Recent Articles

Polarization Transfer in the ${}^{2}H (d, p) {}^{3}H$ Reaction at $θ = 0^{\circ}$

T. B. Clegg, D. D. Armstrong, R. A. Hardekopf, and P. W. Keaton, Jr.

Phys. Rev. C 8, 922 (1973) - Published 1 September, 1973

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Measurements of the polarization transfer coefficients $K_{y}^{y^{'}}$ and $K_{yz}^{x^{'}}$ for the ${}^{2}H (d, p) {}^{3}H$ reaction at an angle of 0° and for six incident deuteron energies from 6 to 15 MeV are reported. The experimental method used is independent of the knowledge of the analyzing tensor $A_{zz}$ and of current integration. Our measurements of $K_{y}^{y^{'}}$ agree well with the results of Simmons et al. for the mirror reaction ${}^{2}H (d, n) {}^{3}{He}$ , supporting their conclusion that the outgoing-nucleon polarization is nearly constant over this energy range but less than that which is calculated for the nucleons in the incident deuterons.

[NUCLEAR REACTIONS ${}^{2}H (d, p)$ , $E = 6 - 15$ MeV, $θ = 0^{\circ}$ ; measured polarization transfer.]

Optimal Separation of Center-of-Mass Motion

C. M. Vincent

Phys. Rev. C 8, 929 (1973) - Published 1 September, 1973

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It is shown that a given approximate eigenfunction $F$ of a many-body Hamiltonian can be improved by exploiting the separation of the exact eigenfunction in internal and c.m. coordinates. The method given leads reliably to a lower energy for the ground state, without requiring the solution of an additional dynamical problem. The best internal wave function (BIWF) is taken to be the internal factor of the separable wave function that has the largest possible overlap with $F$ . The existence, uniqueness, and construction of the BIWF are derived. A canonical expansion $F = Σ a_{i} Ψ_{i} ψ_{i}$ is described, in which $Ψ_{i}$ are orthogonal functions of the c.m. coordinates, and $ψ_{i}$ are orthogonal functions of the internal coordinates. The canonical expansion is shown to converge faster than any other. It is applied to the discussion of the "projection" or "generator-coordinate" treatment of c.m. motion using arbitrary c.m. wave functions. It is shown that the Gartenhaus-Schwartz prescription is not equivalent to the use of any internal wave function, and gives systematically worse results for off-diagonal matrix elements than for expectation values.

Study of $d + {}^{3}H$ and $d + {}^{3}{He}$ Systems with the Resonating-Group Method

F. S. Chwieroth, Ronald E. Brown, Y. C. Tang, and D. R. Thompson

Phys. Rev. C 8, 938 (1973) - Published 1 September, 1973

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The one-channel resonating-group method, together with a phenomenological imaginary potential, is used to study the $d + {}^{3}H$ and $d + {}^{3}{He}$ systems. Our calculations show that, because of the Pauli exclusion principle, these systems possess the following rather interesting feature: In the $s = \frac{3}{2}$ state they behave similar to the $d + α$ system, whereas in the $s = \frac{1}{2}$ state they behave more like the ${}^{3}{He}$ + ${}^{4}{He}$ system. Phase shifts are computed up to $l = 7$ , and an $R$ -matrix analysis of these phases indicates that there are a large number of levels with resonance energies below 12 MeV. Differential scattering and total-reaction cross sections are compared with experimental data at several c.m. energies from about 2 to 14 MeV. This comparison shows that the introduction of an odd-even orbital-angular-momentum dependence into the imaginary potential can improve the agreement of the calculation with experiment.

Recoil Effects in Single-Nucleon-Transfer Heavy-Ion Reactions

R. M. DeVries

Phys. Rev. C 8, 951 (1973) - Published 1 September, 1973

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The effect of recoil in single-nucleon-transfer heavy-ion reactions is found to be twofold: Extra transfer angular momenta are introduced and the radial integrals are changed. Comparison of an exact recoil distorted-wave Born-approximation program with experimental data from heavy-ion reactions induced on ${}^{11}B$ and ${}^{12}C$ gives excellent quantitative agreement. Recoil effects on other targets and energies are considered.

