Results are presented for the damped reaction ${}_{}{}^{144}{}_{}{}^{}\mathrm{Sm}$ + ${}_{}{}^{84}{}_{}{}^{}\mathrm{Kr}$ at laboratory energies of 470, 595, and 720 MeV, corresponding to 1.2, 1.6, and 1.9 times the Coulomb barrier energy, respectively. Correlations between angle, kinetic-energy loss, and charge of the projectile-like fragments are discussed. A probability for orbiting is observed at all bombarding energies, with fragment kinetic energies indicative of large exit-channel deformations. This study shows that energy dissipation (up to $\approx 100$ MeV) and charge exchange are correlated in a manner which appears to be independent of initial relative ion velocity. The results are compared with a classical dynamical transport model.

[NUCLEAR REACTIONS ${}_{}{}^{144}{}_{}{}^{}\mathrm{Sm}({}_{}{}^{84}{}_{}{}^{}\mathrm{Kr},x)$, $E_{\mathrm{lab}}=470,595, \mathrm{and} 720$ MeV; measured $\sigma (\theta ,E,Z)$ damped reaction products for projectile-like fragments; compared with transport model calculations.]

High-resolution measurements of total and fission cross sections for neutrons on ${}_{}{}^{237}{}_{}{}^{}\mathrm{Np}$ in the region of the subthreshold fission structures near 40 and 120 eV have been made with samples cooled to liquid nitrogen temperature. The previously accepted normalization of the fission cross section is found to be too low by a factor of 3. Extensive one- and two-fission-channel multilevel $R$-matrix fits to the data and statistical analyses of the fine structure at 40 and 120 eV fail to differentiate between the two proposed coupling schemes in ${}_{}{}^{238}{}_{}{}^{}\mathrm{Np}$: very weak coupling of the class I states to a narrow class II state or moderately weak coupling of the class I states to a broad class II state.

[NUCLEAR REACTIONS ${}_{}{}^{237}{}_{}{}^{}\mathrm{Np}\mathrm{}(n,f)\mathrm{}$, ${}_{}{}^{237}{}_{}{}^{}\mathrm{Np}(n,\mathrm{total})$; measured ${\sigma }_T\left(E_n\right)$, ${\sigma }_f\left(E_n\right),1\mathrm{}\mathrm{eV}\lesssim E_n\lesssim 600\mathrm{eV}$; samples cooled to LN; $R$-matrix and statistical analyses, class I and class II states.]

Differential cross sections were measured for pion elastic and inelastic scattering from ${}_{}{}^{40,42,44,48}{}_{}{}^{}\mathrm{Ca}$ and ${}_{}{}^{54}{}_{}{}^{}\mathrm{Fe}$ at $T_{\pi }=116.0,180.0, \mathrm{and} 292.5$ MeV. Elastic scattering was analyzed using a Kisslinger potential with and without free parameters. The low-lying collective states were analyzed using distorted wave impulse approximation calculations with no free parameters and with transition densities obtained from electron and proton scattering. These calculations adequately reproduced the pion scattering data. Ground-state neutron density root-mean-square radii were estimated.

[NUCLEAR REACTIONS ${}_{}{}^{40,42,44,48}{}_{}{}^{}\mathrm{Ca}({\pi }^{\pm },{\pi }^{\pm \prime })$ and ${}_{}{}^{54}{}_{}{}^{}\mathrm{Fe}({\pi }^{\pm },{\pi }^{\pm \prime })$; $T_{\pi }=116.0,180.0, \mathrm{and} 292.5$ MeV; measured $\sigma \left(\theta \right)$; optical-model analysis.]

