Free nucleon-nucleon scattering is investigated with the explicit treatment of $\Delta \mathrm{}\left(1236\right)\mathrm{}$ resonances. The coupled channel problem ($NN+N\Delta +\Delta \Delta$) is solved and compared with the results obtained from perturbation calculations (impulse approximation). The nucleon-nucleon potential used is the Reid-soft-core potential which has been modified according to the phase shift and deuteron binding energy requirements. The perturbation treatment turns out to be not a very good approximation as the $\Delta$ admixtures are off up to about 50% in a nonsystematic way.

NUCLEAR REACTIONS nucleon-nucleon scattering, resonance admixtures.

$\gamma$ rays emitted in the decay of 9.4-h ${}_{}{}^{201}{}_{}{}^{}\mathrm{Pb}$ were studied with Ge(Li) and NaI(Tl) detectors in singles, anti-Compton, anticoincidence, and integral coincidence configurations. The construction of a consistent decay scheme including 66 of the 74 $\gamma$ rays observed in the decay of ${}_{}{}^{201}{}_{}{}^{}\mathrm{Pb}$ was also aided by a two-dimensional $\gamma -\gamma$ coincidence experiment of 4096 × 4096 channels using two Ge(Li) detectors. The proposed levels in ${}_{}{}^{201}{}_{}{}^{}\mathrm{Tl}$ lie at 0, 331.15, 692.40, 1098.4, 1134.8, 1157.4, 1238.8, 1277.1, 1290.0, 1330.4, 1401.2, 1420.0, 1445.8, 1479.9, 1575.1, 1617.4, 1639.5, 1672.0, and 1755.3 keV with possible additional levels at 1550.5 and 1712.5 keV. Using existing conversion-electron data, conversion coefficients were calculated and multipolarity assignments proposed for 25 transitions. Based on these multipolarities, photon branching ratios, and $log\mathrm{ft}$ values, unique spin and parity assignments have been made for several states and limits placed on others. The resulting decay scheme is compared with recent theoretical calculations of level energies and $\gamma$ transition probabilities, using the intermediate-coupling unified model.

RADIOACTIVITY ${}_{}{}^{201}{}_{}{}^{}\mathrm{Pb}$ [from ${}_{}{}^{203}{}_{}{}^{}\mathrm{Tl}$($p$,$3n$) and ${}_{}{}^{203}{}_{}{}^{}\mathrm{Tl}$(${}_{}{}^3{}_{}{}^{}\mathrm{He}$,$5n$) ${}_{}{}^{201}{}_{}{}^{}\mathrm{Bi}\stackrel{\epsilon }{\to }{}_{}{}^{201}{}_{}{}^{}\mathrm{Pb}$]; measured $E_{\gamma }$, $I_{\gamma }$, ${\alpha }_K$, ${\alpha }_L$; deduced $log\mathrm{ft}$; ${}_{}{}^{201}{}_{}{}^{}\mathrm{Tl}$ deduced levels, $J$, $\pi$; Ge(Li) singles and $\gamma -\gamma$ coincidence.

