Superconductivity from magnetic order for Van Vleck metals

For Van Vleck metals such as Pr, it is shown in a previous paper that the indirect electron-electron interaction V, mediated by the virtual excitation of the crystal field singlet ground state through the s-f exchange interaction, can give rise to superconductivity. With increase of the s-f exchange interaction or the Van Vleck susceptibility, V increases and makes the superconducting transition temperature T-c increase. Conversely, when V is beyond the critical value V-M, magnetic order appears at a finite temperature T-M. In this case, a problem arises in whether T-c lies above or below T-M. This paper shows that, except for the very vicinity of V-M, the superconductivity appears below T-M contrary to the conclusion of the BCS approximation. This is due to the depairing effect by the scattering of the conduction electrons accompanied by the excitation of the singlet ground state. The phase diagram is obtained by assuming the free electron band and the orthorhombic crystal field.