Optical Model Wave Functions Research Articles

Antisymmetrized, translationally invariant theory of the nucleon optical potential

Earlier work showed how a nucleon optical model wave function could be defined as a projection of a many-nucleon scattering state within a translationally invariant second quantised many-body theory. In this paper an optical potential operator that generates this optical model wave function is defined through a particular off-shell extension of the elastic transition operator. The theory is express explicitly in terms of the many-nucleon Hamiltonian in a mixed representation in which localised target nucleus states feature. No reference to a mean-field concept is involved in the definition. It is shown that the resulting optical model operator satisfies the requirements of rotational and translational invariance and has standard behaviour under the time reversal transformation. The contributions to the optical potential from two different exchange mechanisms are expressed in terms of an effective Hamiltonian involving a nucleon-number conserving one-body interaction. In the weak-binding limit the method reduces to a a version of Feshbach's projection operator formulation of the optical potential with a truncated nucleon-nucleon potential including exchange terms and recoil corrections. Definitions of the nucleon single-particle Green's function and the corresponding Dyson self-energy modified by corrections for translational invariance are presented and different definitions of the optical potential operator are compared.

Translation invariance and antisymmetry in the theory of the nucleon optical model

The first step in any formalism that aims to connect a many-nucleon theory of nucleon-nucleus scattering and the concept of an optical model potential in the sense pioneered by Feshbach is to explain what is meant by the optical model wavefunction. By definition this is a function of a single space coordinate plus a set of single nucleon internal variables. This article gives a critique of the definitio as it is frequently expressed in 2nd Quantisation language and suggest a new definition which is more consistent with the requirements of antisymmetry and translational invariance. A modification of the time-dependent Green's function formalism is suggested.

Direct capture of protons in the medium-energy range

Direct capture of protons to the bound states in 12C and 13N is treated using a previously suggested consistent approach to the capture reactions. In this approach, the initial state is represented by an optical model wavefunction and the final state by a single-particle wavefunction bound to the nuclear potential. The influence of the complicated configurations involved in the process is taken into account by employing a consistency requirement for the electromagnetic transition operator. The approach is tested against the capture of medium-energy protons by the light nuclei, where the direct process strongly dominates. The quality of the results obtained proves the validity of the consistency principle even at medium energies.

Barrier and internal wave contributions to the quantum probability density and flux in light heavy-ion elastic scattering

We investigate the properties of the optical model wave function for light heavy-ion systems where absorption is incomplete, such as $\ensuremath{\alpha}{+}^{40}$Ca and $\ensuremath{\alpha}{+}^{16}$O around 30 MeV incident energy. Strong focusing effects are predicted to occur well inside the nucleus where the probability density can reach values much higher than that of the incident wave. This focusing is shown to be correlated with the presence at back angles of a strong enhancement in the elastic cross section, the so-called ALAS (anomalous large angle scattering) phenomenon; this is substantiated by calculations of the quantum probability flux and of classical trajectories. To clarify this mechanism, we decompose the scattering wave function and the associated probability flux into their barrier and internal wave contributions within a fully quantal calculation. Finally, a calculation of the divergence of the quantum flux shows that when absorption is incomplete, the focal region gives a sizable contribution to nonelastic processes.

Approximate Coulomb distortion effects in (e,e′p) reactions

In this paper we apply a well-tested approximation of electron Coulomb distortion effects to the exclusive reaction (e,e{sup {prime}}p) in the quasielastic region. We compare the approximate treatment of Coulomb distortion effects to the exact distorted wave Born approximation evaluated by means of partial wave analysis to gauge the quality of our approximate treatment. We show that the approximate Mo/ller potential has a plane-wave-like structure and hence permits the separation of the cross section into five terms which depend on bilinear products of transforms of the transition four current elements. These transforms reduce to Fourier transforms when Coulomb distortion is not present, but become modified with the inclusion of Coulomb distortion. We investigate the application of the approximate formalism to a model of {sup 208}Pb(e,e{sup {prime}}p) using Dirac-Hartree single particle wave functions for the ground state and relativistic optical model wave functions for the continuum proton. We show that it is still possible to extract, albeit with some approximation, the various structure functions from the experimentally measured data even for heavy nuclei. {copyright} {ital 1997} {ital The American Physical Society}

