}$
Given the dominance of single-particle excitations in these nuclei, the focus of our calculations will be on single-particle matrix elements. We will proceed to evaluate electromagnetic transitions in one-particle nuclei using the Davidson bases and compare the results to those obtained using the harmonic oscillator basis.
To understand the determination of the polarization constant, it is essential to grasp the reasoning behind effective charges. In electric transitions, the mediator of the transition is the electric charge, e. Protons possess an electric charge, while neutrons do not. Based on this principle, no electric transition should occur for one-neutron nuclei. However, experimental evidence contradicts this assumption. Therefore, the concept of effective charges suggests that the mediator of the transition is not directly proportional to the total electric charge of the mediator. Instead, it may be proportional to a fraction of the electric charge. Since this auxiliary definition should not contradict the original definition of the electric charge, finding small or negligible values for the polarization constant is favorable. For instance, χ ≈ 0.1 is considered acceptable, and we should strive to find values of the constant such that χ → 0, aligning with the original definition of electric charges for nucleons. Table 1 presents three nuclei, with the first column listing the nucleus name. The second and third columns indicate the initial and final states, respectively. The fourth column specifies the mode of the transitions. The numerical values of a, b, χs are presented in the fifth, sixth, and seventh columns. Lastly, the experimental values of the transition are given in the last column. An intriguing observation from this table is that one-neutron nuclei exhibit higher values of χ than one-proton nuclei. This pattern will also be observed for one-hole nuclei. Notably, the polarization constants in this table are remarkably small considering the relevant values for the Davidson bases. As is evident from this table, it is clear that Davidson bases provide much smaller values for the polarization constant while those values for the harmonic oscillator in all cases are larger. It is important to note that negative values of the χ may not be considered.
Table 1. This table presents the relevant information for three one-particle nuclei. The values presented in this table represent the successful reproduction of experimental data using Davidson bases. |
| Nucleus | ∣i⟩ | ∣f⟩ | σ λ | a | b | χD | χHO | $\mathrm{Exp}.\,({e}^{\lambda }{\mathrm{fm}}^{2\lambda })$ |
|---|---|---|---|---|---|---|---|---|
| ${}_{8}^{17}{{\rm{O}}}_{9}$ | 1s1/2 | 0d5/2 | E2 | 0.001 | 0.001 | 0.051 | 0.538 | 6.294 |
| ${}_{9}^{17}{{\rm{F}}}_{8}$ | 1s1/2 | 0d5/2 | E2 | 0.38 | 0.30 | 0.003 | {−1.537, −0.463} | 6.912 |
| ${}_{20}^{41}{\mathrm{Ca}}_{21}$ | 1p3/2 | 0f7/2 | E2 | 0.001 | 0.001 | 0.092 | 0.392 | 48.707 |
Having thoroughly examined the one-particle nuclei, let us now shift our focus to the one-hole nuclei.
4.2. Examination of new bases for one-hole nuclei
The theoretical approach for one-hole nuclei shares similarities with that of one-particle nuclei, albeit with certain distinctions. One-hole nuclei are characterized by the absence of a single nucleon within a closed shell. Examples of such nuclei include 15O, 15N, 39Ca, and 39K. These nuclei possess one proton or neutron less than a closed shell configuration. Consequently, their state can be symbolically represented as [1–7]4.4 ) is employed once again, taking into account the specific considerations applicable to one-hole nuclei. In this case we have
$\begin{eqnarray}| {\pi }^{-1}\rangle ={h}_{\pi }^{\dagger }| \mathrm{CORE}\rangle ,\quad | {\nu }^{-1}\rangle ={{\rm{h}}}_{\nu }^{\dagger }| \mathrm{CORE}\rangle .\end{eqnarray}$
For the calculation of the reduced transition probability, equation ( $\begin{eqnarray}({\xi }_{f}{J}_{f}\parallel {{ \mathcal M }}_{\sigma \lambda }\parallel {\xi }_{i}{J}_{i})={\left(-1\right)}^{{j}_{i}+{j}_{f}+\lambda }(a\parallel {{ \mathcal M }}_{\sigma \lambda }\parallel b).\end{eqnarray}$
An evident observation from the calculations is the presence of a phase difference between the one-particle and one-hole nuclei. While this phase difference may not be of significant consequence in the current context, it assumes paramount importance in applications involving configuration mixing.Our focus now shifts to the transitions observed in experiments. For 15O and 15N, the ${(3/{2}^{-})}^{-1}\to {(1/{2}^{-})}^{-1}$ transition has been experimentally observed and its strength measured. Similarly, for 39Ca and 39K, the ${(1/{2}^{+})}^{-1}\to {(3/{2}^{+})}^{-1}$ transition has been measured. Analogous to table 1, we have tabulated the results obtained using Davidson and harmonic oscillator bases to facilitate a comparison between these two different approaches.
