Kisslinger Potential Research Articles

Different Methods for Pion-nucleus Optical Potential

Elastic and inelastic cross-sections for pion scat tering on 12 C at pion kinetic energy ranging from 50 to 260 MeV are computed using three independent methods of π ± -nucleus optical potential, the 3 α-particle model of the nucleus, the equivalent loca l Kisslinger potential and the Laplacian one. Reasonable fits to the measured values are obtained for 12 C without adjusting free parameters. The ability of these methods to account for elastic, in elastic, total and reaction cross-section data are somewhat similar. The Kisslinger-based local potent ial is the more suitable for describing the elastic and inelastic cross-sections of π ± -nucleus scattering . It seems that the 3 α -particle model of 12 C is not useful in the description of pion scattering on 12 C at least in the ∆-resonance region.

Perturbation theory for velocity-dependent potentials

In the presence of a velocity-dependent Kisslinger potential, the partial-wave, time-independent Schrodinger equation with real boundary conditions is written as an equation for the probability density. The changes in the bound-state energy eigenvalues due to the addition of small perturbations in the local as well as the Kisslinger potentials are determined up to second order in the perturbation. These changes are determined purely in terms of the unperturbed probability density, the perturbing local potential, as well as the Kisslinger perturbing potential and its gradient. The dependence on the gradient of the Kisslinger potential stresses the importance of a diffuse edge in nuclei. Two explicit examples are presented to examine the validity of the perturbation formulas. The first assumes each of the local and velocity-dependent parts of the potential to be a finite square well. In the second example, the velocity-dependent potential takes the form of a harmonic oscillator. In both cases the energy eigenvalues are determined exactly and then by using perturbation theory. The agreement between the exact energy eigenvalues and those obtained by perturbation theory is very satisfactory.

Bound and scattering wave functions for a velocity-dependent Kisslinger potential for l > 0

Using formal scattering theory, the scattering wave functions are extrapolated to negative energies corresponding to bound-state poles. It is shown that the ratio of the normalized scattering and the corresponding bound-state wave functions, at a bound-state pole, is uniquely determined by the bound-state binding energy. This simple relation is proved analytically for an arbitrary angular momentum quantum number l > 0, in the presence of a velocity-dependent Kisslinger potential. The extrapolation relation is tested analytically by solving the Schrodinger equation in the p-wave case exactly for the scattering and the corresponding bound-state wave functions when the Kisslinger potential has the form of a square well. A numerical resolution of the Schrodinger equation in the p-wave case and of a square-well Kisslinger potential is carried out to investigate the range of validity of the extrapolated connection. It is found that the derived relation is satisfied best at low energies and short distances.

S-wave bound and scattering state wave functions for a velocity-dependent Kisslinger potential

A relation linking the normalized s-wave scattering and the corresponding bound state wave functions at bound state poles is derived. This is done in the case of a non-local, velocity-dependent Kisslinger potential. Using formal scattering theory, we present two analytical proofs of the validity of the theorem. The first tackles the non-local potential directly, while the other transforms the potential to an equivalent local but energy-dependent one. The theorem is tested both analytically and numerically by solving the Schrodinger equation exactly for the scattering and bound state wave functions when the Kisslinger potential has the form of a square well. A first order approximation to the deviation from the theorem away from bound state poles is obtained analytically. Furthermore, a proof of the analyticity of the Jost solutions in the presence of a non-local potential term is also given.

Phenomenological local potentials for - + 12C scatteringfrom 120 to 766 MeV

Pion-nucleus scattering cross sections are calculated by solving a Schrödinger equation reduced from the Klein-Gordon equation. Local potentials are assumed, and phenomenological potential parameters are searched energy-dependently for a - + 12C system so as to reproduce not only elastic differential cross sections but also total elastic, reaction and total cross sections at 13 pion incident energies from 120 to 766 MeV. The real and imaginary parts of the local potentials thus obtained are shown to satisfy the dispersion relation. The imaginary part of the potentials as a function of the pion energy is found to peak near the (1232)-resonance energy. The strong absorption radius of the pion projectile with incident energies near the -resonance region is found to be about 1.6A1/3 fm, which is consistent with previous studies of the region where the decay of the takes place in nuclei. The phenomenological local potentials are then compared with the local potentials exactly phase-shift equivalent to Kisslinger potentials for pion energies near the -resonance.

