It is shown that the amplitude for the scattering of a $\theta$ particle from an arbitrary number of fixed $N$ particles in the Lee model can be obtained from Watson's multiple scattering theory if the elementary interaction is taken to be the full $\theta -N$ $T$ matrix, including the $V$-particle pole, and the propagator is taken to be a free particle propagator.

NUCLEAR REACTIONS Watson multiple scattering theory for $\theta$-many $N\text{'}\mathrm{s}$ sector of Lee model.

A comprehensive analysis of the problem of deuteron smearing corrections is given. Some problems of the usual approach (the conventional approach) are discussed. It is shown that the formalism adopted by this approach leads to (i) the existence of the so-called West $\beta$ correction and (ii) the disagreement of the sum rules with the quark parton model expectations in leptonic scattering. It is also shown that in this formalism the electroproduction expression is not inconsistent. Following West for the identification of the wave function, it is found that the smearing correction ${\sigma }_s$ of the high-energy (50-370 GeV) pion and nucleon process is non-negligible, and hence, in contradiction with the usual expectations. Another test for this approach is suggested.

NUCLEAR REACTIONS Deuteron, Fermi motion, smearing correction, West correction.

Described is a procedure for solving in momentum space the Coulomb plus nuclear problem for coupled bound and continuum eigenstates. The method is an extension of the Kwon-Tabakin formulation of the Lande subtraction technique. Complex eigenenergies are now determined after formulating and incorporating the correct momentum space "boundary conditions" into the Schrödinger equation. This exact numerical procedure can be applied to local or nonlocal, complex or real potentials even with relativistic kinematics. Application to the $K^{-}p$ problem and other systems with strongly coupled absorptive channels is indicated.

NUCLEAR STRUCTURE Hadronic atoms, momentum space formulations, coupled channels.

An ab initio method of taking Fermi motion effects in pion scattering into account combined with proper antisymmetrization of the two nucleon final state in $d({\pi }^{\pm } {\pi }^{\pm }, p)n$ shows that the reaction amplitude consists of not only a contribution from pion scattering on a proton, but also pion scattering on a neutron, and that the two amplitudes differ considerably in their $\pi -N$ c.m. energies. The neutron scattering contribution to the differential cross section exceeds 10% in as many as 51 out of 195 ${\pi }^{-}$ events recorded in a recent kinematically complete experiment by Hoftiezer et al. in the $\Delta$-resonance region. In several kinematical situations the neutron contribution exceeds 100% and could even be as high as ∼400%.

NUCLEAR REACTIONS $d({\pi }^{\pm }, {\pi }^{\pm } p)n$, ab initio method for Fermi motion effects, Pauli principle, effective mass, exchange amplitude.

An automated rapid chemistry system has been used to isolate enriched sources of 12.4-min ${}_{}{}^{133}{}_{}{}^{}\mathrm{Te}_{}^g{}_{}{}^{}$ from mixed fission products via the sequence ${}_{}{}^{133}{}_{}{}^{}\mathrm{Sb}\left(\beta \right){}_{}{}^{133}{}_{}{}^{}\mathrm{Te}_{}^g{}_{}{}^{}$. In addition, pure sources of ${}_{}{}^{133}{}_{}{}^{}\mathrm{Te}_{}^g{}_{}{}^{}$ were obtained by using the reduction of ${\mathrm{Te}}^{\mathrm{VI}}$ to ${\mathrm{Te}}^{\mathrm{IV}}$ during the isomeric decay of ${}_{}{}^{133}{}_{}{}^{}\mathrm{Te}_{}^m{}_{}{}^{}$. Singles, Compton suppression singles, and $\gamma -\gamma$ coincidence Ge(Li) spectra were used to identify approximately 210 $\gamma$ rays from the decay of ${}_{}{}^{133}{}_{}{}^{}\mathrm{Te}_{}^g{}_{}{}^{}$. This information was used to establish 38 levels below 3000 keV in ${}_{}{}^{133}{}_{}{}^{}\mathrm{I}$. Cluster-vibration model calculations have been carried out for the $Z=53\mathrm{}$ nucleus ${}_{}{}^{133}{}_{}{}^{}\mathrm{I}$. The calculated level structure and transition probabilities are found to be in good agreement with the experimental results.

