Measurement of higher-order multipole amplitudes in psi(3686) -> gamma chi(c1,2) with chi(c1,2) -> gamma J/psi and search for the transition eta(c)(2S) -> gamma J/psi

Using 106 x 10(6) psi(3686) events collected with the BESIII detector, we measure multipole amplitudes for the decay psi(3686) ->; gamma chi(c1,2) -> gamma gamma J/psi beyond the dominant electric-dipole amplitudes. The normalized magnetic-quadrupole (M2) amplitude for psi(3686) -> gamma chi(c1,2) -> gamma gamma J/psi and the normalized electric-dipole amplitudes for psi(3686) -> gamma chi(c2) -> gamma J/psi and determined. The M2 amplitudes for psi(3686) -> gamma chi(c1) and ; chi(c1,2) -> gamma J/psi are found to differ significantly from zero and are consistent with theoretical predictions. We also obtain the ratios of M2 contributions of psi(3686) and J/psi decays to;2,chi(c1,2,) b(2)(1/)b(2)(2) = 1.35 +/- 0.72 and a(2)(1/)a(2)(2) = 0.617 +/- 0.083,,which agree well with theoretical expectations. By considering the multipole contributions of chi(c1,2), we measure the product branching fractions for the cascade decays psi(3686) -> gamma chi(c 0,1,2) -> gamma gamma J/psi and search for the process eta(c)(2s) -> gamma J/psi through psi(3686) -> gamma eta(c)(2s).The product branching fraction for psi(3686) -> gamma chi(c0) -> gamma gamma J/psi is 3 sigma larger than published measurements, while those of psi(3686) -> gamma chi(c1,2) -> gamma gamma J/psi are consistent. No significant signal for the decay psi(3686) -> gamma eta(c) (2s) -> gamma gamma J/psi is observed, and the upper limit of the product branching fraction at the 90% confidence level is determined.