The high-spin rotational states of ${}_{}{}^{162}{}_{}{}^{}\mathrm{Er}$, ${}_{}{}^{168}{}_{}{}^{}\mathrm{Yb}$, ${}_{}{}^{174}{}_{}{}^{}\mathrm{Hf}$, and ${}_{}{}^{238}{}_{}{}^{}\mathrm{U}$ are calculated microscopically by diagonalizing the cranking Hamiltonian $H-\omega J_x$ using both BCS and fully particle-number projected wave functions. The computational effort in the latter case is greatly reduced due to a newly derived compact formula for the residuum integral. The results show that pairing collapse does not occur in all four nuclei up to spin 20. The moderate increase of the moment of inertia at low spin is due to both higher-order cranking and Coriolis-antipairing effects. The crossing of the decoupled two-quasiparticle band with the ground band is responsible for the rapid increase of the moment of inertia at high spin. The present calculations are able to produce the rotational energies fairly well in general, but the Nilsson single-particle levels have to be adjusted in order to reproduce the backbending behavior in ${}_{}{}^{162}{}_{}{}^{}\mathrm{Er}$.

NUCLEAR STRUCTURE Calculated high-spin rotational states by diagonalizing cranking model. New formula for particle-number projection. $g_R$ factor, backbending, Coriolis-antipairing, and rotational alignment effects.

From the quantum kinetic equation of Bogoliubov, Gurov, Born, and Green, the equations of motion for macroscopic variables are examined in many different ways. First, in the configuration space a hierarchy of generalized fluid-dynamical equations can be obtained by taking the appropriate limits of the quantum kinetic equation. The resultant equations of continuity, of momentum, and of energy are similar in form to those one encounters in classical fluid dynamics with the exception of additional terms proportional to ${\hslash }^2$ Secondly, the quantum kinetic equation is examined in phase space. The same set of equations of continuity, of momentum, and of energy can be derived by taking the first three moments of the quantum kinetic equation. The exact results we obtained are utilized to form the starting point for many simplifying approximations for the investigation of the dynamics of quantum many-body systems such as the elastic response limit, the hydrodynamical limit, Landau's Fermi-liquid theory, and finally the time-dependent Hartree-Fock and the multideterminant time-dependent Hartree-Fock approximations. The fact that all these different dynamical descriptions can be traced to a common origin provides a unifying viewpoint to the present approach with the quantum kinetic equation.

NUCLEAR STRUCTURE Dynamics of nuclear fluid, quantum kinetic equation, and generalized fluid-dynamical equation. Elastic response, hydrodynamical limit, Landau's Fermi-liquid theory, time-dependent Hartree-Fock approximation, and multideterminant time-dependent Hartree-Fock approximation.

Deformed configuration mixing shell model calculations of the spectra and electromagnetic properties of ${}_{}{}^{47,49,51}{}_{}{}^{}\mathrm{V}$ have been carried out. Agreement between the calculations and available observations is good. On the basis of the correspondence between the calculated and observed decay properties, we suggest the spin assignments $J=\frac{1}{2}$ to the 2.21 MeV level in ${}_{}{}^{47}{}_{}{}^{}\mathrm{V}$ and $J=\frac{1}{2},\frac{9}{2},\frac{7}{2},\frac{11}{2}, \mathrm{and} \frac{13}{2}$ to the levels at 1.64, 2.35, 2.41, 2.67, and 2.73 MeV, respectively, in ${}_{}{}^{49}{}_{}{}^{}\mathrm{V}$. An interesting possibility of the existence of an excited highly deformed $K=\frac{3}{2}$ band of states in ${}_{}{}^{51}{}_{}{}^{}\mathrm{V}$ is also suggested.

NUCLEAR STRUCTURE: ${}_{}{}^{47,49,51}{}_{}{}^{}\mathrm{V}$, calculated spectra, $B\left(E2\mathrm{}\right)\mathrm{}$, $B\left(M1\mathrm{}\right)\mathrm{}$, ($\frac{E2\mathrm{}}{M1\mathrm{}}$) mixing ratios, branching intensities, lifetimes, deformed configuration mixing shell model calculations based on projected Hartree-Fock theory in ${\mathrm{}\left(\mathrm{fp}\right)\mathrm{}}^n$ space, modified Kuo-Brown interaction, comparison with experiment, and rotation particle-coupling model calculation.

