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Elastic electron scattering formalism for relativistic nuclear models

L. D. Miller

Phys. Rev. C 14, 706 (1976) - Published 1 August, 1976

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The classical static equation of motion for a relativistic vector field interacting through direct and derivative coupling with a spinor source term is presented in spherical coordinates. The equation is generalized to include the effects of form factors at the vector-spinor vertices. Using appropriate nucleon electromagnetic form factors and nuclear single-particle wave functions resulting from relativistic self-consistent model calculations of spherical nuclei, the contributions of the various components of the nucleon electromagnetic form factors to the effective nuclear charge distributions and the elastic electron scattering cross sections for ${}^{40}{Ca}$ , ${}^{48}{Ca}$ , and ${}^{208}{Pb}$ are analyzed.

NUCLEAR STRUCTURE Electromagnetic form factors, relativistic formalism, $ρ$ and $σ (θ)$ calculated for elastic scattering of electrons off ${}^{40}{Ca}$ , ${}^{48}{Ca}$ , and ${}^{208}{Pb}$ .

A simple nuclear model

R. Serber

Phys. Rev. C 14, 718 (1976) - Published 1 August, 1976

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A nuclear model is considered in which the nuclear forces are taken to be a Yukawa potential acting between nucleons, such as would be produced by a neutral scalar meson, and a repulsive core. The model is very similar to the Fermi-Thomas model of the atom. It is first worked out in the absence of the Coulomb interaction, and it is shown how the binding energy, surface energy, and shape of the nuclear density at the surface of the nucleus are determined. The Coulomb interaction is then included and the model is worked out for a heavy nucleus, the parameters of the model being chosen to give the correct radius and binding energy. One predicts the right value of $\frac{Z}{A}$ , and obtains a charge density distribution not too different from that deduced from electron scattering experiments.

NUCLEAR STRUCTURE Model, neutral scalar meson interaction plus repulsive core. Calculated binding energy, charge, neutron, and proton spatial distributions.

Systematics of nuclear charge distributions in Fe, Co, Ni, Cu, and Zn deduced from muonic x-ray measurements

E. B. Shera, E. T. Ritter, R. B. Perkins, G. A. Rinker, L. K. Wagner, H. D. Wohlfahrt, G. Fricke, and R. M. Steffen

Phys. Rev. C 14, 731 (1976) - Published 1 August, 1976

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The results of precise measurements of the energies of the $2 p_{\frac{3}{2}} \to 1 s_{\frac{1}{2}}$ and $2 p_{\frac{1}{2}} \to 1 s_{\frac{1}{2}}$ muonic x-ray transitions of ${}^{54}{Fe}$ , ${}^{56}{Fe}$ , ${}^{57}{Fe}$ , ${}^{58}{Fe}$ , ${}^{59}{Co}$ , ${}^{58}{Ni}$ , ${}^{60}{Ni}$ , ${}^{61}{Ni}$ , ${}^{62}{Ni}$ , ${}^{64}{Ni}$ , ${}^{63}{Cu}$ , ${}^{65}{Cu}$ , ${}^{64}{Zn}$ , ${}^{66}{Zn}$ , ${}^{68}{Zn}$ , and ${}^{70}{Zn}$ , are reported. Using a highly linear digitally stabilized Ge(Li) spectrometer system, the absolute energies and energy shifts between nuclei were measured with total errors of approximately 40-60 eV (110 eV for ${}^{70}{Zn}$ ). The data were analyzed in terms of the Barrett moments $〈 r^{k} e^{- a r} 〉$ of the nuclear charge distributions from which the equivalent nuclear radii $R_{k}$ and the isotopic and isotonic differences $δ R_{k}$ were computed. Particular attention was given to higher-order corrections of the energies of the muonic states. Appropriate quantum-electrodynamical corrections were calculated to all significant orders. Nuclear polarization corrections for multipole interactions up to and including $L = 4$ were computed for each isotope. The $Δ A = 2$ isotone shifts $δ R_{k}$ for even $A$ isotopes show a strong shell closure effect at $Z = 28$ , which is quite independent of the neutron number. The $Δ N = 2$ isotope shifts between even nuclei decrease smoothly and uniformly with increasing $N$ from $N = 28$ to $N = 40$ and are essentially independent of $Z$ . This unexpected behavior suggests that the added neutrons interact with the entire proton core rather than with the valence protons. The $Δ N = 1$ isotope shift results show a pronounced odd-even staggering effect, which, however, is somewhat smaller than theoretical predictions. The isotone series ${}^{58}{Fe}$ - ${}^{59}{Co}$ - ${}^{60}{Ni}$ , which is just below the $Z = 28$ shell closure, shows strong odd-even staggering, whereas the series ${}^{64}{Ni}$ - ${}^{65}{Cu}$ - ${}^{66}{Zn}$ and ${}^{62}{Ni}$ - ${}^{63}{Cu}$ - ${}^{64}{Zn}$ just above $Z = 28$ exhibit only a very small staggering effect. A comparison of the experimental data of the rms radii ${〈 r^{2} 〉}^{\frac{1}{2}}$ with the results of spherically constrained Hartree-Fock calculations shows good agreement for all four Zn isotopes and the heavier Ni isotopes ( ${}^{60}{Ni}$ , ${}^{62}{Ni}$ , ${}^{64}{Ni}$ ), but poor agreement for the Fe isotopes and ${}^{58}{Ni}$ .

