The order ${\mathrm{eg}}^{ 2}$ and the ${\Delta }_{33}$ excitation parts of the one-pion-exchange neutron-proton electromagnetic current have been included in a calculation of the $\mathrm{np}$ bremsstrahlung cross section. The order ${\mathrm{eg}}^{ 2}$ exchange current was found to have a large effect. We have shown that the one-pion-exchange current operator can be very well approximated by its zero-photon-energy limit for the existing experiments. By comparison with other work which uses this same approximation but which includes all nonlocal and exchange terms of the potential, we have concluded that contributions to the two-body current other than one-pion exchange are also important.

[NUCLEAR REACTIONS Order ${\mathrm{eg}}^2$ and ${\Delta }_{33}$ excitation one-pion-exchange currents included in $\mathrm{np}$ bremsstrahlung calculation at 130 and 208 MeV.]

The photoproduction of neutral pions near threshold from complex nuclear targets in investigated. Momentum dependent terms in the photoproduction operator and a two-nucleon charge exchange production mechanism are included. Various selection rules related to the quantum numbers of the nuclear states involved are discussed. Differential cross sections for pion energies <7.5 MeV from ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ are calculated using realistic shell model wave functions. The effects of distortion of the pion wave function are found to be small, but the two-nucleon charge exchange mechanism is found to dominate the coherent production from $J\ne 0$ targets near threshold.

[NUCLEAR REACTIONS ($\gamma$, ${\pi }^0$) near threshold; realistic photoproduction operator; shell model wave functions; calculated differential cross section on ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$.]

We obtain expressions for proton-nucleus elastic scattering amplitudes in the Glauber approximation which include the effects of spin dependence and Coulomb interactions. These effects are shown to be important for determining the neutron and matter radii for all target nuclei. The results for neutron radii are compared with theoretical predictions.

[NUCLEAR REACTIONS ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{40,48}{}_{}{}^{}\mathrm{Ca}$, ${}_{}{}^{90}{}_{}{}^{}\mathrm{Zr}$, ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}\mathrm{}(p,p)\mathrm{}$, $E=1\mathrm{}$ GeV; calculated $\sigma \left(\theta \right)$, determined radii of neutron density distributions.]

The angular distributions of deuterons for $7^{\circ }\le {\theta }_{\mathrm{c}.\mathrm{m}.\mathrm{}}\le {60}^{\circ }$ have been measured for 56.3 MeV neutrons incident on targets of ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ and ${}_{}{}^7{}_{}{}^{}\mathrm{Li}$. Transitions corresponding to the pickup of $1p$-shell and $1s$-shell protons are strong with the latter exhibiting structure indicative of groups of excited states in ${}_{}{}^5{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^6{}_{}{}^{}\mathrm{He}$. In ${}_{}{}^5{}_{}{}^{}\mathrm{He}$ these have excitation energies of 16.7, 18.5, and 20.5 MeV, and in ${}_{}{}^6{}_{}{}^{}\mathrm{He}$ an average excitation energy of 15.6 MeV. The spectroscopic factors deduced from the measurements are in good agreement with theoretical values where the latter are available.

[NUCLEAR REACTIONS ${}_{}{}^6{}_{}{}^{}\mathrm{Li}\mathrm{}(n,d)\mathrm{}$, ${}_{}{}^7{}_{}{}^{}\mathrm{Li}\mathrm{}(n,d)\mathrm{}$; 56.3 MeV neutrons, measured $\sigma \left(\theta \right)$, deduced spectroscopic factors.]

In a recent paper a general impulse approximation formalism for ($p,\pi$) reactions in nuclei was developed which incorporated distortion, spin, and antisymmetrization effects. Here we discuss two further considerations in connection with this model with particular reference to its application to the reactions $\mathrm{pd}\to t{\pi }^{+}$ and $\mathrm{pd}\to {}_{}{}^3{}_{}{}^{}\mathrm{He}{\pi }^0$. First, realistic wave functions for $t$ and ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ which produce the dip in the electromagnetic form factor are incorporated. We find results qualitatively similar to those found previously when simple analytic wave functions were used. This is in contrast to the work of Locher and Weber who found that the dip in the electromagnetic form factor produced a similar dip in the ($p,\pi$) cross section. In the present model correct antisymmetrization of the wave functions and inclusion of spin components of the wave functions, in particular the $D$ state of the deuteron, fill the dip. Second, we examine the factors which affect the energy at which a bump due to the $\Delta \mathrm{}\left(1232\right)\mathrm{}$ resonance would appear in the ($p,\pi$) cross section. This is motivated by a recent experiment which found no evidence of a bump at 450 MeV where it would be expected from simple kinematic arguments. We find that distortion and form factor effects both tend to push the bump to lower energies and that the net result is that the model predicts a smooth falloff with energy of the fixed angle cross section above 350 MeV, in qualitative agreement with the experiment, and a bump below 350 MeV, where there are no data. The physical mechanisms responsible for the shift, and some possible approaches which might improve quantitative agreement, are then briefly discussed.

