A perturbation theory for three-particle scattering is formulated which in each order of approximation leaves invariant the essential structural features of the scattering amplitudes. The correct primary and secondary singularity structure is reproduced along with unitarity (both off shell and on shell), the correct threshold behaviors, and the correct residues of the double-scattering poles in the 3-to-3 amplitude. The only integral equations to be solved are of the sort which appear in the case of finite-rank two-particle interactions. The formalism, which is related to some recent work by Noyes, depends crucially on the proof of the existence of a decomposition of the two-particle transition operators into so-called essential and residual parts. The special case when the residual part is ignored provides a practical exploitation of the separable expansions found by Fuda and Osborn.

We have evaluated energy- and non-energy-weighted sum rules ($N$ and $M$) for the nuclear charge monopole and dipole modes. Sums are given for different intermediate isospins, using the method in which sums are split into isoscalar, vector, and tensor parts. This facilitates the separation of model-dependent quantities, and we particularly consider possible modifications of shell-model evaluations arising from correlations. Four important specific results are: (i) The shell-model value of $M$ in the dipole case is carefully calculated for ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$, and is 30% larger than the observed value; this difference must arise from correlations. (ii) The previously suggested smallness of the isotensor part of $M$ is not general, but depends on using a shell-model wave function; it is shown theoretically and from experiment that the isotensor part may be large. (iii) The effects of correlations on $M$ can be displayed in a form requiring no new calculations (given published Tamm-Dancoff diagonalizations). (iv) There is a close relation between the charge-exchange part of $N$ in the dipole and monopole problems, so one can use data on the former to evaluate the latter; also, a connection with the effective $M1\mathrm{}$ charge and effective mass is shown.

Levels in ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}$ were populated with the ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}(t,\alpha ){}_{}{}^{18}{}_{}{}^{}\mathrm{O}$ reaction. Extensive data were collected with good $\alpha$-particle energy resolution (40 keV) at $E_t=2.4\mathrm{}$ MeV, using a computer based data acquisition system. Reaction-produced $\alpha$ particles were detected near 180° in a solid-state annular counter and $\gamma$ radiation was detected in a 10.2×10.2-cm NaI(Tl) crystal at detection angles in the interval $0^{\circ }\le {\theta }_{\gamma }\le {90}^{\circ }$. Multiparameter data collection techniques were used to store the matrix of particle-$\gamma$ pulse-height pairs for coincident events. Angular correlations of $\gamma$ rays from levels with $E_x<8\mathrm{}$ MeV were extracted from these data from which level spins, together with $\gamma$-ray branching ratios and $\gamma$-ray multipole mixing ratios, were deduced. Results include the spin assignments $J=1\mathrm{}$ for the 6.19-MeV level, $J=0\mathrm{}\left(1\right)\mathrm{}$ for the 6.86-MeV level, $1\le J\le 4$ for the 7.75-MeV level, and $1\le J\le 5$ for the 7.96-MeV level. Some new $\gamma$-ray decay modes were found. The 7.96-MeV level decays 67% to the 1.98-MeV level, 12% to the 5.09-MeV level, and 21% to the 5.37-MeV level. The 7.75-MeV level decays 50% to the 1.98-MeV level, 11% to the 4.45-MeV level, and 39% to the 5.09-MeV level. The 6.86-MeV level decays 100% to the 4.45-MeV level. The 6.19-MeV level decays 88% to the ground state, 6% to the 4.45-MeV level, and 6% to either the 5.25- or 5.33-MeV level. The 6.34-MeV level decays 35% to the 1.98-MeV level, 53% to the 3.91-MeV level, and 11% to the 4.45-MeV level. A number of $\gamma$-ray multipole mixing ratios are also reported. For the 7.10-MeV level, $\frac{{\Gamma }_{\gamma }}{{\Gamma }_{\alpha }}$ was found to be 0.9 ± 0.1. These results are compared with predictions of the model due to Ellis and Engeland.

