The fine structure of the isoharic analogue resonances

The distribution of an isobaric analogue resonance over the numerous surrounding states of different isobaric spin is studied analytically in the frame of the Fano-Weidenmüller shell-model theory of nuclear reactions. The doorway state wave function is described byLane's coupled equations. The matrix element of the residual nuclear interaction between this doorway state and a complicated state of different isobaric spin exhibits a particular energy dependence. A sum rule is established: the sum of the fine structure widths is larger than the width of the doorway state, as calculated fromLane's model. The residual Coulomb interactions are neglected. When perturbation theory holds, the distribution of the fine structure peaks is shown to have a characteristic asymmetrical shape. The connection withRobson's R-matrix treatment is discussed.     

A perturbation expansion in the tunneling problem of many particle systems

The fine structure observed at Duke University in the total neutron cross section of²⁰⁶Pb is analysed in terms of theK- andS-matrix parametrizations. The distributions of the resonance parameters are in fairly good agreement with the theoretical ones derived from a doorway state hypothesis, combined with the picket fence model. Attention is drawn to the fact that the average cross section usually allows only a very inaccurate determination of the parameters of the doorway state resonance.     

Elastic unitarity and the K-matrix in a Lorentz covariant representation of the NN interaction

A Lorentz covariant representation of the NN t-matrix has been obtaained over the energy range from 150 to 500 MeV by solving the integral equation that connects the t-matrix with the K-matrix. The K-matrix is expanded in a complete set of on-shell Lorentz invariant amplitudes represented phenomenologically by isoscalar and isovector “meson” exchanges. The real part of the K-matrix is fit over the energy range from 150 to 500 MeV using coupling strengths that are allowed to vary quadratically with energy. Above the pion production threshold at T_lab = 280 MeV, the real K-matrix is supplemented by an imaginary part with linear energy dependence. The K-matrix parameters are fit to thesmost recent (January 1999) Arndt amplitudes [R.A. Arndt, D. Roper, VPI and SU Scattering Analysis Interactive Dial-in Program and Data Base]. Direct and exchange contributions to the K-matrix are handled explicitly in the formalism. The resulting t-matrix satisfies elastic unitary below the pion production threshold and contains non-local terms that are not present in direct Love-Franey parameterizations of the t-matrix. Results are given for the NN amplitudes and compared with both the Arndt amplitudes and amplitudes obtained from a direct fit of the t-matrix [O.V. Maxwell, Nucl. Phys. A 600 (1996) 509]. Results are also given for a selected set of np and pp observables.     

s- and p-wave neutron spectroscopy. Xe. Intermediate structure in 90Y and the generalized R-matrix theory

The resonance structure observed in the ⁸⁹Y(n, n)⁸⁹Y total cross section measurements in the range of 0.3 to 1.2 MeV incident energy was investigated using the generalized R-matrix theory of nuclear reactions and the doorway interpretation of intermediate structure. The energies and wave functions of the doorway resonances were calculated in a 2-particle and 3p-1h basis of the shell model. The model space and the parameters of the model calculation chosen were consistent with other shell model calculations in the mass-90 region. Several strong p-wave doorways with J^π = 0⁺, 1⁺, and 2⁺ were predicted by the model in the energy range studied. This is due to proximity of p-wave giant resonance. The escape widths Γ^↑ and the spreading widths Γ^↓ for these states were evaluated using the model wave functions and the R-matrix formalism. The calculated energy dependence of the total cross section shows that most of the predicted doorways are in general agreement with the observed anamolies with similar relative strength. More significantly, the underlying p-wave gross structure representing a grand average is of very similar shape in both theory and experiment. As expected in the mass 90-region, the s- and d-wave doorways contribute less significantly to the calculated resonance structure.     