For calculations involving the deuteron $S$-state wave function it is common to use the asymptotic form for a rough approximation and the ad hoc Hulthen form for a better estimate. The two parameters of the Hulthén function are determined by the deuteron binding energy and the singlet effective range. In like manner we propose an ad hoc analytic form for the $D$-state wave function, which contains two parameters which are determined by the deuteron quadrupole moment and $D$-state percentage. The resultant function coincides very closely with the functions obtained from several widely used potential models. Since the chief value of such an analytic wave function is to calculate matrix elements quickly and easily we explicitly calculate a form of integral that allows a large class of matrix elements to be evaluated immediately.

NUCLEAR STRUCTURE ${}_{}{}^2{}_{}{}^{}\mathrm{H}$$D$-state wave function, calculated electromagnetic moments.

The Nilsson model is combined with the Hill-Wheeler definition of nuclear deformations and with the stationarity condition of Yariv et al. to calculate a new deformed basis. The wave functions of this basis are the same for neutrons and for protons, and for all mass numbers. The energy levels depend on $Z$, $A$ via an isospin-$A$-dependent scaling factor. This basis is combined with an improved theory of pairing to calculate a new deformed-quasiparticle basis. It is shown that without any adjustment of the model parameters from one nucleus to another, this basis leads to reasonable potential energy surgaces for many even-even nuclei (Mg,Zr,Sm,Er,Os,Hg), and reasonable low energy spectra of ${}_{12}{}^{24}{}_{}{}^{}\mathrm{Mg}$, ${}_{40}{}^{102}{}_{}{}^{}\mathrm{Zr}$, and ${}_{68}{}^{168}{}_{}{}^{}\mathrm{Er}$. The Strutinsky method is used to calculate the potential energy surfaces. A modified Kumar-Baranger method is used to calculate the moments of inertia and the mass parameters, and to solve the collective Schrödinger equation.

NUCLEAR STRUCTURE ${}_{}{}^{148,150,152,154}{}_{}{}^{}\mathrm{Sm}$, ${}_{}{}^{186,188,190,192,194}{}_{}{}^{}\mathrm{Os}$, ${}_{}{}^{184,186,188,190}{}_{}{}^{}\mathrm{Hg}$; calculated deformation energy curves. ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}$, ${}_{}{}^{102}{}_{}{}^{}\mathrm{Zr}$, ${}_{}{}^{168}{}_{}{}^{}\mathrm{Er}$; calculated deformation energy curves and collective spectra. Modified Nilsson method. Modified BCS method. Combined Strutinsky method with Kumar-Baranger method.

The phenomenological synthesis of backward fast hadron production for complex nuclei achieved by Frankel is shown to arise from coherent recoil of the residual nucleus required by energy-momentum conservation. The failure of the assumptions needed to interpret these experiments in terms of the nuclear momentum distribution is discussed.

NUCLEAR REACTIONS Scaling behavior of inclusive cross sections, constraint of energy conservation, failure of closure approximation.

Possible ${\beta }^{+}$ decay experiments are discussed with respect to a newly developed polarimeter.

RADIOACTIVITY: Calculated longitudinal ${\beta }^{+}$ polarization for allowed decays.

We give the experimental elastic cross sections for ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ at 757.5 MeV.

NUCLEAR STRUCTURE Electron scattering.

Cross sections have been measured for the ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}(\alpha ,n_{3,4}\gamma ){}_{}{}^{21}{}_{}{}^{}\mathrm{Ne}$ reactions which populate the $J^{\pi }={\frac{1}{2}}^{-},{\frac{1}{2}}^{+}$ doublet at 2.8 MeV excitation in ${}_{}{}^{21}{}_{}{}^{}\mathrm{Ne}$. These states are possible candidates for the study of nuclear parity mixing. Data are presented for $\alpha$-particle energies in the range 4.3 to 6.5 MeV.

NUCLEAR REACTIONS ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}(\alpha ,n\gamma )$, $E=4.3\mathrm{}-6.5$ MeV; measured $\sigma \mathrm{}\left(E\right)\mathrm{}$ for ($\alpha ,n_3$) and ($\alpha ,n_4$). Enriched target.

The widths of the 7.06 and 7.08 MeV states in ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ were determined in a resonance fluorescence experiment to be ${\Gamma }_0=29\mathrm{}\pm 3 \mathrm{and} 16\pm 3$ eV, respectively. The ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}(p,\alpha \gamma ){}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ reaction was used to produce the incident $\gamma$ rays.

NUCLEAR REACTIONS ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}(\gamma ,\gamma )$, $E=6.92\mathrm{} \mathrm{and} 7.12$ MeV from ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}(p,\alpha \gamma )$; measured $\sigma \left(E_{\gamma }\right)$; deduced ${\Gamma }_0$.

