In this work, we describe the fusion mechanism between two heavy ions as penetration through a parabolic barrier in the presence of a proper Coulomb interaction with appropriate boundary conditions. The parabolic potential is matched to the Coulomb potential in order to avoid any discontinuity of the potential surface. The Schrödinger equations in the exterior and interior regions containing, respectively, the Coulomb and the parabolic potentials, are solved and the penetrability function is calculated from the logarithmic derivative at the matching radius. The theory is then applied to calculate the fusion cross sections for the reactions ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$+${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$+${}_{}{}^{28,29,30}{}_{}{}^{}\mathrm{Si}$, ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$+${}_{}{}^{24,26}{}_{}{}^{}\mathrm{Mg}$, ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$+${}_{}{}^{28,29,30}{}_{}{}^{}\mathrm{Si}$, and ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$+${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ and are found to reproduce the data quite well. The theory is valid for energies both below and above the barrier.

NUCLEAR REACTIONS Heavy-ion fusion; effect of Coulomb interaction; analytic expression for the penetrability function; application to various reactions.

Transport model description of the correlation between the energy loss and the variance of the charge or mass distributions in the heavy-ion deep inelastic reactions has been modified to include in the theory the effect of the degree of neutron-proton correlation in the particle exchanges. With the use of the revised theoretical expressions and considering the role played by the presence of correlations, the contradictions and discrepancies seen in the earlier analysis are removed. A better quantitative agreement with the experimental results is obtained by the present formalism. This brings out in an unambiguous way the dominant role of the nucleon exchange process in the energy loss mechanism.

NUCLEAR REACTIONS Heavy-ion reactions, energy loss, nucleon exchanges, neutron-proton correlations.

The systematic behavior of linear momentum transfer from projectile to target in nonrelativistic nucleus-nucleus collisions has been studied using the results of fission-fragment angular-correlation measurements on uranium target nuclei. Data for ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, and ${}_{}{}^{20}{}_{}{}^{}\mathrm{Ne}$ projectiles have been analyzed over an energy range which extends well above the interaction barrier. The data illustrate the division of the total reaction cross section into two primary components: one associated with ∼ 90 percent or greater linear momentum transfer and the other involving much smaller amounts of linear momentum transfer. The former is attributed to fusionlike collisions and the latter to peripheral collisions. The minimum between these two components corresponds to a linear momentum transfer of about 50 percent. It is observed that the ratio of fusionlike collisions to the total reaction cross section decreases regularly as a function of both increasing bombarding energy and projectile mass. From comparison of the experimental fission-fragment angular correlation functions with the predictions of complete fusion kinematics, it is shown that above 10 MeV/nucleon, the experimental definition of complete fusion is complicated by the increasing probability for large, but incomplete, linear momentum transfer collisions. Estimates of critical angular momenta derived from these data do not show any major disagreement with rotating-liquid-drop predictions.

NUCLEAR REACTIONS, FISSION Studied systematics of fission-fragment angular correlation measurements from uranium target nuclei. Deduced linear momentum transfer distributions, fusionlike collision and complete fusion probabilities, and critical angular momenta.

Measurements of the angular distributions of the single charge exchange reactions to isobaric analog states for ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}({\pi }^{+}, {\pi }^0){}_{}{}^{13}{}_{}{}^{}\mathrm{N}$ and ${}_{}{}^{15}{}_{}{}^{}\mathrm{N}({\pi }^{+}, {\pi }^0){}_{}{}^{15}{}_{}{}^{}\mathrm{O}$ at 165 MeV are described. The two angular distributions are very similar. The shapes are reproduced by semiphenomeno-logical isobar-doorway calculations and by second-order coupled channels calculations. The calculated magnitudes are low by about a factor of 2. The measurements are consistent with older angle-integrated cross sections for ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}$.

NUCLEAR REACTIONS ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{15}{}_{}{}^{}\mathrm{N}({\pi }^{+}, {\pi }^0)$, $T_{\pi }=164\mathrm{}$ MeV; measured $\sigma \left(\theta \right)$; IAS transitions; enriched targets; test of reaction models.

