Intermediate Coupling Calculations Research Articles

O17andO19Lifetimes by a Particle-Gamma Coincidence Technique

Lifetimes of the 96-keV and 1.47-MeV levels in ${\mathrm{O}}^{19}$ produced by the ${\mathrm{O}}^{18}(d, p){\mathrm{O}}^{19}$ reaction have been measured from the distribution of time delays between protons and the appropriate decaying gamma rays. A solid-state detector was used both as a timing and as an energy-measuring device for the protons. The time-delay distributions were obtained with a time-to-height converter in the normal fast-slow coincidence arrangement. The measured half-lives of the 96-keV and 1.47-MeV levels in ${\mathrm{O}}^{19}$ are 1.39\ifmmode\pm\else\textpm\fi{}0.05 nsec and \ensuremath{\le}75 psec, respectively. These results are compared with intermediate-coupling shell-model calculations. Using this particle-gamma coincidence technique a remeasurement of the lifetime for the 871-keV level in ${\mathrm{O}}^{17}$ produced by the ${\mathrm{O}}^{16}(d, p){\mathrm{O}}^{17}$ reaction resulted in a half-life of 182\ifmmode\pm\else\textpm\fi{}5 psec.

Nuclear Spectroscopic Studies with High-Energy Boron Ions

High-energy boron beams have been used in studies of proton and deuteron transfer reactions involving a carbon target. These studies, involving examination of reactions leading to isolated states in the residual nuclei, were undertaken to obtain information on the transfer mechanism at high energies, and to ascertain the extent to which such reactions might serve as probes for certain one- and two-nucleon states in the residual nuclei involved. The proton transfer reactions ${\mathrm{C}}^{12}$(${\mathrm{B}}^{11}$, ${\mathrm{Be}}^{10}$)${\mathrm{N}}^{13}$ and ${\mathrm{C}}^{12}$(${\mathrm{B}}^{10}$, ${\mathrm{Be}}^{9}$)${\mathrm{N}}^{13}$, at bombarding energies of 115.5 and 105 MeV, respectively, are observed to populate only a few levels in ${\mathrm{N}}^{13}$. The first excited state, which is known to have a large proton width, is not one of these levels. It is suggested that this reflects an angular-momentum mismatch which acts to inhibit the $S$-wave proton transfer which would populate this state. The deuteron transfer reaction ${\mathrm{C}}^{12}$(${\mathrm{B}}^{11}$, ${\mathrm{Be}}^{9}$)${\mathrm{N}}^{14}$, at 115.5 MeV bombarding energy, is also observed to populate only a few levels in the residual nucleus ${\mathrm{N}}^{14}$. Arguments are presented favoring direct population of ${\mathrm{N}}^{14}$ levels having ${\mathrm{C}}^{12}$+${({d}_{\frac{5}{2}})}^{2}$ and ${\mathrm{C}}^{12}$+(${d}_{\frac{3}{2}}, {d}_{\frac{5}{2}}$) two-particle configurations. In particular, the observed strongly populated levels at approximately 9- and 13-MeV excitation have been tentatively identified as the $J=5$, $T=0$ and $J=4$, $T=0$ members of the above multiplets, respectively. The existing intermediate-coupling calculations pertaining to ${\mathrm{N}}^{14}$ are in excellent accord with such a description but suggest significant (${p}_{\frac{1}{2}}, {f}_{\frac{7}{2}}$) admixtures in the latter state. The angular distributions of the strong groups in all reactions are smoothly varying, exponential functions of angle. These have been fitted, in terms of a surface-diffuseness parameter, using a simple diffraction model. The diffuseness obtained is in excellent accord with other results in the same mass and energy range. By comparing the present data with those on other reactions involving the same incident systems, qualitative conclusions are drawn regarding fractional parentage in the incident systems. In both single- and two-nucleon situations, the experimental results are compared in detail with corresponding data from ${\mathrm{He}}^{3}$- and ${\mathrm{He}}^{4}$-induced reactions.

Spectra of Doubly and Triply Ionized Ytterbium, Yb iii and Yb iv†

The emission spectrum of ytterbium was photographed under varied conditions from 600 to 11 000 Å in order to obtain a complete description of the spectra of the doubly and triply ionized species. The main transition arrays of these species were identified, and where feasible, the analysis was carried through. All 41 energy levels, 28 odd and 13 even, belonging to the 4f14, 4f13 5d, 4f13 6s, 4f13 6p, and 4f13 7s configurations of Yb iii have been experimentally determined along with the appropriate quantum numbers. Eleven of 20 possible 4f13 6d levels and their quantum numbers have also been determined. Many Yb iii levels are supported by intermediate coupling calculations. The ionization potential of Yb iii was determined to be 25.4±0.5 eV from the Rydberg formula. The two 2F 4f13 levels of Yb iv have been determined in addition to 20 of a possible 107 4f12 5d levels, of which a few are expected to be spurious.