We show that it is possible to construct a meson-exchange Hamiltonian for $\mathrm{N},\pi ,\Delta$, and ${\mathrm{N}}^{*}$ (1470 MeV) to describe NN scattering up to 2 GeV. The model consists of: (a) vertex interactions $\pi \mathrm{N}\leftrightarrow \Delta or {\mathrm{N}}^{*}$, and $\pi \Delta \leftrightarrow \Delta or {\mathrm{N}}^{*}$ with which an isobar model is constructed to describe the $P_{33}$ and $P_{11}\pi \mathrm{N}$ scattering phase shifts up to 1 GeV; (b) the transition interactions from $\mathrm{NN}\mathrm{to}\mathrm{N}\Delta ,\Delta \Delta ,\mathrm{N}{\mathrm{N}}^{*}, \mathrm{and} {\mathrm{N}}^{*}{\mathrm{N}}^{*}$ are determined from one-pion and one-rho exchange mechanisms; (c) the $\mathrm{NN}\to \mathrm{NN}$ interaction is directly derived from the Paris potential by using a momentum-dependent procedure to subtract the contributions from intermediate states involving $\Delta$ or ${\mathrm{N}}^{*}$. The $\mathrm{NN}\to \mathrm{NN}$ scattering equation is cast into the familiar coupled-channel form, but with a highly nonlocal isobar self-energy $\Sigma \mathrm{}(E,p)\mathrm{}$ calculated from the vertex interactions $\pi \mathrm{N}\leftrightarrow \Delta or {\mathrm{N}}^{*}$ in a dynamical three-body approach. Both the isospin $T=1\mathrm{}$ and $T=0\mathrm{}$ NN scattering phase shifts of Arndt et al. up to 1 GeV can be described to a very large extent by the model. The fits are, on the average, better than most of the previous NN calculations. The model also describes reasonably well both the magnitudes and signs of the NN total cross sections ${\sigma }^{\mathrm{tot}}$, $\Delta {\sigma }_T^{\mathrm{tot}}$, and $\Delta {\sigma }_L^{\mathrm{tot}}$ up to 2 GeV, except the strong energy dependences in the region near 800 MeV. We discuss the origin of this problem in connection with future necessary improvements of the model and the questions about the dibaryon resonances. The model can be used for a unified approach to study the isobar-nucleus dynamics at both low and intermediate energies.

[NUCLEAR REACTIONS Isobar model for $\Delta$ and ${\mathrm{N}}^{*}$ excitation, mesonexchange theory of NN scattering from 0 to 2 GeV.]

An approximate method of determining the central, the spin-orbit, and the tensor parts of a local nucleon-nucleon interaction from the dependence of the eigenphase shifts and the mixing parameter on the total angular momentum $J$ is studied. With the assumption that the tensor force is zero at the origin, the wave equation is transformed into a differential equation which can accurately be approximated by a matrix difference equation. The finite difference analog of the $\mathcal{R}$ matrix obtained from the solution of this matrix difference equation has a unique continued fraction expansion which enables one to obtain the central, the spin-orbit, and the tensor potentials provided that the exchange characters of all of the forces are specified. Model calculations are presented to show the utility and the accuracy of the method and a brief discussion is given about the possibility of extension of the inverse problem to include certain nonstatic potentials.

[NUCLEAR REACTIONS Construction of local nucleon-nucleon potentials; inverse scattering problem at fixed energy.]

800 MeV inclusive (p,p′) data, spanning the momentum range from that of the quasi-elastic peak to outgoing proton momenta of about $350\frac{\mathrm{MeV}}{c}$, are reported for the target nuclei ${}_{}{}^1{}_{}{}^{}\mathrm{H}$,${}_{}{}^2{}_{}{}^{}\mathrm{H}$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$, and ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$. The data are consistent with simple ideas concerning quasi-free $\pi$ production, and an angle and momentum integration of $\frac{d^2\sigma }{d\Omega \mathrm{dp}}$ across the entire quasi-free region accounts for most of the p + nucleus total reaction cross section.

[NUCLEAR REACTIONS (p,p′) on ${}_{}{}^1{}_{}{}^{}\mathrm{H}$, ${}_{}{}^2{}_{}{}^{}\mathrm{H}$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$, ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$; $E_p=800\mathrm{}$ MeV, ${\theta }_L=5\mathrm{}°-30\mathrm{}°$; measured $\frac{d^2\sigma }{d\Omega \mathrm{dp}}$ for quasi-elastic and quasi-free pion production.]

