Selection rules for 4f²→4f² electric dipole transitions by considering mixing with 4f5d components of Pr³⁺ in SrAl₁₂O₁₉

A set of selection rules is deduced for the 4f² transitional levels that mix with explicit 4f5d components. Based on these selection rules the dipole-allowed transitions between the 4f² crystal field levels are determined in Pr³⁺ doped SrAl₁₂O₁₉. Meanwhile, the main 4f5d components mixed into 4f² transitional levels are identified by determining the nonzero matrix elements of both the odd-rank crystal-field and the electric dipole operator. The selection rules have been applied to analyze the transitions from ¹S₀ and ³P₀ to lower crystal field levels.