In this paper it is shown how the fundamental quantum mechanical constraints of unitarity and analyticity in three-body final states can be realized when finite-range effects, like centrifugal barriers, are important. The resulting equations are remarkably similar to the usual separable-potential scattering equations.

NUCLEAR REACTIONS Three-body scattering theory, implementation of unitarity and analyticity constraints with finite-range effects.

The $g$ factor of the 3.830 MeV level of ${}_{}{}^{41}{}_{}{}^{}\mathrm{Ca}$ has been measured by the time-differential perturbed-angular-distribution method to be $g=+0.29\mathrm{}\pm 0.02$. The reaction ${}_{}{}^{27}{}_{}{}^{}\mathrm{Al}({}_{}{}^{16}{}_{}{}^{}\mathrm{O}, p, n){}_{}{}^{41}{}_{}{}^{}\mathrm{Ca}$ was used to populate the state in a magnetic field of 58 kG. The measured $g$ factor can be understood within the configuration ${\left({d_{\frac{3}{2}}}^{-1\mathrm{}}{f_{\frac{7}{2}}}^2\right)}^{\frac{15}{2^{+}}}$. The calculated $g$ factor depends sensitively on the isospin character of the particle-hole interaction.

NUCLEAR REACTIONS ${}_{}{}^{27}{}_{}{}^{}\mathrm{Al}({}_{}{}^{16}{}_{}{}^{}\mathrm{O}, \mathrm{pn}\gamma )$, $E=48\mathrm{}$ MeV; measured $\gamma (\theta , B, t)B=58\mathrm{}$ kG deduced $g\left(3.830\mathrm{} \mathrm{MeV}\right)$.

The $0^{+}$′ state at 937 keV in ${}_{}{}^{72}{}_{}{}^{}\mathrm{Se}$ is unusual in that it is only 75 keV above the first excited $2^{+}$ state. The half-life $T_{\frac{1}{2}}$ of the $0^{+}$′ state was determined to be 15.8±0.9 nsec by delayed coincidence techniques. The ratio of the $\frac{E0\mathrm{}}{E2\mathrm{}}$ decays was measured to be 0.37±0.23 and the reduced monopole matrix element $\rho$ was calculated to be ${0.176}_{-0.070\mathrm{}}^{+0.048\mathrm{}}$. The $B\left(E2\mathrm{}\right)\mathrm{}$ value of the 75-keV transition was deduced to be 36±7 single-particle units. This large $B\left(E2\mathrm{}\right)\mathrm{}$ value indicates that the $0^{+}$′ state is collective. These data are interpreted in terms of a coexistence of spherical and deformed shapes.

RADIOACTIVITY ${}_{}{}^{72}{}_{}{}^{}\mathrm{Br}$ [from ${}_{}{}^{58}{}_{}{}^{}\mathrm{Ni}({}_{}{}^{16}{}_{}{}^{}\mathrm{O}, pn)$]; measured $\gamma -\gamma$ delayed $T_{\frac{1}{2}}$; calculated $B\left(E2\mathrm{}\right)\mathrm{}$, $\rho$, Nal-Ge(Li) detectors.

Results of the ${}_{}{}^{10}{}_{}{}^{}\mathrm{B}({}_{}{}^7{}_{}{}^{}\mathrm{Li}, p){}_{}{}^{16}{}_{}{}^{}\mathrm{N}$ reaction suggest $J=2\mathrm{}$ if the 3.520-MeV "level" is a single state and $J=0\mathrm{}$ and 1 if it is an unresolved doublet.

NUCLEAR REACTIONS ${}_{}{}^{10}{}_{}{}^{}\mathrm{B}({}_{}{}^7{}_{}{}^{}\mathrm{Li}, p)$, $E=16\mathrm{}$ MeV; measured $\sigma (E_p, \theta )$; deduced $J\left(3.520\right)\mathrm{}$.

