Cumulant approach to spin echoes

Abstract

The cumulant method, originally applied to NMR lineshape problems by Kubo, (1962) can be extended to treat spin echoes. The echo relaxation is shown to depend on the conventional spin torque correlation function and the symmetry properties of the coupling Hamiltonian. For the dipole-dipole interaction the author demonstrates that the echo decays identically to the free induction decay, while for diffusion in a field gradient the echo decays in the familiar cubic form. The same result is shown to follow from spin diffusion produced by the Heisenberg exchange Hamiltonian. In the motional narrowing region both the free induction decay and the echo decay exponentially with the same relaxation time assuming a perfect NMR magnet.