New analytical results in solid state physics using the Lambert W function

Abstract

Analytical solutions in physics are always preferred for the sake of mathematical completeness. For this, using the Lambert W function, I derive closed-form analytical expressions for the equilibrium interionic distance in an ionic crystal, the formation energy of a vacancy in a crystal, the zero-temperature energy gap of a clean-limit superconductor and the critical Kosterlitz–Thouless temperature for the phase transition in 2D-XY model. Besides their theoretical interest, some of the present results suggest alternative experimental determinations of the relevant physical quantities. In addition, I similarly derive an explicit analytical formula based on the Lambert W function for a determination of the Boltzmann constant. This formula is proposed here as a theoretical basis for an experimental method to measure this constant. This method is suitable for undergraduate physics students.