PACS: 25.80.Dj; 11.80.-m; 24.10.Ht.

1. Introduction

It is well-known that pion-nucleus elastic scattering has been of great importance in nuclear physics and other related disciplines. The pion’s properties of having a spin zero, an isospin one and a mass between the electron and the nucleon masses, make it the most useful probe of nuclear research ^{1}. Contrary to other probes, as electrons and protons, the pion has both charges which makes it special in describing Coulomb effects and studying the charge-exchange scattering experiments. In particular, low-energy pions are of special interest and importance as they can penetrate deeply into the nucleus, and information about the nuclear interior can be obtained ^{2}. This attribute is very important in gleaning nuclear information as nuclear structure and densities; in addition to other subtler aspects ^{3}. Nevertheless, the low-energy pion-nucleus elastic scattering process contributes to the solution of pionic-atom problem, especially the explanation of the shifts and widths of the energy levels in pionic atoms ^{4}. The significance of the low-energy pion-nucleus elastic scattering measurements remains incomplete without the complementary theoretical treatments. As alluded to in a previous study ^{5}, several theoretical optical models, used in explaining the pion-nucleus elastic scattering data, faced a limited success. This provides a strong inducement to use our potential adopted in explaining successfully the measured differential and reaction cross sections for the elastically scattered low-energy negative pions from the same nucleus ^{12}C ^{6}.

When phase shift analyses are available for elastically scattered pions from certain nuclei, one can use the inverse scattering theory as a guide to extract potential points from available phase shifts ^{7}. Then one can search for better potential parameters, by adjusting one parameter or more, that gives a reasonable agreement between the analytical forms of the potentials and the extracted potential points. This will be in conjunction with providing a nice explanation for the experimental angular distributions. The accuracy of the extracted potential points is affected by errors that obligatory exist in the phase shift analysis of the scattering data, and usually given as error bars. Alam ^{8} has highlighted all possible errors and provided valuable suggestions for reducing, but not eliminating, their effects.

So either the nonexistence of phase shift analysis or the unavoidable errors in the available phase shifts for pionnucleus scattering data limits, if not prevents, the use of the inverse scattering theory. This creates a strong motivation to search for an alternative method that gives approximate values, if not exact ones, for the potential parameters. For nucleus-nucleus and alpha-nucleus cases, the scaling method ^{9} has shown a reasonable success. With this in mind, we are trying to test the extent of success for a scaling method, or relatively a similar one, in obtaining approximate values for the potential parameters by scaling from one pionnucleus system to another nearby one in the same energy region. In this investigation, we try this method in obtaining the *π*^{+}−^{16}O potential parameters from the *π*^{+}−^{12}C potential parameters.

The following section briefs the theory. In its subsequent section, results and discussions for *π*^{+}−^{16}O and *π*^{+}−^{12}C systems, and the scaling method are outlined. The last section summarizes the conclusions.

2. Theory

The adopted nuclear potential here,
*V*_{N}(*r*), has the
same analytical form used in our recent study ^{6}:

*i.e.*, it consists of a real part composed of an attractive WoodsSaxon and a
repulsive squared Woods-Saxon terms, and an imaginary phenomenological attractive
surface WoodsSaxon term, respectively. For an incident positive pion and a target
nucleus, which is considered of a uniformly charged spherical distribution, the
Coulomb potential term
*V*_{C}(*r*) is given
by:

where *Z*_{T} is the atomic number of the target nucleus, *e*^{2}*/*4*πε*_{0} = 1*.*44 MeV.fm, *ε*_{0} is the permittivity of free space and *R*_{c} is the Coulomb radius. The total potential *V* (*r*) is the sum of nuclear and Coulomb potential terms:

To calculate the scattering amplitude, the differential and reaction cross sections, the potential *V*(*r*) is implemented in the radial part of Klein-Gordon equation:

In order to put Eq. (4) in a more convenient
mathematical form,
*R*_{nl}(*r*) is written
as:

which assures the convergence of
*φ*_{nl}(*r*) at the
origin, *i.e.*
lim_{r→0}*φ*_{nl}(*r*)
= 0, and Eq. (4) becomes,

with *k*^{2} and *U*(*r*) are given by

where *E*, *m*, and *c* are the effective pion
energy, effective pion mass, and the velocity of electromagnetic wave in vacuum,
respectively. Using Numerov’s method, Eq.
(4) is integrated numerically from the origin outward. As such the
logarithmic derivative for the inner solution is obtained at the surface,
*i.e*. at *r* = *R* where the
nuclear part of the potential
*V*_{N}(*r*) vanishes. In
contrast, the outer solution, *i.e.* for *r* ≥
*R*, where the potential is purely Coulomb, is well-known and is
given by

where the relativistic versions of the regular and irregular Coulomb wave functions,
*F*_{ℓ} and
*G*_{ℓ} respectively, are used.
These relativistic Coulomb wave functions are generated when the nuclear part is
turned off in solving the radial part of Klein-Gordon equation ^{10}^{,}^{11}. For a projected positive pion, the Sommerfeld
parameter *η* is the defined by :

with *α* is the fine structure constant.