Measurements of the angular distribution of deuterons from the ${}_{}{}^{58}{}_{}{}^{}\mathrm{Ni}({}_{}{}^6{}_{}{}^{}\mathrm{Li}, d){}_{}{}^{62}{}_{}{}^{}\mathrm{Zn}$ reaction have been extended to large angles. A decrease of about two orders of magnitude is seen between 30 and 170°, in general agreement with distorted-wave Born-approximation (DWBA) predictions. The case for using simple DWBA analysis to extract $\alpha$ particle spectroscopic factors is thus strengthened.

NUCLEAR REACTIONS ${}_{}{}^{58}{}_{}{}^{}\mathrm{Ni}({}_{}{}^6{}_{}{}^{}\mathrm{Li}, d)$, $E=28\mathrm{}$ MeV; measured $\sigma \left(\theta \right)$.

By increasing the cutoff $q_{max}$ from 1.7 ${\mathrm{fm}}^{-1\mathrm{}}$ to a value ≳2.8 ${\mathrm{fm}}^{-1\mathrm{}}$ in momentum-space Faddeev calculations, we eliminate several discrepancies previously existing between momentum- and coordinate-space Faddeev calculations of trinucleon properties.

NUCLEAR STRUCTURE ${}_{}{}^3{}_{}{}^{}\mathrm{He}$; calculated binding energy and charge form factor. Momentum space Faddeev calculation with several different momentum cutoffs.

A recently developed method for efficiently computing off-shell $t$ -matrix elements is applied to the nucleon-nucleon interaction. Sample results are given for the ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ Reid soft core potential.

NUCLEAR SCATTERING Computational methods; off-shell $t$ matrix; Møller wave operator; Reid soft core potential.

The precise isotopic mass-difference values of Smith and Wapstra have been used as a reference in the determination of the energy of the 6129-keV $\gamma$ of ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$. This transition is widely used as an energy standard in nuclear spectroscopy and in muonic x-ray studies. The value of the $\gamma$-ray energy was found to be 6129.170±0.043 keV. The quoted uncertainty is approximately one order of magnitude smaller than that of previous determinations. Data from the measurements also indicate that the new and highly precise mass values form a self-consistent set.

RADIOACTIVITY ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$; measured $E_{\gamma }$ referred to mass scale.

Magnetic dipole and electric quadrupole moments and reduced transition rates of several $1g_{\frac{9}{2}}$ nuclei are calculated using the wave functions generated in a previous paper. The pattern of wave functions is shown to support the application of a fixed parameter Coriolis coupling model to nuclei in this region. By a total inclusion of both diagonal and nondiagonal single-particle contributions, theoretical magnetic moments for ground and first excited states of both parities are brought into exceptional agreement with experiment. In particular, this analysis resolves the essential equality of $\mu \left(\frac{7}{2}\right)$ and $\mu \left(\frac{9}{2}\right)$ in ${}_{}{}^{83}{}_{}{}^{}\mathrm{Kr}$ as opposed to the expected equivalence of their $g$ factors. Quadrupole moments, calculated without resorting to effective charges, are generally well predicted. Theoretical transition rates are calculated in two instances and agree with experiment.

NUCLEAR MOMENTS ${}_{}{}^{73}{}_{}{}^{}\mathrm{Ge}$, ${}_{}{}^{75,77,79,81}{}_{}{}^{}\mathrm{Se}$, ${}_{}{}^{83,85}{}_{}{}^{}\mathrm{Kr}$, ${}_{}{}^{87}{}_{}{}^{}\mathrm{Sr}$; calculated $q$, $\mu$. ${}_{}{}^{73}{}_{}{}^{}\mathrm{Ge}$; $B\left(E2\mathrm{}\right)\mathrm{}$. ${}_{}{}^{83}{}_{}{}^{}\mathrm{Kr}$; $B\left(M1\mathrm{}\right)\mathrm{}$. Deformed Coriolis coupling treatment with pairing. Both parities.

The exactly soluble R(8) model of Krumlinde and Szymański is used as a testing ground for three many-body methods applied to the yrast band: (i) the self-consistent cranking model, (ii) self-consistent cranking plus random-phase approximation, and (iii) variation after projection of particle number. All three methods agree very well with exact results outside of the critical region. The behavior in the critical region is examined in detail.

