Energy Levels, Wave Functions, Dipole and Quadrupole Transitions of Trivalent Gadolinium Ions in Sapphire.

Abstract

A computation is made of energy levels, wave functions and transition matrix elements of the Gd3+ ion in Al2O3. The crystal field parameters are taken from Geschwind and Remeika's paramagnetic resonance experiments. The transition probabilities are given for dipole radiation in three polarization directions. For ultrasonic work we give the real and imaginary parts of the five matrix elements of the quadrupole transition. From these one can easily deduce the transition probabilities in any given direction. The magnetic field directions are described by the angles θ and ϕ, the polar and azimuthal angles with respect to the crystalline c axis. The values of θ go from 0 to π/2 in six steps and two values of π are chosen, 0 and 2 π/3 for which the variation is largest. The magnetic field strengths are from 0 to 0.6 tesla (6000 gauss); beyond this value the spin can be considered as "free." Some consideration is given to the analytical behavior of the energy versus field diagram for the direction θ = ϕ = 0.