Excitations functions for the formation of ${}_{}{}^{126}{}_{}{}^{}\mathrm{I}$ from ${}_{}{}^{127}{}_{}{}^{}\mathrm{I}$ have been determined with 60-350 MeV ${\pi }^{+}$ and ${\pi }^{-}$. The contributions of both direct nucleon knockout and pion inelastic scattering followed by neutron evaporation are visible in the excitation functions. The results are compared with a cascade-evaporation calculation, the Sternheim-Silbar nucleon charge exchange models, and the semiclassical model of Ohkubo and Porile.

NUCLEAR REACTIONS ${}_{}{}^{127}{}_{}{}^{}\mathrm{I}({\pi }^{\pm }, \pi N){}_{}{}^{126}{}_{}{}^{}\mathrm{I}$, excitation functions (activation), $T_{\pi }=60-350\mathrm{}$ MeV. KI targets. Comparison with various models.

The extended Lee model describes the interaction of two fermions, $V$ and $N$, with a scalar boson $\theta$ and its antiparticle $\overline{\theta }$ through the virtual processes $V\rightleftarrows N+\theta$ and $N\rightleftarrows V+\overline{\theta }$. Here it is shown that the amplitudes for the physical processes $V+\theta \to V+\theta$ and $V+\theta \to N+2\mathrm{}\theta$ can be obtained in a reasonable approximation from the solution of an Amado-Lovelace type of three-particle equation. The analysis presented gives some insight into the problem of extending a crossing symmetric, Chew-Low type of $T$ matrix off the energy shell.

NUCLEAR REACTIONS Amado-Lovelace equations for $V-\theta$ sector of extended Lee model.

A novel method of separating strong and electromagnetic contributions to the charge particle scattering on nuclei is presented. Data from nonforward angles are extrapolated to the forward angle to obtain the residue of the Coulomb pole which is proportional to the forward nuclear amplitude $f_N\mathrm{}\left(0\right)\mathrm{}$. Except for the treatment of the Coulombic effects, the method is model independent. Optical polynomial expansion is used to improve the goodness of the fits. Reliable values of the total elastic, total inelastic, and total nuclear scattering cross sections can be obtained. The method is applied to study the elastic scattering of charged pions from ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ nuclei. The differential cross-section results of Binon et al. at five energies ranging from 110 to 260 MeV and of Crowe et al. at four energies ranging from 51 to 75 MeV are separately analyzed. The extracted results agree, not only with those of Binon et al., but also with the values of real parts of the forward scattering amplitude obtained by Wilkin et al. and Batty et al. using forward dispersion relations.

NUCLEAR REACTIONS ${}_{}{}^4{}_{}{}^{}\mathrm{He}({\pi }^{\pm }, {\pi }^{\pm })$, $T^{\mathrm{lab}}=51-75\mathrm{}$ MeV; ${}_{}{}^4{}_{}{}^{}\mathrm{He}({\pi }^{-}, {\pi }^{-})$, $T^{\mathrm{lab}}=110-260\mathrm{}$ MeV; calculated forward nuclear amplitudes, elastic, inelastic, total $\sigma$; pole extrapolation method.

The $S$ matrix for a wide class of complex and nonlocal potentials is studied, with special attention given to the motion of singularities in the complex $k$ plane as a function of the imaginary coupling strength. Modifications of Levinson's theorem are obtained and discussed. Analytic approximations to the $S$ matrix in the vicinity of narrow resonances are exhibited and compared to numerical results of resonating-group calculations. The problem of defining resonances in the case of complex interactions is discussed, making contact with the usual analysis of scattering in terms of Argand diagrams.

NUCLEAR REACTIONS Scattering theory, $S$ matrix for absorptive potentials.

A schematic shell model calculation is used to analyze the adequacy of the interacting boson approximation as a function of the quadrupole interaction strength $V$. It is found that an interacting boson basis truncation approximation provides a good description of the nuclear ground state, but a poor one of moderately high angular momentum states, for all values of $V$. An alternate basis truncation scheme is proposed.

NUCLEAR STRUCTURE Interacting boson approximation, shell model diagonalization.

