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A theory of superconductivity is presented where the effect of anharmonicity,as entailed in the acoustic, or optical, phonon damping, is explicitlyconsidered in the pairing mechanism. The gap equation is solved includingdiffusive Akhiezer damping for longitudinal acoustic phonons or Klemens dampingfor optical phonons, with a damping coefficient which, in either case, can bedirectly related to the Gruneisen parameter and hence to the anharmoniccoefficients in the interatomic potential. The results show that the increaseof anharmonicity has a strikingly non-monotonic effect on the criticaltemperature $T_{c}$. The optimal damping coefficient yielding maximum $T_c$ isset by the velocity of the bosonic mediator. This theory may open upunprecedented opportunities for material design where $T_{c}$ may be tuned viathe anharmonicity of the interatomic potential, and presents implications forthe superconductivity in the recently discovered hydrides, where anharmonicityis very strong and for which the anharmonic damping is especially relevant.

Anharmonic phonon damping enhances the $T_c$ of BCS-type superconductors