Coupled channel calculations for heavy ion inelastic scattering can be alleviated through the use of integrals involving the product of two Coulomb functions, regular or irregular, with some inverse power of the radius, the range of the integration going from the point where the nuclear potential is weak to infinity. These integrals can be obtained by recurrence relations, starting with at most four of them which must be obtained by some method already described in the literature.

NUCLEAR REACTIONS HI inelastic scattering. Treatment of Coulomb interaction above Coulomb barrier.

A macroscopic liquid drop model is considered in order to demonstrate the deep inelastic processes that take place in heavy ion collisions. The colliding nuclei, treated as two liquid drops, are assumed to form a single rotating and vibrating compound drop. This drop will either separate again into the parent nuclei, in addition to some smaller particles (satellites), or will form an equilibrium compound nucleus which decays either by particle evaporation or through fission. The maximum and minimum values of the critical angular momentum between which complete fusion occurs are calculated according to this model for several heavy ion reactions and good agreement with the experimental values is observed.

NUCLEAR REACTIONS Heavy ion scattering; modified model for interaction of two liquid drops, its application to deep inelastic collisions between two heavy nuclei; calculations of critical angular momentum.

The electrofission cross section for ${}_{}{}^{234}{}_{}{}^{}\mathrm{U}$ from 5.5 to 25.4 MeV has been measured. From a combined analysis of it and the previously measured photofission cross section, using virtual-photon spectra calculated in the distorted-wave Born approximation, the $E2\mathrm{}$ photofission cross section has been determined. Parameters of the fission-decay branch of the giant quadrupole resonance for this nucleus have been obtained. A comparison of the $E2\mathrm{}$ and $E1\mathrm{}$ integrated photofission cross sections for the even uranium isotopes shows that the $E1\mathrm{}$ fission channel increases in strength more rapidly with fissility than does the $E2\mathrm{}$ channel.

NUCLEAR REACTIONS Measured $\sigma \mathrm{}(e,f)\mathrm{}$ for ${}_{}{}^{234}{}_{}{}^{}\mathrm{U}$; deduced ${\sigma }^{E2\mathrm{}}(\gamma ,f)$.

Experimental elastic and inelastic angular distributions for 0.8 GeV $p+{}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ are reported for momentum transfers up to $q=7.1\mathrm{}$ ${\mathrm{fm}}^{-1\mathrm{}}$. At the largest angle the elastic cross section is ∼2.0×${10}^{-8\mathrm{}}$ mb/sr. The data are analyzed using the distorted-wave Born approximation and the coupled-channels formalism, where phenomenologically deformed optical potentials are used in the latter analysis. The multipole moments extracted from the optical potentials are found to be in fair agreement with those of the charge densities as obtained from electron scattering.

NUCLEAR REACTIONS ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}(p,p^{\prime })$, $E=0.8\mathrm{}$ GeV; measured $\sigma \left(\theta \right)$; natural targets; resolution 100 to 650 keV, ${\theta }_{\mathrm{c}.\mathrm{m}.\mathrm{}}=3.9\mathrm{}°-67.2°$, $q_{max}=7.1\mathrm{}$ ${\mathrm{fm}}^{-1\mathrm{}}$. Optical model potential, DWBA and coupled channels analyses, rotational model, coupling parameters, multipole moments, inelastic deformation lengths ${\beta }_{l}R$.

A recently developed relativistic theory of nuclear matter and of finite nuclei is extended to allow for consideration of the nucleon-nucleus interaction at positive energy. A new method of reducing the Dirac equation to a nonrelativistic Schrödinger equation is presented. This scheme is used, along with the results of calculations performed earlier for nuclear matter, to determine the parameters of the nuclear optical model. The results are in good agreement with the phenomenological local potentials conventionally used to analyze nucleon-nucleus scattering.

NUCLEAR REACTIONS Relativistic effects and the nuclear optical potential, reduction of Dirac equation to Schrödinger form.

The (${\pi }^{+},p$) reaction has been studied at $T_{\pi }=90\mathrm{} \mathrm{and} 180$ MeV on ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, and at $T_{\pi }=90\mathrm{} \mathrm{and} 170$ MeV on ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}$. The resolution was sufficient to resolve many single levels in the residual nuclei, and angular distributions were obtained between laboratory angles of 10° and 115°. A comparison between the present ($\pi ,p$) data and existing high-energy ($p,d$) data in terms of a model assuming intermediate pion exchange in the ($p,d$) reaction indicates that the ($\pi ,p$) reaction is an important cubprocess in the ($p,d$) reaction. The present data also suggest that pion rescattering is likely to be a fundamental part of the ($\pi ,p$) reaction mechanism.

