Pion Wave Research Articles

Off-Shell Ambiguities and Phenomenological Pion-Nucleus Potentials

The pionic wave functions obtained by fitting phenomenological optical potentials to elastic-scattering data are found to be dependent on the assumed off-shell extrapolation of the $\ensuremath{\pi}N$ amplitude. Using data from low-lying excited states or employing finite range potentials does not eliminate the ambiguities.

Effect of pion distortion on the asymmetry in (p→,π+) reactions on light nuclei

Recent ($\stackrel{\ensuremath{\rightarrow}}{p},{\ensuremath{\pi}}^{+}$) experiments on $^{12}\mathrm{C}$ at 200 MeV proton bombarding energy indicate that the production asymmetry $\ensuremath{\epsilon}$ cannot be explained (within the distorted-wave Born-approximation framework) by proton distortion alone. We examine in a simple model the effect of pion rescattering from the nucleus. The dependence of $\ensuremath{\epsilon}$ on the pion wave function, as well as its sensitivity to nuclear structure and the pion production operator are studied. The implication of the large negative asymmetry observed experimentally is briefly discussed.[NUCLEAR REACTIONS Calculated asymmetries in ($\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}},{\ensuremath{\pi}}^{+}$) reactions on $^{12}\mathrm{C}$ and $^{16}\mathrm{O}$ at ${T}_{p}=200$ MeV.]

Radius perturbation theory and its application to pionic atoms

Radius perturbation theory, starting from a variational principle, is developed for a two particle interaction, consisting of two parts, an external and the internal “core” part. The small parameters of the theory are the ratios of the core radius to the characteristic length parameters of the outer potential, such as the radius, the size of the bound state orbits and so on. The simple formulae for core corrections to scattering amplitudes, wave functions and bound state levels of the external potential are obtained. Comparison of our approximation with exact solutions for pionic atoms and the nucléon-nucléon system, as well as with results obtained in other approximation theories, is given. The present theory includes from the beginning the finite size of the core interaction and therefore its first approximation to the energy shifts of pionic atoms yields much better results than the usual zero range approximation to which our theory reduces in the limit of a zero core size. The present theory is able also to reproduce both the form and the magnitude of the pion wave function reasonably well.

C12(e, e′π+)Reaction Leading to Low-Lying States inB12

Energy and angular distributions of photopions from $^{12}\mathrm{C}$, leading to low-lying residual states in $^{12}\mathrm{B}$, have been measured via the ($e, {e}^{\ensuremath{'}}{\ensuremath{\pi}}^{+}$) reaction at ${E}_{e}=195$ MeV. The shape of the patterns agrees with the theoretical estimates deduced from other electron scattering data and also with shell-model calculations. A comparison of the absolute value shows the importance of correct estimation of the pion wave. The result demonstrates that photopion experiments can provide a promising method for studying spin-flip-type electromagnetic transitions in nuclei.

The axial current in extended systems

We establish a connection between the nuclear axial current and pionic properties in the medium. As a strict consequence of PCAC alone, the time component has its matrix elements determined by a wave pion absorption to leading order in the momentum transfer. The properties of the system are governed by scale parameters associated both with its size and with the local interaction structure. Based on this observation we suggest a locality condition for the part of the axial current which has no pion pole. It follows that the full axial current can be explicitly constructed from the local s and p wave pion source functions. A model illustration is presented. Our approach unifies in a model independent way axial β-decay and μ capture with pion processes, so that they all reflect the same phenomenon.

Thresholdπ0photoproduction from complex nuclear targets

The photoproduction of neutral pions near threshold from complex nuclear targets in investigated. Momentum dependent terms in the photoproduction operator and a two-nucleon charge exchange production mechanism are included. Various selection rules related to the quantum numbers of the nuclear states involved are discussed. Differential cross sections for pion energies 7.5 MeV from $^{4}\mathrm{He}$ and $^{6}\mathrm{Li}$ are calculated using realistic shell model wave functions. The effects of distortion of the pion wave function are found to be small, but the two-nucleon charge exchange mechanism is found to dominate the coherent production from $J\ensuremath{\ne}0$ targets near threshold.[NUCLEAR REACTIONS ($\ensuremath{\gamma}$, ${\ensuremath{\pi}}^{0}$) near threshold; realistic photoproduction operator; shell model wave functions; calculated differential cross section on $^{4}\mathrm{He}$ and $^{6}\mathrm{Li}$.]