[NUCLEAR REACTIONS ($p,\pi$) reactions in distorted-wave impulse approximation. Effects of realistic wave functions on cross section and of form factors and distortion on resonance position. Differential cross section and energy dependence of fixed angle cross section for ${}_{}{}^2{}_{}{}^{}\mathrm{H}(p,{\pi }^{+}){}_{}{}^3{}_{}{}^{}\mathrm{H}$ and ${}_{}{}^2{}_{}{}^{}\mathrm{H}(p,{\pi }^0){}_{}{}^3{}_{}{}^{}\mathrm{He}$ at 250-800 MeV.]

The transition operator for threshold charged-pion photoproduction and radiative pion capture is considered as the nonrelativistic limit of the analogous relativistic operator. We demonstrate that the resulting nonrelativistic threshold photoproduction cross sections are consistent with the corresponding relativistic results. Including a model dependent $N^{*}$ anomalous moment contribution to the cross section of order ${\left(\frac{k}{M}\right)}^2$ compared with the static limit, we calculate a rate for ${\pi }^{-}p\to \gamma n$ in reasonable agreement with the known charge-exchange scattering length combination $|a_1-a_3|$ and the measured Panofsky ratio $P_1$.

[NUCLEAR REACTIONS $p({\pi }^{-},\gamma )n$, $n({\pi }^{+},\gamma )p$; calculated threshold cross section, capture rate; nonrelativistic operator.]

The transition operator for threshold charged-pion photoproduction and radiative pion capture is utilized in its nonrelativistic limit to calculate the absolute rate for the ${\pi }^{-}d\to \gamma \mathrm{nn}$ reaction from a $1s$ atomic orbital in the impulse approximation. Sensitivity to assumptions concerning the initial state of the ${\pi }^{-}d$ system and the short-range character of the $n-n$ wave function in the final state is discussed, as is the relevance of the absolute rate to the determination of the $n-n$ scattering length from this reaction.

[NUCLEAR REACTIONS ${}_{}{}^2{}_{}{}^{}H({\pi }^{-},\gamma )2n$; calculated capture rate; $n-n$ scattering length; impulse approximation.]

We discuss the 50 MeV data for the elastic scattering of pions by ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ and show that a careful treatment of off-shell effects and the effects of pion absorption, in the context of a multiple-scattering theory, is able to explain the data. We present a discussion of the importance of nuclear binding effects and describe how the Fermi motion influences the calculation of the optical potential. (In this work we have performed a complete integration over the variables which describe the Fermi motion of the target nucleons.) It may be seen from this discussion how the neglect of these features in the fixed scatterer approximation leads to large discrepancies between the results of theoretical calculations and the experimental data.

[NUCLEAR REACTIONS Low-energy $\pi$-nucleus elastic scattering; 50 MeV $\pi -{}_{}{}^4{}_{}{}^{}\mathrm{He}$ and $\pi -{}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ differential cross sections. Influence of Fermi motion, nuclear binding, and pion absorption.]

