Precise $2p-1s$ muonic x-ray transition energies and energy shifts were measured for the $1f_{\frac{7}{2}}$-shell nuclei ${}_{}{}^{39,41}{}_{}{}^{}\mathrm{K}$, ${}_{}{}^{40,42,43,44,46,48}{}_{}{}^{}\mathrm{Ca}$, ${}_{}{}^{45}{}_{}{}^{}\mathrm{Sc}$, ${}_{}{}^{46,47,48,49,50}{}_{}{}^{}\mathrm{Ti}$, ${}_{}{}^{51}{}_{}{}^{}\mathrm{V}$, ${}_{}{}^{50,52,53,54}{}_{}{}^{}\mathrm{Cr}$, ${}_{}{}^{55}{}_{}{}^{}\mathrm{Mn}$, and ${}_{}{}^{56}{}_{}{}^{}\mathrm{Fe}$. The data were analyzed in terms of the Barrett moments $〈r^k{e}^{-ar}〉$ of the nuclear charge distribution, from which equivalent nuclear radii $R_k$ and isotopic and isotonic differences $\Delta R_k$ were computed. For those nuclei for which electron-scattering data were available, model-independent rms radii and radius differences were deduced from a combined analysis of the present data and elastic electron-scattering data. The $\Delta N=2\mathrm{}$ isotope shifts between even-$A$ nuclei decrease nearly linearly with increasing $N$ and become negative in the second half $24\le N\le 28$ of the $1f_{\frac{7}{2}}$ shell; a strong shell-closure effect is evident at $N=28\mathrm{}$. The isotope shifts are independent of $Z$, suggesting that the added neutrons interact with the entire proton core rather than with the valence protons. The $\Delta Z=2\mathrm{}$ isotone shifts between even-$A$ nuclei decrease smoothly and uniformly with increasing $Z$ for $20\le Z\le 28$ and are essentially independent of $N$. Like the isotope shifts, the isotone shifts display a sudden increase at $Z=28\mathrm{}$. Both the $\Delta N=1\mathrm{}$ isotope shifts and the $\Delta Z=1\mathrm{}$ isotone shifts show pronounced odd-even staggering. A strong correlation is found between nuclear deformation, as revealed by experimental $B\left(E2\mathrm{}\right)\mathrm{}$ values, and the measured isotope and isotone shifts. The measured isotope and isotone shifts, including those for odd-$A$ nuclei, are satisfactorily described by a simple empirical formula. The experimental results were compared with Hartree-Fock calculations; some recent calculations that include ground-state correlations display improved agreement with experiment through the inclusion of these deformation-related effects.

NUCLEAR STRUCTURE ${}_{}{}^{39,41}{}_{}{}^{}\mathrm{K}$, ${}_{}{}^{40,42,43,44,46,48}{}_{}{}^{}\mathrm{Ca}$, ${}_{}{}^{45}{}_{}{}^{}\mathrm{Sc}$, ${}_{}{}^{46,47,48,49,50}{}_{}{}^{}\mathrm{Ti}$, ${}_{}{}^{51}{}_{}{}^{}\mathrm{V}$, ${}_{}{}^{50,52,53,54}{}_{}{}^{}\mathrm{Cr}$, ${}_{}{}^{55}{}_{}{}^{}\mathrm{Mn}$, ${}_{}{}^{56}{}_{}{}^{}\mathrm{Fe}$; measured muonic x-ray spectra; deduced nuclear charge radii, isotope and isotone shifts; performed model-independent combined analysis of muonic x-ray data and elastic electron scattering data; compared charge parameters with Hartree-Fock and other calculations.

Arguments based on recent level width and $\alpha$ branching ratio measurements are presented to show that ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ + ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ resonances may not be populated by the ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}({}_{}{}^{16}{}_{}{}^{}\mathrm{O},\alpha ){}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}$ reaction as reported. Recent $\alpha$ spectra obtained at high bombarding energy are explained assuming ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ projectile excitation followed by $\alpha$ emission.

NUCLEAR REACTIONS ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}({}_{}{}^{16}{}_{}{}^{}\mathrm{O},\alpha ){}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}$, $E=60\mathrm{}-100$ MeV, measured $\sigma \left(\theta \right)$, $\Gamma$, ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}$, deduced levels.

Inelastic scattering spectra have been measured with 245 MeV incident energy ${}_{}{}^9{}_{}{}^{}\mathrm{Be}$ ions on a ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ target. They show large excitation of the ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ giant quadrupole resonance. Distorted-wave Born-approximation calculations are reported and compared with the data.