NUCLEAR REACTIONS ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}({\pi }^{+},p){}_{}{}^{11}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}({\pi }^{+},p){}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, $E_x=0.0\mathrm{} \mathrm{to} 28$ MeV, $T_{\pi }=90\mathrm{} \mathrm{and} 170 or 180$ MeV, $\Delta E=400\mathrm{} \mathrm{to} 600$ keV, enriched targets, magnetic spectrometer, measured $\frac{d\sigma }{d\Omega }({\pi }^{+},p)$ from ${\theta }_{\mathrm{lab}}=10\mathrm{}°$ to ${\theta }_{\mathrm{lab}}=115\mathrm{}°$, comparison with ($p,d$) reaction data, reaction mechanism.

We propose a two-component model for the proton-nucleus collisions. One of the components is a compressed tube of hadronic matter which decays into the fast particles over a long span of time. The other is a partially thermalized remnant nucleus. Its thermalized region evaporates into nucleons, which strike at the undisturbed region of the remnant nucleus and break it into the nuclear fragments. We assume that a fragment is formed by the recombination of an evaporation nucleon with a piece of the remnant nucleus which is at rest in the rest frame of the remnant. In this way we obtain a good description of the fragment mass dependence of the apparent temperature.

NUCLEAR REACTIONS Proton-nucleus collisions; fragment production; fire-balls; evaporation model; nuclear temperature.

We calculate pion-deuteron elastic scattering at 142, 181, and 254 MeV, using four separable models of the off-shell nucleon-nucleon $T$ matrix. The form factors for two of these models were obtained by fitting the low-energy data, as well as the change of sign of the phase shifts at $T_{\mathrm{lab}}\gtrsim 300$ MeV, by means of energy-dependent separable potentials. These form factors have ranges in momentum space which are larger than those of the Yamaguchi potentials. We find that the differential cross section for pion-deuteron scattering is very sensitive to the off-shell behavior of the nucleon-nucleon $T$ matrix. We also find that it is possible to fit the data at 254 MeV by using the long-range form factors that take into account the hard core, although the data at 142 MeV is better described using shorter ranges.

NUCLEAR REACTIONS $d(\pi ,\pi )d$, $E=142\mathrm{}$, 181, 254 MeV; calculated $\frac{d\sigma }{d\Omega }$.

Angular distributions for the elastic and inelastic scattering of 291-MeV ${\pi }^{+}$ and ${\pi }^{-}$ 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 ${}_{}{}^9{}_{}{}^{}\mathrm{Be}(\pi ,{\pi }^{\prime })$, $\mathrm{Si}(\pi ,{\pi }^{\prime })$, ${}_{}{}^{58}{}_{}{}^{}\mathrm{Ni}(\pi ,{\pi }^{\prime })$, ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}(\pi ,{\pi }^{\prime })$, $E_{\pi }=291\mathrm{}$ MeV, measured $\frac{d\sigma }{d\Omega }$; optical potential and DWIA collective analyses.

The effect of the shift of the effective energy of the pion-nucleon amplitude on the pion-nucleus charge exchange due to isospin splitting of the core is investigated. Calculations are done in both plane wave and distorted wave frameworks.

NUCLEAR REACTIONS ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}({\pi }^{+},{\pi }^0){}_{}{}^{13}{}_{}{}^{}\mathrm{N}$, $E=162\mathrm{}$ MeV, calculated $\sigma \left(\theta \right)$.

In-flight radiative capture cross sections have been studied for the process ${}_{}{}^{15}{}_{}{}^{}\mathrm{N}({\pi }^{+},\gamma ){}_{}{}^{15}{}_{}{}^{}\mathrm{O}(\mathrm{g}.\mathrm{s}.\mathrm{})$. Total and differential cross sections have been calculated, and the effects of pion distortions, of higher order photoproduction operators, and of nuclear structure have been examined. The Blomqvist-Laget photoproduction operator has been used in an appropriately time-reversed form, the Stricker, McManus, and Carr pion optical potential has been used to account for pion distortions, and the nuclear structure information has been stipulated by use of simple shell model wave functions or by fitting recent electron scattering form factors. Particularly interesting interference effects due to the pion propagator and enhanced isobar contributions are revealed by this study.