Neutron optical potentials from capture reactions

A new version of the direct-semidirect (DSD) model has been suggested recently. The imaginary part of the form factor is interpreted in terms of consistency requirements for the multipole operators. The calculations of excitation functions and angular distributions of γ-rays from capture reactions show close agreement with experimental values for many different sets of the optical model. Some sets, however, are not so successful. The results of the DSD calculations are sensitive to the optical model wave function in the vicinity of the target nucleus. The elastic scattering of nucleons on the other hand determines the scattering matrix in the asymptotic region of very large radii. In this work we select some global optical model sets which reproduce both elastic scattering and capture reactions well, We believe that such double test of the sets may help in better description of other reactions.

Parity non-conserving effects in neutron-nucleus scattering at thermal energies

The contribution of the neutron-nucleus component to parity non-conserving effects in neutron-nucleus scattering at thermal energies is considered for several nuclei. The wave function relative to the neutron-nucleus component incorporates for the first time physics which is usually accounted for by optical potential models (strength function and existence of a finite potential scattering length). Different models aiming to reproduce these quantities are developed. Some way to use directly information available from optical model wave functions is also proposed. The various results so obtained show that it is impossible to account for several parity non-conserving effects observed in 81Br, 111Cd, 117Sn and 139La with a single weak neutron-nucleus force, as it should be if the dominant parity admixture was taking place at the level of the neutron-nucleus component. Furthermore, its strength is too large compared to usual expectations. The only explanation left to account for these effects assumes that the dominant parity admixture takes place at the level of the compound nucleus.

Theoretical investigation of parity violation in [formula omitted] scattering

Calculations are presented of the parity-non-conserving (PNC) analyzing power A z for longitudinally polarized incoming protons in elastic p α scattering below ∼50 MeV. The results are given in terms of the PNC as well as the regular, parity-conserving (PC) meson-nucleon coupling constants for single π-, ρ- and ω-exchange. The reliability and parameter dependence of the calculations are discussed in detail. The PC on-shell scattering parameters play an important role in particular for the angular dependence of A z , which must be known for the actual analysis of measurements and are taken from experiments directly (phase analysis) rather than model calculations. The PNC scattering amplitudes are calculated in DWBA with optical-model wave functions properly antisymmetrized. Short-range correlations between nucleon pairs are taken into account by a Jastrow factor whose appropriate choice is discussed in detail. While absorption from the elastic channel is taken into account by the optical-model wave function, intermediate breakup channels are not included explicitly in the matrix elements.

Theoretical investigation of parity violation in [formula omitted] scattering

Calculations are presented of the parity-non-conserving (PNC) analyzing power A z for longitudinally polarized incoming protons in elastic p → α scattering below ∼50 MeV. The results are given in terms of the PNC as well as the regular, parity-conserving (PC) meson-nucleon coupling constants for single π-, ρ- and ω-exchange. The reliability and parameter dependence of the calculations are discussed in detail. The PC on-shell scattering parameters play an important role in particular for the angular dependence of A z , which must be known for the actual analysis of measurements and are taken from experiments directly (phase analysis) rather than model calculations. The PNC scattering amplitudes are calculated in DWBA with optical-model wave functions properly antisymmetrized.Short-range correlations between nucleon pairs are taken into account by a Jastrow factor whose appropriate choice is discussed in detail. While absorption from the elastic channel is taken into account by the optical-model wave function, intermediate breakup channels are not included explicitly in the matrix elements.