Vector $i{T}_{11}$ and tensor $Q=\left(\frac{1}{2\sqrt{2}}\right)(T_{20}+\sqrt{6}{T}_{22})$ and $R=\left(\frac{1}{2\sqrt{2}}\right)(T_{20}-\sqrt{6}{T}_{22})$ analyzing powers have been measured in the elastic scattering of 20.4-, 25.2-, and 29.9-MeV polarized deuterons by protons. Comparison is made with Faddeev-equation calculations including spin-dependent nucleon-nucleon interactions. Qualitative agreement is obtained for all the measured quantities only when both the spin-dependent nucleon-nucleon $P$-wave forces and the tensor (${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}-{}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}$) force are taken into account.

The second- and third-order rearrangement contributions to the single-particle energies in ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ are calculated microscopically using the Sprung-de Tourreil force. The $s$-state second-order rearrangement has been evaluated by an energy-averaging procedure proposed by Engelbrecht and Weidenmüller. The second order is as big as the third for $p$ states and much bigger for the $s$ state. Thus it should be included before comparison to experiment. The average rearrangement energy per particle is 10.5 MeV and the $s$-state spreading width 13 MeV.

The photon spectrum from the reaction ${}_{}{}^6{}_{}{}^{}\mathrm{Li}({\pi }^{-},\gamma )$ was measured with high resolution in the 50- to 150-MeV region by using an electron-positron pair spectrometer. The total fraction of pions absorbed radiatively is 4.4 ± 0.6% with branching ratios to the ${}_{}{}^6{}_{}{}^{}\mathrm{He}$ ground state of 0.31 ± 0.04% and to the 1.8-MeV state of 0.15 ± 0.03%, in disagreement with previous experimental results, but in qualitative agreement with theoretical predictions. Evidence for higher excitations is observed in addition to quasifree capture accompanied by neutron emission.

The cross sections for the knockout reactions ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}({\pi }^{-},{\pi }^{-}n){}_{}{}^{11}{}_{}{}^{}\mathrm{C}$ and ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}({\pi }^{+},{\pi }^{+}n+{\pi }^{0}p){}_{}{}^{11}{}_{}{}^{}\mathrm{C}$ were measured over the energy range of 30 to 90 MeV by an activation method. These data together with previous data at higher energies are compared to existing impulse-model calculations. These calculations give general agreement with the shape of the excitation curve for $\sigma \left({\pi }^{-12\mathrm{}}\mathrm{C}\right)$ and poorer agreement for $\sigma \left({\pi }^{+12\mathrm{}}\mathrm{C}\right)$. In the energy range of 30 to 90 MeV, the ratio $\frac{\sigma \left({\pi }^{-12\mathrm{}}\mathrm{C}\right)}{\sigma \left({\pi }^{+12\mathrm{}}\mathrm{C}\right)}$ of the cross sections is 2 to 3 times smaller than the corresponding ratio for pions on nucleons, contrary to the expectations of the simple impulse approximation. This discrepancy has previously been known only near the energy of the (3, 3) resonance.

A systematic investigation of the Glauber multiple-scattering theory is made using eikonal expansion methods developed in two previous papers and shown there to be valid for large-angle scattering. The multiple-scattering eikonal expansion of the $T$ matrix is developed for scattering of a fast projectile interacting with $A$ target particles through local, two-particle interactions. Throughout this study, the fixed scatterer form of the impulse approximation is employed. The expansion starts from the most naive form of the Glauber theory based on the simple eikonal approximation and is shown to produce an equivalent form of the sophisticated version of the Glauber theory in which empirically determined scattering amplitudes are generally used. The multiple-scattering eikonal expansion also produces corrections to the Glauber theory which arise from overlapping interactions, i.e., the fact that the projectile may interact with more than one target particle at a time during its passage through the target. All of these effects are summarized in terms of a generalized phase-shift function for multiple scattering which is a product of two-particle Fourier-Bessel impact parameter amplitudes, $\Pi {i=1\mathrm{}}^A\left[S_F\mathrm{}\right(|\mathrm{}\overrightarrow{\mathrm{b}}-{\overrightarrow{\mathrm{b}}}_i\mathrm{}|\left)\mathrm{}\right]$, one for each target particle, times an overlap factor, $\mathcal{O}\left(\overrightarrow{\mathrm{B}}\right)$, which contains multiparticle phase-shift contributions. The overlap factor depends on the set of coordinates $\overrightarrow{\mathrm{B}}=\{\overrightarrow{\mathrm{b}},{\overrightarrow{\mathrm{b}}}_1,\dots ,{\overrightarrow{\mathrm{b}}}_A\}$ of the projectile and the $A$ fixed scatterers and represents nonadditive contributions to the phase shift. Some approximate forms of the three-particle contribution to the phase shift in $\mathcal{O}\left(\overrightarrow{\mathrm{B}}\right)$ are developed for hadron-nucleus scattering and also for atomic collisions. For nonoverlapping interactions, we find that the sophisticated version of the Glauber theory appears as the sum of the eikonal expansion rather than its leading term, the eikonal approximation. This removes the usual small-angle limitation.

