Matrix Elements Of Tensor Research Articles

Coupling coefficients and matrix elements of arbitrary tensors in the elliott projected angular momentum basis

An explicit expression for the transformation brackets between the basis for an arbitrary irreducible representation of U(3) labeled in the U(3) ⊃ R(3) scheme introduced by Elliott and the Gel'fand U(3) ⊃ U(2) ⊗ U(1) scheme is obtained. The results are applied in deriving an expression for both the U(3) ⊃ R(3) coupling coefficients and projected tensor matrix elements.

Irreducible tensors for the groupSU 3

The explicit determination of the matrix elements of theSU3 irreducible tensors is carried out by a purely algebraic method. These expressions may be used to compute the Clebsch-Gordan coefficients by orthogonalisation. For the special case of (0,q) tensors simple formulas are derived.

Electrostatic Interactions in Complex Electron Configurations

Simplified expressions for the matrix elements of electrostatic interactions both within and between several types of complex electron configurations have been obtained by the application of angular-momenta recoupling techniques. The use of these recoupling techniques avoids the usual extensive calculation of the sums of products of the matrix elements of tensors of the types Vκk and Uk. The derived expressions involve the sums of products of coefficients of fractional parentage and n − j symbols, and as such, are amenable to machine computation.

Intensities of Crystal Spectra of Rare-Earth Ions

Magnetic and electric dipole transitions between levels of the 4fx configuration perturbed by a static crystalline field are treated. The expression obtained for the pure-electronic electric-dipole transition probability involves matrix elements of an even-order unit tensor between the two 4fx states involved in the transition. The contributions to the transition probability from interactions, via the crystalline field, with the nd94fx−1, 4fx−1nd, 4fx−1ng configurations are shown to add linearly, in such a manner as to multiply each odd k crystal-field parameter Akq by a constant. If ``J mixing'' in the 4fx configuration is neglected ΔJ between the upper and lower 4fx levels is restricted to six units or less. If ``L mixing'' is neglected then ΔL is also restricted to six units or less. Application is made to the fluorescence spectra of PrCl3 and EuCl3. Many of the missing and weak transitions are explained.

Tensor Coupling and the Vector Shell Model

Because of the large tensor force contribution to the nuclear binding energy, which results in a situation of near degeneracy for the low-lying states, the tensor force may produce a large mixing of the low states even in the absence of tensor matrix elements between them. The type of coupling among the low states resulting from this near degeneracy is investigated by perturbation theory; the high-lying states, which are considered largely responsible for the effects of the tensor force, are eliminated by applying closure. An intermediate-coupling model for the low states emerges which is very similar to the customary one based on a vector force in that the effective nuclear potential for the low states is shown to consist of a central two-body force (with a spin-dependence different from that of the elementary two-body central force), plus a strong vector force, plus a tensor force which is probably weaker than the elementary two-body tensor force. The effective vector force is principally a three-body force, and hence may be expected to show a quite different "hole"-particle relationship than the one- or two-body vector forces usually assumed in the shell model. Because of the neglect in the wave function of the high-lying states which are mixed in directly by the tensor force, the model is expected to be valid for light nuclei only. The $\ensuremath{\beta}$ decay of ${\mathrm{B}}^{12}$ is discussed briefly, and is shown to be compatible with the tensor force as the sole spin-orbit force in the elementary two-body interaction.