1. Introduction

Due to the mechanism of pion-nucleon interaction, with the pion’s mean free path associated with the pion’s incident kinetic energy Tπ, the pion-nucleus elastic-scattering has been classified into three major energy bins¹, namely (I) the high energy region, Tπ>400 MeV, (II) the delta resonance energy region, 100<Tπ≤400 MeV, and (III) the low energy region, Tπ≤100 MeV.

In region I, the pion is the best and most sensitive probe of spatial distributions in a nucleus. The pion is spinless; and it penetrates deeply inside the nucleus because pion-nucleon two-body interaction is weak. In fact, it is one of the most penetrating of the strongly interacting particles. Also it is a preferred probe, compared to the proton, as no need to make antisymmetrization with the nucleons inside a nucleus. In addition, it is the best spectroscopic tool to investigate the many overlapping resonances that do exist. Nevertheless, the presence of unconventional phenomena as mesonic current contributions and modified nucleon properties in the medium could be tested².

In region II, the pion’s mean free path is small and it is comparable to the inter-nucleon distance³. So the pion is completely absorbed at the surface of the target nucleus, i.e. the scattering involves the nuclear surface. As such, the pion is an adequate probe for gleaning important nuclear peripheral information.

To facilitate the study of region III, it has been subdivided into two energy sub-regions: (III-a) the transition energy region, 50<Tπ≤100 MeV, and (III-b) the very low energy region, Tπ≤50 MeV. Here we are investigating the elastic scattering of the nuclear systems π± - ¹²C, ¹⁶O, ⁴⁰Ca at Tπ=20, 30, 40 MeV, π± - ⁹⁰Zr at Tπ=20, 30 MeV, π± - ⁵⁸Ni and π± - ⁹⁰Zr at Tπ=40 MeV, i.e. in the very low energy region. In this sub- energy region, the pion mean free path is much greater than the inter-nucleon distance which allows the pion to penetrate deeply inside the nucleus⁴. As such the pion is considered the most suitable and best informative probe for studying nuclear structure and revealing nuclear structure effects. It is worthy to point out that nuclear structure effects are obscured in the resonance region.

Recently we have analyzed low-energy pion-nucleus elastic scattering data using a suggested scaling method, based on the results of the inverse scattering theory using available phase shifts for both π± - ¹²C and π± - ¹⁶O nuclear systems⁵. It was obvious that the three changed potential parameters, R0, V1 and W3, depend on the atomic mass of the target nucleus. As such, the scaling procedure for the three potential parameters, from a certain pion-nucleus system to a nearby one, became visible. Using the scaled potential parameters for π± - ¹⁶O system from the potential parameters of π± - ¹²C system at several bombarding energies below 100 MeV, namely at Tπ=13.9, 20.0, 30.0, 35.0, 40.0, 50.0, 60.0, 65.0, 80.0 MeV in our adopted potential, our results have shown a nice agreement between the calculated differential and reaction cross sections and the measured values. Moreover, the real and imaginary parts of the scaled potential agree with the inverted potential points obtained from available phase shifts using the inverse scattering theory⁶. The obtained good results were a strong stimulus to investigate the scattering of both charged pions from different target nuclei and isotopes, namely ¹⁶O, ²⁸Si, ³⁰Si, ³²S, ³⁴S, ⁴⁰Ca, ⁴⁸Ca, ⁵⁶Fe, ⁵⁸Ni, ⁶⁴Ni, and ⁹⁰Zr, at one bombarding energy, Tπ=50.0 MeV⁷. The obtained good results of this investigation have supported the use of the scaling method, which is our new proposed method, in determining low-energy pion-nucleus reliable potentials. These determined potentials proved the strength, capability and reliability of the scaling method as they provided nice fits to the differential and integral cross sections; and showed a very reasonable agreement with the inverted potential points. As far, the uniqueness of the scaling method appears clearly in obtaining the correct potentials for low-energy pion-nucleus systems with no available shifts. At this stage, and for integrity and completeness, the achieved successful results formed a strong inducement to extend using our suggested scaling method to analyze several pion-nucleus elastic scattering data at few low energies, namely Tπ=20.0, 30.0, 40.0 MeV. Once more, this allows for an important additional test for the correctness, strength and reliability of this new method in obtaining the scaled potential parameters and, then, the correct potentials for other pion-nucleus scattering cases. Here in this investigation, we will use the scaling method to obtain the optical potentials, with real and imaginary parts, required to explain the π± - ¹²C, ¹⁶O, ⁴⁰Ca, ⁵⁶Fe, ⁵⁸Ni, and ⁹⁰Zr available data at the three energies 20, 30 and 40 MeV. Having in mind to submit our scaling method as a patent, we strongly believe that this study forms a valuable test and a complementary witness for the success of our scaling method.

