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Revista mexicana de física

versión impresa ISSN 0035-001X

Rev. mex. fis. vol.64 no.3 México may./jun. 2018

 

Research

Research in Nuclear Physics

Analyses of π±-nucleus elastic scattering data at Tπ = 40, 30, 20 MeV using a suggested scaling method

Z.F. Shehadeha  * 

R.M. El-Shawafa 

a Physics Department, Taif University, Taif, Zip Code 21974, Saudi Arabia.


ABSTRACT

The data for elastically scattered charged pions from few nuclei, namely 12C, 16O, 28Si, 32S, 40Ca, 56Fe, 58Ni, and 90Zr have been analyzed by obtained potentials using a suggested scaling procedure. Originally the π± 12C elastic scattering data at 50 MeV was nicely fitted by a parameterized simple local optical potential extracted from available phase shifts using inverse scattering theory. The potential parameters of the π± 12C systems were scaled to π± 16O systems and then successively to other few systems covering the scattering of charged pions from target nuclei, namely π± 28Si, π± 32S, π± 40Ca, π± 56Fe, π± 58Ni and π± 90Zr. The obtained scaled potentials showed a remarkable success in explaining the available measured elastic differential cross sections, and in predicting other ones for the systems under consideration. The reaction cross sections have been calculated for all these systems at the three incident pion’s kinetic energies, Tπ = 40, 30, 20 MeV. Unfortunately, experimental reaction cross sections are totally absent or cloudy and unconfident. As such, and at this stage, we consider our calculated values useful and pending for future investigations. For the systems and energies considered herein, simple scaling relations are well established. This will be beneficial in analyzing similar nuclear scattering data, as low-energy pion-nucleus and kaon-nucleus elastic scattering data; and, hopefully, in explaining pionic atom data.

Keywords: Pion-nucleus potential; Klein-Gordon equation; elastic scattering; inverses cattering theory; phase shift analysis; scaling method; low-energy physics

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

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<Tπ400 MeV, and (III) the low energy region, Tπ100 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<Tπ100 MeV, and (III-b) the very low energy region, Tπ50 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 work5,7, 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):

V(r)=VN(r)+VC(r). (1)

The analytical form of the nuclear part is given by :

VNr=V01+expr-R0a0+V1[1+expr-R1a1]2

+iW3expr-R3a3[1+expr-R3a3]2 (2)

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:

VC(r)=±ZTe28πε0RC3-r2RC2rRC±ZTe24πε0rr>RC (3)

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:

d2dr2+2(l+1)rddr+k2-U(r)φnl(r)=0 (4)

with k2 and U(r) are given by

k2=(E2-m2c4)/2c2 (5)

U(r)=2E2c2[V(r)V2(r)/2E] (6)

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, 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:

dσ/dΩ=|f(θ)|2 (7)

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

fθ=fc+12ikl=0(2l+1)e2iσγ

×e2iδl-1Pl(cosθ) (8)

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

σγ=argΓγ+12+iη-12πγ-12-l (9)

with the dimensionless parameter γ:

γ=l+122-ZT2α2 (10)

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

σr=πk2l=0(2l+1)[1-|Sl|2] (11)

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 rR, considered here is the usual and familiar Coulomb wave function:

φnl=1(kr)l+1{Flη,kr+exp2iδl-12i

×[Glη,kr+iFlη,kr]} (12)

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

η=ZTαEk (13)

where α is the fine structure constant.

3. Results and Discussion

As pointed out in our previous work2,5,7, 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-nucleus12,13 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 available14,15. 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 ones16-21 in Figs. 4, 5, and 6 for Tπ=20, 30, 40 MeV, respectively. The underestimated measured differential cross sections for θc.m30 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.

Figure 1 The real and imaginary parts of the scaled potentials, drawn as solid and dashed lines, respectively, used in analyzing π±- 12C, O, 40Ca and 90Zr elastic scattering data at 20 MeV. Unfortunately, there are no available phase shifts. 

Figure 2 Same as Fig. 1 but for 30 MeV. The inverted real and imaginary potential points, obtained from available phase shifts14,15, are represented by solid circles and empty triangles, respectively. 

Figure 3 Same as Fig. 2 but for 40 MeV. Available phase shifts are taken from Refs. 14 and 15

Figure 4 The calculated differential cross sections, drawn as solid and dashed lines for positive and negative pions, respectively, compared to the experimental data, represented by solid circles and empty triangles16,17, as a function of center of mass angle θc.m. for 20 MeV incident charged pions scattered off the nuclei 12C, 16O, 40Ca, and 90Zr. The calculations are made by using the scaled potentials, real and imaginary parts, drawn as solid and dashed lines in Fig. 1 for both positive and negative pions, respectively. 

