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.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.

Physical Review AAug 1, 199839 citations

Theory of two-dimensional polarized Fermi gases

Brandon Anderson, Chien‐Te Wu, Rufus Boyack +1 more

Bulletin of the American Physical SocietyMay 26, 20160 citations

Limits of sympathetic cooling of fermions: The role of heat capacity of the coolant

Lincoln D. Carr, Yvan Castin

The sympathetic cooling of an initially degenerate Fermi gas by either an ideal Bose gas below ${T}_{c}$ or an ideal Boltzmann gas is investigated. It is shown that the efficiency of cooling by a Bose gas below ${T}_{c}$ is by no means reduced when its heat capacity becomes much less than that of the Fermi gas, where efficiency is measured by the decrease in the temperature of the Fermi gas per number of particles evaporated from the coolant. This contradicts the intuitive idea that an efficient coolant must have a large heat capacity. In contrast, for a Boltzmann gas a minimal value of the ratio of the heat capacities is indeed necessary to achieve $T=0$ and all of the particles must be evaporated.The sympathetic cooling of an initially degenerate Fermi gas by either an ideal Bose gas below ${T}_{c}$ or an ideal Boltzmann gas is investigated. It is shown that the efficiency of cooling by a Bose gas below ${T}_{c}$ is by no means reduced when its heat capacity becomes much less than that of the Fermi gas, where efficiency is measured by the decrease in the temperature of the Fermi gas per number of particles evaporated from the coolant. This contradicts the intuitive idea that an efficient coolant must have a large heat capacity. In contrast, for a Boltzmann gas a minimal value of the ratio of the heat capacities is indeed necessary to achieve $T=0$ and all of the particles must be evaporated.

arXiv:cond-mat/0310671Physical Review AApr 20, 20049 citations

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.

The sympathetic cooling of an initially degenerate Fermi gas by either an ideal Bose gas below

{T}_{c}

or an ideal Boltzmann gas is investigated. It is shown that the efficiency of cooling by a Bose gas below

{T}_{c}

is by no means reduced when its heat capacity becomes much less than that of the Fermi gas, where efficiency is measured by the decrease in the temperature of the Fermi gas per number of particles evaporated from the coolant. This contradicts the intuitive idea that an efficient coolant must have a large heat capacity. In contrast, for a Boltzmann gas a minimal value of the ratio of the heat capacities is indeed necessary to achieve

T=0

and all of the particles must be evaporated.