In Sec. 2, theory is briefly overviewed. In Sec. 3, results and discussion are presented. In the last section, Sec. 4, the conclusions are summarized.

2. Theory

This work is a continuation to our recent work^5,7, and, as such, the theoretical background and theoretical relations are as already outlined and discussed before. Accordingly, a brief overview is provided here for completeness.

The potential adopted here, V(r), is the well-known one which is composed of the nuclear part, VN(r), and the Coulomb part, VC(r):

The analytical form of the nuclear part is given by :

where the three terms are the attractive Woods-Saxon (WS), the repulsive Squared Woods-Saxon (SWS), and the surface Woods-Saxon, respectively. The Coulomb term, VC(r), is given by:

which represents the interaction between a point charge, i.e. the incident charged pion, and a uniformly insulating charged sphere, i.e. the target nucleus of atomic number ZT. The appearing symbols e2, ε0 and RC in Eq. (3) are the squared electron charge, permittivity of free space, and the Coulomb radius. In nuclear units e2/4πε0=1.44 MeV. As usual, RC is numerically obtained from RC=1.2A1/3 with A is the target atomic mass in atomic mass units.

To achieve our goals in obtaining differential and reaction cross sections, dσ/dΩ and σr, respectively, without facing mathematical singularities or computational difficulties, V(r) is inserted into the transformed radial part of Klein-Gordon equation:

with k2 and U(r) are given by

All potential quantities, constants and scaled potential parameters, appearing explicitly or implicitly in the above equations are substituted by their numerical values. For emphasis, the quantities E, m, ℏ and c are the actual pion total energy, effective pion mass, reduced Planck constant and the velocity of electromagnetic wave in vacuum, respectively.

It is well-know that the phase shift, δl, for each contributing partial waves l, is an important ingredient in the definitions of the elastic differential cross section, dσ/dΩ:

where f(θ) is the full elastic scattering amplitude at angle θ in the center of mass system. Mathematically, it is expressed as:

where fc(θ) is the pure Coulomb scattering amplitude, Pl(cosθ) are Legendre polynomials, and σγ is the Coulomb phase shift defined by⁸:

with the dimensionless parameter γ:

Also δl appears solely in the definition of the reaction cross section, σr:

where Sl=e2iδl is the S-matrix, and δl is complex.

The determination of δl is usually done by matching the inner and outer solutions of Eq. (4) at the matching radius r=R, i.e. at the surface of the nucleus. The inner solution is obtained by integrating the equation numerically from the origin r=0 to r=R using Numerov’s method⁹. On the otherhand, the outer solution, i.e. for r≥R, considered here is the usual and familiar Coulomb wave function:

where Fl and Gl are the relativistic regular and irregular Coulomb wave functions¹⁰, respectively; and the parameter η is Sommerfeld parameter defined as:

where α is the fine structure constant.

3. Results and Discussion

As pointed out in our previous work^2,5,7, the scaling method has been successful in obtaining the correct potential parameters capable of providing nice agreements between theory and experiment for alpha-nucleus¹¹ and nucleus-nucleus^12,13 nuclear systems. In addition, the scaling method is supported by Energy-Density Functional (EDF) theory¹³. Recently it has been also successful in determining the correct potential parameters for pion-nucleus systems in the low energy region. The extent of success, strength, capability and reliability of the scaling method are examined, here, by explaining the elastic-scattering angular distributions data for the pion-nucleus systems under consideration, namely π± - ¹²C, ¹⁶O, ⁴⁰Ca at Tπ=20, 30, 40 MeV π± - ⁹⁰Zr at Tπ=20, 30 MeV, π- - ⁵⁸Ni and π- - ⁹⁰Zr at Tπ=40 MeV. It is of importance to notice that these systems include a) bombarding pions with different polarities, b) different incident pion’s kinetic energies, and c) different target nuclei spanning the periodic table. Nice fits, using scaled parameters embedded in our potential within the framework of full Klein-Gordon equation, are achieved and, as such, support our suggested scaling method as a credible benchmark. The general trends of the potential parameters make it possible to predict the parameters for other systems as π± - ²⁸Si, ³²S, ⁵⁶Fe at the three energies 20, 30 and 40 MeV, π± - ⁵⁸Ni at 20 and 30 MeV, π+ - ⁵⁸Ni and π- - ⁹⁰Zr at 40 MeV. With no doubt, one can use this method in determining potential parameters for other-like nuclear systems as kaon-nucleus systems, which will be thoroughly investigated in the nearest future. To the best of our knowledge, this method is new in pion physics and it deserves special attention and appreciation. The crux of its importance is in explaining low-energy pion-nucleus and, hopefully, kaon-nucleus systems with no available phase shifts.