At the matching radius *r* = *R*, which is taken 5.4 fm and 6.0 fm for *π*^{+}−^{12}C and *π*^{+}−^{16}O, respectively, the inner and outer logarithmic derivatives are equated and complex phase shifts, *δ*_{ℓ}, for all contributing partial waves, *ℓ*, are obtained. This makes it feasible to calculate the scattering amplitude *f*(*θ*), at angle *θ* in the center of mass system, using the formula:

where *P*_{ℓ}(cos*θ*) is the Legendre
polynomial, and *σ*_{γ} is the Coulomb phase
shift defined by ^{12}:

and the parameter *γ* is given by :

Knowing *f*(*θ*), the elastic differential cross section, *dσ/d*Ω, can be calculated using the formula:

In addition, one can easily calculate the reaction cross sections, *σ*_{r}, defined as:

where *S*_{ℓ} =
*e*^{2iδℓ} is the S-matrix.

Benefiting from pion-^{12}C, ^{16}O available phase shifts ^{13}^{,}^{14} and the use of inverse scattering theory, with a complete relativistic treatment, in conjunction with obtaining a nice fit for available differential and reaction cross sections data within the framework of the relativistic Klein-Gordon equation, the real and imaginary parameters of the potentials were determined. The inverse scattering theory, used as a guide for extracting potential points from available phase shifts, was fully explained ^{15} and, as such, will not be repeated here.

3. Results and Discussion

**3.1. π**

^{+}

**−**

^{12}

**C Case**

In a recent study ^{6} the *π*^{−}−^{12}C elastic scattering data has been successfully analyzed by using a three-term simple local optical potential obtained by using inverse scattering theory, as a guide, from available phase shifts. As such, and following the same strategy, we started by analyzing the *π*^{+}−^{12}C elastic scattering data in the same low-energy region. As indicated in Table I, six potential parameters, *V*_{0} = −37*.*0 MeV, *a*_{0} = 0*.*324 fm, *R*_{1} = 3*.*00 fm, *a*_{1} = 0*.*333 fm, *R*_{3} = 1*.*70 fm and *a*_{3} = 0*.*370 fm, were kept fixed and the other three parameters, *R*_{0}, *V*_{1} and *W*_{3}, were changed with the pion’s incident kinetic energy, *T*_{π}, to provide a nice agreement between measured and calculated differential and reaction cross sections. The analytical forms of these potentials, along with the extracted potential points from available phase shifts ^{13}^{,}^{16} are shown in Fig. 1 at all energies considered herein. In Fig. 2, and by using these potentials, our calculated differential cross sections are compared with the experimental ones ^{16}^{,}^{17}^{,}^{18}^{,}^{19}^{,}^{20}^{,}^{21}^{,}^{22}^{,}^{23}^{,}^{24}. The agreements are very satisfactory. In addition, our calculated reaction cross sections at all energies under consideration, compared to the available experimental ones, are shown in Table I. Unfortunately, the available experimental and theoretical reaction cross sections ^{4}^{,}^{25}^{,}^{26}^{,}^{27}^{,}^{28} are cloudy and uncertain; and even with no agreement on their increasing behavior with energy ^{29}. In addition, the situation is more corrupted if one compares the reported reaction cross sections at 30 MeV ^{4}^{,}^{30}^{,}^{31} shown in Tables I and II as the value for the carbon case is almost two to three times as that for the oxygen case.

Unfortunately this is an outrageous contradiction, but still acceptable compared to the more valuable and positive results. As such, the priority here is given for the ability of the potential to provide a nice agreement between predictions and experiment for elastic differential cross sections rather than reaction cross sections. In principle, this forms a stringent test for the correctness and success of the potential. Nevertheless, and in our investigation, one can also notice that the calculated reaction cross sections simulate the corresponding changes in the strength of the imaginary pat, *W*_{3}. This is highly expected as both represent the opening of inelastic channels.

Concerning the three potential parameters (*R*_{0}, *V*_{1}, *W*_{3}) that change with *T*_{π} , it is found that both *R*_{0} and *V*_{1} change linearly, while *W*_{3} changes quadratically, with *T*_{π} as clearly represented in Figs. 6, 7 and 8. By applying a spline fit for each of these graphs, the following corresponding relations are obtained:

It is clear that the *V*_{1}−*T*_{π} relation in (17) is different from the corresponding one obtained for the *π*^{−}−^{12}C scattering case ^{6}. This consolidates the need for different, although close, potentials in explaining the elastic scattering data for pions with both charges. This is also supported by the obtained values, and the corresponding relations, for *R*_{0} and *W*_{3}.

