Dear Ilias,

The role of the spin-orbit interaction on the electron-phonon coupling is quite subtle and it was studied in the superconductivity in different forms. For instance see the thesis (and their references)

https://www.ifw-dresden.de/userfiles/groups/itf_folder/Theses/Dissertation_Katja_Paerschke.pdf

and the papers

https://core.ac.uk/download/pdf/36054254.pdf

Article Effects of spin-orbit coupling on the electron-phonon superc...

Daniel Baldomir thanks professor for the answer that's help me to make the connection between the electron phonon coupling and spin orbit coupling to arrive on superconductivity

Best regards professor

Dear Serifi, the particles that interact with phonons in a Rashba metal (I prefer to call it a Rashba metal and no to call it a normal metal) (above or below Tc, it won´t matter, because their spin already is no a good quantum number) They are called helicity states and indeed, it is a difficult question from the theoretical point of view: how helicity states interact with phonons.

Pedro L. Contreras E.

Thank you professor for the answer I know, in theory, this is not clearly approved and defined, but you know, in superconductivity, we are interested in the Tc before the break of the cooper's electron pair; If we see the SOC interaction globally, what can this process do in the general view ?

in the article which professor Daniel Baldomir send to me they told that the electron phonon coupling increase if we apply the SOC on the system.

Thanks for your answer Dear Serifi. Well, the question that was given to me, 12 years ago; it was to tackle a different approach without involving more than the gamma constant of the coupling with some symmetry invariant properties, but no the whole expression for the SOC interaction (it was for the normal state).

But I reviewed the literature, and I can remember this very useful for me a that time paper: PhysRevB.55. pp .15523 Theory of screening of the phonon-modulated spin-orbit interaction in metals, Grimaldi, Claudio and Fulde, Peter, 1997. You could check that one for the normal state. For the supèrconducting properties you could review the lastest publications of Profs. Samokhin and Mineev for noncentrosymetric superconductors. However I don´t know how much progress has been done since 2007. My Best Regards.

Pedro L. Contreras E.

Thanks a lot dear professor i will review all what you give to me

it's a pleasure to interact with you professor.

Best Regards

You are welcome Dear Serifi. And sorry I can´t send some papers I had from that time. But now, I am far away from office and it seems that the server there is down. I couldn´t connect to my account. My Best Regards.