NUCLEAR STRUCTURE Cluster vibration model, energy levels, branching ratios.

RADIOACTIVITY ${}_{}{}^{133}{}_{}{}^{}\mathrm{Te}_{}^g{}_{}{}^{}$ (from Autobatch chemical isolation from fission); measured $E_{\gamma }$, $I_{\gamma }$, $\gamma -\gamma$ coin, Compton suppression; ${}_{}{}^{133}{}_{}{}^{}\mathrm{I}$ deduced levels $J$, $\pi$; deduced $\beta$ branch $log\mathrm{ft}$, HPGe and Ge(Li) detectors.

A comprehensive investigation of the excited states of ${}_{}{}^{82}{}_{}{}^{}\mathrm{Kr}$ has been carried out through the $\gamma$-ray spectroscopy study of the radioactive decay of ${}_{}{}^{82}{}_{}{}^{}\mathrm{Br}_{}^g{}_{}{}^{}(t_{\frac{1}{2}}=35.3\mathrm{} \mathrm{h}, J^{\pi }=5^{-})$, ${}_{}{}^{82}{}_{}{}^{}\mathrm{Br}_{}^m{}_{}{}^{}(t_{\frac{1}{2}}=6.1\mathrm{} min, J^{\pi }=2^{-})$, ${}_{}{}^{82}{}_{}{}^{}\mathrm{Rb}_{}^g{}_{}{}^{}(t_{\frac{1}{2}}=1.273\mathrm{} min, J^{\pi }=1^{+})$, and ${}_{}{}^{82}{}_{}{}^{}\mathrm{Rb}_{}^m{}_{}{}^{}(t_{\frac{1}{2}}=6.2\mathrm{} \mathrm{h}, J^{\pi }=5^{-})$ and in-beam studies using the ${}_{}{}^{80}{}_{}{}^{}\mathrm{Se}(\alpha , 2n\gamma ){}_{}{}^{82}{}_{}{}^{}\mathrm{Kr}$ reaction. We use the ${}_{}{}^{82}{}_{}{}^{}\mathrm{Kr}$ level structure we determined, to test the interacting boson model. We show that the interacting boson model calculation can account for the observed levels below 3 MeV provided the underlying shell structure is taken into account. A low energy four quasiparticle $\nu \left\{p_{\frac{1}{2}}{{\left(g_{\frac{9}{2}}\right)}_{\frac{7}{2}}}^3\right\}{4}^{-}$ band is proposed and its mixing with the octupole band is discussed.

NUCLEAR REACTIONS ${}_{}{}^{80}{}_{}{}^{}\mathrm{Se}(\alpha , 2n\gamma )$, measured $E_{\gamma }$, $I_{\gamma }$, $\gamma -\gamma$ coincidence, deduced level energy and $J^{\pi }$. RADIOACTIVITY Decay of ${}_{}{}^{82}{}_{}{}^{}\mathrm{Br}_{}^g{}_{}{}^{}$, ${}_{}{}^{82}{}_{}{}^{}\mathrm{Br}_{}^m{}_{}{}^{}$, ${}_{}{}^{82}{}_{}{}^{}\mathrm{Rb}_{}^g{}_{}{}^{}$, and ${}_{}{}^{82}{}_{}{}^{}\mathrm{Rb}_{}^m{}_{}{}^{}$. Measured $E_{\gamma }$, $I_{\gamma }$, $\gamma -\gamma$ coincidence, deduced level energy and $J^{\pi }$.