**3.2. π**

^{+}

**−**

^{16}

**O Case**

The success of our potential in explaining the *π*^{+}−^{12}C elastic scattering data forms an inducement to use it in analyzing the *π*^{+}−^{16}O. This has been carried out in a similar fashion as phase shift analyses are available at most, if not all, energies under consideration. This enables the use of the wellestablished inverse scattering theory, for the elastic scattering of two non-identical particles, to be used as a guide in determining the potential parameters. The obtained potential parameters consist of six fixed parameters with the values *V*_{0} = −37*.*0 MeV, *a*_{0} = 0*.*324 fm, *R*_{1} = 3*.*00 fm, *a*_{1} = 0*.*333 fm, *R*_{3} = 1*.*87 fm and *a*_{3} = 0*.*370 fm; and the three changed parameters *R*_{0}, *V*_{1} and *W*_{3}. All these parameters are indicated in Table II. The analytical forms of the potentials, real and imaginary parts, are plotted in Fig. 3. These potentials, real and imaginary parts, are reasonably compared with the potential points, solid circles and empty triangles, respectively, extracted from available phase shifts ^{14}^{,}^{31} using inverse scattering theory. Using these potentials, with only three free parameters, our calculated differential cross sections are in very good agreement with the experimental values at 30, 40 and 50 MeV ^{16}^{,}^{20} but slightly differ at forward angles for 20 MeV ^{19} and at large angles for 80 MeV ^{1} as clearly depicted in Fig. 4. It is worth to mention that all the fixed six parameters have the same values as for *π*^{+}−^{12}C case, but with a slight change in the absorption radius *R*_{3}. As in the *π*^{+}−^{12}C case, our calculated reaction cross sections, compared to the experimental ones where available, are tabulated in Table II; and again the evidence on available experimental and theoretical values ^{4}^{,}^{16}^{,}^{26}^{,}^{31} is scanty. This is very obvious in the very large discrepancy in the total reaction cross sections of 166 and 284 mb, at 50 MeV for *π*^{+}−^{16}O case, reported by Meirav *et al*. ^{26} and Malbrough *et al*. ^{31}, respectively. Hence, our calculated reaction cross sections are proposed pending further related studies.

Again, applying a spline fit to each of the three changed parameters *R*_{0}, *V*_{1} and *W*_{3} with *T*_{π} reveals the following relations:

These three relations for *π*^{+}−^{16}O case are different from their counterparts given by Eqs. (16), (17) and (18) for *π*^{+}−^{12}C case. This strongly indicates the dependence of the parameters of these potentials on the atomic mass, *A*, of the target nucleus. So in comparison, and in connection, with the scaling method for none relativistic alpha-nucleus and nucleus-nucleus scattering cases, does a scaling method exist for pion-nucleus cases?

3.3. The Scaling Method

Scaling potential parameters of a certain nuclear system to another nearby one has been used with a reasonable success. It has been established that scaled potential parameters can predict the general feature of the cross section. It is worthwhile mentioning that the derived scaling relations rely on a strong theoretical background as they are connected to the well-known energy density functional (EDF) theory ^{7}. In fact, Haider and Malik ^{9} obtained the ^{12}C - ^{12}C potential parameters by scaling the ^{16}O - ^{16}O potential parameters ^{32}. Also Malik and Reichstein ^{33} have obtained the *α* - ^{32}S, ^{34}S potentials by scaling from the *α* - ^{28}Si potential parameters ^{34}. In addition, Shehadeh and his coworkers ^{7}^{,}^{35}^{,}^{36} have used the scaling procedure as a guide in determining the potential parameters for ^{32}S - ^{64}Ni, ^{36}S ^{58,64} Ni, ^{40}Ca - ^{48}Ca and ^{27}Al - ^{58}Ni nuclear systems. Nevertheless, Sabra *et al*. ^{37} have used scaling to obtain the potential used in their study. In all mentioned cases, the goals behind using the obtained potentials by scaling procedure are achieved; and a remarkable success was very obvious. This is also compelling to use the scaling method in analyzing high-energy alpha-nucleus scattering data from a recent successful determined potential ^{11}.

Here, and in this new preliminary study, we are investigating the possibility of scaling from
one pion-nucleus system to another; namely from
*π*^{+}−^{12}C into
*π*^{+}−^{16}O. So far, we have obtained
systematic trends for the changed potential parameters with energy rather than
exact scaling relations. This is clearly indicated in the Figs. 6, 7 and 8. This is also legal as the possibility of
parameterization of pion-nucleus optical potential has been previously drawn to
attention ^{38}.