Application of K-matrix CMSR'S to single charged pion photoproduction

The background terms, required in addition to Regge poles with some definite properties, in order to describe the high energy behavior of the t-channel single pion photoproduction amplitudes F_t^(?)(v, t), i = 1, …, 4, are investigated in the framework of a reggeized K-matrix model. To this end continuous moment sum rules (CMSR's) are applied, not to the full amplitudes, but just to the two-particle K-matrix amplitudes assumed to show pure Regge pole asymptotic behavior. The K-matrix amplitudes are defined by taking into account in the Heitler equation the elastic (πN) intermediate state alone. Such a definition corresponds to a weak-cut model with only final-state interactions included. It turns out that strong background contributions are still present on the high energy side of the K-matrix CMSR's. The Regge cut contributions generated within our formalism are found to be too weak to account for the background terms in the full amplitudes. This is presumably to a great extent due to the neglect of the (?N) diffraction-dissociation intermediate state in the definition of the K-matrix amplitudes.     

Average neutron resonance parameters and radiative capture cross sectins for the isotopes of molybdenum

The neutron capture cross sections of the stable molybdenum isotopes have been measured with high energy resolution (ΔE/E ? 0.2 %), between 3 and 90 keV neutron energy, at the 40 m station of ORELA. Average resonance parameters are extracted for s- and p-wave resonances. The s-wave neutron strength function is close to 0.5 × 10^?4 for all isotopes, but the p-wave strength function exhibits a well defined peak near At~ 95.Both s- and p-wave radiative widths decrease markedly as further neutrons are added to the closed shell. The p-wave radiative widths are generally greater than the s-wave widths showing the presence of non-statistical γ-decay mechanisms.Valence neutron theory fails to explain the magnitude of the p- to s-wave radiative width disparity and doorway state processes are invoked. In particular, the data for ⁹⁸Mo appear to violate the usual valence theory, since the correlations between radiative and neutron strengths are small. Further, the radiative widths are smaller than can be explained on the valence model. An explanation for the loss of valence strength is advanced.Interpolated resonance parameters allow an estimate for the unknown cross section for ⁹⁹Mo(n, γ).     

Valence-doorway model for radiative capture

A model of radiative capture is presented in which the T-matrix for valence capture is expressed as the sum of three terms, viz., a direct capture term, a term that describes the coupling between a single-particle resonance and the many possible continuum channels, and a pure single-particle resonance term. Expressions for the target excitations are not explicitly included, and projection operators are used to break up the single-particle space into resonant and nonresonant parts. The spreading of single-particle resonances is described through the coupling to particle-vibration doorways. The model is applied to the cases of ⁹⁸Mo(n, γ) ⁹⁹Mo and ⁹²Mo(n, γ) ⁹³Mo for an energy range of 0–3 MeV. In these cases the single-particle resonance amplitude is overwhelmingly dominant, and doorway effects are found to be relatively unimportant. Radiative widths obtained from our calculated average cross sections are compared to the corresponding widths obtained by averaging over the detailed fine structure measurements of Chrien et al. for ⁹⁹Mo. Good agreement is obtained. The valence capture cross section in ⁹³Mo is considerably smaller than in ⁹⁹Mo. This is consistent with the measurements of Chrien et al. and in good agreement with a recent analysis by Soloviev and Voronov.     

A modified K-matrix approach to many-channel processes

The definition of a K-matrix, that does not have the direct channel singularities of the scattering amplitude, is not unique. Different choices lead to different integral equations for K with different physical content. We discuss the choice where the intermediate states have a simple off-shell behaviour for each partial wave. Crossing leads to an identification of the singularities of the K-matrix elements and to an iterative method of finding the scattering amplitude starting from a “bare” world.     

