Many-body theory of indirect nuclear interactions

We derive an expression for the indirect nuclear coupling tensor ($A_{\mathrm{jj}\mathcal{’}}^{\alpha \beta }$) including spin-orbit and many-body effects. We use a finite-temperature Green’s-function method where the thermodynamic potential is expressed in terms of the exact one-particle propagator and the proper self-energy, and derive a general expression for $A_{\mathrm{jj}\mathcal{’}}^{\alpha \beta }$ in the Bloch representation. While the effects of spin-orbit interaction appear in our expression through the modification of the one-particle eigenvalues and eigenstates and through a change in the orbital hyperfine interaction via the modification of the electronic momentum operator, the many-body effects are more subtle. We find that the many-body corrections to $A_{\mathrm{jj}\mathcal{’}}^{\alpha \beta }$ in the quasiparticle approximation are cancelled in part by the inclusion of exchange and correlation effects. We also show, by making drastic assumptions while solving the matrix integral equations for the nuclear spin-dependent part of the self-energy, that the exchange enhancement effects in a band model on $A_{\mathrm{jj}\mathcal{’}}^{\alpha \beta }$ are different for different terms. The remarkability of the theory is that for the first time a systematic effort has been made to study the effects of electron-electron interaction on the various contributions to $A_{\mathrm{jj}\mathcal{’}}^{\alpha \beta }$. We also discuss the importance of relativistic and electron-electron interaction effects in the calculation of the coupling constant in real systems. The theory is general and can be applied to metals, semiconductors, and insulators.