Formulas are presented which are useful for predicting the number of deuterons from cascade calculations of relativistic heavy ion collisions.

NUCLEAR REACTIONS Relativistic heavy ions, phase space, deuteron production.

It is shown that volume integrals of all the equivalent real potentials are equal in the tail region. This implies the crossing of all these potentials around the sensitive radius.

NUCLEAR REACTIONS Heavy ion optical model, real potential volume integral, sensitive radius.

Virtual photon spectra, corrected for finite nuclear size and charge are used to evaluate integrated cross sections for the scattering of 60 MeV muons from ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{58}{}_{}{}^{}\mathrm{Ni}$, ${}_{}{}^{63}{}_{}{}^{}\mathrm{Cu}$, and ${}_{}{}^{197}{}_{}{}^{}\mathrm{Au}$ nuclei. The ratios of the cross sections $\sigma ({\mu }^{-}, {\mu }^{-} n)$ to $\sigma ({\mu }^{+}, {\mu }^{+}, n)$ agree with the ${\mathrm{experiment}}^1$; the integrated cross sections individually are consistently a little less than experimental values.

NUCLEAR REACTIONS Muon scattering, virtual photon method, $\sigma ({\mu }^{+}, {\mu }^{+} n)$ for ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{58}{}_{}{}^{}\mathrm{Ni}$, ${}_{}{}^{63}{}_{}{}^{}\mathrm{Cu}$, ${}_{}{}^{197}{}_{}{}^{}\mathrm{Au}$, and $\sigma ({\mu }^{-}, {\mu }^{-} n)$ for ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{63}{}_{}{}^{}\mathrm{Cu}$, ${}_{}{}^{197}{}_{}{}^{}\mathrm{Au}$ at $E_{\mu }=60\mathrm{}$ MeV evaluated using second order Born approximation.

We reply to a recent comment concerning our work on nuclear hydrodynamics. A previous numerical error is corrected, and a sample of corrected results is presented.

NUCLEAR REACTIONS Heavy-ion reactions, nuclear hydrodynamics.

A measurement of the circular polarization of the $\gamma$ rays following the $\beta$ decay of ${}_{}{}^{56}{}_{}{}^{}\mathrm{Co}$ reveals an anisotropy coefficient of $A=0.017\mathrm{}\left(13\right)\mathrm{}$. A Fermi to Gamow-Teller ratio, $\frac{M_F{C}_{\mathrm{V}}}{M_{\mathrm{GT}}{C}_A}=-0.13\mathrm{}\left(2\right)\mathrm{}$, was derived demonstrating a sizable interference between the two amplitudes.

NUCLEAR STRUCTURE Fermi and Gamow-Teller amplitudes in ${}_{}{}^{56}{}_{}{}^{}\mathrm{Co}$ $\beta$ decay.

The interactions between a $K^{\pi }=5^{-}$ aligned two-quasineutron band and two octupole bands have been observed in ${}_{}{}^{164}{}_{}{}^{}\mathrm{Er}$. Band crossings occur in both even- and odd-spin sequences. The members of all three bands are observed from the bandheads ($I^{\pi }=1^{-}, 2^{-}, \mathrm{and} 5^{-}$) to about ${12}^{-}$.

NUCLEAR REACTIONS ${}_{}{}^{162}{}_{}{}^{}\mathrm{Dy}(\alpha , 2n\gamma ){}_{}{}^{164}{}_{}{}^{}\mathrm{Er}$, $E=24\mathrm{}$ MeV; measured $E_{\gamma }$, $I_{\gamma }(\theta , t)$, $\gamma -\gamma$ coin. Deduced levels, $K^{\pi }$, $I^{\pi }$, configurations. Two-band mixing analysis.

It is shown that the short-range character of the point-proton distribution in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ cannot be determined unambiguously from experimental data, and hence it cannot serve as a basis for determining aspects of the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ structure.

NUCLEAR STRUCTURE ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, ambiguities in charge density.

