A cluster-phonon model is proposed for odd-mass nuclei to describe simultaneously positive- and negative-parity states, in which octupole as well as quadrupole vibrations of the core are allowed. The cluster states include one-particle and two-particle-one-hole excitations. The residual interaction is of the quadrupole-quadrupole plus pairing type. The model is applied to ${}_{}{}^{89}{}_{}{}^{}\mathrm{Y}$, whose energy levels, eigenvectors, spectroscopic factors, and electromagnetic properties are computed and compared with experiment.

NUCLEAR STRUCTURE Theory, ${}_{}{}^{89}{}_{}{}^{}\mathrm{Y}$; calculated levels, $J$, $\Pi$, $S$, $B\left(E2\mathrm{}\right)\mathrm{}$, $B\left(M1\mathrm{}\right)\mathrm{}$, $q$, $\mu$, $b$, $\tau$, and $\delta$. Cluster-phonon model; quadrupole and octupole vibrations; 2p-1h excitations.

Monopole and quadrupole charge distributions of ${}_{}{}^{154}{}_{}{}^{}\mathrm{Gd}$, ${}_{}{}^{155}{}_{}{}^{}\mathrm{Gd}$, ${}_{}{}^{156}{}_{}{}^{}\mathrm{Gd}$, ${}_{}{}^{157}{}_{}{}^{}\mathrm{Gd}$, ${}_{}{}^{158}{}_{}{}^{}\mathrm{Gd}$, and ${}_{}{}^{160}{}_{}{}^{}\mathrm{Gd}$ were investigated by muonic-atom $K$ and $L$ x-ray measurements. The model-independent Barrett charge radii $R_k$ and the isotope shifts $\Delta R_k$ were measured, and values of $〈r^2〉$ and $\Delta 〈r^2〉$ were deduced. A pronounced even-odd staggering effect of the nuclear charge radii was observed for the series ${}_{}{}^{156-158}{}_{}{}^{}\mathrm{Gd}$. The quadrupole moments of the first excited states of the even-$A$ Gd nuclei were determined to be $Q^{154}\left(2^{+}\right)=-1.82\mathrm{}\left(4\right)\mathrm{} e$ b, $Q^{156}\left(2^{+}\right)=-1.93\mathrm{}\left(4\right)\mathrm{} e$ b, $Q^{158}\left(2^{+}\right)=-2.01\mathrm{}\left(4\right)\mathrm{} e$ b, and $Q^{160}\left(2^{+}\right)=-2.08\mathrm{}\left(4\right)\mathrm{} e$ b, and the quadrupole moments of the ${\frac{3}{2}}^{-}$ ground states of the odd-$A$ ${}_{}{}^{155,157}{}_{}{}^{}\mathrm{Gd}$ nuclei were determined to be $Q^{155}\left({\frac{3}{2}}^{-}\right)=1.27\mathrm{}\left(3\right)\mathrm{} e$ b and $Q^{157}\left({\frac{3}{2}}^{-}\right)=1.35\mathrm{}\left(3\right)\mathrm{} e$ b. Comparison with a separate measurement of the odd-$A$ ground-state quadrupole moments based on the static hyperfine splitting of the muonic $M$ x rays showed that the model error in the extracted quadrupole moments of these nuclei is less than 2 percent. The quadrupole moments and the $B\left(E2\mathrm{}\right)\mathrm{}$ values obtained in the present experiment for the low-lying Gd states are in satisfactory agreement with the axially symmetric rotational model. However, the ${}_{}{}^{154}{}_{}{}^{}\mathrm{Gd}$ nucleus exhibits a considerable softness as indicated by the isomer shift of the $2^{+}$ excited state and by the experimental value of the ratio $\frac{Q\left(2^{+}\right)}{B(E2:0^{+}\to 2^{+})}$.

NUCLEAR STRUCTURE ${}_{}{}^{154-158,160}{}_{}{}^{}\mathrm{Gd}$; measured muonic x-ray spectra; deduced monopole and quadrupole charge parameters; isotope and isomer shifts.

The ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ and ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}-{}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}$ Sussex matrix elements are modified by the addition of the matrix elements of a potential whose radial form is that of a momentum-dependent delta function. The strength parameters of the potential are found by fitting to the binding energy and quadrupole moment of the deuteron. Shell model calculations in a $4\hslash \omega$ model space using the modified matrix elements and a size parameter of 1.5 fm give binding energies of 7.55, 6.79, and 27.77 MeV for ${}_{}{}^3{}_{}{}^{}\mathrm{H}$, ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, and ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, respectively. The energies of the negative parity states in ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ are also calculated using the modified matrix elements in a $5\hslash \omega$ model space. It is found that their calculated absolute energies are much improved as compared to calculations using the unmodified matrix elements. The relative energies of these states are not significantly changed by the modifications, as expected.