Decay schemes of 45.3-d ${}_{}{}^{237}{}_{}{}^{}\mathrm{Pu}$ and 1.55×${10}^5$ yr ${}_{}{}^{236}{}_{}{}^{}\mathrm{Np}$ have been investigated by measuring their $\gamma$ ray and conversion electron spectra with a 5-${\mathrm{cm}}^2$×1-cm Ge detector and a cooled Si(Li) spectrometer, respectively. Intensities of $\gamma$ rays, x rays, and conversion electrons have been determined and these have been used to derive $log\mathrm{ft}$ values for EC and ${\beta }^{-}$ transitions. In addition to confirming previously known level schemes in ${}_{}{}^{237}{}_{}{}^{}\mathrm{Np}$ and ${}_{}{}^{236}{}_{}{}^{}\mathrm{U}$, the energies of the 4 + →2 + and 6 + →4 + transitions in the ${}_{}{}^{236}{}_{}{}^{}\mathrm{Pu}$ ground state band have been determined. Relative intensities of x rays in the decay of ${}_{}{}^{235}{}_{}{}^{}\mathrm{Np}$ have also been measured.

RADIOACTIVITY ${}_{}{}^{237}{}_{}{}^{}\mathrm{Pu}$ [from ${}_{}{}^{235}{}_{}{}^{}\mathrm{U}(\alpha , 2n)$], ${}_{}{}^{235}{}_{}{}^{}\mathrm{Np}$ [from ${}_{}{}^{235}{}_{}{}^{}\mathrm{U}(d, 2n)$], ${}_{}{}^{236}{}_{}{}^{}\mathrm{Np}$ [from ${}_{}{}^{23s}{}_{}{}^{}\mathrm{U}(d, 4n)$]; measured $E_{\gamma }$, $I_{\gamma }$, $E_{\mathrm{ce}}$, $I_{\mathrm{ce}}$. ${}_{}{}^{237}{}_{}{}^{}\mathrm{Np}$, ${}_{}{}^{236}{}_{}{}^{}\mathrm{U}$, ${}_{}{}^{236}{}_{}{}^{}\mathrm{Pu}$ deduced levels, $log\mathrm{ft}$ (EC), $log\mathrm{ft}$ (${\beta }^{-}$), $\gamma$ multipolarity, $I$, $\pi$. Mass-separated ${}_{}{}^{237}{}_{}{}^{}\mathrm{Pu}$. Cooled Si(Li) electron spectrometer.

The electron-capture decay of ${}_{85}{}^{204}{}_{}{}^{}\mathrm{At}$ (9 min) has been studied by means of $\gamma$ and $e^{-}$ multiscaled singles and $\gamma -\gamma$ coincidence measurements with Ge(Li) and Si(Li) detector systems. The samples were produced by $\mathrm{Ir}({}_{}{}^{16}{}_{}{}^{}\mathrm{O}, xn)$ reactions and on-line isotope separation. About 50 $\gamma$-ray transitions have been assigned to the ${}_{}{}^{204}{}_{}{}^{}\mathrm{At}$ decay, ∼35 of which have been placed in a proposed level scheme for ${}_{84}{}^{204}{}_{}{}^{}\mathrm{Po}$. The energy of the $8^{+}$→$6^{+}$ isomeric transition has been deduced to be 12.1 keV, and the half-life of the $8^{+}$ isomer has been measured to be 150±10 ns. From these data, the $B\left(E2\mathrm{}\right)\mathrm{}$ value of this isomeric transition is found to be 3.7 Weisskopf units. The ${}_{}{}^{204}{}_{}{}^{}\mathrm{At}$ ground state is assigned an $I^{\pi }$ value of $7^{+}$, which eliminates the need for some low-energy levels in ${}_{}{}^{204}{}_{}{}^{}\mathrm{At}$ and ${}_{}{}^{200}{}_{}{}^{}\mathrm{Bi}$ that were tentatively proposed in recent data evaluations, where a $5^{+}$ assignment was made.

RADIOACTIVITY ${}_{}{}^{204}{}_{}{}^{}\mathrm{At}$ from $\mathrm{Ir}({}_{}{}^{16}{}_{}{}^{}\mathrm{O}, xn)$; measured $E_{\gamma }$, $I_{\gamma }$, ICC, $\gamma -\gamma -t$, isomeric lifetime in ${}_{}{}^{204}{}_{}{}^{}\mathrm{Po}$; ${}_{}{}^{204}{}_{}{}^{}\mathrm{Po}$ deduced levels, $J$, $\pi$; ${}_{}{}^{204}{}_{}{}^{}\mathrm{At}$ ground state deduced $J$, $\pi$; Ge(Li) and Si(Li) detectors; on-line isotope-separated sources.