NUCLEAR STRUCTURE, MOMENTS ${}^{54, 56, 57, 58}{Fe}$ , ${}^{59}{Co}$ , ${}^{58, 60, 61, 62, 64}{Ni}$ ${}^{63, 65}{Cu}$ , ${}^{64, 66, 68, 70}{Zn}$ ; measured muonic x-ray spectra; deduced nuclear charge parameters, isotope and isotone shifts. Calculated quantum-electrodynamic and nuclear-polarization corrections. Compared charge parameters with Hartree-Fock calculations.

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Comparison of measured neutron spectra with predictions of an intranuclear-cascade model

Aaron Galonsky, R. R. Doering, D. M. Patterson, and H. W. Bertini

Phys. Rev. C 14, 748(R) (1976) - Published 1 August, 1976

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Neutron spectra resulting from bombardment of targets of ${}^{48}{Ca}$ , ${}^{90}{Zr}$ , ${}^{120}{Sn}$ , and ${}^{208}{Pb}$ with 45-MeV protons have been measured at many angles between 0° and 160°. Intranuclear-cascade, Monte Carlo calculations predict too many high-energy neutrons in the forward direction and too few neutrons, particularly high-energy neutrons, at angles greater than ∼45°. Beyond 90° the underprediction is by factors of 10 to 100. For angle-integrated spectra, however, there is reasonable agreement between theory and experiment.

NUCLEAR REACTIONS ${}^{48}{Ca}$ , ${}^{90}{Zr}$ , ${}^{120}{Sn}$ , ${}^{208}{Pb} (p, n x)$ ; $E = 45$ MeV; measured $σ (E_{n}, θ)$ and $\int d Ω σ (E_{n}, θ)$ ; enriched targets. Intranuclear-cascade model.

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${}^{12}C (p, p n) {}^{11}C$ cross section at 800 MeV

Kenneth R. Hogstrom

Phys. Rev. C 14, 753(R) (1976) - Published 1 August, 1976

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( $α, n$ ) cross sections in the region of anomalous $α$ scattering

