Spin fluctuations in conventional superconductors and anomalous isotope effect in PdH and PdD

Spin fluctuations have been suggested to be an influence to the superconducting phenomenon, either favorably or disruptively. Consequently, we study the role of magnons in the thermodynamics of conventional superconductors proposing a relationship between magnetization and electron-magnon coupling (${\lambda }_{em}$) for vanadium, niobium, lead, palladium, and the PdX (where $X=$ hydrogen, deuterium, and tritium) systems. We suggest a different relationship for the Coulomb pseudopotential (${\mu }_N^{\ast }$), which is influenced by the characteristic frequency (${\omega }_{ln}$). In this way, the critical temperatures ($T_c$) of the elements studied and the PdX compounds are very close to those reported experimentally. The PdH, PdD and PdT compounds have shown critical temperatures of 9.05 K, 12.20 K and 12.90 K, respectively. Finally, we determine the behavior of $T_c$ for PdX under hydrostatic pressures up to 10 GPa, showing an alternative explanation for the anomalous isotope effect in PdH and PdD compounds. Our study was carried out by numerically solving the linearized Migdal-Eliashberg equations coupling the Bogoliubov-de-Gennes method, while the magnetic excitation was calculated using linear response time dependent within the Density Functional Theory, framework, both implemented in the Elk code.