The problem of alpha decay is discussed on the basis of a theory which describes discrete and continuous states in a unified manner. A formula for numerical calculations is given in which configurational mixing as well as channel coupling is taken into account. The $R$-matrix approximation is shown to be justified if the width is spread over a small number of decay channels. Generally, renormalization of the wave function is necessary if a factorization of the width is assumed. The importance of channel coupling for the case of a small reduced width is discussed.

RADIOACTIVITY Continuum shell model, spectroscopic factors, influence of channel coupling on absolute values of partial widths.

The $\sigma {\tau }_{+}$ strength function is studied with the quasiparticle random phase approximation. The residual interaction modifies the pairing theory strength function in two ways. The particle-hole interaction reduces the overall strength by about a factor of 2, without shifting strength between different levels. The particle-particle interaction does not affect the overall strength, but shifts part of the strength from the lowest excitation to a higher excitation energy region. By comparing the theory with the observed $log\mathrm{}\left(\mathrm{ft}\right)\mathrm{}$ values of the ${\beta }^{+}$ decay in medium heavy nuclei, we find that an additional quenching is required for the $\sigma {\tau }_{+}$ mode, similar in magnitude to the additional quenching present in other isovector spin-flip transitions. Finally, we predict that there is a large concentration of the $\sigma {\tau }_{+}$ strength at higher excitation energy which cannot be observed by the ${\beta }^{+}$ decay.

NUCLEAR STRUCTURE $\sigma {\tau }_{+}$ states in odd-odd mass nuclei between $A=100-150\mathrm{}$. QP-RPA calculation with zero-range interaction. Calculated $log\mathrm{}\left(\mathrm{ft}\right)\mathrm{}$.

The isotope shifts of the radii of the distributions of neutron and nucleon matter in the samarium isotopes are deduced by combining data from x-ray, muonic x-ray, and optical isotope shift measurements of charge (proton matter) radii with data for the isotope shifts of Coulomb displacement energies. The average isotope shifts from $A=144\mathrm{}$ (spherical) to $A=154\mathrm{}$ (deformed) are nearly the same for proton and neutron matter with ${\gamma }_N\approx 1.3$ or ${〈r^2〉}^{\frac{1}{2}}\propto A^{0.43}$. The neutron matter radii increase more rapidly than the proton matter radii in the region of the light transitional Sm isotopes. The transition from spherical to deformed ground states between $N=88\mathrm{} \mathrm{and} 90$ is accompanied by a strong increase in the radius of the proton matter distribution (${\gamma }_N\approx 1.9$), whereas the radius of the neutron-excess matter distribution remains almost constant (${\gamma }_N\approx 0.0$). The data suggest strong even/odd staggering effects for both proton and neutron matter radii which are out of phase. The droplet model of the atomic nucleus when combined with experimental deformation parameters describes the isotope shift of the charge radii very well. The influence of zero-point vibrations in the light transitional Sm isotopes is apparent. The isotope shift of the neutron matter radii appears to be overestimated. The interacting boson model is found to describe the isotope shift of neutron matter radii satisfactorily but fails to reproduce the isotope shift of the charge radii.

NUCLEAR STRUCTURE Deduced isotope shifts of neutron-excess, neutron, and nucleon matter radii. Interpretation of isotope shifts of proton and neutron matter radii with droplet model and interacting boson model.

