We find a simple analytic expression which approximates the relativistic Coulomb scattering of a spinless particle from the Coulomb field of a point source. We discuss the application of our result for various aspects of pion-nucleus scattering. The results are easily extended for spinless heavy ions.

NUCLEAR REACTIONS Coulomb scattering, Klein-Gordon equation, found analytic expression for $\sigma \left(\theta \right)$, application to $\pi$ scattering.

Using a chiral Lagrangian we have evaluated the tree diagrams for ${\pi }^{-}p\to \gamma \gamma n$. With this model we have studied the feasibility of using an experiment which detects only the final state gammas to resolve two matters of theoretical interest: the off-mass-shell behavior of the $\pi N$ amplitude and the sign of the ${\pi }^0\to \gamma \gamma$ amplitude. Our results suggest that this reaction is not a practicable way to distinguish between different models of the off-mass-shell amplitude, but that a reasonably accurate experiment could determine the sign of the ${\pi }^0$ decay amplitude.

NUCLEAR REACTIONS $p({\pi }^{-},\gamma \gamma )n$; inflight capture; $T_{\pi }^{\mathrm{lab}}<~100$ MeV; chiral Lagrangian model; calculated $\sigma \left(\theta \right)$; two photon detection. Studied effects of off-mass-shell $\pi N$ amplitude and sign of ${\pi }^0\to \gamma \gamma$ amplitude.

Within the general framework of multiple scattering theory for particle production in nucleon-nucleus and nucleus-nucleus collisions we investigate the results of two limiting situations: independent multiple collisions with Fermi motion effects versus the collective tube model with the assumption of universality for the elementary cross sections. Dramatically different predictions may be obtained. We summarize the results for pion multiplicity distributions in nucleus-nucleus collisions to assess the possible evidence for a coherent-collective mechanism and find that no strong conclusion is possible with the currently available pion multiplicity data. Attention is then focused on potentially more conclusive tests—subthreshold (in the nucleon-nucleon kinematics) production of massive particles, $\overline{p}$, $\frac{\psi }{J}$, and $W$ as examples. Evidence for a collective mechanism is found by examining subthreshold $\overline{p}$ production data in particle-nucleus collisions. However, owing to the limited scope of these data a firm conclusion is probably premature. Further $\overline{p}$ production experiments are proposed. With the assumption that the collective mechanism can be proved, it is worth studying the possibility of producing other massive particles, especially in nucleus-nucleus collisions where the collective effects have large subthreshold consequences. Within the (limited) framework of universality we then present predictions for the subthreshold production of $\frac{\psi }{J}$ and the $W$ boson. Owing to the qualitative nature of our assumptions and since these approaches bracket the $\overline{p}$ data, we expect that the collective tube model provides an optimistic limit and independent multiple collisions a pessimistic limit for $W$ production cross sections.

NUCLEAR REACTIONS Nucleon-nucleus and nucleus-nucleus at high energies; particle production subthreshold; pion, antiproton, $\frac{\psi }{J}$, $W$ production, Fermi motion and collective effects.

Dynamical nuclear polarization occurs in pionic atoms when a nuclear excitation of appropriate multipolarity is nearly degenerate with de-excitation of a pion atomic level. This phenomenon has been studied in several nuclei, one goal being to test the pion optical potential for pion atomic states normally "hidden" because of pion absorption. We find that, in addition to Coulomb mixing of the atomic and nuclear levels; strong interaction mixing and nuclear excitations above the lowest collective quadrupole mode are important for understanding the experimental results. All cases except ${}_{}{}^{110}{}_{}{}^{}\mathrm{Pd}$ can then be understood. For ${}_{}{}^{110}{}_{}{}^{}\mathrm{Pd}$, additional nuclear structure information is needed to determine whether or not the conventional pion optical potential will suffice again. We discuss the sensitivity of dynamical polarization measurements to the parameters of the optical potential and to various aspects of nuclear structure. In particular, we find that pionic ${}_{}{}^{150}{}_{}{}^{}\mathrm{Sm}$ provides a test of the interacting boson model and that the difference in neutron and proton radii predicted by Hartree-Fock calculations affects the mixing appreciably.