We present and discuss here the results of a calculation of the real part of the potential energy of a proton within nuclear matter, making use of the realistic Reid soft-core nucleon-nucleon interaction, which we justify in the intermediate-energy range. The problem of evaluating the volume of the real part of the proton-nucleus optical-model potential as a function of incident energy is solved with the help of a number of techniques previously developed in the literature of nuclear-matter theory. We find good agreement with results of various phenomenological studies over an energy range of 100 to 1050 MeV incident proton energy for targets ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ and ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$, which we adopt as test cases. Furthermore, we consider a number of higher-order corrections to our initial result; all of these have minimal effect in the energy regime we have considered except for the third-order correction proposed by Rajaraman, as we demonstrate and discuss. Finally, we show that the conventional approach to the nucleon-nucleus potential at medium energy, the so-called Rayleigh-Lax potential, which makes use of a straightforward impulse approximation and an empirical nucleon-nucleon scattering amplitude, is in gross disagreement both with phenomenological findings concerning the energy dependence of the optical potential and with results of the present calculations. We make an effort to shed some light on the physical reasons for the failure of the Rayleigh-Lax approach, as opposed to the success of other approaches relying upon realistic nucleon-nucleon interactions or phase shifts rather than an empirical medium-energy nucleon-nucleon $T$ matrix.

[NUCLEAR REACTIONS ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}\mathrm{}(p,p)\mathrm{}$, ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}\mathrm{}(p,p)\mathrm{}$, $E=100-1050\mathrm{}$ MeV; nuclear matter approach used to estimate volume of real part proton-nucleus optical model potential, multiple-scattering formalism, numerical estimates of various higher-order terms.]

The ${}_{}{}^{239}{}_{}{}^{}\mathrm{Pu}$ half-life was measured both by $\alpha$ counting with an intermediate geometry proportional counter and by mass spectrometric measurements of the rate of ${}_{}{}^{235}{}_{}{}^{}\mathrm{U}$ formation, using isotopic dilution with titrated ${}_{}{}^{233}{}_{}{}^{}\mathrm{U}$ and ${}_{}{}^{238}{}_{}{}^{}\mathrm{U}$ spikes. The plutonium concentration was measured by oxidation-reduction titration, using NBS-certified plutonium as a reference standard. The results from the two measurements were statistically consistent and their average gives for the ${}_{}{}^{239}{}_{}{}^{}\mathrm{Pu}$ half-life the value 24131 ± 16 yr, corresponding to a specific activity of (1.3758 ± 0.0009) × ${10}^{11}$ dis/min per gram ${}_{}{}^{239}{}_{}{}^{}\mathrm{Pu}$ (one standard deviation error). This result agrees with some recent measurements, but is in marked disagreement with a number of earlier ones.

[RADIOACTIVITY ${}_{}{}^{239}{}_{}{}^{}\mathrm{Pu}$ half-life, specific activity; measured: $\alpha$ activity, intermediate geometry $\alpha$ counter; ${}_{}{}^{235}{}_{}{}^{}\mathrm{U}$ growth rate, isotope dilution and mass spectrometer; plutonium concentration, titration.]

${}_{}{}^{59}{}_{}{}^{}\mathrm{Mn}$ was produced using the ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}({}_{}{}^{13}{}_{}{}^{}\mathrm{C},pn){}_{}{}^{59}{}_{}{}^{}\mathrm{Mn}$ reaction at $E_{{13}_{\mathrm{C}}}=26\mathrm{}$ MeV. $\gamma$ rays from the decay of ${}_{}{}^{59}{}_{}{}^{}\mathrm{Mn}$ were observed in singles and coincidence experiments using large Ge(Li) detectors after the targets were transferred to a shielded counting area by a multiple-target pneumatic transfer system. Assignment of $\gamma$ rays to transitions in ${}_{}{}^{59}{}_{}{}^{}\mathrm{Mn}$ was confirmed by performing an ${}_{}{}^{58}{}_{}{}^{}\mathrm{Fe}(d,p\gamma ){}_{}{}^{59}{}_{}{}^{}\mathrm{Fe}$ experiment. The half-life of ${}_{}{}^{59}{}_{}{}^{}\mathrm{Mn}$ was determined to be 4.6 ± 0.1 s. The results of this work limit the ground-state spin of ${}_{}{}^{59}{}_{}{}^{}\mathrm{Mn}$ to ${5/2}^{-}$ or ${3/2}^{-}$. Theoretical and systematic considerations favor ${5/2}^{-}$. The total $\beta$-decay energy, $Q_{\beta }$, was measured to be 5.2 ± 0.1 MeV, which corresponds to a mass excess of -55.5 ± 0.1 MeV.