Measurement of the ${}^{2}H (p, 2 p) n$ Reaction over a Large Volume of Phase Space

J. G. Rogers, Mahavir Jain, and J. D. Bronson

Phys. Rev. C 8, 961 (1973) - Published 1 September, 1973

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A kinematically complete measurement of the deuteron breakup cross section has been made using a 39.5-MeV proton beam. The cross section was simultaneously measured at approximately 2000 angle pairs chosen to uniformly sample the entire phase-space volume available to the two protons. A portion of the data in the form of a partial Dalitz plot is compared with a simple model calculation. In the model, the breakup cross section is expressed as an incoherent sum of terms to separately represent the dependence of the cross section on statistical processes, $p p$ quasifree scattering, and nucleon-nucleon final-state interactions. The model calculation does not reproduce the minimum in the partial Dalitz plot, consistent with the previously published hypothesis that the minimum is due to a destructive interference effect. The results of the model calculation are also compared with spectra at a representative set of individual angle pairs.

Energy Levels of ${}^{14}N$ from the Elastic Scattering of $α$ Particles by ${}^{10}B$

T. Mo and H. R. Weller

Phys. Rev. C 8, 972 (1973) - Published 1 September, 1973

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The level structure above 13-MeV excitation energy in ${}^{14}N$ was investigated through the reaction ${}^{10}B (α, α) {}^{10}B$ in the energy range from $E_{α} = 2.0 to 4.3$ MeV. Excitation functions for elastically scattered $α$ particles were measured in 20-keV steps at six c.m. angles of 67.8, 89.7, 113.5, 141.8, 154.9, and 169.1°. Besides the two strong resonances observed at $E_{α} ≃ 2.21 and 4.26$ MeV, respectively, two additional resonances were required to fit the excitation functions. An $R$ -matrix calculation was used to obtain spin and parity assignments of $3^{+}$ , $1^{+}$ , $3^{+}$ , and $2^{-}$ to the levels at excitation energies of $E_{x} = 13.19, 13.72, 14.25, and 14.66$ MeV, respectively. The partial widths of $α$ -particle decay and total widths of these states were also determined. In addition, the excitation function from $E_{α} = 1.4 to 2.0$ MeV was measured at $θ_{c . m .} = {89.7}^{\circ}$ , and no obvious resonances were observed in this energy region. Possible correlations of the strong $α$ -particle levels to the $α$ -particle core-excited threshold-state model are also discussed.

Study of Neutron and Alpha-Particle Transfer Mechanisms in the ${}^{7}{Li} (d, t) {}^{6}{Li}$ Reaction

Michael F. Werby and Steve Edwards

Phys. Rev. C 8, 978 (1973) - Published 1 September, 1973

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The ${}^{7}{Li} (d, t) {}^{6}{Li}$ reaction is analyzed at a bombarding energy of 12 MeV using a finite-range, multi-interaction, two-mode distorted-wave Born-approximation (DWBA) formalism. Extracted direct-mode spectroscopic factors agree with theoretical predictions only when knockout components are included in the analysis, showing the usual DWBA cancellation assumption to be inappropriate for this reaction. Contributions from the exchange-mode mechanisms are found to be necessary for a satisfactory description of the complete angular distribution. It is also found that two of the three optical-model-parameter sets used can be eliminated and thus the optical-model ambiguities resolved by examining the exchange-mode spectroscopic factors.

Proton Pickup from ${}^{86}{Sr}$ and the ${}^{85}{Rb}$ Level Structure

R. C. Ragaini, J. D. Knight, and W. T. Leland

Phys. Rev. C 8, 988 (1973) - Published 1 September, 1973

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The levels of ${}^{85}{Rb}$ have been investigated by the ${}^{86}{Sr} (t, α)$ reaction with 15-MeV tritons. $α$ -particle spectra were measured by a $Δ E - E$ counter telescope system with an over-all resolution of 50 keV. Angular distributions were measured from 12.5 to 90°, and comparison with distorted-wave Born-approximation calculations permitted angular momentum assignments for 11 of the 20 levels observed. The data are compared with previous information on the ${}^{85}{Rb}$ level structure.