NUCLEAR REACTIONS Dynamical nuclear polarization in pionic atoms ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ti}$, ${}_{}{}^{104}{}_{}{}^{}\mathrm{Ru}$, ${}_{}{}^{110}{}_{}{}^{}\mathrm{Pd}$, ${}_{}{}^{112}{}_{}{}^{}\mathrm{Cd}$, ${}_{}{}^{150}{}_{}{}^{}\mathrm{Sm}$; nuclear structure effects.

The independent multiple collision model for pion production in nuclear collisions is extended to include the effects of Fermi motion. We have developed an analytic expression for the distribution of invariant mass for colliding nucleons whose momentum distributions are in relative motion. This expression is tractable if the colliding distributions are described in terms of Gaussian functions. We make use of this formalism to calculate Fermi averaged cross sections for pion production, using as input multi-Gaussian fits to momentum distributions obtained from nuclear structure calculations. We have calculated charged pion multiplicity distributions and inclusive cross sections for proton-nucleus and nucleus-nucleus collisions. We use a phenomenological method for incorporating the effects of energy loss by participant nucleons and pion reabsorption. Sample calculations are presented and compared with available experimental data. We discuss the striking effect of Fermi motion on pion production at bombarding energies below the nucleon-nucleon pion production threshold.

NUCLEAR REACTIONS Particle production, pion multiplicities, pion inclusive cross sections, Fermi motion effects, absorption and energy loss effects, intermediate energy particle-nucleus and nucleus-nucleus collisions.

Motivated by pion productions in high-energy heavy-ion collisions, we study the $p$-wave pion absorption in highly excited nuclear matter. The basic absorption mechanism is the two-nucleon model with both pion and rho-meson rescattering. The excited nuclear matter is modeled by a finite temperature Fermi gas. We deduce that for a given pion kinetic energy, the mean free path for pion absorption depends weakly on the temperature of the nuclear matter, but strongly on the density. Its relevance to high-energy heavy-ion collisions is discussed.

NUCLEAR REACTIONS intermediate energy: high-energy heavy-ion collisions, pion production and absorption, highly excited nuclear matter.

The measurement of invariant cross sections for the production of protons by 400 GeV protons from a variety of nuclear targets is described. Results are given for ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$, Be, C, A1, Cu, and Ta at 70°, 90°, 118°, 137° and 160° for detected protons from 0.4 to 1.4 GeV/c. Some angular distributions are given over a wider selection of angles. Comparisons are made with observed experimental results at other energies.

NUCLEAR REACTIONS Inclusive cross section, 400 GeV, ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$, Be, C, A1, Cu, and Ta.

We discuss nuclear reactions induced by nucleons or heavy ions at incident energies in the range of 0.5 to 1 GeV/nucleon. One-particle spectra and the correlations of two nucleons emitted are studied in the statistical limit: Subgroups of nucleons acquire a random occupation of their available phase space. As opposed to the thermal limit the finite number of nucleons gives rise to sizable correlations among the nucleons via the conservation laws. A surprising reproduction of the observed spectra—in particular at high momentum transfers (as in proton backward scattering)—is achieved by the model.

NUCLEAR REACTIONS Proton and heavy ion induced; $E=0.5-1.0\mathrm{}$ GeV/nucleon; calculation of inclusive one- and two-proton (coincidence) spectra by statistical model.

