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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?

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  • $\begingroup$ see en.wikipedia.org/wiki/Cooper_pair to understand that a medium should exist so that the attractive force between electrons can appear. In the vacuum of the tube there is no medium. $\endgroup$ – anna v May 11 at 11:41
  • $\begingroup$ I did read the article, but the medium was very briefly mentioned so it slipped my mind although I tried to figure out what electron-phonon interaction was told about, which makes more sense in a medium yes. Thanks for clarifying. $\endgroup$ – Exocytosis May 11 at 11:44
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    $\begingroup$ Take a TEM at normal beam conditions, say a few nanoamps. Take 200 keV electrons. Calculate how many are actually in the microscope column at any one time. Note that the answer is close to 1 electron at a time. $\endgroup$ – Jon Custer May 11 at 18:24
  • $\begingroup$ @Jon Custer: This puts everything in perspective. Thank you for this enlightening comment. $\endgroup$ – Exocytosis May 11 at 18:27
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The electrons in a Cooper pair form a bound state because there is an attractive force between them. Since they form a bound state they pair up with opposite spins in the lowest energy level of the bound state. This bound state then has a zero net spin.

In an electron beam there is no attractive force between the electrons so there is no way for pairs of electrons to form any bound states analogous to a Cooper pair.

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  • $\begingroup$ I see. Can there be superposition though, since different spins mean no exclusion principle? $\endgroup$ – Exocytosis May 11 at 11:02
  • $\begingroup$ You'll have to define what you mean by superposition in this context. $\endgroup$ – John Rennie May 11 at 11:06
  • $\begingroup$ Occupy the same space. $\endgroup$ – Exocytosis May 11 at 11:24
  • $\begingroup$ @Exocytosis The electrons in the beam are approximately plane waves. They could only clash if they had exactly the same frequency and wave vector and in practice this will never happen. So there is no problem in many electrons occupying the same space. $\endgroup$ – John Rennie May 11 at 11:28
  • $\begingroup$ Are you saying the wave nature of electrons sort of discard their electrostatic mutual repulsion? $\endgroup$ – Exocytosis May 11 at 11:33

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