A semiclassical model is developed to give a first estimate of the expected cross section for observation of the recoil proton in quasielastic pion scattering. Advantages of detecting the proton, rather than the scattered pion, are discussed.

NUCLEAR REACTIONS calculated $\sigma \left(\theta \right)$ for ($\pi , p$), quasielastic scattering; semiclassical model.

Measurements of proton total reaction cross sections for ${}_{}{}^9{}_{}{}^{}\mathrm{Be}$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, and ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$ have been made in the energy range between 20-44 MeV. The cross sections show irregular energy variation for ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ at about 23.8 and 25.9 MeV, and for ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$ at 30.3 and 33.5 MeV; irregularities were not observed clearly for ${}_{}{}^9{}_{}{}^{}\mathrm{Be}$ or ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$.

NUCLEAR REACTIONS ${}_{}{}^9{}_{}{}^{}\mathrm{Be}$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$: $20 \mathrm{MeV}<T<\mathrm{}44\mathrm{} \mathrm{MeV}$, between 23 and 51 measurements for each nucleus; measured ${\sigma }_R$; natural targets, except for ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ gas and Si${\mathrm{O}}_2$ used for ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ measurements.

Off-energy-shell $T$ matrix elements for any partial wave are calculated using realistic local two-nucleon potentials with soft or hard cores. The off-shell behaviors are discussed in relation to on-shell values. Off-shell cross sections involved in the impulse approximation of nuclear reactions are analyzed. The validity of some usual on-shell approximations is examined.

NUCLEAR REACTION Calculated off-energy-shell $T$ matrix for local realistic potential with soft or hard cores.

A multipartition version of the Feshbach unified nuclear reaction theory is formulated. The new theory can deal with rearrangement processes as well as with elastic and inelastic processes. The generalization results from replacing the Lippmann-Schwinger equation for the transition operator by the Baer-Kouri or Kouri-Levin coupled integral equations.

NUCLEAR REACTIONS Unified nuclear reaction formalism of Feshbach generalized to include rearrangement channels.

Two effects of a direct reaction on compound processes are discussed: the enhancement of the average compound cross section that competes with the direct process and the cross correlations in the fluctuations of cross sections involving the directly coupled channels. For the case of two directly coupled channels it is shown that both effects are maximized at the causality limit of the average $S$ matrix where the penetration matrix $P$ is singular. Computer experiments demonstrate that both effects fall off sharply when $\overline{S}$ is not near this causality limit and similar reductions in the effects are expected when more than two channels are strongly coupled to one another. A formula based upon the Engelbrecht-Weidenmüller transformation and the $M$-cancellation principle gives an excellent account of the enhancements obtained from the computer experiments. It also gives a qualitative account of the magnitudes of the cross section cross correlation coefficients. A somewhat more complicated formula due to Hofmann et al. also gives a good account of the enhancements. Both of these formulas agree with that of Kawai et al. in the limit of large width-to-spacing ratios.

NUCLEAR REACTIONS Average compound cross sections, cross correlations, direct reaction effects.

The inelastic scattering of 35-MeV protons by ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ was measured with a resolution of 5-8 keV. Angular distributions of states up to an excitation energy of about 8 MeV were obtained. Collective model calculations enabled the $l$ transfers of many states to be identified. The possibility of excitation of $1^{+}$ states is discussed. Microscopic model calculations were made with both phenomenological and theoretical information. The large number of observed unnatural parity states permitted the role of noncentral forces in these transitions to be investigated.

NUCLEAR REACTIONS ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}(p, p)$, $E=35\mathrm{}$ MeV; measured $\sigma \left(\theta \right)$, $\theta =10-100\mathrm{}°$. Deduced $L$, ${{\beta }_L}^2$. Microscopic DWBA analysis.

The scattering of 79.1 MeV $\alpha$ particles from ${}_{}{}^{40,48}{}_{}{}^{}\mathrm{Ca}$ has been measured with high relative accuracy and analyzed with a folding model for the optical potential. The result for $\Delta R$, the difference between the ${}_{}{}^{40,48}{}_{}{}^{}\mathrm{Ca}$ rms matter radii, is $\Delta R=0.05\mathrm{}\pm 0.04$ fm. This result appears smaller than most previous measurements which employed Woods-Saxon potentials or diffraction analyses. Most of the discrepancies are removed when the definition of the matter radius used in each analysis is scrutinized. Combining $\Delta R$ with the results of electromagnetic experiments yields the relative neutron-proton radius difference in ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}$; $R_n\mathrm{}\left(48\right)-R_p\mathrm{}\left(48\right)=0.03\mathrm{}\pm 0.08$ fm. These results are compared with Hartree-Fock calculations which tend to produce larger differences. Possible reasons for this discrepancy are discussed.

