Pion-Nucleus Microscopic Optical Potential at Intermediate Energies and In-Medium Effect on the Elementary πN Scattering Amplitude
E. V. Zemlyanaya, V. K. Lukyanov, K. V. Lukyanov, E. I. Zhabitskaya, M. V. Zhabitsky
aa r X i v : . [ nu c l - t h ] O c t NUCLEAR THEORY, Vol. 31eds. A. Georgieva, N. Minkov, Heron Press, Sofia
Pion-Nucleus Microscopic Optical Potential atIntermediate Energies and In-Medium Effecton the Elementary πN Scattering Amplitude
E.V. Zemlyanaya , V.K. Lukyanov , K.V. Lukyanov , E.I. Zhabitskaya , M.V. Zhabitsky
Joint Institute for Nuclear Research, 141980 Dubna, Russia
Abstract.
Analysis is performed of calculations of the elastic scattering dif-ferential cross sections of pions on the Si, Ca, Ni and
Pb nuclei atenergies from 130 to 290 MeV basing on the microscopic optical potential (OP)constructed as an optical limit of a Glauber theory. Such an OP is defined by thecorresponding target nucleus density distribution function and by the elemen-tary πN amplitude of scattering. The three (say, “in-medium”) parameters ofthe πN scattering amplitude: total cross section, the ratio of real to imaginarypart of the forward πN amplitude, and the slope parameter, were obtained byfitting them to the data on the respective pion-nucleus cross sections calculatedby means of the corresponding relativistic wave equation with the above OP.A difference is discussed between the best-fit “in-medium” parameters and the“free” parameters of the πN scattering amplitudes known from the experimentaldata on scattering of pions on free nucleons. There is a great number of papers on pion-nucleus scattering at different en-ergies. In theoretical study two approaches are usually employed. First, themicroscopic Kisslinger potential is based on s − , p − , and d − πN scattering am-plitudes having six and more parameters obtained from phase analysis of πN data [1].Second approach is the Glauber high-energy approximation (HEA) that usesanalytic form of the πN amplitude inherent in high energy scattering [2]. Suchapproach was employed, for example, in [3].Here we utilize our HEA-based microscopic optical potential (OP) [4] forcalculation of π -nucleus elastic scattering. This potential is constructed as anoptical limit of a Glauber theory. Such an OP is defined by the known densitydistribution of a target nucleus and by the elementary πN amplitude of scatter-ing.The πN amplitude itself depends on three parameters: total cross section σ ,the ratio α of real to imaginary part of the forward scattering πN amplitude, andthe slope parameter β . For π -scattering on “free” nucleons they are known, in1emlyanaya,Lukyanov,Lukyanov,Zhabitskaya,Zhabitskyprinciple, from the phase analysis of the pion-nucleon scattering data. However,if one studies the pion-nucleus data then respective “in-medium” pion-nucleonamplitudes can be extracted. Thus the established best-fit “in-medium” πN pa-rameters can be compared with the corresponding parameters of the “free” πN scattering amplitudes.The aim of our study is an explanation of the experimental pion-nucleus datain the region of (3 3)-resonance energies and estimation of the “in-medium”effect on the elementary pion-nucleon amplitude. The differential cross sections are calculated by solving the relativistic waveequation [5] with the help of the standard DWUCK4 computer code [6]: (cid:0) ∆ + k (cid:1) ψ ( ~r ) = 2¯ µU ( r ) ψ ( ~r ) , U ( r ) = U H ( r ) + U C ( r ) . (1)Here k is relativistic momentum of pion in c.m. system: k = M A k lab p ( M A + m π ) + 2 M A T lab , k lab = q T lab ( T lab + 2 m π ) , (2) ¯ µ = EM A / [ E + M A ] – relativistic reduced mass, E = p k + m π – totalenergy, m π and M A – the pion and nucleus masses, T lab and k lab – kineticenergy and momentum of pion in the laboratory system.The HEA-based microscopic optical potential U consists of nuclear andCoulomb parts. The nuclear part is as that derived in [4]: U H = − σ ( α + i ) · ~ cβ c (2 π ) Z ∞ dq q j ( qr ) ρ ( q ) f π ( q ) , f π ( q ) = e − βq , (3)where ~ c = 197 . MeV · fm, β c = k/E, j is the spherical Bessel function, f π ( q ) – formfactor of πN -scattering amplitude, ρ ( q ) – formfactor of the nucleardensity distribution in the form of symmetrized Fermi-function: ρ SF ( r ) = ρ sinh ( R/a )cosh (
R/a ) + cosh ( r/a ) , ρ = A . πR h πaR ) i − (4)Parameters of radius R and diffuseness a are known from electron-nucleus scat-tering data.Three parameters of the πN scattering amplitude are obtained by fitting tothe experimental πA differential cross sections: • σ , total cross section πN , • α , ratio of real to imaginary part of the forward πN amplitude, • β , the slope parameter.2ion-NucleusMicroscopicOpticalPotential...
