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RKKYRudermann-Kittel-Kasuya-Yoshida interaction The mechanism was proposed initially by Rudermann and Kittel to explain broad NMR lines in Ag metal, invoking a nuclear spin-spin interaction mediated by hyperfine coupling to unpolarized conduction electrons. It describes therefore a hyperfine interaction mechanism active in metals. This is described in Y. Yafet, Phys. Reb. B 36, 3948 It supposes a contact (Fermi) hyperfine interaction between an electron of spin {$\boldsymbol S_i$} at {$\boldsymbol x_i$} and a nucleus of spin {$\boldsymbol I_n$} at {$\boldsymbol x_n$}, of the type {${\mathcal H}_{in} = A \boldsymbol S_i\cdot\boldsymbol I_n \, \delta (\boldsymbol x_i-\boldsymbol x_n)$}. This interaction, mediated over a Fermi electron gas by the factor {$$F(\boldsymbol q) = P \frac 8 {L^3} \sum_{\boldsymbol k} \frac {n_{\boldsymbol k} - n_{\boldsymbol k + \boldsymbol q}} {E_{\boldsymbol k} - E_{\boldsymbol k + \boldsymbol q}}$$} results in an indirect nuclear coupling {${\mathcal H}_{nm} = -A^2 \boldsymbol I_n\cdot\boldsymbol I_m \Phi( x_n- x_m)$}, where the range function {$\Phi(x) = FT\left[F(\boldsymbol q)\right] $} depends on the modulus the vector {$\boldsymbol x$} as {$$\Phi(x) \approx \frac 1 2 \left[\frac \pi 2 - \mathrm{Si}(2k_F x) - \frac {\cos 2k_F x} {2k_F x} + \frac {\sin 2k_F x} {2k_F x}\right] $$} By extension the same indirect mechanism drives an exchange interaction among conduction electrons, as described, more qualitatively e.g. in P. Bruno and C. Chappert, Phys. Rev. Lett. 67, 1602 Remember that the Sine integral function is {$\mathrm {Si}(z) = \int_0^z \frac {\sin t} t dt$} |