The half-life of ${}_{}{}^{242}{}_{}{}^{}\mathrm{Pu}$ was measured relative to the half-lives of ${}_{}{}^{239}{}_{}{}^{}\mathrm{Pu}$ and ${}_{}{}^{238}{}_{}{}^{}\mathrm{Pu}$ in two independent sets of experiments. Synthetic mixtures were prepared using enriched isotopes of ${}_{}{}^{238}{}_{}{}^{}\mathrm{Pu}$, ${}_{}{}^{239}{}_{}{}^{}\mathrm{Pu}$, and ${}_{}{}^{242}{}_{}{}^{}\mathrm{Pu}$. The $\alpha$ activity ratios in these mixtures were determined by $\alpha$ spectrometry on electrodeposited sources using a silicon surface barrier detector while the atom ratios were obtained by mass spectrometry. The $\alpha$ activity ratios and the atom ratios were so chosen that these could be measured with a high precision and accuracy. A half-life value of (3.742±0.024) × ${10}^5$ yr is obtained using the half-life of ${}_{}{}^{239}{}_{}{}^{}\mathrm{Pu}$ as 24110 yr. The value obtained using the half-life of ${}_{}{}^{238}{}_{}{}^{}\mathrm{Pu}$ as 87.74 yr is (3.766±0.025) × ${10}^5$ yr. The average of these two independent sets of experiments leads to a half-life value of ${}_{}{}^{242}{}_{}{}^{}\mathrm{Pu}$ as (3.754±0.025) × ${10}^5$ yr. The uncertainty given on the values is a combination of the one standard deviation on the average value and the error evaluated from estimates on various error components. The discrepancy existing in the earlier published values for the half-life of ${}_{}{}^{242}{}_{}{}^{}\mathrm{Pu}$ determined relative to the half-lives of ${}_{}{}^{239}{}_{}{}^{}\mathrm{Pu}$ and ${}_{}{}^{238}{}_{}{}^{}\mathrm{Pu}$ is resolved.

RADIOACTIVITY ${}_{}{}^{242}{}_{}{}^{}\mathrm{Pu}$ half-life, relative to the half-life of ${}_{}{}^{239}{}_{}{}^{}\mathrm{Pu}$; measured: ${}_{}{}^{239}{}_{}{}^{}\mathrm{Pu}$/${}_{}{}^{242}{}_{}{}^{}\mathrm{Pu}$ $\alpha$ activity ratio by $\alpha$ spectrometry, ${}_{}{}^{239}{}_{}{}^{}\mathrm{Pu}$/${}_{}{}^{242}{}_{}{}^{}\mathrm{Pu}$ atom ratio by mass spectrometry; relative to the half-life of ${}_{}{}^{238}{}_{}{}^{}\mathrm{Pu}$; measured ${}_{}{}^{238}{}_{}{}^{}\mathrm{Pu}$/${}_{}{}^{242}{}_{}{}^{}\mathrm{Pu}$ $\alpha$ activity ratio by $\alpha$ spectrometry, ${}_{}{}^{238}{}_{}{}^{}\mathrm{Pu}$/${}_{}{}^{242}{}_{}{}^{}\mathrm{Pu}$ atom ratio by mass spectrometry.

The projected quasiboson model developed in a previous paper is extended to triaxial nuclei. Energy levels and electromagnetic properties are calculated and results presented for even-even Ba isotopes, $A=130-136\mathrm{}$.

NUCLEAR STRUCTURE Angular momentum projected triaxial quasiboson formalism, ${}_{}{}^{130,132,134,136}{}_{}{}^{}\mathrm{Ba}$, calculated levels, $J$, $\pi$, $B\left(E2\mathrm{}\right)\mathrm{}$, quadrupole moments.

The structure of the levels of ${}_{}{}^{212}{}_{}{}^{}\mathrm{Po}$ has been investigated within the model space of the lowest seven neutron and six proton single particle orbits; the matrix elements of Herling and Kuo were used. Agreement between experiment and theory for the lowest six levels, including the recently discovered $8^{+}$, is excellent. The isomer of 2.93 MeV is suggested to have spin ${16}^{+}$; the ${10}^{+}$ is not expected to be an isomer.

NUCLEAR STRUCTURE ${}_{}{}^{212}{}_{}{}^{}\mathrm{Po}$; calculated levels, $J^{\pi }$ realistic interaction.