W. Fitz, F. Kienle, R. Maschuw, B. Zeitnitz, W. Scobel, and K. A. Eberhard

Phys. Rev. C 14, 755(R) (1976) - Published 1 August, 1976

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Cross sections for the production of neutrons from the bombardment of ${}^{40}{Ca}$ , ${}^{44}{Ca}$ , and ${}^{40}{Ar}$ targets with 24.1 MeV $α$ particles at 30° (lab) have been measured. For ${}^{40}{Ar}$ and ${}^{44}{Ca}$ smooth evaporation spectra are observed with a cross section of about 100 mb; for ${}^{40}{Ca}$ pronounced structures appear in the spectrum which correspond to high-spin states in the residual nucleus ${}^{43}{Ti}$ . The observed cross section for ${}^{40}{Ca}$ is about one order of magnitude smaller than for ${}^{40}{Ar}$ and ${}^{44}{Ca}$ ; it is, however, more than 3 times larger than expected from evaporation calculations.

NUCLEAR REACTIONS ${}^{40, 44}{Ca} (α, n)$ , ${}^{40}{Ar} (α, n)$ , $E = 24.1$ MeV; measured $σ (E, θ)$ . Statistical model calculations.

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Level structure of ${}^{85, 87}Y$

K. Krishan and S. Sen

Phys. Rev. C 14, 758(R) (1976) - Published 1 August, 1976

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The odd- $A$ Y isotopes ${}^{85}Y$ and ${}^{87}Y$ are studied in a semimicroscopic model which couples the proton quasiparticle motion to the quadrupole vibrations of the neighboring even core. The calculated results are in good agreement with recent ( ${}^{3}{He}, d$ ) data.

NUCLEAR STRUCTURE ${}^{85, 87}Y$ ; calculated levels, $J$ , $π$ , $S$ , $B (E 2)$ , $B (M 1)$ . Quasiparticle-phonon coupling.

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Extension of the interacting boson model to odd- $A$ nuclei

A. Arima and F. Iachello

Phys. Rev. C 14, 761(R) (1976) - Published 1 August, 1976

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Evaluation of the fluctuation enhancement factor

P. A. Moldauer

Phys. Rev. C 14, 764(R) (1976) - Published 1 August, 1976

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The width fluctuation correction enhances, sometimes by large factors, small compound cross sections that compete with channels having much larger transmission coefficients. The effect is demonstrated by the results of statistical computer experiments. A simple numerical quadrature of the width fluctuation integral describes the effect much more accurately than the formula proposed by Tepel .

NUCLEAR REACTIONS Average compound cross sections.

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Energy levels of ${}^{62}{Co}$

F. Ajzenberg-Selove, E. R. Flynn, O. Hansen, J. D. Sherman, N. Stein, and J. W. Sunier

Phys. Rev. C 14, 767(R) (1976) - Published 1 August, 1976

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The ${}^{62}{Ni} (t, {}^{3}{He}) {}^{62}{Co}$ reaction has been used to study the level structure of ${}^{62}{Co}$ using 24 MeV tritons and a Q3D spectrometer with a helix detector system. At least 22 excited states of ${}^{62}{Co}$ have been observed with $E_{x} < 2.2$ MeV. These are discussed and compared with previous work. A new value for the mass of ${}^{62}{Co}$ is proposed.

NUCLEAR REACTIONS ${}^{62}{Ni} (t, {}^{3}{He})$ , $E = 24.0$ MeV; measured $σ (θ)$ . ${}^{62}{Co}$ deduced levels. New mass of ${}^{62}{Co}$ .

Erratum: Conversion electron studies on the new high-spin isomer in ${}^{154}{Eu}$

Y. Y. Chu and E. -M. Franz

Phys. Rev. C 14, 772 (1976) - Published 1 August, 1976

Erratum: Electrostatic potentials for nucleus-nucleus optical model

A. K. Jain, M. C. Gupta, and C. S. Shastry

Phys. Rev. C 14, 772 (1976) - Published 1 August, 1976

Additional $1^{+}$ states in ${}^{20}F$

R. Medoff, L. R. Medsker, S. C. Headley, and H. T. Fortune

Phys. Rev. C 14, 1 (1976) - Published 1 July, 1976

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Observation of a dominant $L = 0$ component in the ${}^{18}O ({}^{3}{He}, p) {}^{20}F$ angular distribution to the 3.97-MeV state, when combined with previous restrictions, assigns $J^{π} = 1^{+}$ . An apparent $L = 0$ component for the 3.17-MeV state gives a tentative ( $1^{+}$ ) assignment for that state. Both these states appear to consist largely of core excitation.