NUCLEAR REACTIONS ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}(\alpha , \alpha )$ $E=79.1\mathrm{}$ MeV; measured $\sigma \left(\theta \right)$; $\theta =10-45\mathrm{}°$, enriched targets. Microscopic optical model analysis; deduced rms matter radii.

Vibrational states in ${}_{}{}^{246}{}_{}{}^{}\mathrm{Cm}$ and ${}_{}{}^{248}{}_{}{}^{}\mathrm{Cm}$ have been investigated by inelastic deuteron scattering. The $K^{\pi }=0^{-}, 1^{-}, \mathrm{and} 2^{-}$ octupole states are identified in ${}_{}{}^{246}{}_{}{}^{}\mathrm{Cm}$ at 1251, 1080, and 842 keV, respectively. In ${}_{}{}^{248}{}_{}{}^{}\mathrm{Cm}$ several excited states above 1 MeV have been identified; however, definite assignments for these states, except a $J^{\pi }=3^{-}$ state at 1095 keV and a $J^{\pi }=2^{+}$ state at 1050 keV, could not be made. Differences between the ${}_{}{}^{246}{}_{}{}^{}\mathrm{Cm}$ and ${}_{}{}^{248}{}_{}{}^{}\mathrm{Cm}$ level spectra are interpreted in terms of their microscopic components. Reduced transition probabilities $B\left(E3\mathrm{}\right)\mathrm{}$ are extracted for the observed $J^{\pi }=3^{-}$ levels and are compared with theoretical values available in the literature.

NUCLEAR REACTIONS ${}_{}{}^{246,248}{}_{}{}^{}\mathrm{Cm}(d, d^{\prime })$,, $E_d=15\mathrm{}$ MeV; measured $\sigma (E_{d^{\prime }}, \theta )$. ${}_{}{}^{246,248}{}_{}{}^{}\mathrm{Cm}$ deduced levels, $J$, $K$, $\pi$, $B\left(E3\mathrm{}\right)\mathrm{}$. Enriched targets.

The electrostatic potential between nuclei having arbitrary charge distributions is expressed as a triple integral. For two nuclei having uniform distributions, an algebraic expression for the electrostatic potential is obtained. This is then compared with the electrostatic potentials normally used in various nucleus-nucleus optical model calculations. The results of the former are found to be significantly different from the latter. For Fermi-Fermi types of charge distributions, the nuclei may be assumed to be uniformly charged spheres without much loss of accuracy. If one of the charge distributions has either Gaussian or harmonic well shape, the triple integral is reduced to a single integral. For a number of nucleus-nucleus pairs, we have calculated the electrostatic potentials assuming them to be uniformly charged spheres and compared them with those derived from realistic charge distributions. With a proper choice of radius parameters, the former can be used to a good degree of accuracy to generate the electrostatic potential between the nuclei.

NUCLEAR REACTIONS Realistic Coulomb potential for optical model.

The ${}_{}{}^{50}{}_{}{}^{}\mathrm{Cr}(p, {}_{}{}^3{}_{}{}^{}\mathrm{He}){}_{}{}^{48}{}_{}{}^{}\mathrm{V}$ reaction has been studied at 44.7 MeV for levels with an excitation energy up to 3.7 MeV. A distorted-wave Born approximation analysis permitted several $L$ assignments. A comparison with available information on ${}_{}{}^{48}{}_{}{}^{}\mathrm{V}$ is made.

NUCLEAR REACTION ${}_{}{}^{50}{}_{}{}^{}\mathrm{Cr}(p, {}_{}{}^3{}_{}{}^{}\mathrm{He})$,, $E_p=44.7\mathrm{}$ MeV; measured $\sigma (E_{{}_{}{}^3{}_{}{}^{}\mathrm{He}}, \theta )$; enriched target. Deduced energies, $L$ values of ${}_{}{}^{48}{}_{}{}^{}\mathrm{V}$ levels.