NUCLEAR REACTIONS ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$(${}_{}{}^9{}_{}{}^{}\mathrm{Be}$, ${}_{}{}^9{}_{}{}^{}\mathrm{Be}\prime$) $E=245\mathrm{}$ MeV; measured giant quadrupole resonance $\sigma \left(\theta \right)$; DWBA analysis.

The nuclide ${}_{}{}^{259}{}_{}{}^{}104$ has been produced in the ${}_{}{}^{249}{}_{}{}^{}\mathrm{Cf}({}_{}{}^{13}{}_{}{}^{}\mathrm{C},3n)$ reaction. Alpha particle groups of 8.77±0.01 MeV and 8.87±0.01 MeV were attributed to the decay of ${}_{}{}^{159}{}_{}{}^{}104$, and the measured half-life was found to be 3.0±1.3 s. The branching ratio for spontaneous fission decay was determined to be 0.063±0.037.

RADIOACTIVITY, FISSION ${}_{}{}^{259}{}_{}{}^{}104\left(\mathrm{sf} \mathrm{and} \alpha \right)$, measured $T_{\frac{1}{2}}$, $E_{\alpha }$, $I_{\alpha }$, $I_{\mathrm{sf}}$, $\sigma$ for ${}_{}{}^{249}{}_{}{}^{}\mathrm{Cf}({}_{}{}^{13}{}_{}{}^{}\mathrm{C},3n)$ and ${}_{}{}^{249}{}_{}{}^{}\mathrm{Cf}({}_{}{}^{13}{}_{}{}^{}\mathrm{C},\sigma 2n)$ reactions; deduced $\frac{\mathrm{sf}}{\alpha }$ for ${}_{}{}^{259}{}_{}{}^{}104$; enriched target.

We calculate the binding energy of tritium using two relativistic versions of the Faddeev equations with separable potentials, and compare with the nonrelativistic results. We find that the relativistic effects increase the binding energy by less than 0.5 MeV. We do our calculations considering only $S$ waves for the two-body interactions, but take into account the tensor force for different values of the deuteron $D$-state probability.

NUCLEAR STRUCTURE Tritium binding energy, relativistic effects.

A calculation is made of the elastic scattering of pions from ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ at an energy of 450 MeV. A single scattering optical potential is used with nuclear and nucleon form factors as determined elsewhere and the $\pi N$ scattering amplitude expressed in terms of phase parameters. It is found that several of the $N$ phase parameter sets can be readily distinguished by means of the $\pi$-nuclear scattering which they predict at this energy.

NUCLEAR REACTIONS ${\pi }^{-}-{}_{}{}^4{}_{}{}^{}\mathrm{He}$, ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ elastic scattering, calculated $\sigma \left(\theta \right)$, compared phase shift sets, $E=450\mathrm{}$ MeV.

Assuming time-reversal invariance of the many-body Hamiltonian, it is proven that the kernels in a general coupled-channels formulation are symmetric, to within a specified spin-dependent phase, under the interchange of channel labels and coordinates. The theorem is valid for both Hermitian and suitably chosen non-Hermitian Hamiltonians which contain complex effective interactions. While of direct practical consequence for nuclear rearrangement reactions, the reciprocity relation is also appropriate for other areas of physics which involve coupled-channels analysis.

NUCLEAR REACTIONS Symmetry relation for multichannel coupling kernels. Non-Hermitian operators. Time-reversal invariance and reciprocity.

We report a calculation of the nucleon potential energy in nuclear matter based upon a one-boson-exchange potential which is obtained from a fit to the two-nucleon scattering data. We evaluate the potential energy in the Hartree-Fock approximation and also perform a calculation which includes the effects of nucleon-nucleon correlations. We find large correlation corrections and conclude that calculations making use of the Hartree (or Hartree-Fock) approximation require the use of effective coupling constants which differ from those determined in a fit to scattering data. These effective constants would include effects of correlations in an implicit fashion.

NUCLEAR STRUCTURE Nucleon potential energy in nuclear matter; one-boson-exchange potential; validity of the Hartree-Fock approximation.

Major difficulties associated with attempts to describe the $\mathrm{YN}$ (hyperon-nucleon) interaction in all $\Lambda$-hypernuclear systems having $A\le 4$ by simple, effective $\Lambda p$ and $\Lambda n$ potentials are examined. The explicit $A$ dependence of the $\mathrm{YN}$ interaction associated with each system is explored, and the necessity of using exact four-body theory (in the $A=4\mathrm{}$ system) to calculate small charge-symmetry-breaking effects in terms of $\Lambda N$ potentials fitted to free $\Lambda$ scattering data is discussed.