[RADIOACTIVITY ${}_{}{}^{59}{}_{}{}^{}\mathrm{Mn}$: measured $T_{\frac{1}{2}}$, $E_{\gamma }$, $I_{\gamma }$, $\gamma -\gamma$ coin, $E_{\beta }$, $\beta -\gamma$ coin; deduced decay scheme, $log\mathrm{ft}$ limits, $Q_{\beta }$, g.s. $J^{\pi }$ limit. Enriched targets, Ge(Li) detectors, plastic scintillator, multiple rabbit.]

Mineral monazite from Malaysia has been investigated by energy dispersive x-ray analysis employing three different methods: x-ray, relativistic electron and proton induced excitation. An upper limit of 15 ppm has been placed on the possible concentration of the superheavy element with $Z=126\mathrm{}$.

[RADIOACTIVITY Monazite (Ce, La, Th)P${\mathrm{O}}_4$; measured x ray, electron and proton induced x-ray spectra. Deduced upper limit for concentration of element $Z=126\mathrm{}$.]

Radioactive sources of ${}_{}{}^{164}{}_{}{}^{}\mathrm{Lu}$ were produced in ${}_{}{}^{14}{}_{}{}^{}\mathrm{N}$ bombardments of ${}_{}{}^{155}{}_{}{}^{}\mathrm{Gd}$. A helium-jet capillary system was used to transport product nuclei to a shielded area suitable for $\gamma$-ray counting. Both singles and coincidence measurements were made. The half-life of ${}_{}{}^{164}{}_{}{}^{}\mathrm{Lu}$ was found to be 3.17 ± 0.03 min, in agreement with earlier measurements. This state in ${}_{}{}^{164}{}_{}{}^{}\mathrm{Lu}$ is thought to have a low spin, most probably a value of 2; there is also some evidence for the existence of a high-spin isomer, possibly of shorter half-life, based on weak population of high-spin states in ${}_{}{}^{164}{}_{}{}^{}\mathrm{Yb}$. Fifteen levels were placed in ${}_{}{}^{164}{}_{}{}^{}\mathrm{Yb}$, 12 of them previously unknown. The $2^{+}$ and $3^{+}$ members of the $\gamma$ vibrational band are located at 863.9 and 1004.2 keV; there are also candidates for the $4^{+}$ and $5^{+}$ members. An even-odd shift in the $\gamma$ band, both in level energies and mixing parameters, is observed with a magnitude greater than in the previously studied ${}_{}{}^{166,168}{}_{}{}^{}\mathrm{Yb}$ cases. A $K=0\mathrm{}$ $2^{+}$ level is placed at 1073.8 keV, and its effect on the $\gamma$ band is discussed. From an analysis of the population trends of levels in ${}_{}{}^{164,166,168}{}_{}{}^{}\mathrm{Yb}$ from Lu decays, we make arguments concerning the available proton and neutron Nilsson orbitals.

[RADIOACTIVITY ${}_{}{}^{164}{}_{}{}^{}\mathrm{Lu}$ [from ${}_{}{}^{155}{}_{}{}^{}\mathrm{Gd}({}_{}{}^{14}{}_{}{}^{}\mathrm{N},5n)$, $E=79\mathrm{}$ MeV]. Measured $T_{\frac{1}{2}}$, $E_{\gamma }$, $I_{\gamma }$, $\gamma \gamma$ coin. Deduced ${}_{}{}^{164}{}_{}{}^{}\mathrm{Yb}$ levels, $J$, $\pi$, $log\mathrm{ft}$.]

A fairly detailed analysis is given of the available experimental data on ${}_{}{}^{12}{}_{}{}^{}\mathrm{B}\to {}_{}{}^{12}{}_{}{}^{}\mathrm{C}{e}^{-}{\overline{\nu }}_e$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{N}\to {}_{}{}^{12}{}_{}{}^{}\mathrm{C}{e}^{+}{\nu }_e$, and ${\mu }^{-}{}_{}{}^{12}{}_{}{}^{}\mathrm{C}\to {\nu }_{\mu }{}_{}{}^{12}{}_{}{}^{}\mathrm{B}$ making use of the "elementary-particle" treatment. Our results indicate that the variation with energy of the asymmetry coefficients obtained by Sugimoto et al. cannot be reconciled with the observed value of the muon capture rate on the hypothesis (A) of gross violation of ("strong") conserved vector current and absence of second-class axial currents and can be reconciled only with great difficulty on the alternative hypothesis (B) of validity of (strong) conserved vector current and presence of relatively large second-class axial currents. On the other hand, a good fit to the observed muon capture rate can be obtained on the hypothesis (C) of (strong) conserved vector current, ("suitably corrected") partially conserved axial vector current, and absence of second-class axial currents. This last hypothesis leads, of course, to a prediction for the variation with energy of the asymmetry coefficients in contradiction to the results of Sugimoto et al. We also find that the magnitude of the recoil ${}_{}{}^{12}{}_{}{}^{}\mathrm{B}$ polarization obtained by Possoz et al. agrees with that calculated on hypothesis (C) but not with those calculated on hypotheses (A) or (B).