The effects of the 2p-2h ground state correlation, isobar-hole excitation, pion pair current, and pionic current on the electromagnetic $M1\mathrm{}$ strength are calculated as a first order perturbation correction, for ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}$, ${}_{}{}^{90}{}_{}{}^{}\mathrm{Z}$, and ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ using Hartree-Fock wave functions. The isobar-hole contribution is the dominant quenching mechanism in heavier nuclei, though the enhancement effect of the pion pair current is not at all negligible. We also comment on the Gamow-Teller strength.

NUCLEAR STRUCTURE Calculated $M1\mathrm{}$ strength ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}$, ${}_{}{}^{90}{}_{}{}^{}\mathrm{Zr}$, ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ including ground state correlation, isobar-hole excitation, meson exchange currents.

We have measured $\alpha$ + ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ total reaction cross sections (${\sigma }_R$) at $E_{\mathrm{lab}}=94\mathrm{} \mathrm{and} 173$ MeV/N using the attenuation method. Our data indicate a decreasing ${\sigma }_R\mathrm{}\left(E\right)\mathrm{}$ at intermediate incident energies. These results, along with existing measurements, are in quantitative agreement with theoretical predictions for ${\sigma }_R\mathrm{}\left(E\right)\mathrm{}$ which are based on nucleon-nucleon total cross sections.

NUCLEAR REACTIONS Attenuation measurements of $\alpha$ + ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ total reaction cross sections at $E_{\mathrm{lab}}=94\mathrm{} \mathrm{and} 173$ MeV/N.

A large discrepancy is found between the value for the integrated total photonuclear cross section for ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ determined from the sum of the partial cross sections and that obtained from a total photon absorption measurement.

NUCLEAR REACTIONS ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}(\gamma , \mathrm{tot}:E_{\gamma })$, ($\gamma , \mathrm{partial}:E_{\gamma }$): integrated cross sections; sum rule; meson-exchange factor $K$, $A$ dependence.

An effective boson Hamiltonian is derived from the original Hamiltonian of a many-fermion system. The results show that when the intrinsic motion of bosons is collective and the excitation energies of the system are small, the effective boson-boson interaction can be reduced to a two-body interaction while for odd systems or systems with uncorrelated fermions, there is an additional two-body boson-fermion interaction. Both these interactions can be calculated from the original fermion-fermion ones. Thus, the microscopic foundation of the interacting boson model is established without recourse to mapping procedures.

NUCLEAR STRUCTURE Interacting boson model, boson fermion hybrid representation, microscopic foundation.

The multichannel generalizations of the Feshbach formalism are considered. These generalizations are nonunique. We demonstrate the equivalence of two such generalizations and show that they reduce to the familiar coupled reaction channel formalism under certain approximations. The relationship to the many-body formulations of the multichannel problem proposed recently is also discussed.

NUCLEAR REACTIONS Coupled reaction channels, Feshbach formalism, and effective interactions.

The off-shell $t$ matrix is expressed as a sum of one nonseparable and one separable term so that it is useful for applications to more-than-two-body problems. All poles are involved in this one separable term. Both the nonseparable and the separable terms of the kernel $G_{0}t$ are regular at the origin. The nonseparable term of this kernel vanishes at large distances, while the separable term behaves asymptotically as the spherical Hankel function. These properties make our expression free from defects inherent in the Jost- or the $K$-matrix expressions, and many applications are anticipated. As the application, a compact expression of the many-level formula is presented. Also the application is suggested to the breakup three-body problem based on the Faddeev equation. It is demonstrated that the breakup amplitude is expressed in a simple and physically interesting form and we can calculate it in coordinate space.

NUCLEAR REACTIONS Off-shell $t$ matrix with one nonseparable and one separable term; many-level formula; three-body breakup amplitude; the Faddeev equation in coordinate space.

Elastic scattering of polarized protons of 80-180 MeV from a number of target nuclei ($24<A<\mathrm{}208\mathrm{}$) has been measured. The systematic energy and target mass dependence of the cross section and analyzing power angular distributions is discussed. The data have been analyzed within the framework of a conventional optical potential model, and the behavior of the potential parameters as a function of energy is investigated. Comparisons are made with results obtained at energies greater than 180 MeV and with predictions obtained from microscopic models.