NUCLEAR REACTIONS ${}_{}{}^{15}{}_{}{}^{}\mathrm{N}({\pi }^{+},\gamma ){}_{}{}^{15}{}_{}{}^{}\mathrm{O}(\mathrm{g}.\mathrm{s}.\mathrm{})$; in-flight capture 50 MeV pions; distorted waves; shell model wave functions; fit to $M1\mathrm{}$ form factor; total and differential cross sections.

Formulas for pion-nucleus inelastic scattering are presented in a form that may suggest experiments to isolate various contributions to the reaction, including $S$-wave, $P$-wave spin and nonspin flip, and effects of nucleon Fermi motion. Adopting a form of the distorted wave impulse appoximation, we obtain an expression for inelastic cross sections that clearly separate the pion laboratory energy $E$, three-momentum transfer $q$, and scattering angle $\theta$ dependences. The result is similar to the separation of longitudinal and transverse form factors in inelastic electron scattering. By varying the energy of the incident pion, but working at fixed $q$, one can determine whether a given nuclear excitation has natural or unnatural parity. By working at fixed $\theta$, and varying $E$ and thus $q$, one can isolate different reaction contributions—spin, scalar, and "convection current." We also discuss the potential usefulness of studying the energy dependence of angle-integrated differential cross sections at fixed energy loss. The predictions of our formulas are in good agreement with recent data on natural and unnatural parity excitations in ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}(\pi ,{\pi }^{\prime }){}_{}{}^{12}{}_{}{}^{}\mathrm{C}_{}^{*}{}_{}{}^{}$. Thus, this approach may be useful in analyzing future data in which the final nuclear spin is uncertain. Future experiments with selective $q$, $E$, and $\theta$ variations to separate nuclear structure from reaction-mechanism uncertainities are suggested.

NUCLEAR REACTIONS, NUCLEAR STRUCTURE Pion inelastic scattering for $T_{\pi }\simeq 100 \mathrm{to} 300$ MeV; simple formulas for calculating $\frac{d\sigma }{d\Omega }(q,E,\theta )$ either at fixed $E$ or fixed $q$; effects of Fermi motion.

The magnetic hyperfine splitting frequencies ${\nu }_M=\frac{g{\mu }_N{B}_{\mathrm{HF}}}{h}$ of ${}_{}{}^{105}{}_{}{}^{}\mathrm{Rh} (j^{\pi }=\frac{7}{2^{+}};T_{\frac{1}{2}}=35.4\mathrm{} h)$ as dilute impurities in Fe and Ni have been measured with nuclear magnetic resonance on oriented nuclei as ${\nu }_M\left({}_{}{}^{105}{}_{}{}^{}\mathrm{Rh}Ni\right)=218.06\mathrm{}\left(5\right)\mathrm{}$ MHz and ${\nu }_M\left({}_{}{}^{105}{}_{}{}^{}\mathrm{Rh}Fe\right)=539.62\mathrm{}\left(3\right)\mathrm{}$ MHz. With the hyperfine field $B_{\mathrm{HF}}\left({}_{}{}^{105}{}_{}{}^{}\mathrm{Rh}Fe\right)=-556.6\mathrm{}\left(1.2\right)\mathrm{}$ kG the $g$ factor of the ${}_{}{}^{105}{}_{}{}^{}\mathrm{Rh}$ ground state is derived as $\mathrm{}\left|g\right|=1.272\mathrm{}\left(2\right)\mathrm{}$. In combination with earlier nuclear orientation measurements $J^{\pi }={\frac{7}{2}}^{\mathrm{}(+)\mathrm{}}$ can be uniquely assigned. The $g$ factor fits well into the systematics of $\pi g_{\frac{9}{2}} g$ factors in this mass region and confirms the interpretation of this state as ${{\left(\pi g_{\frac{9}{2}}\right)}_{{\frac{7}{2}}^{+}}}^3$ shell model configuration. The hyperfine field of ${}_{}{}^{105}{}_{}{}^{}\mathrm{Rh}$Ni is deduced as - 224.9(5) kG for $T=0\mathrm{}$ K. Hyperfine anomalies, the abnormal resonance shift with $B_0$ and the overall systematics of $\pi g_{\frac{9}{2}}g$ factors are discussed.