Based on our previous successful results, and guided by the corresponding obtained scaling relations, we are following the same strategy in determining the potential parameters and in obtaining the associated scaling relations for systems under investigation. The six potential parameters, V0, a0, R1, a1, R3 and a3, are kept as in ^{Ref. 7}. The starting values for the other three parameters, R0, V1 and W3, for π± - ¹²C and π± - ¹⁶O were taken as reported before⁵. For other target nuclei, the values for R0 and V1 were first taken following the (A2/A1)^1/3 rule, and W3 values following the (A2/A1) rule. To obtain the best possible agreements between measured and calculated differential cross sections, the values of the three free parameters, R0, V1 and W3, were slightly adjusted. For pion-nucleus scattering cases with available phase shifts, one may notice the good agreements between the scaled potentials, real and imaginary parts, with the inverted potential points obtained by using inverse scattering theory, where available^14,15. This is illustrated in Figs. 1, 2 and 3 for Tπ=20, 30, 40 MeV respectively. The potential parameters, phenomenological and scaled, are noted in Tables I, II, and III for Tπ=20, 30, 40 MeV, respectively. The calculated differential cross sections, using these potentials, are reasonably compared with measured ones^16-21 in Figs. 4, 5, and 6 for Tπ=20, 30, 40 MeV, respectively. The underestimated measured differential cross sections for θc.m≤30∘ may be attributed to Coulomb scattering effects. In general the achieved simultaneous results are good and they make our suggested scaling method very confident. In addition, our calculated reaction cross sections are tabulated for all pion-nucleus systems at all three pion’s incident kinetic energies considered herein. The values of these calculated cross sections have a certain trend with both incident pion’s kinetic energy, Tπ, and the atomic mass of the target nucleus, A(u), in atomic mass units. Unfortunately the available measured values, and the ones calculated by other authors, are uncertain, with no trends, and with a noticeable contradiction⁵.

Our results show clearly that scaled potential parameters depend mainly on the incident pion’s kinetic energy, Tπ, and the atomic mass of the target nucleus, A(u), in atomic mass units. It is very informative, essential, and of great interest to investigate deeply such a dependence. Figures 7, 8, and 9 show a linear increase of R03, V13, -W3, respectively with A(u) for the three energies considered herein in addition to the values at 50 MeV⁷. This allows to estimate any of these scaled potential parameters for any target nucleus in a low- energy pion-nucleus scattering case. On the contrary, Figs. 10 and 11 show a systematic decrease of R0 and V1, respectively, with Tπ for π+ scattered off the four target nuclei ¹²C, ¹⁶O, ⁴⁰Ca, and ⁹⁰Zr. Moreover, Fig. 12 shows a curve-like, somewhat quadratic, behavior for -W3 versus Tπ. In all these figures, the values at 50 MeV are taken from our recent previous study⁷. Moreover the nine W3-values, at Tπ=65, 70, 80 MeV for the three target nuclei ⁴⁰Ca, ⁵⁸Ni and ⁹⁰Zr, were extrapolated from those for ¹²C and ¹⁶O⁵ as W3 is connected to A2/A1. Although our manuscript is concerned with the three low energies, Tπ=20, 30, 40 MeV, it is worthy to point out that the inclusion of the values at 50 MeV and the nine W3-values serves for integrity, completeness, compatibility and comparison reasons. Overall, our results show a good qualitative agreement with our previous ones^5,7.

For positive pions with Tπ=20, 30, 40, 50 MeV, the obtained scaling relations for the three potential parameters with the atomic mass number are, respectively:

and for negative pions with Tπ=20, 30, 40, 50 MeV the obtained scaling relations for the three potential parameters with the atomic mass number are, respectively:

For positive pions, the obtained scaling relations for the three potential parameters with the pion’s incident kinetic energy, for the five target nuclei under consideration, are, respectively :

4. Conclusions

This study confirms the suitability of the scaling method in obtaining potential parameters for a certain pion-nucleus system from a nearby one. This method is unique for low-energy pion-nucleus systems with no available shifts. The good agreements between calculated and measured differential cross sections, for different pion-nucleus systems at three low energies, represent a strong indication on the success of the method. Nevertheless, the analytical forms of our potentials agree with the inverted potential points where available. On the otherhand, and even with no available experimental reaction cross sections for most of the cases under consideration, or uncertain values for few of them, our calculated values are included, pending for future measured precise values. We believe that the results of this study, accompanied with the results of our two recent related studies, based on the scaling method are suitable to be submitted for a patent. In view of this, our scaling method will be used in analyzing low-energy kaon-nucleus elastic-scattering data. It is also tempted that this study, especially at 20 MeV, will positively contribute in accounting for pionic atom data and, then, in investigating pionic atom physics.