Primary $\gamma$-ray transitions in ${}_{}{}^{162}{}_{}{}^{}\mathrm{Dy}$ and ${}_{}{}^{164}{}_{}{}^{}\mathrm{Dy}$ have been studied following average resonance neutron capture with neutron beams of mean energy 2 and 24 keV. The averaging process guarantees the observation of the complete set of $J^{\pi }=1^{-(+)\mathrm{}}, 2^{\pm }, 3^{\pm }, 4^{-(+)\mathrm{}}$ states in ${}_{}{}^{162}{}_{}{}^{}\mathrm{Dy}$ (${}_{}{}^{164}{}_{}{}^{}\mathrm{Dy}$) up to an excitation energy of 2 MeV. Use of the results of recent Monte Carlo calculations has been made to establish semiquantitative criteria for the $J^{\pi }$ assignments. Further spin limitations have been imposed by the study of primary transitions following capture in the 2.72 and 3.68 eV resonances in ${}_{}{}^{162}{}_{}{}^{}\mathrm{Dy}$ and the 1.71 eV resonance in ${}_{}{}^{164}{}_{}{}^{}\mathrm{Dy}$. The known completeness of the set of observed levels has been used to establish a correspondingly complete band structure in both nuclei. The results are discussed in the framework of both the interacting boson approximation and the geometrical model.

NUCLEAR REACTIONS ${}_{}{}^{161,163}{}_{}{}^{}\mathrm{Dy}(n, \gamma )$, $E_n=\mathrm{r}\mathrm{e}\mathrm{s},2\mathrm{k}\mathrm{e}\mathrm{V},24\mathrm{keV}$; measured $E_{\gamma }$, $I_{\gamma }$; ${}_{}{}^{162,164}{}_{}{}^{}\mathrm{Dy}$ deduced levels, $J$, $\pi$, band structure. Ge(Li) detectors, enriched targets, ARC spectroscopy. Comparison with predictions of interacting boson approximation and geometrical model.

Low-spin states in even Po and Rn isotopes are studied within the framework of the interacting-boson-approximation + two-quasiparticle model. The irregular behavior of the $4^{+}$ levels is shown to result from an interplay between collective, quasiproton, and quasineutron states. Some predictions are made for the corresponding Ra isotopes.

NUCLEAR STRUCTURE ${}_{}{}^{200-210}{}_{}{}^{}\mathrm{Po}$, ${}_{}{}^{202-212}{}_{}{}^{}\mathrm{Rn}$, ${}_{}{}^{204-214}{}_{}{}^{}\mathrm{Ra}$; calculated levels, $B\left(E2\mathrm{}\right)\mathrm{}$. IBA + two-quasiparticle model.

The Hartree-Fock and Hartree-Fock-Bogolyubov methods are used for the description of nuclei with triaxial (i.e., ellipsoidal) shapes. The Gogny $D1\mathrm{}$ finite range density dependent interaction is employed. A comparison between various prescriptions for the calculation of pairing correlations in nuclei indicates the necessity for the self-consistent treatment of pairing. Using this last method we obtain results concerning nuclear shapes and deformation energy surfaces for nuclei ranging from the lightest ones to the fission barrier of ${}_{}{}^{240}{}_{}{}^{}\mathrm{Pu}$.

NUCLEAR STRUCTURE Density-dependent Hartree-Fock-Bogolyubov approximation applied to the description of nuclei with triaxial shapes: pairing correlation, comparison HFB-HFBCS, potential energy surface, density distributions.

Upper bounds, lower bounds, and estimates based on moments are compared for the ground state energy of a soluble Hamiltonian due to Lipkin. Several bounding methods based on variational approaches and one based on the method of moments are employed. We show that lower bounds are of comparable quality with upper bounds and are no more difficult to obtain. No single bounding method is found superior over the entire range of parameters we study for the soluble Hamiltonian. However, in one region we find an upper bound from one method which coincides with the lower bound of another method, thus yielding the exact ground state energy. We also show that estimation methods based on the easily calculated first and second moments of the eigenvalue distribution are surprisingly accurate for a wide range of coupling parameters and number of active particles.

NUCLEAR STRUCTURE Upper and lower bounds to ground state energy with soluble model; spectral properties of Lipkin model; variational methods; moment methods.

The single particle energy is calculated self-consistently in nuclear matter using the Paris potential. Binding energy and wave functions are compared for this choice and the zero particle spectrum. A gain in binding energy of 6 MeV is obtained at normal density of which 5 MeV is due to the ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}-{}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}$ channel alone. The wave function for the ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}-{}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}$ channel shows marked enhancement in the 0.9-3.0 fm region. The saturation point is shifted to 1.6 ${\mathrm{fm}}^{-1\mathrm{}}$ with a binding energy of about 21 MeV. Real parts of optical model parameters are also deduced and discussed.