We compare the utility of the generalized unitary pole expansion and the energy-dependent pole expansion for the three-body subsystem amplitudes in four-body bound state calculations for a variety of separable and local nucleon-nucleon interactions. It is found that with the energy-dependent pole expansion the four-body binding energy is well reproduced with only two terms each for the (2 + 2) and the (3 + 1) subsystems, respectively, while the generalized unitary pole expansion requires three terms for the (3 + 1) channel and four terms for the (2 + 2) channel. We thus conclude that pole dominance is of greater importance for the generalized unitary pole expansion than for the energy-dependent pole expansion, which works equally well for both types of subsystems. It is found that both methods, in particular the energy-dependent pole expansion, converge more rapidly with increasing repulsion in the two-body interaction, i.e., the more realistic the interaction becomes. Both expansions require similar computing times for a converged calculation and are about 15-20 times faster than the widely used Hilbert-Schmidt expansion. The respective merits of the two pole expansions are discussed and compared with the Hilbert-Schmidt expansion.

NUCLEAR STRUCTURE Four-body bound state calculations for various nucleon-nucleon interactions. Comparison of the generalized unitary pole expansion and energy-dependent pole expansion.

A bounded operator is introduced so that rigorous constrained variational calculations can be performed. With this operator the calculations are no more complex than previous, nonrigorous calculations and no dependence on unphysical parameters is introduced. It is shown that results previously obtained in a mathematically invalid way are, to a large extent, reproduced by these valid calculations.

NUCLEAR STRUCTURE Modified constraint introduced for calculation of nuclear deformation energy.

The number conservation as introduced by Lipkin and Nogami has been incorporated in the modified Tamm-Dancoff approximation method. The necessary relevant formulas arising due to this modification are derived and applied to the even Ni isotopes employing the Kuo-Brown effective interaction. A general improvement in agreement with experiments is obtained.

NUCLEAR STRUCTURE MTDA method using Lipkin-Nogami approach, applied to Ni isotopes. Kuo-Brown interaction.

Measurements have been made of the spin depolarization parameters $D_{\mathrm{NN}}$, $D_{\mathrm{SS}}$, and $D_{\mathrm{LS}}$ ($27°<~{\theta }_{\mathrm{c}.\mathrm{m}.\mathrm{}}<~90°$) and the spin transfer parameters $K_{\mathrm{NN}}$, $K_{\mathrm{SS}}$, and $K_{\mathrm{LS}}$ ($56°<~{\theta }_{\mathrm{c}.\mathrm{m}.\mathrm{}}<~90°$) for $\mathrm{pp}\to \mathrm{pp}$ at 647 MeV. Typical uncertainties are about 0.03 compared with about 0.1 for previous data. Previous data are reviewed. The present data are in agreement with corrected previous data, and are in agreement with Arndt's phase-shift solutions.

NUCLEAR REACTIONS ${}_{}{}^1{}_{}{}^{}\mathrm{H}(p, p,){}_{}{}^1{}_{}{}^{}\mathrm{H}$, $E=647\mathrm{}$ MeV, measured $D_{\mathrm{NN}}$, $D_{\mathrm{SS}}$, $D_{\mathrm{LS}}$, $K_{\mathrm{NN}}$, $K_{\mathrm{SS}}$, $K_{\mathrm{LS}}$, $\theta =27\mathrm{}° \mathrm{to} 90°$ c.m.

A remeasurement of the half-life of ${}_{}{}^6{}_{}{}^{}\mathrm{He}$ has been carried out by forming the activity in the ${}_{}{}^9{}_{}{}^{}\mathrm{Be}(n, \alpha ){}_{}{}^6{}_{}{}^{}\mathrm{He}$ reaction, transferring samples to a plastic detector, and multiscaling $\beta$-ray counts. Activities were produced in either Be metal or pure Be crystal. No significant differences were noted in results depending on the type of sample, $\beta$-ray bias, initial counting rate, or starting channel in the analysis. The adopted value, $T_{\frac{1}{2}}=805.4\mathrm{}\pm 2.0$ ms, is in agreement with a previous Brookhaven result but disagrees with recent work by Barker et al.

RADIOACTIVITY ${}_{}{}^6{}_{}{}^{}\mathrm{He}$: measured $T_{\frac{1}{2}}$.