The hyperspherical or $K$-harmonics method is used to investigate the properties of ${}_{\Lambda }{}^3{}_{}{}^{}\mathrm{H}$ and ${}_{\Lambda }{}^9{}_{}{}^{}\mathrm{Be}$ (in the $\alpha -\alpha -\Lambda$ model). A set of coupled integral equations for functions of a single variable is obtained for determining the eigenenergy. Retaining nine harmonics, the ground-state energy of ${}_{\Lambda }{}^3{}_{}{}^{}\mathrm{H}$ is calculated for two-body potentials of square-well, Gaussian, exponential, and Yukawa shape. The volume integrals of the total $\Lambda N$ potential in ${}_{\Lambda }{}^3{}_{}{}^{}\mathrm{H}$ which yield the experimental energy are obtained. These are utilized to find the $\Lambda N$ scattering length and effective range. For ${}_{\Lambda }{}^9{}_{}{}^{}\mathrm{Be}$, suitable forms for the $\alpha -\alpha$ and $\alpha -\Lambda$ interactions are employed to evaluate the $\Lambda$ separation energy. Comparison is made with available experimental data.

NUCLEAR STRUCTURE hyperspherical harmonics, $K$-harmonics, ${}_{\Lambda }{}^3{}_{}{}^{}\mathrm{H}$, ground-state energy, ${}_{\Lambda }{}^9{}_{}{}^{}\mathrm{Be}$, $\alpha -\alpha -\Lambda$ model.

A fundamental problem in cosmic-ray physics is the production and propagation of Be. We have measured the cross section at 740 MeV for ${}_{}{}^7{}_{}{}^{}\mathrm{Be}$ production in N(10.9 mb ± 6%), C(9.82 mb ± 6%), ${}_{}{}^{11}{}_{}{}^{}\mathrm{B}$(4.17 mb ± 7%), and ${}_{}{}^{10}{}_{}{}^{}\mathrm{B}$(6.28 mb ± 7%). The monitor reactions were C($p$,$x$) ${}_{}{}^{11}{}_{}{}^{}\mathrm{C}$ = 29.9 mb ± 5% and the C($p$,$x$)${}_{}{}^7{}_{}{}^{}\mathrm{Be}$ cross section listed above.

NUCLEAR REACTIONS N, C, ${}_{}{}^{11}{}_{}{}^{}\mathrm{B}$, ${}_{}{}^{10}{}_{}{}^{}\mathrm{B}$($p$,$x$)${}_{}{}^7{}_{}{}^{}\mathrm{Be}$, $E=740\mathrm{}$ MeV: measured $\sigma$. Enriched targets. NaI, Ge(Li) detector, 2.2 keV at 478 keV.

The isospin-forbidden beta-decay branching ratio of the $0^{+}$, $T=2\mathrm{}$ ${}_{}{}^{28}{}_{}{}^{}\mathrm{Mg}$ ground state to the 972-keV $0^{+}$, $T=1\mathrm{}$ level of ${}_{}{}^{28}{}_{}{}^{}\mathrm{Al}$ is measured to be (0.31 ± 0.04)% corresponding to $ft=(9.20\mathrm{}\pm 1.1)\times {10}^7$s, $logft=7.96\mathrm{}\pm 0.06$. A charge-dependent matrix element between the $0^{+}$, $T=2\mathrm{}$ analog state at 5992 keV in ${}_{}{}^{28}{}_{}{}^{}\mathrm{Al}$ and the anti-analog level at 972 keV of 20.6 ± 1.6 keV is deduced from this branching ratio. This is in good agreement with a previous result.

RADIOACTIVITY ${}_{}{}^{28}{}_{}{}^{}\mathrm{Mg}$ [from ${}_{}{}^{26}{}_{}{}^{}\mathrm{Mg}$($t$,$p$)]; measured $\beta -\gamma$ coin; deduced branching ratio to ${}_{}{}^{28}{}_{}{}^{}\mathrm{A}1\mathrm{}$(0.972), $log\mathrm{ft}$, isospin mixing matrix element. Magnetic spectrometer, NaI(T1) detector.

