Thermodynamic properties of an ideal Fermi gas in an external potential with U=brt in any dimensional space

The properties of an ideal Fermi gas in an external potential in any dimensional space are studied, based on the semiclassical (Thomas-Fermi) approximation. The general analytical expressions of the total particle number $N,$ density of states $D(\ensuremath{\varepsilon}),$ Fermi energy ${E}_{F},$ total energy $E,$ and heat capacity $C$ have been derived, where $N,$ $E,$ and $C$ are expressed by the Fermi integration. Moreover, the analytical expressions of the total energy $E,$ chemical potential \ensuremath{\mu}, and heat capacity $C$ in the high- and low-temperature approximations are given. From these results, how the characteristics of the Fermi gas depend on an external potential and the dimension of space is discussed.