Thermal vibration of single-walled carbon nanotubes with quantum effects.

The thermal vibration of a single-walled carbon nanotube (SWCNT) is investigated by using the models of Euler beam and Timoshenko beam with quantum effects taken into consideration when the law of energy equipartition is unreliable. The relation between temperature and the root of mean-squared (RMS) amplitude of thermal vibration at any cross section of the SWCNT is derived via the beam models in simply supported case and cantilevered case. The RMS amplitude of thermal vibration of SWCNT predicted by using Timoshenko beam is higher than that predicted by using Euler beam. The RMS amplitude of thermal vibration of an SWCNT predicted by the quantum theory is lower than that predicted by the law of energy equipartition. The quantum effect is more important for the thermal vibration of an SWCNT in the cases of higher-order modes, short length and low temperature.