[RADIOACTIVITY ${}_{}{}^{12}{}_{}{}^{}\mathrm{B}\to {}_{}{}^{12}{}_{}{}^{}\mathrm{C}{e}^{-}{\overline{\nu }}_e$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{N}\to {}_{}{}^{12}{}_{}{}^{}\mathrm{C}{e}^{+}{\nu }_e$, and ${\mu }^{-}{}_{}{}^{12}{}_{}{}^{}\mathrm{C}\to {\nu }_{\mu }{}_{}{}^{12}{}_{}{}^{}\mathrm{B}$; theoretical analysis of available experimental data. Are there second-class currents and is CVC valid?]

The decay of 5.2-min ${}_{}{}^{128}{}_{}{}^{}\mathrm{La}$ to levels in ${}_{}{}^{128}{}_{}{}^{}\mathrm{Ba}$ has been investigated by $\gamma$-ray and conversion-electron spectroscopy. Of 78 $\gamma$ rays assigned to the decay, 69 have been placed in a decay scheme consisting of 31 excited states. The ${}_{}{}^{128}{}_{}{}^{}\mathrm{La}$ ground state is probably $5^{-}$ and is interpreted as the ($\nu {\frac{7}{2}}^{-}\mathrm{}\left[523\right]+\pi {\frac{3}{2}}^{+}\mathrm{}\left[411\right]\mathrm{}$) configuration. $\mathrm{EC}+{\beta }^{+}$ decays with $log\mathrm{ft}$ values in the allowed range to states at 2425.5 and 2878.3 keV are explained by ${\left(\nu {\frac{7}{2}}^{-}\mathrm{}\right[523]+\nu {\frac{1}{2}}^{+}\mathrm{}[411\left]\right)}_{4-}$ and ${\left(\nu {\frac{7}{2}}^{-}\mathrm{}\right[523]+\nu {\frac{3}{2}}^{+}\mathrm{}[402\left]\right)}_{5-}$ configurations, respectively. A quasi-$\gamma$ band is proposed with levels at 884.5 keV ($2^{+}$), 1324.5 keV ($3^{+}$), 1372.4 keV ($4^{+}$), and 1931.4 keV ($5^{+}$). A state at 1939.4 keV is possibly the $6^{+}$ band member. The properties of ${}_{}{}^{128}{}_{}{}^{}\mathrm{Ba}$ were calculated using the microscopic boson-expansion code of Kishimoto and Tamura and found to be in rather good agreement with experiment. The small experimentally observed splitting (48 keV) between the $3_{\gamma }$ and $4_{\gamma }$ states is a signature of $\gamma$ instability. The prolate-oblate difference, consistent with the experimental data, is estimated to be 0.5 MeV, a value in good agreement with potential-surface calculations.

[RADIOACTIVITY ${}_{}{}^{128}{}_{}{}^{}\mathrm{La}$ [from ${}_{}{}^{118}{}_{}{}^{}\mathrm{Sn}({}_{}{}^{14}{}_{}{}^{}\mathrm{N},4n)$], measured $T_{\frac{1}{2}}$, $E_{\gamma }$, $I_{\gamma }$, $I_{\mathrm{ce}}$, $\gamma -\gamma$ coin; deduced $log\mathrm{ft}$. ${}_{}{}^{128}{}_{}{}^{}\mathrm{Ba}$ deduced levels, ICC, $J$, $\pi$. Enriched target, Ge(Li) and Si(Li) detectors.]

