Iron impurities are believed to act as deep acceptors that can compensate for the n-type conductivity in as-grown Ga₂O₃, but several scientific issues, such as the site occupation of the Fe heteroatom and the complexes of Fe-doped β-Ga₂O₃ with native defects, are still lacking. In this paper, based on first-principle density functional theory calculations with the generalized gradient approximation approach, the controversy regarding the preferential Fe incorporation on the Ga site in the β-Ga₂O₃ crystal has been addressed, and our result demonstrates that Fe dopant is energetically favored on the octahedrally coordinated Ga site. The structural stabilities are confirmed by the formation energy calculations, the phonon dispersion relationships, and the strain-dependent analyses. The thermodynamic transition level Fe³⁺/Fe²⁺ is located at 0.52 eV below the conduction band minimum, which is consistent with Ingebrigtsen’s theoretical conclusion, but slightly smaller than some experimental values between 0.78 eV and 1.2 eV. In order to provide direct guidance for material synthesis and property design in Fe-doped β-Ga₂O₃, the defect formation energies, charge transitional levels, and optical properties of the defective complexes with different kinds of native defects are investigated. Our results show that V_Ga and O_i can be easily formed for the Fe-doped β-Ga₂O₃ crystals under O-rich conditions, where the +3 charge state Fe_GaGa_i and −2 charge state Fe_GaO_i are energetically favorable when the Fermi level approaches the valence and conduction band edges, respectively. Optical absorption shows that the complexes of Fe_GaGa_i and Fe_GaV_Ga can significantly enhance the optical absorption in the visible-infrared region, while the energy-loss function in the β-Ga₂O₃ material is almost negligible after the extra introduction of various intrinsic defects.

We investigate the constraint problem of a small ring on non-smooth elliptical orbits. The influence mechanism of the orbital shapes on the motion velocity and constraint force of the small ring are analyzed. Taking circular orbit as an example, the influence of gravity and sliding friction on the orbital motion is given, and the results are verified by kinetic energy theorem. Finally, the characteristics of the orbital motion of the small ring at different initial velocities are analyzed.

Orbit constraint problems can be encountered in mechanical equipment and amusement equipment. Mechanics exercises generally consider the ideal physical model, and the practical problems also consider the influence of friction, which makes the problem more complex and and practical. The problem of the force and oscillation of objects on orbit needs to be deeply discussed. In order to simulate the orbital motion of objects more realistically and help students expand their theoretical mechanics beyond class, we study the orbit constraint of a small bead on a rotating large circular hoop in a horizontal plane about an axis passing through a point on the circumference. The coupling equations followed by the bead on the hoop are derived using Newton’s second law in a planar polar coordinate system and solved by numerical methods. We found that under the action of friction, when the initial angular velocity of the bead is greater than the critical angular velocity, the bead will rotate on the ring, and the number of rotations is related to the initial angular velocity and influenced by the friction coefficient. At different initial angular velocities, the number of oscillations of the bead on the hoop is basically the same and ultimately stops near the fixed point.

We study the orbit constraint of a small bead on a rotating large circular hoop in a horizontal plane about an axis passing through a point on the circumference. Firstly, the oscillation of the bead on the smooth rotating hoop is studied. Secondly, the influence mechanism of friction coefficient on angular velocity, constraint force and work of the bead on the rotating hoop is studied. The calculation results are consistent with the kinetic energy theorem. Finally, the influence mechanism of initial angular velocity on the kinematics of the bead on the hoop is studied.