Couplage d'un nucléon à des vibrations de surface anharmoniques

The spectrum of Cu63 has been computed by an intermediate coupling calculation using the model of Kerman and Shakin to describe the anharmonic vibrations of the Ni62 core. The wave functions are compared with those obtained with the assumption of a harmonic core and are found to be not much different from them. As a consequence the discrepancy between the experimental and theoretical E 2 de-excitation probabilities to the ground state remains unchanged.

Ground-State Properties and Low-Lying States of theN14Nucleus

An approximate Hartree-Fock self-consistent-field calculation is carried out on the /sup 14/N nucleus using the concept of symmetry and equivalence restrictions. The configuration interaction matrices belonging to a definite value of J, II, T are constructed using the jj coupled configurations arising from (1p)/sup -2/ shell. The approximate HartreeFock binding energy turns out to be 38.5 Mev too small, while it was only 13 Mev too small for the /sup 16/O. This is attributed to an inadequate choice of the tensor interaction or to the omission of a two-body spin-orbit interaction. The static magnetic dipole moment is calculated to be 0.39 nuclear magneton and the rms radius is found to be 2.14 F The results are compared with the earlier intermediate coupling calculations. A number of low-lying even-parity states are calculated and compared with the experimental values. (auth)

ElectronicgFactors of thep2Configuration in Ge I and Sn I

Electronic $g$ factors have been obtained by the atomic-beam magnetic-resonance technique for all $J\ensuremath{\ne}0$ states arising from the ${p}^{2}$ electronic configuration in Ge i and Sn i. The results for the $^{1}D_{2}$, $^{3}P_{2}$, and $^{3}P_{1}$ states for Ge are 1.00639(8), 1.49458(9), 1.50111(7); and for Sn they are 1.05230(8), 1.44878(9), and 1.50110(7), respectively. The results are compared with intermediate-coupling calculations.

Analysis of the Optical Absorption Spectrum of Neodymium Magnesium Nitrate

The optical absorption spectrum and magnetic properties of Nd2 Mg3 (NO3)12·24H2O have been interpreted in the light of crystalline field theory, using intermediate coupling calculations within each level of the free Nd3+ ion. A field of C3v symmetry has been used to account successfully for most of the observed crystal field energy levels and their Zeeman splittings. The possibility of using icosahedral symmetry as an approximation has also been investigated.

An investigation of the energy levels of A 38 and K 38

The energy levels of A 38 and K 38 have been investigated by means of the reaction K 39 (t, α)A 38 at an incident triton energy of 5.48 MeV and the reaction Ca 40(d, α)K 38 at an incident deuteron energy of 5.46 MeV. The reaction products have been analysed using a broad range magnetic spectrograph. Eleven energy levels have been assigned to A 38 up to an excitation energy of 6.06 MeV, and twelve energy levels have been assigned to K 38 up to an excitation energy of 3.98 MeV. Absolute values of differential cross sections have been determined. The observed energy level spectra have been compared with the spectrum predicted from a recent intermediate coupling calculation of the mass-38 system.

Optical Spectrum of K3IrCl6

The complete matrices of the spin-orbit interaction for the d4, d6 configurations are given in a strong cubic field scheme. Intermediate coupling calculations have been carried out for the energy levels of K3IrCl6, and lead to a satisfactory interpretation of the observed optical spectrum.

Ground-State Energy of the Nucleon

Halpern's method of moments has been applied to the intermediate-coupling reduced Hamiltonian, whose lowest eigenvalue is a variational upper bound to the ground-state energy of the nucleon in the fixed-source model of meson theory. The results compare favorably with an earlier intermediate-coupling calculation of Friedman, Lee, and Christian, and agree with direct moment-method results. A discussion of the relationship between the present work and a Tamm-Dancoff approximation for the reduced Hamiltonian is included.

Positive parity states of Be 9 and C 13

The states produced by weakly coupling a 2s- or Id-neutron to a core consisting of the ground or first excited state of C 12 are mixed by means of a two-body central interaction and a one-body spin-orbit interaction of the type normally used in intermediate-coupling calculations, to provide the low-lying positive-parity eigenstates of C 13. The positions and neutron widths of the states agree satisfactorily with experimental data, as do the calculated cross sections for photoneutron disintegration of C 13; however, in order to obtain agreement with other data involving E1 radiative widths the single-particle shell-model wave functions must be modified so as to have the correct asymptotic form. Similar calculations are performed for the Be 9 positive-parity states, about which little is known experimentally.

Pairing plus quadrupole force model for odd-mass single closed shell nuclei

An intermediate coupling calculation is performed for odd-mass nuclei with the neutrons or protons at a closed shell. The Hamiltonian, an adiabatic approximation to the pairing force plus quadrupole Hamiltonian, is diagonalized to include all states containing up to two phonons of quadrupole vibration. The validity of the approximations is discussed. The level energies and wave functions resulting from the calculation are compared with experimental results and with the results of ref. 1).