Determination of neutron and proton multipole matrix elements in 208Pb from pi - and pi + scattering at 180 MeV.

New data for elastic and inelastic ${\mathrm{\ensuremath{\pi}}}^{+}$ and ${\mathrm{\ensuremath{\pi}}}^{\mathrm{\ensuremath{-}}}$ scattering by $^{208}\mathrm{Pb}$ at 180 MeV are presented. The elastic scattering data were analyzed with a standard Kisslinger potential in a coordinate space (r space) calculation using the free \ensuremath{\pi}-N amplitudes to calculate the potential coefficients. Momentum-space (p-space) calculations were also performed using the free \ensuremath{\pi}-N amplitudes to calculate on-shell matrix elements and the Landau-Tabakin model for the off-shell. The same ground-state proton and neutron densities were used in these two spaces and the fits are good. The distorted-wave impulse approximation calculations in both spaces, with the same collective model transition densities, reproduce the inelastic differential cross sections quite well. The proton and neutron transition multipole matrix elements and their ratios were calculated from the deformation parameters extracted from fits to inelastic scattering data. It is found that the ratios calculated in these two spaces are quite close to each other and close to those found in a comparison of (e,e') and (p,p') experiments, while the absolute value of matrix elements found in the p-space calculations are \ensuremath{\sim}15--20 % smaller than those from the r-space calculations. The r-space proton matrix elements are generally in good agreement with those obtained from electron scattering. The effects of the pion-nucleon interactions and form factors used in these calculations were investigated.

Coupled channels analysis of 12C

Existing data for the ${0}_{2}^{+}$ state in $^{12}\mathrm{C}$ at 50 and 67.5 MeV are compared with coupled channels calculations. Calculations are done with a realistic potential which includes the Lorentz-Lorenz-Ericson-Ericson p-wave correction and terms second order in the nuclear density. These are compared to results obtained using a phenomenological Kisslinger potential to investigate the sensitivity to the details of the \ensuremath{\pi}A interaction. The suppression of the forward angle cross sections by the Lorentz-Lorenz-Ericson-Ericson effect is insufficient to bring the direct transition into agreement at both energies. The interference between the one- and two-step excitation mechanisms is required to reproduce the forward angle data. The dependence of the cross section on the choice of monopole transition density is examined.

Pion elastic and inelastic scattering from 51V at 180 MeV.

Differential cross sections were measured for pion elastic and inelastic scattering from $^{51}\mathrm{V}$ at ${T}_{\ensuremath{\pi}}$=180 MeV. Elastic scattering data were analyzed using a standard Kisslinger potential. Inelastic scattering data were analyzed using distorted-wave impulse-approximation calculations with collective-model transition densities. Ground-state neutron-density parameters and matrix elements for some of the transitions were extracted.

Pion elastic and inelastic scattering from 48,50Ti, 52Cr, and 54,56Fe at 180 MeV: Determination of neutron and proton multipole matrix elements.

Differential cross sections were measured for pion elastic and inelastic scattering from /sup 48,50/Ti, /sup 52/Cr, and /sup 54,56/Fe at T/sub ..pi../ = 180 MeV. Elastic scattering data were analyzed using a standard Kisslinger potential. Inelastic scattering data were analyzed using distorted-wave impulse-approximation calculations with collective-model transition densities. Ground-state neutron density parameters and matrix elements for some of the transitions were extracted and ..pi../sup +//..pi../sup -/ differences investigated.

Kemmer-Duffin-Petiau equation approach to pionic atoms.

We lay down the formalism for the treatment of pionic atoms level widths and shifts using the Kemmer-Duffin-Petiau equation. Interactions are introduced in a Lorentz invariant way. An analytical transformation is shown that connects between different sets of interactions. The Kisslinger potential is obtained as a special case of a scalar-tensor optical potential. Results are shown in the accompanying paper.