NUCLEAR REACTIONS ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}(\overrightarrow{p}, p)$, $E_p=134.7\mathrm{}$ MeV; ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}(\overrightarrow{p}, p)$, $E_p=134.5\mathrm{}$ MeV; ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}(\overrightarrow{p}, p)$, $E_p=80.2, 181.5$ MeV; ${}_{}{}^{90}{}_{}{}^{}\mathrm{Zr}(\overrightarrow{p}, p)$, $E_p=79.6, 98.7, 134.8, 160$ MeV; ${}_{}{}^{92}{}_{}{}^{}\mathrm{Zr}(\overrightarrow{p}, p)$, $E_p=104\mathrm{}$ MeV; ${}_{}{}^{120}{}_{}{}^{}\mathrm{Sn}(\overrightarrow{p}, p)$, $E_p=104\mathrm{}$ MeV; ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}(\overrightarrow{p}, p)$, $E_p=79.8, 98, 182$ MeV; measured $\sigma \left(\theta \right)$, $A_y\left(\theta \right)$; optical-model analysis, deduced energy dependence, comparison with microscopic models.

A separable potential is proposed for the $n-p$ interaction in ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ and ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}-{}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}$ partial waves. In the singlet $S$ state the potential fits new phenomenological phases rather accurately in the laboratory kinetic energy range 0-600 MeV; it is also capable of reproducing singlet effective-range parameters in close agreement with experiment. In the coupled state ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}-{}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}$ the potential provides a correct description of the deuteron data ($E_D$, $p_D$, $Q$, $\eta$), while at the same time it adequately fits modern phenomenological phases up to $E_{\mathrm{lab}}\gtrsim 500$ MeV in both the $S$ and the $D$ wave; triplet effective-range parameters are also in agreement with experimental data. Only the mixing parameter ${\epsilon }_1$ deviates from predictions of phase-shift analyses. In the construction of the potential care was taken that its off-shell behavior be reasonable. As an eminent property it thus exhibits an off-shell behavior similar to the one of the Paris potential. In particular the ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}-{}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}$ potential yields a deuteron $S$-state wave function, which in momentum space shows a zero like the Paris potential. Still the model is simple enough to be of good use in modern computer codes for few-body systems and other nuclear applications.

NUCLEAR REACTIONS Separable potential proposed for $n-p$ interaction; fits new on-shell data; off-shell behavior compared to Paris potential.

A simple method is proposed for the calculation of the position and residue of low-energy resonances and virtual states located in the unphysical sheet associated with the lowest two body scattering threshold. Instead of analytically continuing the scattering equation for the $t$ matrix into the unphysical sheet we propose an approximate solution of the scattering equation with known analytic properties and continue it analytically into the unphysical sheet. The poles of the analytically continued solution in the unphysical sheet correspond to virtual states and resonances. The present method is applied to study the ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ virtual state of the two nucleon system, the Efimov virtual states in the three-boson Amado model, and the spin doublet virtual state of the three-nucleon system.

NUCLEAR REACTIONS Resonances and virtual states, nucleon-nucleon ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ virtual state, Efimov virtual states, three-nucleon spin doublet virtual state.

We present the ratio of experimental 0° ($p, n$) cross sections with the measured $B\left(M1\mathrm{}\right)\mathrm{}$ values for transitions in several nuclei. Taking the strong spin-flip transition in ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ for normalization, we find overall good agreement with shell-model predictions for ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}$, ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$, and ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}$. The values of the ratios vary considerably and are very sensitive to the relative amounts of spin and current contributions to each transition. Experimental ratios are also presented for ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ and ${}_{}{}^{26}{}_{}{}^{}\mathrm{Mg}$.

NUCLEAR REACTIONS Charge exchange ($p, n$) at 0°, $E_p=62-160\mathrm{}$ MeV, cross sections; targets ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}$, ${}_{}{}^{26}{}_{}{}^{}\mathrm{Mg}$, ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$, ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}$. Compared ${\sigma }_{p,n}\left(0\mathrm{}°\right)$ with $B\left(M1\mathrm{}\right)\mathrm{}$ values from inelastic electron scattering.