Nucleon-Transfer Reactions and Coulomb Displacement Energies

E. Friedman

Phys. Rev. C 8, 996 (1973) - Published 1 September, 1973

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Differential cross sections for several nucleon-transfer reactions on ${}^{40}{Ca}$ , ${}^{42}{Ca}$ , and ${}^{48}{Ca}$ are analyzed in distorted-wave Born approximation. It is shown that the values of the root-mean-square radius of the $1 f_{\frac{7}{2}}$ proton in ${}^{41}{Sc}$ and the $1 f_{\frac{7}{2}}$ neutrons in ${}^{48}{Ca}$ are quite well determined. These values are in a reasonable agreement with theoretical results obtained from shell-model and Hartree-Fock calculations, but imply Coulomb displacement energies which differ from experimental results by about 0.6 MeV. It is concluded that the anomaly observed in nuclear matter radii, as derived from Coulomb displacement energies, is probably due to difficulties in calculating these energies.

Accelerator Production of ${}^{8}B$ Neutrinos

R. E. Marrs, D. Bodansky, and E. G. Adelberger

Phys. Rev. C 8, 427 (1973) - Published 1 August, 1973

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The ${}^{8}B$ yield from the reaction ${}^{6}{Li} ({}^{3}{He}, n) {}^{8}B$ has been measured from 8.9 to 26.5 MeV (lab) by counting the annihilation $γ$ rays from the 0.762-sec positron decay of ${}^{8}B$ . The measurement was motivated by the negative results of the Davis solar neutrino experiment, and the consequent possible usefulness of direct tests of neutrino detection systems with neutrinos from the decay of ${}^{8}B$ . The present upper limit on the capture of solar neutrinos in ${}^{37}{Cl}$ is 1 SNU, corresponding to a flux of 0.74 × $10^{6}$ ${}^{8}B$ neutrinos per ${cm}^{2}$ sec. It was found that a ${}^{3}{He}^{+}$ current of 30 mA at 24 MeV is required to produce this ${}^{8}B$ neutrino flux at a distance of 8 m from the target, indicating that the contemplated test is not impossible.

${}^{4}{He} (γ, p) t$ Reaction for Photon Energies Between 180 and 320 MeV

S. E. Kiergan, A. O. Hanson, and L. J. Koester, Jr.

Phys. Rev. C 8, 431 (1973) - Published 1 August, 1973

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The differential cross section for the reaction ${}^{4}{He} (γ, p) t$ was measured in a spark chamber experiment for photon energies between 180 and 320 MeV and proton c.m. angles between 60 and 120°. The results show that the cross section continues to decline with increasing energy. No evidence of a peak was found in the region of the first pion-nucleon resonance. An empirical function, involving the nucleon momentum distribution in the nucleus, was derived from a simple model of the photodisintegration process, and was used to correlate the observed differential cross sections over a large range of energy and angle.

Center-of-Mass Motion and Angular Momentum Projection

D. J. Ernst, C. M. Shakin, and R. M. Thaler

Phys. Rev. C 8, 440 (1973) - Published 1 August, 1973

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The results of calculations of rotational energies contain unphysical contributions from the rotation of the center-of-mass part of the wave function. A simple method is presented to remove those unphysical contributions in the case of projected Hartree-Fock calculations. Corrections to several existing Hartree-Fock calculations are computed.

Mechanisms in ${}^{3}H$ + ${}^{3}{He}$ and ${}^{3}H$ + ${}^{2}H$ Reactions

Ivo Slaus, R. G. Allas, L. A. Beach, R. O. Bondelid, E. L. Petersen, J. M. Lambert, and D. L. Shannon

