Levels in ${}_{}{}^{81}{}_{}{}^{}\mathrm{Kr}$ have been investigated using the ${}_{}{}^{81}{}_{}{}^{}\mathrm{Br}(p,n\gamma )$ reaction. The first excited state of ${}_{}{}^{81}{}_{}{}^{}\mathrm{Kr}$ has been confirmed at 49.6 keV by observation of the $\gamma$-ray transition to the ground state. A level scheme has been constructed consistent with present Si(Li)-Ge(Li) coincidence measurements and with previous ${}_{}{}^{80}{}_{}{}^{}\mathrm{Kr}\mathrm{}(d,p)\mathrm{}$ and radioactive decay data.

NUCLEAR REACTIONS ${}_{}{}^{81}{}_{}{}^{}\mathrm{Br}(p,n\gamma )$, $E=5.0\mathrm{}$ MeV; measured $E_{\gamma }$, $I_{\gamma }$, $\gamma -\gamma$ coin, ${}_{}{}^{81}{}_{}{}^{}\mathrm{Kr}$ deduced levels, $J$, $\pi$. Enriched target.

Various results concerning the necessary inclusion of two-cluster partitions in $N$-particle channel coupling schemes in order to avoid manifestly nonunique scattering integral equations are derived and clarified. An adaptation of the Foldy-Tobocman argument in $T$-matrix form to this case is employed.

NUCLEAR REACTIONS Scattering theory, $N$-particle scattering, channel coupling schemes.

Previous results for high-spin decay schemes in ${}_{}{}^{38}{}_{}{}^{}\mathrm{Ar}$ and ${}_{}{}^{39}{}_{}{}^{}\mathrm{K}$ obtained by the Brook-haven group from heavy-ion fusion-evaporation $\gamma$-ray spectroscopy are revised and some new results are given. In particular, guided by recently published particle-$\gamma$ coincidence measurements, we find that in both nuclei the time sequences of two $\gamma$ transitions in cascade were wrongly ordered. Thus, in ${}_{}{}^{38}{}_{}{}^{}\mathrm{Ar}$ a 7609-keV level is replaced by a $J^{\pi }=\left(7^{-}\right)$ 8972-keV level, while in ${}_{}{}^{39}{}_{}{}^{}\mathrm{K}$ a 4831-keV level is replaced by a 7142-keV level. In addition, new mean lives of 17 ± 3 and 20 ± 1.5 ps were extracted for the ${}_{}{}^{39}{}_{}{}^{}\mathrm{K}$ 6476- and 8029-keV levels, respectively.

NUCLEAR REACTIONS ($\mathrm{HI},xn,yp,z\alpha$) studies for $A\sim 40$ nuclei, ${}_{}{}^{14}{}_{}{}^{}\mathrm{N}$ projectile, ${}_{}{}^{27}{}_{}{}^{}\mathrm{Al}$ target, $E=40\mathrm{}$ MeV, measured $\sigma (E_{\gamma },\theta )$ and $\gamma \gamma$ coin; deduced ${}_{}{}^{38}{}_{}{}^{}\mathrm{Ar}$ and ${}_{}{}^{39}{}_{}{}^{}\mathrm{K}$ levels; measured recoil distance; deduced ${\tau }_m$.

The angular correlation of photons emitted in double internal bremsstrahlung associated with electron capture has been investigated. A ${}_{}{}^{37}{}_{}{}^{}\mathrm{Ar}$ source was studied and the correlation investigated for relative emission angles which ranged from 60 to 140°. The results are in reasonable agreement with an existing theoretical prediction.

RADIOACTIVITY ${}_{}{}^{37}{}_{}{}^{}\mathrm{Ar}$: measured angular correlation of DIB.

An orthonormal set of single-nucleon orbital functions, explicity satisfying the Hartree-Fock equations for occupied states, is introduced and shown to display, assuming several parameterizations of the Skyrme interaction, $\alpha$-clustering effects at low densities in nuclear matter.

NUCLEAR MATTER Hartree-Fock; nuclear structure.

