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A very simple treatment of paramagnetic shift and the corresponding conduction electron relaxation in metals may be performed by back-of-the-envelope calculations that give the correct order of magnitude. More precise calculations with the Drude-Lorenz-Sommerfeld model yield different numerical prefactors, but experimental data from real metals display further numerical deviations.
The main point is that for simple metals the paramagnetic shift is temperature independent, reflecting the Pauli susceptibility. The relaxation rate is linear in temperature and this is again obtained straightforwardly from Sommerfeld, with the use of the Fermi Golden rule for the transitions induced by an electron spin on the nuclear spin.