NUCLEAR REACTIONS ${}^{18}O ({}^{3}{He}, p) E = 19.0$ MeV, measured angular distributions for $1^{+}$ states; assigned $L = 0 + 2$ and hence $J^{π} = 1^{+}$ for 3.175-, 3.966-MeV states.

High-spin states in ${}^{68}{Zn}$

J. F. Bruandet, B. Berthet, C. Morand, A. Giorni, J. P. Longequeue, and Tsan Ung Chan

Phys. Rev. C 14, 103 (1976) - Published 1 July, 1976

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Yrast levels of ${}^{68}{Zn}$ have been investigated via measurements of excitation functions and angular distributions of single $γ$ rays and of $γ$ - $γ$ coincidences following the ${}^{65}{Cu} (α, p γ) {}^{68}{Zn}$ reaction with $α$ particle energies between 12-21 MeV. Spins up to $8 ℏ$ were assigned to observed states.

NUCLEAR REACTIONS ${}^{65}{Cu} (α, p γ)$ , $E_{α} = 12 - 21$ MeV; measured $γ$ , $γ$ - $γ$ , $σ (E_{γ}, θ_{γ})$ , deduced ${}^{68}{Zn}$ decay scheme, $J$ , $π$ , and $γ$ mixing. Enriched target, Ge(Li) detectors.

Gamma decay of high-spin states in ${}^{47}{Ti}$

L. Meyer-Schützmeister, Gerald Hardie, and Terrence Sjoreen

Phys. Rev. C 14, 109 (1976) - Published 1 July, 1976

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The compound nucleus reaction ${}^{45}{Sc} (α, n p γ) {}^{47}{Ti}$ was used to confirm the high spins of states, observed previously in the ( $α, d$ ) reaction, by studying their $γ$ decay. Measurements of $γ$ - $γ$ coincidences, $γ$ -ray excitation functions, and $γ$ -ray angular distributions for several target thicknesses have been performed. These measurements allowed us to determine the energies and the spins for some of the observed levels, as well as the branching ratios and the multipolarities for the emitted $γ$ rays. Lifetimes for a few states were deduced from the Doppler shifts which were exhibited in the angular distributions. Shell-model calculations were made assuming an inert ${}^{40}{Ca}$ core and all active nucleons in the $1 f_{\frac{7}{2}}$ shell. Agreement between the measured and calculated branching ratios and multipolarities is excellent for all the yrast levels, and even the lifetimes of these states are predicted rather well by the theory. For two other levels, at 2.684- and 3.289-MeV energy, disagreement between the measured and calculated $γ$ decay is observed. The lifetime of the lower of these two states was found to be unusually slow for an $M 1$ transition. This result suggests that perhaps the earlier spin assignments for both states are wrong. For the 2.684-MeV state a spin of either ${\frac{7}{2}}^{+}$ or ${\frac{11}{2}}^{+}$ is in agreement with all the available experimental data.

[NUCLEAR REACTIONS ${}^{45}{Sc} (α, np γ)$ , $E = 23$ MeV; measured $γ$ spectra, $γ$ - $γ$ coincidences. Deduced decay scheme, multipolarities, $T_{\frac{1}{2}}$ , $J^{π}$ . Shell-model calculations].