Pion-nucleus optical potential in the α-particle model

The theoretical π-nucleus optical potential is constructed based on the α-particle model of the nucleus. This potential has only two necessary inputs, namely, the α-particle wave function of the nucleus and the π-α amplitudes, and has otherwise no adjustable parameters. The differential cross sections for π- 12 C elastic scattering are calculated using this potential over a wide energy region (260-50 MeV). The calculated results show better agreement with the experimental data, particularly in the low-energy region, than those calculated with the Kisslinger potential.

Pion elastic and inelastic scattering fromCa40,42,44,48andFe54

Differential cross sections were measured for pion elastic and inelastic scattering from $^{40,42,44,48}\mathrm{Ca}$ and $^{54}\mathrm{Fe}$ at ${T}_{\ensuremath{\pi}}=116.0,180.0, \mathrm{and} 292.5$ MeV. Elastic scattering was analyzed using a Kisslinger potential with and without free parameters. The low-lying collective states were analyzed using distorted wave impulse approximation calculations with no free parameters and with transition densities obtained from electron and proton scattering. These calculations adequately reproduced the pion scattering data. Ground-state neutron density root-mean-square radii were estimated.[NUCLEAR REACTIONS $^{40,42,44,48}\mathrm{Ca}({\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}},{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}\ensuremath{'}})$ and $^{54}\mathrm{Fe}({\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}},{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}\ensuremath{'}})$; ${T}_{\ensuremath{\pi}}=116.0,180.0, \mathrm{and} 292.5$ MeV; measured $\ensuremath{\sigma}(\ensuremath{\theta})$; optical-model analysis.]

Systematics of pion-nucleus optical potentials from analysis of elastic scattering

Most of the available data on the elastic scattering of pions by nuclei between 14 and 80 MeV are analyzed with a complex local optical potential which is parametrized by a bias-free Fourier-Bessel series, and is used as a means of extracting as much information from the data as possible without the explicit use of a model for the interaction. A typical shape of a negative real potential at large radii with a positive one at smaller radii is obtained and a systematic dependence on energy, charge of the pion, and isospin of the target is observed. The Kisslinger potential is shown to reproduce all these systematics, whereas the Laplacian potential fails to do so. Some results for energies of up to 230 MeV are also analyzed. The dependence of reaction cross sections on the real potential is discussed.NUCLEAR REACTIONS Pion-nucleus "model-independent" optical potentials 14-230 MeV on various targets.

Elastic and inelastic scattering of 291-MeV pions byBe9, Si,Ni58, andPb208

Angular distributions for the elastic and inelastic scattering of 291-MeV ${\ensuremath{\pi}}^{+}$ and ${\ensuremath{\pi}}^{\ensuremath{-}}$ by $^{9}\mathrm{Be}$, Si, $^{58}\mathrm{Ni}$, and $^{208}\mathrm{Pb}$ have been measured. The characteristics of two first-order optical potentials have been investigated, and reasonable agreement with the elastic scattering data has been obtained. Macroscopic distorted-wave impulse approximation calculations of the inelastic scattering with Kisslinger potentials are reasonably successful in reproducing the experimental data. The inelastic scattering results for $^{9}\mathrm{Be}$ provide evidence for the importance of radial localization effects in pion inelastic scattering.NUCLEAR REACTIONS $^{9}\mathrm{Be}(\ensuremath{\pi},{\ensuremath{\pi}}^{\ensuremath{'}})$, $\mathrm{Si}(\ensuremath{\pi},{\ensuremath{\pi}}^{\ensuremath{'}})$, $^{58}\mathrm{Ni}(\ensuremath{\pi},{\ensuremath{\pi}}^{\ensuremath{'}})$, $^{208}\mathrm{Pb}(\ensuremath{\pi},{\ensuremath{\pi}}^{\ensuremath{'}})$, ${E}_{\ensuremath{\pi}}=291$ MeV, measured $\frac{d\ensuremath{\sigma}}{d\ensuremath{\Omega}}$; optical potential and DWIA collective analyses.

A unified analysis of pionic atoms and low-energy pion-nucleus scattering

The nature of the insensitivity of low-energy (⩽ 50 MeV, including π --atoms) data to be detailed structure of the optical potential is established. By means of the effective nuclear density for the pion interaction, a phenomenological Kisslinger potential for the pionic atoms is microscopically shown to be extrapolated successfully to the low-energy scattering.