NUCLEAR STRUCTURE ${}_{}{}^3{}_{}{}^{}\mathrm{H}$, ${}_{}{}^{3,4}{}_{}{}^{}\mathrm{He}$. Calculated binding energies, negative parity states. Shell model, Sussex matrix elements.

We calculate the momentum-transfer dependence of the effective nuclear interaction in a conventional model and in a recently developed relativistic model which emphasizes self-consistent-field effects. These latter effects lead to major modifications in the $T=0\mathrm{}$ interaction for $S=0\mathrm{}$ and $S=1\mathrm{}$ and less dramatic changes for the $T=1\mathrm{}$ interaction. The calculational methods are similar to those used previously and involve considering only the first iteration of the self-consistent field equations. We review these methods of calculation and present some new results for the (off-shell) nucleon self-energy.

NUCLEAR STRUCTURE Effective force in nuclei; momentum transfer dependence in a relativistic theory; one-boson-exchange models.

The total fusion cross sections for ${}_{}{}^9{}_{}{}^{}\mathrm{Be}$+${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ were measured in the ${}_{}{}^9{}_{}{}^{}\mathrm{Be}$ bombarding energy range 35-60 MeV in 5 MeV steps by detecting the evaporation residues with a gas ionization $E-\Delta E$ detector. The measured fusion cross sections are significantly smaller than the total reaction cross sections calculated using optical model parameters obtained from fitting the elastic scattering cross sections in the same energy range. Similar results were obtained earlier for the case of ${}_{}{}^9{}_{}{}^{}\mathrm{Be}$+${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$ fusion. The fact that ${}_{}{}^9{}_{}{}^{}\mathrm{Be}$ is weakly bound and is easily dissociated in the Coulomb and nuclear fields of either the ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ or ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$ nucleus is believed to have a strong influence on the observed fusion and elastic scattering cross sections. Sources of possible discrepancy between fusion cross sections measured by determining the yield of heavy evaporation residues and those by summing the cross sections for the emission of light particles such as protons, neutrons, and alpha particles from the compound system are discussed.

NUCLEAR REACTIONS ${}_{}{}^9{}_{}{}^{}\mathrm{Be}$+${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ fusion, $E\left({}_{}{}^9{}_{}{}^{}\mathrm{Be}\right)=35, 40, 45, 50, 55, 60$ MeV, detected evaporation residues, deduced total fusion cross sections, calculated ${}_{}{}^9{}_{}{}^{}\mathrm{Be}$+${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ total reaction cross sections.

A search has been made for indications of resonant behavior in forward angle ($20°<{\theta }_{\mathrm{lab}}<40\mathrm{}°$) ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$+${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}$ inelastic scattering. No evidence of resonant structure has been observed over the bombarding energy range $25 \mathrm{MeV}<E_{\mathrm{c}.\mathrm{m}.\mathrm{}}<39\mathrm{} \mathrm{MeV}$. An upper limit for the reduced partial width in the ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}$($2_1^{+}$) +${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ channel is deduced.

NUCLEAR REACTIONS ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}$(${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$)${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}$* ($2_1^{+}$, 1.37 MeV); excitation function $41<\mathrm{}{E}_{\mathrm{lab}}<65\mathrm{}$ MeV, $20°<{\theta }_{\mathrm{lab}}<40\mathrm{}°$, search for resonant structures.

We calculate the effect of the Coulomb interaction of the ejected $\beta$ ray with the bound atomic electron in the $\beta$ decay of a tritium atom. The excited state probabilities of the residual helium ion are changed by at most 0.17% from the usual sudden approximation.

RADIOACTIVITY ${}_{}{}^3{}_{}{}^{}\mathrm{H}$; atomic final states, neutrino mass.

Sum rules that hold in the case of a self-consistent charge-exchange random-phase approximation theory are used here to derive transition densities for the charge-exchange components of isovector giant resonances. The derived transition densities are then employed in a calculation of (${\pi }^{\pm }, {\pi }^0$) cross sections.

NUCLEAR STRUCTURE Transition densities to giant resonances are obtained from sum rules.