A separable approximation to a local square-well potential is examined. For $s$-wave scattering it is shown that a rank-2 separable potential can be constructed which fits the square-well phase shifts below 300 MeV, and also reproduces the off-shell $T$-matrix elements over the entire region where the $T$ matrix is, itself, appreciable.

Using a three-dimensional reduction of the Bethe-Salpeter equation, we calculate the nucleon-nucleon phase parameters in the energy range of 0-1 GeV of incident laboratory energy. In the elastic energy region the generalized one-boson-exchange potential is assumed for driving terms and in the inelastic energy region an imaginary potential is added. Ingredients of the one-boson-exchange potential are $\pi$, $\rho$, $\omega$, $\eta$, $\delta$, and $S^{*}$ mesons. The dipole form factor is adopted for all the interactions of mesons with nucleons. The broad mass distribution is also taken into account for the $\rho$ and $S^{*}$ resonances. Our imaginary potential which is given by a superposition of regularized Yukawa potentials which have an attractive long-range part and a repulsive core, can describe the peripheral nature of the absorption found in phase-shift analyses. Comparisons of our theoretical phase shifts with those derived by the phase-shift analyses, and of our theoretical observables with experiments at 660 and 970 MeV are made in detail and physical implications are discussed.

[NUCLEAR REACTIONS $N-N$ interaction, 0-1000 MeV; calculated phase parameters, observables.]

The elastic and inelastic proton differential cross sections for ${}_{}{}^{14}{}_{}{}^{}\mathrm{N}$ have been measured at 8.6, 10.6, 12.6, and 14.6 MeV. The variation with energy of the cross sections, as well as the shape of the angular distributions, indicated that the contribution of compound processes to some of the inelastic cross sections was not negligible at these energies. Fits to the elastic scattering data yielded proton optical parameters, which were used in calculating transmission co-efficients for a Hauser-Feshbach calculation of the compound nuclear contribution to the inelastic cross sections. These data corrected for compound effects, and measurements for scattering to the 2.31- and 3.95-MeV levels at 18.0, 21.0, 23.0, and 26.0 MeV reported in the literature, were analyzed with a microscopic coupled-channel calculation. The purpose of the analysis was to find out how sensitive the calculations were to the wave functions used for ${}_{}{}^{14}{}_{}{}^{}\mathrm{N}$, to the values of the effective two-body force, and to the channels coupled. The results were also compared to a simple distorted-wave Born-approximation (DWBA) calculation. Calculations for the 3.95- and 7.03-MeV level were also carried out assuming a weak-coupling model for ${}_{}{}^{14}{}_{}{}^{}\mathrm{N}$, two $p_{\frac{1}{2}}$ nucleons coupled to a ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ core. This collective model gave over-all better agreement with the measured differential cross sections for these levels than the DWBA or the microscopic coupled-channel calculation.

[NUCLEAR REACTIONS ${}_{}{}^{14}{}_{}{}^{}\mathrm{N}\mathrm{}(p,p)\mathrm{}$, ${}_{}{}^{14}{}_{}{}^{}\mathrm{N}(p,p^{\prime })$, $E=8.6,10.6,12.6, \mathrm{and} 14.6$ MeV; measured $\sigma (E,\theta )$; $\theta =15-{165}^{\circ }$; coupled-channel analysis. ${}_{}{}^{14}{}_{}{}^{}\mathrm{N}(p,p^{\prime })$, $E=18,21,23,26\mathrm{}$ MeV, previous data compared with coupled-channel calculation.]

