A method for expanding a nonlocal, nonsymmetric potential as the sum of separable terms is presented. The method of calculating the bound state and the scattering amplitude is described. A numerical example of the nonlocal potential related to $n-d$ elastic scattering is given.

Models of quantum mechanical many-body systems are presented such that the Hartree-Fock equations for them admit analytical solutions, which simulate an isolated atomic nucleus or two nuclei colliding with each other.

NUCLEAR REACTIONS Scattering theory, time-dependent Hartree-Fock method.

Exchange symmetry is used to eliminate explicit reference to physically indistinguishable channels in the Baer, Kouri, Levin, and Tobocman coupled-equations nuclear reaction formalism. The resulting equations are similar in structure to the original ones except they are fewer in number and the interactions have a more complex structure. The reduction procedure presented here differs from an earlier one by Tobocman in that it preserves the kernel connectivity property of the original Baer-Kouri-Levin-Tobocman coupled integral equations.

NUCLEAR REACTIONS Coupled-equations nuclear reaction formalism of Baer, Kouri, Levin, and Tobocman is explicitly antisymmetrized in a manner that preserves kernel connectivity.

The ($t,p$) two nucleon transfer reaction has been studied on ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ and ${}_{}{}^{14}{}_{}{}^{}\mathrm{C}$ at a beam energy of 23 MeV with an overall final resolution of better than 30 keV. A new mass excess of + 13.692 ± 0.004 MeV is inferred for the ${}_{}{}^{16}{}_{}{}^{}\mathrm{C}$ ground state. Seven final states of ${}_{}{}^{16}{}_{}{}^{}\mathrm{C}$ are noted, and angular distributions are measured for six of these.

NUCLEAR REACTIONS ${}_{}{}^{12,14}{}_{}{}^{}\mathrm{C}\mathrm{}(t,p)\mathrm{}{}_{}{}^{14,16}{}_{}{}^{}\mathrm{C}$, $E_t=23\mathrm{}$ MeV; measured $\sigma (E_p;\theta )$; Enriched targets, DWBA analysis.

The connected kernel equations of Bencze, Redish, and Sloan for the scattering operators in the $N$-body problem are derived by a new and more direct method. It is seen that the equations can be obtained by distributing the final state's residual interaction over all partitions in a particular way. A different resolvent equation is then applied to each term of the sum. For the specific distribution chosen the disconnected parts of the kernel vanish exactly. The validity of the off-shell transformation which simplifies the inhomogeneous term is demonstrated.

NUCLEAR REACTIONS $N$-body problem, scattering theory, Bencze-Redish-Sloan equations, Kouri-Levin method, Faddeev equations.

The (${}_{}{}^4{}_{}{}^{}\mathrm{He}$, ${}_{}{}^8{}_{}{}^{}\mathrm{He}$) and (${}_{}{}^3{}_{}{}^{}\mathrm{He}$, ${}_{}{}^8{}_{}{}^{}\mathrm{Li}$) reactions have been employed at detection angles near 8° on gas targets of ${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$ and ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ to produce and measure the masses of the nuclei ${}_{}{}^{16}{}_{}{}^{}\mathrm{Ne}$, ${}_{}{}^{15}{}_{}{}^{}\mathrm{F}$, and ${}_{}{}^{12}{}_{}{}^{}\mathrm{O}$. The (${}_{}{}^4{}_{}{}^{}\mathrm{He}$, ${}_{}{}^8{}_{}{}^{}\mathrm{He}$) reactions were performed at an incident energy of 117 MeV and the (${}_{}{}^3{}_{}{}^{}\mathrm{He}$, ${}_{}{}^8{}_{}{}^{}\mathrm{Li}$) reaction at 88 and 75 MeV. The mass excesses of ${}_{}{}^{16}{}_{}{}^{}\mathrm{Ne}$, ${}_{}{}^{15}{}_{}{}^{}\mathrm{F}$, and ${}_{}{}^{12}{}_{}{}^{}\mathrm{O}$ were determined to be 23.92±0.08, 16.67±0.18, and 32.10±0.12 MeV, respectively. Estimated ground state decay widths were ${\Gamma }_{\mathrm{c}.\mathrm{m}.\mathrm{}}=0.2\mathrm{}\pm 0.1, 0.8\pm 0.3, \mathrm{and} 0.40\pm 0.25$ MeV, for ${}_{}{}^{16}{}_{}{}^{}\mathrm{Ne}$, ${}_{}{}^{15}{}_{}{}^{}\mathrm{F}$, and ${}_{}{}^{12}{}_{}{}^{}\mathrm{O}$, respectively. A $d$ coefficient of 8±5 keV is indicated for the isobaric multiplet mass equation description of the mass 16 multiplet. Consideration of the possible decay mechanisms of ${}_{}{}^{16}{}_{}{}^{}\mathrm{Ne}$ and ${}_{}{}^{12}{}_{}{}^{}\mathrm{O}$ indicates that both nuclei probably have an appreciable diproton decay width.

