Three-body ($\alpha \mathrm{NN}$) models of the ${}_{}{}^6{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ ground states are used to investigate their shell structure. Three models for each nucleus are considered: simple, full ($\mathrm{nn}$), and full ($\mathrm{np}$) for ${}_{}{}^6{}_{}{}^{}\mathrm{He}$, and simple, full (0%), and full (4%) for ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$. The full models in both cases are obtained by including the $S_{\frac{1}{2}}$, $P_{\frac{1}{2}}$, and $P_{\frac{3}{2}}$ partial waves of the $\alpha N$ interaction, whereas the simple model truncates to only the strongly resonant $P_{\frac{3}{2}}$ wave. The ${}_{}{}^6{}_{}{}^{}\mathrm{He}$ full models distinguish between use of the $\mathrm{nn}$ or $\mathrm{np}$ parameters for the ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ $\mathrm{NN}$ interaction, while the ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ full models have either a pure ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ $\mathrm{NN}$ interaction (0%) or a ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}-{}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}$ interaction that leads to a 4% $d$-wave component in the deuteron (4%). These models are used to calculate the probabilities of the orbital components of the wave functions, the configuration-space single-particle orbital densities, and the configuration-space two-particle wave function amplitudes in $j-j$ coupling with the nucleon coordinates referred to the alpha particle as the "core" or "center of force." The results are then compared with those from phenomenological and realistic-interaction shell models. Major findings of the comparison are the following: None of the shell models considered have a distribution of orbital probabilities across shells like that predicted by three-body models; the orbital rms radii from three-body models indicate an ordering of the orbits within shells, i.e., $p_{\frac{1}{2}}$ outside $p_{\frac{3}{2}}$, unlike oscillator shell models with a single oscillator parameter where the $p$-shell orbitals have the same shape; and, as expected, three-body orbital densities decay at large radial distances as exponentials rather than the too compact Gaussian falling off of oscillator shell models.

NUCLEAR STRUCTURE ${}_{}{}^6{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$, three-body models, shell structure.

A new relativistic three-dimensional two-body equation is presented. If the negative energy poles of the two-body propagator are neglected, this equation is exactly equivalent to the Bethe-Salpeter equation. In the derivation of this equation from the Bethe-Salpeter equation, retardation effects in the two-body interaction are treated carefully by taking fully into account contributions of the exchanged-"meson" singularities. The results are presented for the two cases: (a) the driving term in the Bethe-Salpeter equation is given by the relativistic one-meson exchange diagram (the ladder approximation) and (b) the driving term also includes relativistic crossed diagrams. Using these results it is shown that the Bethe-Salpeter equation in the ladder approximation goes over to the Lippmann-Schwinger equation in the nonrelativistic limit. But unlike the Blankenbecler-Sugar and Gross equations, the two-body potential is not the nonrelativistic Yukawa potential. It is given by an infinite sum of terms corresponding to instantaneous multimeson exchange contributions in the nonrelativistic two-body interaction. A method is presented which in the low energy limit greatly simplifies the calculation with crossed diagrams. The theory is applied to the problem of two scalar "nucleons" exchanging scalar "mesons," and comparisons are made with other relativistic equations. Also a simple model calculation is performed to demonstrate the sensitivity of phenomenological meson masses and meson-nucleon coupling constants to the theoretical reduction scheme chosen.

NUCLEAR STRUCTURE Three-dimensional reduction of the Bethe-Salpeter equation. Ladder approximation. Crossed diagrams. Nonrelativistic limit. Instantaneous multimeson exchange.

The energetic shift caused by the two-pion-exchange three-nucleon force in the triton is estimated in first order perturbation theory. The triton wave function $\psi$ is gained through a Faddeev calculation using the Reid potential. Within the restricted presentation of $\psi$ strong cancellations between terms of the order of 1 MeV lead to an unimportant small effect.

NUCLEAR STRUCTURE Energy expectation value of the two-pion-exchange three-nuclear force, Faddeev calculation.

Formal aspects of boson expansions in the Tamm-Dancoff representation are investigated in detail. This is carried out in the framework of the extended commutator method by solving in complete generality the coefficient equations, searching for Hermitian as well as non-Hermitian boson expansions. The solutions for the expansion coefficients are obtained in a new form, called the square root realization, which is then applied to carry out an analysis of the relationship between the type of expansion and the boson space in which the expansion is defined. It is shown that this new realization is reduced to various well-known boson theories when the boson space is chosen in an appropriate manner. Further discussed, still on the basis of the square root realization, is the equivalence, on a practical level, of a few boson expansion approaches when the Tamm-Dancoff space is truncated to a single quadrupole collective component.