The mean lifetimes of the $6_1^{+}$ states in ${}_{}{}^{42}{}_{}{}^{}\mathrm{Ti}$ at 3041 keV and ${}_{}{}^{46}{}_{}{}^{}\mathrm{Ca}$ at 2974 keV have been measured with the pulsed beam method to be $\tau =5\mathrm{}\pm 2 \mathrm{ns}\mathrm{and} 15.2\pm 0.8 \mathrm{ns}$, respectively. The states were populated via the reactions ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}\left({}_{}{}^3{}_{}{}^{}\mathrm{He}n\right){}_{}{}^{42}{}_{}{}^{}\mathrm{Ti}$ at $E_{{}_{}{}^3{}_{}{}^{}\mathrm{He}}=12\mathrm{}$ MeV and ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}(\alpha , \alpha 2n){}_{}{}^{46}{}_{}{}^{}\mathrm{Ca}$ at $E_{\alpha }=50\mathrm{}$ MeV. The deduced $B(E2, 6^{+}\to 4^{+})$ values result in proton and neutron effective charges which are in good agreement with other ${\left(1\mathrm{}{f}_{\frac{7}{2}}\right)}^{\pm 2}$ nuclei. However, the present lifetime result for ${}_{}{}^{42}{}_{}{}^{}\mathrm{Ti}$ is in disagreement with another measurement for this nucleus. A comparison of $E2\mathrm{}$ effective charges extracted from all presently available $B\left(E2\mathrm{}\right)\mathrm{}$ data on the $6^{+}$→$4^{+}$→$2^{+}$→$0^{+}$ transitions in ${\left(1\mathrm{}{f}_{\frac{7}{2}}\right)}^{\pm 2}$ nuclei is presented.

NUCLEAR REACTIONS ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}({}_{}{}^3{}_{}{}^{}\mathrm{He}, n\gamma )$, $E=12\mathrm{}$ MeV, ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}(\alpha , \alpha 2n\gamma )$, $E=50\mathrm{}$ MeV; measured pulsed beam electronic timing; deduced $T_{\frac{1}{2}}$, $B\left(E2\mathrm{}\right)\mathrm{}$, effective charges. Natural and enriched targets, Ge(Li) detectors.

The populations of levels in ${}_{}{}^{184}{}_{}{}^{}\mathrm{W}$ in the ($n, \gamma$) reaction are studied. On the basis of the technique of population systematics it is argued that the 1713 and 1775 keV levels are not the $1^{+}$ and $2^{+}$ members of a $K=1\mathrm{}$ rotational band as recently proposed. Rather it is shown that the 1775 keV level is most likely a $K=2\mathrm{}$ bandhead. Arguments from earlier charged particle reaction data support these conclusions whose implications are important in the study of the breakdown of the Nilsson model in this mass region. An assessment is made of current interpretations of the states between 1600 and 1900 keV in ${}_{}{}^{184}{}_{}{}^{}\mathrm{W}$.

NUCLEAR REACTIONS ${}_{}{}^{183}{}_{}{}^{}\mathrm{W}(n, \gamma )$, $E=\mathrm{thermal}, 7.6$ eV. ${}_{}{}^{184}{}_{}{}^{}\mathrm{W}$ deduced level populations, $J$, $\pi$. Interpretation of 1775 keV level, rotational band assignments from 1600-1900 keV, fragmentation of Nilsson strength.

In a measurement of the angular distribution of proton groups from the ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}(d, p){}_{}{}^{49}{}_{}{}^{}\mathrm{Ca}$ reaction at $E_d=20\mathrm{}$ MeV, particular emphasis is placed on the observation of seven weak transitions to states below 5 MeV excitation. These distributions are analyzed using distorted-wave Born-approximation (DWBA) parameters fixed by fitting, at $E_d=13.0, 16.0, \mathrm{and} 19.3$ MeV, the angular distributions of the elastically scattered deuterons and the protons populating the single-particle $2p_{\frac{3}{2}}$, $2p_{\frac{1}{2}}$, and $1f_{\frac{5}{2}}$ states. The DWBA model also fits the ($d, p$) data of other investigators at $E_d=7.0\mathrm{} \mathrm{and} 10.0$ MeV. The same model is then applied to the seven weak transitions, for which earlier data are available at 7.5 and 10.0 MeV in addition to the new $E_d=20\mathrm{}$ MeV data. New partial angular distributions at 9.0 MeV are helpful in resolving some discrepancies. For three of the weak transitions, older $l$ assignments are confirmed, for two of them new $l$ assignments are made, and for two transitions the observed behavior is incompatible with any reasonable DWBA model. All seven transitions can be explained as populating 2p-1h states formed by coupling a neutron to excited states of the ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}$ core.

