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Character of the low-energy $E1$ excitations is investigated by analyzingtransition densities obtained from the RPA calculations in the doubly-magicnuclei. We propose a decomposition method of the $E1$ excitations into the $pn$mode (i.e. oscillation between protons and neutrons) and the skin mode(oscillation of the neutron skin against the inner core) via the transitiondensities, by which their mixing is handled in a straightforward manner.Crossover behavior of the $E1$ excitations is found, from the skin mode at lowenergy to the $pn$ mode at higher energy. The ratio of the skin-mode strengthto the full strength turns out to be insensitive to nuclide and to effectiveinteractions in the energy region of the crossover. Depending on the excitationenergy, the observed low-energy $E1$ excitations are not necessarily dominatedby the skin mode, as exemplified for $^{90}$Zr. fied for $^{90}$Zr.

Crossover from skin-mode to proton-neutron-mode in $E1$ excitations of neutron-rich nuclei