1. Introduction
The Schrödinger equation, being a source of information in quantum mechanics, can only be solved exactly for few potentials in all partial waves and energies. They are Coulomb, harmonic oscillator, square well and ${r}^{-4}$ potentials [1]. However, the study of exponential type potentials like the Hulthén potential [2], Eckart potential [3], Manning–Rosen potential [4], Rosen-Morse potential [5, 6], multi-parameter exponential type potentials [7–10], etc, applicable for various quantum mechanical systems, cannot be solved exactly for all partial waves and at all energies. Rather, these exponential potentials are exactly solvable for a particular partial wave $\left({\ell }=0\right).$ However, approximate solutions in all partial waves have been derived by several authors [11–15] with a screened centrifugal barrier.
For low-energy NN scattering the long range part of the interaction is described by certain degrees of freedom which are related to spin and isospin symmetries. However, for the intermediate and high-energy region of NN scattering, meson and quark degrees of freedom play a significant role [16, 17]. With the discovery of heavy mesons, some potential such as the Partovi-Loman model [18], Stony Brook-group [19], Paris group [20], Nijmegen group [21] and Bonn group [22] interactions were suggested to describe the NN system. But there were still problems with these potentials. Afterwards, lots of theoretical efforts were made to develop high-precision parameterized potentials such as the Paris potential [23], Nijm 93, Nijm-I, Nijm-II [24], CD-Bonn [25], and many others [26–29]. These phenomenological potentials involve several free parameters to be fitted to the experimental scattering data and phase shifts. Their ability to describe the practical facts of NN interaction, their flexibilities and convenience for using nuclear structure calculation are remarkable.
The Manning–Rosen potential [4, 14, 15], a three-parameter one, is in general used in molecular dynamics. In the recent past we have successfully applied it in the nuclear domain [30, 31] for treating both deuteron and triton. For these systems both the bound and scattering state observables were reproduced quite efficiently. Curiously enough, we propose to study the charged hadron system using Manning–Rosen plus electromagnetic interaction.
The Manning–Rosen potential has become a source of information for many researchers, for example: Diaf et al [32] through their investigation on the S-wave Schrödinger equation by the path integral method, Dong and Ravelo [33] for a bound state solution for S-wave using this potential, and Sameer et al by the Nikiforov-Uvarov numerical method [34] for approximate solution of ${\ell }\,th$ -states solutions of the D-dimensional Schrödinger equation. Besides the bound state solutions of the Schrödinger equation using the Manning–Rosen potential as reported above, the scattering state solutions are also an important point of interest to the physicist working in this direction. By using this potential, Chen et al [35] have obtained the exact scattering state solution for the S-wave only. With appropriate approximation of the centrifugal term, Wei et al [36] have found an approximate analytical scattering state solution of the ${\ell }\,th$ –partial wave Schrödinger equation for the Manning–Rosen potential [4]. All these works are being completed by using this potential in the context of their applications to atomic physics. In the present text, we apply this potential to the field of nuclear scattering theory for computing phase shifts, by proper utilization of the phase function method (PFM) [37], for the nucleon–nucleon and the alpha-nucleon systems. In section 2 we briefly outline the phase function method. In section 3 we present the results and summary.
2. Phase function method
The ${\ell }\,th$ partial wave Manning–Rosen potential [4], which is considered as the nuclear part of the interaction, is given by $ \begin{eqnarray}\begin{array}{lll}{V}_{N}\left(r\right) & = & {b}^{-2}\left[\displaystyle \frac{\alpha \left(\alpha -1\right)}{{\left(1-{{\rm{e}}}^{-r/b}\right)}^{2}}{{\rm{e}}}^{-2r/b}\right.\\ & & \left.+\displaystyle \frac{{\ell }\left({\ell }+1\right)}{{\left(1-{{\rm{e}}}^{-r/b}\right)}^{2}}{{\rm{e}}}^{-2r/b}-\displaystyle \frac{{{A}{\rm{e}}}^{-r/b}}{1-{{\rm{e}}}^{-r/b}}\right],\end{array}\end{eqnarray}$ where A and α are two dimensionless parameters and b has the dimension of length. Here we consider a screened centrifugal barrier. For a charged hadron system one must add an electromagnetic potential to the nuclear part. For electromagnetic interaction we chose a screened Coulomb potential, the Hulthén one [2]. Therefore, the effective potential is $ \begin{eqnarray}V\left(r\right)\,=\,{V}_{N}\left(r\right)+{V}_{H}\left(r\right).\end{eqnarray}$
The atomic Hulthén potential is written as $ \begin{eqnarray}{V}_{H}\left(r\right)\,=\,{V}_{o}\displaystyle \frac{{{\rm{e}}}^{-r/a}}{{1 \mbox{-} {\rm{e}}}^{-r/a}}.\end{eqnarray}$ The quantities ${V}_{o}$ and $a$ have their usual meaning. For the potentials ${V}_{N}\left(r\right)$ and $V\left(r\right)$ the phase shifts will be computed for the systems under consideration by applying the standard prescription, the phase function method (PFM) [37].