A theorem is proved which states that the energies $E_i$ of a general translation-invariant system of $N$ identical particles interacting by pair potentials and obeying nonrelativistic quantum mechanics are bounded below one by one by the energies $E_i^L$ of a related model system consisting of ($N-1\mathrm{}$) noninteracting particles of the same symmetry type bound to a fixed center. The theorem is applied to a three-fermion harmonic oscillator test system whose exact eigenvalues are found. The ratios $\frac{E_i^L}{E_i}$ for this problem satisfy $\frac{E_i^L}{E_i}\ge \frac{E_0^L}{E_0}=\frac{\sqrt{3}}{4}$ for at least the first twenty three-body states.

NUCLEAR STRUCTURE Lower energy bounds provided by exactly soluble atomlike model. Justification for shell theory. Application to harmonic oscillator test system.

Collective $K^{\pi }=0^{+}$ multiphonon states are studied in a realistic case and compared to those obtained by use of two different boson expansions.

NUCLEAR STRUCTURE Multiphonon $K^{\pi }=0^{+}$ states in even-even deformed nuclei. Application to a realistic case. Comparison with the results of boson expansions.

Nuclear magnetic resonance experiments with oriented nuclei have been performed on ${}_{}{}^{127}{}_{}{}^{}\mathrm{Te}$, ${}_{}{}^{129}{}_{}{}^{}\mathrm{Te}$, and ${}_{}{}^{131}{}_{}{}^{}\mathrm{Te}$ implanted in iron, using the $A_1$ asymmetry from $\beta$ decay as resonance indication. From the resonance frequencies the following moments have been extracted: $\left|\mu \mathrm{}\right|=0.635\left(4\mathrm{}\right)\mathrm{}$ for ${}_{}{}^{127}{}_{}{}^{}\mathrm{Te}$, $\left|\mu \mathrm{}\right|=0.702\left(4\mathrm{}\right)\mathrm{}$ for ${}_{}{}^{129}{}_{}{}^{}\mathrm{Te}$, and $\left|\mu \mathrm{}\right|=0.696\left(9\mathrm{}\right)\mathrm{}$ for ${}_{}{}^{131}{}_{}{}^{}\mathrm{Te}$, in units of ${\mu }_N$. These are compared to theoretical calculations in the pairing-plus-quadrupole model, using experimental occupation coefficients.

NUCLEAR STRUCTURE ${}_{}{}^{127}{}_{}{}^{}\mathrm{Te}$, ${}_{}{}^{129}{}_{}{}^{}\mathrm{Te}$, and ${}_{}{}^{131}{}_{}{}^{}\mathrm{Te}$: isotope separation, implantation in Fe. NMR/ON: Si(Li) and Ge(Li) detectors. Deduced $\mu$: calculation $P$ and $Q$ model.

Let $F\left(q^2\right)=\left(\frac{4\pi }{\mathrm{Zq}}\right)\int 0^{\infty }rsin\mathrm{qr}\rho \mathrm{}\left(r\right)\mathrm{}\mathrm{dr}$ be a charge density form factor for an atomic nucleus. Then it is shown how rigorous upper and lower bounds can be imposed on a variety of indefinite integrals involving $\rho \mathrm{}\left(r\right)\mathrm{}$ starting from information about $F\left(q^2\right)$ which may itself derive from experimental measurements. Examples are provided in the context of the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ nucleus to demonstrate the feasibility of the method.

NUCLEAR STRUCTURE Moment theory, nuclear form factors.

The ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$(${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$,$t$)${}_{}{}^{21}{}_{}{}^{}\mathrm{Na}$ reaction has been studied at $\theta =0\mathrm{}°$ over an energy range from 37 to 50 MeV. Rapid fluctuations in the cross sections to individual final states were observed, but no conclusive evidence for pronounced resonant structure was observed, either at the 19.3 MeV (c.m.) anomaly or elsewhere.

NUCLEAR REACTIONS ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$(${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$,$t$); $E=37-50\mathrm{}$ MeV, measured $\sigma (E,E_t, \theta =0\mathrm{}°)$. ${}_{}{}^{21}{}_{}{}^{}\mathrm{Na}$ deduced levels.