NUCLEAR STRUCTURE Paris potential, self-consistent particle spectrum.

The unification of the nuclear collective and single particle models with the help of a transformation, first published by the author, is presented. The Schrödinger equation with the new coordinates exhibits all relevant features of the so-called unified nuclear model, but it is no longer an artificial combination of the continuum, used in the collective model, and the conventional shell model. As a first application a sum rule for the quadrupole moments of mirror nuclei is derived. Estimates are given for the deformations of the charge and mass distributions of ${}_{}{}^{17}{}_{}{}^{}\mathrm{F}$ and ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}$. The approximation of strong coupling of the nucleon motions to a symmetry axis gives good results, while the weak coupling approximation does not work at all.

NUCLEAR STRUCTURE Genuine definition of single particle and collective states from $A$ nucleon Schrödinger equation. Sum rule for $q$ of mirror nuclei.

The 164-MeV angular distribution for the non-analog double charge exchange reaction ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}({\pi }^{+}, {\pi }^{-}){}_{}{}^{16}{}_{}{}^{}\mathrm{Ne}(\mathrm{g}.\mathrm{s}.\mathrm{})$ has been measured and is found to be basically diffractive. This is contrasted with the previously measured analog distribution for ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}({\pi }^{+}, {\pi }^{-}){}_{}{}^{18}{}_{}{}^{}\mathrm{Ne}$ (double isobaric analog) which is not diffractive.

NUCLEAR REACTIONS ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}({\pi }^{+}, {\pi }^{-}){}_{}{}^{16}{}_{}{}^{}\mathrm{Ne}(\mathrm{g}.\mathrm{s}.\mathrm{})$; $E_{\pi }=164\mathrm{}$ MeV, $\theta =0\mathrm{}°-50\mathrm{}°$; measured $\sigma \left(\theta \right)$, non-analog transition, eikonal fit to angular distribution.

Strong and systematic anomalies in the angular distributions for transitions to the first $2^{+}$ levels in ${}_{}{}^{96,98,100}{}_{}{}^{}\mathrm{Mo}$ via the ($d$, ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$) reaction are reproduced by coupled channels calculations. Disturbingly large effects of the coupled channels calculations on the strength of the ground-state transitions can be explained by assuming modified distorted waves in the exit channel.

NUCLEAR REACTIONS ${}_{}{}^{100,102,104}{}_{}{}^{}\mathrm{Ru}(d, {}_{}{}^6{}_{}{}^{}\mathrm{Li})$ $E=45\mathrm{}$ MeV. ${}_{}{}^{96,98,100}{}_{}{}^{}\mathrm{Mo}$ measured $\sigma \left(\theta \right)$. DWBA, CCBA analysis, enriched targets, magnetic spectrograph.

The depth of the asymmetry potential for nucleons interacting with ${}_{}{}^{90}{}_{}{}^{}\mathrm{Zr}$ is obtained at sub-Coulomb proton energies, from the observed isospin splitting of the $3s_{\frac{1}{2}}$ proton single-particle resonance. The splitting is described by the Lane model with an asymmetry potential depth of 31.4 MeV and with other parameters adopted from analyses of measured, sub-Coulomb energy ${}_{}{}^{92,94}{}_{}{}^{}\mathrm{Zr}(p, n)$ and ($p, p$) cross sections. The resulting value of 31.4 MeV is significantly larger than the value obtained at higher energies by proton-scattering and charge-exchange measurements, indicating that the asymmetry potential must be energy dependent. The present result and the results at higher energies are shown to be in excellent agreement with the energy-dependent asymmetry potential of Giannini et al.

NUCLEAR REACTIONS Lane model analysis of isospin splitting of $3s_{\frac{1}{2}}$ proton state in ${}_{}{}^{90}{}_{}{}^{}\mathrm{Zr}$+$p$ deduced asymmetry potential depth.