Judd–Ofelt theory

Theory

Judd-Ofelt theory may be used to predict and analyze the intensities of intra 4f electronic dipole transitions. Such transitions are disallowed by Electronic Dipole Transition Selection rules in free space, as the initial and final states have the same parity. However, intra 4f transitions have been observed. The transition strengths and the transition changing the orbital angular momentum quantum number are not consistent with them being Magnetic Dipole Transitions.

This apparent discrepancy is reconciled by treating the crystal field an ion in a solid experiences as a perturbation to the free space Hamiltonian. This perturbation mixes free space electronic states of opposite parity (namely the rare earth ion's ground 4fⁿ electronic configuration with the opposite parity 4f^n-15d). Thus Electronic Dipole transitions between these crystal field perturbed electronic states do not violate this parity change selection rule.

The theory quantitatively describes this mixing using three phenomenological parameters particular to the host crystal, denoted as ${\displaystyle {\mathrm{\Omega }}_{\lambda }}$ (where ${\displaystyle \lambda =2,4,6}$). These parameters account for the asymmetric nature of the crystal field and enable the calculation of transition probabilities, oscillator strengths, and radiative lifetimes of excited states, which are crucial for the development of various photonic devices such as lasers and optical amplifiers.

Like Russell-Saunders Coupling (LS-Coupling), Judd-Ofelt theory can be simplified to a list of selection rules. The rules for Judd-Ofelt Induced Electric dipole transitions are listed in the following table and compared to LS-coupling Magnetic and Electric dipole transitions.

Term symbols in LS-Coupling describe the total orbital angular momentum (${\displaystyle L}$), total spin (${\displaystyle S}$), and total angular momentum (${\displaystyle J=S+L}$). ${\displaystyle \pi }$ is the parity of an electronic configuration.

Example: Europium (III)

For example let us apply these selection rules to triply ionized Europium (Eu³⁺). Eu³⁺ has an electronic configuration of [Xe]4f⁶. The ground term symbol for this configuration (according to Hund's Rules) is ⁷F₀.

Applying the above selection rules, transitions from this state to ⁵D₂, ⁵D₄, and ⁵D₆ are allowed, but not to ⁵D₁, ⁵D₀, or ⁵L₇ (violating the restrictions on ${\displaystyle J}$).