A currently popular Dirac model of nuclear wave functions is examined to determine whether its ratio of "small" to "large" components agrees with recent data on the asymmetry in polarized $A=12\mathrm{}$ nuclear beta decay, as well as with $A=15\mathrm{}$ magnetic moments. The magnetic moments agree well, and no clear inconsistency is found with the beta-decay asymmetry, but future experiments on the latter might reveal a problem since theory and experiment disagree by about 1 standard deviation.

NUCLEAR STRUCTURE ${}_{}{}^{15}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{15}{}_{}{}^{}\mathrm{N}$; calculated magnetic moments. Relativistic single-particle wave functions, exchange current. $A=12\mathrm{}$; calculated axial dipole matrix element. Relativistic treatment, Cohen-Kurath configurations.

A revised version of the two-fireball model for relativistic light ion collisions is presented. This version utilizes our knowledge from nucleon-nucleon or nucleon-nucleus collisions to give an essentially parameter free transparency formulation. Calculations are done for proton and pion spectra.

NUCLEAR REACTIONS Relativistic collisions of ions; fireball; proton and pion spectra.

We consider the recently developed relativistic mean-field theory of nuclei. Exchange of scalar and vector mesons between slowly moving nucleons leads to an effective static potential interaction. We solve the Bethe-Goldstone equation for this static interaction in order to estimate the order of magnitude of the corrections to the theory due to short-range nucleon-nucleon correlations. We find that the masses of the scalar and vector mesons must be made quite small before the Hartree approximation is a reasonable first approximation.

NUCLEAR STRUCTURE Relativistic mean-field theory; role of short-range correlations; criteria for use of mean-field approximations.

Angular distributions to the low-lying states of ${}_{}{}^{38}{}_{}{}^{}\mathrm{K}$ have been measured in the ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$($p$,${}_{}{}^3{}_{}{}^{}\mathrm{He}$)${}_{}{}^{38}{}_{}{}^{}\mathrm{K}$ reaction at 42.5 MeV. Distorted-wave Born approximation calculations for these states, using spectroscopic amplitudes from the wave functions of Widenthal and Chung, underestimate the $1^{+}$, $T=0\mathrm{}$ cross sections, particularly for the 0.46 MeV state.

NUCLEAR REACTIONS ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$($p$,${}_{}{}^3{}_{}{}^{}\mathrm{He}$)${}_{}{}^{38}{}_{}{}^{}\mathrm{K}$, $E=42.5\mathrm{}$ MeV, measured $\sigma \left(\theta \right)$, microscopic DWBA analysis.

The $Q_{2+}$ value of ${}_{}{}^{98}{}_{}{}^{}\mathrm{Mo}$ is revised in the light of the existence of a fast $2^{+}$ →$0^{+}$′ transition. For constructive interference via the second $2^{+}$ excited state the determined static quadrupole moment of ${}_{}{}^{98}{}_{}{}^{}\mathrm{Mo}$ is $(-0.26\mathrm{}\pm 0.09) e$b. This $Q_{2+}$ corresponds to about 55% of that deduced from the rotational model.

NUCLEAR REACTIONS ${}_{}{}^{98}{}_{}{}^{}\mathrm{Mo}$($\alpha$,${\alpha }^{\prime }$), $E=8.0\mathrm{}$ MeV; ${}_{}{}^{98}{}_{}{}^{}\mathrm{Mo}$ (${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$′), $E=36.5\mathrm{}$ MeV; ${}_{}{}^{98}{}_{}{}^{}\mathrm{Mo}$ deduced $Q_{2^{+}}$ and $B (E2; 0^{+}\to 2^{+})$.

The analyzing power for the reaction ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$($\overrightarrow{p}$,$t$)${}_{}{}^{206}{}_{}{}^{}\mathrm{Pb}$ leading to the unnatural parity $3^{+}$ state at 1.34 MeV was measured at $E_p=22.0\mathrm{}$ MeV. The cross section of this reaction was fitted well by a two-step ($p$,$d$)($d$,$t$) calculation, but the analyzing power was not at all. The reaction mechanism of this transition is not explained by only the sequential transfer processes.

