Product operator (NMR)

In the various fields of nuclear magnetic resonance, the product operator mathematical formalism is often used to simplify both the design and the interpretation of often very complex sequences of radio frequency electromagnetic pulses applied to samples under study. Basically, it is a short hand mathematical construct, a set of equations, that is used in place of more complex, although equivalent, matrix multiplication. The formalism uses a rotating frame of reference, in which the central irradiation frequency, say 800 MHz, is fixed on the X- or Y-axis, and the magnetic field, by convention, points towards the postive Z-axis. By convention, I and S indicate magnetic vectors associated with protons or heteroatom, respectively. Subscripts are used to indicate the axial orientation of the magnetic vector. At equilibrium, the net proton magnetic vector is thus I_z.

Single Pulses (rotations)

Arbitrary pulses (rotations)

I_x -->($\theta$_x) --> I_x

I_x -->($\theta$_y) --> I_xcos($\theta$) + I_zsin($\theta$)

I_x -->($\theta$_z) --> I_xcos($\theta$) - I_ysin($\theta$)

I_y -->($\theta$_x) --> I_ycos($\theta$) - I_zsin($\theta$)

I_y -->($\theta$_y) --> I_y

I_y -->($\theta$_z) --> I_ycos($\theta$) + I_xsin($\theta$)

I_z -->($\theta$_x) --> I_zcos($\theta$) + I_ysin($\theta$)

I_z -->($\theta$_y) --> I_zcos($\theta$) - I_xsin($\theta$)

I_z -->($\theta$_z) --> I_z

90 degree pulses

So called 90 degree ($\pi$/2) pulses, in which magnetization is rotated from one axis to another, are the most widely used single pulses in NMR spectroscopy and the above equations simplify to the following for such pulses.

I_x -->(90_x) --> I_x

I_x -->(90_y) --> I_z

I_x -->(90_z) --> -I_y

I_y -->(90_y) --> I_y

I_y -->(90_z) --> I_x

I_y -->(90_x) --> -I_z

I_z -->(90_z) --> I_z

I_z -->(90_x) --> -I_y

I_z -->(90_y) --> -I_x