The Dirac impulse approximation formulation of intermediate energy, proton nucleus scattering is extended to include the inelastic excitation of collective degrees of freedom. In this approach, the deformed spin-orbit interaction appears naturally and is found to have a large effect on the inelastic spin observables. The agreement with experimental data for the collective inelastic observables is comparable to that obtained for the corresponding elastic ones.

The $\pi -{}_{}{}^3{}_{}{}^{}\mathrm{He}-{}_{}{}^3{}_{}{}^{}\mathrm{H}$ form factor $G_{\pi }\left(q^2\right)$ is calculated from a connection between the elementary particle method and the impulse approximation using a realistic wave function with an asymptotic tail. An analytic continuation is used for the timelike region. The $\pi -{}_{}{}^3{}_{}{}^{}\mathrm{He}-{}_{}{}^3{}_{}{}^{}\mathrm{H}$ coupling constant $G_{\pi }(-m_{\pi }^2)$ is found to be ≅ -1.7, in absolute value higher than the elementary nucleon value $g_{\pi }=1.41\mathrm{}$ because of the nuclear extension. The form factor is in very good agreement with results of dispersion relations. A determination which led to $G_{\pi }\approx -1.0\mathrm{}$ is shown to be wrong.

[NUCLEAR STRUCTURE ${}_{}{}^3{}_{}{}^{}\mathrm{He}$; calculated $\pi -{}_{}{}^3{}_{}{}^{}\mathrm{He}-{}_{}{}^3{}_{}{}^{}\mathrm{H}$ coupling constant.]

General differences between the impulse approximation and the elementary particle method in the treatment of two-step processes like pion photoproduction and radiative muon and pion capture are discussed. Important numerical differences arise, exemplified by pion photoproduction on ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. The reason is that the Born approximation with low-energy theorems leaves terms undetermined in the elementary particle method. In particular it is shown that in pion photoproduction in pseudoscalar coupling the impulse approximation accounts for numerically sizable contributions of intermediate nuclear excited states not contained in the elementary particle method. In pseudovector coupling no unique and unambiguous relation between the pion-nucleus coupling constant and pion photoproduction holds. An earlier calculation of radiative muon capture by ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is improved by including in a semiempirical way exchange contributions to the muon radiating diagram.

[NUCLEAR REACTIONS ${}_{}{}^3{}_{}{}^{}\mathrm{He}(\gamma ,{\pi }^{+}){}_{}{}^3{}_{}{}^{}\mathrm{H}$; calculated $\sigma \left(\theta \right)$. ${}_{}{}^3{}_{}{}^{}\mathrm{He}({\mu }^{-},{\nu }_{\mu }\gamma ){}_{}{}^3{}_{}{}^{}\mathrm{H}$; calculated photon spectrum.]

The deuteron electric polarizability is calculated and the various contributions are analyzed theoretically. Upper limits are constructed from zero-range approximations. These analytic approximations explicate the smallness of the deviation of the exact numerical results from the zero-range approximation using no odd-parity forces and no deuteron $D$ state.

[NUCLEAR STRUCTURE Electric polarizability.]

The time-dependent variational principle of the Schrödinger equation is applied to a formally exact solution of the Schrödinger equation whose variational elements are operators which define a collective subspace of the many-body system under study. This generalizes the procedure employed to derive time-dependent Hartree-Fock theory. Four distinct formally exact time-independent variational principles, including several familiar forms, are derived. Application to a class of exactly soluble models, studied in the vibrational regime, illustrates the different modes of implementation of two of the principles. The general theory of large amplitude collective motion is derived. It is shown that the principles can be applied equally well, using either boson or fermion pair degrees of freedom. Some aspects of the relation to the semiclassical limit are discussed.

[NUCLEAR STRUCTURE Time-dependent variational principle for collective motion. Invariance principle of Schrödinger equation. Generalized coherent state trial function. Exact time-independent variational principles. Variational principle for intrinsic state. Cranking variational principle. Large amplitude collective motion. Semiclassical limit.]