Tensor analyzing power measurements in the ${}^{16}O (\vec{d}, α) {}^{14}N$ reaction at 0°

D. T. Petty, J. A. Kuehner, J. Szücs, P. W. Green, and G. D. Jones

Phys. Rev. C 14, 12 (1976) - Published 1 July, 1976

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The tensor analyzing power near 0° is measured in the ${}^{16}O (\vec{d}, α) {}^{14}N$ reaction at beam energies of 11, 12, 13, and 14 MeV. Spin-parity assignments for states in ${}^{14}N$ from 4.91 to 9.13 MeV are deduced. In particular, $J^{π} = 0^{-}$ assignments are made to the level at 4.91 MeV as well as one member of the doublet at 9.13 MeV. The method is discussed, with emphasis on the technique of energy averaging of the cross sections and on the effects of small deviations of the detection angle from 0°. Comparisons of these predictions are made also with data for the ${}^{12}C (\vec{d}, α) {}^{10}B$ reaction.

NUCLEAR REACTIONS ${}^{16}O (\vec{d}, α)$ , $E = 11, 12, 13, 14$ MeV; measured $T_{20} (0 °)$ , deduced $J$ , $π$ .

Effects of nonlocal potentials in heavy-ion reactions

John G. Cramer and R. M. DeVries

Phys. Rev. C 14, 122 (1976) - Published 1 July, 1976

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A nonlocal optical model program has been written in order to investigate the effects of Gaussian nonlocal potentials on heavy-ion scattering and reactions. We have tested the two principal approximations which have been used to handle nonlocal effects in light-ion reactions, the Perey-Buck nonlocal-to-local potential transformation and the local-energy approximation. We find that both of these approximations work very well for the heavy-ion reactions studied up to nonlocal ranges of about $β = 0.4$ fm, but fail for larger ranges. Using the local-energy approximation we have tested the sensitivity of distorted-wave Born-approximation calculations to the nonlocal range $β$ . For strongly absorbing potentials, values of $β ≳ 0.4$ fm cause negligible effects while weakly absorbing potentials with $β = 0.4$ fm allow strong modifications to the predicted cross sections.

NUCLEAR REACTIONS Effects of nonlocal optical potentials in heavy-ion scattering and reactions, exact nonlocal optical model calculations, tested approximations to nonlocal effects, nonlocal DWBA calculations.

Diffractive, Regge-pole, and distorted-wave Born-approximation description of the ${}^{26}{Mg}$ ( ${}^{16}O$ , ${}^{14}C$ ) ${}^{28}{Si}_{g . s .}$ transition between 33 and 128 MeV

C. K. Gelbke, J. D. Garrett, M. J. LeVine, and C. E. Thorn

Phys. Rev. C 14, 127 (1976) - Published 1 July, 1976

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Angular distributions for the ${}^{26}{Mg}$ ( ${}^{16}O$ , ${}^{14}C$ ) ${}^{28}{Si}_{g . s .}$ transition have been measured at incident energies of 33 and 40 MeV using a magnetic spectrometer. These data together with previous measurements for this same transition at 45, 60, and 128 MeV incident energies have been analyzed in terms of a diffractive model, Regge-poles, and the distorted-wave Born approximation. The shapes of the angular distributions at all incident energies are reproduced in each of the three analyses. Parameters required to fit the diffractive and Regge-pole models to the data are compared to equivalent quantities extracted from the distorted-wave Born-approximation analysis, and the relevant parameters are found to be consistent in all three models. This agreement is due principally to the peripheral nature of these reactions. To simultaneously reproduce the angular shape at the lowest and highest energies in the distorted-wave Born-approximation analysis, it was necessary to assume an energy dependent absorption in the vicinity of the nuclear surface. The use of such potentials removes much of the energy dependence of the normalization between the predicted and measured cross sections.

NUCLEAR REACTIONS ${}^{26}{Mg}$ ( ${}^{16}O$ , ${}^{14}C$ ) ${}^{28}{Si}$ ; $E = 33 and 40$ MeV measured $σ (θ)$ ; diffractive model, Regge pole, and DWBA analysis at $E = 33, 40, 45, 60, and 128$ MeV.