The intensity ratio of the ground state ($2^{-}$) to the 0.30 MeV ($3^{-}$) state of ${}_{}{}^{16}{}_{}{}^{}\mathrm{N}$ as measured in the ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}(d,{}_{}{}^3{}_{}{}^{}\mathrm{He})$ reaction and the intensity ratio of their analog states at 12.97 and 13.26 MeV in ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ as measured in the ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}(d,t)$ and ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}({}_{}{}^3{}_{}{}^{}\mathrm{He},\alpha )$ reactions consistently differ by about 50%. A largely model independent analysis of this isospin mixing effect in ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ suggests ≥ 17% mixing of the $2^{-}$ states at 12.97 ($T=1\mathrm{}$) and 12.53 MeV ($T=0\mathrm{}$) as its dominant source, in which case a lower limit of the isospin mixing matrix element of (155 ± 30) keV is obtained.

NUCLEAR REACTIONS ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}(d,{}_{}{}^3{}_{}{}^{}\mathrm{He})$, ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}(d,t)$, $E=52\mathrm{}$ MeV; ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}({}_{}{}^3{}_{}{}^{}\mathrm{He},\alpha )$, $E=36\mathrm{}$ MeV. Measured $\sigma \left(\theta \right)$ and relative population of the ground state ($2^{-}$) and the 0.30 MeV ($3^{-}$) state of ${}_{}{}^{16}{}_{}{}^{}\mathrm{N}$, and their analog states in ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ at 12.97 and 13.26 MeV, respectively.

NUCLEAR STRUCTURE ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, deduced isospin mixing of the $2^{-}$ states at 12.97 MeV ($T=1\mathrm{}$) and 12.53 MeV ($T=0\mathrm{}$).

The strength and range parameters of simple separable potentials for $n-\alpha$ scattering are determined from knowledge of the resonance-pole locations in the $P_{\frac{1}{2}}$ and $P_{\frac{3}{2}}$ $n-\alpha$ scattering amplitudes. The phase shifts associated with these potentials are compared with experiment. Agreement with the experimental phase shift is good for the $P_{\frac{3}{2}}$ case where the resonance is sharp; however, the predicted phase shift for the broad $P_{\frac{1}{2}}$ "resonance" is poor. The results are interpreted on the basis of the resonance-pole proximity to the physical region.

NUCLEAR REACTIONS Separable potentials, resonance poles, $n-\alpha$ scattering.

Gamma-ray measurements have been performed on enriched ${}_{}{}^{144}{}_{}{}^{}\mathrm{Nd}$ targets following resonance neutron capture. The $\gamma$ rays, measured with a Ge(Li) detector, have been incorporated in a level scheme for ${}_{}{}^{145}{}_{}{}^{}\mathrm{Nd}$. The neutron separation energy for ${}_{}{}^{145}{}_{}{}^{}\mathrm{Nd}$ was determined to be 5756.9±2.0 keV.

NUCLEAR REACTIONS ${}_{}{}^{144}{}_{}{}^{}\mathrm{Nd}(n,\gamma )$ $E=0.04\mathrm{}-11$ keV; measured $E_{\gamma }$, $I_{\gamma }$. ${}_{}{}^{145}{}_{}{}^{}\mathrm{Nd}$ deduced levels, neutron separation energy. Enriched target. Ge(Li) detector, neutron time of flight.

Single-particle motion and spin-orbit splitting in finite nuclei are described in a relativistic Dirac-Hartree model for nucleons interacting through exchange of scalar and vector bosons. Parameters are adjusted to reproduce properties of nuclear matter as described by Walecka. The model, applied to ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, successfully explains the sign and size of the spin-orbit interaction. The sensitivity of this result to variations in the parameters of the effective scalar boson is discussed.

NUCLEAR STRUCTURE ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$; calculated single-particle energies. Relativistic Hartree method, spin-orbit coupling.

A survey of electromagnetic transitions excited in ${}_{}{}^{27}{}_{}{}^{}\mathrm{Al}$ by 180° inelastic electron scattering has been made at incident energies of 37, 51, and 61 MeV. Transitions to levels at 2.23, 2.98, 4.42, 8.74, and 11.69 MeV are dominantly $M1\mathrm{}$ with some possible $E2\mathrm{}$ contributions, Tentative $M1\mathrm{}$ assignments are given transitions at 7.57 and 8.05 MeV, but the possibility of their being $E1\mathrm{}$ cannot be excluded. Transitions to levels at 10.68 and 12.30 MeV are probably $M1\mathrm{}$ but with considerable $E2\mathrm{}$ admixture. A transition at 6.50 MeV is also probably $M1\mathrm{}$, but both $E1\mathrm{}$ and $E2\mathrm{}$ are possible alternative assignments. Model-dependent radiative widths ${\Gamma }_0\mathrm{}\left(M1\mathrm{}\right)\mathrm{}$ for the $M1\mathrm{}$ transitions are given based on an analysis of the data using simple but representative harmonic-oscillator wave functions to calculate cross sections.