The phase equation of Calegero [37] for a local potential is written as $ \begin{eqnarray}\begin{array}{r}{{\delta }_{{\ell }}}^{^{\prime} }\left(k,r\right)\,=\,-{k}^{-1}V\left(r\right)\left[\cos \,{\delta }_{{\ell }}\left(k,r\right){\hat{j}}_{{\ell }}\left(kr\right)\right.\\ \,\,\,{\left.-\sin {\delta }_{{\ell }}\left(k,r\right){\hat{\eta }}_{{\ell }}\left(kr\right)\right]}^{2}\end{array},\end{eqnarray}$ where ${\hat{j}}_{{\ell }}\left(kr\right)$ and ${\hat{\eta }}_{{\ell }}\left(kr\right)$ are the Riccati–Bessel functions [38]. Equation (4 ) for ${\ell }\,=\,0,\,1,\,2$ takes the form $ \begin{eqnarray}{{\delta }_{0}}^{{\prime} }\left(k,r\right)=-{k}^{-1}V\left(r\right)\,{\left[\sin \left({\delta }_{0}\left(k,r\right)+kr\right)\right]}^{2},\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{lll}{{\delta }_{1}}^{^{\prime} }\left(k,r\right) & = & -\displaystyle \frac{V\left(r\right)}{{k}^{3}{r}^{2}}\left[\sin \left({\delta }_{1}\left(k,r\right)+kr\right)\right.\\ & & {\left.-kr\cos \left({\delta }_{1}\left(k,r\right)+kr\right)\right]}^{2}\end{array}\end{eqnarray}$ and $ \begin{eqnarray}\begin{array}{lll}{{\delta }_{2}}^{^{\prime} }\left(k,r\right) & = & -{k}^{-1}V\left(r\right)\,\left[\left(\displaystyle \frac{3}{{k}^{2}{r}^{2}}-1\right)\sin \left({\delta }_{2}\left(k,r\right)+kr\right)\right.\\ & & {\left.-\displaystyle \frac{3}{kr}\cos \left({\delta }_{2}\left(k,r\right)+kr\right)\right]}^{2},\end{array}\end{eqnarray}$ here prime denotes differentiation with respect to $r.$ The quantity $k$ stands for the center of mass momentum and is related to the center of mass energy by $k=\sqrt{2{\rm{mE}}}/\hslash .$ The laboratory energy ELab is related to the center of mass energy E by standard relation. To solve these equations numerically, we develop the solution ${\delta }_{{\ell }}\left(k,r\right)$ starting from $r=0$ to infinity to have the scattering phase shifts for the associated partial wave states.
3. Results and summary
The parameters for the various systems with different states are given in table 1.