Phys. Rev. C 8, 444 (1973) - Published 1 August, 1973

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The study of ${}^{3}H$ + ${}^{3}{He}$ reactions at $E_{{}^{3}{He}} = 50$ MeV, ${}^{3}H$ + ${}^{2}H$ reactions at $E_{d} = 35$ MeV, as well as the reanalysis of ${}^{2}H ({}^{3}{He}, p t) p$ reaction demonstrates that the quasifree reaction mechanism plays an important role. Evidence for the quasifree reaction mechanism is found in the processes ${}^{3}H ({}^{3}{He}, d d) d$ , ${}^{3}H ({}^{3}{He}, p t) d$ , ${}^{3}H ({}^{3}{He}, d t) p$ , ${}^{3}H ({}^{3}{He}, p α) n$ , ${}^{2}H ({}^{3}{He}, p t) p$ , and ${}^{3}H ({}^{2}H, p t) n$ . Evidence for $d - d$ , $p - {}^{3}{He}$ , $d - {}^{3}{He}$ , and $p - t$ quasifree scattering is found in the processes ${}^{3}H ({}^{2}H, d d) n$ , ${}^{3}H ({}^{3}{He}, p {}^{3}{He}) 2 n$ , ${}^{3}H ({}^{3}{He}, d {}^{3}{He}) n$ , and ${}^{3}H ({}^{2}H, p t) n$ , respectively. Comparing the absolute cross sections of these reactions, one concludes that though the shapes of the correlation spectra indicate the significance of quasifree reaction mechanisms, the pole diagram is not dominant.

Multi-Interaction, Finite-Range, Two-Mode, DWBA Analyses of Direct Nuclear Reactions

Steve Edwards, D. Robson, Thurman L. Talley, William J. Thompson, and Michael F. Werby

Phys. Rev. C 8, 456 (1973) - Published 1 August, 1973

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The finite-range distorted-wave Born approximation (DWBA) theory of direct nuclear reactions is modified to facilitate inclusion of both direct and exchange modes of transfer reactions and all effective interaction potentials for single-cluster transfers. While spin-orbit terms are neglected many other of the usual approximations are not made. Application of the formalism using the computer code FANLU2 is demonstrated in several examples designed to demonstrate useful features of such analyses. Contributions from all reaction mechanisms and both modes are shown to be necessary for an adequate description of the reaction $d (α, t) {}^{3}{He}$ at 82 Mev with no parameter adjustment. The reaction ${}^{9}{Be} (p, α) {}^{6}{Li}$ at 6 MeV illustrates angular momentum dependence of the reduced widths in a more complex reaction. Finite-range effects are demonstrated in a simplified analysis of the ${}^{6}{Li} ({}^{3}{He}, p) {}^{8}{Be}$ reaction at 17 MeV. Various aspects of such analyses with the code FANLU2 are discussed.

Truncation of Configuration Space and the Nature of Effective Two-Body Interaction

Raj K. Gupta

Phys. Rev. C 8, 472 (1973) - Published 1 August, 1973

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Using the renormalized $G$ matrix (the $G_{r}$ ) results of Kuo (for the $d - s$ shell) and of Kuo and Brown (for the $f - p$ shell), a re-renormalized effective $G$ matrix (the $G_{eff}$ ) for only two active orbits is calculated numerically, which would be valid for use in ${(d_{\frac{5}{2}} s_{\frac{1}{2}})}^{n}$ and ${(f_{\frac{7}{2}} p_{\frac{3}{2}})}^{n}$ configurations. The operators $G_{r}$ and $G_{eff}$ as well as the bare $G$ matrix (the $G_{b}$ ) of Kuo and Brown, which thus define the transformation chain $G_{b} \to G_{r} \to G_{eff}$ , are then analyzed in the light of truncation of the shell-model space. The analysis shows that the effect of this truncation can largely be taken into account by the addition of a pairing-force term to the effective two-body interaction. The relative change in $T = 1$ and $T = 0$ parts of the interaction due to the truncation of configuration space in various ways is also discussed. Finally, a comparison between the available empirically fitted effective interactions and our theoretical $G_{eff}$ interaction is carried out.

Interpretation of Volume Conservation in the Nilsson Model

W. H. Bassichis

Phys. Rev. C 8, 480 (1973) - Published 1 August, 1973

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The condition of constant volume for the deformed potential of the Nilsson model is shown to follow from the relationship of the model to constrained Hartree Fock. Essential to the demonstration are two assumptions, one relating the deformation and rms radius in the Nilsson model and the second concerning the separation of the Nilsson potential into an internal and external part. This analysis yields insight into the various schemes for obtaining the energy from the Nilsson eigenvalues.

Production of the Light Elements Lithium, Beryllium, and Boron by Proton-Induced Spallation of ${}^{14}N$

Helmut Laumer, Sam M. Austin, Lolo M. Panggabean, and Cary N. Davids

Phys. Rev. C 8, 483 (1973) - Published 1 August, 1973

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Astrophysically interesting cross sections for the production of isotopes of lithium, beryllium, and boron in the proton-induced spallation of ${}^{14}N$ were measured for proton energies between 17 and 42 MeV. A time-of-flight method was used for mass identification, supplemented at one energy by radioactivity analysis. Gas targets were used throughout. The astrophysical significance of the results is discussed.