NUCLEAR REACTIONS ${}_{}{}^{27}{}_{}{}^{}\mathrm{Al}(e,e^{\prime })$, $\theta =180\mathrm{}°$, $E=37.3, 50.9, 60.5$, MeV; measured $\sigma \mathrm{}\left(E\right)\mathrm{}$; deduced ${\Gamma }_0$, multipolarities for transitions excited.

Branching ratios have been obtained for the electromagnetic decays of the lowest ${\mathrm{O}}^{+}$ $T=2\mathrm{}$ states in ${}_{}{}^{20}{}_{}{}^{}\mathrm{F}$, ${}_{}{}^{28}{}_{}{}^{}\mathrm{Al}$, ${}_{}{}^{32}{}_{}{}^{}\mathrm{P}$, ${}_{}{}^{56}{}_{}{}^{}\mathrm{Co}$, and ${}_{}{}^{60}{}_{}{}^{}\mathrm{Cu}$. In all cases the (${}_{}{}^3{}_{}{}^{}\mathrm{He},p$) reaction was used to populate the levels. Comparisons are presented with theoretical predictions of the shell model and with analogous Gamow-Teller $\beta$ decays leading to the same final states.

NUCLEAR REACTIONS ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}({}_{}{}^3{}_{}{}^{}\mathrm{He},p\gamma )$, ${}_{}{}^{26}{}_{}{}^{}\mathrm{Mg}({}_{}{}^3{}_{}{}^{}\mathrm{He},p\gamma )$, ${}_{}{}^{30}{}_{}{}^{}\mathrm{Si}({}_{}{}^3{}_{}{}^{}\mathrm{He},p\gamma )$, ${}_{}{}^{54}{}_{}{}^{}\mathrm{Fe}({}_{}{}^3{}_{}{}^{}\mathrm{He},p\gamma )$, ${}_{}{}^{58}{}_{}{}^{}\mathrm{Ni}({}_{}{}^3{}_{}{}^{}\mathrm{He},p\gamma )$ $E=11\mathrm{}-15$ MeV, measured $p-\gamma$ coincidences, ($E_p,E_{\gamma },{\theta }_p=0\mathrm{}°,{\theta }_{\gamma }$) deduced $\gamma$ branching ratios for $T=2\mathrm{}$ states.

Neutron inelastic scattering cross sections have been determined for six levels in ${}_{}{}^{23}{}_{}{}^{}\mathrm{Na}$ over the incident neutron energy range 0.52 to 4.23 MeV by measuring the $\gamma$-ray production cross sections for 10 deexcitation $\gamma$ transitions in the ${}_{}{}^{23}{}_{}{}^{}\mathrm{Na}(n,n^{\prime }\gamma )$ reaction. $\gamma$-ray angular distributions from the first two excited levels were obtained at various incident neutron energies. From these, the magnetic substate populations were evaluated for the 0.440-MeV (${5/2}^{+}$) first excited level and the 2.078-MeV (${7/2}^{+}$) second excited level in ${}_{}{}^{23}{}_{}{}^{}\mathrm{Na}$. Comparisons have been made between the distributions and magnetic substate populations calculated from compound-nuclear and direct-interaction theory, the former yielding better agreement in the interpretation of experimental results.

NUCLEAR REACTIONS ${}_{}{}^{23}{}_{}{}^{}\mathrm{Na}(n,n^{\prime }\gamma )$, $E_n=0.52\mathrm{}-4.23$ MeV; measured $\sigma (E_n,{\theta }_{\gamma })$; deduced $\sigma \left(E_n\right)$ and magnetic substate populations; calculated $\sigma (E_n,{\theta }_{\gamma })$, $\sigma \left(E_n\right)$, magnetic substate populations; CN and DWBA analysis; natural (Na) target.