An optical potential depending linearly on the energy is transformed into a phase-equivalent energy-independent potential which depends on the angular momentum via $l(l+1\mathrm{})\mathrm{}$. The transformation is exact and can be performed explicitly for an arbitrary shape of the potential. It is illustrated for the case of a Woods-Saxon potential.

NUCLEAR REACTIONS Optical potentials, equivalence of energy and angular momentum dependence.

Finite rank expansions for two- and three-body $t$ matrices are analytically continued to the unphysical sheet of the complex energy plane associated with the lowest two-body scattering threshold in order to obtain the position and residue of the virtual state and resonance poles. The present method is applied to study the ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ virtual state of two nucleons, the Efimov virtual states of three identical bosons, and the doublet virtual state of three nucleons.

NUCLEAR REACTIONS Separable expansions for $t$ matrices, resonances and virtual states, virtual states for two and three nucleons, Efimov virtual states.

In a recently published paper, Cassing and collaborators discussed the general tendencies of the motion of the complex poles of the $S$ matrix of a complex potential with increasing absorption. In this Comment, it is pointed out that their conclusions are, in fact, model dependent and therefore are not valid for as wide a class of complex potentials as is suggested. Because of this model dependence of these conclusions, questions about the generalized Levinson's theorem proposed by these authors for complex interaction are raised.

NUCLEAR REACTIONS Scattering theory. $S$ matrix for absorptive potentials. Levinson's theorem.

A number of ($p, n$), ($n, p$), and (${}_{}{}^3{}_{}{}^{}\mathrm{He}$, $p$) reactions have been interpreted on the basis of the statistical multistep compound emission mechanism. Good agreement with experiment is found both in spectrum shape and in the value of the coherence widths.

NUCLEAR REACTIONS ${}_{}{}^{25}{}_{}{}^{}\mathrm{Mg}({}_{}{}^3{}_{}{}^{}\mathrm{He}, p)$, ${}_{}{}^{27}{}_{}{}^{}\mathrm{Al}({}_{}{}^3{}_{}{}^{}\mathrm{He}, p)$, ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}(n, p)$, ${}_{}{}^{59}{}_{}{}^{}\mathrm{Co}(p, n)$, ${}_{}{}^{89}{}_{}{}^{}\mathrm{Y}(p, n)$, ${}_{}{}^{103}{}_{}{}^{}\mathrm{Rh}(p, n)$, $E_{\mathrm{inc}}=13, 13, 14.5, 14.7, 14.8, 18$ MeV. Statistical multistep compound emission mechanism. Calculated differential cross sections and coherence widths.

Cross sections ${\sigma }_{\kappa }$ for pair creation by photons on Ta and Bi nuclei were deduced with 1% uncertainty between 3 and 28 MeV from total absorption measurements which were combined with data already published from the same apparatus. The gamma ray spectrometer consisted of a photoneutron time-of-flight detector and a liquid deuterium target. Energy resolution varied from 0.6% at 6 MeV (40 keV) to 1.9% at 25 MeV. At 10 MeV, ${\sigma }_{\kappa }\left(\mathrm{Bi}\right)$ is (13.04±0.07) b, exceeding by 1.7% a value calculated for Bi from a theoretical result for uranium which assumes a point nucleus but is otherwise exact. New tables of atomic cross sections agree closely with our measurements for Ta and Bi up to 10 MeV. The agreement in the case of Ta remains good at higher energies where older tables significantly underestimate the pair cross sections of heavy elements. For Bi the experimental ${\sigma }_{\kappa }$ exceed even the new tabulated values by (0.8±0.7)% around 20 MeV.

NUCLEAR REACTIONS ${}_{}{}^{181}{}_{}{}^{}\mathrm{Ta}$, ${}_{}{}^{209}{}_{}{}^{}\mathrm{Bi}$; measured total photon absorption ${\sigma }_{\gamma }\mathrm{}\left(E\right)\mathrm{}$; observed GDR; deduced electron pair creation ${\sigma }_{\kappa }\mathrm{}\left(E\right)\mathrm{}$; $E=3.0\mathrm{} \mathrm{to} 28.0$ MeV; resolution 40 keV at 6 MeV, 480 keV at 25 MeV; comparison of ${\sigma }_{\kappa }$ with calculated values; Coulomb distortion factors of Ta and Bi; comparison of integrated photonuclear cross section with sum rule; ${}_{}{}^2{}_{}{}^{}\mathrm{H}(\gamma , n)$ L ${\mathrm{D}}_2$/TOF spectrometer.