Table 2. This table presents the relevant information for three one-hole nuclei. The values presented in this table demonstrate the successful reproduction of experimental data using Davidson bases. For further details regarding the values marked with *, please refer to the main text. |
| Nucleus | ∣i⟩ | ∣f⟩ | σ λ | a | b | χD | χHO | Exp. (eλfm2λ) |
|---|---|---|---|---|---|---|---|---|
| ${}_{7}^{15}{{\rm{N}}}_{8}$ | 1p3/2 | 1p1/2 | E2 | 0.33 | 2.37 | 0.000 | 0.221 | 6.394 |
| ${}_{8}{}^{15}{\rm{O}}_{7}$ | 1p3/2 | 1p1/2 | E2 | 0.01 | 250 | 0.055 | 1.266 | >0.615* |
| ${}_{19}^{39}{{\rm{K}}}_{20}$ | 1s1/2 | 0d3/2 | E2 | 0.039 | 0.010 | 0.000 | 0.355 | 40.852 |
| ${}_{20}^{39}{\mathrm{Ca}}_{19}$ | 1s1/2 | 0d3/2 | E2 | 0.001 | 0.001 | 0.156 | 0.311 | 38.294* |
For 15O, the exact values of B(E2) or B(M1) for the considered transition are unavailable, while its δ parameter has been experimentally determined to be δ = +0.125. Therefore, the theoretical values presented in table 2 for the 15O row have been obtained by reproducing the experimental δ parameter.
Table 3. Probabilities for electromagnetic transitions of multipolarity λ, denoted by Eλ, are presented. The energy values are expressed in megaelectronvolts (MeV). |
| Eλ | T(Eλ) (s−1) |
|---|---|
| E1 | 1.587 × 1015E3 B (E1) |
| E2 | 1.223 × 109E5 B (E2) |
| E3 | 5.698 × 102E7 B (E3) |
| E4 | 1.694 × 10−4E9 B (E4) |
| E5 | 3.451 × 10−11E11 B (E5) |
A similar situation exists for 39Ca. While experimental data for the considered transition is lacking, the half-life of the initial state is known to be 162 × 10−15 s. For our purposes, we assume that this transition proceeds exclusively via the E2 mode. Consequently, the value of B(E2) for the transition is estimated to be 38.294 e2fm4. This estimation is consistent with values reported in various nuclear physics textbooks, as indicated in table 3.
Furthermore, akin to the one-particle case, the polarization constant for each one-hole nucleus is significantly smaller when evaluated using Davidson bases compared to harmonic oscillator bases. This remarkable achievement is attributed to the incorporation of a hardcore in the Davidson bases.
After evaluating applications of the Davidson bases in the gamma decay, we are ready to explore another applications in the subsequent section.
4.3. First-forbidden unique β decay in one-hole nuclei
Another compelling application of Davidson bases is to investigate the β-decay of a nucleus in the shell model. This type of decay typically manifests in three distinct scenarios, two of which exhibit identical outcomes, while the third produces unique results. The difference between the kinetic energies of the particles before and after the decay is termed the Q-value of the decay. To provide a theoretical foundation for this section, we offer a concise review of this decay process.
In β− decay, a neutron undergoes transformation into a proton, an electron, and an antineutrino. This decay is energetically feasible for a free neutron as the neutron is heavier than the proton, resulting in a positive Q-value. Conversely, β+ decay involves a proton's conversion into a neutron, accompanied by the emission of a positron and an electron neutrino. The negative Q-value for β+ decay signifies that the process requires additional energy, which can only be provided within the confines of a nucleus. Thus, β+ decay is exclusively observed in nuclear environments. The third form of β decay, electron capture (EC), bears similarities to β+ decay. In EC, a proton captures an orbiting electron, resulting in the formation of a neutron and an electron neutrino. Regarding the effect of β decay on the atomic number of a nucleus, β− decay leads to an increase in atomic number by one, while both β+ decay and EC result in a decrease in atomic number by one.
