Scaling and time reversal of spin couplings in zero-field NMR.

Among the most extraordinary phenomena in nuclear magnetic resonance (NMR) is the spin echo [1]. Although the original spin echo and its analogs in other areas of spectroscopy [2] result from reversing the dephasing due to independent, "inhomogeneous" interactions, a true many-body spin echo resulting from a reversal of the (seemingly irreversible) decay due to "homogeneous" spin-spin couplings has been demonstrated [3]. The phenomenon occurs in high magnetic field and can be thought of as arising from a reversal of the sign of the Hamiltonian describing the truncated spin-spin couplings [4], thereby reversing the evolution of the spins. Such time-reversal eAects, induced by coherent averaging under sequences of radio-frequency pulses [5], have made possible a number of novel experiments including selec tive excitation of n-quantum transitions [6]. An interesting question is whether the high magnetic field normally present in NMR experiments is essential to the possibility of time reversal of the spin-spin couplings. Could such experiments be done in zero-field NMR, where there is no truncation or privileged direction and the full, untruncated, Hamiltonian is responsible for the decay of spin order [7]? In this Letter, we show theoretical and experimental results involving general schemes of coherent averaging in zero field using dc magnetic-field pulse sequences that conform to cubic and icosahedral symmetry. From these schemes emerges the possibility of scaling, decoupling, and time reversal in zero field. The most general constraint that we impose upon coherent-averaging schemes in zero field is that the energy levels of the coherently averaged local interactions remain "zero-field-like, " i.e., independent of the orientation of the sample with respect to the laboratory frame. Although there may be some privileged directions in the laboratory frame, along which the dc magnetic-field pulses are applied, for instance, the overall eA'ect of the process must be independent of the orientation of the local interaction axis with respect to these directions. The process or the sequence is then called isotropic. It can be shown that isotropic processes are reduced to the scaling of the local interactions, i.e., multiplication by k, a constant scaling factor. For a given isotropic sequence, the scaling factors may not be identical for all types of interactions: heteronuclear or homonuclear, first rank (due k( =&@((to;));/(2l+ I ) . (2)