NUCLEAR REACTIONS ${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$(${}_{}{}^4{}_{}{}^{}\mathrm{He}$, ${}_{}{}^8{}_{}{}^{}\mathrm{He}$), ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$(${}_{}{}^4{}_{}{}^{}\mathrm{He}$, ${}_{}{}^8{}_{}{}^{}\mathrm{He}$), ${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$(${}_{}{}^3{}_{}{}^{}\mathrm{He}$, ${}_{}{}^8{}_{}{}^{}\mathrm{Li}$). Measured reaction $Q$ values, ground state ${\Gamma }_{\mathrm{c}.\mathrm{m}.\mathrm{}}$ deduced mass excesses. Deduced coefficients for IMME. Discuss diproton decay widths.

The ${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$(${}_{}{}^3{}_{}{}^{}\mathrm{He}$, ${}_{}{}^8{}_{}{}^{}\mathrm{Li}$) reaction has been used to study the particle-unstable nucleus ${}_{}{}^{15}{}_{}{}^{}\mathrm{F}$. The ground state of ${}_{}{}^{15}{}_{}{}^{}\mathrm{F}$ is observed as a broad peak at a mass excess of 16.9±0.2 MeV with a width greater than 900 keV. A well defined narrower peak is observed at a mass excess of 18.088±0.025 MeV and with a width of 240±30 keV. It is shown to be the mirror of the ${5/2}^{+}$ first excited state of ${}_{}{}^{15}{}_{}{}^{}\mathrm{C}$. The analog of the ${5/2}^{+}$ state was observed in ${}_{}{}^{15}{}_{}{}^{}\mathrm{O}$ at $E_x=12.255\mathrm{}\pm 0.013$ MeV. This state is shown to proton decay to the $T=1\mathrm{}$ first excited state of ${}_{}{}^{14}{}_{}{}^{}\mathrm{N}$ and to have a width of 135±15 keV.

NUCLEAR REACTIONS ${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}({}_{}{}^3{}_{}{}^{}\mathrm{He},{}_{}{}^8{}_{}{}^{}\mathrm{Li}){}_{}{}^{15}{}_{}{}^{}\mathrm{F}{}_{}{}^{17}{}_{}{}^{}\mathrm{O}\mathrm{}(p,t)\mathrm{}{}_{}{}^{15}{}_{}{}^{}\mathrm{O}$ ($T=\frac{3}{2}$) measured reaction $Q$ values, ${\Gamma }_{\mathrm{c}.\mathrm{m}.\mathrm{}}$; deduced mass excesses, excitation energies. Deduced coefficients IMME.

An earlier formulation rougly relating core-polarization effects for different external fields, electromagnetic ($p,p^{\prime }$), ($\alpha ,{\alpha }^{\prime }$), etc., through a 2 × 2 core-polarization matrix has been generalized to include chargeexchange reactions under the assumption that the isospin of the nuclear states is pure. The pure isovector field of the direct charge-exchange reaction to an excited-analog state has both an isovector and an isoscalar field for the model space as in inelastic scattering. A crude schematic representation of the giant resonances, virtually excited by interaction with the model space and responsible for the core polarization, is used to estimate the effect on the core-polarization parameters of imposing isospin conservation. The results of the earlier formulation are changed only slightly. Direct excitation of giant-quadrupole resonances in ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ by ($p,p^{\prime }$) and ($n,p$) are compared. Energy shifts of the $T_{>}$ and $T_{<}$ isovector giant-quadrupole resonances are compared with the corresponding dipole shifts.

NUCLEAR REACTIONS Low lying $0^{+}\to 2^{+}$ transitions; extension of core-polarization effects to include charge exchange by application of isospin conservation. Giant-quadrupole transition rates and isovector splitting estimated.

Angular distributions have been measured at a bombarding energy of 28 MeV for the reaction ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}(d,{}_{}{}^6{}_{}{}^{}\mathrm{Li}){}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$, populating most of the known levels below 9.1 MeV excitation. Results were compared with distorted wave Born-approximation calculations, and α spectroscopic factors extracted. Those for the g.s. band are in reasonable agreement with theoretical predictions. Members of the 7.20-MeV band are found to possess very large spectroscopic factors, in agreement with the core-excited nature of this band.