Figure 5 The calculated differential cross sections, drawn as solid and dashed lines for positive and negative pions, respectively, compared to the experimental data, represented by solid circles and empty triangles18,19, as a function of center of mass angle θc.m. for 30 MeV incident charged pions scattered off the nuclei 12C, 16O, 40Ca, and 90Zr. The calculations are made by using the scaled potentials, real and imaginary parts, drawn as solid and dashed lines in Fig. 2 for both positive and negative pions, respectively. 

Figure 6 The calculated differential cross sections, drawn as solid and dashed lines for positive and negative pions, respectively, compared to the experimental data, represented by solid circles and empty triangles17,20,21, as a function of center of mass angle θc.m. for 40 MeV incident charged pions scattered off the nuclei 12C, 16O, 40Ca, 58Ni and 90Zr. The calculations are made by using the scaled potentials, real and imaginary parts, drawn as solid and dashed lines in Fig. 3 for both positive and negative pions, respectively. 

Table I The optical potential parameters R0 (in fm), V1 (in MeV), W3(in MeV), R1(in fm), R3(in fm) and a3(in fm) used in Eq. (2) for 20 MeV incident charged pions on target nuclei noted in column one. Other optical potential parameters, given in Eq. (2), are kept fixed with the values V0 = -37.0 MeV, a0 = 0.324 fm and a1 = 0.333 fm. Our calculated reaction cross sections, σr (theor) in millibarns, for both π+ and π- , are listed in columns 8. 

Nucleus R0 V1 W3 R1 R3 a3 σr (theor)
π+ π- π+ π- π+ π- π± π+ π-
Carbon-12 4.00 4.03 150.0 140.0 -210.0 -70.0 3.00 1.70 0.370 100.4 227.4
Oxygen-16 4.15 4.18 180.0 142.0 -300.0 -75.0 3.00 1.87 0.370 123.1 360.4
Silicon-28 4.70 4.75 214.0 171.0 -400.0 -83.0 3.25 2.25 0.420 248.4 641.7
Sulfur-32 4.82 4.92 223.0 179.0 -444.0 -88.0 3.25 2.36 0.420 324.6 754.3
Calcium-40 5.20 5.30 250.0 193.0 -543.0 -100.0 3.54 2.54 0.420 345.4 842.6
Iron-56 5.45 5.74 267.0 218.0 -680.0 -107.0 3.85 2.84 0.420 381.4 1003.0
Nickel-58 5.54 5.80 271.0 221.0 -742.0 -118.0 3.92 2.86 0.420 405.1 1004.7
Zirconium-90 6.20 6.25 310.0 256.0 -1100.0 -150.0 4.50 3.30 0.420 494.4 1129.6

Table II The optical potential parameters R0 (in fm), V1 (in MeV), W3(in MeV), R1(in fm), R3(in fm) and a3(in fm) used in Eq. (2) for 30 MeV incident charged pions on target nuclei noted in column one. Other optical potential parameters, given in Eq. (2), are kept fixed with the values V0 = -37.0 MeV, a0 = 0.324 fm and a1 = 0.333 fm. Our calculated reaction cross sections, σr (theor) in millibarns, for both π+ and π- , are listed in columns 8. 

Nucleus R0 V1 W3 R1 R3 a 3 σ r (theor)
π+ π- π+ π- π+ π- π± π+ π-
Carbon-12 3.85 3.95 129.0 110.0 -120.0 -72.0 3.00 1.70 0.370 107.1 169.5
Oxygen-16 4.00 4.10 155.0 122.0 -175.0 -84.0 3.00 1.87 0.370 137.1 256.3
Silicon-28 4.55 4.69 185.0 147.0 -245.0 -111.0 3.25 2.25 0.420 278.6 473.4
Sulfur-32 4.64 4.82 194.0 154.0 -276.0 -122.0 3.25 2.36 0.420 347.0 563.9
Calcium-40 5.11 5.20 210.0 166.0 -340.0 -140.0 3.54 2.54 0.420 393.0 634.2
Iron-56 5.14 5.48 236.0 186.0 -458.0 -180.0 3.85 2.84 0.420 426.0 769.3
Nickel-58 5.20 5.56 239.0 188.0 -479.0 -193.0 3.92 2.86 0.420 435.9 774.1
Zirconium-90 5.70 6.25 277.0 217.0 -720.0 -280.0 4.50 3.30 0.420 503.3 961.0

Table III The optical potential parameters R0 (in fm), V1 (in MeV), W3(in MeV), R1(in fm), R3(in fm) and a3(in fm) used in Eq. (2) for 40 MeV incident charged pions on target nuclei noted in column one. Other optical potential parameters, given in Eq. (2), are kept fixed with the values V0 = -37.0 MeV, a0 = 0.324 fm and a1 = 0.333 fm. Our calculated reaction cross sections, σr (theor) in millibarns, for both π+ and π- , are listed in columns 8. 