The high-energy part of the spectrum of internal bremsstrahlung (IB) photons accompanying the allowed electron-capture (EC) transition from ${}_{}{}^7{}_{}{}^{}\mathrm{Be}$ to the ground state of ${}_{}{}^7{}_{}{}^{}\mathrm{Li}$ has been measured with a Ge(Li) $\gamma$-ray spectrometer supplied with a pileup rejector. The probability for emission of internal bremsstrahlung photons with energies higher than 523.7 keV per groundstate EC decay of ${}_{}{}^7{}_{}{}^{}\mathrm{Be}$ was determined to be (9.0 ± 0.6)×${10}^{-5\mathrm{}}$. Comparable theoretical results for radiative capture of $1s+2s$ orbital electrons are 8.61 × ${10}^{-5\mathrm{}}$, deduced from the theory of Martin and Glauber, and 8.60 × ${10}^{-5\mathrm{}}$, deduced from recent improved calculations of Intemann. The confirmation of the predicted rate of radiative to ordinary $1s+2s$ capture in ${}_{}{}^7{}_{}{}^{}\mathrm{Be}$ is in good agreement with our recent similar experiment on ${}_{}{}^{51}{}_{}{}^{}\mathrm{Cr}$, but is partly at variance with previous IB $\gamma$-coincidence experiments on ${}_{}{}^7{}_{}{}^{}\mathrm{Be}$. The transition energy of ${}_{}{}^7{}_{}{}^{}\mathrm{Be}$ was determined to be 851 ± 12 keV, in agreement with accepted atomic mass differences.

[RADIOACTIVITY ${}_{}{}^7{}_{}{}^{}\mathrm{Be}$; measured internal bremsstrahlung, $\gamma$; deduced $I_{\mathrm{IB}}$, $Q$. Ge(Li) detector, pileup rejection, extrapolation method.]

A quantitative estimate is made of the changes in pion-nucleus scattering caused by incorporating the Pauli principle in a realistic optical potential. A modification is hypothesized in which the theoretical momentum-space potential is simply multiplied by a Pauli suppression factor. Several forms for this factor are derived, combined with the optical potential, and the Schrödinger equation including these potentials is solved in momentum space. It is found that including the Pauli principle in a realistic optical potential can have significant albeit not drastic effects on calculated low- and intermediate-energy elastic scattering. The effects on differential cross sections are not readily noticeable, except at large angles, whereas the approximately 15% changes in total cross sections can significantly improve the agreement with data.

A method of solving the integral equation of the generator-coordinate method which makes use of the degenerate form of the overlap kernel is applied to three simple solvable problems. When a practical prescription to exclude numerical errors is used, good results are obtained for the lowest and the first excited states. Generator wave functions are presented explicitly as well as the modified generator wave functions by the broadening kernel defined by Griffin and Wheeler. The generator wave functions in the biorthogonal representation are also calculated. The generator wave function modified by using the broadening kernel rather than the original generator wave function is well behaved and seems to reflect the physical nature of the wave function. The numerical version of the Rayleigh-Ritz minimization method is discussed.

The $E\lambda$ effective charge and its momentum transfer dependence in electroexcitation are studied in the framework of the particle-vibration coupling model. In particular, it is predicted that the effective charge for $E2\mathrm{}$ transitions depends strongly on the value of the momentum transfer.

Energy level widths and level shifts of $K^{-}$-mesonic atomic levels corresponding to terminal atomic transitions preceding nuclear capture are calculated for ${}_{}{}^{10}{}_{}{}^{}\mathrm{B}$, ${}_{}{}^{11}{}_{}{}^{}\mathrm{B}$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{31}{}_{}{}^{}\mathrm{P}$, ${}_{}{}^{32}{}_{}{}^{}\mathrm{S}$, and ${}_{}{}^{35}{}_{}{}^{}\mathrm{Cl}$. In this calculation we have considered the effects on the $\overline{K}N$ system arising from the presence of the $Y_0^{*}\mathrm{}\left(1405\right)\mathrm{}$ quasibound state below threshold as well as the momentum distribution of the nucleons in the nucleus. The mass difference between the ${\overline{K}}^{0}n$ and $K^{-}p$ channels is also taken into account. The Klein-Gordon equation for the $K^{-}$ meson is solved by an iterative procedure in the presence of the nuclear optical potential and the Coulomb potential. Results are seen to compare quite favorably with experimental values given by Backenstoss et al.