NUCLEAR REACTIONS ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}(d, p)$, $E_d=9, 13, 16, 19.3, 20$ MeV; measured $\sigma (E_p, \theta )$; deduced levels, $J$, $\pi$, $l$, spectroscopic factors. Enriched target.

The ${}_{}{}^{107,109}{}_{}{}^{}\mathrm{Ag}(p, t){}_{}{}^{105,107}{}_{}{}^{}\mathrm{Ag}$ reactions have been studied at 30 MeV bombarding energy. Tritons were detected with a 60 cm position-sensitive wire proportional counter backed by a plastic scintillator in the focal plane of a quadrupole-dipole-dipole-dipole (QDDD) spectrograph. Multiplet structure, interpretable as the coupling of a $2p_{\frac{1}{2}}$ proton to vibrational core states, was observed in both nuclei. In addition, some 50 levels in each nucleus were seen below about 3 MeV of excitation with a resolution of 10 keV. Distorted wave Born approximation calculations with simple two particle configurations worked rather well and permitted the determination of $L$ transfers. A considerable amount of ($p, t$) strength in the region from 2-3 MeV of excitation in each nucleus was observed, not all of which could be associated with expected weak coupling to the $3^{-}$ core state.

NUCLEAR REACTIONS ${}_{}{}^{107}{}_{}{}^{}\mathrm{Ag}$, ${}_{}{}^{109}{}_{}{}^{}\mathrm{Ag}(p, t)$, $E=30\mathrm{}$ MeV; measured $\sigma \left(\theta \right)$ and level energies; DWBA analysis, deduced $L$, $J^{\pi }$. Resolution 10 keV. Enriched targets.

The influence of a double-humped fission barrier on the photofission cross section far below the top of the barrier is considered. In the region about 2 MeV below the top of the outer barrier and at a cross section in the region of ${10}^{-9\mathrm{}}$-${10}^{-6\mathrm{}}$ b the photofission is expected to become almost entirely isomeric or delayed fission. When this occurs a "shelf" should appear in the photofission cross section where the cross section changes with energy far more slowly than at higher $\gamma$-ray energy. The cross section on the shelf can therefore be orders of magnitude higher than otherwise expected. While the angular distribution near the top of the barrier is expected to have a well-defined and nonisotropic angular dependence, the angular distribution on the shelf is expected to be isotropic. The cross sections, although small, appear to be measurable with bremsstrahlung beams in the 3- to 5- MeV range using a several hundred $\mu \mathrm{A}$ electron beam. Successful experiments of this type should provide information on the shape of the inner and outer barriers at much lower excitation energies than possible by other techniques.

NUCLEAR REACTIONS, FISSION Theory of photofission in the $E_{\gamma }=2-6\mathrm{}$ MeV region, a prominent shelf, experiments proposed.

The photofission cross section of ${}_{}{}^{238}{}_{}{}^{}\mathrm{U}$ was measured in the 2.75- to 5.75-MeV range. Cross sections as small as 2×${10}^{-11\mathrm{}}$ b were detected. A shelf was observed which had been predicted to occur in the cross section owing to the dominance of radiative decay of levels in the second well. Barrier penetrabilities for both inner and outer barriers were determined from these measurements and together with penetrabilities from other measurements were used to infer a potential shape for ${}_{}{}^{238}{}_{}{}^{}\mathrm{U}$. The total $\gamma$-ray strength function in the second well at 4.25 MeV was found to be 1.3×${10}^{-5\mathrm{}}$. From comparison of expected and measured cross sections at the lowest energies, the density of levels effective in photofission at 3 MeV was found to be greater than 4 per MeV.

NUCLEAR REACTIONS, FISSION Measured photofission cross section for ${}_{}{}^{238}{}_{}{}^{}\mathrm{U}$ in the 2.75-6-MeV interval. Deduced details of fission barrier shape and $\gamma$-ray decay properties in both inner and outer wells.

Established properties of the relativistic intrinsic and orbital angular momentum tensors are reviewed, and relationships between the components of these tensors are derived. These relationships reduce the number of unknowns that must be determined for the calculation of the relativistic spin of the heavy fragments in high-energy heavy-ion interactions, in the limit where quantum mechanics may be ignored. An example using the intranuclear-cascade model is presented.

NUCLEAR REACTIONS Heavy particle induced; angular momentum; theoretical; relativistic energies.