NUCLEAR REACTIONS ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}(d,{}_{}{}^6{}_{}{}^{}\mathrm{Li})$, $E=28.0\mathrm{}$ MeV; measured $\sigma \left(\Theta \right)$. ${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$ deduced levels, $S_{\alpha }$. DWBA analysis.

The ${}_{}{}^{31}{}_{}{}^{}\mathrm{P}({}_{}{}^3{}_{}{}^{}\mathrm{He},d){}_{}{}^{32}{}_{}{}^{}\mathrm{S}$ reaction was investigated at 25 MeV incident energy. One-hundred and eleven levels up to an excitation energy of 12.5 MeV were observed using a split-pole magnetic spectrograph. The experimental angular distributions were analyzed with the distorted-wave Born approximation. The optical model parameters used in the distorted-wave Born-approximation calculations were obtained from a fit to elastic ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ scattering data taken on ${}_{}{}^{31}{}_{}{}^{}\mathrm{P}$ at 25 MeV. Gamow functions were used as form factors for the transferred proton in the case of unbound states. Values of the transferred orbital angular momenta $l$ and spectroscopic strengths were obtained for sixty levels, with many odd-parity levels being observed above 9 MeV excitation. Spin and parity assignments were made upon the basis of the $l$ values obtained from the shapes of the angular distributions and upon comparison with the results of other reactions. Isospin assignments were made by comparison with ${}_{}{}^{32}{}_{}{}^{}\mathrm{P}$ levels. Except for the $l_p=1\mathrm{}$, $T=0\mathrm{}$ transfers, most of the observed spectroscopic strength is concentrated into a few levels. The existence of a $T$-mixed doublet of levels, $J^{\pi }=1^{-}$, is suggested in the 11 MeV region of excitation. The excitation energies and spectroscopic strengths are compared with results of a recent shell-model calculation.

NUCLEAR REACTIONS ${}_{}{}^{31}{}_{}{}^{}\mathrm{P}({}_{}{}^3{}_{}{}^{}\mathrm{He},d)$, ${}_{}{}^{31}{}_{}{}^{}\mathrm{P}$(${}_{}{}^3{}_{}{}^{}\mathrm{He}$, ${}_{}{}^3{}_{}{}^{}\mathrm{He}$), $E=25\mathrm{}$ MeV; measured $\sigma (E_d,\theta )$, ${}_{}{}^{32}{}_{}{}^{}\mathrm{S}$ deduced levels, $l$, $J$, $\pi$, $T$, spectroscopic strengths; measured $\sigma \left(\theta \right)$, deduced optical model parameters. DWBA analysis using Gamow functions as form factors.

We present the details of a novel approach to the treatment of Coulomb effects in atomic and nuclear reactions of the three-body type in which two of the particles are charged. Based on three-body integral equations the formalism allows the parctical calculation of elastic, inelastic, rearrangement, and breakup processes with full inclusion of the Coulomb repulsion or attraction in a mathematically correct way. No restrictions need to be made concerning the form of the short-range interactions between the three pairs. A particular virtue of our method lies in the fact that it corroborates, and gives precise meaning to, the intuitively anticipated conception of how to describe such reactions.

NUCLEAR REACTIONS Three-body scattering theory. Coulomb effects when two of the particles charged. Screening and renormalization approach. Quasiparticle method. Defined scattering amplitudes for elastic, inelastic, rearrangement and breakup processes, and scattering wave functions. Derived practical intergral equations for these quantities.

The precession of the plane of polarization of $\gamma$ radiation in an external magnetic field was observed time differentially. As this experiment is in full analogy to that done by Hanle in 1925 on atoms, we called this behavior nuclear Hanle effect. The measurement was performed with a pulsed proton beam of 5.5 MeV, which defined the moment of excitation of the 197 keV level in the ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$ nucleus. The result is well described by a simple theory in accordance with the theory of perturbed angular distributions.

NUCLEAR REACTIONS ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}(p,p^{\prime })$, $E=5.5\mathrm{}$ MeV; measured nuclear Hanle effect of ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$ (197 keV level), deduced $g$.

The nuclear field theory is applied to the inelastic scattering of structureless particles by nuclei. It is found that the coupling Hamiltonian between scattering and collective variables is linear. Therefore for one-step processes the only anharmonicities present correspond to the description of the bound state.

