The mechanism of angular dependence of ferromagnetic resonance linewidth of dilute Tb and Nd doping in permalloy thin films with various thicknesses of adjacent copper layer are investigated by experimental approach and theoretical fitting by considering the contributions from intrinsic spin-orbit coupling, inhomogeneous broadening and two-magnon scattering. The results show that the damping coefficient α, by intrinsic contribution extracted from ferromagnetic resonance linewidth, increases from 0.0153 to 0.0218 for NiFe-Nd films and from 0.0193 to 0.0261 for NiFe-Tb films resulting from the spin pumping effect at the interface of NiFe-Nd(or Tb)/Cu as the thickness of copper layer increases from 1 nm to 15 nm. The surface magnetic anisotropy constant K1 is obtained and shows an decreasing trend from positive to negative, which implies that the copper layer could reduce the surface perpendicular anisotropy. The fitting spin mixed conductivity is (2.72±0.18)×1015cm−2 at NiFe-Tb/Cu interface and (2.4±0.2)×1015cm−2 at Cu/NiFe-Nd interface, respectively.