Six of seven cases of dynamic calculation of Negele et al. lead to results in disagreement with the experimental results for the number of prompt neutrons or the spin of fission products. The seventh, based on one-body dissipation according to the Fermi gas model, is free from such contradiction, gives time and kinetic energy in agreement with the statistical theory, dissipates enough energy, implies statistical equilibrium, and therefore may be considered as the dynamical interpretation of the statistical theory of fission.

NUCLEAR REACTIONS, FISSION Statistical theory.

The isotopic dispersions have been calculated for fission processes listed in the Introduction below by using recommended fission yield data. Those dispersions in which the neutron numbers of isotopes lying in the vicinity of the most probable mass fit a strong shell have been found to be pronouncedly discontinuous; the discontinuity is proved to occur exactly at the most probable mass. The influence of neutron shells on mass dispersion with special reference to discontinuity is discussed on the basis of shell correction vs deformation and neutron number reported by Wilkins, Steinberg, and Chasman. In discontinuous dispersions on the narrower branch lie the isotopes fitting the strong spherical and deformed neutron shells. Apart from recommended data, the relative isotopic fragment yields predicted by the Wilkins-Steinberg model for the complementary pairs $z=50-42, 51-41, 54-38, \mathrm{and} 56-36$, in thermal-neutron fission of ${}_{}{}^{235}{}_{}{}^{}\mathrm{U}$ have also been used to calculate the corresponding dispersions. These have been found to be in good agreement with those calculated from recommended data, and discontinuities were observed, in general, wherever expected.

NUCLEAR REACTIONS, FISSION ${}_{}{}^{232}{}_{}{}^{}\mathrm{Th}$, ${}_{}{}^{233}{}_{}{}^{}\mathrm{U}$, ${}_{}{}^{235}{}_{}{}^{}\mathrm{U}$, ${}_{}{}^{238}{}_{}{}^{}\mathrm{U}$, ${}_{}{}^{239}{}_{}{}^{}\mathrm{Pu}$, ${}_{}{}^{252}{}_{}{}^{}\mathrm{Cf}$; calculated isotopic mass dispersions from recommended yields and Wilkins-Steinberg model. Discontinuity as neutron-shell effect.

We present the differential cross section for proton-proton bremsstrahlung at an incident beam momentum of 1.38 GeV/c (730 MeV). The measurements are made with an overconstrained detection system to reliably exclude all $\mathrm{pp}\to \mathrm{pp}{\pi }^0$ events. We have obtained differential cross sections at 16 photon angles at photon energies up to 200 MeV for protons scattered at 50° (lab). For each of the photon angles, $\frac{d^5\sigma }{d{\Omega }_{p}d{\Omega }_{\gamma }d{E}_{\gamma }}$ decreases smoothly with increasing photon energy up to $E_{\gamma }\left(\mathrm{lab}\right)\simeq 80$ MeV. Above this photon energy the cross sections rise again. Our results are compared with the soft photon approximation and the external emission dominance calculations. The soft photon approximation describes most spectra fairly well up to $E_{\gamma }\simeq 100$ MeV and external emission dominance up to $E_{\gamma }\simeq 60$ MeV (lab), but both predict cross sections which are too small at the highest photon energies.

NUCLEAR REACTIONS Proton-proton bremsstrahlung, $E=730\mathrm{}$ MeV; measured $\sigma (\theta ,E_{\gamma })$; compared with soft-photon calculations.

A simple model for total reaction cross sections based on the impulse approximation and the use of an eikonal propagator is derived. Semiquantitative approximations are used to demonstrate that total reaction cross sections are determined by the radius at which the profile function is equal to one mean free path for the incident particle. A simple quantitative relation between the density and the profile function is also provided. The model is tested against existing data for proton-nucleus total reaction cross sections for ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{40,44,48}{}_{}{}^{}\mathrm{Ca}$, and ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ targets at energies from 100 MeV to 1 GeV. The total reaction cross sections are generally found to be in agreement (with several exceptions) with neutron densities determined either theoretically or from other experiments.

NUCLEAR REACTIONS Total reaction cross sections, matter densities and radii.

