Evaluation of the nuclear charge form factor requires the evaluation of a many-body matrix element. In this paper it is shown how this many-body matrix element can be reduced to an integral over single-particle matrix elements when hyperspherical coordinates are used.

Protons from the ${}_{}{}^{14}{}_{}{}^{}\mathrm{C}\mathrm{}(d,p)\mathrm{}{}_{}{}^{15}{}_{}{}^{}\mathrm{C}$ reaction were analyzed with a magnetic spectrograph. Data were taken at the bombarding energies of 12.0, 13.0, and 14.0 MeV at the observation angles of 60, 90, and 120°. Nine states were found in ${}_{}{}^{15}{}_{}{}^{}\mathrm{C}$ between the ground state and the 7.35-MeV region of excitation. Excitation energies and uncertainties in keV are 744.1±2, 3105.3±5, 4221.1±3, 6428.1±7, 6539.8±5, 6844.9±5, 6882.4±5, 7097.2±6, and 7351.3±6. Widths of approximately 42 and 61 keV were measured for the levels at 3105.3 and 6428.1 keV, respectively. Uncertainties in excitation energies are reduced by almost an order of magnitude from previous work.

For nucleons scattered by odd-mass nuclei, a spin-spin interaction consisting of spherical and tensor terms is added to the optical potential. The effect of this interaction on the scattering amplitude is calculated using distorted-wave Born approximation. The shape and strength of the spherical term are based on an approximate microscopic treatment, while the tensor term is treated phenomenologically. Theoretical calculations are carried out for several observable quantities. The calculated depolarizations are compared with the limited data available for 50- and 20-MeV protons. Because of large uncertainties in the 50-MeV data it is not possible to clearly assess the role played by each of the spin-spin terms. However, for the 20-MeV data, the depolarizations calculated using the spherical term alone are found to be consistently different from the measured values. These suggest that contributions from the tensor term are important. The effects of the spin-spin interaction on the differential cross section, polarization, asymmetry, and rotation parameters are found to be small. It is shown that measurements of the angular distribution of the depolarization parameter could be useful in determining the strength of the tensor interaction.

The simultaneous study of the ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$+${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$+${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, and ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$+${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ total reaction cross sections at low energy strongly suggests that the optical potential of the heavy ions has a central soft repulsive core. Conventional Woods-Saxon potentials can fit the ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$+${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ and ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$+${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ reactions on the one hand or the ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$+${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ reaction on the other but only the potentials with the repulsive core can fit all three reactions with the parameters varying smoothly from reaction to reaction. These reactions also determine accurately the surface thickness of the imaginary part of the potential; it varies from 0.35 fm for the ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$+${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ reaction to 0.55 fm for the ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$+${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ reaction. At very low energies most of the absorption takes place while the two nuclei are far from each other.

The total and differential cross sections were calculated for ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}({\pi }^{+},{\pi }^0){}_{}{}^{13}{}_{}{}^{}\mathrm{N}$, considering the effects of double and triple scatterings as well as pion absorption. An attempt is also made to explain the extremely small total cross section for ${}_{}{}^{14}{}_{}{}^{}\mathrm{N}({\pi }^{+},{\pi }^0){}_{}{}^{14}{}_{}{}^{}\mathrm{O}$ compared with other cases ${}_{}{}^{10}{}_{}{}^{}\mathrm{B}$, ${}_{}{}^{11}{}_{}{}^{}\mathrm{B}$, ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}$, and ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}$.

A Nilsson-like calculation of single-particle energies is carried out in a manner so that their sum may be exactly related to the total energy. The proposal of Strutinsky that the fluctuations in the energy are contained mainly in this sum, is shown to be quantitatively inaccurate. This is achieved by an explicit calculation of the fluctuations of other terms in the energy.

