K.W. McVoy

Inclusive projectile fragmentation in the spectator model

Abstract Grazing-angle singles spectra for projectile fragments from nuclear collisions exhibit a broad peak centered near the beam velocity, suggesting that these observed fragments play only a “spectator” role in the reaction. Using only this spectator assumption (but not DWBA), we find that a “prior form” formulation of the reaction leads, via closure, to a 〈ψ∥W∥ψ〉-type estimate of the inclusive spectator spectrum, thus relating it to the reaction cross section for the “participant” with the target. We show explicitly that this expression includes an improved multi-channel version of the Udagawa-Tamura formula for the “breakup-fusion” or incomplete-fusion cross section, and identifies it as the fluctuation part of the participant-target reaction cross section. A Glauber-type estimate of the distorted wave functions which enter clearly shows how the width of the peak in the spectator spectrum arises from the “Fermi motion” within the projectile, as in the simpler Serber model, but is modified by the “overlap geometry” of the collision.

Nuclear rainbows and heavy-ion scattering

Abstract We discuss refractive effects on scattering in the presence of strong absorption, and conclude that a residuum of a true nuclear rainbow has been seen only for light ions. Such a rainbow seems to be unlikely for heavy ions because of their stronger absorption. Nonetheless, the structureless falloff in angular distribution that has been observed in some heavy-ion measurements to follow a “Fraunhofer crossover” region of diffraction oscillations indicates that the farside scattering amplitude has become dominant at these larger angles. This “farside tail” itself is a refractive effect and its observation already carries important information on the optical potential.

Asymmetric deflection functions and the extinction of rainbows: A comparison of α-particle scattering from 40Ca and 44Ca

Abstract A decomposition of elastic scattering amplitudes into their nearside and farside components is employed to exhibit the presence of a striking farside (“nuclear”) rainbow in the angular distributions for α + 40 Ca scattering at energies from 36 to 61 MeV. The rainbow is identified by a deep “Airy minimum” which is present in the farside components of the angular distributions throughout this energy region. Over the same energy range, the angular distributions for the more absorptive α + 44 Ca scattering exhibit no Airy minima, and, in this sense, show no nuclear rainbow. However, a steepening of the slope of their smooth farside component, which appears at energies above that (∼60 MeV) for which the rainbow angle for α + 40 Ca becomes less than 180°, may serve to identify a residual rainbow effect, even in α + 44 Ca, at energies above 60 MeV. The manner in which the Airy minima of this and other nuclear rainbows are extinguished by absorption is examined in detail and is found to depend critically on the asymmetry in the shape of their deflection functions at low energies.

Exploratory studies of the elastic scattering of 11Li + 12C

Abstract Calculations have been made of the elastic scattering of 11Li + 12C at E A = 30 and 85 MeV as an example of the scattering to be expected for “exotic” neutron-rich light nuclei. The real optical potential was generated from a folding model using Hartree-Fock densities for 11Li. A variety of assumptions for the absorptive imaginary potential was explored, subject to the important constraint that the predicted reaction cross section remain reasonable. Contrary to some expectations, we provide evidence that these weakly bound systems do not exhibit the diffractive type of scattering that is characteristic of strong absorption. If anything their more diffuse real potentials can result in an enhancement of the refractive features.

Virtual states and resonances

Abstract The distinction between an s-wave virtual state and a resonance is explained by exhibiting the characteristic phase shifts and scattering cross-section shapes attributable to each. They are described in both R-matrix and S-matrix terms, and the relation between these two parametrizations is explained. As an application, in the two succeding papers we investigate the energy dependence of the s-wave neutron strength function and show how it is affected by the presence of a virtual state in the complex optical potential. Such a state is expected to occur for A ≈ 50 and A ≈ 150, where the 3s and 4s optical states are slightly unbound.

Analysis of an unusual potential ambiguity for 16O+16O scattering

Abstract Woods-Saxon optical-potential fits to recent measurements of 16 O+ 16 O elastic scattering at 350 MeV have been interpreted as providing clear evidence for a nuclear rainbow. Three model-independent potentials have also recently been found to reproduce these data. Two of them were indeed found to be members of a rainbow series, in which a distinctive minimum near 44° is the first or second Airy minimum of a rainbow pattern, but the scattering by the third (shallower) potential was less easy to interpret. We show here that this potential achieves its fit to the data by a different and somewhat unconventional mechanisms. This serves as a reminder that ambiguities in interpretation can persist, even with as extensive and accurate a data set as this measurement provides.

Nuclear rainbows overlapping resonances

Abstract Stimulated by a controversy which has recently arisen concerning the identification of nuclear rainbows in the angular distributions of elastic heavy-ion scattering, we present a detailed examination of the properties of such rainbows in this and the following article. Realistic nuclear rainbows are found to deviate substantially from the classic Airy function shape (even for purely real potentials), primarily because of the asymmetry of their deflection functions. This asymmetry (which is essential to an understanding of the influence of absorption on the rainbows) occurs because nuclear rainbows are actually an overlapping-resonance phenomenon, with broad resonances in low- l waves and narrow ones near the grazing l . The distinction between “orbiting” and rainbows is elucidated in the course of the discussion, and the following two articles extend previous suggestions by Goldberg, to explain how various features of rainbows can uniquely determine specific properties of both the real and imaginary parts of the corresponding optical potential.

The 90° excitation function for elastic 12C + 12C scattering: The importance of Airy elephants☆

Abstract The 90° excitation function for elastic 12 C + 12 C scattering, at laboratory energies between the Coulomb barrier and 130 MeV, exhibits a complex structure of peaks and valleys whose nature has remained an unsolved mystery for more than 20 years. The problem has primarily been caused by the difficulty of choosing from a plethora of discretely ambiguous optical potentials. However, data accumulated above 150 MeV over the last decade have determined unique potentials at these higher energies, and the requirement of continuity downward in energy has recently permitted the determination of a unique set of potentials for angular distributions at energies below 130 MeV, where the excitation-function data exist. These new potentials are used to provide a mean-field (i.e., nonresonant) interpretation of the structure in the 12 C + 12 C 90° excitation function between 70 and 130 MeV. Its most prominent minima are found to be Airy minima from nuclear rainbows, with the remaining structure arising primarily from more elementary optical phenomena related to Fraunhofer diffraction. These same potentials are also successful in explaining the details of excitation functions measured very recently at other angles by Morsad.

Coulomb-nuclear interference in elastic heavy-ion scattering

Abstract Elastic heavy-ion angular distributions at small angles, forward of the quarter-point angle, have often been characterized as examples of either a Coulomb rainbow or of Fresnel diffraction. Using the modern high-quality data of Hostachy et al. as an example, we argue that the truth is intermediate between these two extremes, and indeed so far from them as to make both models quite inappropriate descriptions of nuclear reality. We suggest retreating to the more general label of “Coulomb-nuclear interference”, which includes both of these (inapplicable) extremes as special cases. A semiclassical analysis of optical-model fits is found to offer helpful insight into the nature of this interference.

R-matrix and S-matrix strength functions

Abstract Strength functions are defined for both R - and S -matrices as the energy average of the ratio of pole residues to pole spacings. The R -matrix strength functions exhibit no threshold effects, while the S -matrix strength function for channel c considered as a function of the energy has a branch point at that channel threshold. Although the R strength function for l = 0 neutrons may exhibit a giant resonance as a function of energy just above neutron threshold (e.g. for A ≈ 55 and 155), the S strength function does not as a direct consequence of its threshold branch point.