The State of the Nd3+ Ion as Derived from the Absorption and Fluorescence Spectra of NdCl3 and Their Zeeman Effects

The absorption and fluorescence spectra of Nd3+ in NdCl3 diluted by LaCl3 have been examined with their Zeeman effect at 4.2° and less completely also at 77°K. The empirical energy-level diagram is complete up to 28 000 cm−1 except for the location of the 2K term. Crystal Stark splittings, magnetic properties, and intensities are compared to calculated values obtained from an intermediate coupling calculation of the 4f3 configuration.

Theoretical Predictions for the Spectra of the Odd-Mass Xenon and Tellurium Isotopes

The spectra of the isotopes ${\mathrm{Te}}^{123,125}$, ${\mathrm{Xe}}^{127,129}$ are calculated by assuming that the odd neutron, having available the $3{s}_{\frac{1}{2}}$ and $2{d}_{\frac{3}{2}}$ states, is coupled to collective surface vibrations of the core. Good agreement is obtained with the known levels in these nuclei using a reasonable value for the coupling parameter. To obtain the agreement, the ${d}_{\frac{3}{2}}\ensuremath{-}{s}_{\frac{1}{2}}$ splitting, $\ensuremath{\epsilon}$, must be regarded as a function of neutron number. The manner in which $\ensuremath{\epsilon}$ varies, as found in the intermediate coupling calculation is compared with the predictions of the pairing correlation theory originally introduced in connection with superconductivity. Agreement as to the general trend is found. This may be regarded to some extent as an indication of the applicability of the pairing correlation theory to nuclear structure calculations.

On the coupling of a [formula omitted] particle to nuclear quadrupole surface oscillations

An intermediate coupling calculation is performed for a nucleon confined to move in a j = 3 2 orbit and coupled to a nuclear surface capable of performing quadrupole oscillations. It is shown that each term in the hamiltonian commutes with a symplectic group of transformations. This feature is used to define a representation in which the hamiltonian matrix takes an especially simple form. It is possible to follow the transition from weak to strong coupling, at which there appear sets of rotational bands built on equidistant vibrational excitations. These rotational bands have some special features due to the γ-instability of the problem.

Magnetic Moment Calculation forLi8

BS>An intermediate-coupling model calculation of the Li/sup 8/ nuclear g factor is compared with a recent measurement. The calcalation was carried out as a function of the spinorbit coupling parameter for the relative rate of nuclear forces given by L/K = 6.8. The resulting values of the magnetic moment are given. (W.D.M.) Particle evaporation experiments at energies in excess of 20 Mev have shown a disappointing fit to the theory of energy level densities in excited nuclei. Temperatures deduced from particle evaporation spectra are too low. This fact is to be expected from actual nuclei because of the diffuseness of the actual nuclear potential well, and level densities of highly excited nuclei are a sensitive measure of the details of the shape of the nuclear potential well. (W.D.M.)

Simplified intermediate-coupling procedure in the 1p-shell

Intermediate-coupling wave functions are constructed by applying Elliott's generating procedure to intrinsic wave functions based on Nilsson's deformed-potential calculation. These are compared with wave functions from an intermediate-coupling calculation with a two-body central interaction. A strong overlap is found between the groups of low-lying states obtained by these two methods of calculation.

The structure of the nuclei of mass 18 and 19

An intermediate coupling calculation has been carried out for nuclei of mass 18 and 19, taking full account of the configurational mixing of the 2 s and 1 d shells. The spin-orbit splitting of the d -level and the relative position of the s -level are taken from the observed 17 O spectrum, while the strength of the two-body central force is treated as a free parameter. No excitation of the 16 O closed shell core is considered. Excellent agreement is found for the positions of the even-parity energy levels and their properties, including the reaction data. The configurational mixing is very large, the strength of central force is similar to that required for the deuteron, and the mode of coupling is similar to that in the lithium nuclei.

Mobility of Slow Electrons in Polar Crystals

An intermediate-coupling calculation of polaron-phonon scattering has been carried out by expressing the scattering amplitude directly in terms of matrix elements between exact eigenstates of the Hamiltonian corresponding to the initial and final states of the scattered particle. The basic approximation made is in the use of the rather accurate polaron wave functions of Lee, Low, and Pines for these initial and final states. The result for the mobility, which is only valid in the low-temperature region ($T\ensuremath{\ll}\ensuremath{\theta}$) is $\ensuremath{\mu}=\frac{1}{2\ensuremath{\alpha}\ensuremath{\omega}}\left(\frac{e}{m}\right)\frac{1}{{[1+(\frac{\ensuremath{\alpha}}{6})]}^{3}}f(\ensuremath{\alpha})\mathrm{exp}(\frac{\ensuremath{\omega}}{\ensuremath{\kappa}T}),$ where $\ensuremath{\alpha}$ is the coupling constant for the electron-phonon interaction and $f(\ensuremath{\alpha})$ is a slowly varying quantity of order $\frac{5}{4}$ for $3l\ensuremath{\alpha}l6$.