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Using the ground-state of Eu(III), how can I identify the possible (number of) luminescent transitions from the $^{5}D_{0}$ excited state?

I identified the ground-state term symbol for Eu(III) using Russel-Saunders coupling as $^{7}F_{0}$. I understand the selection rules and allowed transitions, but I'm not sure where to start on listing all of the possible transitions. I know that using RS coupling the $J$ states are $J=6,5,4,3,2,1,0$ but the transition from the $^{5}D_{0}$ state to the $^{7}F_{0}$ ground-state has $0\rightarrow 0$ with respect to $J$.

The only thing I could think of was to take the possible combinations of $\Delta S = 0,1$ and $\Delta L = 0,1$ and identify which of these transitions were allowed; is this along the right lines?

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