The ${}_{}{}^{206}{}_{}{}^{}\mathrm{Pb}(\overrightarrow{t}, p)$ reaction has been studied over the angular range from 10° to 60° at a triton energy of 17 MeV. Cross sections and analyzing powers were measured for 11 states between 2.6 and 5.0 MeV excitation, including the $4^{-}$ state at 3.475 MeV. The two-neutron transfer reaction to this state was represented by a sequential transfer calculation. Data for three natural parity states, in particular the analyzing powers, were not fitted well by one-step distorted-wave Born approximation calculations. Numerous sequential transfer calculations with different transfer configurations were tried for each of these states. While most of the calculations produced results similar to the distorted-wave Born approximation results, at least one calculation was found for each state which gave an improved representation over that of the distorted-wave Born approximation. The failure of the distorted-wave Born approximation documents the need to apply sequential transfer reaction processes in general descriptions of two-nucleon transfer reactions.

NUCLEAR REACTIONS: ${}_{}{}^{206}{}_{}{}^{}\mathrm{Pb}(\overrightarrow{t}, p){}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$, $E=17\mathrm{}$ MeV; measured $\sigma \left(\theta \right)$, $A_y\left(\theta \right)$. DWBA and sequential transfer calculations.

The ${}_{}{}^{194,196,198}{}_{}{}^{}\mathrm{Pt}(\overrightarrow{t}, \alpha ){}_{}{}^{193,195,197}{}_{}{}^{}\mathrm{Ir}$ reactions have been studied with a 17 MeV polarized triton beam. Based on the measured angular distributions of cross sections and analyzing powers, several new $I^{\pi }$ assignments were made in ${}_{}{}^{195}{}_{}{}^{}\mathrm{Ir}$ and new levels were identified in ${}_{}{}^{197}{}_{}{}^{}\mathrm{Ir}$. The change in the distribution of $d_{\frac{3}{2}}$ spectroscopic strength as a function of neutron number is observed to be quite dramatic. Two large fragments are observed in ${}_{}{}^{193}{}_{}{}^{}\mathrm{Ir}$, all of the $d_{\frac{3}{2}}$ strength is concentrated in the ground state of ${}_{}{}^{197}{}_{}{}^{}\mathrm{Ir}$, and ${}_{}{}^{195}{}_{}{}^{}\mathrm{Ir}$ shows a situation intermediate between these two. The empirical $d_{\frac{3}{2}}$ strength in ${}_{}{}^{193}{}_{}{}^{}\mathrm{Ir}$ is well reproduced by the predictions of the $\mathrm{U}\left(\frac{6}{4}\right)$ supersymmetry proposed for the Os-Ir-Pt nuclei. The change in the $d_{\frac{3}{2}}$ strength as a function of neutron number is well reproduced by a perturbation of this symmetry.

NUCLEAR REACTIONS ${}_{}{}^{194,196,198}{}_{}{}^{}\mathrm{Pt}(\overrightarrow{t}, \alpha )$, $E_t=17\mathrm{}$ MeV, enriched targets; measured $\sigma (E_{\alpha }, \theta )$, $A_y(E_{\alpha }, \theta )$. ${}_{}{}^{193,195,197}{}_{}{}^{}\mathrm{Ir}$ deduced levels, $I$, $\pi$, spectroscopic strengths.

NUCLEAR STRUCTURE Interacting boson fermion model, $\mathrm{U}\left(\frac{6}{4}\right)$ supersymmetry, spin(6) symmetry.

An efficient calculational method for distorted wave impulse approximation analyses of quasifree knockout reactions is described. Since the usual Racah algebra is eliminated, it is straightforward to include spin-orbit distortions. Results are presented for 150 MeV ($p, 2p$) and ($p, \mathrm{pn}$) reactions. In some cases the effects of the spin-orbit terms are large.

NUCLEAR REACTIONS Effect of spin-orbit distortions in distorted wave impulse approximation calculation for ($p, 2p$) and ($p, \mathrm{pn}$) reactions.

