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Korringa< Knight shift | Index | Hebel-Slichter peak in relaxation? > Relaxation rate from conduction electrons Let us perform a quick rough calculation. The energy exchanged by nuclei with electrons during thier mutual spin flip is negligible , compared to the Fermi energy {$\epsilon_F$}. Therefore only electrons in a narrow crust of width {$k_BT$}, centered at the surface of the Fermi sphere, will contribute to nuclear spin flipping, since only these electrons have available occupied and empty states which allow the conservation of energy. The electron spins may be thought as fluctuating on a time-scale {$\tau$} determined by the uncertainty principle {$ \tau = \frac {\hbar} {\epsilon_F}, $} which is very short compared to the nuclear precession frequency in the instantaneous electron field, {$\omega=\gamma B_{hf}$}. Therefore the relaxation rate will be rather well approximated by the motional narrowing limit, {$\omega^2\tau$}. However this expression overestimates the electron spin flips because only a fraction {$k_B T/\epsilon_F$} can contribute, by the usual Fermi electron gas argument. We must evaluate the instantaneous hyperfine field to get {$\omega=\gamma B_{hf}$}. We may relate this to the Knight shift, as shown at the bottom of the previous page. {$ B_{hf}\approx K \frac {\epsilon_F} {\mu_B}.$} Taking all these ingredients a rough estimate of the so-called Korringa relaxation rate is {$ \frac 1 T_1 \approx \omega^2\tau \approx \frac{ $k_B T} \,{\epsilon_F}\gamma^2 K^2 \left(\frac {\epsilon_F} {\mu_B}\right)^2 \, \frac {\hbar} {\epsilon_F} = {k_B T} \, \frac {\gamma^2} {\gamma_e^2}\, K^2 \, \frac {4} {\hbar}\, , $} which implies that {$T_1 T$} and its reciprocal are both constant with temperature. The value of {$1/T_1TK^2$} is actually a universal quantity. We obtained {$\frac {\gamma^2} {\gamma_e^2} \, \frac {4 k_B}{\hbar}$} for it, however more precise calculations with the correct Sommerfeld model, by means of the Fermi Golden rule, yield {$ \frac 1 {T_1TK^2} = \frac {\gamma^2} {\gamma_e^2} \frac {4\pi k_B} {\hbar}\, .$} This is not really important since often, in real metals, further corrections take place, e.g. due to electronic correlation, and unless more sophisticated band calculations are implementes also the Sommerfeld value falls short of the experimental one by a numerical factor (the Stoner factor), quite easily of order 2-3. < Knight shift | Index | Hebel-Slichter peak in relaxation? > |