Proton total reaction cross sections for 700 MeV protons on ${}_{}{}^{40,44,48}{}_{}{}^{}\mathrm{Ca}$ have been measured by detecting inelastically scattered particles at angles greater than 10° (lab). The measurements were performed with overall uncertainties of ±6.2% and relative uncertainties of ±2.5%. These measurements yield the values of ${\sigma }_R\left({}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}\right)=614\mathrm{}$ mb, ${\sigma }_R\left({}_{}{}^{44}{}_{}{}^{}\mathrm{Ca}\right)=643\mathrm{}$ mb, and ${\sigma }_R\left({}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}\right)=736\mathrm{}$ mb. Matter radii differences $\Delta \mathrm{}(44-40)=(0.05\mathrm{}\pm 0.09)$fm and $\Delta \mathrm{}(48-40)=(0.36\mathrm{}\pm 0.09)$fm were extracted based on a theoretical calculation which takes advantage of the geometrical nature of the cross section. The results are compared to other available measurements of total reaction cross sections on medium-weight nuclei at medium energies and to other determinations of the matter radii differences for the calcium isotopes.

NUCLEAR STRUCTURE ${}_{}{}^{40,44,48}{}_{}{}^{}\mathrm{Ca}$; measured proton total reaction cross sections, $E=700\mathrm{}$ MeV. Deduced matter radii differences.

The experimental production of $p$, $d$, $t$, ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, and ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ particles at 150° from 720-MeV $\alpha$ bombardment of Al, Ag, and Ta nuclei are shown to agree with Frankel's phenomenological "quasi-two-body-scaling" formalism. However, we show our data to also be qualitatively consistent with multiple-scattering, thermodynamic models. More theoretical work is needed, especially to understand composite-particle production.

NUCLEAR REACTIONS Al, Ag, Ta ($\alpha$,$X$), $E=720\mathrm{}$ MeV; $X=p, d, t, {}_{}{}^3{}_{}{}^{}\mathrm{He}, {}_{}{}^4{}_{}{}^{}\mathrm{He}$; measured $\sigma (E, 150°)$. Natural targets. Coalescence, fireball, firestreak, and thermodynamic model calculations.

The results of an optical model analysis of medium energy $p$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ elastic scattering cross section and analyzing power measurements are reported. The analysis is based on the use of the Dirac equation with a mixture of a Lorentz scalar potential and the timelike component of a four-vector potential.

NUCLEAR REACTIONS Nucleon-nucleus scattering. Relativistic optical model analysis of $p$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ scattering. Calculated cross sections and analyzing powers.

The closure approximation is necessary for the evaluation of the double scattering term, for the reduction of the multiple scattering from a dynamic nuclear system to the scattering from fixed scattering centers, and for the derivation of the Glauber formalism from the Watson multiple scattering formalism. If closure is valid, there should be no double scattering term in the optical potential for an uncorrelated target nucleus. A simple Gaussian model, in which the intermediate nuclear states are scattering states, is used to calculate the binary scattering term in the optical potential for such a nucleus. The nonlocal binary potential is used, together with the local single-scattering potential, to predict the differential scattering cross section of 1 GeV protons incident on ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$. The results confirm the validity of the closure approximation for the heavier nucleus at all scattering angles. The results for the lighter nucleus confirm its use for small scattering angles but cast doubt on its use in the region of the first diffraction minimum. In the case of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, the results also show that double scattering may improve the optical model fits to the scattering data.

NUCLEAR REACTIONS Optical potential model, double scattering effects; relation to Watson, Glauber, theories.

We have estimated the parameters of the optical potential for low energy pion-nucleus scattering from the values obtained empirically in the analysis of pionic atom data. The ambiguities in these parameters are discussed. The shape of the low energy elastic scattering cross section for light elements is particularly sensitive to the relative strength of the repulsive and velocity-dependent attractive parts of the optical potential, and is well reproduced by pionic atom parameters. Absorption is the dominant reaction channel at low energies. Pion nucleon phase shifts are used to extrapolate optical model parameters to higher energies. The behavior of the elastic and inelastic angular distributions and of various partial cross sections is discussed, and compared with experiment.

NUCLEAR REACTIONS Calculated pion-nucleus optical potential, elastic scattering, inelastic scattering and partial cross sections. 30, 40, 50 MeV on various targets. 116-220 MeV on individual targets.

The fourth-order noniterative diagrams involving nucleon intermediate states and including $\pi$ exchange are calculated in momentum space, in the framework of noncovariant perturbation theory. The effect of these so-called crossed-box and stretched-box diagrams on $\mathrm{NN}$-scattering phase shifts is studied by adding these diagrams to a one-boson-exchange potential, likewise derived from noncovariant perturbation theory, and solving a corresponding scattering equation.