NUCLEAR STRUCTURE Boson expansion, commutator method, Tamm-Dancoff representation, particle-pair representation, Hermitian and non-Hermitian boson expansions, square root realization, truncation of the fermion and boson spaces.

The ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}(\mathrm{p}, \alpha ){}_{}{}^9{}_{}{}^{}\mathrm{B}$ and ${}_{}{}^{32}{}_{}{}^{}\mathrm{S}(\mathrm{p}, \alpha ){}_{}{}^{29}{}_{}{}^{}\mathrm{P}$ reactions have been studied at incident energies of 42.77 and 41.9 MeV, respectively. The experimental (p, $\alpha$) relative cross sections are well reproduced by distorted wave direct pickup calculations with a semimicroscopic form factor and current shell model wave functions. A comparison between (p, $\alpha$) and (${}_{}{}^3{}_{}{}^{}\mathrm{He}$,d) spectra on $1p$ and $2s-1d$ shell nuclei, leading to the same final nucleus, shows a clear evidence of a dominant pickup process over the knockout mechanism in the dynamics of the (p, $\alpha$) reaction.

NUCLEAR REACTIONS, NUCLEAR STRUCTURE ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}(\mathrm{p}, \alpha ){}_{}{}^9{}_{}{}^{}\mathrm{B}$, $E=42.77\mathrm{}$ MeV, ${}_{}{}^{32}{}_{}{}^{}\mathrm{S}(\mathrm{p}, \alpha ){}_{}{}^{29}{}_{}{}^{}\mathrm{P}$, $E=41.9\mathrm{}$ MeV, measured $\sigma (E_x, \theta )$; natural targets. DWBA analysis. Calculated ${}_{}{}^9{}_{}{}^{}\mathrm{B}$ energy levels, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}(\mathrm{p}, \alpha ){}_{}{}^9{}_{}{}^{}\mathrm{B}$ and ${}_{}{}^{32}{}_{}{}^{}\mathrm{S}(\mathrm{p}, \alpha ){}_{}{}^{29}{}_{}{}^{}\mathrm{P}$ spectroscopic strengths.

The inclusive energy spectra of ${}_{}{}^{6,7}{}_{}{}^{}\mathrm{Li}$ ejectiles from the reaction ${}_{}{}^9{}_{}{}^{}\mathrm{Be}$ + Au at 27.4 MeV/nucleon have been measured. The quasielastic peaks for ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ and ${}_{}{}^7{}_{}{}^{}\mathrm{Li}$ have reduced momentum widths ${\sigma }_0$ of 65 and 60 MeV/c, respectively. When plotted in conjunction with similar measurements, these new results clarify the behavior of the reduced momentum width in the important region between 10-100 MeV/nucleon.

NUCLEAR REACTIONS ${}_{}{}^9{}_{}{}^{}\mathrm{Be}+\mathrm{Au}\to {}_{}{}^{6,7}{}_{}{}^{}\mathrm{Li}+X$. $E\left({}_{}{}^9{}_{}{}^{}\mathrm{Be}\right)=27\mathrm{}$ MeV/nucleon. Measured momentum widths of ${}_{}{}^{6,7}{}_{}{}^{}\mathrm{Li}$ ejectiles. Deduced reduced width ${\sigma }_0$. Discuss ${\sigma }_0$ over energy range 10-2000 MeV/nucleon.

The mean life of the first excited state of ${}_{}{}^{10}{}_{}{}^{}\mathrm{B}$ at 718 keV has been measured as 1020±5 ps. The result is compared to previous measurements, and its relevance to an understanding of second order effects in Coulomb excitation of ${}_{}{}^{10}{}_{}{}^{}\mathrm{B}$ is commented upon.

NUCLEAR REACTIONS ${}_{}{}^{10}{}_{}{}^{}\mathrm{B}(p, p^{\prime })$, $E=4.5\mathrm{}$ MeV; measured $p-\gamma$ coincidences, deduced $T_{\frac{1}{2}}$. Enriched target.

The half-life of ${}_{}{}^{218}{}_{}{}^{}\mathrm{Ac}$ has been determined by use of a pulsed beam and a recoil technique. The reduced $\alpha$ width, inferred from the new lifetime value $T_{\frac{1}{2}}=1.12\mathrm{}\pm 0.11$ μs, agrees with the systematics for $N=129\mathrm{}$ isotones.

