Nb$_3$Sn is currently the most promising material other than niobium for future superconducting radiofrequency cavities. Critical fields above 120 mT in pulsed operation and about 80 mT in CW have been achieved in cavity tests. This is large compared to the lower critical field as derived from the London penetration depth, extracted from low field surface impedance measurements. In this paper direct measurements of the London penetration depth from which the lower critical field and the superheating field are derived are presented. The field of first vortex penetration is measured under DC and RF fields. The combined results confirm that Nb$_3$Sn cavities are indeed operated in a metastable state above the lower critical field but are currently limited to a critical field well below the superheating field.
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