NUCLEAR REACTIONS Nucleon-nucleon interaction, noniterative $\pi$-exchange, crossed-box diagrams, $\mathrm{NN}$ phase shifts.

The pion-nucleon vertex function with an off-mass-shell nucleon is obtained through sideways dispersion relations with the $P_{11}$ and $S_{11}$ pion-nucleon phase shifts as only input. Contrary to the recent calculation of Nutt and Shakin, we find that the proper and improper vertex functions behave quite differently, indicating the importance of the nucleon propagator dressing. In particular the proper vertex function is found to have two poles in the unphysical region.

NUCLEAR REACTIONS pion-nucleon vertex functions, sideways dispersion relations, nucleon self-energy corrections.

Activation cross sections were measured for the nucleides ${}_{}{}^7{}_{}{}^{}\mathrm{Be}$, ${}_{}{}^{11}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{13}{}_{}{}^{}\mathrm{N}$, ${}_{}{}^{18}{}_{}{}^{}\mathrm{F}$, ${}_{}{}^{22}{}_{}{}^{}\mathrm{Na}$, and ${}_{}{}^{24}{}_{}{}^{}\mathrm{Na}$ formed by the interaction of 300 GeV protons with aluminum. The measurements were made relative to the production of ${}_{}{}^{11}{}_{}{}^{}\mathrm{C}$ in polyethylene and graphite foils. The cross section ratios are the same within experimental error as those observed at 28 GeV.

NUCLEAR REACTIONS Al($p$,$x$)${}_{}{}^7{}_{}{}^{}\mathrm{Be}$, ${}_{}{}^{11}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{13}{}_{}{}^{}\mathrm{N}$, ${}_{}{}^{18}{}_{}{}^{}\mathrm{F}$, ${}_{}{}^{22}{}_{}{}^{}\mathrm{Na}$, ${}_{}{}^{24}{}_{}{}^{}\mathrm{Na}$, $E=300\mathrm{}$ GeV; measured $\sigma$ relative to ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$($p$,$\mathrm{pn}$)${}_{}{}^{11}{}_{}{}^{}\mathrm{C}$.

The low-energy theorem for emission of soft photons from two nucleons is derived in a three-dimensional relativistic theory. It is shown that if the free nucleon currents are defined in terms of the Foldy-Wouthuysen transformation, the exchange current in the soft-photon limit is exactly determined by the commutator of the nucleon-nucleon interaction with the electric dipole operator. The constraints imposed by this theorem are used to examine the conventional method of calculating the pair-excitation currents.

NUCLEAR REACTIONS Low-energy theorem for the exchange current, relativistic effects.

Angular distributions for elastic and inelastic scattering of ∼ 50 MeV ${\pi }^{+}$ projectiles from targets of ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, 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 $b_1$ is close to the value predicted from free $\pi N$ models, while the $s$-wave parameter $b_0$ 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 good description of both elastic and inelastic scattering.

NUCLEAR REACTIONS Elastic and inelastic scattering of 50 MeV ${\pi }^{+}$ from ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, Si and Fe. Angular distributions: $25°<{\ominus }_L<145\mathrm{}°$. Optical model and DWBA of cross-section data.

Using the short pulses of a linac a measurement has been performed of partial ${\mu }^{-}$-capture rates in ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ leading to the ${\mathrm{O}}^{-}$ and $1^{-}$ bound excited states of ${}_{}{}^{16}{}_{}{}^{}\mathrm{N}$. The experimental ${\mathrm{O}}^{+}$→${\mathrm{O}}^{-}$ transition rate supports the assumption of a large mesonic exchange effect in the time part of the weak axial current.

NUCLEAR REACTIONS Muon capture ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$($\mu$,$\nu$)${}_{}{}^{16}{}_{}{}^{}\mathrm{N}$. Measured $I_{\gamma }$, Ge(Li) detector. Deduced partial capture rates. Mesonic exchange correction to capture rate calculations.