Table 1. List of parameters for various systems and states. |
| System | State | A | b (fm) | α |
|---|---|---|---|---|
| 1S0 | 0.952 | 1.152 | −0.0043 | |
| 3S1 | 1.57 | 1.213542 | 0.005 | |
| 1P1 | 0.05 | 1.1 | 2.34 | |
| 3P0 | 1.32 | 1.1 | 0.005 | |
| n-p | 3P1 | 0.05 | 1.1 | 2.5099 |
| 3P2 | 0.75 | 1.01 | 0.005 | |
| 1D2 | 1.05 | 1.59 | 0.82 | |
| 3D1 | 0.05 | 1.59 | 3.2 | |
| 3D2 | 2.45 | 1.59 | 0.82 | |
| 3D3 | 0.95 | 1.59 | 0.96 | |
| 1S0 | 0.952 | 1.152 | −0.0043 | |
| 3P0 | 1.32 | 1.1 | 0.005 | |
| p-p | 3P1 | 0.05 | 1.1 | 2.5099 |
| 3P2 | 0.75 | 1.01 | 0.005 | |
| 1D2 | 1.05 | 1.59 | 0.82 | |
| 1/2(+) | 2.189 | 1.215 | 0.005 | |
| 1/2(−) | 3.71 | 1.215 | 0.005 | |
| α-n | 3/2(−) | 5.72 | 1.03 | 0.005 |
| 3/2(+) | 1.15 | 1.03 | 0.005 | |
| 5/2(+) | 1.85 | 1.03 | 0.005 | |
| 1/2(+) | 2.529 | 1.215 | 0.005 | |
| 1/2(−) | 3.85 | 1.215 | 0.005 | |
| α-p | 3/2(−) | 5.857 | 1.03 | 0.005 |
| 3/2(+) | 2.535 | 1.03 | 0.005 | |
| 5/2(+) | 2.95 | 1.03 | 0.005 |
The parameters in table 1 along with equations (2 )–(4 ) are utilized to compute scattering phase shifts for the nucleon–nucleon and the alpha-nucleon systems up to partial waves ${\ell }=2.$ We calculate phase shifts with $\tfrac{{\hslash }^{2}}{{m}_{p}}=41.47\,{\rm{MeV}}\,{{\rm{fm}}}^{2}$ and ${V}_{o}a=2k\eta =0.03472\,{{\rm{fm}}}^{-1}\,\&\,0.1117\,{{\rm{fm}}}^{-{\rm{1}}}$ for (p-p) and (α-p) systems respectively [39–42]. Here the screening radius $a$ of the Hulthén potential is considered to be 20 fm. The parameters for the 3S1 (n-p) state have been fixed by fitting deuteron binding energy [30, 31]. The 1S0 (n-p) state parameters are also taken from [30, 31]. For other states for both nucleon–nucleon and alpha-nucleon systems, we have given free running to the parameters in our numerical routines to fit the proper phase shifts as far as possible. The nucleon–nucleon and the alpha-nucleon scattering phase shifts are presented in figures 1–6. It is noticed that our results for the phase shifts agree quite well with those of Arndt et al [43], Gross and Stadler [44] and Satchler et al [45]. Looking at figure 2 it is seen that our 1P1 and 3P1 results show slight a difference from those of [43, 44]. These differences are restricted within 2°. From figure 6 it is seen that our phase shift values for 1/2+ state of (α-p) system differ by 5°–6° at very low energies (up to 3 MeV), and beyond that they are in close agreement with those of [45], while for the (α-n) system our results for 1/2+ state are in exact agreement with Satchler et al [45] at low energies. Similarly, for the (α-p) case small differences are observed at the peak values for the 3/2− state compared to the (α-n) case. This may be attributed to the fact that at low energies the Coulomb force plays a dominant role over the nuclear one. Although, the phase shift results differ by a narrow margin in numerical values, reproducing the correct trends.
The potentials for the associated partial wave states are portrayed in figures 7–11 and we found that they are fully consistent with the phase shifts produced. From the foregoing discussion it is clear that the parameterization of the Manning–Rosen potential for nuclear systems is in order and can easily be extended for nucleus-nucleus elastic scattering. We can conclude that our study on the three-parameter Manning–Rosen potential in the context of nuclear physics is expected to explore new possibilities for pure theoretical and experimental physicists, because the results are in close agreement with experimental/standard data obtained from several-parameter interactions.
Figure 7. S-wave (n-p) and (p-p) potentials as a function of r. |
Figure 8. P-wave (n-p) and (p-p) potentials as a function of r. |
Figure 9. D-wave (n-p) and (p-p) potentials as a function of r. |
Figure 10. Alpha-n and alpha-p potentials for the 1/2+ states as a function of r. |
Figure 11. Alpha-n and alpha-p potentials for the 3/2−, 1/2− states as a function of r. |