The neutron total cross section of oxygen is measured in the energy regions from 0.6 to 0.9, 1.12 to 1.16, and 1.39 to 4.33 MeV. More than half of this region is surveyed with energy resolution and energy steps of about 2.5 keV. On the basis of these data and supplementary published data on the ($n,n$) angular distributions and the ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}(\alpha ,n)$ reaction, the ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}$ level energies in MeV and, in parenthesis, $J^{\pi }$ values and center-of-mass total widths, ${\Gamma }_T={\Gamma }_n+{\Gamma }_{\alpha }$ in keV, are: 5.696 (${\frac{7}{2}}^{-}$, 3.4); 5.731 (not ${\frac{1}{2}}^{+}$, <1.0); 5.867 (${\frac{3}{2}}^{+}$, 6.6); 5.937 (${\frac{1}{2}}^{-}$, 32); 6.354 (${\frac{1}{2}}^{+}$, 124); 6.860 (not ${\frac{1}{2}}^{+}$, <1.0); 6.970 (not ${\frac{1}{2}}^{+}$, <1.0); 7.163 (${\frac{5}{2}}^{-}$, 1.4); 7.20 (${\frac{3}{2}}^{+}$, 280); 7.377 (${\frac{5}{2}}^{+}$, 0.5); 7.380 (${\frac{7}{2}}^{-}$, 1.2); 7.56 (${\frac{3}{2}}^{-}$, 500); 7.685 (${\frac{7}{2}}^{-}$, 18); 7.955 (${\frac{1}{2}}^{+}$, 90); 7.99 (${\frac{1}{2}}^{-}$, 260); 8.058 (${\frac{3}{2}}^{+}$, 86); 8.18 (${\frac{1}{2}}^{-}$, 69); and 8.20 (${\frac{3}{2}}^{-}$, 52). Resonances are not observed for known ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}$ levels at 5.215 and 7.573 MeV; these levels have ${\Gamma }_T<0.1\mathrm{}$ keV. The previously reported narrow ${\frac{3}{2}}^{+}$ level at 7.694 MeV does not exist. Many of the results are based on a multilevel $R$-matrix analysis; this is also extended to the published ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}\mathrm{}(n,n)\mathrm{}$ and ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}(\alpha ,n)$ cross sections for energies corresponding to ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}$ excitations up to 9.5 MeV. This extension shows that there are two levels rather than only one near 8.47 MeV and that the 9.42-MeV level has $J^{\pi }={\frac{3}{2}}^{-}$. Other assignments are confirmed. The most accurate excitation energies below 9.5 MeV are reviewed from the literature and the ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}$-${}_{}{}^{17}{}_{}{}^{}\mathrm{F}$ mirror structure is reviewed.

The energy spectra of normal-parity states of Ni isotopes are studied in the conventional shell model. The effective two-body interactions are determined by a least-squares fit between the calculated and observed low-lying energy levels. The agreement between the experimental and calculated energy spectra is quite remarkable. Single-nucleon spectroscopic factors for some low-lying states of odd-mass Ni isotopes are calculated. The resulting spectroscopic factors are similar to those calculated in other effective interaction calculations.

The ${}_{}{}^{46}{}_{}{}^{}\mathrm{Ca}$ nucleus was studied using the ($p,t$) reaction. A number of spin-parity assignments are clarified for the low-lying states. Many new energy levels are reported up to an excitation energy of 6.3 MeV including an additional $0^{+}$ state at 4.76 MeV. Distorted-wave calculations with $f-p$ shell wave functions give reasonable agreement for the first $0^{+}$, $2^{+}$, $4^{+}$, and $6^{+}$ states and for two weak $0^{+}$ states near 5.5 MeV which were strongly excited in the ${}_{}{}^{44}{}_{}{}^{}\mathrm{Ca}\mathrm{}(t,p)\mathrm{}$ reaction.

Intranuclear cascade studies have been made of low-energy pion-induced nuclear reactions which include the production and subsequent interactions of (3,3) isobars. The results are in fair to excellent agreement with the sparse amount of experimental information now available. Several experiments are outlined which could test the validity of some of the assumptions which were made in these calculations.

The spallation of copper with 65-MeV ${\pi }^{+}$ and ${\pi }^{-}$ has been investigated radiochemically by the determination of the yields of 12 product nuclides. These yields are generally comparable to those from the interaction of 205-MeV protons with copper; however, there is evidence of a higher probability of very large energy deposition by pions than by protons. For product nuclides not too far removed from the target there are often large differences in yields with ${\pi }^{+}$, $p$, and ${\pi }^{-}$, reflecting the different charge-to-mass ratios of the initial systems of incident particle plus target. Most of the characteristics of pion spallation are adequately reproduced by intranuclear cascade simulation calculations.

