I read Wikipedia articles "Cooper pair" and "Electron pair" among others.
In an atom electrons can be in the same orbital only if they have a different spin (Pauli exclusion principle). In Cooper pairs, the interaction is far more distant, almost in the micrometer range, but I am unsure of the reason: I assume this is because electrostatic repulsion is stronger than magnetic attraction, combined with the absence of a positive charge that could force them to be closer (as protons in an atom). Cooper pairs happening in superconductors, I assume very low temperatures and thus very little effect of temperature on electrons distance.
Back to free electrons in a cathode ray. It is clear that due to particles acceleration and electromagnetic lensing a beam can be highly collimated. This is why electron microscopy (nanoscopy, actually) is so successful. Here the distance between electrons is way smaller than in Cooper pairs.
So is there a remote possibility of pair formation among spin opposed electrons inside the collimated beam? And if not, why not?