RADIOACTIVITY ${}_{}{}^{209}{}_{}{}^{}\mathrm{Bi}({}_{}{}^{12}{}_{}{}^{}\mathrm{C}, 3n){}_{}{}^{218}{}_{}{}^{}\mathrm{Ac}$, $E_{\mathrm{lab}}=72\mathrm{}$ MeV. Measured pulsed beam $\alpha$ timing. Deduced $T_{\frac{1}{2}}$ ${}_{}{}^{218}{}_{}{}^{}\mathrm{Ac}$ ground state.

Structure is observed in the yield curve for the 871-keV $\gamma$ rays of ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}$ from the ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}$ + ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}$ reaction but absent for other observable transitions. The single-particle nature of the ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}$ transition suggests an interpretation in terms of the nuclear Landau-Zener effect.

NUCLEAR REACTIONS ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}$ + ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}$, $E_{\mathrm{c}.\mathrm{m}.\mathrm{}}=15-26\mathrm{}$ MeV, Ge(Li) detectors, measured $I_{\gamma }\mathrm{}\left(E\right)\mathrm{}$.

In a recent article, we have proposed a theory of three-body reactions. The theory is constructed in a manner in which, even for the breakup process, the numerical calculations need to be performed only in a finite spacial region, and in which the calculations are performed in a perturbative way, where the zeroth order term includes all pole terms. The use of the partially separable $t$ matrix plays a central role in this theory. In the present paper, we extend the partially separable $t$ matrix to the optical potential so that the theory of three-body reactions may be applicable to nuclear reactions.

NUCLEAR REACTIONS Off-shell $t$ matrix with one nonseparable and one separable term. Optical potential. Many-level formula.

Several recent analyses of neutron induced reactions have necessitated the use of different potential parameters for different partial waves. We suggest that this parametrization of the optical potential can be alleviated if the proper dynamic deformations of the target nucleus are introduced. In an application to the ${}_{}{}^{32}{}_{}{}^{}\mathrm{S}$ + n reaction, a coupled channel calculation yields good fits to elastic and compound cross sections with a unique set of real optical potential parameters.

NUCLEAR REACTIONS ${}_{}{}^{32}{}_{}{}^{}\mathrm{S}$ + n reaction; coupled channel calculation.

A full two-body shell-model calculation has been used to estimate the pion-exchange contribution to the parity-violating $\alpha$ decay of the $0^{+}$, $T=1\mathrm{}$ state of ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$. For a neutral-current-induced weak pion amplitude equal to the "best" value given by Desplanques, Donoghue, and Holstein, the $\alpha$ width is 1.3×${10}^{-8\mathrm{}}$ eV. There is no contribution in lowest order from the intrinsic parity impurity in the $\alpha$ particle, but there is a small contribution from the deuteron.

NUCLEAR STRUCTURE ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$. Calculation of parity-violating $\alpha$ decay of $0^{+}$, $T=1\mathrm{}$ state.

Self-consistent cranked BCS calculations based on a monopole pairing field indicate that configuration-dependent changes in the nuclear shape cannot reproduce observed correlations between band-crossing frequencies in odd-$N$, rare-earth nuclei and the quadrupole moment of the orbit of the valence quasineutron. This conclusion strengthens the previously suggested necessity for shape-dependent (e.g., quadrupole) components in the pairing field.

NUCLEAR STRUCTURE ${}_{}{}^{161}{}_{}{}^{}\mathrm{Er}$, ${}_{}{}^{173}{}_{}{}^{}\mathrm{W}$, ${}_{}{}^{181}{}_{}{}^{}\mathrm{Os}$, calculated deformations and band crossings as function of configuration, Nilsson-Strutinsky and cranked shell model method. ${}_{}{}^{161}{}_{}{}^{}\mathrm{Er}$, calculated band crossing as function of configuration, self-consistent cranked BCS method.

The inverse scattering problem for the off-shell $T$ matrix is formulated within the framework of a combined variable-phase—off-shell scattering theory. A simple expression is constructed for the $s$-wave half-off-shell $T$ matrix. The input information consists of binding energies and phase shifts at all energies.

NUCLEAR REACTIONS Scattering theory, off-shell generalization of the variable-phase approach, inverse problem for the $T$ matrix.

A quick way of calculating pion distorted waves in the ${\Delta }_{33}$ resonance region using eikonal approximation to the pion wave function is presented. The pion distorted waves thus obtained are in good agreement with the exact solutions of the Klein-Gordon equation for pion at backward angles in the energy range 200 MeV < pion kinetic energy < 300 MeV.