The 45.3-day nuclide ${}_{}{}^{237}{}_{}{}^{}\mathrm{Pu}$ decays primarily via electron capture. The only information available up to now concerning the nuclide's $\alpha$-decay properties has been the observation of two $\alpha$ groups with energies of 5.65 and 5.36 MeV. In the present study, the photon spectrum of ${}_{}{}^{237}{}_{}{}^{}\mathrm{Pu}$ was investigated with x- and $\gamma$-ray Ge(Li) detectors. A total of 21 transitions was found to belong to ${}_{}{}^{237}{}_{}{}^{}\mathrm{Pu}$ decay; of these, 16 were assigned to its $\alpha$-decay branch. An $\alpha$-decay scheme was constructed on the bases of transition energy fits and known ${}_{}{}^{233}{}_{}{}^{}\mathrm{U}$ levels (including some observed only in reaction studies). The amounts of direct $\alpha$-decay feedings were then calculated from intensity balances, and a total $\alpha$-branching of (4.2 ± 0.4) × ${10}^{-3\mathrm{}}$% was deduced. $\alpha$-decay hindrance factors identify the band heads of the $\left(\frac{5}{2}\right)\left[752\right]\mathrm{}$ and $\left(\frac{7}{2}\right)\left[743\right]\mathrm{}$ orbitals in ${}_{}{}^{233}{}_{}{}^{}\mathrm{U}$ at 298.9 and 503.5 keV, respectively. Contrastingly, the ${}_{}{}^{233}{}_{}{}^{}\mathrm{Pa}$ and ${}_{}{}^{233}{}_{}{}^{}\mathrm{Np}$ decay studies had assigned the $\left(\frac{7}{2}\right)\left[743\right]\mathrm{}$ state to the level at 298.9 keV.

RADIOACTIVITY ${}_{}{}^{237}{}_{}{}^{}\mathrm{Pu}$, measured $E_{\gamma }$, $I_{\gamma }$; deduced $\alpha /\mathrm{electron}$-capture ratio. ${}_{}{}^{233}{}_{}{}^{}\mathrm{U}$, deduced levels, $J^{\pi }$.

It is shown, in the "elementary-particle" treatment, that, if the source to produce the background ${}_{}{}^{12}{}_{}{}^{}\mathrm{B}$(g.s.) comes only from the cascade processes ${\mu }^{-}{}_{}{}^{12}{}_{}{}^{}\mathrm{C}(\mathrm{g}.\mathrm{s}.\mathrm{})\to {\nu }_{\mu }{}_{}{}^{12}{}_{}{}^{}\mathrm{B}_{}^{*}{}_{}{}^{}\mathrm{}\left(2.62\right)\mathrm{}\to {\nu }_{\mu }{}_{}{}^{12}{}_{}{}^{}\mathrm{B}(\mathrm{g}.\mathrm{s}.\mathrm{}) \gamma$, the average polarization of the recoil ${}_{}{}^{12}{}_{}{}^{}\mathrm{B}$(g.s.) produced by the direct polarized-muon capture ${\mu }^{-}{}_{}{}^{12}{}_{}{}^{}\mathrm{C}(\mathrm{g}.\mathrm{s}.\mathrm{})\to {\nu }_{\mu }{}_{}{}^{12}{}_{}{}^{}\mathrm{B}(\mathrm{g}.\mathrm{s}.\mathrm{})$ can be extracted reliably from the observed angular distribution of the electrons in the subsequent beta decays of ${}_{}{}^{12}{}_{}{}^{}\mathrm{B}$(g.s.). Using the recent data of Possoz et al, our result for the average polarization $P_{\mathrm{av}}$ of the recoil ${}_{}{}^{12}{}_{}{}^{}\mathrm{B}$(g.s.) from the direct polarized-muon capture is $P_{\mathrm{av}}=0.47\mathrm{}\pm 0.05$, which agrees very well with the prediction of the standard picture, namely, conservation of vector current, partial conservation of axial vector current, and absence of second-class axial currents.