Low-lying states of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ have been studied by calculating the trajectories of the first two eigen-values of the kernels of the Yakubovskii four-body equation as the total energy $E$ increases from $-\infty \mathrm{to} \infty +i\epsilon$. The two-particle interactions used are of the separable Yamaguchi type and include spin-dependent forces. The integral equations are derived for each state with values of spin $S$, isospin $T$, and total angular momentum $L$. To obtain a set of single variable integral equations, the Schmidt expansion is applied. The deformation of the integration contour is performed for the complex eigenvalues, and the eigenvalue problems for these equations are solved to determine the binding energy or the resonance energy including $l=1\mathrm{}$ subamplitudes for 3 + 1 subsystems. The binding energies for the ground and the first excited state are -45.009 MeV and -11.529 MeV, respectively. A resonance state is found to be about -4.889 MeV in the state with $ST=10\mathrm{}$, $L=1\mathrm{}$ corresponding to the degenerate state of the second, third, and ninth excited states of the ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ nucleus.

[NUCLEAR STRUCTURE ${}_{}{}^4{}_{}{}^{}\mathrm{He}$; calculated levels. Four-body, separable potential model.]

The effective mass and pole strength for protons and neutrons are evaluated in neutron matter for different densities. The source for the N-N interaction is the nonstatic one-boson-exchange model, and the many-body problem is treated self-consistently by looking simultaneously at the renormalization of the nucleons and the bosons in the medium. The $\pi$ and $\rho$ meson exchanges in the exchange channel are shown to play an important role in the energy and momentum dependence of the nucleon self-energy. The renormalization of the meson properties in the medium, especially the pion properties, stresses the energy dependence of the interaction, making retardation effects important at high densities. Some astrophysical implications of our results are also discussed.

[NUCLEAR STRUCTURE Neutron matter. Effective mass, strength factor.] Boson exchange model used. Self-consistent treatment.]

Analysis of aligned angular momenta $i_{\alpha }\left(\omega \right)$ in different rotational bands extracted from experimental data with a linear spin term $\sim \mathrm{BI}$ in the formulas for $E_{\mathrm{rot}}\mathrm{}\left(I\right)\mathrm{}$ reveal that, in actinide nuclei in the levels with modest spin $I\le 23$, $i_{\alpha }\left(\omega \right)$ usually is very small ($\lesssim 0.7$); i.e., is much smaller than in rare earth nuclei.

[NUCLEAR STRUCTURE ${}_{}{}^{226}{}_{}{}^{}\mathrm{Ra}$, ${}_{}{}^{232}{}_{}{}^{}\mathrm{Th}$, ${}_{}{}^{232-238}{}_{}{}^{}\mathrm{U}$, ${}_{}{}^{235}{}_{}{}^{}\mathrm{U}$, ${}_{}{}^{237}{}_{}{}^{}\mathrm{Np}$; analyzed intra-band level energies and Coriolis alignment strength.]

Assignments to the $K^{\pi }=0_2^{+},0_4^{+},0_5^{+}, \mathrm{and} 0_1^{-}$ rotational bands of ${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$ (all characterized by large reduced width for alpha decay to the ground state of ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$) are critically discussed. Some past assignments seem to be in error. The $0_5^{+}$ band is new. Promising candidates are suggested for gaps in the $K^{\pi }=1^{-} \mathrm{and} 2^{-}$ bands. Verification of assignments and conjectures needs more work, both experimental and theoretical.

[NUCLEAR STRUCTURE Assignments of ${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$ levels to rotational bands.]

Collective potential energy surface calculations have been performed for ${}_{}{}^{74-80}{}_{}{}^{}\mathrm{Kr}$ nuclei in the framework of the Gneuss and Greiner model. The collective Hamiltonian is diagonalized in the basis of the five-dimensional quadrupole oscillator. Our calculations suggest that the ${}_{}{}^{78,80}{}_{}{}^{}\mathrm{Kr}$ nuclei are $\gamma$ unstable with a triaxial minimum at $\gamma \simeq 40°$. The ${}_{}{}^{74,76}{}_{}{}^{}\mathrm{Kr}$ nuclei have two axially symmetric minima with the $0_1^{+}$ ground state being located mainly in the deformed minimum, whereas the first excited $0_2^{+}$ state is located in the spherical minimum. The calculations also indicate that the ground state wave function for ${}_{}{}^{74}{}_{}{}^{}\mathrm{Kr}$ is spread over both the minima.