Isospin purity of the isobaric analogue resonances for several open channels

The isospin purity of the full and of the doorway state scattering functions in the internal region is investigated, under the assumption of no internal mixing. When several channels are open, it is shown that the isospin is pure at an energyE _> (slightly larger then the resonance energy) only under a restrictive condition, namely that the reduced widths of the analogue state vanish in all channels but one. This may happen either because of isospin selection rules (for the neutron channels) or for reasons of nuclear structure (for some of the proton channels). Then, the corresponding (p, n) and (p, p′) cross sections vanish atE _>. If several channels with non-vanishing reduced widths are open, the isospin of both the full and doorway state wave function is no longer pure atE _>. If two proton channels are open, the doorway state contains three different types of admixtures in the internal region. Two, one of isospinT _> and one of isospinT _<, are excited according to a simple Breit-Wigner probability law. They are both independent of the entrance channel. The third one has isospinT _<, depends upon the entrance channel and is excited according to an asymmetric enhancement factor identical toRobson's.     

Resonances and Fluctuations in Nuclear Reaction Cross Sections

On the basis of the continuum shell model, an expression for the S-matrix is derived. The external mixing between shape resonances and resonance states of complicated nuclear structure is investigated. It is shown that the doorway properties of the shape resonances may lead to an l-dependent transparency of the optical potential and, consequently, to an energy dependence of large back angle scattering if the unitarity of the S-matrix is taken into account.     

Optical-model description of the fragmentation of a single-particle state

Several possible definitions of thel = 0 neutron strength function are examined. Their energy dependence is investigated and particular attention is paid to threshold effects when the single-particle state lies in the vicinity of the elastic threshold. Among other results, it is shown that, in actual physical situations, the S-matrix strength function never exhibits a cusp at threshold.     

Inelastic scattering of20Ne and12C on58Ni and the three-body continuum

Using beams of²⁰Ne at 291 and 392 MeV and¹²C at 300 MeV the inelastic excitation of collective modes in⁵⁸Ni has been studied with a Q3D magnetic spectrometer. In the analysis different contributions are unfolded from the spectra:1. single lines,2. the inelastic continuum,3. the three-body continuum, which mainly originates from one-nucleon pick-up to unbound states followed by the nucleon emission. Special care was taken to calculate the spectral shape and strength of this process. The population of unbound²¹Ne-states is measured in a neutron-²⁰Ne coincidence experiment. Coupled channel calculations have been performed to extract the deformation parameters for the inelastic states, including mutual excitation. For the giant quadrupole resonance 70 % of the energy weighted sum rule is found. The excitation of higher multipoles is calculated and extrapolated to higher incident energies (50 MeV/u). The importance of the in-elastic excitation as a “doorway process” to more complex interactions is discussed. It is found that especially the excitation of the projectile (²⁰Ne) takes a large fraction (20–50 %) of the incoming flux in the first step.     

Pre-equilibrium nuclear reactions: The overlapping-doorway model for multi-step compound processes

Griffin's simple exciton model, designed to describe the spectra of pre-equilibrium particles emitted in compound-nuclear reactions, has recently been developed into a full-fledged doorway theory of multi-step compound reactions by three separate groups: Feshbach et al., Agassi et al., and Friedman et al. All three groups employ a time-independent scattering formalism, but since the time-sequence in which probability diffuses through the system of N doorway classes (e.g., 2p-1h, 3p-2h, etc. in the exciton model) is essential to a full understanding of the process, we have re-analyzed the problem in a time-dependent formalism. This shows explicitly how the statistical assumptions of the theory produce an irreversible flow of probability through the classes, described by a master equation. The solution of this equation demonstrates that the occupation probability of the compound system decays in time like a superposition of exponentials, with decay rates equal to the energy autocorrelation widths of the N “eigenclasses” of the system.Althrough intercomparison of the three theoretical approaches is also given, indicating which ones can be derived from the others and pointing out the differences in their basic statistical assumptions.     

s- and p-wave neutron spectroscopy Xf(i). Intermediate structure in the 28Si + n reaction: R-matrix interpretation of experimental data