RADIOACTIVITY ${\mu }^{-}{}_{}{}^{12}{}_{}{}^{}\mathrm{C}(\mathrm{g}.\mathrm{s}.\mathrm{})\to {\nu }_{\mu }{}_{}{}^{12}{}_{}{}^{}\mathrm{B}_{}^{*}{}_{}{}^{}\mathrm{}\left(2.62\right)\mathrm{}$; effects on the average polarization of ${}_{}{}^{12}{}_{}{}^{}\mathrm{B}$(g.s.); CVC, PCAC, and no second-class axial vector currents.

It is shown in a model-independent fashion that, even without the conserved vector current test by extracting the shape factors from the observed $e^{\mp }$ energy spectra in the beta decays of ${}_{}{}^{12}{}_{}{}^{}\mathrm{B}$ and ${}_{}{}^{12}{}_{}{}^{}\mathrm{N}$, the validity of conservation of vector current and partial conservation of axial vector current and the absence of second-class axial currents have now been individually confirmed by the combination of the muon capture data and the results of the recent asymmetry measurements. On the other hand, it is found that the presence of appreciable meson-exchange corrections does not cause a serious discrepancy between the asymmetry data and the Cohen-Kurath nuclear model after the spin-quadrupole-moment term (from the $L=2\mathrm{}$ component of the lepton wave functions) and all the theoretical uncertainties are also taken into account.

RADIOACTIVITY CVC, PCAC, and no second-class axial vector current, tested individually; meson-exchange corrections, compatibility with the Cohen-Kurath nuclear wave functions.

A unified shell-model description of nuclear deformation valid throughout the periodic table is presented. Microscopic calculations for the Zr and Mo isotopes are carried out in the frameworks of the shell model and the Hartree-Fock-Bogoliubov method, respectively, to study the shape transition in these nuclei. It is shown that deformation is produced by the isoscalar component of the neutron-proton ($n-p$) interaction in this region, as in the lighter ($2s$,$1d$)-shell region. Deformation sets in when the $T=0\mathrm{}$ $n-p$ interaction dominates over the sphericity-favoring pairing interaction between $T=1\mathrm{}$ pairs of nucleons. When shell effects are important, as for the light and medium-weight regions mentioned above, the simultaneous occupation of neutrons and protons of spin-orbit "partner" orbitals plays a crucial role in determining the onset of deformation. However, their effect is probably less important in the rare-earth and transuranic regions due to the rapid accumulation of single-particle orbitals.

NUCLEAR STRUCTURE Microscopic description of nuclear deformation; shell-model calculations of ${}_{}{}^{96}{}_{}{}^{}\mathrm{Zr}$, ${}_{}{}^{98}{}_{}{}^{}\mathrm{Zr}$, and ${}_{}{}^{100}{}_{}{}^{}\mathrm{Zr}$; HFB calculations of ${}_{}{}^{98}{}_{}{}^{}\mathrm{Mo}$-${}_{}{}^{110}{}_{}{}^{}\mathrm{Mo}$; discussion of light and heavy deformed nuclei; relation to interacting boson approximation.

Conventional shell-model calculations of the iron isotopes with $A=56-60\mathrm{}$ are discussed. The calculated level spectra are in good agreement with experiment. The calculations of $E2\mathrm{}$ observables are in reasonable agreement with experiment when added effective charges of $1.0e$ are used for both neutrons and protons. The neutron effective charge is significantly different from one appropriate ($1.9e$) for ${}_{}{}^{56}{}_{}{}^{}\mathrm{Ni}$. There is a strong state dependence in the effective $M1\mathrm{}$ operator, and calculations of $M1\mathrm{}$ observables are in general disagreement with experiment. An analysis of the structure of ground-state bands and yrast levels is made in terms of a simple rotational model. In all even isotopes, the intrinsic quadrupole moments in the ground-state "band" shrink dramatically with increasing $J$. There is evidence of a band crossing in all three even isotopes, but no evidence is found for an abrupt shape change in the ground-state bands of these nuclei. It is concluded that no simple rotational model gives a good description of the low-lying states of the iron isotopes.