NUCLEAR STRUCTURE $\Lambda N$ interactions; ${}_{\Lambda }{}^3{}_{}{}^{}\mathrm{H}$, ${}_{\Lambda }{}^4{}_{}{}^{}\mathrm{H}$, ${}_{\Lambda }{}^4{}_{}{}^{}\mathrm{He}$; $\Lambda -\Sigma$ coupling.

Cross sections for the ($p,p\alpha$) reaction at 101.5 MeV have been measured for nine nuclei ranging from ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ to ${}_{}{}^{66}{}_{}{}^{}\mathrm{Zn}$. Distorted-wave impulse approximation analyses of the ground state transitions provide relative alpha-cluster spectroscopic factors in qualitative agreement with (${}_{}{}^6{}_{}{}^{}\mathrm{Li},d$) studies, although quantitative differences exist. The calculations are sensitive to the bound alpha-cluster parametrization, so that the experimental data suggest limits on the rms radius of the cluster-core wave function.

NUCLEAR REACTIONS ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$, ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}$, ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$, ${}_{}{}^{32}{}_{}{}^{}\mathrm{S}$, ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$, ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ti}$, ${}_{}{}^{54}{}_{}{}^{}\mathrm{Fe}$, ${}_{}{}^{66}{}_{}{}^{}\mathrm{Zn}$ ($p,p\alpha$), $E=101.5\mathrm{}$ MeV; Measured ($E_p,E_{\alpha },{\theta }_p,{\theta }_{\alpha }$); DWIA analysis; deduced spectroscopic factors.

Differential cross section angular distributions have been measured for levels below 2.5 MeV in each residual nucleus in the ${}_{}{}^{124,126,128,130}{}_{}{}^{}\mathrm{Te}\mathrm{}(t,d)\mathrm{}{}_{}{}^{125,127,129,131}{}_{}{}^{}\mathrm{Te}$ reactions at $E_t=16\mathrm{}$ MeV. The reaction products were analyzed in an Enge split-pole magnetic spectrograph and were detected by a delay-line counter at the focal plane. The deuteron energy resolution obtained was between 13 and 15 keV full width at half maximum depending on the target used. The deuteron spectra from these $\mathrm{Te}\mathrm{}(t,d)\mathrm{}$ reactions are quite similar to the proton spectra obtained in the ($d,p$) reactions on the same target nuclei. Several new levels were, however, populated and most of them have $L$ values ≥2. Distorted-wave Born-approximation analyses of these $\mathrm{Te}\mathrm{}(t,d)\mathrm{}$ reaction data have been performed and spectroscopic factors extracted. These compare well with the spectroscopic factors obtained from previous ($d,p$) experiments except for levels in ${}_{}{}^{125}{}_{}{}^{}\mathrm{Te}$. In this case the present values are ∼25% lower than those obtained in the ($d,p$) reaction; however, they are in good agreement with the spectroscopic factors obtained in ($t,d$) and ($d,p$) reactions on the other tellurium nuclei.

NUCLEAR REACTIONS ${}_{}{}^{124,126,128,130}{}_{}{}^{}\mathrm{Te}\mathrm{}(t,d)\mathrm{}{}_{}{}^{125,127,129,131}{}_{}{}^{}\mathrm{Te}$, $E_t=16\mathrm{}$ MeV, measured $\sigma (E_d,\theta )$, resolution 13-15 keV, DWBA calculations, deduced $L$ transfers, spectroscopic factors.

The results of Coulomb excitation experiments using $\alpha$ particles and ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ ions and employing both particle spectroscopy and particle gamma coincidence techniques are presented. The intrinsic quadrupole moments of the quasi-gamma band and of the $K=0\mathrm{}$ octupole band have been determined. It was found that these excited band moments are within experimental uncertainties equal to the ground band intrinsic moment.

NUCLEAR REACTIONS ${}_{}{}^{238}{}_{}{}^{}\mathrm{U}(\alpha ,{\alpha }^{\prime })$ $E=16.0,17.0$ MeV, ${\theta }_{\mathrm{lab}}=130\mathrm{}°, 171.5°$; ${}_{}{}^{238}{}_{}{}^{}\mathrm{U}(\alpha ,{\alpha }^{\prime },\gamma )$, $E=16.0,17.0$ MeV, ${\theta }_{\alpha }=160.8\mathrm{}°$, ${\theta }_{\gamma }=55\mathrm{}°,65°,90°$; ${}_{}{}^{238}{}_{}{}^{}\mathrm{U}({}_{}{}^{16}{}_{}{}^{}\mathrm{O},{}_{}{}^{16}{}_{}{}^{}\mathrm{O}_{}^{\prime }{}_{}{}^{},\gamma )$, $E=52.0,51.2$ MeV, ${\theta }_{{}_{}{}^{16}{}_{}{}^{}\mathrm{O}}=168.0\mathrm{}°$, ${\theta }_{\gamma }=55\mathrm{}°,65°,70°$; measured $\frac{d{\sigma }_{\mathrm{inel}}}{d{\sigma }_{\mathrm{el}}}$, particle-$\gamma$ coincidence yield; deduced electromagnetic matrix elements in quasi-$\gamma$ band and $K=0\mathrm{}$ octupole band. Depleted ${}_{}{}^{238}{}_{}{}^{}\mathrm{U}$ metal target. Ge(Li) detector. Semiclassical and quantum-mechanical coupled channel analysis.

