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.

中文翻译：

---

使用 Lambert W 函数的固体物理新分析结果


出于数学完整性的考虑，物理学中的解析解始终是首选。为此，使用 Lambert W 函数，推导了离子晶体中平衡离子间距离、晶体中空位的形成能、清洁极限超导体的零温能隙和临界临界值的闭式解析表达式。 2D-XY 模型中相变的 Kosterlitz-Thouless 温度。除了理论上的兴趣之外，目前的一些结果还提出了相关物理量的替代实验测定。此外，类似地基于Lambert W函数导出用于确定玻尔兹曼常数的显式解析公式。这里提出这个公式作为测量该常数的实验方法的理论基础。此方法适合本科物理专业学生。