We suggest that two-nucleon transfer reactions be treated within the framework of the SU(6) boson model. We derive the corresponding intensity rules in the vibrational, SU(5), and rotational, SU(3), limit. We show that the failure of the pair vibrational model in accounting for the observed intensities is due to the neglecting of the finite dimensionality of the proton and neutron shells.

NUCLEAR REACTIONS Two-nucleon transfer, boson model, intensity rules given in vibrational and rotational limit.

A statistical study is made of 1131 ${}_{}{}^4{}_{}{}^{}\mathrm{He}\mathrm{}(p,p)\mathrm{}{}_{}{}^4{}_{}{}^{}\mathrm{He}$ scattering data from 91 experiments in the range 0 to 18 MeV. A goodness-of-fit test is made comparing the 91 data subsets to the $R$-matrix predictions of Dodder et al., who use these data. The result is generally satisfactory indicating a proper random fluctuation of the experimental data sets; however, there is some evidence of an overestimation of errors in several experiments.

NUCLEAR REACTIONS ${}_{}{}^4{}_{}{}^{}\mathrm{He}\mathrm{}(p,p)\mathrm{}$, 0 to 18 MeV, statistical study of $R$-matrix analysis.

A simple method based on a linearized Hartree-Fock technique is proposed to calculate effective charges and isoscalar nuclear shifts. As an application, the change in rms radii near closed shell nuclei is calculated using a Skyrme interaction.

NUCLEAR STRUCTURE Hartree-Fock calculation of effective charge, isoscalar nuclear shift, Skyrme interaction.

The $\alpha Q$ values of even-even nuclei are analyzed in terms of the $\alpha$-(nucleus $A$) separation energy. Assuming a simple Woods-Saxon potential with given diffuseness and radius parameters $a$ and $R$, the potential depth is determined from the $\alpha Q$ value for each node quantum number $N_r$. It is found that the depth obtained is generally a smooth function of $A$ having a jump at the closed shell nucleus. It is also empirically found that the depth for closed shell plus $\alpha$-particle-type nuclei is given by $\left[\frac{4A}{\mathrm{}(A+4\mathrm{})\mathrm{}}\right]V_{n-A}$ for the $N_r\text{'}\mathrm{s}$ appropriate for the shell model, and is consistent with intermediate-energy $\alpha$-scattering data.

NUCLEAR REACTIONS Systematics of the $\alpha$-nucleus potential depths obtained from $\alpha Q$ values.

Scattering formalisms that take antisymmetrization into account may result in nonsymmetric nonlocal potentials for the effective interaction. Furthermore, when the incident particle state is antisymmetrized with respect to single-particle states of the target, these states appear as redundant states in the scattering spectrum. This requires the Fredholm determinant associated with the kernel of the integral equation for the physical wave function to be zero for all wave numbers of the incident particle. Under these circumstances, a scattering solution may not exist. The conditions for existence are examined, and a consistency condition is established. Evidence is presented that the nonlocal potential in the full Hartree-Fock equation including target excitations is not symmetric, and thus that the equation will not exhibit a scattering solution unless certain consistency requirements are met in its construction. Since the scattering solution must be orthogonal to each of the single-particle states of the target, the standard procedure in Hartree-Fock scattering formalisms is to drop all terms in the potential which project onto these states. This results in the reduced Hartree-Fock equation usually considered. It is demonstrated that constructing the reduced equation without reference to the full equation may result in a failure of the solution of the reduced equation to meet the consistency condition required for it to be a solution of the full equation.

NUCLEAR REACTIONS Antisymmetrization, scattering by a nonlocal potential, nonsymmetric nonlocal potentials, Hartree-Fock scattering, redundant states.