The Efimov effect in the $S$ wave spin doublet trinucleon system is studied and the virtual state of this system is found to be an Efimov virtual state. In the Efimov limit this state becomes a bound state. But as the strength of the spin triplet nucleon-nucleon interaction is increased, it first moves away from the lowest physical scattering threshold, then stops and turns around and eventually becomes a physical bound state. The movement of other Efimov states of this system were studied and once they become a virtual state they move into the left-hand cut with the increase of the strength of the triplet nucleon-nucleon potential and do not become resonances.

NUCLEAR REACTIONS Efimov effect, trinucleon system, Efimov virtual states, Amado model.

A broad resonance has been observed by inelastic scattering of 200 MeV protons from ${}_{}{}^{90}{}_{}{}^{}\mathrm{Zr}$, ${}_{}{}^{92}{}_{}{}^{}\mathrm{Zr}$, ${}_{}{}^{94}{}_{}{}^{}\mathrm{Zr}$, and ${}_{}{}^{96}{}_{}{}^{}\mathrm{Zr}$. This resonance has a sharply forward peaked angular distribution and an excitation energy and strength which strongly suggest that it is the $M1\mathrm{}$ giant resonance. Microscopic distorted wave impulse approximation calculations match the shape of the angular distribution reasonably well. The strength, however, is only about 30% of that predicted.

NUCLEAR REACTIONS ${}_{}{}^{90,92,94,96}{}_{}{}^{}\mathrm{Zr}(p, p^{\prime })$, $E=200\mathrm{}$ MeV, measured $\sigma \left(\theta \right)$, DWBA analysis, enriched targets, deduced levels, resolution 70 keV.

The vibrational states of ${}_{}{}^{156,158}{}_{}{}^{}\mathrm{Dy}$, ${}_{}{}^{162,164}{}_{}{}^{}\mathrm{Er}$, and ${}_{}{}^{168}{}_{}{}^{}\mathrm{Yb}$ have been studied via Coulomb excitation. Thin high purity targets were prepared in an isotope separator. The Coulomb excitation of these nuclei was studied by the scattering of 12-16 MeV $\alpha$ particles, which were detected in an Enge split-pole spectrograph. The reduced transition probabilities $B(E\lambda ;0_{\mathrm{g}.\mathrm{s}.\mathrm{}}^{+}\to I^{\pi }K)$ were obtained for $I^{\pi }K=2^{+}2, 2^{+}0$, and $I^{\pi }=3^{-}$ states in each nucleus. Limits on $B\left(E\lambda \right)$ values and values extracted from published ($d, d^{\prime }$) cross sections are also presented. Trends in $B\left(E2\mathrm{}\right)\mathrm{}$ values and energy levels for the Dy, Er, and Yb nuclei show that the $K^{\pi }=2^{+}$ levels are reasonably constant in energy for only Dy and Er. In all three nuclei $B\left(E2\mathrm{}\right)\mathrm{}$ values decrease with increasing neutron number. Except for the Dy isotopes, the lightest isotope has the largest $B\left(E2\mathrm{}\right)\mathrm{}$ value for the lowest lying $I^{\pi }K=2^{+}0$ state. $I^{\pi }=3^{-}$ states level energies and $B\left(E3\mathrm{}\right)\mathrm{}$ values are found to be in fair agreement with microscopic calculations. Values of $B\left(E2\mathrm{}\right)\mathrm{}$ for $K^{\pi }=2^{+}$ states are in fair agreement with microscopic calculations when these exist.

NUCLEAR REACTIONS ${}_{}{}^{156,158}{}_{}{}^{}\mathrm{Dy}(\alpha , {\alpha }^{\prime })$, ${}_{}{}^{162,164}{}_{}{}^{}\mathrm{Er}(\alpha , {\alpha }^{\prime })$, ${}_{}{}^{168}{}_{}{}^{}\mathrm{Yb}(\alpha , {\alpha }^{\prime })$, $E=12-16\mathrm{}$ MeV; measured $\sigma$, deduced $B\left(E\lambda \right)$. Enriched targets.