Nucleus R0 V1 W3 R1 R3 a3 σr (theor)
π+ π- π+ π- π+ π- π± π+ π-
Carbon-12 3.75 3.95 103.0 90.0 -77.0 -74.0 3.00 1.70 0.370 127.3 141.4
Oxygen-16 3.89 4.05 125.0 100.0 -110.0 -96.0 3.00 1.87 0.370 170.7 201.6
Silicon-28 4.33 4.59 154.0 123.0 -178.0 -134.0 3.25 2.25 0.420 313.7 393.9
Sulfur-32 4.43 4.73 162.0 128.0 -203.0 -151.0 3.25 2.36 0.420 385.5 471.3
Calcium-40 4.75 4.95 180.0 138.0 -245.0 -170.0 3.54 2.54 0.420 403.3 519.4
Iron-56 4.93 5.43 199.0 156.0 -351.0 -234.0 3.85 2.84 0.420 505.1 678.6
Nickel-58 5.00 5.52 200.0 160.0 -366.0 -250.0 3.92 2.86 0.420 512.6 683.2
Zirconium-90 5.55 6.20 235.0 180.0 -563.0 -395.0 4.50 3.30 0.420 633.8 878.0

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 ones5,7.

Figure 7 The cube of R0-values versus the atomic mass of the target nucleus in atomic mass units, A(u) for Tπ = 20, 30, 40 MeV and 50 MeV. The R0-values at 50 MeV are taken from Ref. 7. The solid and dashed lines, for positive and negative pions, respectively, are just to guide the eye. 

Figure 8 The cube of V1-values versus the atomic mass of the target nucleus in atomic mass units, A(u) for Tπ = 20; 30, 40 and 50 MeV. The V1-values at 50 MeV are taken from Ref. 7. The solid and dashed lines, for positive and negative pions, respectively, are just to guide the eye. 

Figure 9 The W3-values versus the atomic mass of the target nucleus in atomic mass units, A(u) for Tπ = 20, 30, 40 and 50 MeV. The W3-values at 50 MeV are taken from Ref. 7. The solid and dashed lines, for positive and negative pions, respectively, are just to guide the eye. 

Figure 10 The R0-values versus the incident pion’s kinetic energy, Tπ (MeV) for five different target nuclei. The R0-values at 50 MeV are taken from Ref. 7. The solid and dashed lines are just to guide the eye. 

Figure 11 The V1-values versus the incident pion’s kinetic energy, Tπ (MeV) for positive pions scattered off five different target nuclei. The V1-values at 50 MeV are taken from ref.7. The solid and dashed lines are just to guide the eye. 

Figure 12 The W3-values versus the incident pion’s kinetic energy, Tπ (MeV) for five different target nuclei. The W3-values at 50 MeV are taken from Ref. 7 and the W3-values at energies greater than 50 MeV are extrapolated.. The solid and dashed lines are just to guide the eye. 

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:

R03=2.25A+39.9fm3

V13=3.33×105A+5.34×105 (MeV)3

W3=-11.1A-98.00 (MeV) (14a)

R03=1.64A+45.5fm3

V13=2.42×105A-4.41×105 (MeV)3

W3=-7.55A-40.70 (MeV) (14b)

R03=1.52A+37.5fm3

V13=1.50×105A-5.08×105 (MeV)3

W3=-6.19A-4.740 (MeV) (14c)

R03=1.40A+33.8fm3

V13=0.767×105A-4.52×105 (MeV)3

W3=-4.89A-2.630 (MeV) (14d)

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:

R03=2.92A+25.6fm3

V13=1.84×105A+1.46×105 (MeV)3

W3=-1.02A-58.50 (MeV) (15a)

R03=2.35+35.8fm3

V13=1.14×105A-0.06536×105 (MeV)3

W3=-2.67A-38.00 (MeV) (15b)