NUCLEAR STRUCTURE Many-body methods, solvable model, high-spin states.

The expansion coefficients of a fourth-order collective Hamiltonian for the low-lying quadrupole vibrations are derived from the microscopic fermion Hamilton operator by a modified Marumori boson-expansion method. Their dependence on the phonon structure, on the parameters of the two-body (surface $\delta$) interaction, and on the single-particle energies is numerically investigated. For the isotopes ${}_{}{}^{76}{}_{}{}^{}\mathrm{Se}$ and ${}_{}{}^{100}{}_{}{}^{}\mathrm{Ru}$, ${}_{}{}^{102}{}_{}{}^{}\mathrm{Ru}$ the results are compared with coefficients that are obtained from phenomenological fits to low-lying levels. Quadrupole moments and $B\left(E2\mathrm{}\right)\mathrm{}$ values are calculated in lowest order.

NUCLEAR STRUCTURE ${}_{}{}^{76,78}{}_{}{}^{}\mathrm{Se}$, ${}_{}{}^{100,102}{}_{}{}^{}\mathrm{Ru}$ collective states microscopically by boson expansion.

The 41-cm double-focusing mass spectrometer at the University of Minnesota has been used to measure the atomic masses of ${}_{}{}^{152}{}_{}{}^{}\mathrm{Sm}$ and ${}_{}{}^{154}{}_{}{}^{}\mathrm{Sm}$ and the isotopic mass differences of most Sm and Gd isotopes. Through the use of a new generalized error signal technique, measurements on the narrow doublets of Sm and Gd chlorides gave improved mass differences over previous work in this mass region. Where intense peaks were available, the results are consistent with errors of 0.4 to 1.0 μu. The conventional wide doublet technique was used to measured Sm-hydrocarbon doublets, yielding the masses of ${}_{}{}^{152}{}_{}{}^{}\mathrm{Sm}$ and ${}_{}{}^{154}{}_{}{}^{}\mathrm{Sm}$ with errors of less than 4 μu. This new mass data, when combined with other mass measurements and reaction data of comparable accuracy, is used in a least-squares mass adjustment in the range of 141 to 160 u.

NUCLEAR STRUCTURE ${}_{}{}^{152}{}_{}{}^{}\mathrm{Sm}$, ${}_{}{}^{154}{}_{}{}^{}\mathrm{Sm}$ measured atomic mass. Sm, Gd measured mass differences. Mass adjustment for $A=141\mathrm{} \mathrm{to} 160$.

The results from the three- and the four-parameter expressions of the angular velocity expansion (AVE) and the angular momentum expansion (AME) are compared with the available experimental information on transition energies for yrast band levels in even-even nuclei. It is found that from the point of view of range of applicability, degree of fit to experimental transition energies, and high spin behavior, AME gives distinctly better results than AVE. The critical spin for backbending calculated from three-parameter AME is found to be in a fair agreement with the experimental data.

NUCLEAR STRUCTURE Even-even rare earth nuclei, calculated yrast band transition energies, three- and four-parameter expansions in angular velocity and angular momentum, compared with experimental data.

The shape of the lowest ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$+$p$ resonance has been studied in a high-resolution ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}(d, n){}_{}{}^{13}{}_{}{}^{}\mathrm{N}$ measurement. The shape displays a pronounced asymmetry, in quantitative agreement with a corresponding observation in elastic proton scattering.

NUCLEAR REACTIONS ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}(d, n)$, $E_d=6.3\mathrm{}$ MeV; measured $\sigma \left(E_n\right)$. ${}_{}{}^{13}{}_{}{}^{}\mathrm{N}$ deduced peak shape.

For nuclei near $A=208\mathrm{}$, comparisons have been made between formation of isobaric analog states by the ($p, n$) reaction and their decay by proton emission. The discrepancy in cross sections is removed if the isobaric analog state formed in the ($p, n$) reaction is taken to have a Lorentzian rather than Gaussian line shape, and if the background of evaporation and pre-equilibrium protons is subtracted from the proton spectra. The latter procedure should also resolve the width anomaly.