20 40 60 8010 −2 Si; 291 MeV × π + π − Θ c.m. [deg] d σ / d Ω [ m b / s r ]
20 40 60 8010 −2 Ni; 291 MeV × π + π − Θ c.m. [deg] d σ / d Ω [ m b / s r ]
20 40 60 8010 −2 Pb; 291 MeV × π + π − Θ c.m. [deg] d σ / d Ω [ m b / s r ] Figure 1. Comparison of the calculated pion-nucleus elastic scattering differential crosssections at T lab = 291 MeV with experimental data from [12]. The best-fit “in-medium”parameters σ , α , and β are given in the Table 1. We minimize the function χ = f ( σ, α, β ) = k X i ( y i − ˆ y i ( σ, α, β )) ( s as i ) , (5)where y i = log [ dσd Ω ] i and ˆ y i = log [ dσd Ω ( σ, α, β )] i are, respectively, experimen-tal and theoretical differential cross sections of elastic scattering. Asymmetricexperimental errors s as i are calculated at each i -th experimental point as follows s as i = ( y (+) i − y i if ˆ y i > y i y i − y ( − ) i if ˆ y i < y i (6)where y (+) i and y ( − ) i are, respectively, maximal and minimal estimations of theexperimental value y i .The fitting technique is based on the asynchronous differential evolution al-gorithm [7, 8]. 3emlyanaya,Lukyanov,Lukyanov,Zhabitskaya,Zhabitsky
20 40 60 80 10010 −2 Si; 162 MeV × π + π − Θ c.m. [deg] d σ / d Ω [ m b / s r ]
20 40 60 80 10010 −2 Ni; 162 MeV × π + π − Θ c.m. [deg] d σ / d Ω [ m b / s r ]
20 40 60 80 10010 −2 Pb; 162 MeV × π + π − Θ c.m. [deg] d σ / d Ω [ m b / s r ] Figure 2. The same as in Figure 1 but for T lab = 162 MeV. The experimental data arefrom [13].
20 40 60 80 10010 −1 Θ c.m. [deg] d σ / d Ω [ m b / s r ] Si; 130 MeV × π + π −
20 40 60 80 10010 −2 −1 Θ c.m. [deg] d σ / d Ω [ m b / s r ] Ca; 130 MeV × π + π − Figure 3. The same as in Figure 1 but for T lab = 130 MeV. The experimental data arefrom [14] and [15].
20 40 60 80 100 12010 −2 Θ c.m. [deg] d σ / d Ω [ m b / s r ] Si; 180 MeV π − × π +
20 40 60 80 100 12010 −2 Θ c.m. [deg] d σ / d Ω [ m b / s r ] Ca; 180 MeV π − × π + Figure 4. The same as in Figure 1 but for T lab = 180 MeV. Experimantal data arefrom [14] and [15].
Figures 1–5 show the differential cross sections of elastic pion-nucleus scatteringat energies between 291 and 130 MeV calculated using the OP presented inSection 2. Parameters (radius R and diffuseness a ) of the target nuclear densitydistribution are following: R = 3 . fm and a = 0 . fm for Si [9]; R =4 . fm and a = 0 . fm for Ni [10]; R = 3 . fm and a = 0 . fmfor Ca [9]; R = 6 . fm and a = 0 . fm for Pb [11]. Calculatedbest-fit parameters σ , α , and β of the in-medium pion scattering amplitude andrespective χ values are given in the Table 1.It is seen that our results are in a reasonable agreement with experimentaldata. Some dissimilarity is observed only at large angles (discussed below).Figure 6 shows the averaged values X = ( X π + + X π − ) / where X = σ, α, β for the “free” π ± N -scattering parameters from [16] in comparison withthe obtained, also averaged, best-fit “in-medium” parameters in dependence on T lab .Note, the bell-like forms of σ free and σ eff ( T lab ) have maximum at thesame T lab . The dark gray (blue) domain σ eff is located below the light gray(yellow) σ free region. This means that the “in-medium” π ± N -interaction be-5emlyanaya,Lukyanov,Lukyanov,Zhabitskaya,Zhabitsky
20 40 60 8010 −2 Si; 226 MeV × π + π − Θ c.m. [deg] d σ / d Ω [ m b / s r ]
20 40 60 80 10010 −2 Ca; 230 MeV × π + π − Θ c.m. [deg] d σ / d Ω [ m b / s r ] Figure 5. The same as in Figure 1 but for T lab = 226 and 230 MeV. Experimental dataare, respectively, from [14] and [15]. comes weaker as compared with that for “free” π ± N -scattering.“In-medium” α eff ( T lab ) behavior indicates that refraction increases at en-ergy T lab > T labres ≃ MeV. It can be seen also that dark gray (blue) and lightgray (yellow) regions become closer at T lab > MeV.In our study we met two numerical problems which should be accounted forin future investigations. First problem is already mentioned disagreement be-tween calculated and experimental cross sections at large angles, and the visibledissimilarities increased with decreasing the energy. This effect can be explainedby the fact that the standardly applied Gaussian form of πN form factor f π (seeEg.(3) is not realistic in the region of large angles. Indeed, as experimentallyestablished in [17] the pion-nucleon cross section does not follow down but in-creases at angles over 80-100 degrees. Our calculation shows that agreementwith experimental data is improved as we remove, in our fitting procedure, afew experimental points at large angles. It is demonstrated on Figure 7(a) for thecase of π + + Si scattering at 180 MeV.The other remark that should be pointed here is an ambiguity problem aris-ing because the χ function (5) has more than one minimum in the region ofphysically realistic parameters. In some cases two minima provide almost thesame agreement with experimental data and additional information (such as to-tal reaction cross sections) is needed to make a choice. This is demonstrated onthe Figure 7(b) for the case π − + Si at 130 MeV.6ion-NucleusMicroscopicOpticalPotential...