The reaction ${}_{}{}^{142}{}_{}{}^{}\mathrm{Nd}(e,e^{\prime }p){}_{}{}^{141}{}_{}{}^{}\mathrm{Pr}$ has been used to study the $T_{>}$ giant resonance in ${}_{}{}^{142}{}_{}{}^{}\mathrm{Nd}$. Cross sections for the ${}_{}{}^{142}{}_{}{}^{}\mathrm{Nd}(\gamma ,p){}_{}{}^{141}{}_{}{}^{}\mathrm{Pr}$ and ${}_{}{}^{142}{}_{}{}^{}\mathrm{Nd}(\gamma ,p_0+p_1){}_{}{}^{141}{}_{}{}^{}\mathrm{Pr}$ reactions were measured for excitation energies from 15.9 to 26.0 MeV. These cross sections are compared with previous ${}_{}{}^{141}{}_{}{}^{}\mathrm{Pr}(p,{\gamma }_0){}_{}{}^{142}{}_{}{}^{}\mathrm{Nd}$ data. The angular distributions for the larger resonances at 17.3, 19.7, and 22.9 MeV were measured. Analysis indicates that the 22.9-MeV state has about a 13% $E2\mathrm{}$ component. The decay proton spectra from the 19.7- and 22.9-MeV resonances were analyzed with a schematic shell model and indicates that the core excitation process is dominant in these isobaric analog resonances. The measured ($\gamma ,p$) cross section together with the ($\gamma ,n$) cross section is found to be fairly consistent with the predictions of isospin splitting theory.

NUCLEAR REACTIONS ${}_{}{}^{142}{}_{}{}^{}\mathrm{Nd}(e,e^{\prime }p)$, $E=15.9\mathrm{}-26$ MeV; measured $\sigma (E;E_p)$; deduced $\sigma (\gamma ,p)$, $\sigma (\gamma ,p_0+p_1)$. $E=18.0, 19.8, 20.8, 22.9, 23.9$ MeV: measured $\sigma (E_p,\theta )$. ${}_{}{}^{142}{}_{}{}^{}\mathrm{Nd}$ deduced isobaric analog resonances. Enriched target.

92.5-MeV electrons were used to study in ${}_{}{}^{89}{}_{}{}^{}\mathrm{Y}$ the excitation range between 2 and 55 MeV. Above neutron threshold, broad electric resonances have been seen at 14.0 ($63A^{-\frac{1}{3}}$) MeV ($E2, \Delta T=0\mathrm{}$) and 28 (${125\mathrm{A}}^{-1/3\mathrm{}}$) MeV ($E2, \Delta T=1\mathrm{}$). The total width of the isoscalar $E2\mathrm{}$ resonance is (4.5 ± 0.4) MeV and its strength (56 ± 6)% energy-weighted sum rule ($E2, \Delta T=0\mathrm{}$). For the isovector $E2\mathrm{}$ resonance only a minimal value of 7 MeV for the width can be given which is connected with (48 ± 5)% of the isovector sum rule. The strength of the $E1\mathrm{}$ resonance [$T_{<},(104\mathrm{}\pm 10)\%$ of the Thomas-Reiche-Kuhn sum rule] agrees well with ($\gamma ,n$) measurements, thus giving a check for the accuracy of the evaluating methods employed. A structure around 19-20 MeV, believed to be the $T_{>}$ part of the giant dipole resonance, carries (8 ± 3)% of the $E1\mathrm{}$ sum rule. In addition to these generally well known states, clustering of $E3\mathrm{}$ strength, (13 ± 2)% energy-weighted sum rule ($E3, \Delta T=0\mathrm{}$), was found at 6.75 and 8.05 MeV; the enveloping line shape of these clusters was best described by a Breit-Wigner term. Other concentrations of strength include $E3\mathrm{}$ at 2.6 MeV, (15 ± 3)% energy-weighted sum rule ($E3, \Delta T=0\mathrm{}$), $E2\mathrm{}$ at 4.0 MeV, (11 ± 3)% energy-weighted sum rule ($E2, \Delta T=0\mathrm{}$), and $E3\mathrm{}$ strength at 13.5 MeV. No resonance other than the $E1\mathrm{}$ was found around 17 MeV, thus ruling out in ${}_{}{}^{89}{}_{}{}^{}\mathrm{Y}$ the existence of a monopole state with 100% sum rule proposed for ${}_{}{}^{90}{}_{}{}^{}\mathrm{Zr}$ from ($\alpha ,{\alpha }^{\prime }$) and ($e,e^{\prime }$) measurements. In contrast to heavy nuclei, no resonant $3\hslash \omega E3\mathrm{}$ strength could be located.

NUCLEAR REACTIONS ${}_{}{}^{89}{}_{}{}^{}\mathrm{Y}(e,e^{\prime })$, $E_0=92.5\mathrm{}$ MeV. Measured $\frac{d^2\sigma }{d\Omega d{E}_x}$, bound and continum states. Deduced multipolarity $\lambda$, reduced matrix element $B\left(E\lambda \right)$, radiative width ${\Gamma }_{\gamma }^0$, sum rule exhaustion, single particle strength, and total width of the continuum and clustered states.