In addition to the three primary forms of β decay, a further classification scheme exists based on the decay's $\mathrm{log}{ft}$ value. This quantity, experimentally determined and reported, serves as an indicator of the decay's half-life. From a theoretical perspective, the $\mathrm{log}{ft}$ value can be reproduced by analyzing the nucleus's structure through intricate mathematical manipulations. The table 4 provides a comprehensive overview of the β-decay classification based on $\mathrm{log}{ft}$ values.
Table 4. Classification of β decay transitions according to their $\mathrm{log}{ft}$ values |
| Type of transition | $\mathrm{log}\,{ft}$ |
|---|---|
| Superallowed | 2.9–3.7 |
| Unfavoured allowed | 3.8–6.0 |
| l-forbidden allowed | ≥5.0 |
| First-forbidden unique | 8–10 |
| First-forbidden non-unique | 6–9 |
| Second-forbidden | 11–13 |
| Third-forbidden | 17–19 |
| Fourth-forbidden | >22 |
As extensively discussed in the literature [37–40], allowed transitions in β decay primarily depend on the properties of the initial and final nuclear states, with the wave functions playing a negligible role. However, the overlap of wave functions becomes crucial in non-unique β decays, where the transition involves multiple operators. This class of transitions presents a significant challenge due to the involvement of six operators. While calculating matrix elements to reproduce the ft value remains complex, a significant simplification arises when only one operator is active, leading to ΔJ = 2 transitions. In this scenario, the matrix elements assume the following form [37–40]
$\begin{eqnarray}\begin{array}{l}{{ \mathcal M }}_{1{\rm{u}}}=\displaystyle \frac{{m}_{{\rm{e}}}{c}^{2}}{\sqrt{4\pi }}\zeta ({\xi }_{f}\,{J}_{f}\parallel {[{\boldsymbol{\sigma }}{\boldsymbol{r}}]}_{2}\parallel {\xi }_{i}\,{J}_{i})\\ \,=\,\displaystyle \sum _{{ab}}{{ \mathcal M }}^{(1{\rm{u}})}({ab})({\xi }_{f}\,{J}_{f}\parallel {[{c}_{a}^{\dagger }{\tilde{c}}_{b}]}_{2}\parallel {\xi }_{i}\,{L}_{i}),\end{array}\end{eqnarray}$
where the Pauli spin matrices are represented by σ, and ζ = 1 in the Condon-Shortly convention. Consequently, the reduced probability and the ft value can be expressed as follows $\begin{eqnarray}{f}_{K{\rm{u}}}{t}_{1/2}=\displaystyle \frac{\kappa }{{B}_{K{\rm{u}}}},\quad {B}_{K{\rm{u}}}=\displaystyle \frac{{1.25}^{2}}{2{J}_{i}+1}| {{ \mathcal M }}_{K{\rm{u}}}{| }^{2},\end{eqnarray}$
in which the parameter κ takes a value of 6147 s in this case. The general form of the shape function is employed to describe this type of transition, capturing the intricacies of the transition process and providing insights into the nuclear structure $\begin{eqnarray}\begin{array}{l}{S}_{K{\rm{u}}}^{(\mp )}({Z}_{f},\varepsilon )\approx {F}_{0}(\pm {Z}_{f},\varepsilon )p\varepsilon {\left({E}_{0}-\varepsilon \right)}^{2}\displaystyle \sum _{{k}_{{\rm{e}}}+{k}_{\nu }=K+2}\\ \displaystyle \frac{{\left({\varepsilon }^{2}-1\right)}^{{k}_{{\rm{e}}}-1}{\left({E}_{0}-\varepsilon \right)}^{2({k}_{\nu }-1)}}{(2{k}_{{\rm{e}}}-1)!(2{k}_{\nu }-1)!},\end{array}\end{eqnarray}$
where K is the order of forbidden, kν, ke are integer starting from zero and $\begin{eqnarray}{F}_{0}({Z}_{f},\varepsilon )\approx \displaystyle \frac{\varepsilon }{p}{F}_{0}^{(\mathrm{PR})}({Z}_{f}),\quad {F}_{0}^{(\mathrm{PR})}({Z}_{f})=\displaystyle \frac{2\pi \alpha {Z}_{f}}{1-\exp \left(-2\pi \alpha {Z}_{f}\right)},\end{eqnarray}$