[NUCLEAR STRUCTURE ${}_{}{}^{74,76,78,80}{}_{}{}^{}\mathrm{Kr}$: Calculated potential energy surfaces; compared with experimental energy levels and $B\left(E2\mathrm{}\right)\mathrm{}$ values.]

Single-channel resonating group calculations give a poor account of backward angle nucleon-${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ scattering. In an attempt to resolve this discrepancy we have included inelastic coupling to the 2.19 MeV $3^{+}$ level of ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ and a nucleon-nucleon spin-orbit potential. Only small effects on the backward angle elastic scattering were found, but we could account reasonably well for the inelastic cross section and elastic polarization data.

In the present paper a general survey of the mathematical formalism for microscopic nuclear structure calculations in configuration spaces consisting of arbitrary spin- and number-projected Hartree-Fock-Bogoliubov—type quasiparticle determinants is given. On the basis of this formalism, various levels of approximation are then discussed. These lead to a number of microscopic nuclear structure models in between the standard Hartree-Fock-Bogoliubov theory and the complete diagonalization of a given effective many nucleon Hamiltonian. For all these models variational equations are derived and possibilities for their numerical application are estimated. The second part of the present series of two papers will then present initial results of the applications of the simplest of these models to several nuclei in various mass regions.

In the first part of the present series of two papers we discussed several nuclear structure models all working in configuration spaces consisting of spin- and number-projected quasiparticle determinants. In the present paper a particular version of the numerically simplest of these models is presented. This model approximates the nuclear wave functions by linear combinations of the angular momentum- and particle number-projected Hartree-Fock-Bogoliubov vacuum and the equally spin- and number-projected two quasiparticle excitations with respect to it. The model allows the use of realistic two body interactions and rather large model spaces. It can hence be applied to a large number of nuclear structure problems in various mass regions. First applications have been performed for the nuclei ${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$, ${}_{}{}^{22}{}_{}{}^{}\mathrm{Ne}$, ${}_{}{}^{46}{}_{}{}^{}\mathrm{Ti}$, and ${}_{}{}^{164}{}_{}{}^{}\mathrm{Er}$. In all these cases the results are very encouraging.

[NUCLEAR STRUCTURE ${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$, ${}_{}{}^{22}{}_{}{}^{}\mathrm{Ne}$, ${}_{}{}^{46}{}_{}{}^{}\mathrm{Ti}$, ${}_{}{}^{164}{}_{}{}^{}\mathrm{Er}$; calculated spectra and transitions. Spin- and number-projected Hartree-Fock-Bogoliubov and shell model methods.]

Angular distributions of the differential cross section and of the vector analyzing power were measured over a wide range of angles at 9 MeV for ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ and ${}_{}{}^{58}{}_{}{}^{}\mathrm{Ni}$ and at 11 MeV for ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{51}{}_{}{}^{}\mathrm{V}$, and ${}_{}{}^{58}{}_{}{}^{}\mathrm{Ni}$. Optical-model parameters which could serve as input into ${}_{}{}^7{}_{}{}^{}\mathrm{Li}$ folding model calculations were determined from the data.

[NUCLEAR REACTIONS ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}, {}_{}{}^{58}{}_{}{}^{}\mathrm{Ni}(\overrightarrow{\mathrm{t}},\mathrm{t})$, $E=9\mathrm{}$ MeV, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}, {}_{}{}^{51}{}_{}{}^{}\mathrm{V}, {}_{}{}^{58}{}_{}{}^{}\mathrm{Ni}(\overrightarrow{\mathrm{t}},\mathrm{t})$, $E=11\mathrm{}$ MeV; measured $\sigma \left(\theta \right)$, $A_y\left(\theta \right)$; optical-model analysis.]