NUCLEAR REACTIONS ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}(p, n)$ and ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}(p, n\tilde{p})$, $E_p=21-35\mathrm{}$ MeV measured $\sigma$.

The anisotropy of $\mathrm{Ll}$ x rays and $\gamma$ rays of ${}_{}{}^{239}{}_{}{}^{}\mathrm{Pu}$ following the decay of ${}_{}{}^{243}{}_{}{}^{}\mathrm{Cm}$ has been investigated. A Ge(Li) crystal with 250 eV resolution at 5.9 keV was used to detect $\mathrm{Ll}$ x rays. A NaI(T1) crystal was used to detect six $\gamma$ rays between 210 and 285 keV. The measured anisotropy is $A=(3\mathrm{}\pm 6)\times {10}^{-3\mathrm{}}$. This is less than 2% of the maximum predicted anisotropy assuming that the magnetic substate alignment has been completely retained in the coupling of cascade $\gamma$ rays and x rays.

RADIOACTIVITY ${}_{}{}^{243}{}_{}{}^{}\mathrm{Cm}$; measured $\mathrm{Ll}$ x-$\gamma \left(\theta \right)$. Ge(Li) and NaI(T1) detectors.

Isomeric states in ${}_{}{}^{194}{}_{}{}^{}\mathrm{Au}$ with half-lives of 600±8 and 420±10 ms have been studied by observing the delayed $\gamma$ rays following the ${}_{}{}^{194}{}_{}{}^{}\mathrm{Pt}(p, n)$ and ${}_{}{}^{194}{}_{}{}^{}\mathrm{Pt}(d, 2n)$ reactions. $\gamma$-ray singles and $\gamma -\gamma$ coincidence measurements were made and isomer cross section ratios were calculated. It is argued that the isomer at 80.5 keV has the shell model configuration $(\pi d_{\frac{3}{2}}, \nu i_{\frac{13}{2}})5^{+}$,, while the higher-lying isomer is believed to decay by an unobserved isomeric transition.

RADIOACTIVITY ${}_{}{}^{194}{}_{}{}^{}\mathrm{Au}$ (isomeric levels) [from ${}_{}{}^{194}{}_{}{}^{}\mathrm{Pt}(d, 2n)$,, ${}_{}{}^{194}{}_{}{}^{}\mathrm{Pt}(p, n)$]; measured $E_{\gamma }$, $I_{\gamma }$, $T_{\frac{1}{2}}$, $\gamma -\gamma$ coin, isomer ratios. Natural and enriched targets. Calculated isomer ratios, deduced shell model assignments.

In the rare-earth deformed region, shifts from systematic Nilsson ground-state assignments are suggested for ${}_{}{}^{177}{}_{}{}^{}\mathrm{W}$, ${}_{}{}^{181}{}_{}{}^{}\mathrm{Os}$, and for ${}_{}{}^{171,173}{}_{}{}^{}\mathrm{Ta}$. Other proposed changes in the regular ordering of Nilsson ground states in the $71\le Z\le 77$ region far from the $\beta$-stability line are discussed.

NUCLEAR STRUCTURE ${}_{}{}^{171,173}{}_{}{}^{}\mathrm{Ta}$, ${}_{}{}^{177}{}_{}{}^{}\mathrm{W}$, ${}_{}{}^{181}{}_{}{}^{}\mathrm{Os}$, deduced ground-state Nilsson assignments.

The aim of the present letter is to show that there exist two different longitudinal coordinations of the collective vibrational levels built on several quadrupoles phonons ($d$ bosons). The first one reproduces that given earlier by Sakaǐ. The second one explains and generalizes the $X$, $Y$, $Z$, $X^{\prime }$, $Z^{\prime }$, $\beta$, and $\Delta$ bands recently introduced by Iachello and Arima.

NUCLEAR STRUCTURE Classification of excited states in even-even nuclei.