NUCLEAR STRUCTURE ${}_{}{}^{56}{}_{}{}^{}\mathrm{Fe}$, ${}_{}{}^{57}{}_{}{}^{}\mathrm{Fe}$, ${}_{}{}^{58}{}_{}{}^{}\mathrm{Fe}$, ${}_{}{}^{59}{}_{}{}^{}\mathrm{Fe}$, ${}_{}{}^{60}{}_{}{}^{}\mathrm{Fe}$, calculated levels. ${}_{}{}^{56}{}_{}{}^{}\mathrm{Fe}$, ${}_{}{}^{58}{}_{}{}^{}\mathrm{Fe}$, ${}_{}{}^{60}{}_{}{}^{}\mathrm{Fe}$, calculated $B\left(M1\mathrm{}\right)\mathrm{}$, $B\left(E2\mathrm{}\right)\mathrm{}$, $\mu$, branching ratios. Shell model, Vervier interaction, Horie-Ogawa interaction, effective operators. Comparison with experiment and rotational model.

Isospin mixing in the yrast $I=\frac{5}{2^{+}}$ $T=\frac{1}{2}$ level has been calculated assuming an isoscalar nucleonnucleon interaction and the single-particle level spacings observed in ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}$ and ${}_{}{}^{17}{}_{}{}^{}\mathrm{F}$. Results are presented for both the ($d_{\frac{5}{2}}$,$s_{\frac{1}{2}}$) and two variants of the ($d_{\frac{5}{2}}$,$s_{\frac{1}{2}}$,$d_{\frac{3}{2}}$) model spaces. The experimental value does not require the introduction of a charge symmetry or charge independence breaking nucleon-nucleon interaction.

NUCLEAR STRUCTURE Isospin mixing in ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$ calculated.

The variational derivation of the time-dependent Hartree-Fock equation is reviewed. When norm-violating variations are included, a unique time-dependent Hartree-Fock Hamiltonian, which differs from that customarily used in time-dependent Hartree-Fock analyses, is implied. This variationally "true" Hartree-Fock Hamiltonian has the same expectation value as the exact Hamiltonian, equal to the average energy of the system. Since this quantity remains constant under time-dependent Hartree-Fock time evolution, we suggest the label "constant $〈H〉$" for this form of time-dependent Hartree-Fock theory.

NUCLEAR REACTIONS Unique time-dependent Hartree-Fock Hamiltonian obtained from variational principle by norm-violating variations.

This paper reports a reanalysis of the data of an earlier reported experiment in which ${}_{}{}^{110}{}_{}{}^{}\mathrm{Ag}_{}^m{}_{}{}^{}$ was oriented as an impurity in an iron foil and the asymmetry of the beta rays was observed. This new analysis gives for the ratio of the Fermi to Gamow-Teller matrix elements in the 531 keV $6^{+}$→$6^{+}$ $\beta$ decay $x=0.0029\mathrm{}\pm 0.0106$.

RADIOACTIVITY ${}_{}{}^{110}{}_{}{}^{}\mathrm{Ag}_{}^m{}_{}{}^{}$ (oriented); reanalysis 531 keV $\beta$ anisotropy; deduced ratio Fermi/Gamow-Teller matrix elements.

${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ + ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}$ elastic scattering angular distributions have been measured at 12 bombarding energies over a range $20 \mathrm{MeV}\le E_{\mathrm{lab}}\le 36 \mathrm{MeV}$. The elastic $S$ matrix elements were parametrized in the framework of a complex Woods-Saxon potential. The measured angular distributions exhibit some irregular structure at surprisingly low energies—and no optical potential has been found to reproduce this low energy structure. Three surface-transparent optical potentials (one of which has energy-independent well depths) have been found to give a good overall account of the data. All three of these potentials predict essentially identical cross sections for angles ≲ 120°; additionally, they predict strong but strikingly different oscillations in the 180° elastic scattering excitation function.

NUCLEAR REACTIONS ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}$(${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$)${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}$, measured $\sigma (E,\phi )$ $E=20-36\mathrm{}$ MeV; deduced optical model parameters.