For superconductors the bosons involved are the so-called Cooper pairs which are quasiparticles formed by two electrons. Hence m = 2m and q = −2e where m and e are the mass of an electron and the elementary charge. It follows from Eq. () that
Integrating Eq. () over a closed loop gives
As in the case of helium we define the vortex strength
and use the general relation
where Φ is the magnetic flux enclosed by the loop. The so-called fluxoid is defined by
In general the values of κ and Φ depend on the choice of the loop. Due to the single-valued nature of the wave function and Eq. () the fluxoid is quantized
The unit of quantization is called the flux quantum
The flux quantum plays a very important role in superconductivity. The earth magnetic field is very small (about 50 μT), but it generates one flux quantum in an area of 6 μm by 6 μm. So, the flux quantum is very small. Yet it was measured to an accuracy of 9 digits as shown in Eq. (). Nowadays the value given by Eq. () is exact by definition.
In Fig. 3 two situations are depicted of superconducting rings in an external magnetic field. One case is a thick-walled ring and in the other case the ring is also thick-walled, but is interrupted by a weak link. In the latter case we will meet the famous Josephson relations. In both cases we consider a loop inside the material. In general a superconducting circulation current will flow in the material. The total magnetic flux in the loop is the sum of the applied flux Φ and the self-induced flux Φ induced by the circulation current