where PR indicate to the Primakoff–Rosen approximation.The integration of the shape function, conventionally referred to as the phase-space factor, takes the following form
$\begin{eqnarray}{f}_{K{\rm{u}}}^{(\pm )}={\left(\displaystyle \frac{3}{4}\right)}^{K}\displaystyle \frac{(2K)!!}{(2K+1)!!}{\int }_{1}^{{E}_{0}}{S}_{K{\rm{u}}}^{(\mp )}({Z}_{f},\varepsilon )\,{\rm{d}}\varepsilon ,\end{eqnarray}$
with the following relation with the first-forbidden unique type $\begin{eqnarray}{f}_{K=1,{\rm{u}}}^{(\mp )}=\displaystyle \frac{1}{12}{f}_{1,{\rm{u}}}^{(\mp )}.\end{eqnarray}$
For the EC the phase factor is $\begin{eqnarray}{f}_{K{\rm{u}}}^{(\mathrm{EC})}=\displaystyle \frac{2(2K)!!}{(2K+1)!!(2K+1)!}\pi {\left(\alpha {Z}_{i}\right)}^{3}{\left({\varepsilon }_{0}+{E}_{0}\right)}^{2(K+1)}.\end{eqnarray}$
with ${\varepsilon }_{0}\approx 1-\tfrac{1}{2}{(\alpha {Z}_{i})}^{2}$, and the endpoint energy E0.To assess the Davidson bases' efficacy, we analyze the first-forbidden unique transition of ${}_{20}^{39}{\mathrm{Ca}}_{19}$, a one-hole nucleus since no non-unique transitions for corresponding one-particle nuclei were reported in [36]. We focus on the first-forbidden unique β+/EC decay in 39Ca. This transition, represented by ${}^{39}\mathrm{Ca}(3/{2}^{+}){\mathop{\longrightarrow }\limits^{{\beta }^{+}/\mathrm{EC}}}^{39}{\rm{K}}(7/{2}^{-})$, possesses an experimentally determined $\mathrm{log}{ft}$ value of $\mathrm{log}{ft}(\exp )\gt 9$ [36]. Employing the harmonic oscillator bases, we calculate $\mathrm{log}{ft}(\mathrm{HO})=8.44803$, indicating a deviation from the experimental value. To better approximate the experimental data using Davidson bases, we set the parameters, for instance, a = 2, b = 1.8. This yields $\mathrm{log}{ft}({\rm{D}})=9.1309$, significantly closer to the experimental $\mathrm{log}{ft}$ value. This point should be mentioned if one continues investigating different values other than what we found, obliviously, some other combinations may lead to the $\mathrm{log}{ft}$ values greater than 9. We mention that using the Davidson base has the flexibility and capability to reproduce such high values of $\mathrm{log}{ft}$.
5. Conclusion
The primary objective of this research was to develop new bases for shell model calculations. To achieve this goal, we initially considered the harmonic oscillator potential and subsequently incorporated a centrifugal term. This modified potential gave rise to new bases that surpassed the capabilities of the harmonic oscillator bases. As a preliminary exploration of these new bases, we examined a selection of one-particle and one-hole nuclei. Experimental data for these nuclei were successfully reproduced using both the Davidson and harmonic oscillator bases. The results obtained with the Davidson bases exhibited a more consistent physical interpretation compared to those obtained with the harmonic oscillator bases. This favorable outcome can be attributed to the inclusion of a hardcore term in the mean-field potential, which is effectively perceived by the single particle. This study demonstrates the potential of the Davidson bases to replace the harmonic oscillator bases, and we anticipate that further investigations will yield even more accurate results. To illustrate the another application of Davidson bases, we investigated the first-forbidden β-decay in 39Ca and contrasted the outcomes with those obtained using harmonic oscillator bases. This comparison revealed the superior performance of Davidson bases over harmonic oscillator bases in modeling β-decay phenomena.