Analyzing-power angular distributions of the ($\overrightarrow{p}, d$) reaction on ${}_{}{}^{60}{}_{}{}^{}\mathrm{Ni}$, ${}_{}{}^{86}{}_{}{}^{}\mathrm{Sr}$, and ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ have been measured at 94-MeV bombarding energy. A distinctive $j$ dependence was observed for the pickup of neutrons from the $p_{\frac{1}{2}}$, $p_{\frac{3}{2}}$, $f_{\frac{5}{2}}$, and $f_{\frac{7}{2}}$, orbitals which is independent of target mass and of the number of nodes in the neutron bound-state radial wave function. These characteristic $j=l\pm \frac{1}{2}$ signatures may thus be used as experimental templates for reliable spin determinations over the whole periodic table at this bombarding energy. Conventional distorted-wave Born approximation calculations using standard optical model parameters describe the experimental analyzing power angular distributions and the associated $j$ dependence rather poorly.

NUCLEAR REACTIONS ${}_{}{}^{60}{}_{}{}^{}\mathrm{Ni}$, ${}_{}{}^{86}{}_{}{}^{}\mathrm{Sr}$, ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}(\overrightarrow{p}, d)$, $E=94\mathrm{}$ MeV; measured $A_y\left(\theta \right)$. Enriched targets. DWBA analysis.

The solution to a system of coupled equations is examined with regard to the effect of the long range part of the inelastic potentials upon the elastic phase shifts. It is found that those parts of the inelastic potentials which occur beyond the range of the elastic to inelastic transition potentials affect the elastic phase shifts in only a minor way. The proof is given theoretically by means of a Green's function formulation which includes the long range part of the inelastic potentials perturbatively. When applied to the calculation of the effect of breakup on the deuteron-nucleus elastic scattering, the argument confirms the finding that errors in the long range part of the potentials in the breakup channels do not sensitively affect the elastic deuteron scattering cross section. This result explains why the elastic scattering is not very sensitive to the choice of the discretization procedure of the breakup space.

NUCLEAR REACTIONS Coupled equations, Green's functions, effect of long range inelastic potentials on elastic phase shifts.

Low energy nuclear fragments ($12\le A\le 140$) from the bombardment of Au by high energy protons, ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, and ${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$ projectiles are measured to yield information on the breakup of the target nucleus. We discuss the shapes of the fragment energy spectra across this broad range of fragment mass and pick out the fission component by its binary breakup. Multiplicities of coincident fast charged particles are presented as a measure of the degree of violence of the initial stage of the collision. Angular distribution of light fragments ($8\le Z\le 11$) give the average velocities imparted to the residue in central collisions, and these values are compared to the results of cascade calculations. Measurements of slow fragment multiplicities from central events reveal a many-fragment final state. In the light of this finding we develop a simple calculation of the Coulomb features of the observed spectra. The applicability of a statistical breakup model is discussed.

NUCLEAR REACTIONS Au(${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$, heavy fragment) $E=5, 8, 21, \mathrm{and} 42$ GeV; Au(${}_{}{}^4{}_{}{}^{}\mathrm{He}$, heavy fragment) $E=5\mathrm{}$ GeV; Au($p$, heavy fragment) $E=4.9\mathrm{}$ GeV. Measured $\frac{d^2\sigma }{\mathrm{dEd}\Omega }$, associated multiplicities of fast charged particles, binary coincidences, slow fragment multiplicities.

The ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$+ ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ elastic scattering has been measured for $E_{\mathrm{c}.\mathrm{m}.\mathrm{}}=14.6-31.3\mathrm{}$ MeV, ${\theta }_{\mathrm{c}.\mathrm{m}.\mathrm{}}=30\mathrm{}°-110\mathrm{}°$. The elastic data have been analyzed via a phase shift analysis, enabling the extraction of model independent sets of phase shift parameters. The extracted $J^{\pi }$ values for the intermediate structure resonances at $E_{\mathrm{c}.\mathrm{m}.\mathrm{}}=18.4, 19.3, \mathrm{and} 20.3$ MeV are ${12}^{+}$, ${12}^{+}$, and ${12}^{+}$ or ${14}^{+}$, respectively. The questions of ambiguities in the phase shift analysis and the comparison with $J^{\pi }$ values deduced from other experiments are discussed. Evidence is presented for the existence of gross structure resonances. The elastic scattering has also been analyzed using the sum-of-differences method to directly extract the total reaction cross section. The results of these analyses are compared to existing models of the origin of intermediate structure resonances.

NUCLEAR REACTIONS Measured the ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$+ ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ elastic scattering, $E_{\mathrm{c}.\mathrm{m}.\mathrm{}}=14.6-31.3\mathrm{}$ MeV, ${\theta }_{\mathrm{c}.\mathrm{m}.\mathrm{}}=30\mathrm{}°-110\mathrm{}°$. Phase shift analysis, sum-of-differences analysis.