Low energyπ+scattering from light nuclei

Angular distributions for elastic and inelastic scattering of approx. 50 MeV ..pi../sup +/ projectiles from targets of /sup 12/C, /sup 16/O, Si, and Fe are reported. Systematic differences between the experimental results and predictions of some first-order theories constructed from free ..pi..-N amplitudes are observed for all cases studied. These discrepancies are similar to those reported earlier. Although several improved optical models exist which take into account better treatments of the first-order potential as well as some second-order effects such as ..pi.. absorption and the Lorentz-Lorentz effect, good agreement with the data cannot be achieved without a phenomenological adjustment of the potential parameters. A purely phenomenological Kisslinger potential when used to fit the data yields potential parameters which are independent of mass number over the A region studied. The resulting p-wave parameter b/sub 1/ is close to the value predicted from free ..pi..N models, while the s-wave parameter b/sub 0/ requires severe adjustment of both real and imaginary parts from the value predicted from free ..pi..N models. The inelastic results are compared to distorted-wave Born-approximation calculations, which are seen to manifest characteristics similar to those seen in the elastic channel. A common optical potential is found to provide a goodmore » description of both elastic and inelastic scattering.« less

Why does the first-order π-nucleus optical potential work in the region of the (33) resonance?

The sensitivity of elastic π-nucleus scattering with respect to various corrections to the Kisslinger potential is investigated near the (33) resonance. Due to the strong absorption of the first order optical potential and to the density-dependence of these corrections, it seems unlikely that new information about nuclear structure can be obtained in this particular energy region.

Acausality of the kisslinger potential

Acausality of the kisslinger potential

Positive pion production by 185 MeV protons

The 12C(p, π +) 13C and 9Be(p, π +) 10Be reactions at 185 MeV are examined. We start with a model in which the pion is produced by an incident proton which then becomes a bound neutron. A non-relativistic reduction of the pseudoscalar operator results in a non-Galilean invariant pion production operator. The recent Uppsala data are then fit with a DWBA calculation which takes the pion's distortion from the Kisslinger potential. In this calculation, which uses the free pion-nucleon coupling constant at the interaction vertex, the same pion optical potential accounts for both the elastic and (p, π +) data. The reaction is sensitive to details of nuclear structure because of the large momentum transfer involved. The final nuclear state has components which consist of a single particle of high angular momentum (or high radial quantum number) coupled to an excited state of the target. The amplitudes of these components are calculated in perturbation theory. Such configurations participate in the reaction in three ways (i) a (p, p') reaction, in which the target is excited, followed by pion emission from the p'; (ii) pion emission from the proton followed by a (π, π') reaction; and, (iii) the proton emitting the pion first, and then exciting the target (pre-emission). The size of these effects is not negligible but can be absorbed into the freedom of choosing the wave function of the bound neutron.

Improved theoretical pion-nucleus optical potentials

An approach which makes the first order pion-nucleus optical potential theoretically sound is presented. This study should permit higher order improvements to the potential to be more meaningful and the nuclear structure information extracted from pi-nucleus scattering to be more reliable. Based on multiple scattering theory, three optical potentials are constructed and studied in momentum space. These models are the popular Kisslinger potential, the local “Laplacian” potential, and an “improved off-shell potential;” the latter one is derived from absorptive separable pion-nucleon potentials which exactly reproduce on-shell πN scattering. By working in momentum space and explicitly including πN resonances and off-shell effects in the definition of the optical potential, the approach described here is capable of handling any number of pi-nucleon partial waves, is applicable over a very wide energy region, is based on a physical model for off-shell behavior, and is extended easily to include higher order effects. The optical potentials are inserted into two different relativistic wave equations to determine the total cross section and elastic differential cross section for pi-nucleus scattering. It is found that the various models for off-shell πN scattering determine significantly different πC 12 scattering, with the improved off-shell model preferred on theoretical grounds. Also discussed is the importance of properly transforming πN scattering to the pi-nucleus c.m. system, the origin of the shift in the peak position of the π − C total cross section, and the reason for the increased diffractive nature of the differential cross section at 180 MeV.