Perturbed angular distribution measurements in a Pb target have been performed for the ${}_{}{}^{206,208,210}{}_{}{}^{}\mathrm{Po}$($8^{+}$) and ${}_{}{}^{209}{}_{}{}^{}\mathrm{Po}$(${17/2}^{-}$) isomeric levels. Target temperatures were varied from 80 to 575 K and magnetic fields were applied from 0.1 to 6.1 T. The data can be explained with a combined strong magnetic and weak quadrupole interaction, with the quadrupole interaction arising from radiation damage in the host correlated in position with the implanted ion. A perturbation theory treatment of the combined magnetic and quadrupole interaction was developed and provides the basis for the interpretation of the results. The data are consistent with a Lorentzian distribution of electric field gradients centered around zero with half width $\frac{\Gamma }{2}=0.9\mathrm{}\times {10}^{17}$ V/${\mathrm{cm}}^2$ at room temperature. At higher temperatures the gradients are weaker indicating that before the measurement begins at ${10}^{-8\mathrm{}}$ sec after the implantation, some annealing of the initial damage takes place.

NUCLEAR REACTIONS ${}_{}{}^{204,206,207,208}{}_{}{}^{}\mathrm{Pb}(\alpha ,2n){}_{}{}^{206,208,209,210}{}_{}{}^{}\mathrm{Po}_{}^{*}{}_{}{}^{}$, $E=24\mathrm{}-25.5$ MeV. Enriched targets measured $\sigma (E_{\gamma };\theta ,H,t,T)$ for isomeric levels. Deduced electric field gradients from radiation damage.

The photoneutron cross section for ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ has been measured from threshold up to 47 MeV using monoenergetic photons and a high-pressure gas sample. The results agree with earlier monoenergetic-photon and certain photoneutron time-of-flight (liquid-sample) results at the lower energies, and consequently disagree with the results of other measurements at these energies. At the higher energies, however, the present results are essentially in agreement with all previous results.

NUCLEAR REACTIONS ${}_{}{}^4{}_{}{}^{}\mathrm{He}(\gamma , n)$; measured $\sigma \left(E_{\gamma }\right)$, 21 to 47 MeV, with monoenergetic photons, $4\pi$ neutron detector, high-pressure gas sample; four-nucleon system; isospin mixing; charge asymmetry.

The ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}({}_{}{}^3{}_{}{}^{}\mathrm{He}, d){}_{}{}^{13}{}_{}{}^{}\mathrm{N}$ stripping reaction has been studied at 43.6 MeV for all states of ${}_{}{}^{13}{}_{}{}^{}\mathrm{N}$ up to 10.78 MeV. Reliable spectroscopic factors are extracted for the low-spin states. The 7.16 MeV ($\frac{7}{2^{+}}$) and 7.38 MeV ($\frac{5}{2^{-}}$) states are shown to be populated by sequential scattering to the quadrupole states and $d_{\frac{5}{2}}$ or $p_{\frac{1}{2}}$ stripping in a rotational model. The data for a state at 9.0 MeV are shown to be accounted for within the same model for a $\frac{9}{2^{+}}$ spin assignment, confirming the results of pion scattering experiments to the analogous 9.50 MeV state in ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}$.

NUCLEAR REACTIONS ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}({}_{}{}^3{}_{}{}^{}\mathrm{He}, d){}_{}{}^{13}{}_{}{}^{}\mathrm{N}$, $E_{3_{\mathrm{He}}}=43.6\mathrm{}$ MeV, measured $\sigma (\theta , E_d)$, deduced spectroscopic factors, two-step reaction paths, spin of 9.0 MeV level.