NUCLEAR REACTIONS Pion distorted waves in eikonal approximation; Kisslinger optical potential.

We compute the magnetic moment of ${}_{}{}^{15}{}_{}{}^{}\mathrm{N}$ by solving the Dirac equation with a mixture of vector and scalar Woods-Saxon components. Our result confirms the failure of the "scalar-vector" model to reproduce experimental magnetic moment data.

NUCLEAR STRUCTURE ${}_{}{}^{15}{}_{}{}^{}\mathrm{N}$; calculated magnetic moments. Relativistic single-particle wave functions.

We find that the shell model and Nilsson-pairing model predictions for the matrix elements mediating the $2\nu$ $\beta \beta$ decay of ${}_{}{}^{76}{}_{}{}^{}\mathrm{Ge}$ are in quantitative agreement when the comparison is made for equivalent model spaces. This suggests that further expansion of shell model bases may exacerbate the existing discrepancies between theory and various geochemical rates.

RADIOACTIVITY Double beta decay; ${}_{}{}^{76}{}_{}{}^{}\mathrm{Ge}$, lepton number conservation; comparison of shell and Nilsson-pairing models.

Momentum widths of fragments produced in the reactions of 320 MeV ${}_{}{}^{10}{}_{}{}^{}\mathrm{B}$ with ${}_{}{}^{181}{}_{}{}^{}\mathrm{Ta}$ have been measured. At forward angles the energy spectra of light particles ($Z=1, 2$) exhibit energy peaks characterized by a reduced momentum width ${\sigma }_0=54\mathrm{}$ MeV/c, which is well below the values near 85 MeV/c observed in the reactions of 43 MeV/u ${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$ with ${}_{}{}^{181}{}_{}{}^{}\mathrm{Ta}$ and in reactions induced by relativistic heavy ions. These results suggest a rapid increase in momentum widths, or apparent temperature, in the 32 to 43 MeV/u range.

NUCLEAR REACTIONS ${}_{}{}^{181}{}_{}{}^{}\mathrm{Ta}({}_{}{}^{10}{}_{}{}^{}\mathrm{B}, X)$, $E=320\mathrm{}$ MeV, projectile fragmentation; measured energy spectra of H and He isotopes at forward angles, extracted momentum widths.

Strong spin-isospin mode isovector analog states were studied in differential cross section measurements ($0.6\lesssim q\lesssim 1.5$ ${\mathrm{fm}}^{-1\mathrm{}}$) of the reaction ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}(\gamma , {\pi }^{+}){}_{}{}^{13}{}_{}{}^{}\mathrm{B}$ up to 12 MeV excitation energy in ${}_{}{}^{13}{}_{}{}^{}\mathrm{B}$. The photopion reaction predominantly excites $M2\mathrm{}$ states at low $q$ and $M4\mathrm{}$ states at high $q$, with both peaking in the energy region 6-7 MeV in ${}_{}{}^{13}{}_{}{}^{}\mathrm{B}$ (21-22 MeV in ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}$). The results are compared with predicted $M2\mathrm{}$ and $M4\mathrm{}$ relative strength distributions and also with the available data on radiative capture of stopped pions and inelastic electron scattering.

NUCLEAR REACTIONS ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}(\mathrm{e}, {\pi }^{+}){}_{}{}^{13}{}_{}{}^{}\mathrm{B}_{}^{*}{}_{}{}^{}$ ($E_x=0, 3.5, 6.4, \mathrm{and} 9$ MeV), $E_{\mathrm{e}}=195\mathrm{}$ MeV; measured $\sigma \left(\theta \right)$, $\theta =30-150\mathrm{}°$, $\Delta \theta =20\mathrm{}°$. Enriched target. Results compared to theoretical $M2\mathrm{}$ and $M4\mathrm{}$ strength distributions.

We discuss the feasibility of radiochemical and geochemical measurements of the rates for two double beta decay reactions, ${}_{}{}^{238}{}_{}{}^{}\mathrm{U}$→${}_{}{}^{238}{}_{}{}^{}\mathrm{Pu}$ and ${}_{}{}^{232}{}_{}{}^{}\mathrm{Th}$→${}_{}{}^{232}{}_{}{}^{}\mathrm{U}$, that produce unstable daughter nuclei.

RADIOACTIVITY Double beta decay, radiochemical techniques, ${}_{}{}^{238}{}_{}{}^{}\mathrm{U}$ and ${}_{}{}^{232}{}_{}{}^{}\mathrm{Th}$, lepton number conservation.