We have measured the high-energy end of the $\gamma$-ray spectrum from radiative muon capture in calcium. This accounted for only about one half of the neutral events detected beyond 57 MeV, the rest apparently being due to neutrons emitted by nuclei after capture. By fitting a theoretical prediction of the photon spectrum to our data, we found a value for the induced pseudoscalar coupling constant which disagrees with that expected.

The effects of inelastic excitations on the ${}_{}{}^{22}{}_{}{}^{}\mathrm{Ne}\mathrm{}(p,t)\mathrm{}{}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$ reaction are investigated. Differential cross sections at a proton energy of 39.8 MeV were measured for ${}_{}{}^{22}{}_{}{}^{}\mathrm{Ne}\mathrm{}(p,t)\mathrm{}{}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$ transitions to the $0^{+}$, $2^{+}$, and $4^{+}$ members of the ground-state rotational band, to the $2^{-}$ and $3^{-}$ members of the excited $K^{\pi }=2^{-}$ band; and to the $0^{+}$ and $2^{+}$ members of the first excited $K^{\pi }=0^{+}$ band. The cross sections to members of the ground-state band are compared with distorted-wave Born-approximation (DWBA) and coupled-channel-Born-approximation (CCBA) calculations using both pure Nilsson and pairing-mixed Nilsson rotational wave functions. Only the CCBA calculation with pairing provides a reasonable description of the experimental data. The effects of multistep processes are found to be as large for neon as for rareearth nuclei. The shape and relative strength of the measured cross section to the $2^{-}$ state, a transition which is forbidden by a direct single-step process, are reproduced well by the CCBA calculation, which allows the excitation of the $2^{-}$ state by inelastic processes in the entrance and exit channels.

Angular distributions of the neutron polarization have been measured for the ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}({}_{}{}^3{}_{}{}^{}\mathrm{He},n){}_{}{}^{26}{}_{}{}^{}\mathrm{Si}$ reaction at $E\left({}_{}{}^3{}_{}{}^{}\mathrm{He}\right)=5.0\mathrm{} \mathrm{and} 5.8$ MeV. The measurements were made using a neutron polarimeter consisting of a high-pressure helium gas scintillator and a spin-precession solenoid. The two angular distributions are similar in shape and have values approaching 0.7 at the backward angles. An attempt was made to describe $\sigma \left(\theta \right)$ and $P\left(\theta \right)$ data with direct reaction calculations made using the multinucleon transfer distorted-wave Born-approximation (DWBA) code JULIE.

Coupled-channel calculations including angular momentum dependence in the absorptive part of the optical potential have been performed for the elastic and inelastic scattering of $\alpha$ particles from ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ where $J$-dependent absorption might be expected. For the case of ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca} (\alpha ,{\alpha }^{\prime })$ at $E_{\alpha }=29\mathrm{}$ MeV it is found that the inelastic cross sections calculated without $J$-dependent absorption fail to predict the experimental differential cross sections by 2 orders of magnitude at backward angles while the inclusion of $J$-dependent absorption leads to predictions which are in good qualitative agreement with the data at all angles.