Deeply inelastic scattering of ${}^{84}{Kr}$ from ${}^{208}{Pb}$

R. Vandenbosch, M. P. Webb, and T. D. Thomas

Phys. Rev. C 14, 143 (1976) - Published 1 July, 1976

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We report here energy spectra and angular distributions for deeply inelastic scattering of ${}^{84}{Kr}$ from ${}^{208}{Pb}$ at laboratory energies of 494, 510, and 718 MeV. The deeply inelastic process is found to be the dominant reaction process at all energies investigated. At the highest bombarding energy the most probable kinetic energy of the deeply inelastic component becomes progressively larger as the grazing angle is approached, in contrast to near barrier energies where the most probable kinetic energy remains more nearly independent of angle. A semiclassical calculation of the deeply inelastic angular distributions, employing the deflection functions calculated from an optical model potential that reproduced the elastic scattering in this system, is shown to reproduce the deeply inelastic scattering at the lower energies.

NUCLEAR REACTIONS ${}^{208}{Pb} ({}^{84}{Kr}, X)$ , $E = 494, 510, and 718$ MeV; measured energy spectra and $σ (θ)$ for quasielastic and deeply inelastic reaction products. Obtained integrated $σ$ ; compared with optical model absorption $σ$ : Deflection function analysis.

Complete fusion mechanism in the ${}^{16}O$ + ${}^{63}{Cu}$ reaction

M. Langevin, J. Barreto, and C. Détraz

Phys. Rev. C 14, 152 (1976) - Published 1 July, 1976

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Absolute values of the cross sections for most evaporation channels consecutive to the complete ${}^{16}O$ + ${}^{63}{Cu}$ fusion have been measured for incident ${}^{16}O$ energies ranging between 40 and 85 MeV, through the observation of the corresponding $γ$ activities. They give evidence for a limitation in the contributing incident partial waves for energies above 50 MeV. The excitation functions of the various evaporation channels are satisfactorily accounted for, in shape and magnitude, by statistical evaporation calculations. They allow a very precise determination of the variation of the critical angular momentum with the excitation energy of the compound nucleus.

[NUCLEAR REACTIONS ${}^{63}{Cu} ({}^{16}O, X)$ , $E = 40 - 85$ MeV. Measured $E_{γ}$ , $I_{γ}$ ; deduced $σ (X, E)$ , critical angular momentum. Enriched target.]

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Elastic electron scattering formalism for relativistic nuclear models

A simple nuclear model

Systematics of nuclear charge distributions in Fe, Co, Ni, Cu, and Zn deduced from muonic x-ray measurements

Comparison of measured neutron spectra with predictions of an intranuclear-cascade model

${}^{12}C (p, p n) {}^{11}C$ cross section at 800 MeV

( $α, n$ ) cross sections in the region of anomalous $α$ scattering

Level structure of ${}^{85, 87}Y$

Extension of the interacting boson model to odd- $A$ nuclei

Evaluation of the fluctuation enhancement factor

Energy levels of ${}^{62}{Co}$

Erratum: Conversion electron studies on the new high-spin isomer in ${}^{154}{Eu}$

Erratum: Electrostatic potentials for nucleus-nucleus optical model

Additional $1^{+}$ states in ${}^{20}F$

High-spin states in ${}^{68}{Zn}$

Gamma decay of high-spin states in ${}^{47}{Ti}$

Tensor analyzing power measurements in the ${}^{16}O (\vec{d}, α) {}^{14}N$ reaction at 0°

Effects of nonlocal potentials in heavy-ion reactions

Diffractive, Regge-pole, and distorted-wave Born-approximation description of the ${}^{26}{Mg}$ ( ${}^{16}O$ , ${}^{14}C$ ) ${}^{28}{Si}_{g . s .}$ transition between 33 and 128 MeV

Deeply inelastic scattering of ${}^{84}{Kr}$ from ${}^{208}{Pb}$

Complete fusion mechanism in the ${}^{16}O$ + ${}^{63}{Cu}$ reaction

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