50 100 150 200 250 3002468101214
Locher et al Si Ca Ni Pb T lab , MeV σ e ff , f m
50 100 150 200 250 300−1−0.500.511.52
Locher et al Si Ca Ni Pb T lab , MeV α e ff , f m
50 100 150 200 250 30000.20.40.60.811.21.41.61.82
Locher et al Si Ca Ni Pb T lab , MeV β e ff , f m Figure 6. (Color online) Light gray (yellow): “free” π ± N -scattering parameters from thepaper of Locher et al [16]. Dark gray (blue): the best fit values X eff = ( X π + + X π − ) / ; X = σ, α, β .
20 40 60 80 100 12010 −4 −2 Θ c.m. [deg] d σ / d Ω [ m b / s r ] π + + Si; 180 MeV π + + Si; 180 MeV π + + Si; 180 MeV (a)
20 40 60 8010 −1 Θ c.m. [deg] d σ / d Ω [ m b / s r ] π − + Si; 130 MeV (b)
Figure 7. (a) Differential cross sections of π + + Si scattering at 180 MeV. Solid curve:calculation with full set of experimental points from [14]. Best-fit parameters are givenin the Table 1. Dashed curve: calculation with reduced set of experimental points; theremoved points are indicated by dark solid circles, and thus the obtained parameters are σ = 7 . , α = 0 . , β = . , χ /k = 5 . . (b) Differential cross sections of π − + Si elastic scattering at 130 MeV. Solid curve: calculation with the best-fit pa-rameters from the Table 1. Dashed curve: calculation with parameters correspondingto the second minimum of the χ function where one gets σ = 10 . , α = − . , β = 0 . , χ /k = 4 . . • We show that the HEA-based three-parametric microscopic OP provides areasonable agreement with experimental data of pion-nucleus elastic scat-tering at intermediate energies between 130 and 290 MeV. • Comparison of σ free and σ eff shows that, at (3 3)-resonance energies,the πN -interaction in nuclear matter is weaker than in the case of free πN collisions. • Behavior of parameter α indicates that the refraction increases at energiesmore than T labres ≃ MeV. • The decrease of the inmedium slope parameter β eff in comparison to thefree one β free means that effective rms radius of the πN -system in nu-clear medium becomes less than in the pion collisions with free nucleons • Total cross section data are desirable to be involved to resolve the ambi-8ion-NucleusMicroscopicOpticalPotential...
Table 1. The best-fit parameters σ , α , β and corresponding χ /k quantities where k isthe number of experimental points. reaction T lab χ /k σ α βπ − + Si 130 2.1 7.08 ± ± ± π + + Si 5.5 9.61 ± ± ± π − + Ca 3.9 6.97 ± ± ± π + + Ca 13.3 8.58 ± ± ± π − + Si 162 3.5 11.02 ± ± ± π + + Si 6.7 8.48 ± ± ± π − + Ni 10.7 10.95 ± ± ± π + + Ni 7.5 9.28 ± ± ± π − + Pb 3.7 9.62 ± ± ± π + + Pb 10.3 6.60 ± ± ± π − + Si 180 10.5 10.03 ± ± ± π + + Si 12.1 10.24 ± ± ± π − + Ca 3.3 9.44 ± ± ± π + + Ca 4.2 5.78 ± ± ± π − + Si 226 13.8 7.36 ± ± ± π + + Si 23.8 9.79 ± ± ± π − + Ca 230 7.56 5.25 ± ± ± π + + Ca 7.70 8.95 ± ± ± π − + Si 291 6.2 5.03 ± ± ± π + + Si 4.9 5.35 ± ± ± π − + Ni 3.8 4.78 ± ± ± π + + Ni 2.6 5.63 ± ± ± π − + Pb 4.1 4.50 ± ± ± π + + Pb 3.0 5.56 ± ± ± • We should note that the usage of isotopically averaged parameters of π ± N -scattering amplitudes in the microscopic OP (3) is available for nu-clei with the same numbers of protons and neutrons Z ≃ A − Z [18].Hence the case of π -scattering on Pb with significant difference be-tween numbers of protons and neutrons requires a special consideration.
Acknowledgements
The work was partly supported by the Program “JINR – Bulgaria”. AuthorsE.V.Z. and K.V.L. thank the RFBR for the partial financial support under grantNo. 12-01-00396a. 9emlyanaya,Lukyanov,Lukyanov,Zhabitskaya,Zhabitsky
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