_{1}

^{*}

A new updated simple local optical potential is proposed for analyzing low-energy
π-
^{-12}C elastic scattering data at 80 MeV and below. This potential is composed of two real terms and an imaginary term. The nature of the real part of the potential is repulsive at smaller radii and attractive at larger ones. In fact, the height of the repulsive term is found to change linearly with the incident pion kinetic energy. On the other hand, the imaginary part of the potential is attractive, shallow and non-monotonic with a dip at about 1.6 fm. Such a nature of the potential makes it feasible to predict
π
^{-}-
^{12}C cross sections at other energies in the energy region considered herein. Coulomb effects are incorporated by following Stricker’s prescription. This study will serve positively in studying both pionic atoms and the role of negative pions in radiotherapy.

Pions play a major role in studying the nucleus and, as such, they are crucial in nuclear physics and other disciplines [_{π} = 200 ± 100 MeV), the pion has a very small mean free path of less than 1 fm and usually faces a complete absorption at the surface of the nucleus. Such a scattering process of pions from different nuclei is usually described by simple optical potentials [

Recently Shehadeh et al. [

In the low energy region (T_{π} < 100 MeV), the pion has a large mean free path of few Fermi which enables it to penetrate deeply in the nucleus. As such, it can be used as a probe for studying the nuclear structure and discovering subtler aspects [^{−}-^{12}C elastic scattering data.

In Section 2, the theory is presented. Section 3 is mainly concerned with results and discussion. The last section draws the conclusions.

The analytical form of our potential, usually used, has the following form:

Which consists of Satchler’s potential form [^{12}C-nucleus), one may disregard the W_{2} term, i.e. W_{2} = 0, in Equation (1). As such, Equation (1) becomes

The parameters in this potential form are determined by implementing the IST outlined in Ref. [

In Equation (3), ^{2} and

where E, m, c, and _{c} is implicitly considered as a constant in^{−}-^{12}C, V_{c} is taken +3.4 MeV for an effective Coulomb radius R_{c} = 2.54 fm.

To sustain the possibility of using available standard optical codes, Zemlyanaya et al. [_{π} and the pion bombarding energy in the laboratory system _{π} and the actual beam energy _{T} is the target mass. The calculated values using Zemlyanaya et al.’s method agree very well with the values obtained from Satchler’s treatment. Zemlyanaya et al.’s method can be considered as another version of Satchler’s treatment. Here we adopt Zemlyanaya et al.’s method, and computer codes are modified accordingly. As such, one can start by calculating the pion’s momentum in the laboratory system

and obtain

The relativistic momentum of the pion in center-of-mass system P_{cm} can then be calculated,

So the total energy of the pion in the center-of-mass system E is given by

Comparing with Satchler’s treatment, E is M_{π}.

Since_{π}, and then the relativistic wave number of the pion k in the center-of-mass system,

where the constant (0.707447967) is not more than the ratio between the true rest mass of the pion m_{π}c^{2} = 139.6 MeV and

One can now easily calculate the pion’s kinetic energy in the center-of-mass system (E_{c.m.}),

where the constant (20.90104704) is_{T} = 12u for ^{12}C and

The drawbacks of several theoretical models, as Kisslinger first-order pion-nuclear optical potential, to give a full understanding of π^{±}-nucleus elastic scattering data obliges theoreticians to solve this problem. As an example, Gmitro et al. [^{2}-de- pendent term to give a reasonable explanation for the data. This was actually hinted by Johnson et al. [

The IST, used in extracting potential points from available phase shifts, has proved to be successful in determining the analytical form of the potential and its parameters. Such a potential has shown a remarkable success in describing nicely the elastic scattering of charged pions from calcium, calcium isotopes, ^{54}Fe [^{12}C above 100 MeV [^{40}Ca below 100 MeV [^{−}-^{12}C elastic scattering data at several energies; namely at 80 MeV and below. Fortunately, and benefiting from the general trend and behavior of the extracted potential points from available phase shifts, the imaginary and real parameters of the potential are kept unchanged in the two energy sub-regions, 50 ≤ T_{π} < 100 and T_{π} < 50, except for the strength of the repulsive real part V_{1} which is found to change linearly with the incident pion kinetic energy T_{π}. The potential parameters in Equation (2) are: V_{0} = 37.0 MeV, R_{0} = 3.75 fm, a_{0} = 0.324 fm, R_{1} = 3.00 fm, a_{1} = 0.333 fm, W_{3} = 100.0 MeV, R_{3} = 1.70 fm, a_{3} = 0.370 fm for T_{π} = 80, 69.5, 50 MeV. For T_{π} = 40, 30, 20 MeV, all these parameters are kept unchanged except R_{0} = 3.95 fm and W_{3} = 70.0 MeV. The parameter V_{1} is found to change with T_{π} as V_{1} = 18, 35, 70, 90, 110, 140 MeV for T_{π} = 80, 69.5, 50, 40, 30, 20 MeV, respectively. These values were fitted to a linear equation form, as shown in

The theoretical predictions of the differential cross sections deduced using our new complex potential defined in Equation (2) are compared with the experimental ones in

The nature of the imaginary part of the potential is the same at all energies. It is non-monotonic and shallow with a minimum at about 1.6 fm. On the other hand, the real part is non-monotonic, shallow with a small repulsive core at 80 MeV which grows linearly as the pion’s incident kinetic energy decreases. In fact, it is repulsive at small radii (r ≤ 2.5 fm) and attractive at large ones (r > 2.5 fm). On the other hand, ^{12}C-nucleus) case with a radius of 2.54 fm. The extracted potential points are also compared with the analytical potential forms in _{1} parameter approximated from Equation (12), are very successful in accounting nicely for the data at all six energies: 80, 69.5, 50, 40, 30, and 20 MeV. Such a nature of the potential makes it feasible to predict the differential cross sections at other energies in a certain energy sub-region. As an example

that successful analyses of data at several energies reduces the ambiguity and selects a unique potential. The use of this successful potential can be tested at energies below 20 MeV for explaining pionic atom data, and a reasonable link may be established between pionic atom results and the scattering data. In contrast, Friedman et al. [

As an important application for this investigation, Raju [

The new updated potential proves to be successful in explaining low energy π^{−}-^{12}C elastic scattering data. In fact, the calculated differential and reaction cross sections are in nice agreements with the measured ones. It is interesting to see that the imaginary part of the potential, in each energy sub-region, is energy independent; and only the strength of the repulsive part V_{1} of the real part decreases linearly as the incident pion kinetic energy T_{π} increases. This, and for the first time, establishes a relation between V_{1} and T_{π} given by Equation (12). It is worth noting that when T_{π} ≈ 0, V_{1} ≈ 172 MeV; and when T_{π} > 87 MeV, V_{1} becomes negative. The nature of the potential makes it possible to predict the differential and reaction cross sections at other energy in the energy range of 20 - 80 MeV. Moreover, this investigation emphasizes the strength of IST in predicting our new updated potential which proves to be very successful in explaining low-energy π^{−}-nucleus elastic scattering data. In addition, it will serve positively in both pionic atom studies and radiotherapy applications.

The author is very pleased to acknowledge the encouragement and financial support of the Deanship of Scientific Research at Taif University for carrying out this investigation. Many thanks go to Prof. F. B. Malik, to whom this paper is dedicated, for fruitful discussions.