Second-order pion-nucleus optical potential

The second-order pion-nucleus optical potential is expressed in momentum space in terms of the $\ensuremath{\pi}N$ off-shell $t$ matrix and the $\mathrm{NN}$ correlation function. The dependence of the $\ensuremath{\pi}N t$ matrix on the total $\ensuremath{\pi}N$ momentum is determined within the framework of relativistic particle quantum mechanics. Numerical results are presented for $\ensuremath{\pi}\ensuremath{-}^{4}\mathrm{He}$ scattering. It is found that the $\mathrm{NN}$ correlations tend to decrease the forward elastic cross sections by a factor of about 1.5 at low energy, ${E}_{\ensuremath{\pi}}\ensuremath{\simeq}60$ MeV. At higher energies the main effect of the $\mathrm{NN}$ correlations is to increase the differential cross section at large angles. The effect of $\mathrm{NN}$ correlations on the total cross section and the pion wave function is also found to be important. A study of the dependence of the calculated cross sections on the range parameters of the $\ensuremath{\pi}N t$ matrix and of the $\mathrm{NN}$ correlations suggests that the $\mathrm{low}$-energy pion-nucleus scattering data cannot be adequately described by an optical model constructed from the first- and second-order terms in an expansion in terms of the $\ensuremath{\pi}N t$ matrix.[NUCLEAR REACTIONS Second-order $\ensuremath{\pi}$-nucleus optical potential studied in the multiple-scattering theory.]

Δ-isobar dynamics in finite nuclei: Pion elastic scattering

Pion elastic scattering from finite nuclei in the 3,3 resonance region is described in terms of the coupling of a pion wave to correlated isobar-hole states. This is first carried out in an optical potential picture and then treated as a many-body response problem within the framework of RPA. Results are presented for the example of pion scattering from 4He; they show appreciable effects of isobar self-energy interactions and isobar-hole correlations other than one-pion exchange.

S-wave production of pions inp+p→π++d

The data for the total cross section for $p+p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}+d$ are reanalyzed to obtain the component for $s$-wave pion production near threshold. This comprehensive analysis yields a value which is compatible with other low-energy pion-nucleon reactions.

Field-theoretic Low equation approach to pion-nucleus scattering

From the pion-nucleus Low equation in the one-meson approximation we obtain an uncoupled, nonlinear, singular integral equation to determine the crossing symmetric elastic pion-nucleus scattering amplitude. The driving term of this equation is evaluated in an impulse single-scattering approximation for (${J}^{\ensuremath{\pi}}={0}^{+}$, $I=0$) ground state nuclei. It is found that unitarity requires that we include the effects of the strong absorption of the low partial waves in the entrance and exit channels. Thus, we introduce distortion of the pion waves by a procedure used in the inelastic peripheral scattering of elementary particles. Numerical results are presented for the interative solution of this equation for $\ensuremath{\pi}$-$^{12}\mathrm{C}$ scattering in the energy region of the (3, 3) resonance.NUCLEAR REACTIONS $\ensuremath{\pi}$-nucleus scattering theory with field-theoretic Low equation. $^{12}\mathrm{C}(\ensuremath{\pi}, \ensuremath{\pi})$, $E$ in (3, 3) resonance region; calculated $\ensuremath{\sigma}$, $\ensuremath{\sigma}(\ensuremath{\theta})$.

Distorted-wave approximation and nuclear correlations in pion reactions

The final- (or initial-) state interactions of pions participating in reactions with a complex nucleus are usually treated in distorted-wave impulse approximation with optical model wave functions. We improve this approximation by using an "effective pion field" which takes into account the hard-core nuclear correlations. This is illustrated here by the example of charged pion photoproduction from a $^{12}\mathrm{C}$ target.NUCLEAR REACTIONS $^{12}\mathrm{C}(\ensuremath{\gamma}, {\ensuremath{\pi}}^{\ensuremath{-}})^{12}\mathrm{N}_{\mathrm{g}.\mathrm{s}.}$, calculated $\ensuremath{\sigma}(\ensuremath{\Omega})$, distorted pion wave, included effects of nucleon-nucleon correlations.

The Lorentz-Lorenz correction

The Ericson-Ericson Lorentz-Lorenz correction in the propagation of pions in matter is derived in a relativistic theory for general pion frequency and wave vector, in terms of the nucleon-nucleon pair correlation function. The relation to the electromagnetic Lorentz-Lorenz correction is described. It is shown that ρ-meson exchange between nucleons can be expected to give contributions to the Lorentz-Lorenz correction comparable to those from π-mesons. The precise magnitude of the ρ-contributions depends sensitively on the detailed behaviour of the two-body correlation function, since the range of the interaction through ρ-exchange is comparable to that of the repulsive correlations.

Inelastic pion-nucleus scattering in distorted-wave impulse approximation

We have examined the inelastic scattering of pions using the distorted-wave impulse approximation formalism, with full inclusion of nucleon spin and isospin variables for the transition operator. A one-particle-one-hole model was used to describe the nuclear excitations, comprising both low-lying surface vibrations and higher-energy giant resonance states. A numerical solution of the Klein-Gordon equation with an appropriate form for the pion-nucleus optical potential generated the distorted pion waves. Numerical results for the cross sections are presented and the relevant dynamic selection rules are discussed.

