relation between lambda and lambda_nk

[ywchoi]

Dear all,

I am wondering what is the relation between the following two quantities.

1. the total electron-phonon coupling strength lambda
which is defined as the integral of Eliashberg function over omega (as Eq. 206 in F. Giustino RMP (2017))

2. the electron-phonon mass enhancements (lambda_nk) as defined in Eq. 189 of the same reference.


I suspect that the former is derived from the latter by ignoring the phonon energy in electron self-energy (so-called double-delta approx.)
but I am not exactly sure about how the derivation goes.
Furthermore, is it possible to use the Fermi-surface-averaged value of lambda_nk instead of lambda, in such a case where double-delta approx. is not so valid?

I would be appreciating any comments or discussions.

Thank you,
Young Woo Choi.