The cross section and analyzing power for ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}(\overrightarrow{p}, p_0){}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}$ have been measured at 13 angles and 11 energies in the vicinity of the 4.0 MeV resonance. A phase-shift analysis of the results yields a spin and parity assignment of ${\frac{3}{2}}^{+}$ in conflict with the accepted value of ${\frac{5}{2}}^{+}$.

NUCLEAR REACTION ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}(p, p)$, $E=3.89-4.17\mathrm{}$ MeV; measured $\sigma (E, \theta )$, $P(E, \theta )$; ${}_{}{}^{25}{}_{}{}^{}\mathrm{Al}$ deduced resonance, $J$, $\Pi$, $\Gamma$, ${\Gamma }_p$; enriched target.

Measurements of the tensor analyzing powers for sub-Coulomb ($\overrightarrow{d}, t$) reactions on ${}_{}{}^{91}{}_{}{}^{}\mathrm{Zr}$ at $E_d=5\mathrm{}$ MeV and ${}_{}{}^{147}{}_{}{}^{}\mathrm{Sm}$ at $E_d=6.5\mathrm{}$ MeV are presented. The measurements are analyzed to obtain values of the triton $D$-state parameter $D_2$. The results are found to be in good agreement with the $D_2$ values derived from previous sub-Coulomb measurements, and lend support to the contention that analyzing power measurements for sub-Coulomb ($\overrightarrow{d}, t$) reactions provide reliable information about the $D$-state components.

NUCLEAR REACTIONS ${}_{}{}^{91}{}_{}{}^{}\mathrm{Zr}(\overrightarrow{d}, t)$, $E_d=5.0\mathrm{}$ MeV, ${}_{}{}^{147}{}_{}{}^{}\mathrm{Sm}(\overrightarrow{d}, t)$, $E_d=6.5\mathrm{}$ MeV; measured polarization parameters $T_{20}\left(\theta \right)$, $T_{21}\left(\theta \right)$, $T_{22}\left(\theta \right)$; deduced $D_2$. Enriched targets, DWBA analysis.

An effective two-nucleon potential for high-energy heavy-ion collisions is presented which resolves the undesirable features reported for the potentials (no simultaneous reproduction of the first two moments of the differential nucleon-nucleon scattering cross-sections at high energies and no saturation properties for normal nuclear densities) used in the classical equation of motion approach for the study of such collisions.

NUCLEAR REACTIONS Heavy ion classical equation of motion; laboratory energy/nucleon=50, 100, 200, 300, and 500 MeV; effective two-nucleon potential.

The longitudinal response functions extracted from the deep-inelastic electron scattering data of Altemus et al. were shown to be in reasonable agreement with calculations based on J. V. Noble's hypothesis that the nucleon properties are radically altered in nuclear matter. Other data are shown in this work to be much less well fitted.

NUCLEAR STRUCTURE Quasielastic electron scattering, nuclear properties, momentum distribution.

The surface incompressibility $K_{\mathrm{sf}}$ is evaluated for the forces $S3\mathrm{}$ and $\mathrm{SkM}$ by performing scaled Hartree-Fock calculations on semi-infinite nuclear matter.

NUCLEAR STRUCTURE Scaled Hartree-Fock calculations on semi-infinite nuclear matter. Determine surface incompressibility for forces $S3\mathrm{}$ and $\mathrm{SkM}$.

The difference between the polarization, $P$, and analyzing power, $A$, in inelastic scattering by a spin ½ projectile is shown to be proportional to the reaction $Q$-value. This follows from parity and time reversal symmetry only. At intermediate energy (500-1000 MeV) analyticity suggests that $Q$ scales with $E_{\mathrm{c}.\mathrm{m}.\mathrm{}}$ so that $\mathrm{}|P-A|\mathrm{}\le \frac{Q}{E_{\mathrm{c}.\mathrm{m}.\mathrm{}}}$, and hence that $\mathrm{}|P-A|\mathrm{}$ is very small.

NUCLEAR REACTIONS General restrictions on polarization minus analyzing power with application to intermediate energy inelastic scattering.