Specific heats of dilute neon inside a long single-walled carbon nanotube

An elegant formula for coordinates of carbon atoms in a unit cell of a single-walled nanotube (SWNT) is presented and the potential of neon (Ne) inside an infinitely long SWNT is analytically derived under the condition of the Lennard-Jones potential between Ne and carbon atoms. Specific heats of dilute Ne inside a long (20, 20) SWNT are calculated at different temperatures. It is found that Ne exhibits three-dimensional (3D) gas behaviour at high temperature but behaves as a 2D gas at low temperature. Especially, at ultra-low temperature, Ne inside (20, 20) nanotubes behaves as a lattice gas. A rough method to determine the characteristic temperature Tc for low density gas in a potential is put forward. If T much greater than T-c, we just need to use the classical statistical mechanics without solving the Schrodinger equation to consider the thermal behaviour of gas in the potential. But if T similar to T-c, we must solve the Schrodinger equation. For Ne in a (20, 20) nanotube, we obtain T-c approximate to 60 K.