Product operator (NMR)

Heteronuclear J-coupling: A Doublet On Resonance

Assuming the proton of interest is exactly on resonance, thus ignoring chemical shift effects, and also ignoring relation and other effects, J-coupling to a single nuclei causes a single resonance (I) to split into typically two, three or four lines, although each of these may then also be split again by addition J-coupling to other nuclei into very complex patterns. Heteronuclear J-coupling operators are designated like I_xS_z. The heteronuclear J-coupling operators for doublets have the following effects:

I_x --> I_xS_z($\tau$) -->

1/2{I_xcos(2$\pi$J$\tau$) + I_ysin(2$\pi$J$\tau$)} + 1/2{I_xcos(-2$\pi$J$\tau$) + I_ysin(-2$\pi$J$\tau$)}