RADIOACTIVITY ${}_{}{}^{105}{}_{}{}^{}\mathrm{Rh}$ from ${}_{}{}^{104}{}_{}{}^{}\mathrm{Ru}(n,\gamma ){}_{}{}^{105}{}_{}{}^{}\mathrm{Ru}\to {}_{}{}^{105}{}_{}{}^{}\mathrm{Rh}$; NMR on oriented nuclei; deduced $j$, $\mu$. Hyperfine fields, hyperfine anomaly. Knight shift.

Biparametric coincidence techniques using low energy photon spectrometer detectors have been used in order to determine energies, intensities, conversion coefficients, and placements of transitions in the level scheme here proposed for ${}_{}{}^{140}{}_{}{}^{}\mathrm{Cs}$, A new 9.4 keV $(I_{\gamma }<0.7\%)\gamma$ ray has been found and a doublet character for 38 keV [38.34 keV ($I_{\gamma }=0.9\mathrm{}\pm 0.2\%$) and 38.33 keV ($I_{\gamma }=0.1\mathrm{}\pm 0.1\%$)] has been established. Ten internal conversion coefficients were measured; three of them, as well as six additional $\frac{K}{L}$ relations, have not been reported previously. Spin and parity assignments were made for 25 levels, based on our assigned multipolarities and $log\mathrm{ft}$ values.

RADIOACTIVITY ${}_{}{}^{140}{}_{}{}^{}\mathrm{Xe}$ from ${}_{}{}^{235}{}_{}{}^{}\mathrm{U}(n,f)$; measured $E_{\gamma }$, $I_{\gamma }$, $\gamma -\gamma$ coin, ICC (for low-energy $\gamma$; deduced multipolarities, levels, $J$, $\pi$, $log\mathrm{ft}$. Mass-separated ${}_{}{}^{140}{}_{}{}^{}\mathrm{Xe}$.

The binding energy of the $\alpha$ particle is calculated to provide an example of the properties of the refined Brillouin-Wigner perturbation procedure. The potential consists of a series of Gaussian functions which are fitted to the Reid soft core potential. The terms in the perturbation series are evaluated in closed form, allowing critical tests of the number of oscillator states needed for precise calculations and the importance of multiparticle terms. The resulting formalism is similar to the separation method of Moszkowski and Scott, but in this case the nuclear force is separated into three parts. The divergent part of the strong short range repulsion is deducted from the perturbation series, even with the multiparticle terms. The relatively weak long range attractive part of the force is well adapted to ordinary perturbation theory. Finally one has an intermediate range component of the interaction which provides most of the binding energy in nuclear bound states.

NUCLEAR STRUCTURE Variational perturbation method. Reid soft core potential. ${}_{}{}^4{}_{}{}^{}\mathrm{He}$.

Calculations including three-particle interactions have been performed using empirical single-particle energies and two-particle interactions. The inclusion of pairing correlations is discussed. In the framework of these calculations, the high-spin negative parity levels in ${}_{}{}^{145}{}_{}{}^{}\mathrm{Eu}$ are interpreted as members of the ${\left(h_{\frac{11}{2}}d_{\frac{5}{2}}^{}{}_{}{}^{-2\mathrm{}}\right)}_J$ and ${\left(h_{\frac{11}{2}}d_{\frac{5}{2}}^{}{}_{}{}^{-1\mathrm{}}g_{\frac{7}{2}}^{}{}_{}{}^{-1\mathrm{}}\right)}_J$ multiplets, and this interpretation is supported by the comparison of experimental and calculated $E2\mathrm{}$ transition probabilities and branching ratios. The interpretation of high-spin positive parity levels of $I^{\pi }={\frac{15}{2}}^{+},{\frac{25}{2}}^{+}, \mathrm{and} {\frac{27}{2}}^{+}$ as members of the five quasiparticle multiplets ($h_{\frac{11}{2}}^{}{}_{}{}^{2}d_{\frac{5}{2}}^{}{}_{}{}^{-3\mathrm{}}$) and ($h_{\frac{11}{2}}^{}{}_{}{}^{2}d_{\frac{5}{2}}^{}{}_{}{}^{-2\mathrm{}}g_{\frac{7}{2}}^{}{}_{}{}^{-1\mathrm{}}$) is proposed.