Extraction ofannfrom the reactionπ−d→γnn

Extraction of a value for the neutron-neutron scattering length by an analysis of the reaction ${\ensuremath{\pi}}^{\ensuremath{-}}+d\ensuremath{\rightarrow}\ensuremath{\gamma}+n+n$ necessitates making assumptions about the reaction amplitude. We have studied the sensitivity of the extracted value of ${a}_{\mathrm{nn}}$ to such an analysis by performing a series of theoretical experiments in which a standard set of assumptions is used to analyze data generated by variations in one or more of these assumptions. For kinematically complete experiments the major sources of uncertainty lie in the energy dependence of the short-range characterization of the $n\ensuremath{-}n$ final-state wave function and to a lesser extent the treatment of the pion wave function in the initial state. We conclude that a value of ${a}_{\mathrm{nn}}$ can be extracted from this experiment with a theoretical uncertainty of less than 0.3 fm.NUCLEAR REACTIONS ${\ensuremath{\pi}}^{\ensuremath{-}}(d, nn)\ensuremath{\gamma}$, scattering length extraction, deuteron wave function, $\mathrm{nn}$ wave function neutron time-of-flight spectra.

Faddeev approach to pion production and pion-deuteron scattering

Some consequences of a new approach to pion production are explored in the threshold region. We treat the $\ensuremath{\pi}\mathrm{NN}$ system as a three-body system, with absorption (or production) occurring because of the existence of a $\ensuremath{\pi}N$ bound state in the ${P}_{11}$ (nucleon) channel. This description of the $\mathrm{NN}\ensuremath{\pi}$ vertex is justified by successfully using the model to calculate the long range part of the $N\ensuremath{-}N$ scattering amplitude. The results of the pion production calculation confirm a sensitivity to the short range behavior of the deuteron wave function and its $D$-state probability reported earlier. We also find a stronger variation of the production amplitude with energy than has previously been reported. This may resolve the apparent incompatibility of some experimental results. Finally, we also discuss the implications of the model for pion-deuteron elastic scattering.NUCLEAR REACTIONS $\mathrm{pp}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}d$, ${E}_{\ensuremath{\pi}}=0\ensuremath{-}30$ MeV; $s$-wave pions, calculated $\ensuremath{\sigma}$ as function of ${E}_{\ensuremath{\pi}}$, also dependence on $N\ensuremath{-}N$ interaction. $d(\ensuremath{\pi}, \ensuremath{\pi})$; calculated scattering length, including absorption. Faddeev calculation, separable interactions.

Partial radiative capture rates of pions in 6Li

The partial radiative capture rates of pions from the 1s and 2p states by 6Li are calculated using the impulse approximation. The possible configuration mixing of the initial and final nuclear wave functions as well as the distorsion of the pion wave function due to the finite size of nucleus and the strong interaction between the pion and the nucleons are taken into account.

Final state interaction in pion electroproduction from nuclei near threshold

We have calculated cross sections for pion electroproduction from nuclei near threshold, including the distortion of the pion wave function by the pion-nucleus optical potential. Distortion effects near threshold are found to be significant, changing the ratio σ π-/σ π+ for production of 10 MeV pions from 16O by nearly fifty percent.

An approach to pion-nucleus scattering near the (3,3) resonance with a dynamically modified optical potential

We develop a new approach to pion-nucleus scattering near the first pion-nucleon resonance which is based upon the modification of the pion Green function due to isobarhole excitation in the nuclear medium. Pion-nucleon scattering in the medium is modified in at least two essential ways: (i) the wave number is changed from the free-space value and (ii) the isobar mass (and width) are modified in the medium. These modifications are important because the elementary scattering amplitude is a resonant, and hence strongly varying, function of these variables. We construct a dynamically modified effective optical potential. The essential results are: (i) the growth of the attractive real potential with nuclear density is limited, and thus so is the growth of the pion wave number in the medium; (ii) the resonance structure is broadened and is shifted downward in energy, in the medium, since the increase in the real part of the wave number in the attractive medium compensates in the P-wave scattering amplitude for the lower beam energy. An essential new quantity appears in the formulation of the effective optical potential denoted by (V ∗ − V) = Δm ∗ . This quantity includes a shift in the effective mass of the composite isobar in the medium due to the interactions of the component pion, and due to the Pauli principle restriction upon the component nucleon. The quantity also includes a possible difference between the binding potential felt by an isobar and by a nucleon in the medium.

Pion production by 600 MeV protons on light nuclei

The forward cross section at E p=600 MeV is calculated for the reaction p+(N, Z)→(N+1, Z) ∗+π + using a single-nucleon mechanism, distorted pion and proton wave functions and bound-state nucleon wave functions with realistic asymptotic behavior. Good agreement with experiment is found for 4He, 12C, and 14N.

Pion production by 600 MeV protons on light nuclei

Abstract The forward cross section at E p = 600 MeV is calculated for the reaction p+( N , Z )→( N +1, Z ) * + π + using a single-nucleon mechanism, distorted pion and proton wave functions and bound-state nucleon wave functions with realistic asymptotic behavior. Good agreement with experiment is found for 4 He, 12C and 14 N.