R03=2.30A+371.6fm3

V13=0.664×105A-0.197×105 (MeV)3

W3=-4.08A-20.10 (MeV) (15c)

R03=1.55A+3939fm3

V13=0.306×105A-1.30×105 (MeV)3

W3=-5.49A-19.80 (MeV) (15d)

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 :

12C :R0=-0.0109Tπ+4.20 fm,

V1=-2.27Tπ+196 MeV,

-W3=0.112Tπ2-13.0Tπ+419 (MeV) (16a)

16O :R0=-0.0119Tπ+4.37 fm,

V1=-2.70Tπ+235 MeV,

-W3=0.160Tπ2-18.7Tπ+603 (MeV) (16b)

40Ca :R0=-0.0274Tπ+5.79 fm,

V1=-3.69Tπ+323 MeV,

-W3=0.261Tπ2-29.2Tπ+1005 (MeV) (16c)

58Ni :R0=-0.0224Tπ+5.93 fm,

V1=-3.75Tπ+349 MeV,

-W3=0.345Tπ2-38.6Tπ+1357 (MeV) (16d)

90Zr :R0=-0.0255Tπ+6.60 fm,

V1=-4.17Tπ+398 MeV,

-W3=0.508Tπ2-56.2Tπ+1993 (MeV) (16e)

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.

Acknowledgments

The authors are very pleased to acknowledge the encouragement and financial support of the Deanship of Scientific Research at Taif University for carrying out this investigation. Our thanks are extended to Professors A. K. Basak and M. S. Abu-Jafar for illuminating discussions, and for sharing their computer expertise.

REFERENCES

1. Z. F. Shehadeh, J. Assoc. Arab Univ. Basic Appl. Sci. 14 (2013) 32-37. [ Links ]

2. R. M. Al-Shawaf and Z. F. Shehadeh, Int. J. Phys. Sci. 11 (2016) 85-95. [ Links ]

3. C. M. Chen, D. J. Ernst and M. B. Johnson, Phys. Rev. C 48 (1993) 84-849. [ Links ]

4. E. Friedman, Acta Physica Polonica B 24 (1993) 1673-1684. [ Links ]

5. Z. F. Shehadeh and R. M. El-Shawaf, Rev. Mex. Fis. 62 (2016) 475-483. [ Links ]

6. Z. F. Shehadeh, Int. J. Mod. Phys. E 18 (2009) 1615-1627. [ Links ]

7. Z. F. Shehadeh and R. M. El-Shawaf, Rev. Mex. Fis. 63 (2017) 230-237. [ Links ]

8. H. M. Pilkuhn, Relativistic Particle Physics, (Springer-Verlag, New York, 1979). [ Links ]

9. W. R. Gibbs, Computations in Modern Physics, (World Scientific, New York, 2006). [ Links ]

10. R. A. Eisenstein and G. A. Miller, Comp. Phys. Commun. 8 (1974) 130-140. [ Links ]

11. P. Manngard, M. Brenner, M. M. Alam, I. Reichstein and F. B. Malik, Nucl. Phys. A 504 (1989) 130-142. [ Links ]

12. Q. Haider and F. B. Malik, J. Phys. G: Nucl. Phys. 7 (1981) 1661-1669. [ Links ]

13. Z. F. Shehadeh, Non-relativistic nucleus-nucleus and relativistic pion-nucleus interactions, Ph. D. Thesis Southern Illinois University at Carbondale, (1995). [ Links ]

14. J. Fröhlich, H. G. Schlaile, L. Streit and H. Zingl, Z. Phys. A- Atoms and Nuclei 302 (1981) 89-94. [ Links ]

15. O. Dumbrajs, J. Fröhlich, U. Klein and H. G. Schlaile, Phys. Rev. C 29 (1984) 581-591. [ Links ]

16. F. E. Obenshain et al., Phys. Rev. C 27 (1983) 2753-2758. [ Links ]

17. G. Burleson et al., Phys. Rev. C 49 (1994) 2226-2229. [ Links ]

18. K. K. Seth et al., Phys. Rev. C 41 (1990) 2800-2808. [ Links ]

19. B. M. Preedom et al., Phys. Rev. C 23 (1981) 1134-1140. [ Links ]

20. M. Blecher et al., Phys. Rev. C 20 (1979) 1884-1890. [ Links ]

21. D. J. Malbrough et al., Phys .Rev. C 17 (1978) 1395-1401. [ Links ]

Received: December 11, 2017; Accepted: February 01, 2018

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