NUCLEAR REACTIONS. Coupling with collective variables in inelastic scattering.

The decomposition into channels of three- and four-body wave functions are fully worked out. It is explicitly shown that the closed channels do not involve any primary singularity.

NUCLEAR REACTIONS Three- and four-body equations. Decompositions into channels. Special attention to closed channels.

The Yakubovski equation for a four-body system is decoupled into a tractable expression suitable for the treatment of reactions. Because of two-body interactions, some channel wave functions collapse in the course of collision processes. A part of the decoupled Yakubovski wave function justifies the coupled cluster approximation. The optical potential is defined by the $L^2$ operator in the four-body space. The lowest order amplitude is the impulse approximation.

NUCLEAR REACTIONS Four-body Yakubovski equation decoupled to handle reactions. Cluster approximation. Relation to optical model and impulse approximation.

The $\gamma$ spectroscopy of ${}_{}{}^{67}{}_{}{}^{}\mathrm{Zn}$ levels has been investigated via the ${}_{}{}^{64}{}_{}{}^{}\mathrm{Ni}(\alpha ,n\gamma ){}_{}{}^{67}{}_{}{}^{}\mathrm{Zn}$ reaction at 8.2 MeV. Further information concerning low-spin states situated between 1 and 2.5 MeV has been obtained from $\gamma -\gamma$ coincidence, $\gamma$ angular distribution, and Doppler shift attenuation measurements. The results are compared with available theoretical calculations.

NUCLEAR REACTIONS ${}_{}{}^{64}{}_{}{}^{}\mathrm{Ni}(\alpha ,n\gamma )$, $E_{\alpha }=8.2\mathrm{}$ MeV; enriched target, measured $E_{\gamma }$, $I_{\gamma }$, $\gamma \gamma$ coin, $I_{\gamma }\left(\theta \right)$, $\Delta E_{\gamma }\left(\tau \right)$; ${}_{}{}^{67}{}_{}{}^{}\mathrm{Zn}$ deduced level, $J$, $\delta$, $\tau$.

The internal conversion coefficients were measured for several transitions in ${}_{}{}^{93}{}_{}{}^{}\mathrm{Tc}$ with an electron spectrometer and a Ge(Li) detector. The ${}_{}{}^{92}{}_{}{}^{}\mathrm{Mo}(\alpha ,p2n)$ reaction was used to study the decay of the isomeric $\tau =15\mathrm{}$ μsec $\frac{17}{2}$ level at 2185.3 keV as well as the decay of the prompt ${\frac{17}{2}}^{+}$ level at 2185.0 keV. Due to parity mixing the wave function for the $\frac{17}{2}$ isomeric level may have the form $|{\frac{17}{2}}^{-}〉+\alpha |{\frac{17}{2}}^{+}〉$ where $\alpha$ is small. The experimental $K$-shell conversion coefficients for the $\frac{17}{2}\to {\frac{13}{2}}^{+}$ and ${\frac{17}{2}}^{+}\to {\frac{13}{2}}^{+}$ transitions imply $\frac{\Gamma (E2,\frac{17}{2}\to {\frac{13}{2}}^{+})}{\Gamma (E2+M2+E3,\frac{17}{2}\to {\frac{13}{2}}^{+})}<~0.33$. The energy difference between the prompt and delayed $\gamma$-ray transitions to the ${\frac{13}{2}}^{+}$ level was measured to be 0.32 ± 0.03 keV. These results imply $\left|〈{\frac{17}{2}}^{+}\right|H_{\mathrm{pv}}\left|{\frac{17}{2}}^{-}〉\right|<~0.13$ eV for the matrix element of the parity violating Hamiltonian.

NUCLEAR REACTIONS ${}_{}{}^{92}{}_{}{}^{}\mathrm{Mo}(\alpha ,p2n)$, $E=43\mathrm{}$ MeV measured pulse beam electronic timing, $I_{\mathrm{ce}}$, $I_{\gamma }$, $E_{\gamma }$; deduced ${\alpha }_K$, $\gamma$ multipolarity, parity mixing. Enriched target, electron spectrometer, Si(Li) and Ge(Li) detectors.