The ($p,d$) and ($p,t$) reactions on ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}$ have been studied using the 24.4-MeV polarized proton beam of the Saclay cyclotron. Differential cross sections and analyzing powers (asymmetries) have been measured for eight deuteron and seven triton groups. These data are also compared with distorted-wave Born-approximation (DWBA) calculations as a test of their validity for a light nucleus. Some of the deuteron angular distributions are not accounted for by the calculations, and in fact for some transitions the results suggest that two-step processes might be involved. The reliability of the spectroscopic information deduced from the ($p,d$) reaction is thoroughly discussed. In the triton optical potential a spin-orbit term with a well depth of about 4 MeV is necessary in order to fit the ($p,t$) analyzing power data for the transitions with $L=0\mathrm{}$ and $L=1\mathrm{}$. For higher angular momentum transfers acceptable fits are obtained only if a radial cutoff is used. The analysis of the transition to the 8.88-MeV ($J^{\pi }=2\mathrm{}$) state in ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ indicates that the $d$ state component in the triton wave function may be important for ($p,t$) reactions leading to unnatural-parity states.

The cross section of the bremsstrahlung process ${}_{}{}^4{}_{}{}^{}\mathrm{He}(\alpha ,\alpha \alpha )\gamma$ has been measured at five energies between 11.4 and 13.5 MeV with detector lab angles of ${\theta }_1={\theta }_2={37}^{\circ }$, and at 9.35 MeV with ${\theta }_1={\theta }_2={35}^{\circ }$. The results agree well with predictions based on the work of Feshbach and Yennie, Signell, and Green. No significant enhancement is found at the final state energy corresponding to the 2.9 MeV resonance in ${}_{}{}^8{}_{}{}^{}\mathrm{Be}$. A single measurement of the process ${}_{}{}^4{}_{}{}^{}\mathrm{He}(\tau ,\alpha \alpha )\gamma$ made at 7.4 MeV with ${\theta }_1={\theta }_2={37}^{\circ }$ is also in good agreement with theory. Details of the experimental technique and of finite geometry corrections are discussed.

[NUCLEAR REACTIONS ${}_{}{}^3{}_{}{}^{}\mathrm{H}\mathrm{e}-\alpha$ and $\alpha -\alpha$ bremsstrahlung, $E=7.4,9.35,11.4-13.5\mathrm{}$ MeV, measured $\frac{d^2\sigma }{d{\Omega }_{1}d{\Omega }_2}$.]

The Strutinsky method for extracting the shell corrections from the Nilsson model has been interpreted in the framework of constrained Hartree-Fock theory. Cast in this form, the shell corrections, the smooth part of the energy and those terms ignored, can all be explicitly calculated. The results of such a calculation are presented here using a procedure that circumvents the necessity for imposing constraints. It is shown that the neglected terms can be as large as 15% of the usual shell corrections, in agreement with the results of related studies.

The ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}(\alpha ,d){}_{}{}^{26}{}_{}{}^{}\mathrm{Al}$ reaction has been studied at energies $E_{\alpha }=24.0-25.5\mathrm{}$ and $E_{\alpha }=50\mathrm{}$ MeV. Excitation functions and angular distributions were obtained from 24.0-25.5 MeV for the first three $T=0\mathrm{}$ levels in ${}_{}{}^{26}{}_{}{}^{}\mathrm{Al}$, and angular distributions for all the $T=0\mathrm{}$ states with $E_x\le 3.074$ MeV were taken at 50 MeV. Nondirect structure was seen in the excitation functions of the low-energy data, and distorted-wave Born-approximation (DWBA) calculations were not successful in reproducing the changes in shape with energy of the observed angular distributions. DWBA calculations with shell-model wave functions were also made for the 50-MeV data and were largely successful in fitting the observed angular distributions. Spins and parities were assigned to ${}_{}{}^{26}{}_{}{}^{}\mathrm{Al}$ levels based on the determination of the $L$ transfers, and in all cases agreed with previous results.

[NUCLEAR REACTIONS ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}(\alpha ,d)$, $E=24.0-25.5\mathrm{}$; 50 MeV; measured $\sigma (E,\theta )$. Enriched target, DWBA analysis, deduced $L$ transfers; $\theta =5-{60}^{\circ }$, $\Delta \theta =5^{\circ }$.]