The lifetime values of 105 ± 15, 8.4 ± 0.4 16 ± 8, and 185 ± 15 fs for the 2313, 3948, 5690, and 6204 keV levels of ${}_{}{}^{14}{}_{}{}^{}\mathrm{N}$, respectively, were measured using the Doppler-shift attenuation method through the ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}(p,\gamma ){}_{}{}^{14}{}_{}{}^{}\mathrm{N}$ reaction at $E_p=1150\mathrm{}$ keV. For the Doppler-shift attenuation analysis the correction factors of the nuclear and electronic stopping powers were determined by measuring Doppler-shift attenuation and line shape of $\gamma$ rays from the 6204→2313 keV transition and by measuring range values of the 20, 40, 60, 80, and 100 keV ${}_{}{}^{15}{}_{}{}^{}\mathrm{N}$ nuclei. The two most commonly used scattering potentials, i.e., Thomas-Fermi and Lenz-Jensen, were used in the estimation of the correction factors and $F\left(\tau \right)$ curves. Both potentials were shown to give the same lifetimes. All calculations were done by the Monte Carlo method. The accuracy of the results and the transition rates in the frame of a weak coupling model are discussed.

NUCLEAR REACTIONS ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}(p,\gamma )$, $E=1.15\mathrm{}$ MeV; measured $E_{\gamma }$, Doppler-shift attenuation. ${}_{}{}^{14}{}_{}{}^{}\mathrm{N}$ levels deduced, $\tau$, Ge(Li) detector, enriched target.

The equality of the two averages is shown for $S$ matrix elements and cross sections (and other quantities of physical interest) by expanding the scattering matrix in powers of the interaction matrix elements (Born series), and by demonstrating the equality of the two averages for each term separately of the resulting simple or double (or multiple) Born series. As an input, we use the equality of ensemble average and level average over all possible combinations of interaction matrix elements and level energies.

NUCLEAR REACTIONS Statistical theory. Calculation of mean values as ensemble averages or energy averages yields identical results.

Angular distributions of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ inelastic scattering from the stable even isotopes of samarium were measured at an incident particle energy of 53.4 MeV. The data for $2^{+}$ and $3^{-}$ collective states extend to 150° for ${}_{}{}^{144,148,150}{}_{}{}^{}\mathrm{Sm}$ and to 100° for $2^{+}$ states in ${}_{}{}^{152,154}{}_{}{}^{}\mathrm{Sm}$. The cross sections span approximately seven orders of magnitude. The data were analyzed with the distorted-wave Born approximation and the strong-coupling approximation. Quadrupole and octopole deformation parameters thus obtained are compared with results from analyses of proton and $\alpha$-particle-scattering data.

NUCLEAR REACTIONS ${}_{}{}^{144,148,150,152,154}{}_{}{}^{}\mathrm{Sm}$ (${}_{}{}^3{}_{}{}^{}\mathrm{He}$, ${}_{}{}^3{}_{}{}^{}\mathrm{He}\prime$) $E_{{}_{}{}^3{}_{}{}^{}\mathrm{He}}=53.4\mathrm{}$ MeV, measured $\sigma \left(\theta \right)$ for $2^{+}$ and $3^{-}$ states, DWBA and SCA analyses, deduced deformation parameters.

A finite expansion for the scattering of a structureless projectile from a complex target of $A$ particles is presented. This development is given as a spectator expansion, in the sense that the first term represents the scattering of the projectile from single target consitutent particles, with all other target particles playing a passive role (i.e., acting as spectators). Similarly, the second term represents the scattering from pairs of target particles with ($A-2\mathrm{}$) spectators, and so on. It is demonstrated that such expansions, one of which has been obtained previously as a resummation of the multiple scattering series, are very general in nature and obtain under circumstances for which the standard multiple scattering treatment is not valid.

NUCLEAR REACTIONS Scattering theory, nonperturbative finite expansion of the transition operator for an $A$-body target.