The levels of ${}_{}{}^{95}{}_{}{}^{}\mathrm{Mo}$ populated in the decay of $20-h{}_{}{}^{95}{}_{}{}^{}\mathrm{Tc}_{}^g{}_{}{}^{}$ and 61-day ${}_{}{}^{95}{}_{}{}^{}\mathrm{Tc}_{}^m{}_{}{}^{}$ decay have been studied using singles and Compton suppression Ge(Li) $\gamma$-ray spectrometers and a Si(Li) conversion electron spectrometer. We find 20 $\gamma$ rays in the decay of ${}_{}{}^{95}{}_{}{}^{}\mathrm{Tc}_{}^g{}_{}{}^{}$ and 26 $\gamma$ rays in the decay of ${}_{}{}^{95}{}_{}{}^{}\mathrm{Tc}_{}^m{}_{}{}^{}$. Using these data we identify 16 levels in ${}_{}{}^{95}{}_{}{}^{}\mathrm{Mo}$ and measure a total positron branch of (0.24 ± 0.03)% for the decay of ${}_{}{}^{95}{}_{}{}^{}\mathrm{Tc}_{}^m{}_{}{}^{}$. A cluster-vibrational-field coupling model calculation has been performed for ${}_{}{}^{95}{}_{}{}^{}\mathrm{Mo}$ and rather good agreement is found between the model predictions and our experimentally observed level ordering and level decay branching ratios.

[RADIOACTIVITY ${}_{}{}^{95}{}_{}{}^{}\mathrm{Tc}_{}^g{}_{}{}^{}$, ${}_{}{}^{95}{}_{}{}^{}\mathrm{Tc}_{}^m{}_{}{}^{}\left[\mathrm{from}\mathrm{Mo}\mathrm{}\right(p,xn), {}_{}{}^{93}{}_{}{}^{}\mathrm{Nb}(\alpha ,2n\left)\right]$; measured $E_{\gamma }$, $I_{\gamma }$, $E_{\mathrm{c}\mathrm{e}-}$, $I_{\mathrm{c}\mathrm{e}-}$, $T_{\frac{1}{2}}$; deduced $log\mathrm{ft}$. ${}_{}{}^{95}{}_{}{}^{}\mathrm{Mo}$ deduced levels, $J$, $\pi$, $B\left(M_1\right)$, $B\left(E_2\right)$ Ge(Li) Si(Li) detectors, enriched and natural targets.]

Experimental and theoretical cross sections are presented for the ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$(${}_{}{}^3{}_{}{}^{}\mathrm{He}$, ${}_{}{}^3{}_{}{}^{}\mathrm{He}$${}_{}{}^3{}_{}{}^{}\mathrm{He}$)${}_{}{}^3{}_{}{}^{}\mathrm{H}$ and ${}_{}{}^6{}_{}{}^{}\mathrm{Li}({}_{}{}^3{}_{}{}^{}\mathrm{He},t{}_{}{}^3{}_{}{}^{}\mathrm{He}){}_{}{}^3{}_{}{}^{}\mathrm{He}$ reactions for the symmetric angle pairs 20°-20°, 28.3°-28.3°, and 35°-35°. The theoretical cross sections are calculated in a three-body model where the trions (i.e., mass-3 nuclei) are treated as elementary particles with ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ being a ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^3{}_{}{}^{}\mathrm{H}$ bound state. The trion-trion interaction is represented by $S$ wave separable potentials with the breakup cross sections calculated with the three-body Haftel-Ebenhöh code. the Coulomb interaction is taken into account by fitting the separable potential parameters to the trion-trion scattering data and is included approximately in the breakup code. The experimental cross sections are compared with both the plane-wave impulse approximation and the three-body model predictions. The plane-wave impulse approximation predicts both the shapes and magnitudes poorly (10 to 20 times experiment). Without Coulomb corrections the three-body model gives good agreement with experiment for the shapes of the spectra with the magnitudes generally being about 40% of experiment for ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$(${}_{}{}^3{}_{}{}^{}\mathrm{He}$, ${}_{}{}^3{}_{}{}^{}\mathrm{He}$${}_{}{}^3{}_{}{}^{}\mathrm{He}$)${}_{}{}^3{}_{}{}^{}\mathrm{H}$ and about 80% for ${}_{}{}^6{}_{}{}^{}\mathrm{Li}({}_{}{}^3{}_{}{}^{}\mathrm{He},t{}_{}{}^3{}_{}{}^{}\mathrm{He}){}_{}{}^3{}_{}{}^{}\mathrm{He}$. The Coulomb corrections improve the magnitudes predicted by the three-body model but not the shapes. It is observed that for these reactions $S$ wave separable potentials describe the breakup data much better than they do the two-body trion-trion scattering data. This result should encourage further three-body treatment of these and similar reactions.