Nuclear Effects in the Production of $π^{+}$ and $π^{-}$ by Protons on Nuclei

Richard R. Silbar and Morton M. Sternheim

Phys. Rev. C 8, 492 (1973) - Published 1 August, 1973

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We have modified an earlier semiclassical model of pion production by protons on nuclei by including various nuclear physics corrections. In addition to allowing for $p n$ charge exchange and realistic nuclear densities, we have corrected the two-body input cross sections for the effects of the Pauli principle, Fermi motion, and nuclear optical potentials. The agreement with experiment is similar to that obtained earlier. Ambiguities arising from the nuclear corrections are found to be larger than the effects of moderate variations of the nuclear surface parameters.

Self-Consistent Radial Density and Electron Scattering

S. Boffi and A. Molinari

Phys. Rev. C 8, 504 (1973) - Published 1 August, 1973

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Three-Nucleon Experiments and ${}^{2}S$ Defects

D. P. Saylor and F. N. Rad

Phys. Rev. C 8, 507 (1973) - Published 1 August, 1973

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Supported by the results of recent theoretical calculations of nucleon-deuteron scattering, it is suggested that most parts of the calculated three-nucleon scattering and breakup amplitudes are model-independent at low energies and, furthermore, that the ${}^{2}S$ -state amplitudes are the important exception to this rule. On the basis of the assumption that it is only differences between theoretical and physical ${}^{2}S$ amplitudes which will remain substantial as calculations improve, a new approach to the experimental three-nucleon problem is proposed. According to this approach, experiments would be chosen on the basis of the sensitivity to ${}^{2}S$ amplitudes and would be analyzed by using theoretical predictions for the amplitudes and empirically adjusted ${}^{2}S$ defects. Experiments expected to be sensitive to the ${}^{2}S$ amplitudes are discussed including angular distributions, spin-correlation measurement, and experiments of the triple-scattering type. Formulas for these parameters are given in terms of amplitudes having the simple structure predicted by the Amado model and it is shown how the ${}^{2}S$ defects will modify such predictions.

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Recent Issues

Polarization Transfer in the ${}^{2}H (d, p) {}^{3}H$ Reaction at $θ = 0^{\circ}$

Optimal Separation of Center-of-Mass Motion

Study of $d + {}^{3}H$ and $d + {}^{3}{He}$ Systems with the Resonating-Group Method

Recoil Effects in Single-Nucleon-Transfer Heavy-Ion Reactions

Measurement of the ${}^{2}H (p, 2 p) n$ Reaction over a Large Volume of Phase Space

Energy Levels of ${}^{14}N$ from the Elastic Scattering of $α$ Particles by ${}^{10}B$

Study of Neutron and Alpha-Particle Transfer Mechanisms in the ${}^{7}{Li} (d, t) {}^{6}{Li}$ Reaction

Proton Pickup from ${}^{86}{Sr}$ and the ${}^{85}{Rb}$ Level Structure

Nucleon-Transfer Reactions and Coulomb Displacement Energies

Accelerator Production of ${}^{8}B$ Neutrinos

${}^{4}{He} (γ, p) t$ Reaction for Photon Energies Between 180 and 320 MeV

Center-of-Mass Motion and Angular Momentum Projection

Mechanisms in ${}^{3}H$ + ${}^{3}{He}$ and ${}^{3}H$ + ${}^{2}H$ Reactions

Multi-Interaction, Finite-Range, Two-Mode, DWBA Analyses of Direct Nuclear Reactions

Truncation of Configuration Space and the Nature of Effective Two-Body Interaction

Interpretation of Volume Conservation in the Nilsson Model

Production of the Light Elements Lithium, Beryllium, and Boron by Proton-Induced Spallation of ${}^{14}N$

Nuclear Effects in the Production of $π^{+}$ and $π^{-}$ by Protons on Nuclei

Self-Consistent Radial Density and Electron Scattering

Three-Nucleon Experiments and ${}^{2}S$ Defects

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