The ${}_{}{}^{76}{}_{}{}^{}\mathrm{Ge}\mathrm{}(t,p)\mathrm{}$ reaction has been studied in order to extend the known level structure in the Ge isotopes to ${}_{}{}^{78}{}_{}{}^{}\mathrm{Ge}$ and to further investigate the proposed shape transition in the Ge nuclei. About 30 states were observed in ${}_{}{}^{78}{}_{}{}^{}\mathrm{Ge}$, which was previously uninvestigated. Angular distributions were measured and were analyzed using zero-range microscopic two-nucleon transfer distorted-wave calculations to extract $L$ values. Based on the energy of the first excited $0^{+}$ level, it would appear that ${}_{}{}^{78}{}_{}{}^{}\mathrm{Ge}$ does not continue the trend toward prolate deformation but begins to return toward the critical or "supersoft" region.

NUCLEAR REACTIONS ${}_{}{}^{76}{}_{}{}^{}\mathrm{Ge}\mathrm{}(t,p)\mathrm{}$, $E=15.0\mathrm{}$ MeV; measured $\sigma (\theta ,E_p)$. ${}_{}{}^{78}{}_{}{}^{}\mathrm{Ge}$ deduced levels, $L$, $\pi$, $J$. DWBA analysis.

The $3p_{\frac{1}{2}}$ isobaric analog resonances in the compound nuclei ${}_{}{}^{207}{}_{}{}^{}\mathrm{Bi}$ and ${}_{}{}^{208}{}_{}{}^{}\mathrm{Bi}$ have been studied by measuring the cross sections and analyzing powers of the reactions ${}_{}{}^{206}{}_{}{}^{}\mathrm{Pb}(\overrightarrow{p},p_0)$ and ${}_{}{}^{207}{}_{}{}^{}\mathrm{Pb}(\overrightarrow{p},p_0)$ The data were fitted with an optical-model background and Breit-Wigner resonance terms. The resonance energies, elastic partial widths, and elastic mixing phases were extracted. The elastic partial widths are significantly different from those previously determined from differential cross-section measurements. The previously measured inelastic partial widths were adjusted to account for this difference. In the case of the isobaric analog state in ${}_{}{}^{208}{}_{}{}^{}\mathrm{Bi}$, the results are inconsistent with a recent theoretical treatment of the isobaric analog state as a single broad pole in the energy-averaged $S$ matrix.

NUCLEAR REACTIONS ${}_{}{}^{206,207}{}_{}{}^{}\mathrm{Pb}(\overrightarrow{p},p)$, $E=10\mathrm{}-13.6$ MeV; measured $\sigma (E;\theta )$, $A(E;\theta )$; deduced optical-model parameters. ${}_{}{}^{207,208}{}_{}{}^{}\mathrm{Bi}$ IAS, deduced energy, partial width, mixing phase.

States in ${}_{}{}^{197}{}_{}{}^{}\mathrm{Pt}$ have been studied using the ${}_{}{}^{196}{}_{}{}^{}\mathrm{Pt}(d,p){}_{}{}^{197}{}_{}{}^{}\mathrm{Pt}$ and ${}_{}{}^{198}{}_{}{}^{}\mathrm{Pt}(d,t){}_{}{}^{197}{}_{}{}^{}\mathrm{Pt}$ reactions with 13.5 MeV deuterons and the ${}_{}{}^{196}{}_{}{}^{}\mathrm{Pt}(n,\gamma )$ reaction using thermal neutrons. The measured $Q$ values were 3.606 ± 0.020, - 1.221 ± 0.020, and 5.8463 ± 0.0004 MeV for the ($d,p$), ($d,t$), and ($n,\gamma$) reactions, respectively. Transferred $l$ values have been determined from angular distributions and spectroscopic factors determined by comparison with distorted-wave Born-approximation calculations. Some of the states can be tentatively assigned as Nilsson states arising from an oblate deformation. These include the ${5/2}^{-}$[532], decoupled ${1/2}^{+}$[600] and ${9/2}^{+}$[615] states. A number of low lying states with $l=1\mathrm{}$ cannot be explained using the Nilsson model or the shell model.

NUCLEAR REACTIONS, NUCLEAR STRUCTURE ${}_{}{}^{196}{}_{}{}^{}\mathrm{Pt}(d,p)$, ${}_{}{}^{198}{}_{}{}^{}\mathrm{Pt}(d,t)$, $E_d=13.5\mathrm{}$ MeV. Measured $\sigma (E_p,\theta )$, $\sigma (E_t,\theta )$, $Q$, DWBA analysis; ${}_{}{}^{196}{}_{}{}^{}\mathrm{Pt}(n_{\mathrm{th}},\gamma )$. Measured $E_{\gamma }$, $I_{\gamma }$, $Q$. Deduced energies, $l$, $J$, $\pi$ of ${}_{}{}^{197}{}_{}{}^{}\mathrm{Pt}$ levels, enriched targets.