The level structure and the decay properties of the levels in ${}_{}{}^{61}{}_{}{}^{}\mathrm{Cu}$ up to 4082 keV of excitation have been carefully investigated via prompt $\gamma \gamma$ and $p\gamma$ coincidence experiments employing Ge(Li) $\gamma$-ray detectors and annular Si surface-barrier charged-particle detectors. From these experiments and from high-resolution singles energy and directional- correlation measurements an improved decay scheme has been obtained which includes 44 levels in ${}_{}{}^{61}{}_{}{}^{}\mathrm{Cu}$ and incorporates 120 $\gamma$ rays. New levels at 2584.5, 3739.4, 3942.4, and 4081.8 keV were postulated, while the $\gamma$ decay of several previously reported levels between 3277-3802 keV was not observed. The singles directional correlations from many transitions were used to obtain (i) reliable branching ratios for the more intense $\gamma$ rays and (ii) multipole mixing ratios $\delta \left(\frac{E2}{M1}\right)$ by employing both a ratio method and an independent analysis of the correlations via the compound statistical theory of nuclear reactions. Branching ratios for the weaker $\gamma$ rays were obtained from spectra taken at 55° to the beam direction. The lifetimes of 31 levels in ${}_{}{}^{61}{}_{}{}^{}\mathrm{Cu}$ were measured by the Doppler-shift attenuation method via the ${}_{}{}^{58}{}_{}{}^{}\mathrm{Ni}({}_{}{}^4{}_{}{}^{}\mathrm{He},p\gamma )$ reaction at bombardment energies between 9.7 and 12.2 MeV. Lifetimes were extracted from singles spectra obtained at 0, 15, 30, 45, 70, 90, and 110° and from $p\gamma$ coincidence spectra obtained at 0° to the beam direction, by employing both a centroid shift analysis for the data from all angles and a line-shape fitting analysis for the data from angles ≤45°. The stopping power of Ni previously calibrated for Cu ions, was used for the line-shape analysis and for obtaining reliable $F\left(\tau \right)$ curves applicable to the present conditions of both the singles and the coincidence experiments. For a large number of transitions in ${}_{}{}^{61}{}_{}{}^{}\mathrm{Cu}$ values for $B\left(E2\right)$ and $B\left(M1\right)$ were obtained. The $B\left(E2\right)$ and $B\left(M1\right)$ values together with the $\delta \left(\frac{E2}{M1}\right)$ values and the branching ratios from the first 10 levels in ${}_{}{}^{61}{}_{}{}^{}\mathrm{Cu}$ are compared with recent detailed calculations for ${}_{}{}^{61}{}_{}{}^{}\mathrm{Cu}$ based on a unified model. Good general agreement is observed between experiment and theory for the $B\left(E2\right)$ values. Only moderate agreement between experiment and theory is observed for the $B\left(M1\right)$ and $\delta \left(\frac{E2}{M1}\right)$ values.

The decays of ${}_{}{}^{39}{}_{}{}^{}\mathrm{Ca}$ and ${}_{}{}^{41}{}_{}{}^{}\mathrm{Sc}$ have been studied because of the related interest in the magnitude of the axial coupling constant in nuclear matter. ${}_{}{}^{39}{}_{}{}^{}\mathrm{Ca}$ was produced in the ${}_{}{}^{39}{}_{}{}^{}\mathrm{K}\mathrm{}(p,n)\mathrm{}{}_{}{}^{39}{}_{}{}^{}\mathrm{Ca}$ reaction at $E_p=10\mathrm{}$ MeV and its half-life was measured as 860.4 ± 3.0 msec resulting in an $\mathrm{ft}$ value of 4283 ± 25 sec. The half-life of ${}_{}{}^{41}{}_{}{}^{}\mathrm{Sc}$, formed in the ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}(p,\gamma ){}_{}{}^{41}{}_{}{}^{}\mathrm{Sc}$ reaction with 3-MeV protons on thick Ca, is 596.3 ± 1.7 msec. From the $\gamma$-ray energies of 2882.6 ± 0.3 keV and 3696.9 ± 0.5 keV measured at the 1.84- and 2.68-MeV resonances the $Q$ value of the ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}(p,\gamma ){}_{}{}^{41}{}_{}{}^{}\mathrm{Sc}$ reaction is found to be 1085.7 ± 1.4 keV. This leads to a ${}_{}{}^{41}{}_{}{}^{}\mathrm{Sc}$ mass excess of -28 642.2 ± 1.7 keV, to a ${}_{}{}^{41}{}_{}{}^{}\mathrm{Sc}$ decay energy of 6494.9 ± 2.0 keV, and to an $\mathrm{ft}$ value of 2849 ± 9 sec. An upper limit of 2×${10}^{-3\mathrm{}}$ is placed on the $\beta$ branching of ${}_{}{}^{41}{}_{}{}^{}\mathrm{Sc}$ to excited states of ${}_{}{}^{41}{}_{}{}^{}\mathrm{Ca}$. If the free-space value of $g_A$ is used to extract the Gamow-Teller matrix element ${〈\sigma 〉}_{\exp }$ from these data we find $\frac{{〈\sigma 〉}_{\exp }}{{〈\sigma 〉}_{\mathrm{s}.\mathrm{p}.\mathrm{}}}$ values of 0.685 ± 0.009 and 0.761 ± 0.006 for ${}_{}{}^{39}{}_{}{}^{}\mathrm{Ca}$ and ${}_{}{}^{41}{}_{}{}^{}\mathrm{Sc}$, respectively, where the ${〈\sigma 〉}_{\mathrm{s}.\mathrm{p}.\mathrm{}}$ are the relevant single-particle matrix elements.

