What is the 'triangle selection rule' in spectroscopy? This paper states that

However, the multiphonon relaxation bridging the $^5\!D_1$ and $^5\!D_0$ levels is a well-known exception because it is formally forbidden by the triangle rule and occurs due to $J$-mixing.

What is the triangle rule? Examples?
 A: The 'triangle rule' seems to be (as used, e.g. in Molecular Spectroscopy by D.J. Millen) the application of the triangle inequality to the addition of angular momentum. More specifically, if you are adding momenta with magnitudes $j_1$ and $j_2$ and magnetic quantum numbers $m_1$ and $m_2$, then the result will only have nonzero amplitude along the state $|j_3,m_3⟩$ (i.e. the Clebsch-Gordan coefficient will be nonzero) if the conditions
$$
|j_k-j_l|\leq j_i \text{ for all distinct }i,k,l,\quad \text{and}\quad m_1+m_2=m_3.
$$
These are justified in depth in any angular-momentum-in-QM book.
It is the first of those conditions that would be understood to be the triangle rule, as it directly embodies the triangle inequality. This is the rule that's at play here: a ${}^5\! D_1\to{}^5 \!D_0$ transition involves a change in $J$ from 1 to 0. However, since phonons have zero spin, the triangle rule forbids any phononic transitions which change the total angular momentum.
It's important to note, though, that for atoms as big as europium, the assumptions behind term symbols tend to break down, and instead of having a bunch of electrons with a well-defined total orbital and total spin angular momentum, the spin-orbit coupling on each electron is so big that you're best off by thinking of each electron as having a well-defined total $j$, and then adding those together. In the middle ground, this simply means that a term symbol like ${}^5\! D_1$ is only an approximate description of the true eigenstate, which includes a nonnegligible amplitude of ${}^5\! D_0$ character. This is what's known as $J$-mixing and it does enable phononic transitions between such states.
