An exact analytical expression for the condensation energy $$E_{\text{cond}} \left( T \right)$$ of a phonon-driven superconductor for all absolute temperatures $$T$$ and for any coupling strength is introduced so as to calculate the Helmholtz free energy difference $$F_{s} \left( T \right) - F_{n} \left( T \right)$$ between superconducting $$\left( s \right)$$ and normal $$\left( n \right)$$ states. This is achieved via a boson–fermion ternary gas theory—called the generalized Bose–Einstein condensation (GBEC) theory—which includes two-hole Cooper pairs, two-electron ones as well as single, free/unbound electrons. The GBEC formalism turns out to be quite useful in dealing with nonzero $$T$$ values of $$E_{\text{cond}} \left( T \right)$$ and reproduces several well-known experimental results. An expression for the condensation energy per atom is also calculated and applied to aluminum and niobium, and both results are compared with experimental data.

中文翻译：

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超导体中所有温度下的凝聚能

对于所有绝对温度 $$T$$ 和任何耦合强度，声子驱动超导体的凝聚能 $$E_{\text{cond}} \left(T\right)$$ 的精确解析表达式被引入，因此至于计算超导$$\left(s \right)$$与超导之间的亥姆霍兹自由能差$$F_{s} \left( T \right) - F_{n} \left( T \right)$$ $$\left( n \right)$$ 状态。这是通过玻色子-费米子三元气体理论（称为广义玻色-爱因斯坦凝聚 (GBEC) 理论）实现的，该理论包括双孔库珀对、双电子对以及单个自由/未束缚电子。事实证明，GBEC 形式主义在处理 $$E_{\text{cond}} \left( T \right)$$ 的非零 $$T$$ 值时非常有用，并重现了几个众所周知的实验结果。