[NUCLEAR REACTIONS ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$(${}_{}{}^3{}_{}{}^{}\mathrm{He}$, ${}_{}{}^3{}_{}{}^{}\mathrm{He}$${}_{}{}^3{}_{}{}^{}\mathrm{He}$)${}_{}{}^3{}_{}{}^{}\mathrm{H}$, ${}_{}{}^6{}_{}{}^{}\mathrm{Li}({}_{}{}^3{}_{}{}^{}\mathrm{He},t{}_{}{}^3{}_{}{}^{}\mathrm{He}){}_{}{}^3{}_{}{}^{}\mathrm{He}$, $E=45\mathrm{}$ MeV, tractable Faddeev calculations $\sigma (E,{\theta }_1{\theta }_2)$, Coulomb corrections.]

The question of validity of the pairing-plus-quadrupole model to well-deformed nuclei is examined. The calculated energy levels, $B\left(E2\right)$ values, $\frac{E2}{M1}$ mixing ratios, and electromagnetic moments are compared with the experimental values of ${}_{}{}^{156}{}_{}{}^{}\mathrm{Gd}$ which is used as a test case. In particular, the structure of the second excited $K=0\mathrm{}$ band and the $K=4\mathrm{}$ band are examined in terms of model wave functions. For the sake of comparison and discussion, some results are also given for ${}_{}{}^{154}{}_{}{}^{}\mathrm{Gd}$.

[NUCLEAR STRUCTURE ${}_{}{}^{156}{}_{}{}^{}\mathrm{Gd}$—calculated level energies, $B\left(E2\mathrm{}\right)\mathrm{}$ values, quadrupole moments, magnetic moments, $E2\mathrm{} \mathrm{and} M1\mathrm{}$ matrix elements, $\delta \left(\frac{E2\mathrm{}}{M1\mathrm{}}\right)$, ${\beta }_{\mathrm{rms}}$, ${\gamma }_{\mathrm{rms}}$. ${}_{}{}^{154}{}_{}{}^{}\mathrm{Gd}$—calculated $B\left(E2\mathrm{}\right)\mathrm{}$ values. Comparison with experiment whenever possible.]

A systematic study of the low-energy properties of odd-mass I nuclei is performed in terms of the Alaga model. Previous theoretical works are updated, according to the present level of experimental information, and extended to lighter isotopes. The residual interaction among the valence protons is approximated by both the pairing force and the surface $\delta$ interaction. We conclude that the refinements introduced by the last interaction are of little importance in the description of low-energy states. Excitation energies, one-body reaction amplitudes, dipole and quadrupole moments, and $B\left(M1\right)$ and $B\left(E2\right)$ values are calculated and compared with the corresponding experimental data. Also, a few allowed $\beta$ transitions are briefly discussed.

[NUCLEAR STRUCTURE ${}_{}{}^{123,125,127,129,131,133}{}_{}{}^{}\mathrm{I}$; calculated levels $J$, $\pi$, $B\left(E2\mathrm{}\right)\mathrm{}$, $B\mathrm{}\left(M1\mathrm{}\right)\mathrm{}$, $Q$, $\mu$, and $S$. Alaga model. Pairing and surface $\delta$ interaction.]

It is shown that when a potential $V\left(r\right)\mathrm{}$, which is local and repulsive for small $r$ and singular at $r=0\mathrm{}$, is truncated into a potential bounded by $V\left(\epsilon \right)$ where $\epsilon$ is small, the corresponding two-body binding energy $E\left(\epsilon \right)$ differs from the exact binding energy by less than $\left(\mathrm{const}\right)\times {\left[V\right(\epsilon \left)\right]}^{\frac{-1\mathrm{}}{2}}$. A variational procedure is derived from this result.

[NUCLEAR STRUCTURE Singular repulsion cutoff parameter is variational parameter for modified Rayleigh-Ritz principle.]