Observation of the decays $chi_{cJ} to phi phi eta$

Using a data sample of $(448.1pm2.9)times10^{6}$ $psi(3686)$ decays collected by the BESIII detector at the Beijing Electron Positron Collider (BEPCII), we observe the decays $chi_{cJ}to phiphieta~(J=0,~1,~2)$, where the $chi_{cJ}$ are produced via the radiative processes $psi(3686)togammachi_{cJ}$. The branching fractions are measured to be $mathcal B(chi_{c0}tophiphieta)=(8.41pm0.74pm0.62)times10^{-4}$, $mathcal B(chi_{c1}tophiphieta)=(2.96pm0.43pm0.22)times 10^{-4}$, and $mathcal B(chi_{c2} to phiphieta)=(5.33pm0.52pm0.39) times 10^{-4}$, where the first uncertainties are statistical and the second are systematic. We also search for intermediate states in the $phiphi$ or $etaphi$ combinations, but no significant structure is seen due to the limited statistics.