A multilevel R-matrix analysis of Si neutron cross-section data measured at NBS has been performed up to about 4.5 MeV neutron energy. Only a small fraction of the p- and s-wave s.p. strength is observed, but both exhibit local concentrations of strength indicative of doorway structure around 1 and 0.2 MeV, respectively. Besides the well-known 180 keV, strong, $\text{1}\text{2}{}^{\mathrm{+}}$ resonance, the s-wave resonance structure is of moderate strength and widely distributed. The f- and d-wave assignments are not unambiguous, but J > 3/2 resonances show strong signs of intermediate structure for d-waves. A possible correlation between neutron and gamma decay channels and the connection between the states observed in (n, n), (d, p), (n, γ), and (γ, n) channels is discussed. A coreparticle doorway interpretation for s and p- waves is presented.     

s- and p-wave neutron spectroscopy Xf(ii). Intermediate structure in the 28Si + n reaction: Doorway state calculation

A core-particle calculation developed to describe the low-lying structure of ²⁹Si is extended above neutron threshold energy to yield information on the structure of doorway states indicated by the ²⁹Si + n reaction. The recent experimental evidence for a $\text{J =}\text{3}\text{2}{}^{\mathrm{?}}$ doorway state common to the ²⁸Si + n and the ²⁹Si + γ channels is supported by the calculation which also reproduces correctly the magnitudes of the neutron escape widths and the E1 radiative strengths of the $\text{3}\text{2}{}^{\mathrm{?}}$ doorway states.     

K −-Proton atomic transitions

The relation between theK ^}-P scattering length and the X-ray spectrum for the 2p → 1s electromagnetic transition inK ^?-P atoms is examined. A coupled-channel potential model is used to explicitly calculate the energy of theS-matrix pole in the 1s channel, which is then compared with the energy obtained from the scattering lengths via the standard equation. The X-ray spectrum is calculated and compared with the Lorentzian shape associated with the complex energy of theS-matrix pole. In addition, theK ^?p branching ratios are compared at threshold and at the complexS-matrix pole energy.     

The Gell-Mann-Goldberger formula for long-range potential scattering

A derivation of the Gell-Mann-Goldberger (GG) formula and cut-off versions of this formula for the T-matrix involving long-range potentials is given. The derivation is based on the time-dependent and recently developed stationary formalisms for scattering via long-range potentials. A stationary S-operator expression for two-body Coulomb-like scattering is derived. Using the well-known expression for the off-energy-shell “T-matrix” for a pure Coulomb potential the energy-shell limit of this stationary expression is shown to yield the pure Coulomb scattering amplitude. A proof of the convergence of the perturbation series corresponding to the Gell-Mann-Goldberger formula for the two-body Coulomb-like T-matrix is given.     

Field redefinition and on-shell renormalisation

The consequences of a field redefinition on the renormalisability character of a theory are thoroughly studied in the special example of the non-linear σ model in two dimensions. Multiplicative renormalisability is lost but it is shown that, at least to one-loop order, an “on-shell renormalisability” property remains which means that S-matrix elements can be made finite by a multiplicative renormalisation of the fields and parameters of the model.     

The quantum theory of nucleon correlation in finite nuclei. II. Higher order correlation effects and additive pair approximation

In the first part of this article it was shown that the variational solution of the Schroedinger equation of a finite Fermion system can be written as a finite sum of A terms (for A particles) the first of which is the Hartree-Fock energy, while the rest represent the correlation effects. In the first part explicit formulas for the 2-particle correlation energy were given. In this paper explicit formulas are given for the higher order correlation energies. It is shown that two different models can be developed depending on the orthogonality condition used. Beginning with the 4th order effects the “linked” and “unlinked” correlation terms are separated. An exact formula is given for the case in which only the 2-particle effects, linked and unlinked are taken into account. The “additive pair approximation” in which the correlation energy is given as the sum of 2-particle energies is investigated and it is shown to be related to the exact formula by a clearly defined set of approximations. Various possible applications of the model are discussed.