Examination of direct radiative capture formalism

Within the direct reaction framework the formalism and approximations used to describe particle radiative capture processes at low to medium energies are reviewed. Survey calculations documenting sizeable cross-section sensitivity to different forms of the electromagnetic interaction (Siegert's theorem versus J · A ) that preserve overall gauge invariance are reported. The source of this large difference is the inconsistent use of different initial and final state model hamiltonians. This defect, common to most reaction theories using optical-model wave functions, generates wave function nonorthogonality effects which are found to be significant for M1 transitions. The importance of the spin-flip contribution to both electric and magnetic transitions is also investigated.

Theory of pion photoproduction in 12C

Differential and total cross sections for the photoproduction of charged pions on 12C are calculated utilizing the Helm model and pion optical-model wave functions, and are compared to recent experimental data. The data are found to be described by our modeli¨n a satisfactory manner.

12C( 6Li, d) 16O at E(Li) = 90 MeV

The 12C( 6Li, d) 16O reaction has been studied at E( 6 Li) = 90.2 MeV with a magnetic spectrometer. Spectra up to E x ≈ 30 MeV in 16O have been obtained and decay line widths, Γ c.m., have been extracted for broad levels. Known α-cluster states ( J π = 2 +, E x = 6.9 MeV; J π = 4 +, E x = 10.35 MeV) are preferentially populated exhibit sharply forward peaked angular distributions in contrast with ( 6Li, d) data obtained at lower bombarding energies. These features are well reproduced by reaction calculations assuming direct alpha-particle transfer provided one utilizes realistic optical model, target, and projectile wave functions. Alpha spectroscopic factors and reduced alpha widths have been extracted for levels E x < 12 MeV including J π = 1 − levels of astrophysical interest. The results are compared with other recent measurements and theoretical calculations. There is generally at least qualitative agreement (within × 2) between the various experiments and theories. The analysis indicates a non-negligible alpha width for the J π = 1 − level at E x = 7.1 MeV in 16O (which determines the 12 C → 16 O stellar helium burning rate) with 0.3 < θ α 2 (7.1 MeV)/ θ α 2 (9.6MeV) < 0.6 and 0.1 < θ α 2 (7.1 MeV)/ θ α 2 (6.9 MeV) < 0.4.

Simplified DWBA for Heavy Ion Induced Transfer at High Energies

The treatment of the distorted wave Born approximation (DWBA) for transfer reactions at high energies between complex nuclei due to Braun-Munzinger and Harney (1974) is simplified by introducing eikonal-like representations for the elastic scattering states. Simple expressions for the differential cross sections are derived. The model includes recoil approximately and takes into account the strong absorption of the nuclear cores. Reasonable values for the parameters of the model wavefunctions are determined by comparison with the usual optical model wavefunctions. Angular distributions calculated for the model are compared with the results of exact finite-range DWBA calculations and experimental data for transitions to the ground state and excited states of 13C in the reaction 12C(14N,13N)13C at several energies. The model reproduces the general features of the exact calculations, giving reasonable fits for the transitions to the ground state and the 1ds/2 (3�85 MeV) state. The transition to the 2S1/2 (3 �09 MeV) state appears to be anomalous as in the case of the full DWBA theory.

Factorized distorted-wave approximation for the (e, 2e) reaction on atoms: Coplanar symmetric

The coplanar symmetric ($e, 2e$) cross section has been studied in the intermediate-energy region for the valence states of the inert gases He, Ar, and Ne. Experimental measurements at 200, 400, 800, and 1200 eV for He, and at 400, 800, and 1200 eV for Ne and Ar, are compared with calculations based on the factorized half-off-shell distorted-wave impulse approximation. Calculations are carried out using partial-wave-expanded optical-model wave functions which describe elastic scattering for the distorted waves, the eikonal approximation, and the plane-wave approximation. The latter two significantly overestimate the cross sections at small angles and the valence $s$-orbital cross section relative to the valence $p$-orbital one. They also predict the small-angle peak in the $p$-orbital cross section to be at an angle significantly smaller than is observed. Of the two, the eikonal approximation provides the better description of the data. The full calculation, on the other hand, underestimates the cross sections at small angles and the cross section for the more tightly bound $s$-orbital relative to the small-angle $p$-orbital cross section. It also underestimates the ratio of the large- to small-angle peak heights in the $p$-orbital cross section. All three approximations improve as the energy is increased. The full calculation shows that the factorized distorted-wave impulse approximation cannot provide an adequate description of the reaction.