NUCLEAR STRUCTURE ${}_{}{}^{145}{}_{}{}^{}\mathrm{Eu}$; calculated high-spin states, three- and five- quasiparticle interactions, energy spectra, $B\mathrm{}\left(E2\mathrm{}\right)\mathrm{}$ values.

Hyperfine structure, spin, and isotope shift measurements have been performed on ${}_{}{}^{76-98}{}_{}{}^{}\mathrm{Rb}$ and ${}_{}{}^{78,81,82,84,86,90}{}_{}{}^{}\mathrm{Rb}_{}^m{}_{}{}^{}$. The Rb nuclei have been produced either by spallation of Nb or by fission of U by a 600 MeV proton beam. They have been separated in mass before being transformed into a thermal atomic beam which interacts at a right angle with the light from a c.w. tunable dye laser. The charge radii changes deduced from the isotope shifts exhibit shell effects at $N=50\mathrm{}$ and deformations at $N=60\mathrm{}$, which are discussed.

NUCLEAR STRUCTURE ${}_{}{}^{76-98}{}_{}{}^{}\mathrm{Rb}$, ${}_{}{}^{78,81,82,84,86,90}{}_{}{}^{}\mathrm{Rb}_{}^m{}_{}{}^{}$. Measured hyperfine constants $A\left(5s{}_{}{}^2{}_{}{}^{}\mathrm{S}_{\frac{1}{2}}^{}{}_{}{}^{}\right)$, $A\left(5p{}_{}{}^2{}_{}{}^{}\mathrm{P}_{\frac{3}{2}}^{}{}_{}{}^{}\right)$, $B\left(5p{}_{}{}^2{}_{}{}^{}\mathrm{P}_{\frac{3}{2}}^{}{}_{}{}^{}\right)$; spins, isotope shifts. Deduced $\mu$, $\mathrm{Qs}$, $\delta 〈r^2〉$. High resolution laser spectroscopy on thermal atomic beams.

The Tamm-Dancoff method is used to give a relativistic description of nuclear bound states; in particular we discuss exchange effects and deformation of the nucleon meson clouds stressing the importance of the self-energy effects. Application to hadronic atoms is also discussed.

NUCLEAR STRUCTURE Relativistic treatment of nucleon interaction, Tamm-Dancoff, self-energy corrections to nucleon binding energy.

A quadrupole-dipole-dipole-dipole spectrograph has been used at 42.34 MeV to observe the ${}_{}{}^{199}{}_{}{}^{}\mathrm{Hg}(p,t){}_{}{}^{197}{}_{}{}^{}\mathrm{Hg}$ reaction using targets prepared with a mass separator. The $Q$ value is measured and a mass excess of -30 485±30 keV is inferred for ${}_{}{}^{197}{}_{}{}^{}\mathrm{Hg}$.

NUCLEAR REACTIONS ${}_{}{}^{199}{}_{}{}^{}\mathrm{Hg}(p,t){}_{}{}^{197}{}_{}{}^{}\mathrm{Hg}$, $E=42.34\mathrm{}$ MeV; $Q$ value measured; mass excess inferred.

The measured element distribution in collisions of ${}_{}{}^{86}{}_{}{}^{}\mathrm{Kr}$ with ${}_{}{}^{166}{}_{}{}^{}\mathrm{Er}$ at $E_{\mathrm{c}.\mathrm{m}.\mathrm{}}=463\mathrm{}$ MeV is examined in the framework of the one-body dissipation model with an empirical potential energy surface deduced from fission. The fragment $Z$ distributions over the entire range of kinetic energy damping are consistent with a driftless nucleon diffusion process. No evidence is found for a fusion-fission component.

NUCLEAR REACTIONS ${}_{}{}^{166}{}_{}{}^{}\mathrm{Er}({}_{}{}^{86}{}_{}{}^{}\mathrm{Kr},A_L)A_H{E}_{\mathrm{c}.\mathrm{m}.\mathrm{}}=326\mathrm{} \mathrm{and} 463$ MeV. Measured $\sigma (A_L, Q \mathrm{value})$; deduced reaction mechanism.