One may notice that the fixed parameter *R*_{3} has
different values for oxygen-16, *R*_{3} =
1*.*87 fm, and carbon-12, *R*_{3} =
1*.*70 fm, cases. It is very interesting to see that the
ratio, 1*.*87*/*1*.*70 ≈
1*.*1, equals the ratio *R*_{1} is the same for
both cases which may tell that the positive incident pion starts facing a
noticeable repulsion at *R*_{1} = 3*.*0
fm. This is legitimate as the two target nuclei are close to each other and each
of them is tightly bound nucleus. For the parameter
*R*_{0}, it decreases as
*T*_{π} increases. The ratio
between the slopes of *R*_{0} given by the relations (16)
and (19), and represented in Fig. 6, is
1.054. Similarly, the parameter *V*_{1} also decreases as
*T*_{π} increases. The ratio between
the slopes of *V*_{1} given in (17) and (20), and plotted
in Fig. 7, is 1.135. Such close ratios
which are, in turn, close to
(*A*_{2})^{1/3}*/*(*A*_{1})^{1/3}
= 1*.*10 makes it feasible to scale the two parameters
*R*_{0} and *V*_{1} with the
pion’s incident kinetic energy *T*_{π}
and the atomic weights, (*A*_{1}) and
(*A*_{2}), of the two target nuclei in the two
pion-nucleus systems. On the contrary, the parameter
*W*_{3} changes quadratically with
*T*_{π}. A deep look at the
*W*_{3}-relations, given in (18) and (21), reveals
that relation (21) is obtained by multiplying (18) with 1.3 which is
approximately the ratio between the atomic weights of the two target nuclei,
*i.e.* 16/12, *i.e*.
*W*_{3} didn’t follow the
*A*^{1/3} rule in scaling.
Moreover, Fig. 8 shows that both
*W*_{3}−*T*_{π}
curves have a minimum at 60 MeV. Such a behavior is in accord with the
calculated cross sections listed in the tables; and may be attributed to the
pion mean free path that allows a minimal number of pion-nucleon inelastic
scattering for *T*_{π} = 60 MeV compared
to other lower and higher energies in the low energy region under consideration.
This is clearly manifested in *W*_{3} as the other two
imaginary parameters, *R*_{3} and
*a*_{3}, remain unchanged.

In Fig. 6, the values of *R*_{0} at 13.9 and 80 MeV are not on the straight line which may suggest that a better scaling procedure is obtained for a certain energy interval as *T*_{π}*<* 20 MeV, 20 ≤ *T*_{π} ≤ 80 MeV and 80 ≤ *T*_{π} ≤ 100 MeV. With a reasonable degree of confidence, especially in the energy region 20 ≤ *T*_{π}*<* 80 MeV, the new suggested scaling method is summarized by a) multiplying the two parameters *R*_{0} and *V*_{1} by 1.05-1.14, b) multiplying the parameter *W*_{3} by 1.30, and c) keeping the other six parameters fixed with *R*_{3} follows the *A*^{1/3} rule. To become sure and more confident, and with these encouraging results, the analyses of elastically scattered positive pions from ^{28}Si, ^{232}S, ^{40}Ca, ^{56}Fe, ^{58}Ni and ^{90}Zr nuclear targets by scaling from *π*^{+} − ^{12}C and *π*^{+} − ^{16}O cases will be investigated in the nearest future.

4. Conclusions

In this investigation, the adopted simple local optical potential shows a remarkable success in explaining low-energy *π*^{+}−^{12}C, ^{16}O elastic differential cross sections data. The role of the inverse scattering theory in guiding to the correct potential parameters is very obvious. For the first time, the use of a scaling method with changes in three free potential parameters *R*_{0}, *V*_{1} and *W*_{3} with *T*_{π} has been established. In fact, linear relations for *R*_{0} and *V*_{1}, and quadratic relations for *W*_{3} for both *π*^{+}−^{12}C and *π*^{+}−^{16}O systems have been obtained. The slopes of *R*_{0} and *V*_{1} are related to (*A*_{2})^{1/3}*/*(*A*_{1})^{1/3} while *W*_{3} is connected to *A*_{2}*/A*_{1}. Also the ratio between *R*_{3} values for *π*^{+}−^{12}C and *π*^{+}−^{16}O cases is found to be the same as (*A*_{2})^{1/3}*/*(*A*_{1})^{1/3} and, as such, to follow the *A*^{1/3} rule. With these important findings, this paper will be a contribution to fit further experimental data in the energy range defined here, and shows the success of the scaling method prediction of potential parameters on similar nuclei to describe the pion plus elastic scattering.