The ${}_{}{}^{36}{}_{}{}^{}\mathrm{Ar}({}_{}{}^3{}_{}{}^{}\mathrm{He},\alpha ){}_{}{}^{35}{}_{}{}^{}\mathrm{Ar}$ reaction has been studied at a bombarding energy of 18 MeV using a multiangle spectrograph and a windowless gas cell. Levels in ${}_{}{}^{35}{}_{}{}^{}\mathrm{Ar}$ have been observed up to 8.3 MeV excitation with an energy resolution of 35 keV. Many new levels are reported. Angular distributions for the strong states have been compared with distorted-wave predictions. Spectroscopic factors have been extracted and are compared with the results of recent shell-model calculations. The agreement is good.

52 levels below $E_x=6.8\mathrm{}$ MeV in ${}_{}{}^{26}{}_{}{}^{}\mathrm{Al}$ were populated with the ${}_{}{}^{27}{}_{}{}^{}\mathrm{Al}({}_{}{}^3{}_{}{}^{}\mathrm{He},\alpha )$ reaction at a bombarding energy of 18 MeV. Angular distributions for 37 levels were extracted and compared with the results of distorted-wave calculations. Of these, 28 were seen to be characteristic of direct pickup; 2 with $l_n=0,3\mathrm{}$ with $l_n=1,10\mathrm{}$ with $l_n=0+2\mathrm{} \mathrm{and} 13$ with $l_n=2\mathrm{}$. A comparison with the results of the ${}_{}{}^{27}{}_{}{}^{}\mathrm{Al}(d,{}_{}{}^3{}_{}{}^{}\mathrm{He}){}_{}{}^{26}{}_{}{}^{}\mathrm{Mg}$ reaction shows good agreement for known $T=1\mathrm{}$ states and allows several new $T=1\mathrm{}$ assignments to be made. Spectroscopic factors for several of the low-lying states are in good agreement with shell-model predictions.

The reactions ${}_{}{}^{26}{}_{}{}^{}\mathrm{Mg}$(${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{15}{}_{}{}^{}\mathrm{N}$)${}_{}{}^{27}{}_{}{}^{}\mathrm{Al}$ and ${}_{}{}^{30}{}_{}{}^{}\mathrm{Si}$(${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{15}{}_{}{}^{}\mathrm{N}$)${}_{}{}^{31}{}_{}{}^{}\mathrm{P}$ have been studied at an ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ beam energy of 42 MeV. Spectroscopic factors for all observed transitions have been extracted using the no-recoil distorted-wave Born approximation. For $J^{\pi }={\frac{5}{2}}^{+} \mathrm{and} {\frac{1}{2}}^{+}$ states, the relative spectroscopic factors agree well with those derived from (${}_{}{}^3{}_{}{}^{}\mathrm{He},d$) studies. For $J^{\pi }={\frac{3}{2}}^{+}$ states, the (${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{15}{}_{}{}^{}\mathrm{N}$) spectroscopic factors disagree by a factor of ∼ 5 with those derived from (${}_{}{}^3{}_{}{}^{}\mathrm{He},d$) studies. As suggested by Kovar et al., this may be a result of neglecting recoil effects. The above factor of ∼ 5 discrepancies are of the same order as those noted ∼ 40 MeV above the Coulomb barrier by Kovar et al.