Recently, precise experimental measurements have been obtained for the reaction ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$($\gamma$,$p$)${}_{}{}^{15}{}_{}{}^{}\mathrm{N}$ for photon energies 50-350 MeV. For photon energies greater than 100 MeV, isobar photoproduction is expected to provide an important part of the ($\gamma$,$p_0$) reaction. We evaluate the isobar contribution to the ($\gamma$,$p$) reaction, and we add this amplitude to a term which corresponds to direct knockout of the proton by the photon. We compare our results in detail with the experimental data from Glasgow and Bates on ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$. Our calculation uses coupling constants and nuclear wave functions determined from independent experiments, and we neglect the isobar-nucleus interaction. Agreement with the experimental data is quite reasonable, and isobar photoproduction provides a very large contribution to the nuclear photoeffect in ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, for photon energies above about 100 MeV. We consider the effects of various corrections to our formalism, such as the effect of distortion of the outgoing nucleon by the nuclear force, center of mass corrections, and the inclusion of $\rho$ mesons in addition to pions at the isobar decay vertex. None of these corrections alters the qualitative features of the isobar amplitude and its importance in the nuclear photoeffect at medium energies.

NUCLEAR REACTIONS, photonuclear reactions, nuclear photoeffect, medium energies, isobars, isobar current, ($\gamma$,$p$) reactions.

Inelastic electron scattering in ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$ between 4 and 50 MeV excitation energy reveals two concentrations of $E2\mathrm{}$ strength in the continuum. One is between 15 and 20 MeV, with a peak at 17 MeV, and can be identified with the giant quadrupole resonance in the ground state oblate well. A broad distribution of $E2\mathrm{}$ strength between 22 and 42 MeV is predominatly isovector in nature. In addition, a small but persistent $E2\mathrm{}$ peak at 24 MeV was found, which may be interpreted as being the corresponding state in the prolate well to the 17 MeV resonance. It is shown that 50% or more of the photon cross section in excess of the classical dipole sum rule between 10 MeV and the pion threshold may be due to $E2\mathrm{}$ absorption.

NUCLEAR REACTIONS ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$($e$,$e^{\prime }$), $E_0=92\mathrm{}$ MeV. Measured $\frac{d^2\sigma }{d\Omega d{E}_x}$, bound and continuum states. Deduced multipolarity $\lambda$, reduced matrix element $B\left(E\lambda \right)$, sum rule exhaustion.

The suppression of the isospin-½, three-body photodisintegration of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is shown to result from two facts: (1) For the $S=\frac{1}{2}$, $I=\frac{1}{2}$ states of three nucleons, the spin-isospin symmetries lead to integral equations dominated by a single effective interaction $V^{+}$, which is the average of the $\mathrm{NN}$ $s$-wave spin-singlet and spin-triplet interactions and (2) This $V^{+}$ supports a bound state due to the nature of the $\mathrm{NN}$ effective-range parameters. In the approximation that the (½,½) states are generated from $V^{+}$ alone, it is clear that a fraction of the isospin-½ bremsstrahlung-weighted sum rule must come from two-body photodisintegration with a corresponding reduction of the three-body channel contribution, since the total isospin-½ sum rule is fixed. The substantial reduction of the actual physical isospin-½, three-body photodisintegration is directly related to the strength of $V^{+}$.

NUCLEAR REACTIONS Photodisintegration of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$; exact three-body calculation; isospin sum rules.

Gordon's method for constructing wave functions for a nonlocal potential is generalized to obtain solutions of the van Leeuwen-Reiner equation. Exact analytical expression for off-shell Jost solutions and Jost functions are derived for the Mongan case IV potential.

NUCLEAR REACTIONS Scattering theory; Generalization of the Gordon's method to off-shell scattering; Jost solutions and Jost functions for the Mongan case IV potential.

A recent work by Rosenberg on cluster states in reaction theory is reexamined and generalized to include energies above the threshold for breakup into four composite fragments. The problem of elastic scattering between two interacting composite fragments is reduced to an equivalent two-particle problem with an effective potential to be determined by extremum principles. For energies above the threshold for breakup into three or four composite fragments effective few-particle potentials are introduced and the problem is reduced to effective three- and four-particle problems. The equivalent three-particle equation contains effective two- and three-particle potentials. The effective potential in the equivalent four-particle equation has two-, three-, and four-body connected parts and a piece which has two independent two-body connected parts. In the equivalent three-particle problem we show how to include the effect of a weak three-body potential perturbatively. In the equivalent four-body problem an approximate simple calculational scheme is given when one neglects the four-particle potential the effect of which is presumably very small.

NUCLEAR REACTIONS Scattering theory. Effective few-particle formulation of multiparticle problems. Approximation schemes.