Proton-proton bremsstrahlung cross sections have been measured at 42 MeV incident beam energy using a wire chamber spectrometer developed for the study of nuclear reactions with three-body final states. Proton-proton bremsstrahlung events from a 22 cm long gaseous target were detected concurrently over polar angle ranges from 16° to 40° and all kinematically allowed values of event noncoplanarity. The data have been analyzed and compared in two different ways to a theoretical calculation based on the Hamada-Johnston potential. In a conventional analysis, the data were divided into eighteen independent polar angle regions and the $\frac{d^3\sigma }{d{\Omega }_{1}d{\Omega }_{2}d{\psi }_{\gamma }}$, $\frac{d^2\sigma }{d{\Omega }_{1}d{\Omega }_2}$, and $\frac{d^2\sigma }{d{\theta }_{1}d{\theta }_2}$ cross sections were determined and compared with the theoretical calculations. In a "global" analysis, the data were compared directly with a theoretically weighted, simulated set of events. The latter comparison was made in terms of a wide range of variables, both in the laboratory and in the center of mass systems. A significant discrepancy between experiment and theory was found in certain parts of the observed phase space.

NUCLEAR REACTIONS Bremsstrahlung $p(p, p\gamma )p$, $E=42.0\mathrm{}$ MeV; measured $\frac{d^3\sigma }{d{\Omega }_{1}d{\Omega }_{2}d{\psi }_{\gamma }}$, $\frac{d^2\sigma }{d{\Omega }_{1}d{\Omega }_2}$, $\frac{d^2\sigma }{d{\theta }_{1}d{\theta }_2}$; global analysis; compared calculations Hamada-Johnston potential.

A multiple scattering description of elastic scattering is formulated in terms of impulsive scatterings from single target nucleons and pairs of target nucleons. In this description, distortion effects on the projectile from the residual medium are also described by multiple scattering in terms of the same single and pair amplitudes. At the level of single scattering, this procedure yields the first order optical potential result of Kerman, McManus, and Thaler. When scattering from both single nucleons and pairs of nucleons is included, the method leads to a one-body integral equation which requires the physical projectile-nucleon and projectile-pair transition amplitudes as input. This input is similar, but not exactly equivalent to that required by the spectator expansion for the optical potential truncated at second order. A principal advantage of the present formulation is that there need be no explicit dependence upon the projection operator $Q$ which projects off the target ground state. This feature introduces a scaling which appears to be a direct extension of the first order Kerman, McManus, and Thaler type of scaling. We follow up suggestions arising in the foregoing to show that the exact optical potential to second order in the spectator expansion can also be cast into a form having no explicit dependence upon $Q$, and requiring physical projectile-nucleon and projectile-pair transition amplitudes as input.

NUCLEAR REACTIONS Multiple scattering from single nucleons, pairs of nucleons in nucleus. Distortion from residual medium. Optical potential. spectator expansion.

The energies and relative intensities of the $\gamma$-ray transitions connecting the bound levels of ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}$ are determined. Level energies and $\gamma$-ray branching ratios are derived from these measurements. It is proposed that a mixed ${}_{}{}^{10}{}_{}{}^{}\mathrm{B}$-Pu source—emitting ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}$ $\gamma$ rays of 3683.921±0.023 and 3853.170±0.022 keV—can provide a valuable $\gamma$-ray calibration standard. The measured branching ratios are combined with previous lifetime measurements to give electromagnetic matrix elements which are compared to the predictions of a large basis shell-model calculation.

NUCLEAR REACTIONS ${}_{}{}^{10}{}_{}{}^{}\mathrm{B}(\alpha , p){}_{}{}^{13}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}(d, p){}_{}{}^{13}{}_{}{}^{}\mathrm{C}$, $E_{\alpha }=1.66\mathrm{}$ MeV, $E_d=2.51\mathrm{}$ MeV. measured $E_{\gamma }$; deduced level energies; measured branching ratios; deduced ${\mathrm{}\left|M\right(ML\left)\right|}^2$ and ${\mathrm{}\left|M\right(EL\left)\right|}^2$. Ge(Li) detectors.

Many-body scattering theory is used to derive coupled reaction channel-type equations. With the choice of the Alt, Grassberger, and Sandhas off-shell extension for the rearrangement transition operators, the effective interactions which enter these equations are shown to be free of all relevant two-cluster channel unitarity cuts and satisfy well-behaved dynamical equations which possess an explicit multiple scattering structure. Several approximations to these equations are presented and the importance of various nonorthogonality effects is discussed. Symmetrized forms of these equations which take into account the effects of particle identity are found. This permits the inclusion of the Pauli principle in coupled reaction channel calculations in a simple and practical manner.

NUCLEAR REACTIONS Generalized coupled reaction channel and resonating group formalisms. Dynamical connected-kernel integral equations for effective potentials. Pauli principle and channel nonorthogonality effects.