Differential cross sections and vector analyzing powers were measured at an incident deuteron energy of 16 MeV for the ${}_{}{}^{70}{}_{}{}^{}\mathrm{Ge}(d,t){}_{}{}^{69}{}_{}{}^{}\mathrm{Ge}$ reaction and at 12 MeV for the ${}_{}{}^{70}{}_{}{}^{}\mathrm{Ge}(d,p){}_{}{}^{71}{}_{}{}^{}\mathrm{Ge}$ reaction. The data are generally well reproduced by the distorted-wave Born approximation except for the forward-angle analyzing-power data for $d,t$ transitions to ${1/2}^{-}$ states in ${}_{}{}^{69}{}_{}{}^{}\mathrm{Ge}$. Several spin assignments in ${}_{}{}^{69}{}_{}{}^{}\mathrm{Ge}$ are made from the vector analyzing power of the $\mathrm{Ge}(\overrightarrow{d},t)$ reaction. A high-resolution study of the ${}_{}{}^{70}{}_{}{}^{}\mathrm{Ge}(d,p){}_{}{}^{71}{}_{}{}^{}\mathrm{Ge}$ reaction identified over 50 ${}_{}{}^{71}{}_{}{}^{}\mathrm{Ge}$ levels as well as a number of additional states corresponding to contaminants in the target. Assignments of $l_n$ were made to 44 levels in ${}_{}{}^{71}{}_{}{}^{}\mathrm{Ge}$. On the basis of analyzing-power measurements at 16° and 20°, values of $J^{\pi }$ could be assigned to 29 levels, of which 15 are new assignments. A number of tentative $J^{\pi }$ assignments are also made. Some of the results of the present work are relevant to the weak-coupling interpretation of ${}_{}{}^{69,71}{}_{}{}^{}\mathrm{Ge}$ by Eberth et al. Some $l=0\mathrm{} \mathrm{and} 2$ transitions were observed in ${}_{}{}^{70}{}_{}{}^{}\mathrm{Ge}(d,t)$ but the $l_n=0\mathrm{}$ cross-section data are not well fitted by the distorted-wave Born approximation. A sum-rule analysis was made and results are compared with previous work and with the predictions of the simple pairing theory.

NUCLER REACTIONS ${}_{}{}^{70}{}_{}{}^{}\mathrm{Ge}(\overrightarrow{d},t){}_{}{}^{69}{}_{}{}^{}\mathrm{Ge}$. Measured $\sigma \left(\theta \right)$ and $A_y\left(\theta \right)$, ${\theta }_{\mathrm{lab}}=15\mathrm{}°-80°$, $E=16\mathrm{}$ MeV. ${}_{}{}^{70}{}_{}{}^{}\mathrm{Ge}(\overrightarrow{d},p){}_{}{}^{71}{}_{}{}^{}\mathrm{Ge}$. Measured $\sigma \left(\theta \right)$, ${\theta }_{\mathrm{lab}}=15\mathrm{}°-90°$, $A_y\left(\theta \right)$, ${\theta }_{\mathrm{lab}}=16\mathrm{}°, 20°$. $E=12\mathrm{}$ MeV. DWBA analyses. Extracted $J^{\pi }$ and $S_j$.

The ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ar}(p,d){}_{}{}^{39}{}_{}{}^{}\mathrm{Ar}$ reaction has been studied at a bombarding energy of 35 MeV. A thin planar gas cell target allowed an energy resolution of 30 keV to be achieved. Excitation energies of levels in ${}_{}{}^{39}{}_{}{}^{}\mathrm{Ar}$ up to 5.6 MeV were obtained, angular distributions being taken from 6° to 40° (laboratory). Distorted-wave Born approximation fits were made to the observed angular distributions of the stronger states and spectroscopic factors extracted. The results are compared with previous experiments and to shell-model calculations. The data support the contention that much of the antianalog state strength is contained in the 3.379 MeV state.

NUCLEAR REACTIONS ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ar}(p,d)$, $E_p=35\mathrm{}$ MeV, $\Delta E=30\mathrm{}$ keV, measured $\sigma (E_d,\theta )$; DWBA analysis, deduced $l$ values and spectroscopic factors. Planar gas cell target.

