Equivalence of Different Pair-Breaking Mechanisms in Superconductors

Paramagnetic impurities in a superconductor and a magnetic field parallel to a thin superconducting film with short electronic mean free path are known to have similar effects on many physical quantities. By investigating the properties of vertex functions and their renormalization factors we have derived necessary and sufficient conditions for such a physical quantity to be equally affected by the two pair-breaking mechanisms. We find that different expressions are obtained in the two theories only for those quantities which involve $s$-wave, spin-triplet vertices with momentum transfer $q\le {\left(l{\xi }_0\right)}^{-\frac{1}{2}}$, where $l$ is the electron mean free path and ${\xi }_0$ is the coherence distance. With the help of those results different response functions are analyzed and it is shown that the spin susceptibility is the only one of the quantities usually considered which satisfies the above requirements and hence gives different results for the field and paramagnetic-impurity cases.