NUCLEAR REACTIONS ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$($\overrightarrow{p}$,$t$), $E=22.0\mathrm{}$ MeV; measured $\sigma (E_t,\theta )$, $A_y(E_t,\theta )$: First- and second-order DWBA analyses.

The (${}_{}{}^7{}_{}{}^{}\mathrm{Li}$, ${}_{}{}^8{}_{}{}^{}\mathrm{Be}$) reaction has been shown to be useful for studying proton pickup. States in ${}_{}{}^{39}{}_{}{}^{}\mathrm{K}$ have been studied at $E_{\mathrm{Li}}=34\mathrm{}$ MeV using a ${}_{}{}^8{}_{}{}^{}\mathrm{Be}$ detection system. Measured angular distributions from approximately 20° to 60° in the center of mass have been analyzed with finite range distorted-wave Born-approximation calculations in order to determine spectroscopic strengths. The data are compared with the results of another pickup reaction.

NUCLEAR REACTIONS ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$(${}_{}{}^7{}_{}{}^{}\mathrm{Li}$, ${}_{}{}^8{}_{}{}^{}\mathrm{Be}$)${}_{}{}^{39}{}_{}{}^{}\mathrm{K}$, measured $\sigma \left(\theta \right)$ at $E_{\mathrm{lab}}\left({}_{}{}^7{}_{}{}^{}\mathrm{Li}\right)=34\mathrm{}$ MeV for ${\theta }_{\mathrm{c}.\mathrm{m}.\mathrm{}} \mathrm{from} \approx 20° \mathrm{to} 60°$, $\Delta {\theta }_{\mathrm{c}.\mathrm{m}.\mathrm{}}\approx 2°$. Finite range DWBA analyses, deduced $C^{2}S$, and ${}_{}{}^8{}_{}{}^{}\mathrm{Be}$ optical parameters.

This paper derives cluster expansions for the three-body scattering problem. We determine, by computation, the utility of the exact and approximate descriptions that emerge from the cluster approach. In general, cluster expansions can give simple approximate solutions to the scattering process that are accurate whenever clusters dominate the underlying physical states of the system. The approach to the problem taken here is to employ the Karlsson-Zeiger integral equations to provide a theoretical framework that is natural for a cluster expansion. Eventually one can restate the scattering problem in terms of effective intercluster potentials. We construct integral equations whose solutions are the effective potentials. The cluster expansion for this problem leads to successively more exact effective potentials. For systems composed of either three bosons or fermions and interacting through separable potentials we compare exact three-body solutions in the bound state and elastic scattering sectors with those obtained by the cluster-expansion techniques.

NUCLEAR REACTIONS Three-body problem. Cluster representations. Approximation schemes.

Calculations based upon the Faddeev equations and measurements of proton-induced deuteron disintegration cross sections have been made at 23.0 and 39.5 MeV for two types of constant-relative-energy loci. In the first configuration the variation of the breakup cross section was studied for fixed values of the final state $\mathrm{NN}$ relative energies and a fixed momentum of one of the emerging protons. In the second configuration the variation of the breakup cross section was studied for equal relative energies between all three pairs of nucleons with the two protons emerging with equal momenta symmetrically with respect to the incident beam direction. At 39.5 MeV calculations and measurements were made of both types of loci while at 23.0 MeV only the second kinematical configuration was studied.

NUCLEAR REACTIONS ${}_{}{}^2{}_{}{}^{}\mathrm{H}\mathrm{}(p,2p)n$ at $T_p=23.0\mathrm{} \mathrm{and} 39.5$ MeV, measured two types of constant-relative-energy loci; comparisons with three-body calculations using the Reid soft-core potential.