Angular distributions have been measured for transitions populating residual states of ${}_{}{}^{167}{}_{}{}^{}\mathrm{Er}$ through the ${}_{}{}^{168}{}_{}{}^{}\mathrm{Er}\mathrm{}(d,t)\mathrm{}$ reaction at $E_d=17\mathrm{}$ MeV. Many of these angular distributions have shapes which are well reproduced by distorted-wave Born-approximation calculations. A small but significant number of angular distributions are anomalous; i.e., either they do not resemble any reasonable distorted-wave Born-approximation calculation or they can only be fitted with a calculation which assumes an $l$ value incompatible with the previous Nilsson model assignment for the residual state. While the summed spectroscopic factors are in good agreement with Nilsson model expectations, there is considerable evidence of fragmentation of the single-quasihole strength and the spectroscopic factors for several levels deviate significantly from Nilsson model predictions, even though Coriolis coupling has been included in the model calculation. The observation of several strongly anomalous angular distributions almost certainly indicates that the assumption of a simple one-step direct reaction mechanism breaks down for some of these transitions. The discrepancies between the model prediction and the experimental spectroscopic factors could arise either from a multistep mechanism or from mixing of the simple quasihole states with more complicated states of the same spin-parity. The observed fragmentation of strength is strong evidence for such mixing.

NUCLEAR REACTIONS ${}_{}{}^{168}{}_{}{}^{}\mathrm{Er}\mathrm{}(d,t)\mathrm{}$, $E_d=17\mathrm{}$ MeV measured $\sigma \left(\theta \right)$; DWBA analysis, deduced levels, $l$ values, spectroscopic factors. Enriched targets.

Differential cross sections for elastic and inelastic scattering of neutrons from ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ and ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ targets have been measured at 11, 20, and 25.7 MeV incident energies. The elastic scattering data were used to provide optical model parameters for use in macroscopic and microscopic model analyses of the data using the distorted wave method. The macroscopic model analysis resulted in deformation parameters consistent with those obtained in analogous ($p,p^{\prime }$) reactions which excite the $3^{-}$ and $5^{-}$ collective states in these nuclei. The macroscopic model provides a good description of the differential cross sections. The microscopic model analysis used random-phase approximation detailed wave vectors for the states involved. Two "realistic" effective interactions were studied, one with and one without density dependence. The imaginary contribution to the effective interaction was found to be important in reproducing the shape of the differential cross section in ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$. Exchange amplitudes were calculated exactly. The magnitudes of the $3^{-}$ cross sections for both nuclei were well described by these calculations.

NUCLEAR REACTIONS ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}(n,n)$ and ($n,n^{\prime }$), $E_n=11\mathrm{} \mathrm{and} 25.7$ MeV, $\mathrm{Ca}(n,n)$ and ($n,n^{\prime }$), $E_n=11\mathrm{} \mathrm{and} 20$ MeV. Measured $\sigma \left(\theta \right)$. Distorted wave analysis.

The spin-spin dependence of the total cross section of ${}_{}{}^{59}{}_{}{}^{}\mathrm{Co}$ for neutrons was measured at seven energies. A polarized neutron beam with energies of 8.2, 11.1, 12.8, and 14.1 MeV from the ${}_{}{}^2{}_{}{}^{}\mathrm{H}(d,\overrightarrow{n}){}_{}{}^3{}_{}{}^{}\mathrm{He}$ reaction and of 23.0, 27.5, and 30.6 MeV from the ${}_{}{}^3{}_{}{}^{}\mathrm{H}(d,\overrightarrow{n}){}_{}{}^4{}_{}{}^{}\mathrm{He}$ reaction was transmitted through a cobalt sample. The cobalt target was cooled with a ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ dilution refrigerator and polarized in a magnetic field. The transmission rates of the neutrons were compared for parallel and antiparallel spin orientations. The values obtained for the spin-spin cross section are much smaller than those found below 3 MeV. Deduced strengths of spin-spin terms in the optical model are in the range of 1 MeV. This is of the same order of magnitude as estimated from microscopic calculations.

NUCLEAR REACTIONS ${}_{}{}^{59}{}_{}{}^{}\overrightarrow{\mathrm{Co}}+\overrightarrow{n}$, $E_n=8\mathrm{}-31$ MeV; measured spin-spin dependent transmission; deduced ${\sigma }_{\mathrm{ss}}$; deduced optical model spin-spin terms.