RKKY interaction: why use the same subscript for electron and ion spins?

  • Level: Graduate 
  • Thread starter Thread starter LagrangeEuler
  • Start date Start date
Join the discussion
Registration is free. Start your own thread to ask a follow-up.
Sign up Log in
4 replies · 2K views
LagrangeEuler
Messages
711
Reaction score
22
Hamiltonian of interaction between fixed ion and conducting electron
##\hat{H}=-J\sum_{i=1}^2\{\hat{s}_i^z\hat{S}_i^z+\frac{1}{2}(\hat{s}_i^+\hat{S}_i^{-}+\hat{s}_i^-\hat{S}_i^{+})\}##
Why both electron and ion are defined with subscript ##i##?
 
Physics news on Phys.org

Discussion

...and why does i run from 1 to 2?

Looks like a bug to me.

One would think that you need separate indices i and j to label the atoms and the (conduction) electrons, and that these would run over the number of particles in question.
 
There is two fixed ions. One ion polarised electron and that electron interract with secound ion. So between ions is indirect interraction.
 
OK, that makes sense. However, I would still expect the electron index to be independent of the ion. Remember that RKKY occurs in the conduction electrons of metals.
 
Well I think that if electron is near ion you could approximate that it is on the same lattice site ion lattice site. I asked question about calculating matrix element in this problem.
Here is that question.
https://www.physicsforums.com/showthread.php?t=646675

I would be grateful if you could look and maybe answer this question.