B. G. Wybourne

Spectral Intensities of the Trivalent Lanthanides and Actinides in Solution. I. Pr3+, Nd3+, Er3+, Tm3+, and Yb3+

We have correlated the experimentally measured intensities of the solution absorption spectra of Pr3+, Nd3+, Er3+, Tm3+, and Yb3+ in three different solvents with a theoretical expression derived by Judd. The expression, P=Σ lim λTλσ(fNψJ‖∪(λ)‖fNψJ′′)2λ=2,4,6 is derived from a theory in which the ligand‐field interactions with the central 3+ ion cause a mixing of higher configurations of opposite parity into the fN configuration and gives rise to induced electric‐dipole transitions. A prerequisite for the intensity calculations is the computation of complete intermediate coupling eigenvectors for the energy levels investigated. Other possible mechanisms such as magnetic‐dipole and electric‐quadrupole transitions are considered. The oscillator strengths of important magnetic‐dipole transitions are tabulated.The results give the first experimental verification of the ability of the theory to account for large changes in certain bands of the same lanthanide observed in different solvents. Conclusions are dra...

Use of Relativistic Wavefunctions in Crystal‐Field Theory

The purpose of this paper is to indicate some of the consequences of using relativistic wavefunctions in crystal‐field calculations. Substitutional expressions are developed which permit the calculation of relativistic crystal‐field matrix elements using LS basis states following the method of Sandars. In these expressions the angular parts of the matrix elements are evaluated using LS basis states while the radial part uses the radial integrals computed from relativistic Hartree—Fock‐type wavefunctions. It is demonstrated that the crystal‐field matrix elements are no longer diagonal in the spin quantum number S when relativistic wavefunctions are used. Recognition of this point leads to an important, and hitherto unnoticed, second‐order contribution to the ground‐state splitting of S‐state ions.

Analysis of the Solid‐State Spectra of Trivalent Neodymium and Erbium

A term analysis of the solid‐state spectra of trivalent neodymium and erbium has been made and this has allowed terms to be assigned to most of the observed levels of these ions. The complete spin‐orbit matrices, with the addition of the electrostatic interaction energies, corresponding to the f3 and f11 configurations have been diagonalized to calculate the energy level schemes for the neodymium and erbium ions, respectively. The appropriate parameters were calculated by a least‐squares method.

Configuration Interaction in Crystal Field Theory

The effect of configuration interaction on the validity of the usual method of expanding the crystal field potential in terms of spherical harmonics is examined using second‐order perturbation theory. It is found that for a configuration of equivalent electrons lN most mechanisms of configuration interaction lead to a simple scaling of the crystal field parameters Bqk. However, one‐electron excitations, either from the lN shell to unfilled orbitals or from closed shells into the lN shell, result in a scaling of the Slater integrals Fk and in overt effects that cannot be accommodated by the Bqk parameters alone. As a result, the crystal field parameters can be expected to vary from one multiplet to another, even in the limit of Russell—Saunders coupling. The consequences of these overt configuration‐interaction effects are discussed.

Effective Operators and Spectroscopic Properties

Perturbation summations that are characteristic of a wide range of spectroscopic problems are shown to be simply expressible in terms of effective operators involving the zero‐order states of the preponderant configuration. The methods developed involve the use of generalized unit tensor operators. Detailed applications to configuration‐interaction effects in lN configurations and to third‐order contributions to spectral intensities of ions in crystal fields are considered.

Kronecker products for compact semisimple Lie groups

A review is given of the application of S-function techniques to the evaluation of Kronecker products of irreducible representations of compact semisimple Lie groups. Explicit formulae are derived for all irreducible representations of all such groups. Recent developments involving composite Young diagrams are brought to fruition and the vexed problem of SO2k is dealt with completely. New branching rules for the classical groups are given in an appendix. These are exploited in the evaluation of Kronecker products by means of a technique which is applied to both classical and exceptional groups. A discussion is made of various modification rules which are needed to express the final results in standard form.

Structure of fn Configurations. II. f5 and f9 Configurations

The complete energy matrices of the f5 configuration have been calculated using the methods of Racah. Energy level schemes have been constructed for the trivalent samarium, dysprosium, and plutonium ions. A preliminary analysis of the observed spectra of these ions is made.

Composition of the Electronic States of Nd(IV) and Er(IV)

The eigenvectors of the electronic states of the trivalent neodymium and erbium ions are tabulated for use in calculating the crystal field and Zeeman splittings of the ground and excited states.

The Lie group of Newton's and Lagrange's equations for the harmonic oscillator

Lies theory of differential equations is applied to the equation of motion of the classical one-dimensional harmonic oscillator. The equation is found to be invariant under a global Lie group of point transformations that is shown to be SL(3, R). The physical significance of the analysis and the results is considered. It is shown that the periodicity of the motion is a local topological property of the equation, while the length of the periods depends upon global properties.

New branching rules induced by plethysm

We derive group branching laws for formal characters of subgroups of leaving invariant an arbitrary tensor T? of Young symmetry type ? where ? is an integer partition. The branchings and fixing a vector vi, a symmetric tensor gij = gji and an antisymmetric tensor fij = ?fji, respectively, are obtained as special cases. All new branchings are governed by Schur function series obtained from plethysms of the Schur function s? ? {?} by the basic M series of complete symmetric functions and the L = M?1 series of elementary symmetric functions. Our main technical tool is that of Hopf algebras and our main result is the derivation of a coproduct for any Schur function series obtained by plethysm from another such series. Therefrom one easily obtains ?-generalized Newell?Littlewood formulae and the algebra of the formal group characters of these subgroups is established. Concrete examples and extensive tabulations are displayed for and , showing their involved and nontrivial representation theory. The nature of the subgroups is shown to be in general affine and in some instances non-reductive. We discuss the complexity of the coproduct formula and give a graphical notation to cope with it. We also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly nontrivial 2-cocycles. The algebra of subgroup characters is identified as a cliffordization of the algebra of symmetric functions for formal characters. Modification rules are beyond the scope of the present paper, but are briefly discussed.