The crystallinity and the structure of heteroepitaxially grown $Y_2O_3$ films on the silicon substrates deposited by Ultra High Vacuum Ionized Cluster Beam(UHV-ICB) were investigated by Back-scattering Spectroscopy(BS)/channeling. The channeling minimum values, $X_{min}$, of the $Y_2O_3$ films deposited by other methods were 0.8~0.95 up to the present, which indicates amorphous or highly polycrystalline nature of the $Y_2O_3$ films. On the contrary, the channeling minimum value of heteroepitaxially grown $Y_2O_3$ films on Si(100) and Si(111) deposited by UHV-ICB are 0.28 and 0.25 respectively. These results point out fairly good crystalline quality. It is also observed that the top region of $Y_2O_3$ films have less crystalline defects than the bottom region regardless of the crystal direction of the Si substrates. The axis of $Y_2O_3$<111> epitaxially grown on Si(111) is tilt by $0.1^{circ}$ with respect to Si<111>. That of $Y_2O_3$<110> on Si(100) is parallel to the Si<001>. The $Y_2O_3$ film on Si(100) grew with single domain structure and that on Si(111) grew with double domain structure. From the result of oxygen resonance BS/channeling, the oxygen atoms in heteroepitaxially grown $Y_2O_3$ film on Si(111) substrate have the crystallinity, but that on Si(100) shows almost channeling amorphous state.