Factorized distorted-wave approximation for the (e, 2e) reaction on atoms: Noncoplanar symmetric

The coplanar symmetric (e,2e) cross section has been studied in the intermediate-energy region for the valence states of the inert gases He, Ar, and Ne. Experimental measurements at 200, 400, 800, and 1200 eV for He, and at 400, 800, and 1200 eV for Ne and Ar, are compared with calculations based on the factorized half-off-shell distorted-wave impulse approximation. Calculations are carried out using partial-wave-expanded optical-model wave functions which describe elastic scattering for the distorted waves, the eikonal approximation, and the plane-wave approximation. The latter two significantly overestimate the cross sections at small angles and the valence s-orbital cross section relative to the valence p-orbital one. They also predict the small-angle peak in the p-orbital cross section to be at an angle significantly smaller than is observed. Of the two, the eikonal approximation provides the better description of the data. The full calculation, on the other hand, underestimates the cross sections at small angles and the cross section for the more tightly bound s-orbital relative to the small-angle p-orbital cross section. It also underestimates the ratio of the large- to small-angle peak heights in the p-orbital cross section. All three approximations improve as the energy is increased. The full calculationmore » shows that the factorized distorted-wave impulse approximation cannot provide an adequate description of the reaction.« less

The calculation of reactions of the type (a, ab) using the plane-wave expansion method: (I). General formalism for the direct one-step process

A formalism for the evaluation of the transition matrix element for reactions of the type (a, ab) in the direct distorted-wave approximation is presented. The plane-wave expansion method is used to represent the optical model wave function for the entrance and exit channels. Because of the use of the plane-wave expansion method the completely off-the-energy-shell behavior is easily demonstrated when distorted waves are used. The use of this expansion method also shows how difficult it is to extract the momentum dependence of the bound particle. We also apply the plane-wave expansion method to evaluate the transition matrix element for the effective f-operator DWTA approach. We also discuss an approximation to the complete transition matrix element which seems equivalent to the DWTA but yet makes use of completely off-the-energy-shell matrix elements.

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.

Optical model wavefunctions for atomic scattering

The absolute magnitudes and phases of the optical model wavefunctions for 20 eV and 400 eV electron elastic scattering on argon are plotted in order to show how the waves are distorted by the potential. The physical understanding so obtained enables an analytic eikonal approximation for the wavefunction to be developed and verified for use in the calculation of non-elastic reactions. This approximation has already proved valid for (e,2e) reactions at 400 eV on argon.

Positive pion reactions with 12C

Positive pion scattering and absorption by spherical nuclei are considered. The opticalmodel treatment of the former process includes a non-linear dependence upon the density of scattering centers originally suggested by Kroll. Explicit results are given for the 12C nucleus and compared with those obtained with the Kisslinger model. Pion optical-model wave functions, both with and without the Kroll modification, are used in a distorted wave calculation of differential cross sections for proton emission following pion absorption, and the results are compared with each other and with experiment for the case of 12C.

A microscopic description of the (γ,pn)-reaction

The quasideuteron process has been investigated for photon energies 40 MeV<Eγ< 160 MeV on the basis of a shell model picture modified by short range nucleon-nucleon correlations. It turns out that the cross section for the (γ,pn)-reaction depends sensitively on the details of the correlation function, i.e. on the exchange of high momenta between otherwise independently moving nucleons. The final state interaction has been consistently taken care of by using optical model wave functions for the outgoing nucleons. The results of the calculations for16O indicate that precise measurements of the (γ,pn) cross section do contain information on the properties of the nuclear wave functions for small internucleonic distances.